STANDARD COVER PAGE FOR ESA STUDY CONTRACT REPORTS ESA STUDY CONTRACT REPORT - SPECIMEN. Subject: Height System Unification with GOCE

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1 STANDARD COVER PAGE FOR ESA STUDY CONTRACT REPORTS ESA STUDY CONTRACT REPORT - SPECIMEN ESA Contract No: /11/NL/EL Subject: Height System Unification with GOCE Contractor: Technische Universität München, Institut für Astronomische und Physikalische Geodäsie (IAPG) *ESA CR( ) No. No. of Volumes: 1 This is Volume No.: 1 Contractors Reference: GO-HSU-PL-0021 Abstract: The outstanding performance of the global GOCE geoid enables new applications like unification of height systems worldwide, for which the global geoid needs to be known with high quality. This study aims at identifying the impact of the GOCE mission for height system unification starting from a review of the state-of the art of existing height systems, continuing with a summary of the existing methodologies including possible improvements to be implemented, further continuing with the application of these algorithms to the two test regions North America and Europe and the ocean in between, and finalizing with an assessment of the impact of GOCE in these test areas. As outcome of the study a list of recommendations for the realization of globally consistent and accurate height systems is formulated addressing specifically different situations related to terrestrial data coverage and applications. The work described in this report was done under ESA Contract. Responsibility for the contents resides in the author or organisation that prepared it. Names of Authors: Thomas Gruber, Reiner Rummel: Technische Universität München Johannes Ihde, Gunter Liebsch, Axel Rülke, Uwe Schäfer: BKG Frankfurt/Main Michael Sideris, Elena Rangelova: University of Calgary Philip Woodworth: Chris Hughes, NOC Liverpool ** NAME OF ESA STUDY MANAGER: Roger Haagmans DIV: Mission Science Division DIRECTORATE: Earth Observation Programmes ** ESA BUDGET HEADING STSE and GOCE

2 Page: 1 of 173 STSE GOCE+ Height System Unification with GOCE Summary and Final Report Doc. No.: GO-HSU-PL-0021 Issue: 1 Revision: 0

3 Authors Information Summary and Final Report Page: 2 of 173 Authors: Reiner Rummel, Thomas Gruber, Technical University Munich Johannes Ihde, Gunter Liebsch, Axel Rülke, Uwe Schäfer, BKG Michael Sideris, Elena Rangelova, University of Calgary Philip Woodworth, Chris Hughes, NOC Liverpool Contributions from: Christian Gerlach, Bavarian Academy of Sciences and Humanities Document Change Record ISSUE /REV. DATE REASON FOR CHANGED PAGES / PARAGRAPHS CHANGE Final Issue Initial Version

4 Abbreviations and Acronyms Summary and Final Report Page: 3 of 173 ATBD Algorithm Theoretical Baseline Document BGI... International Gravimetric Bureau CBN... Canadian Base Network CGG2005 and Canadian Gravimetric Geoid of 2005 and 2010 CGSN... Canadian Gravity Standardization Net CGVD28... Canadian Geodetic Vertical Datum of 1928 DEM Digital Elevation Model D/O... Spherical Harmonic Degree and Order DOT..Dynamic Ocean Topography EPS...Equipotential Surface GBVP.... Geodetic Boundary Value Problem GGOS Global Geodetic Observing System GIA... Glacial Isostatic Adjustment GLOSS.... Global Sea Level Observing System GNSS.. Global Navigation Satellite System GOCE..... Gravity Field and Steady-State Ocean Circulation Explorer GPM Global Geopotential Model CGVD28....Canadian Geodetic Vertical Datum of 1928 HSU.....Height System Unification IAG International Association of Geodesy IAR. Impact Assessment Report IAU International Astronomical Union IGLD85...International Great Lakes Datum of 1985 IGS... International GNSS Service IGSN71...International Gravity Standardization Net 1971 IHF International Height Frame IHS... International Height System ILRS.... International Laser Ranging Service IMU.... Inertial Measurement Unit ITRF International Terrestrial Reference Frame ITRS... International Terrestrial Reference System JGS. Journal of Geodetic Science LDC..... Less Developed Country LIT.. Literature to be provided LSA... Least-Squares Adjustment MSL..... Mean Sea Level NAD83... North American Datum of 1983 NAVD North American Vertical Datum of 1988 NGA...National Geospatial Intelligence Agency NGVD29 National Geodetic Vertical Datum of 1929 OCM Ocean Circulation Model PGR Post Glacial Rebound PSMSL.... Permanent Service for Mean Sea Level RB... Requirements Baseline Document RMS... Root Mean Square RTM.. Residual Terrain Model SH Spherical Harmonics SLR... Satellite Laser Ranging SoW... Statement of Work STD... Standard Deviation UELN.. Unified European Levelling Network USGG United States Gravimetric Geoid of 2009 VCM. Variance-Covariance Matrix WARDEN..... Western Arctic Deformation Network WCDA... Western Canada Deformation Array WHS World Height System

5 Page: 4 of 173 Table of Content 1 Introduction Purpose Applicable Documents Scientific Project Publications Abstract Preparatory Work Scientific Requirements Consolidation Introduction Project Objectives Scientific Requirements Overview Methods for Height System Unification Geometric Levelling and Gravimetry Geodetic Boundary Value Problem (GBVP) Approach Ocean Levelling Standards and Data Requirements Data, Standards and Conventions Data GOCE Gravity Field Models and Performance GNSS Levelling Data Oceanographic Data Sets Standards and Conventions Newtonian Constant of Gravitation G GRS80 Ellipsoid and Normal Gravity Field Permanent Tide System Height Systems and Conversions ITRS Time System Treatment of the Atmosphere Algorithms and Preliminary Analysis Spherical Harmonic Synthesis Error Budgeting of Potential Differences from GPM s Algorithm for Propagation of GPM Errors to Geoid Heights Algorithm for Computation of the Omission Error Optimal Combination of Heterogeneous Data Estimation of Height System Offsets by the GBVP Solution of the GBVP Least-Squares Estimation of Datum Offsets Combined Solution for the Geoid Preliminary Results Review of Height Systems in Europe Evaluation of National Vertical Reference Frames in Europe Evaluation of the European Vertical Reference Frame EVRF Comparison of Observed Geoid Heights with Global Geopotential Models Review of Height Systems in North America Vertical Reference Frames in North America Evaluation of GOCE-based Geoid Models in North America Preliminary Computations of the North American Datum Offsets DOT Determination... 91

6 Page: 5 of Experiments and GOCE Impact Assessment Overview and Approach for Study Experiments Experiments for HSU with GOCE Height Offset Impact on GBVP Equipotential Surface Omission Error Impact on Equipotential Surface Impact of Propagated Error on Equipotential Surface GOCE versus GBVP Equipotential Surface Experiment for Europe Experiment for North America GOCE versus GNSS Levelling Equipotential Surface Oceanic and Tide Gauge/Datum Levelling Oceanic and Tide Gauge/GNSS-Geoid Levelling Oceanic and Altimeter/Geoid Levelling Experiments for Connection of HSU Approaches Connecting GBVP Solution and GNSS Levelling Connecting Tide Gauges to GBVP Solutions Connecting Tide Gauges to GNSS Levelling Connecting Tide Gauges by Ocean Levelling GOCE Impact Assessment Assessment of GOCE Impact on HSU Methods GOCE Geoid to Check Levelling Characteristics of GNSS/Levelling Geoid Residuals Ocean Levelling for Ocean Model Quality Assessment Impact of GOCE Based Combined Models vs. pre-goce Models Impact of GOCE for Regional Geoid Modelling Impact of Omission Error at Tide Gauges Local Geoid in North America at Tide Gauges and Impact of Height Offset Local Geoid in Germany at Tide Gauges and Impact of Height Offsets Local Geoid in Europe at Tide Gauges and Impact of Height Offsets Impact of Local Geoid at Tide Gauges on Ocean Levelling Scientific Roadmap Roadmap Summary Global Height Unification Next Generation Regional/National Height Systems Well Surveyed Regions Sparsely Surveyed Regions Ocean Levelling versus GNSS-Levelling Detailed Roadmap Introduction Global Height Unification Next Generation Regional/National Height Systems Well Surveyed Regions Sparsely Surveyed Regions Ocean Levelling versus GNSS Levelling References

7 Page: 6 of Introduction 1.1 Purpose The aim of the HSU project is to assess the use of GOCE products for regional and/or global realization of a unified height datum. GOCE is expected to provide potential differences with an accuracy corresponding to 1-2 cm height differences between arbitrary points on the Earth. The objective is to make full use of this level of accuracy for global height system unification. The Final Summary Report (this document) shall provide an overview of the activities and results obtained within this project from a technical point of view. It is based on the deliverable technical documents created within this study [AD-7], [AD-10], [AD-14] and [AD-16]. The study logic has been structured into preparatory analyses (chapter 3), experiments and impact assessment (chapter 4) and concluding recommendations in section 5. Apart from the technical documents a number of management and supporting reports has been written as well. In particular these are [AD-6], [AD-12] and the project web site. Further-on, as a major outcome of the project, a number of refereed publications has been published in various journals. Specifically, one should mention the Special Issue on Regional and Global Geoid-based Vertical Datums of Journal of Geodetic science published in December The project team contributed with a significant number of manuscripts (8) to this issue. For details see chapter Applicable Documents [AD-1] Invitation to Tender AO/1-6367/10/NL/AF STSE GOCE+, Item No , [AD-2] Statement of Work, Support to Science Elements (STSE) GOCE+, EOP-SM/2048, Issue 1, Revision 0, March 2010 [AD-3] Proposal to ITT AO/1-6367/10/NL/AF STSE GOCE+, Dated (Vol. 1 Cover Letter, Vol. 2 Technical proposal, Vol. 3 Management Proposal). [AD-4] GO-HSU-MN-0001, Date : Minutes of Meeting, HSU Negotiation Meeting. [AD-5] GO-HSU-MN-0003, Date : Minutes of Meeting, HSU Kick-Off [AD-6] GO-HSU-PL-0002, Issue 1.0, Date , Project Management Plan. [AD-7] GO-HSU-PL-0005, Issue 1.0, Date , Requirements Baseline Document [AD-8] GO-HSU-MN-0007, Date : Minutes of Meeting, PM1 [AD-9] GO-HSU-MN-0009, Date : Minutes of Meeting, PM2 [AD-10] GO-HSU-PL-0010, Issue 2.0, Date , Preliminary Analysis Report, Development and Validation Plan, Validation Report [AD-11] GO-HSU-MN-0011, Date , Minutes of Meeting Mid-Term Review [AD-12] GO-HSU-PL-0012, Issue 2.0, Date , Promotion Plan [AD-13] GO-HSU-MN-0013, Date , Minutes of Meeting, PM3 [AD-14] GO-HSU-PL-0014, Issue 2.2, Date , Algorithm Theoretical Basis Document (ATBD) Impact Assessment Report (IAR) [AD-15] GO-HSU-MN-0015, Date , Minutes of Meeting, PM4 [AD-16] GO-HSU-PL-0016, Issue 2.1, Date , Scientific Roadmap (SR) [AD-17] GO-HSU-MN-0017, Date / , Minutes of Meeting, PM5 [AD-18] GO-HSU-MN-0019, Date 15./ , Minutes of Meeting, Final Meeting of Main Contract

8 1.3 Scientific Project Publications [P-1] [P-2] [P-3] [P-4] [P-5] [P-6] [P-7] [P-8] [P-9] Summary and Final Report Page: 7 of 173 Amjadiparvar, B., Rangelova, E. V., Sideris, M. G. & Véronneau, M. (2013) North American height datums and their offsets: Evaluation of the GOCE-based global geopotential models in Canada and the USA. In J Appl Geodesy, 7(3), pp , DOI: /jag Gerlach, C. & Rummel, R. (2013) Global height system unification with GOCE: A simulation study on the indirect bias term in the GBVP approach. In: Journal of Geodesy, Volume 87, Issue 1, pp 57-67, DOI: /s y. Gerlach, C. & Fecher, T. (2013) Approximations of the GOCE error variance-covariance matrix for least-squares estimation of height datum offsets. In Journal of Geodetic Science, 2(4), pp , DOI: /v Gruber, T., Gerlach, C. & Haagmans, R. H. N. (2013) Intercontinental height datum connection with GOCE and GNSS-levelling data. In Journal of Geodetic Science, 2(4), pp , DOI: /v y Hayden, T., Amjadiparvar, B., Rangelova, E. & Sideris, M. G. (2013) Estimating Canadian vertical datum offsets using GNSS/levelling benchmark information and GOCE global geopotential models. In Journal of Geodetic Science, 2(4), pp , DOI: /v Hayden, T., Rangelova, E., Sideris, M. G. & M., Véronneau. (2013) Evaluation of W0 in Canada using tide gauges and GOCE gravity field models. In Journal of Geodetic Science, 2(4), pp , DOI: /v Ince, E. S., Sideris, M. G., Huang, J. & Véronneau, M. (2012) Assessment of the GOCE- Based Global Gravity Models in Canada. In Geomatica, Hayden, T., Rangelova, E., Sideris, M. G. & Véronneau, M. (2013). Contribution of tide gauges for the determination of W0 in Canada. Proceedings of IAG Symposium on Gravity, Geoid and Height Systems, accepted. Rangelova, E., W. van der Wal, & Sideris, M. G. (2013) How Significant is the Dynamic Component of the North American Vertical Datum? In Journal of Geodetic Science, 2(4), pp , DOI: /v [P-10] Rangelova, E., Sideris, M.G., Amjadiparvar, B., Hayden, T., (2014). Height datum unification by means of the GBVP approach using tide gauges. VIII Hotine-Marussi Symposium, Rome, Italy, June 17-21, 2013 (in a second review cycle). [P-11] Rülke, A., Liebsch, G., Sacher, M., Schäfer, U., Schirmer, U. & Ihde, J. (2013) Unification of European Height System Realizations. In Journal of Geodetic Science, 2(4), pp , DOI: /v [P-12] Rummel, R. (2012) Height unification using GOCE. In Journal of Geodetic Science, 2(4), pp , DOI: /v [P-13] Sideris, M. G., Rangelova, E. & Amjadiparvar, B. (2014). First results on height systems unification in North America using GOCE. Proceedings of IAG Symposium on Gravity, Geoid and Height Systems, accepted [P-14] Woodworth, P., Hughes, C. W., Bingham, R. J. & Gruber, T. (2012) Towards Worldwide Height System Unification using Ocean Information. In Journal of Geodetic Science, 2(4), pp , DOI: /v

9 Page: 8 of Abstract GOCE provides a completely new view of the static part of the global gravity field as it observes the geoid quasi globally with an accuracy of 1-2 cm at spatial scales of 100 km and larger. Before GOCE, the knowledge of the global geoid at these scales was very heterogeneous as along with the available satellite data from the GRACE mission, terrestrial gravity observations had to be used, which often were of bad quality or which had not been observed at all. This implies that in many regions of the world GOCE (in combination with GRACE) is a unique source for describing the geoid with reasonable accuracy. This is nicely shown by comparing the GOCE geoid with independently observed geoid heights at GNSS-Levelling points (a complete description of the procedure is described in Gruber et al, 2011). Figure 3-12 and Figure 3-13 show such comparisons for Germany and Brazil, respectively, applying different degrees of truncation of the GOCE model and using the EGM2008 model for estimating the omitted signal above the truncation degree. While in Germany, at degree 200 (corresponding to 100 km spatial resolution), EGM2008, which incorporates the high quality and dense German gravity anomaly data set, performs slightly better than the most recent GOCE models the opposite is observed for Brazil, where only partially good ground gravity data were available for the EGM2008 model. In Brazil, the most recent GOCE models perform significantly better than EGM2008, which is shown by the lower RMS values of the differences to the ground based GNSS-Levelling geoid heights. The outstanding performance of the global GOCE geoid enables new applications like unification of height systems worldwide, for which the global geoid needs to be known with high quality. This study aims at identifying the impact of the GOCE mission for height system unification starting from a review of the state-of the art of existing height systems, continuing with a summary of the existing methodologies including possible improvements to be implemented, further continuing with the application of these algorithms to the two test regions North America and Europe and the ocean in between, and finalizing with an assessment of the impact of GOCE in these test areas. As outcome of the study a list of recommendations for the realization of globally consistent and accurate height systems is formulated addressing specifically different situations related to terrestrial data coverage and applications. Worldwide more than 100 height systems exist. Most of them refer to an adopted value of mean sea level at a reference marker. The connection of height systems located on one continent is straightforward in principle; it is done by geodetic levelling in combination with gravimetry. Height systems separated by sea cannot be unified in this manner and the height offsets between their reference markers are unknown. They do not refer to one common level (i.e., equipotential) surface. As an example in Figure 3-3 the relation between European national height reference systems is shown as they were estimated from the Unified European Levelling Network (UELN). In North America, large differences between the heights in the two official vertical datums, CGVD28 in Canada and NAVD88 in USA, constitute a significant problem for engineering projects in several areas along the Canadian and USA border. Moreover, unexplained discrepancies of 1.5 m from the Atlantic to the Pacific coast exist even in the NAVD88 readjustment. Determination of the height offsets will result in globally consistent height information, it will eliminate complications in civil constructions associated with linking data from two height zones, and it will facilitate interpretation of sea level records at tide gauges within studies of global and regional sea level change. In essence, GOCE will allow computing gravity potential differences (or corresponding differences in physical heights) between arbitrary points on the Earth s surface with accuracy of m 2 /s 2 (corresponding to 1-2 cm in terms of physical heights). Applying the basic approximate relation Hh N, where H is a physical, h an ellipsoidal and N is the geoid height, one can convert between geometrical heights h, as determined with GNNS and physical heights H, as derived from geodetic levelling combined with gravimetry, if one knows the geoid height with sufficient accuracy (compare Figure 3-4 for a schematic view). The quoted precision of height connection can only be achieved if

10 Page: 9 of 173 two additional principle conditions are met: (1) The positions of the height reference markers have all to be expressed in one global geocentric terrestrial reference frame with compatible accuracy and they have to be processed according to well-defined standards. (2) The omission error has to be quantified. Gravity field models of GOCE are limited to about degree and order 200, which leaves the contribution of the short wavelength part of the gravity field neglected. This omission part varies geographically in size and it has a global RMS value of about 30 cm. If data coverage allows, the omitted part has to be substituted by adding complementary information from regional gravity and topographic data sets. The basic methods for height system unification are: (1) Direct height connection inside a closed land area (continent) by geodetic (spirit) levelling in combination with gravimetry, which is the classical approach. It cannot connect islands separated by sea, but GOCE could be used to detect systematic errors in large levelling networks. (2) The geodetic boundary value problem (GBVP) approach, which represents a rigorous geodetic method of height unification and which is able to deal with the direct effect of height offsets and the indirect effect introduced by the use of gravity anomalies biased due to these offsets. A summary of the GBVP theory for height system unification is given in [Heck and Rummel, 1990]. It is based on the solution of the GBVP and it makes combined use of a set of reference points on land with coordinates derived from space geodetic methods, a high degree and order gravity model such as one of those provided now by GOCE, and regional gravity and topographic data sets. (3) The ocean levelling approach, which is a method suited to directly connect coastal tide gauges. Two tide gauges can be connected knowing the change in mean dynamic topography (MDT) along a line connecting the gauges. In theory, ocean levelling is able to connect all available tide gauges directly. In practice, the accuracy of the levelling is limited by the accuracy of determination of MDT gradients. These are derived either from ocean circulation modelling, from measurements of currents along a level path (usually the ocean surface), from a combination of ocean altimetry and the GOCE geoid (geodetic MDT), or a combination of the three. The most recent GOCE and GOCE-based combined geoid solutions are applied in the methods described above in order to identify their impact to the selected test regions Europe, North America and North Atlantic. Method (1): The GOCE geoid has been applied to check the levelling and the gravity data in Canada. In summary it was observed, that after correcting for the omission error applying the EGM2008 model, areas of subsidence and uplift in the Southern and Northern areas, respectively, can be clearly identified (compare Figure 4-12) and that significant gravity anomaly differences in various areas exist, which can be attributed to differences between topographic information applied to compute the gravity anomalies. Similar results are observed when the European levelling networks are checked against GOCE enhanced geoid models. One clearly can identify several countries where levelling networks exhibit systematic distortions with reference to this geoid. Method (2): Applying the GBVP approach the impact of the GOCE global models on regional geoid solutions in Canada, Europe and Germany is investigated. From these studies it can be concluded that the GOCE geoid provides improved long to medium wavelength contributions in the regional combination procedure, but due to the high quality of terrestrial data in most of the areas selected for our study its impact is small but visible. Further investigations show that with GOCE the indirect effect of height offsets in terrestrial gravity data can be neglected and that a largely simplified error propagation can be applied to estimate geoid errors from the GOCE gravity field model. Using the GOCE geoid for a recomputation of the German gravimetric geoid the overall error budget can be decreased by about 25% compared to applying a pre-goce model. For assessing the impact of GOCE on the European gravimetric geoid optimal spectral weights are derived for the terrestrial gravity data, the GOCE, a GRACE-GOCE combination and a pre-goce geoid before the spectral combination is performed. As a result it is concluded that specifically the combined GRACE-GOCE geoid is able to improve the performance of the European gravimetric geoid by about 15% to 25%. For some areas in Canada with insufficient data coverage like Yukon and Canadian Arctic there are differences of cm between the EGM2008 and the GOCE based geoid. Most likely, these differences can be attributed

11 Page: 10 of 173 to improvements from the GOCE geoid. An independent validation in these areas is difficult as not enough high quality ground based geoid observations are available. But from quality analyses performed in other areas in the world with insufficient terrestrial data coverage one can assume that indeed the geoid is improved there when GOCE is incorporated (compare e.g. Figure 3-13 for the Brazil GNSS levelling points). For areas with rugged terrain in Canada like the Pacific Cordillera and the Rockies the omission error plays a significant role and hides possible improvements from GOCE. There, hardly any improvement is visible when applying the GOCE geoid as reference model. Method (3): Similar to continents the enhanced GOCE geoid also can be applied to assess the quality of ocean models. From this analysis it is concluded that along the North Atlantic, North American Pacific and Mediterranean coastlines all mean ocean dynamic topography models are consistent at the sub-decimetre level, which implies that ocean levelling will be feasible at this level of accuracy. It also is observed that along these coastlines the accuracy of comparisons at point tide gauge locations depends greatly on the short scale component, which comes from in situ gravity information, which is included to a great extent in EGM2008. Nevertheless, for these areas, added information from the GOCE model can be identified as compared to the pure EGM2008 geoid, which shows up in smoother coastal mean ocean dynamic topographies. When extending this investigation to regions with strong gradients of ocean dynamic topography, a number of strong jets, particularly in the Southern Ocean, appear sharper and more realistic when using the GOCE based geoid as compared to EGM2008 (see Figure 4-23). From the study results summarized above a number of recommendations for the realization of globally consistent and accurate height systems is derived. These recommendations are specifically elaborated on the situation for densely surveyed areas, poorly surveyed areas like a development country with little geodetic infrastructure and across the ocean. In addition the required tasks for global height system unifications and the diagnosis of existing height systems are addressed. Figure 5-1 summarizes the recommendation for the different situations. The main conclusions can be formulated as follows: (1) The primary geodetic method for height determination and global height unification will be GNSS-levelling, i.e. the derivation of potential differences between points with known position given in one global Earth fixed, geocentric reference frame. It requires precise geometric positioning by GNSS and a best possible GRACE/GOCE geopotential model complemented with short scale geopotential information from terrestrial data. (2) In well surveyed regions the assumption is the availability of a network of permanent, high quality GNSS stations, records of past and present levelling campaigns and a geopotential/geoid model with highest possible resolution, which enables highly accurate GNSS-levelling. Here, there is the possibility of combining GNSS-levelling over long distances with the high precision of spirit levelling over short distances. (3) Sparsely surveyed regions may be characterized by missing GNSS infrastructure, low accuracy or lack of regional gravity data and lack of surveying and mapping infrastructure. This implies that most likely one has to rely on the best GRACE/GOCE geoid with insufficient knowledge of the omission part due to the missing terrestrial data. The error level of GNSS-levelling in such areas might be at a level of a few decimeter, which is a great progress compared to the current situation. This progress mostly can be addressed to the availability of the GOCE geoid in areas without good geoid knowledge nowadays. (4) A globally consistent and accurate height reference along coastlines and across ocean basins and straits is of high importance for ocean circulation studies and sea level research. At tide gauges the principle of GNSS-levelling can be applied for this purpose. It requires GNSS positioning at tide gauges, the essential connection of the GNSS marker to the tide gauge height reference and the use of a GRACE/GOCE geopotential or geoid model. This corresponds to a geodetic determination of mean dynamic ocean topography (MDT) at tide gauges. Oceanic MDT models are a useful tool for validation of geodetic height systems along coastlines. (5) The GOCE geoid opens the possibility to systematically check levelling and gravity data worldwide. This includes also the possibility to identify height offsets of existing height systems to the GOCE geoid.

12 Page: 11 of Preparatory Work Preparatory work for this study was structured in three main tasks. These were the identification and consolidation of scientific requirements for height system unification, a preliminary analysis including a development and validation plan, and finally the collection and description of the required data sets. The following sub-chapters provide a detailed description of each task. 3.1 Scientific Requirements Consolidation Introduction The first task of the study was to identify a consolidated, coherent and complete view of the scientific requirements for obtaining the project objectives in terms of methodology and necessary input data sets, with special emphasis on processing standards and data requirements [AD-7]. Starting from the objectives of GOCE one of the main science objectives of the mission was to provide a significant contribution to the global unification of height systems. This aim implies that all national and regional height systems shall refer to one global height datum, i.e. to one physically well-defined reference level. This specifically applies to the global system of tide gauges, which usually act as reference point(s) of individual height datums Project Objectives The RB is based on the following list of project objectives as provided on page 9 of [AD-2]. 1) To collect information and review the state-of-the-art in height systems and local/regional/global initiatives to unify height systems. 2) To quantify the currently known differences (without a precise GOCE geoid) between height systems in Europe and other parts of the world based on available literature and existing results. 3) To review, evaluate and improve the methodology and/or existing algorithms for height determination and height system unification. 4) To attempt global height unification and tide gauge unification for those parts of the world where the data sets are available. 5) To select two test regions for height system unification and to demonstrate the benefit of using GOCE data for this purpose (to the resolution and accuracy of the new data). 6) To assess the impact of a unified height system on local gravity information (systematic errors in local gravity anomalies and alike) and topographic heights and vice versa. 7) To geophysically interpret height differences (compared to the GOCE geoid) at one or more tide gauges in terms of ocean dynamic topography and currents, and to compare the results to altimetry. 8) To involve local/national organisations maintaining height systems. 9) To provide a roadmap for future work required to define a world height system and vertical datum exploiting GOCE data.

13 3.1.3 Scientific Requirements Overview Summary and Final Report Page: 12 of 173 The mission objectives of GOCE are the determination of the Earth s gravity field with an accuracy of 1 ppm and geoid heights with an accuracy of 1-2 cm, both with a spatial resolution of about 100 km, which corresponds to a spherical harmonic expansion complete up to degree and order (D/O) 200; compare [ESA, 1999, p.89]. From the analysis of the actual GOCE mission data our expectation is that a geoid precision of 2 cm to 3 cm will be reached by the end of 2012 (see Figure 3-1). Figure 3-1: Performance estimates for GOCE stand-alone solutions (after Pail, personal communication) The science objectives of the GOCE mission are described in (ibid). They belong to the fields of oceanography, solid Earth physics, geodesy, glaciology and sea level research. An important objective is the contribution of GOCE to the global unification of height systems. The aim is thereby to refer all national and regional height systems to one height datum. This also applies to the global system of tide gauges and should make them refer to one physically well defined reference level (ibid, ch.3.5. and 3.6; see also [Plag and Pearlman, 2009, ch. 3.7]), see Figure 3-2. Figure 3-2: GOCE data and their use for navigation, the unification of height systems, precise orbits and leveling by GNSS, with a wide range of applications in practice and science (from [ESA, 1999]). Worldwide more than 100 height systems exist. Most of them refer to an adopted value of mean sea level at a reference marker. The connection of height systems located on one continent is

14 Page: 13 of 173 straightforward in principle; it is done by geodetic levelling in combination with gravimetry. Height systems separated by sea cannot be unified in this manner and the height offsets between their reference markers are unknown. They do not refer to one common level (i.e., equipotential) surface. As an example, in preparation for the construction of the tunnel connecting France with the United Kingdom, one had to find a realistic value for the potential difference between France and the UK [Radcliffe, 1995; Varley et al., 1992]. Today we know that the offset between the official height systems ODN (UK) and NGF/IGN69 (France) is 40 +/- 2 cm (Greaves et al., 2007; Rebischung et al., 2008). This value is in excellent agreement with a value determined by oceanographic leveling by Cartwright and Crease (1963) many years before, and yet it differs significantly from the 52 cm value assumed by the UELN until relatively recently (Figure 3-3). Such offsets can be easily taken into account. In Figure 3-3, estimated offsets between European height systems are shown. In North America, large differences between the heights in the two official vertical datums, CGVD28 in Canada and NAVD88 in USA, constitute a significant problem for engineering projects in several areas along the Canadian and USA border. Moreover, unexplained discrepancies of 1.5 m from the Atlantic to the Pacific coast exist even in the NAVD88 readjustment [Véronneau and Héroux, 2006]. Determination of the height offsets will result in globally consistent height information, it will eliminate complications in civil constructions associated with linking data from two height zones, and it will facilitate interpretation of sea level records at tide gauges as being part of a global process of sea level change. Figure 3-3: Relations between European national height reference systems and EVRF2007 in cm. (from: www. bkg.bund.de/evrs > Related Projects > Height Datum Relations) In essence, GOCE will allow computing gravity potential differences (or corresponding differences in physical heights) between arbitrary points on the Earth s surface with accuracy of m 2 /s 2. The basic approximate formula is H hn, where, H is a physical height, h an ellipsoidal one and N is the geoid height. The problem arises from the fact that h is a purely geometric quantity derived from satellite positioning and given in a chosen terrestrial reference system. N is derived from GOCE, it refers to the chosen gravimetric reference system and it is incomplete, i.e., the omission part is missing. H is a physical height (orthometric,

15 Page: 14 of 173 normal, normal-orthometric), which refers to a reference marker and a certain, well defined height datum. It is derived from geodetic levelling combined with gravimetry and it may contain systematic errors. Alternatively, the situation can be discussed exactly in terms of potential differences: W U T, with the normal potential U, the disturbing potential T and the gravity potential W U T. Here geometry (from satellite positioning) enters not via the ellipsoidal height h, but via the required position coordinates at which W, U and T are evaluated. Figure 3-4: Schematic overview of basic relations between ellipsoidal and physical heights referring to different datum realizations. The quoted precision of height connection can only be achieved if two additional principle conditions are met: 1. The positions of the height reference markers have all to be expressed in one global terrestrial reference frame (e.g., ITRF), with compatible accuracy. They have to be processed according to well-defined standards, such as those defined by the IGS and the ILRS. The corresponding processing standards have been revised and improved over the years. The improvement of the recent reprocessing of IGS stations is shown, e.g., in Figure 3-5. The reference frame has to coincide with the coordinate system in which the gravity potential is expressed. In other words, geometry and gravity have to refer to the same geocentric reference frame, with the same origin and orientation. As an experiment one can test the effect of a difference in origin of the geometric and gravimetric reference frames (see Figure 3-6). 2. Gravity field models of GOCE are limited to about D/O 200 to 210. This leaves the contribution of the short wavelength part of the gravity field, above D/O 200, neglected. This omission part varies geographically in size and it has a global RMS value of about 30 cm. For some less developed countries (LDC) this may be acceptable. But in general the omitted part has to be substituted by adding complementary information from regional gravity and topographic data sets. However, these complementary data sets do in general not refer to a globally uniform height reference. There exist height offsets among them which have to be taken into account. Some numerical estimates of the geoid omission error are contained in Table 3-1.

16 Page: 15 of 173 Figure 3-5: Mean weekly RMS values of 3-day orbit fits through 1-day orbits for the different types of GNSS satellites: (a) CODE final orbits (RMS values only available starting with GNSS week 782), (b) reprocessed orbits; after [Steigenberger et al., 2009] Figure 3-6: Effect of non-geocentric reference frame on the geoid. An offset of 1 cm is applied separately in x-direction (upper left), y-direction (upper right), z-direction (lower left) and in all three directions at the same time (lower right) Table 3-1: Geoid omission error as function of maximum spherical harmonic degree computed from the degree variance models of Kaula and Tscherning/Rapp Maximum spherical harmonic degree Kaula [m] Tscherning/Rapp [m]

17 Page: 16 of 173 The theory for height system unification, dealing with these conditions is well established; a summary is given in [Heck and Rummel, 1990]. It is based on the solution of the so-called geodetic boundary value problem (GBVP) and it makes combined use of a set of reference points on land with coordinates derived from space geodetic methods, a high D/O gravity model such as those provided now by GOCE, and regional gravity and topographic data sets. The regional gravity anomaly data sets may contain the effect of unknown height offsets. Various test computations have been performed in the past that demonstrate which the problem can be solved and how the associated uncertainties can be estimated [Khafid, 1998; Xu, 1992; Xu and Rummel, 1991; Zhang et al., 2008]. See also [Colombo, 1980] and [Haagmans and De Min, 1999]. The underlying assumption of these studies is, however, knowledge about the involved height zones, i.e., the geographical boundaries of individual height datums. The corresponding concept for connecting tide gauges is "ocean levelling". Two tide gauges can be connected knowing the change in mean dynamic topography (MDT) along a line connecting the gauges. In theory, ocean levelling is able to connect all available tide gauges directly. In practice, the accuracy of the levelling is limited by the accuracy of determination of MDT gradients. These are derived either from ocean circulation modelling, from measurements of currents along a level path (usually the ocean surface), from a combination of ocean altimetry and the GOCE geoid (geodetic MDT), or a combination of the three. If a path can be found along which ocean currents are negligible on all but vertical sections of the path, then "steric levelling" allows an analogous procedure using density measurements instead of currents. In all cases, shallow water regions near to tide gauges have to be treated carefully as the dynamics in these regions differs from that in the open ocean, leading to a requirement for detailed regional information. Classic papers on this topic include[cartwright and Crease, 1963; Fischer, 1977; Rummel and Ilk, 1995; Sturges, 1967, 1974]. Temporal Variations: Apart from the two principal conditions (consistency of reference frames and consideration of omission part) given above, one has to deal with the fact that both the geometry of land and ocean surfaces and gravity are changing with time. GIA, land subsidence and tectonic processes are causing vertical motions of the reference markers. Also, sea level changes globally and locally due to, e.g., the melting of ice shields and glaciers, thermo-haline expansion, or changes in bathymetry. Atmospheric and ocean loading affect both the sea level and the land surface. Almost all of these processes are associated with temporal changes in gravity. Thus, a concept is needed that allows distinction between the spatial and temporal consistency of height systems. It should be emphasised that the treatment of temporal variations in height systems does not belong to the primary goals of this study (see the list of project objectives in chapter of this document). This however is important for North America because GIA causes significant changes in the heights of the benchmarks and the geoid that need to be accounted for in view of the targeted one-centimeter accuracy of the vertical datum. Recommendations and a methodology for incorporating the temporal changes in the North American vertical datum have been developed. Datum Definition: The unified vertical height system can be related to one single reference point, the datum point. Then, all potential differences will refer to this point. The reference point should be as stable as possible (whatever this may imply on a dynamic Earth). The resulting concept of potential values and of error propagation is straightforward and clear in this case, as is the transformation from one to another choice of height datum. Alternatively, and in analogy with the definition of the ITRF, the datum could be defined as a weighted mean (minimum norm) of a set of several (stable) reference points. The appropriate theory of datum transformation and S-transformation will be formulated in this study. Quality Description: Height unification as discussed here and based on the outcome of GOCE has to be accompanied by a quality description derived from a rigorous variance-covariance propagation and taking into account the quality (or lack of quality) of all used data sets. As an example, Figure 3-7shows geoid height errors computed by propagation of the error variance-covariance matrix of the global potential model EGM96. The result varies with the level of approximation of the matrix. The upper left panel shows the results without taking any approximations into account (see also

18 Page: 17 of 173 [Haagmans and van Gelderen, 1991]), while in the upper right panel correlations between coefficients of different order m are neglected and in the lower panel only the main diagonal of the matrix is used, i.e., all correlations between the coefficient errors are neglected. Figure 3-7: Geoid height errors from variance-covariance propagation of EGM96 errors (up to D/O 70) using the full error covariance matrix (upper left), using blocks of constant order only (upper right) or using the main diagonal only (lower left). Operational Aspects: It needs consideration whether a mean sea level (MSL) can be uniquely related to the globally unified height system. The same holds true for a reference equipotential surface, W 0. The chosen definition has to be operational (knowing a W 0 value is not equivalent to knowing the level surface it refers to). The corresponding definitions of the International Astronomical Union (IAU) must be taken into account. The latter has to be seen before the background of global time synchronization with a projected precision at a to level. Due to temporal variations in geometry and gravity, global height system unification on the level of a few centimetres becomes a continuous process. Thus a reference epoch has to be introduced. The present task must be seen in terms of long-term datum monitoring. This aspect will be part of the roadmap for future work required to define and realize a world height system (see list of project objectives in section of this documents). International Activities: Several initiatives are currently under way towards modernization of regional and global height systems in North America, South America, Europe and globally. The latter, the World Height System (WHS) was a project of the International Association of Geodesy (IAG; see [Ihde, 2007a, 2007b]) up until the summer of Currently it continues as one of the three major unifying products of IAG s Global Geodetic Observing System (GGOS) [Ihde and Sánchez, 2005; Plag and Pearlman, 2009]. A global unification of height systems should be based on geopotential numbers (or physical heights) of ground markers consistent also with their three dimensional coordinates. It will give precise information about the up or down of any two points belonging to this system. One may say: it provides one global reference level, everywhere. Sample applications are: 1. Precise GNSS levelling with many applications in engineering, geo-informatics, exploration, etc. 2. Digital elevation models (DEMs) referring globally to one single reference level. 3. Precise height information over large distances for large civil engineering projects. 4. Tide gauge records referring all to the same reference ocean level.

19 Page: 18 of 173 The underlying assumptions of this proposal are the mission objectives of GOCE, i.e., a geoid accuracy of 1 to 2 cm (corresponding to 0.1 to 0.2 m 2 /s 2 in terms of potential differences) with a spatial resolution corresponding to a spherical harmonic expansion to D/O 200. We shall investigate the theoretical and operational requirements resulting from global height system unification at the same level of accuracy, say 2 to 4 cm, corresponding to 0.2 to 0.4 m 2 /s 2 and with a negligible omission error. This includes the following aspects, which will be treated in more detail in chapter of this document: 1. Selection and description of an appropriate method of height system unification (boundary value approach and ocean levelling). 2. Data requirements (regional gravity and heights, geometric positions at reference points). 3. Test computations with special focus on GOCE. 4. Role and quality of the international terrestrial reference frame. 5. Recommendation about the operational treatment of temporal variations (land surface, sea level and gravity/geoid). 6. Aspect of local realization of height and sea level as well as their connection (local tie). 7. State-of-the-art of European, North American and world height datum and system as well as of permanent sea level monitoring. Synthesis of the state-of-the-art in Europe, North America, permanent sea level monitoring and of the world height datum. 8. Practical and scientific application of height system unification (examples). 9. Road map of the realization of a globally unified height system and its integration into the Global Geodetic Observing System (GGOS) Methods for Height System Unification The basic methods will be presented in terms of standards, compatibility and data quality from an analysis of the proposed methods of height connection. We have three methods available: - direct height connection inside a closed land area (continent) by geodetic (spirit) levelling in combination with gravimetry: the classical approach; it cannot connect islands separated by sea; GOCE could be used, however, to detect systematic errors in large levelling networks; - geodetic boundary value problem (GBVP) approach: a rigorous geodetic method of height unification; it is able to deal with the direct effect of height offsets and the indirect effect introduced by the use of gravity anomalies biased due to these offsets - ocean levelling: method suited to directly connect coastal tide gauges Geometric Levelling and Gravimetry Levelling networks of national or continental dimension are commonly processed in terms of potential differences between benchmarks. The potential difference between points A and B is derived from geodetic levelling in combination with gravimetry. It is B WAB g d H gini, (3-1) A cf. [Heiskanen and Moritz, 1967, ch. 4], with infinitesimal height increment dh, levelled height increment ni and gravity g. Any pair of points can be connected in this way. From W physical heights such as normal or orthometric height differences can be deduced. In low land countries measured gravity may be substituted by normal gravity without significant loss of precision. There exist national and international standards of precision (norms) for first and second order levelling. Usually there are standards for the i

20 Page: 19 of preparation and accomplishment of the measurements (e.g. maximal distance between levelling instrument and the levelling staff, the sequence of forward and backward readings, temperature measurement for the temperature reduction of the levelling staff, minimum height of readings to minimize the influence of refraction) - the calibration and daily inspection of the instruments (e.g. methods for the calibration of levelling staff and the frequency of calibrations, method and frequency for daily inspections) - maximal permissible differences and errors (e.g. maximal permissible difference between forward and backward levelling between two neighbouring benchmarks, the maximal error of a levelling loop), - determination of gravity values along the levelling lines. which shall ensure the desired precision of the levelling network. Currently, a re-measurement of the first order levelling network of Germany is under way (epoch of observations ). The official error limits (in millimeter) in terms of maximal differences between the forward and backward levellings are 0, 5L 1, 5 L (L single length between the benchmarks in kilometre) and in terms of the maximal errors of levelling loops 2 Akm (A length of the levelling loop in kilometre). Gravity values for each benchmark have to be determined 5 2 with a precision of 1 10 ms [AdV, 2009]. Levelling networks are internally not well controlled. Systematic errors are therefore not easily visible. They can be detected by comparing W between A and B with the sum of U and T, assuming a comparable precision of the two latter. The basic formula is: km km W U T, (3-2) where the gravity potential W is the sum of the normal potential U and the disturbing (or anomalous) potential T: W U T. (3-3) The above model is only applied between a small and selected number of primary levelling benchmarks, e.g., 1 st order points. Any misclosure in the above condition equation indicates: - systematic errors in the levelled potential differences W, and/or - systematic errors in space positioning, and/or GPM - systematic errors of T (either of the GOCE GPM part T or of the residual contribution res T above the maximum D/O of the GOCE GPM). It is assumed that there are no offsets between the height systems. Nevertheless, a critical item in this res respect is systematic errors in T caused by unknown height offsets outside the considered land area entering via gravity anomalies. We will deal with this aspect in the following section. Application of the method requires the availability of (1) precise 3D coordinates of the selected benchmarks, measured by GNSS and/or satellite laser ranging (SLR), (2) a normal gravity field, such as the Geodetic Reference System 1980 (GRS80), (3) a global geoid model from GOCE up to high D/O, e.g., D/O = 200, and (4) high precision and high resolution regional residual geoid heights (above D/O = 200), either from gravimetric geoid computation or automated astronomical levelling and (5) consistency of terrestrial reference systems used for positioning (and astronomical levelling) and for geoid determination [see, e.g., Kotsakis, 2008; Ihde, 2007c; Ihde et al., 2007]. As long as we deal with local or regional differences, large scale effects due to inconsistencies between the geometric and gravity field reference systems, such as differences in scale, origin or orientation, will almost cancel out.

21 Page: 20 of 173 So far we were dealing with estimable quantities, i.e. with potential differences. The method can be tested and applied without worrying about datum related issues. Even datum comparison inside of a closed land area can be done in this manner. We introduce now a datum. A fixed potential (or height) value W a is assigned to one fundamental benchmark at O. This definition is called the choice of a height datum and indicated by the superscript a. It is common practise to connect such a reference benchmark to mean sea level (MSL) at an adjacent tide gauge. All potential differences can now be referenced to this reference value and heights expressed in terms of geopotential numbers C a a a C C ( P, O) W ( O) W( P), (3-4) P all referring to the datum point O. [This definition generates positive geopotential values with increasing altitude while at the same time the actual potential relative to the Earth s center of mass is decreasing.] The geopotential numbers expressed in datum system a can be transformed to a second datum system b by an S-transformation. The S-transformation is applied to the geopotential numbers and to their error VCM. Datum transformation: Height datum transformation between datum a with reference point reference point gives O b Oa and datum b with a a W ( O ) W ( O ) C (3-5) b a ab and b a b a b W ( O ) W ( O ) W W ( O ) C W. (3-6) b b a a ab a b b W ( O ) W ( O ) C (3-7) a b ba gives where a b a b a W ( O ) W ( O ) W W ( O ) C W, (3-8) W b a a a b b ba b W and Cab Cba. a b The choice of the reference point is arbitrary. Here we assume in addition that also the choice of the reference potential value is arbitrary, i.e. a datum parameter. This is only correct if it is not an estimable quantity. See below. The choice of the value W a is arbitrary, in principle. With the coordinates of O known, we may define: a GRS 80 GOCE res W ( O) U( O) T( O) U T T. (3-9) This is possible based on the requirements listed above. In general it will hold: a W ( O) U W, (3-10) a 0 0 a where U0 is the ellipsoidal normal potential, e.g. defined by the constants of GRS80, W0 is the unknown potential difference between the level surface through O and the ellipsoidal reference potential: Thus the potential at an arbitrary point P on a levelling line is W ( P) U C W, (3-11) a a a 0 P 0 While the corresponding telluroid points P are defined via

22 Page: 21 of 173 a U P U0 C P ( '). (3-12) For clarity, the fundamental benchmark at O will be close to sea level and its geopotential number or physical (normal) height (in datum zone a ) is zero by definition; the corresponding telluroid point O is at the same location on the reference ellipsoid Geodetic Boundary Value Problem (GBVP) Approach This approach is a general method of global height system unification. It is widely discussed and tested in the literature; see, e.g., [Sansò and Venuti, 2002]. Our discussion here is based on [Rummel and Teunissen, 1988; Rummel and Ilk, 1995]. It deals with the following situation: - There exist L+1 non-overlapping height zones with 0,1,...L and the Earth s surface divided into E L. - In each height zone there exists at least one point with precise geodetic coordinates measured by GNSS or SLR; all together there are k 1,2,... K of these geodetic stations. - We have a gravity model available up to high degree and order n max, either from GRACE or GOCE; this model is considered unbiased, i.e. unaffected of height system offsets. It allows computation of disturbing potential or geoid height values (or differences) at the geodetic stations. - The missing residual part (omission part) will be computed by the solution of the GBVP using the available terrestrial global set of gravity anomalies g. - The solution of the GBVP will be dealt with in spherical and constant radius approximation. This level of approximation is clearly sufficient if a high D/O satellite gravity model is available. For a more refined model we refer again to [Sansò and Venuti, 2002]. Formulation of the GBVP approach: In each of the L+1 height zones one of the level surfaces defines the reference level W = const. One of them is chosen as reference height datum zone 0. The potential differences to the other zones are denoted C 0 W with 1,2,...L. In the datum zone 0 it holds: C W (3-13) From this and the above definitions we arrive at the linear observation equation of potential anomalies, cf. [Rummel and Teunissen, 1988]. It is U W0 C 0 N T N T, (3-14) n which leads to the well known Bruns equation T W0 C N 0. (3-15) Analogously the linear model of gravity anomalies becomes 2 CP g g( P) H ' g( P) ' P0 P0 n r g T 2 T N N n n r r (3-16) We insert Bruns equation and get the so-called fundamental equation of physical geodesy, which serves as boundary condition: g W0 C 0 T. (3-17) r r r r

23 It should also be pointed out that Summary and Final Report Page: 22 of g g C 0 (3-18) r is the unbiased gravity anomaly. The solution of the GBVP is then: ( GM ) R 2 T( P) St( PQ) g C 0 d Q, R 4 R (3-19) Q which can be re-organized into L ( GM ) R 1 T( P) St( ) g ( Q) d C 0 St( ) d r PQ Q PQ Q 1 L ( GM ) Stokes 1 T C St ( PQ ) d Q. r 2 After insertion into Bruns equation we find for points in datum zone : Stokes L ( GM ) W0 C 0 T ( P) 1 N ( P) C j0 St( PQ ) dq R 2 0 L Stokes P 0 0 ( ) j0 j0 j1 N N N P f C j1 j (3-20) (3-21) P The constant term is N0 ( GM ) R W0. The terms f j0 hold for a computation point P and are derived from an integration of the Stokes integral kernel over all points Q of datum zone j: P 1 f j0 St( PQ ) d Q. 2 (3-22) j P We will refer to N 0 C 0 as direct bias term and fj0cj0 as indirect bias term. We may now distinguish four linear models: A) Potential in point P k (coordinates perfectly known) of datum zone : W U T (3-23) with and k k k W U W C C (3-24) 0 0 k 0 ( GM ) T T T f C. (3-25) L k R GOCE res P j0 j1 j0 B) Potential difference between points Pk and in datum zones and j, respectively: ki ki ki P i (coordinates of both stations perfectly known) W U T (3-26) with W C C C C (3-27) j ki k i 0 j0 and

24 Page: 23 of 173 L L GOCE res k i ki ki ki m0 m0 n0 n0 m1 n1 T T T f C f C. (3-28) C) Heights in point P k : hk Hk N (3-29) k with the ellipsoidal height hk from GNSS or SLR, the orthometric height H C g (3-30) and k L GOCE res k k 0 0 k k j0 j0 j1 k N N N N N f C. (3-31) D) Height difference between points Pk and Pi belonging to height zones and j, respectively (= levelling with GNSS ): j j h H N (3-32) with and ki ki ki hki hk hi, (3-33) H C g C g (3-34) j j ki k i L L j GOCE res k j ki 0 j0 ki ki m0 m0 n0 n0 m1 n1 N N N N N f C f C. (3-35) Discussion: - The four cases formulated above are the point of departure for the formulation of corresponding linear least-squares adjustment (LSA) models. - Cases (A) and (C) deal with the absolute determination of potential or height values. Thus apart from the determination of height offsets also the estimation of W0 and ( GM ) are 8 included. The value of GM 0 is known already to better than 10 and may be assumed to be known. - Cases (B) and (D) deal with relative quantities. These two cases are closest to the actual needs. - In section the control of a geodetic height system inside one closed land area is discussed. Thereby we assumed a situation without height offsets. One can see from the formulas of cases (B) and (C) that even in the absence of height offsets inside the continent the effect of unknown offsets from the outside area, e.g., the oceans may have an effect. The question is their size and geographical distribution. - All models would be simpler without the indirect effect. A key question is therefore whether this part can be neglected and if yes, under what conditions. This depends on the quality and spatial resolution of the available satellite gravity model. An article by Gatti, Reguzzoni and Venuti (2011; submitted for publication) indicates that unbiased gravity model from GRACE or GOCE may lead to a negligible indirect effect Ocean Levelling Ocean levelling describes the use of oceanographic data and/or models to determine the difference in the height of the ocean surface (relative to a level surface or geoid) between different locations. The technique can be used in order to bring coastal tide gauges into one unified vertical datum, thereby

25 Page: 24 of 173 serving the purpose of making sea level records comparable (see e.g., also [ESA, 1999, ch. 3.6]), and thereby can be used in the unification of continental height systems. The sources of information required for an exercise in ocean levelling are (1) sea surface height information from either tide gauges or satellite altimetry, with heights measured relative to appropriate land levels or within modern geodetic reference frames, (2) insight into the way that the level of the ocean surface varies from place to place as a result of the ocean circulation and as a response to forcing from winds and air pressures. This insight can, in principle, be provided from a comprehensive set of ocean measurements of currents and water properties throughout the water column [cf. Cartwright and Crease, 1963 discussed above]. However, such comprehensive information is seldom available in a region, and never on a global basis. As an alternative, one can make use of the known relationships between sea level and other readily-observable ocean parameters (e.g. thermal content or wind stress), as described by the equations of motion in the ocean, within numerical ocean models. The sources of information are: Tide Gauges Tide gauges measure sea level at coastal stations relative to benchmarks on the land on which they are situated. Mean Sea Level (MSL) is an average value of sea level (relative to land) recorded over an extended period such as a month or year. An 'MSL Datum' at a site is often defined with the use of as long a set of data as possible, traditionally chosen to be a lunar nodal period of 18.6 years, although in practice much shorter periods are usually employed. These MSL Datums have been adopted as National Datums, the reference levels for most practical surveying purposes, in many countries. Examples include Ordnance Datum Newlyn (ODN) in the UK, Normaal Amsterdams Peil (NAP) in the Netherlands and Nivellement Général de la France (NGF) in France. All of these datums, when originally established a century or more ago, were close approximations of the then MSL at Newlyn, Amsterdam and Marseille, respectively. Even though MSL will have increased, relative to a land datum, since their establishment (typically by 20 cm at these locations), the national datums remain as important reference levels in each country even though they might be (and might always have been) at a different geopotential because of the variation of the ocean surface (relative to the geoid) from place to place. National datums remain of great practical (and legal) importance and cannot be disregarded now, even if we know that they do not represent exactly the same geopotential (or level ). The establishment of methods for relating them to each other is the topic of the present project. Values of MSL recorded by a tide gauge can be expressed as ellipsoidal heights with the use of a campaign, or more preferably continuous, GNSS receiver installed at a short distance from the gauge, and with its benchmark connected to the Tide Gauge Benchmark by conventional levelling. The GNSS positioning also serves the purpose of monitoring vertical crustal movement. The Permanent Service for Mean Sea Level (PSMSL, is responsible for the collection of MSL records worldwide and defines tide gauge operating procedures and standards as part of the Global Sea Level Observing System (GLOSS, The GLOSS Manuals published by the Intergovernmental Oceanographic Commission [e.g. IOC, 2006] give the requirements for GNSS monitoring at gauge sites including the construction of an adequate local benchmark network. Satellite Altimetry The technique of satellite radar altimetry has revolutionized oceanography since the launch of the first precise missions in the early 1990s. Primary precise altimetry data sources are from the TOPEX/Poseidon (T/P) mission (launched in 1992) and continued later by Jason-1 (2001) and Jason-2 (2008). With these missions a quasi-global ground track between 66 N/S is repeated every 10 days, allowing the monitoring of temporal variations in sea surface height on timescales of months and longer. The T/P/Jason ground track grid is rather sparse, being more than 300 km between tracks at the Equator, but spatial coverage can be densified with the addition of information from other, lowerflying and slightly less precise, satellite missions including ERS-1, ERS-2 and ENVISAT. Their 35 day repeat produces approximately three times higher spatial resolution than for T/P but at the expense of lower temporal resolution (and, more extreme, the geodetic missions of ERS-1 and Geosat with

26 Page: 25 of 173 long, essentially non-, repeat repetitions of ground track). The combination of sea surface height information from many of these missions has enabled the computation of Mean Sea Surface (MSS) products [e.g. Andersen and Knudsen, 2009] where mean represents an average over a given epoch (e.g ) with heights h relative to the ellipsoid. The determination of a precise MSS from altimetry requires the adoption of rigorous processing standards, with uncertainties in the MSS likely to be larger in coastal areas than for the open sea. Both tide gauges and altimetry provide ellipsoidal values of sea surface height h which can be compared to geoid values N relative to the same ellipsoid, such that one obtains a mean dynamic ocean topography (MDT) (H) determined at every location at the coast or in the deep ocean: Ocean Models h H N, (3-36) Ocean models come in many forms. The ones we are concerned with here are numerical models that provide time series of temperature and salinity, and thereby density, throughout the water column (and so are called 3-D models) together with sea surface height relative to an implicit flat geopotential surface or geoid. The averaging of sea surface heights over a given period provides a model value of the MSS for that period. A model MSS is equivalent to a model MDT. All the numerical models we are concerned with here are forced by global wind fields obtained from meteorological centres such as ECMWF or NCEP (and some also by changes in air pressure) together with 3-D fields of temperature and salinity based on hydrographic observations. Each model can be run for as many years as there are reliable forcing fields available (e.g for the MIT- Liverpool model). In the present study, it is important that the models simulate correctly the ocean dynamics from the deep ocean, across the shelf edge and on to the shallow water of the shelf, so that sea level can be accurately reproduced at the model coastline. Therefore, spatial resolution is an important model consideration. The differences between ocean circulation and shelf models tend to be primarily terminological ones. Both employ the same primitive equations that represent flow in the ocean [Gill, 1982]. Shelf models tend to be restricted to a particular region (and so need boundary conditions from a global ocean model), have higher resolution (e.g. of order 100 m rather than km) and pay special attention to fresh water inputs from the land, estuarine mixing processes etc. Such local processes clearly have little role in the gyre or global scale studies that ocean models are usually used for. In the present project, a number of ocean and shelf models will be employed. It is important to realize that while several of them may be state of the art, no one single model can be considered definitive, especially with regard to shelf-deep ocean dynamics where much remains to be learned. Another class of numerical models are 2-D (or depth averaged ) ones which do not concern themselves with density-related changes in the ocean interior but attempt to simulate the barotropic changes in sea level due to changes in the wind and air pressure. Such models employ the depthaveraged equations of motion and are used routinely in forecast and hindcast mode to simulate storm surges over continental shelves [e.g. Flather, 2000]. There is an important technical difference between the use of 2- and 3-D models, aside from the extra dimension, which relates to the forcing fields. In the 2-D case, models are fast to run and can be forced with hourly (or more frequently sampled) meteorological fields. So, they can simulate time series of storm surge at that frequency which can be averaged to produce an MSS. Many ocean models are run with monthly-averaged forcing fields which is a timescale adequately matched to that of the internal ocean variability; those wind fields do simulate the monthly surge at the coast to some extent but not as reliably as using higher frequency forcing (there being non-linearities in the way winds result in surges). As a consequence, the ability of an ocean model to simulate the MSS over the shelf, and MSL at a tide gauge, depends on the relative amplitudes of lower-frequency deep ocean processes and the wind-driven surge-derived MSS. In most of the North Atlantic cases studied in this project, the

27 Page: 26 of 173 former is of greater importance, but it is important to keep the limitations of each model in mind in any analysis. It is true to say that some ocean models do have higher frequency forcing than monthly, for example 6-hourly. There is no technical reason to avoid hourly, but such forcing data are hard to obtain globally. A further difference between 2- and 3-D modeling relates to the treatment of tides, very few deep ocean models having tides in them, although 3-D shelf models will have them. The word model is used in many ways in oceanography and sometimes a better word would be parameterisation. For example, [Rossiter, 1967] studied the relationship between changes in winds and air pressures and those of sea level at different tide gauge locations in northern Europe. He then applied the determined relationships to the mean wind conditions in order to infer the spatial variations in the MSS around the coast. This work, which was used within REUN (Réseau Européen Unifié de Nivellement) studies (the predeceesor to the UELN, United European Levelling Network) to relate national datums around Europe, demonstrated for example that there must be an increase in the MSS in the German Bight of the order of 12 cm due to the mean wind setup. A similar finding has since been obtained using 2-D numerical models. The words model and parameterisation might even be applied to, for example, the [Cartwright and Crease, 1963] determination of sea level difference across the English Channel using a range of ocean data and informed appreciation of the main processes involved. To return to the 3-D models that will provide our main source of model information, how are we able to form an independent impression of how good they are? As explained above, in ocean leveling one can use ocean circulation modeling of various kinds to provide an MDT. The models may be free running runs or, considerably more preferable, constrained by hydrographic and possibly other oceanographic information. For example, the Liverpool version of the MIT model is a global model run for which simply provides dynamical consistency to hydrographic fields (temperatures and salinities) provided by the UK Met Office. It can be seen that in this case we use an ocean circulation model (OCM) as a tool to convert density data into an MDT i.e. a data-based approach which just uses a more sophisticated model than in, for example, an analysis of drifter information based on simple geostrophic arguments [Maximenko et al., 2009]. Our approach is to use as many models as possible; for example, aside from the MIT-Liverpool model, the ECCO and GECCO models can be employed. These particular ones can be described as a way of calculating MDT from ocean dynamics plus the assimilation of as many observations as possible (including altimetry and some tide gauge data). The POLGCOMS model approaches data assimilation from a shelf rather than ocean perspective, as outlined above. Our recommendation then is to obtain a number of MDTs based on data and models in different ways (from simple geostrophy to full OCM). We can make an assessment of errors by comparing results with mutually exclusive data inputs (e.g. drifters plus simple local model vs. density and full OCM). It can be seen that, in this way of doing things, density is not used directly, as in the classical steric leveling method which requires the assumption of a level of no motion, nor are other ocean parameters such as currents as in [Cartwright and Crease, 1963], but via an ocean model which effectively adds an estimate for the velocity at the presumed level of no motion (as well as correcting for a geostrophic effects). The use of oceanographic information, whether as sets of data or from ocean models, can be called the ocean method which can be compared to the geodetic methods described below. The tests of compatibility between these sources of information is what lends credence to results from ocean levelling. For a recent example of such tests of compatibility, see [Thompson et al., 2009]. (1) The Traditional Method for Datum Connection using Tide Gauges Let us suppose that we have two countries with one tide gauge station, each country defining sea level at its station in terms of its national datum which, as explained above, may have been derived from historical sea level measurements itself. The two countries could be adjacent to each other or on opposite sides of the world.

28 Page: 27 of 173 Let us also define an epoch for the datum connection study (e.g ). And let us also assume that we have an ocean model that we believe is accurate which, independently of the tide gauge data, can provide an estimate of sea level difference between the sites. If MSL (measured to a local national datum) in the two countries averaged over the epoch is HL1 and HL2 respectively, and if the model suggests that a sea level difference ΔH exists between them, then the datum in country 2 must lie below that in country 1 by (HL2 - HL1 - ΔH) (Figure 3-8). Figure 3-8: Difference in sea levels = ΔH = (HL2 + D2) - (HL1 + D1) = (h2 N2) (h1 N1) This simple method, which requires no geoid model, was the basis of the European REUN/UELN studies and demonstrates the importance of an ocean model (the word used in its general sense) to the computed datum offset. It implies that the datum offset will be time dependent if h1 and h2 change differently with time (e.g. due to differential land movement). (2) The Geodetic Method for Datum Connection using Tide Gauges This method is essentially the same as method 1 but uses gravity information to provide a leveling connection between sites. In this method, GNSS measurements at benchmarks near to each tide gauge, and leveling connection between each GNSS and tide gauge benchmark, provide values of ellipsoidal height h1 and h2 over the chosen epoch. One also has values of N at each site from a geoid model based on satellite and/or terrestrial gravity information. An essential validation of the model, and other parameters in the system (such as the accuracy of the geoid model), follows from a test that (h2 N2) (h1- N1) - ΔH is equal to zero. The national datums at the two sites are then known to be offset as for method 1. In practice, from knowledge of the various error sources involved, this validation is possible only insofar as there are no geoid omission errors, as discussed further below. It is, therefore, desirable for any national coastline to have a number of tide gauges, and thereby a set of (possibly) independent omission errors which might average down to a much smaller quantity. The defect of this optimistic approach is that the national datums might themselves contain systematic (decimetric) spatial errors (tilts), as also mentioned further below. (3) The Geodetic Method for Datum Connection using Altimetry This method is similar to method 2 but in this case the ellipsoidal height h of sea level comes from satellite altimetry. Once again, N comes from a geoid model. Over two decades of precise altimetry now exist, with many groups in Europe, USA and other parts of the world working on the determination of global and regional sea level change. As mentioned above, several groups [e.g. Andersen and Knudsen, 2009] have computed global MSS models by making time averages of h relative to the reference ellipsoid (or geocentre). Most altimeter missions to date have employed the same reference ellipsoid (the so-called TOPEX ellipsoid), which differs in mean equatorial radius from the GRS80 ellipsoid employed in GNSS processing by 70 cm, but with an almost identical flattening coefficient. One can also see that altimeter information are automatically provided in a 'mean tide' system with regard to the permanent tide component of the Earth's shape, unlike the 'tide free' convention for expression of GNSS heights

29 Page: 28 of 173 including the ellipsoidal heights of sea level at gauges [McCarthy and Petit, 2003]. Before an altimetric MSS can be compared to a geoid model, the two sets of information have to be transformed so as to be related to the same ellipsoid and tide system. Altimetry adds to the datum connection exercise in the same way as GNSS added to the tide gauge data, by providing confidence in and validation of the ocean model used to provide the sea level difference between locations at the coast. Several groups are now attempting a validation of the new geoid models available from the GRACE and GOCE missions by subtracting them in various ways from MSS models from altimetry and then comparing the difference to the MDT values from ocean models. In the present project, the validation must take place along the continental shelf as well as in the deep ocean, and in fact must attempt a comparison as close to the coast as the altimeter data allow. Attention must be paid to the different spectral content of altimetric and geoid model fields, especially near to the shelf edge and coast where both fields could contain spatial high-frequency components [cf. Bingham et al., 2008; Albertella and Rummel, 2009]. Nevertheless, it is possible that one might be able to smooth through the spatial noise in such a subtraction, thereby validating the model MDTs on a regional basis (e.g. on the spatial scale of the coastline of each country in question). The importance of a good ocean model can be appreciated further here, in that altimeter data can never be obtained exactly at the coast (e.g. the footprints of the radiometers used for tropospheric correction are of the order of 50 km and their data are significantly affected by the presence of land) and so cannot provide a measurement of Δh exactly where we need it. However, a case can be made that the ocean model can be used with more confidence at the coast once it has been validated in the nearby coastal waters. The reliability of this assumption clearly depends on the particular coastline; for example, the presence of major rivers, whose fresh water input is not included in the model, could result in decimetric errors in a model-derived MDT. What if we have no ocean model, or none that we believe is accurate in a particular region? In this case, it can be seen that ellipsoidal tide gauge (or altimeter) height minus geoid provides comparable information on sea level difference to a model. However, while any sea level difference in method 2, based on measurement of sea level at discrete tide gauge locations, will clearly contain uncertainties from geoid model omission errors, those for method 3, based on altimetry minus geoid averaged along a coastline, should provide more reliable estimates of sea level difference. Geoid Model Omission Errors The two most important questions in any of the analyses described above are: (1) what spatial scales are the new geoid modes accurate over?; and (2) what spatial scales in the real ocean have the largest amplitude? These questions have been addressed extensively previously [e.g. ESA, 1999] but it is worth recapping them briefly here. With regard to (1), any geoid model derived from the present space gravity mission will have omission errors of many 10s of cm for spatial scales less than 200 km. In the present work, we have assumed that values of geoid height N will be available, expressed as a series of spherical harmonics and truncated at, say, D/O equal to 200 in the case of GOCE (commission error currently 6 to 8 cm and ultimately 2 to 3 cm) or D/O equal to 100 to 120 in the case of GRACE (commission error 2 to 3 cm). In principle, omission errors could be reduced with the use of marine gravity complemented by terrestrial gravity measured near to the coast, although the quality and quantity of marine gravity in particular is limited in many regions. In this way, one might imagine regional geoid models might be developed, precise to high D/O, incorporating all the best features of space and in situ measurements. In practice, and in the medium term, we will have to live with the large omission errors in any analysis. With regard to (2), as discussed in [ESA, 1999, p.53], a generally adopted oceanographic assumption is that the shortest important spatial scales of the steady-state ocean topography are given by the Rossby radius of deformation. That publication states: At scales larger than this, the ocean circulation

30 Page: 29 of 173 is largely described in terms of geostrophic balance in response to applied forcings (e.g. wind stress) and is therefore accessible to measurement by observations of sea level. At shorter scales than this, the circulation tends to have the character of turbulent cascades which are not in geostrophic balance, [Gill, 1982]. At mid-latitudes, the radius associated with the first baroclinic mode (most currents at these latitudes have primarily baroclinic components) is 40km, increasing towards the equator. See also [Pickard and Pond, 1983, section ]. From this it follows that even the high spatial resolution of GOCE will not suffice for the separation of an MSS from the geoid due to the missing omission part of the geoid model. Therefore, depending on the analysis, determination of GOCE H h N will almost certainly require some low-pass filtering of the high spatial resolution altimetric h values before comparison to the GOCE model in order to derive MDT information [Bingham et al., 2008; Albertella and Rummel, 2009]; analysis at higher spatial scales remaining problematical. The ESA (1999) words were, however, written from a primarily ocean circulation perspective. It is important to realise that on the shelf and at the coast there are many processes which result in MSS fluctuations at short spatial scales. These will be (i) represented inadequately or not at all in MSS models such as Andersen and Knudsen (1999) due to altimetry being unable to measure exactly up to the coast and also to spatial and sampling limitations, (ii) represented poorly in ocean models e.g. due to river runoff as mentioned above, which could be decimetric in amplitude and (iii) there will be the geoid omission errors. Working at the coast, therefore, presents challenges due to our imperfection of information at the shortest scales. Spatially-Dependent Biases in National Datums It is well known that some national datums (e.g. UK, France) contain tilts of order decimetres per 1000 km with respect to a true level surface. These long-standing and poorly-understood tilts have primarily a latitudinal dependence in most countries [e.g. Thompson, 1980], although the Canadian datum (CGVD28) provides an example of a tilt of the order of a metre between the Atlantic and Pacific coasts. When large tilts are known to exist, then it would be sensible to consider the studies described above to be performed between sections of a national coastline Standards and Data Requirements In Table 3-2 we give all quantities relevant for height unification and a list of items which have to be defined in terms of requirements and standards. For the modelling of the gravitational field and the unification of height systems, the positions of all observations and the observations themselves have to be given in uniform and consistent reference systems. It is desirable (but often practically not feasible), that all observations can be related to a single realization of these reference systems (reference frame). In order to describe the data sets used for the height system unification, one has to know the basic parameters of i) the reference system (e.g. datum definition, conventions of the reference system especially with respect to the tidal system), ii) information about the reference frames (e.g. datum realization, period of observations, epoch of coordinates, accuracy estimates), and, iii) information about the coordinate systems According to ISO19111 Spatial Referencing by Coordinates, coordinates are referenced to coordinate reference systems. A coordinate reference system comprise of the datum and the coordinate system. The order of the coordinates within a coordinate tuple and their unit(s) of measure are also part of the coordinate reference system.

31 Page: 30 of 173 Table 3-2: Requirements / Standards Table Data Levelling order Vertical Datum Epoch Height Type Gravity Permanent Tides Coord. System Accuracy Tidal Correct. Horizontal Datum H h N(GOCE) X X X X X X X X X X - - X - - X X X X X X - X X X - - N(residual) or Gravity - X X - X X X X X X Tide Gauges Ocean Models Altimetry - X X X X X X - - X - X X X X - X X X X - Remark: The columns may seem partly redundant and some of them related to one type of observation only. The main items are the datum, the coordinate system, the epoch, tidal correction especially with respect to the permanent tides and the accuracy. These items are essential for all kind of observations we are dealing with. All other columns may be explained by these items, e.g.: - Column Levelling order is only marked/valid for row H and can be explained under item accuracy - one can combine columns horizontal datum and vertical datum to datum - gravity is only marked for H ; it is used to define the different physical height types ; the different types of heights are 1D-equivalents to coordinate systems for 2D or 3D (ellipsoidal, spherical); therefore, one can combine the columns height type and gravity with coordinate system - permanent tides are one part of the tidal corrections In the following paragraphs each observation quantity as shown in Table 3-2 is specified in more detail. A detailed analysis of all standards applicable to the observation quantity is provided in [AD-7]. H (Physical Heights of national height systems obtained by spirit levelling) In case of the one dimensional vertical reference systems, the vertical datum is defined by the reference tide gauge, e.g. NAP, Marseille, Newlyn. The realization of the vertical datum will be described by the datum point/points and the height of the datum point/points used in the adjustment of the levelling network. Currently, most of the national height systems are still realized by spirit levellings and gravity observations along the levelling lines. The basic quantities obtained from these observations are geopotential numbers, i.e. differences of the geopotential between neighbouring benchmarks. The kind of physical heights (comparable with the coordinate system for two or three dimensional reference systems) depends on the kind of gravity values used for the scaling of these geopotential numbers. One distinguishes dynamic heights, orthometric heights, normal heights and normal-orthometric heights. The conventions for these gravity values have to be known, e.g. the applied formula for the determination of the mean normal vertical value in case of normal heights. Usually, levelling observations are not corrected for solid earth tides. Due to the small distances between the levelling instrument and the levelling staff, the temporal variable part of the tidal variations is almost cancelled out. On the contrary, the part of the permanent tides is of importance in larger levelling networks. If no tidal corrections were applied, one assumes that the heights are mean tide values. Up to know, only a few countries have related their height system to zero tide according to the IAG resolution [Mäkkinen and Ihde, 2008].

32 Page: 31 of 173 Generally, the observation period for national levelling networks last several years or even decades. The heights obtained from the network adjustment refer to the mean epoch of observations, if no comprehensive precise models of temporal height changes, e.g. like the post glacial uplift models, can be applied. Summarising, the basic pieces of information to describe the realisation of height systems are i) the order of the levelling network, ii) the period of the observations or the epoch of the heights, iii) as well as parameters of the levelling network adjustment (e.g. the accuracy and the redundancy) iv) type of heights (normal, orthometric,...) h (Ellipsoidal Coordinates / Heights) Ellipsoidal Coordinates, resulting from observations of the different space techniques, are referenced to the International Terrestrial Reference System (ITRS) which is defined by the IERS conventions (IERS2010). In general, it is described as a three dimensional spatial reference system co-rotating with the Earth in its diurnal motion in space. The origin being the centre of mass for the whole Earth including oceans and atmosphere, the orientation is equatorial and the unit length is meter (SI). The datum is realized through a combination of the different observation techniques used for the computation of the International Terrestrial Reference Frames (e.g. European data sets are mostly referenced to the European Terrestrial Reference System 1989 (ETRS89). ETRS89 coincides with the ITRS at the epoch and is fixed to the stable part of the Eurasian Plate ( The reference frame of GNSS ellipsoidal heights in Canada has a hierarchy structure [Craymer, 2006] that consists of active and passive components. On the top is the Canadian Active Control System, part of the global ITRF and IGS networks. One level below are the regional ACP networks installed in support of crustal motion studies and sea level rise monitoring, e.g., WARDEN and WCDA. The passive component consists of 151 stations (highly stable, forced-centering pillars) of the Canadian Base Network (CBN) and Provincial High Precision Networks. The density of CBN varies from 200 km in southern Canada to 1,000 km in the northern areas. A network of all GNSS surveys conducted by GSD at vertical control stations are integrated in the SuperNet using the CBN frame [Craymer and Lapelle, 1997] and the ellipsoidal heights are used in the validation of the regional gravimetric geoid models. The coordinates of the ITRS are referenced to a rectangular Cartesian coordinate system. These Cartesian coordinates can be converted as well into an ellipsoidal coordinate system (ellipsoidal longitude, latitude and height) or into a spherical coordinate system, providing equivalent representations used for various other applications. The ellipsoidal heights are usually referred to the reference ellipsoid defined by the Geodetic Reference System 1980 (GRS80). The IERS conventions describe the main corrections and reductions which have to be applied to the data (current version: IERS conventions 2010). The solid earth tide model for the displacement of reference points contains the time-dependent tidal variations as well as a time-independent part. The coordinates computed with this model are conventional tide free values. The conventions also contain the formula for conversion of the coordinates to mean tide values, whereas they are usually not used in the analysis of space geodetic data in general (IERS conventions 2010). The terrestrial reference system will be realized by global networks of GNSS, SLR and VLBI stations (ITRF), i.e. by a reference frame. Continental networks (e.g. EPN) or country specific networks should be connected to the global networks and in that way they represent a densification of former ones. The coordinates of these stations are known with the highest possible accuracy. After some years of observation, the velocities of these stations can be derived as well, describing the temporal evolution.

33 Page: 32 of 173 The majority of observation stations are occupied only temporally within a GNSS campaign. The observation period per station is typically a few hours or days (e.g. GNSS observations at levelling points). The coordinates of these stations have a very high internal accuracy of about a few millimetres only. The datum realization in such networks is usually less precise. A determination of station velocities is not possible due to the short observation period. Therefore, the coordinates are usually related to the mean epoch of the campaign. The coordinate determination using a real time positioning service is a very efficient way for georeferencing, if an accuracy of a few centimetres seems to be sufficient (e.g. gravity surveys). N (GOCE Geoid) N(GOCE) are geoid heights relative to an adopted reference ellipsoid and normal gravity field. The geoid heights are computed from a spherical harmonic representation of one of the available GOCE gravity field models up to a certain maximum degree and order (d/o). The gravity models may be based on GOCE data only (SST and gradiometry); they may be computed with or without some a priori gravity information or they are a combination of GOCE and GRACE (and satellite laser ranging) data. Due to the fact that GOCE does not cover the two polar areas (with an opening angle of 6.5 degrees) some type of regularization has to be applied. The expansion of the spherical harmonic series of GOCE gravity models has usually a maximum d/o of between 210 and 250. A detailed description of the adopted gravity field model and of the applied formulas and parameters is needed in our case. Official GOCE gravity models are computed according to the GOCE-standards, see ESA-document GO-TN-HPF-GS The necessary background is given in the GOCE Level-2 Data Handbook (ESA-document GO-MA-HPF-GS-0110). N (Residual Geoid) N res is the residual geoid height above the resolution of the GOCE derived geoid height N(GOCE) and corresponds to the omission error of GOCE (see section ). It can be computed via Stokes integration from a regional set of gravity anomalies. Regional means, that it is necessary to use gravity anomalies in a certain spherical cap around the computation points. It is necessary to determine the necessary size of this cap also considering suitable modifications of the Stokes kernel in order to keep the truncation error (contribution of residual gravity anomalies outside the chosen cap) small enough not to degrade the quality of N(GOCE). Gravity anomalies given in different datum zones are affected by the corresponding height datum and are therefore biased due to height datum offsets. In section , this was called the indirect bias term. It needs to be determined, under which conditions it is possible, to neglect the indirect bias term. This might have an impact on the requirements listed below. An alternative method for quantifying the residual geoid height N res is to make use of a very high degree and order geopotential model like EGM2008 and to compute the residual geoid height from this model. The omission error above the maximum degree of EGM2008 needs to be quantified. Gravity Anomalies Terrestrial, marine and aero-gravimetric observations are the main data source to determine the omission part of the earth gravity field which is not covered by satellite based gravity models. They are used for the estimation of combined global gravity models and of regional gravity models with high spatial resolution. Gravity anomaly databases usually consist of point free-air anomalies and national or regional grids of Faye and/or Bouguer anomalies. The free-air anomaly is computed from the gravity measurement at a point of the topographic surface and the normal gravity at the telluroid point that corresponds to the surface point. The gravimetric measurement is corrected for the solid Earth tides, gravimeter drift, local atmospheric effects, and scale of gravimeter [Torge, 1989]. The corrected measurement is tied to gravity base stations. It should be noted that there are no international standards for the gravimetric measurements and correction procedures. The gravity anomalies are computed according to the

34 Page: 33 of 173 definitions and procedures described in [NGA, 2008] and accessed through the International Gravimetric Bureau (BGI) website. Modern gravity reference frames are realized by absolute gravimeter measurements. The absolute gravimeters realising technically their own datums, since they measure gravity directly without relative ties to any other station. The datum realized by these instruments is controlled due to a careful calibration and comparison of the instruments ( Relative gravity observations are connected to the absolute gravity stations and densify these networks. Older gravity observation are usually related to the datum of the International Gravity Standardization Network 1971 (IGSN71) or the Potsdam Gravity System. IGSN71 and the absolute gravity measurements agree within their error limits. The Potsdam gravity system has an offset of -14mGal with respect to IGSN71 ( ). Beginning in the 1950s, different kinds of tidal correction for gravity observations have been used over the time. With respect to the permanent tides, the initial correction led to conventional non-tidal values. The Honkasalo correction restores the effect of the permanent tides, which was removed in the tidal correction procedure before. The resulting mean-tidal values were used for the computation of IGSN71. Because the mean tide gravity values are not compatible for the geoid determination with the Stokes formula without additional corrections, the IAG decided 1979 in Canberra, to neglect the Honkasallo correction and thus to revert to the non-tidal or tide free system [Mäkkinen and Ihde, 2008, In 1983, the IAG revisited this decision again and recommended that one remove all tidal effects (tide-free) but then restore only the permanent tidal deformation and the associated changes in the Earth s potential associated with that restored deformation ( The corresponding correction led to zero tide gravity values. Typically, the gravity measurements needed for a gravity field model with high spatial resolution were carried out only ones during a period of several decades. Therefore, corrections to a common time epoch are not possible. Today, with GNSS techniques the 3D position of gravity observation points can easily determined with sub-decimetre accuracy. The coordinates (ellipsoidal latitude, longitude and height) can be referred directly to a global reference system with a high precision. Positioning data of older gravity measurements are related to the various national vertical and horizontal references systems. Usually, the positioning was carried out with simple techniques, e.g. the coordinates were obtained from charts. Therefore, the horizontal and vertical coordinates may have a low accuracy, even if the necessary transformations to a global reference system have been considered. As the geoid is a comparatively smooth surface (geoid variations are in the range of centimetre or decimetre per kilometre), errors in the horizontal position coordinates are apparently of minor importance. But nevertheless, the horizontal coordinates should have an accuracy of a few meters to avoid a loss of accuracy (especially in the mountainous areas), since today s digital elevation models used for the computation of topographic reductions have already a very high spatial resolution. N (Observed Geoid Undulations / Height Anomalies at GNSS Levelling Points) From GNSS observations at benchmarks of the national levelling networks one obtains observed geoid undulations or height anomalies. One has to bear in mind that these height anomalies are not universal but are confined to the realisation of the national height system. They imply the relation between the global reference system and the adopted reference ellipsoid on the one hand and the zero level of the national height system on the other hand. A typical application of this information is the determination of the reference surface of the national height system which will be used in order to transform ellipsoidal heights into physical heights with respect to the national height system. Another application is the validation of global gravity models and the height system unification. In this regard, the different standards for space geodetic techniques, physical heights and gravity field modelling (especially with respect to the permanent tide correction) have to be taken into account. The negligence of the different

35 Page: 34 of 173 permanent tidal corrections implies an additional latitude dependent variation of the height system offsets. Tide Gauges, Ocean Models & Altimetry Details about standards and their impact are summarized in [AD-7]. 3.2 Data, Standards and Conventions Data A project web site has been created under: This web site has an open section where the available data sets are described in terms of meta-data. In addition project presentations given at conferences are available as well and an extensive bibliography about height systems and their unification is maintained at the open part of this web site. In the following characteristics of some data sets are described in more detail GOCE Gravity Field Models and Performance Overview of GOCE based Gravity Field Models Figure 3-9 provides an overview of all GOCE based gravity field models available at the time of writing of this document. It defines how many data from different satellites, as well as if terrestrial/altimetric information have been used for the model computation. The GOCE-DIR, GOCE- TIM and GOCE-SPW models are result of ESA s High level Processing Facility (HPF), while EIGEN, GOCO and DGM models are produced by individual groups or consortia applying different processing strategies or by adding additional data from other satellites or by incorporating terrestrial and altimetric information. Most recent models available from GOCE are the GOCE-DIR4 and the GOCE-TIM4 models. Table 3-3 summarizes the characteristics of the two recent GOCE models produced by the HPF.

36 Page: 35 of 173 Figure 3-9: Overview of GOCE models (status December 2013). Colour code indicates which data have been used for the models: blue = GOCE, green = GRACE, yellow = CHAMP, grey = SLR to Lageos, red = terrestrial/altimetric information. Table 3-3: ESA Release 4 GOCE Model Characteristics DIR4 TIM4 Maximum D/O GOCE Volume Data ; ~2.3yrs (net) ; ~2.2yrs (net) Gravity Gradients V xx, V yy, V zz, V xz ~288 Mio. Obs. V xx, V yy, V zz, V xz ~279 Mio. Obs. Gradient Filter Band-pass filter ARMA filter per segment GOCE SST (GNSS) - Short arc approach (d/o 130) GRACE SST (K- Band) GRGS RL02 (d/o 55), GFZ RL05 (d/o ) - LAGEOS 1/2 (SLR) , ~25 yrs - Regularization Iterative spherical cap (d/o 260) based on GRACE/LAGEOS. Kaula zero constraint (d/o > 200) Kaula zero constraint (near zonals and for d/o > 180)

37 SQRT Signal Degree Variance in Geoid [m] Performance of GOCE Models and Geoid Summary and Final Report Page: 36 of 173 Figure 3-10 shows the signal degree variances for the most recent GOCE gravity field solutions compared to their predecessors. It can be clearly identified that with more data the signal content has been increased. From the comparison with the terrestrial/altimetric based EGM2008 model one can assume that full signal of the Earth gravity field can be determined solely from GOCE up to degree and order 190 to 200 corresponding to spatial resolution of 100 km or slightly higher. With more data and specifically with data from the low orbit mission one can assume that the signal strength of the GOCE models can be further increased. Figure 3-11 shows the estimated errors in terms of cumulative degree variances. These error estimates include all coefficients errors (also the zonal and near zonals affected by the polar gap). which implies that they might be a little bit pessimistic. When comparing the two models one can clearly identify that they deliver quite different estimates for the mean geoid errors at degree 200 (1.2 cm for GOCE-DIR4, 4.0 cm for GOCE-TIM4) Tscherning-Rapp EGM2008 GOCE-DIR3 GOCE-DIR4 GOCE-TIM3 GOCE-TIM Degree Figure 3-10: Signal Degree Variances (GOCE RL4 vs. GOCE RL3 Models) Figure 3-11: Cumulative Geoid Errors from model error estimates

38 Page: 37 of 173 In order to identify how realistic these errors are one needs to compare the GOCE signals with external independent information. For this purpose geoid heights determined from GNSS levelling in well observed areas can be applied. Figure 3-12 shows the results obtained for Germany when applying different degrees of truncation. First one can observe that with the GOCE models the EGM2008 model cannot be improved in this area. The reason is that high quality terrestrial data have been used for this and surrounding areas for the EGM2008 development. Looking at the geoid error at degree 200 we can identify that both models are at a level of 3.8 cm RMS. This is very close to the 4 cm as estimated by the errors of the GOCE-TIM4 model. Taking all error components into account (levelling errors in Figure 3-12, polar gaps in Figure 3-11) one can assume that the most recent GOCE models are at a level of about 3 cm at spatial resolution of 100 km (corresponding to degree and order 200). This is not yet the goal of 2 cm, but one can assume that with at least one more year of data at lower altitude one can reach this level. Figure 3-12: RMS of Geoid Differences at GNSS-Levelling Points in Germany after planar Fit (Omission Error estimated from EGM2008) In order to identify the impact of the new GOCE models in areas where not such good terrestrial information is available a similar study has been performed for Brazil. Figure 3-13 shows the results of the differences for various truncation degrees. As it can be identified for this area there is a significant improvement with respect to EGM2008 (at a level of 6 cm RMS). This means that with GOCE the geoid can be much better determined as using the EGM2008 model incorporating some terrestrial data of low quality. When comparing these results with the results obtained in Germany and assuming that GOCE performs equally well in all areas of the world, there are good reasons to assume that the higher RMS values in Brazil are caused by less accurate levelling and not by the GOCE geoid. These are very encouraging results and show that GOCE has big impact in areas with not such good and dense geodetic infrastructure like in Germany or other countries.

39 Page: 38 of 173 Figure 3-13: RMS of Geoid Differences at GNSS-Leveling Points in Brazil after planar Fit (Omission Error estimated from EGM2008) GNSS Levelling Data German GNSS/Leveling Points (BKG_GNSS_NIV_geoid_points) The data set contains observed height anomalies/quasigeoid undulations ζ determined by GNSS measurements at 950 leveling benchmarks, according to RB 3.3., p.40. The majority of points (about 700) have been occupied during the corresponding GNSS campaigns carried out between 1994 and 2001 within 19 campaigns by the German Federal States (Bundesländer) for at least 24 hours, most of them by 48 hours. In 2004 the station coordinates of the so-called SAPOS stations (the permanent GNSS network of the German Federal States for satellite positioning) were newly adjusted yielding a new realization of the ETRS89 in Germany. This realization of ETRS89 was applied to the data set by the German Federal States. Relevant parameters for the data set are summarized in Table 3-4. Table 3-4: Description of German GNSS/Leveling data set (BKG_GNSS_NIV_geoid_points) Number of points 950 Average spacing between points ~ 30 km Ellipsoidal heights h Spatial reference system ETRS89 Spatial reference frame ETRS89/DREF91 Type of heights Ellipsoidal heights related to GRS80 ellipsoid Treatment of permanent tides Tide-free Epoch Physical heights H Vertical datum NAP Reference frame DHHN92 Type of heights Normal heights Treatment of permanent tides Mean-tide Epoch Reference frame of gravity values DSGN94, DHSN96 European GNSS/Leveling Points (EUVN_DA)

40 Page: 39 of 173 The European Unified Vertical Network (EUVN) was successfully completed in 2000 (Ihde et al, 2000). It is a set of high quality height anomalies derived from GNSS/leveling on benchmarks. Due to the growing need for more detailed GNSS/leveling information, the EUREF Technical Working Group (TWG) initiated the EUVN Densification Action (EUVN_DA) in 2003, targeting at the establishment of a dense, homogeneous continental GNSS/leveling database. EUVN_DA relies on the experiences of EUVN; it also functions as a continental GNSS/leveling reference network, but it was built up from separate national contributions, where the submitted data were homogenized and added to a uniform database. The European National Mapping Agencies generously supported the project; they provided existing, updated or new measurement data sets. As of December 2009, 25 countries participated, and the database contained more than 1400 GNSS/leveling points. The submitted GNSS and leveling data were transformed into the respective common reference frames. The GNSS coordinates referred to the different realizations of ETRS89, while the leveling data were transformed to the actual realization of the European Vertical Reference System (EVRS). During the project's lifetime, the EVRS realization changed from EVRF2000 to EVRF2007 (Sacher et al, 2009). The introduction of the EVRF2007 in 2008 also changed the handling of the permanent tide. Formerly, the permanent tide was inconsistently handled in leveling (mean tide) and GNSS analysis (tide free). Due to the change from the mean to zero tide system in EVRF2007, the GNSS-derived ellipsoidal heights were also transformed from the tide free to the zero tide system. The leveling data validation and the sequential UELN adjustments were done at the UELN/EUVN Data Centre at BKG in Leipzig. To ensure the homogeneity of all leveling data, only those countries could be used, which are part of the UELN. A detailed description of the transformation procedures and parameters applied to the whole data is given in the final report (Kenyeres et al, 2010). A PDF copy of the final report can be downloaded from: Related Projects EUVN-DA Introduction. Table 3-5 summarizes the relevant parameters for the EUVN-DA data set. Table 3-5: Description of EUVN-DA data set Number of points >1400 (Dec. 2009) Average spacing between points ~ 100 km Treatment of PGR For Nordic countries Ellipsoidal heights h Spatial reference system ETRS89 Spatial reference frame National realizations of ETRS89 Type of heights Ellipsoidal heights related to GRS80 ellipsoid Treatment of permanent tides zero-tide Epoch Physical heights H Vertical datum NAP Reference frame EVRF2007 (zero-tide) Type of heights Normal heights and geopotential numbers Treatment of permanent tides zero-tide Epoch (see column 7 in Table 1 in 2.4.1) Reference frame of gravity values National reference frames According to the transformation and homogenization of the national data sets, carried out by the current best knowledge, the EUVN_DA data set represents more than 1400 homogenized GNSS/leveling points over Europe. Canadian GNSS/Leveling Points

41 Page: 40 of 173 Figure 3-14 shows the distribution of 2579 GNSS BMs data points from Nov07 data set used for geoid validation in Canada. These data come from different adjustments that include the mainland, islands (Vancouver Island, Prince Edward Island, Newfoundland, and Anticosti) and other independent local levelling networks which include a tide gauge. From this data set, a subset of 308 Level A points (Figure 3-15) is selected that have orthometric heights in Nov07, NAVD88, and CGVD28. These 308 GNSS BMs sample fairly the Canadian landmasses, and therefore can be used for computing the offset of the vertical reference surfaces using GNSS/levelling data. Figure 3-14: Distribution of 2579 Nov07 GNSS BMs in Canada Figure 3-15: Distribution of a subset of 308 GNSS BMs in Canada (Nov07, NAVD88, and CGVD28) GNSS ellipsoidal heights come from the newest adjustment of the SuperNet network in Canada (Craymer and Lapelle, 1997) in ITRF2005, epoch It should be noted that the ellipsoidal heights were corrected for the effect of glacial isostatic adjustment of the Earth crust using a crustal velocity model developed by GSD of Canada. Among the 2759 points, there are GNSS BMs and GNSS TGs points with ellipsoidal heights given in ITRF2000 ( nominal epoch) and ITRF97 ( nominal epoch). Three sets of GNSS TGs were included in the computation of the offset of the vertical reference frames in Canada: 95 points with Nov07 orthometric heights, 72 points with CGVD28 normal-orthometric heights and 30 points with Helmert orthometric heights. US GNSS/Leveling Points The US GNSS BMs data set consists of points from the continental USA made available by NGS at (Figure 3-16). The GNSS ellipsoidal heights are given in NAD83 (CORS96) and were transformed to ITRF2005 (2006.0) using software developed by NGS. The Canadian GNSS BMs points in Figure 3-16 were replaced by the data set described above. Ellipsoidal heights are classified in 5 orders and 2 classes for each order and a 00 code for points with a given absolute error. All flagged outliers were removed.

42 Page: 41 of 173 Figure 3-16: GNSS BMs in USA Oceanographic Data Sets The main oceanographic data sets used in the unification of height systems are listed below. The objective of any Preliminary (and subsequent) Analysis is to obtain consistency between the several sets of information and thereby be able to related national datums to each other reliably: (1) Ellipsoidal sea level from tide gauges. This data set comes from tide gauges alongside which have been operated campaign or continuous GNSS receivers, with conventional levelling having been undertaken between tide gauge and GNSS benchmarks, such that the gauge data can be expressed as ellipsoidal heights. This information can be provided for any epoch for tide gauge exist in the database of the Permanent Service for Mean Sea Level, and for which ellipsoidal heights of the GNSS benchmarks have been computed. These tide gauge data are contributed to the PSMSL by national agencies and measured sea levels can also be expressed relative to the national datums. (2) Satellite altimeter data. These data come primarily in the form of mean sea surface information, with MSS values expressed relative to a reference ellipsoid. For example, suitable data sets will be DNSC10 obtained from the Technical University of Denmark and CLS2011 obtained from the French AVISO data distribution service. (3) Models of the geoid. These data sets can either be based entirely on space gravity data (e.g. GOCO02S) or blended with terrestrial and marine gravity (e.g. EIGEN06). They can be readily obtained from the ICGEM, Potsdam. (4) Estimates of ocean dynamic topography over a defined period obtained from ocean numerical models and from combinations of ocean observations. Models to be employed include: MIT/Liverpool 1/5 1/6 model (for the N Atlantic, up to 1 degree globally). The Liverpool version of the MIT model provides dynamical consistency to hydrographic fields (temperatures and salinities) provided by the UK Met Office. This model has been run for the period OCCAM 1/12 global ocean model. The model has been run from 1985 to 2006 inclusive. It is initiated with climatological ocean temperature and salinity data, and forced with wind stresses and atmospheric fluxes from meteorological reanalyses, but is otherwise unconstrained by observational data. ECCO2 is a data assimilating ocean model at approximately 18 km resolution ( cubed sphere configuration). It assimilates a wide variety of ocean observations using a Green s Function approach. Data are available on a regular ¼ degree grid, for the period inclusive. ECCO-Godae is a 11 data assimilating ocean model covering the region 80S to 80N. It assimilates a wide variety of ocean observations using an adjoint method. Data are available for the period inclusive. GECCO is a model very similar to the ECCO-Godae model, with data available from 1952 to 2001 inclusive

43 Summary and Final Report Page: 42 of 173 Maximenko 2009 is a dynamic topography derived from a combination of satellite altimeter data, GRACE geoid data, drifter measurements, and wind stress observations, combined using a reduced dynamics approach with a cost-function based weighting to balance the relative strengths of (altimetry-geoid) on long scales and (drifter + winds) on short scales. It uses no ocean temperature or salinity data. The data are available on a regular 1/2 grid, and represent a mean over the period 14 October 1992 to 9 October Altimetry data can be used to change the effective period over which the mean is taken. CLS-09 is a dynamic topography derived from a combination of satellite altimeter data, GRACE geoid data, drifter measurements, wind stress observations and ocean temperature and salinity data, combined using a reduced dynamics approach (different from the Maximenko approach), via an inverse technique. The data are available on a regular 1/4 grid, and represent a mean over the period 1993 to 1999 inclusive. Altimetry data can be used to change the effective period over which the mean is taken. Data from depth-averaged (2-D) models from the European and American shelves may also be employed (5) Other model data may also be used as appropriate. The word model is used in many ways in oceanography and sometimes a better word would be parameterisation, for example of the relationship between changes in winds and air pressures and those of sea level at tide gauge locations Standards and Conventions Besides the theoretical issues of the various approaches for height determination (and subsequent unification) from a practical point of view the heterogeneity of the terrestrial input data is an important issue as well, since one deals with data from different epochs and of varying quality. Furthermore one has to obey whether different references, standards and conventions were applied in the data (pre-) processing and treatment. The standards and conventions used and applied for the unification of height systems have to be consistent. Since various gravity and geodetic data will be incorporated it has to be ensured that in all applied procedures the same standards and conventions are applied. They must be not only aligned with the conventions and standards used for the GOCE mission (GOCE Standards, 2010), but also with those published by other IAG bodies, like IERS Conventions (Petit & Luzum, 2010) or by BGI, ICGEM, etc. At the moment there are certain disagreements between existing standards and conventions used in the geodetic-gravimetric community. These divergences have been analyzed and should be harmonized. At least, they have to be taken into account in the data treatment procedures.

44 Newtonian Constant of Gravitation G Summary and Final Report Page: 43 of 173 The Newtonian gravity constant G is currently the poorest known universal physical constant with a large uncertainty (cf. e.g. Mohr & Taylor, 2000). Although the product GM is known much better from satellite observations within the last 50 years, where M denotes the mass of the Earth (including the mass of the atmosphere) it follows, that also the mass of the Earth defined as M = GM / G is known only with relative uncertainty of 10-4 : M kg, with an uncertainty of about M ± kg. CODATA 1 recommended values of the fundamental physical constants G are given in Table 3-6. Hence, the current (2010) best estimate of G is: ( ) x m 3 kg -1 s -2. Table 3-6: CODATA Recommended Values of the gravitational constant G [unit: m 3 kg -1 s -2 ] Year Value of G (concise form) Standard uncertainty (estimated standard deviation) Relative uncertainty (85) (10) (10) (67) (80) Differing from this best estimate the current (2011) recommended G values according to the GOCE and IERS standards are given in Table 3-7. Table 3-7: Recommended G values according to the GOCE and IERS standards GOCE / IERS Standards G [10-11 m 3 kg -1 s -2 ] Remark GOCE Standards (2010), p = CODATA 1986 value IERS Standards (2003), p = CODATA 1998 value IERS Standards (2010), p = CODATA 2006 value The G value published in the GOCE standards is referred to Groten (2004) who specified G to be equal to ( ± 0.30 ) x m 3 kg -1 s -2 which is equal to the CODATA 1986 value, although with an understated standard uncertainty. The value of G is widely used in gravimetry, e.g. for gravity anomaly computations. More generally, it is employed in all fields of geosciences where the attraction of (topographic, oceanic, atmospheric) masses or the influence of mass redistributions has to be modeled. Using different values for G might be a source for slight data inconsistencies and should be avoided. The usage of different G values, e.g. the G value of the IERS 2010 standards instead of the G value given by the GOCE standards (both differ from the current best estimate!), yields a difference in the mass estimate for the whole Earth of about kg, which is nearly 300 times larger than the estimated mass of the whole atmosphere itself which is about M A (cf. e.g. Ahrens, 1995, Gruber et al. 2009). Since the mass of the atmosphere matters, for instance in atmospheric corrections (see below), the usage of an 1 CODATA Committee on Data for Science and Technology, cf. also NIST National Institute of Standards and Technology,

45 Page: 44 of 173 unique G value is a necessity for a consistent modeling of the Earth gravity field based on terrestrial and GOCE satellite data GRS80 Ellipsoid and Normal Gravity Field The GRS80 is uniquely defined (Moritz, 1984) by the following parameters: a = m GM = m³/s² J 2 = E-03 Ω = E-05 rad/s Derived parameters consistent with the defining parameters of the GRS80 and exact to more than 30 digits (in the mantissa) are: e 2 = E-03 e = E-02 E = m e = E-02 b = m f = E-03 1/f = U 0 = m 2 /s 2 These parameters should be used with at least 16 digits in the mantissa to be consistent with the precision of nowadays computers that have double precision representation of floating point numbers. A smart way to compute the normal gravitational potential everywhere outside the GRS80 ellipsoid, U GRS80, is to use the simple and closed-form expression in frame of ellipsoidal coordinates u, β (Heiskanen & Moritz, 1967, sec. 2-7., p.64-67). E arctan ' GRS q U ( u, ) u U0 a a P20 (sin ) 3 E arctan 3 q0 b 2 2 with E a b and 2 1 u E u q 1 3 arctan 3, 2 2 E u E 2 1 b E b q0 1 3 arctan E b E (3-37) (3-38) Based on the four GRS80 parameters (a, b, U 0, ω) this representation is straightforward connected to Stokes theorem uniquely defining a harmonic function U everywhere in the exterior of a given single-connected body by its values on the body s surface. Since the GRS80 ellipsoidal surface is an equipotential one by definition, it is sufficient to specify the constant U 0. U 0 is connected with the GRS80 defining value GM through an asymptotical far-field (point mass) approximation by: U GM E E b arctan a (3-39) The transformation formulas from the Cartesian (x,y,z) or spherical coordinate systems (r,θ,λ) into the one-parametric ellipsoidal systems of coordinates (u,β,λ) are given e.g. in Vanicek & Krakiwsky (1986), chapter An alternative approach to compute the normal gravitational potential U GRS80 is to use an expansion of the GRS80 gravitational field into a truncated series of spherical harmonics (SH).

46 Page: 45 of 173 GRS 80 GRS 80 N 1 max GRS 80 n ' GRS 80 GM a GRS 80 GRS 80 n0 n0 a n0(2) r U (,, r) c P (cos ) (3-40) Such series converges quickly. In case computations carried out using double precision floating point numbers a truncation at degree 12 seems to be sufficient, since it yields the same result as the closedform expression mentioned above. Due to the rotational symmetry of the GRS80 ellipsoid only the even zonal SH coefficients are nonzero numbers. For the relevant even zonal SH coefficients c n,0 one gets the values (SH coefficients in the definition of Ferrers-Neumann, e.g. Kautzleben (1965) or Heiskanen & Moritz (1967), (2-89) with (2-58) and (1-78)) in Table 3-8. If i) the expansion of U GRS80 into SH is truncated not before d/o 12, and, if ii) one employs the even zonal SH coefficient values given in table 6, then both representations produce numerically the same (double precision) calculated potential values everywhere outside the GRS80 ellipsoid (regardless whether one deals with points at the Earth s topographical surface, at airborne trajectories, or at satellite altitudes of a few hundreds of kilometers). Furthermore, both analytical presentations of U allow easily either analytically or numerically - to compute other normal field elements (like higher derivatives) everywhere at the Earth s surface and in outer space, without applying a spherical approximation. In case that vector gravity data are at disposal the expressions can be used also to replace a scalar (e.g. free-air) gravity anomaly or gravity disturbance by the three anomalous vector components in a chosen coordinate system. Table 3-8: First even zonal SH coefficients in the definition of Ferrers-Neumann (standard and fully normalized) of the GRS80 normal gravitational field exactly computed to about 22 digits; a 2,n is a defining parameter of the GRS80 degree n standard coefficients a n,0 fully normalized coefficients c n, E E E E E E E E E E E E-16 Due to these universal properties both analytical expressions for the potential U are preferable compared to Taylor series expansions with respect to height h of individual elements of the normal gravity field, e.g. of the scalar normal gravity γ = γ in connection with Somigliana s formula for γ 0 (cf. e.g. Torge (1989), formulas (2.66) or (3.5), p. 39 and p.53, respectively). The latter are valid only in the proximity of the GRS80 ellipsoid and cannot be consistently applied to gravity data scattered over a wide range of altitudes, i.e. when one is going to combine terrestrial, airborne and satellite gravity measurements. For the computation of anomalous gravity functionals (e.g. disturbing potential T, gravity disturbance δg, etc.) one can compute the SH series separately: i) for the reference field - the GRS80 normal gravity model, and ii) for the GPM: GPM GPM N n 1 max GPM n GPM GM a GPM GPM GPM nm nm nm a n0 r m0 V (,, r) P (cos )( c cos m s sin m ) (3-41) In this case the anomalies formed at the level of the corresponding functionals not at the level of SH coefficient differences. That means, in case of the disturbing potential one has: GPM GPM GRS 80 GPM ' GRS 80 GPM ' GRS 80 T W U ( V Z) ( U Z) V U (3-42)

47 Page: 46 of 173 Here the centrifugal potential Z cancels out (since the nominal mean angular velocity of the Earth ω is the same for the GRS80 and the GPMs) and for T GPM remains only the difference of the gravitational potentials. In practice, however, it is common to apply computer programs with a merged anomaly computation by means of one single SH series that uses subtracted coefficients (model minus reference). In this case it is inadmissible to subtract simply the SH coefficients of the GRS80 model from the GPM coefficients in one and the same series expansion. Since the coefficients of various GPM differ due to different GM and a values, i.e. due to different scale factors, the GPM coefficients have to be rescaled in compliance with the constants GM and a of the reference potential, e.g. of the GRS80 (cf. e.g. GOCE Level 2 Product Data Handbook, 2009, p. 29) Permanent Tide System When comparing or combining quantities from different sources, the tide system in which these quantities are provided has to be identical in order not to insert systematic distortions. Three tide systems are in use, which distinguish how the time invariant component of solar and lunar tides is taken into account. These are: 1. The mean tide system (MT): This system represents the Earth with presence of direct and indirect components of permanent tides. In other words, this is the system, which implicitly is adopted when observing geometric or gravity field quantities. 2. The zero tide system (ZT): This system represents an Earth without the presence of direct permanent tidal effects, but with the indirect permanent deformation of the Earth due to the permanent tides. 3. The tide free system (TF): This system represents an Earth without presence of sun and moon, i.e. without the effect of permanent tides. It has to be distinguished between the geometric and gravitational impact of permanent tide systems. For joint analysis of different quantities it is essential to work in the same tide system, which implies that conversions between them need to be done. The following equations describe the conversion routines between the three tide systems. Conversion of Gravitational Quantities: The impact of permanent tides is latitude dependent, because sun and moon are moving fairly close to the equator (Ekman, 1989). This implies that in case of the Earth gravity field only zonal terms of the spherical harmonic series are affected. Traditionally, only the second degree zonal coefficient is regarded for the conversion between the three tide systems. The following formulas are applied for conversion (see McCarthy and Petit, 2004; Rapp, 1989 and Ekman, 1989): C C C MT ZT C C k C ZT TF C C 1 k C MT TF (3-43) where: C k average value of the second degree zonal tidal correction for sun and moon (McCarthy and Petit, 2004). loading Love number for second degree zonal coefficient (McCarthy and Petit, 2004).

48 Page: 47 of 173 In order to convert gravity observations (g in m/s 2 ) or geoid heights (N in m) between the tide systems the formulas and constants derived by Ekman can be applied (Ekman, 1989): g g sin 10 MT ZT 2 8 g g sin 10 ZT TF 2 8 g g sin 10 MT TF N N sin 10 MT ZT 2 2 N N k sin 10 ZT TF 2 2 N N 1 k sin 10 MT TF 2 2 (3-44) (3-45) Note: When computing the disturbing potential by subtracting the ellipsoidal potential from the gravity potential both have to be in the same tide system. Conversion of Geometric Heights: Models describing the displacements of geometric coordinates due to various effects are described in (McCarthy and Petit, 2004) from where parts of the the following paragraphs are extracted. These include conventional displacements of reference markers on the crust relating the regularized positions to their conventional instantaneous positions caused by tidal motions (mostly near diurnal and semidiurnal frequencies) and other accurately modelled displacements (mostly at longer periods) as well as other displacements caused by non-tidal motions associated with changing environmental loads (very broad spectral content). Both are described by geophysical models or gridded convolution results derived from geophysical models. Non-tidal load displacements in general are not corrected, as the available models are regarded as not being accurate enough. Instead they remain as signals embedded in the geodetic time series and can be eliminated e.g. by averaging. The time variable effect of tidal motions in general are corrected by tide models, while the permanent effect needs to be taken into account, specifically when combining quantities from different sources. The tidal models applied in GNSS positioning in principle contain a time-independent part so that the coordinates obtained by taking into account this model will be conventional tide free values. The following equations allow the computation of mean tide from conventional tide free coordinates. The corrections need to be added to the conventional tide free position to obtain the mean tide position. The radial component of this amounts to about 12 cm at the poles and about +6 cm at the equator. Radial component of the permanent displacement [m]: P (sin ) P (sin ) (3-46) 2 2 Transverse component of the permanent tide displacement in [m] northwards 2 with: P (sin ) sin 2 (3-47) P (sin ) 3sin 1 / 2 The situation is slightly different in satellite altimetry, where sea surface heights usually only are corrected for time dependent tidal effects. This means that altimetric derived sea surface heights intrinsically are provided in the mean tide system. Finally, it shall be noted, that for all analyses it is important to understand how the permanent tides are treated in a data set. In case quantities from various sources shall be used in a combined analysis they

49 Page: 48 of 173 have to be converted into a common tide system in order to avoid systematic effects entering into the results Height Systems and Conversions Physical heights are defined by the potential difference between a point on the Earth surface and the potential at the local height system reference point. These potential differences are converted to heights by dividing them by the gravity acceleration. There are several height systems in use, which basically are defined by the gravity acceleration used for the conversion of potential differences to heights. These are: Geopotential Number C: It uniquely defines the flow of water, i.e. where is up and where is down. Points with identical C are at the same equipotential surface. Unit is [m 2 /s 2 ]. Levelled Heights H L : They don t take into account that equipotential surfaces are not parallel and their distance is changing. Unit is [m]. Orthometric Heights H: Height along the plumb line above the reference equipotential surface (Unit [m]). Points with same H in general are not located at the same equipotential surface. Heights are dependent on density assumptions in the Earth s interior. Normal Heights H N : Height above the quasi equipotential surface through the reference point. Points with same H N in general are not located at the same equipotential surface. Heights are dependent from the chosen normal gravity field. Dynamic Heights H D : Conversion of geopotential numbers into height values by division with a constant normal gravity. Heights have no physical meaning they just represent a conversion into other units. Normal-Orthometric Heights H NO : For conversion of geopotential numbers into heights the real gravity acceleration along the plumb lines (orthometric heights) is replaced by the normal gravity. The following set of equations describes the definition of the height systems. Most of the height systems are defined via geopotential numbers, which are defined as follows: Orthometric Heights H: H H H 1 C Wo WP dw gdh H gdh Hg H (3-48) C H (3-49) g The mean gravity along the plumb line can be approximated by use of Prey s reduction. Normal Heights H N H 1 1 g 1 g g(h)dh g H g (2 G )H H (3-50) 2 h 2 h 0 for ρ= 2.67 g/cm 3 we get: g g g H, where mgal/m (3-51) H H N C (3-52) N H N N 2 2 N 0 1 H H (h)dh [1 (1 f m 2f sin ) ( ) ] H (3-53) a a

50 Page: 49 of 173 with a,f = semi major and flattening of the reference ellipsoid Dynamic Heights H D m = ω 2 a/γ a H D C (3-54) where γ 45 is the normal gravity at 45 latitude (constant). Normal Orthometric Heights H NO 45 Normal orthometric heights are not based on exact geopotential numbers, but on an approximation when no gravity observations were made during the levelling survey. The approximative geopotential number K is defined by the following equation, where the real gravity is replaced by the normal gravity: H H H 1 K Wo WP dw gdh H dh H H H NO (3-55) K (3-56) Conversion of Height Systems NO H NO NO 2 2 NO 0 1 H H (h)dh [1 (1 f m 2f sin ) ( ) ] H (3-57) a a It is obvious that for height conversions between orthometric, normal and dynamic heights the geopotential number is used as common quantity. Conversions are derived from the different gravity acceleration scaling factors used for the different height systems. Conversions between normal orthometric and all other heights are more complicated because in principle one would need the gravity observations along the levelling lines in order to convert the approximate geopotential number K to the real geopotential number C. In conclusion, this conversion can only be performed approximately. Another aspect when converting height systems, which are defined with a normal gravity field is that for all quantities involved the same normal ellipsoid has to be used, i.e. the ellipsoid parameters have to be identical in all definitions ITRS Time System There are also issues having an indirect effect on height systems through positioning errors. For instance, there is a discrepancy in the ITRS time system between the recommendation and its realization, influencing the position determination by GNSS. Table 3-9: Standards applied for ITRS time system ITRS definition ITRS realization Geocentric Coordinate Time (TCG) Terrestrial Time (TT) GM = m 3 /s 2 GM = m 3 /s 2 IERS Conventions 2010 ITRF, EGM, EIGEN, Generally, such discrepancies are not acceptable and should be overcome by changing either the recommendation (definition) or its realization.

51 Treatment of the Atmosphere Summary and Final Report Page: 50 of 173 Especially for regional combination models that are based on terrestrial data as well as on satellite (e.g. GOCE) data a consistent treatment of the atmospheric masses is a crucial point. The whole theoretical apparatus for harmonic functions is based on the assumption that LAPLACE equation is fulfilled. Strictly speaking, it is applicable to the outer gravitational potential fields V or U only, if there are no masses at all outside the Earth or the GRS80 ellipsoid, respectively. In reality, terrestrial gravity measurements are carried out below the atmospheric masses, at the lower boundary of a domain filled with atmospheric masses having a density of about 1.3 kg/m -3 near the Earth s surface. The density of a standard atmosphere is quickly decreasing with height: density values in the upper stratosphere are by five orders of magnitude less compared to the surface value (cf. Figure 3-17). At a first stage one can surely neglect lateral or time variations of atmospheric density (see Figure 3-18 and Figure 3-19), since they are about ten to twenty times less compared to the whole atmospheric acceleration signal. According to Figure 3-17 the overwhelming part of the atmospheric masses is located below 100 km altitude. Hence, for a stringent treatment of the gravity field in the atmosphere and lower stratosphere at least one should be aware that it does not fulfill LAPLACE equation, but is governed (and described by) POISSON second order differential equation. Concerning real terrestrial gravity measurements it is obvious, that the resulting gravity acceleration vector - measured by a terrestrial gravimeter located on the Earth surface - is a superposition of i) the downward directed acceleration vector caused by the solid Earth body including the water masses plus ii) the opposite upward directed acceleration vector caused by the atmospheric masses. Contrary to that, for a gravimetric sensor onboard a LEO spacecraft at about 250 km altitude nearly all masses of the Earth (solid Earth, hydrosphere, atmosphere) are below the spacecraft. Therefore, the total acceleration felt by the instrument has only downward towards the Earth directed components. Figure 3-17: Height dependency of atmospheric density for the MSIS-E-90 Standard Atmosphere Model (ccmc.gsfc.nasa.gov/modelweb/models/miss_vitmo.php)

52 Page: 51 of 173 Figure 3-18: Latitude dependency of atmospheric density at ground for the MSIS-E-90 Standard Atmosphere Model (ccmc.gsfc.nasa.gov/modelweb/models/miss_vitmo.php) Figure 3-19: Seasonal variation of atmospheric density at ground for the MSIS-E-90 Standard Atmosphere Model (ccmc.gsfc.nasa.gov/modelweb/models/miss_vitmo.php) The GRS80 gravity reference field defines a GM value that includes the mass M of the whole Earth including the mass of the atmosphere. Hence, application of the GRS80 reference field to terrestrial measurements will subtract an atmospheric term which is not contained. Moreover, it is already subtracted by nature. Therefore, before combining terrestrial and satellite-derived anomalous gravitational elements a foregoing atmospheric correction has to be applied to the terrestrial observation. This correction should be done by adding a priori twice the gravitational attraction of atmospheric masses to the terrestrial observations when they are referred to the GRS80 normal gravity field. Wenzel (1985) presented an atmospheric correction formula for gravity anomalies or scalar gravity using an approximate approach based on a spherical symmetric density distribution of a standard atmosphere. This correction (in [mgal]) is valid for terrestrial gravity points with ellipsoidal height h [m] in the interval 0 < h < 8000 m: g A h h (3-58) When taking into account the space-time variation of atmospheric masses (provided by some meteorological services) one can apply also more advanced 3D modeling of atmospheric attraction for correction purposes for e.g. time series of stationary superconducting gravimeters (Klügel & Wziontek, 2009; cf. atmacs.bkg.bund.de).

53 Page: 52 of 173 Recently it is not common practice to correct terrestrial gravity measurements for the influence of atmospheric masses. When combining terrestrial gravity and GOCE satellite data however, it seems absolutely necessary to do so, in order to get consistent models, which might contribute to height system unification. 3.3 Algorithms and Preliminary Analysis Spherical Harmonic Synthesis Geoid values derived from global gravity field models (GPM s) are required for various purposes in this study. Here we follow the conventions as specified in detail in Gruber et al (2010) and write the formulas in a general form. The gravity potential expressed by the spherical harmonic series represents the total effect of the solid Earth mass, the atmospheric and the oceanic masses. The constants of the spherical harmonic series GM (gravitational constant times mass) and a (equatorial radius of the Earth) are provided together with the model coefficients. The atmosphere is assumed to be condensed at the surface of the sphere with radius a. The header of the global gravity field model includes an indicator whether permanent tides are included in the model (mean-tide MT, zero-tide ZT) or not (tide free TF). For the conversion formulas between both see chapter ). From the gravity potential derived gravity quantities such as geoid heights, gravity anomalies and deflections of the vertical are directly computed using the spherical harmonic series after subtracting the spherical harmonic coefficients of the adopted reference potential. Before subtraction, the gravity field spherical harmonic series has to be scaled to the constants of the reference potential. This includes the coefficient C 00 representing the zero-order part of the gravitational potential. ELL MODEL MODEL n MODEL C nm GM a C nm = ELL REF REF MODEL S GM a nm Snm (3-59) with REF GM Factor GM for the reference ellipsoid potential MODEL GM Factor GM used for the gravity field model REF a Equatorial radius for the reference ellipsoid MODEL a Equatorial radius used for the gravity field model ELL ELL C nm,s nm Normalized spherical harmonic coefficients referring to the constants of the reference ellipsoid MODEL MODEL C nm,s nm Normalized spherical harmonic coefficients referring to the constants of the gravity field model After rescaling the spherical harmonic coefficients the disturbing potential T at a point P can be computed by subtracting the normal potential from the gravity potential. with nm nm REF N REF n n GM a T(r,θ,λ)= ΔC cosmλ+δs sinmλ P cosθ r n=0 r m=0 max nm nm nm (3-60) C, S Residual coefficients of the spherical harmonic series after subtracting the coefficients of the normal potential from the gravitational potential. N max Maximum degree of spherical harmonic series r,, Geocentric coordinates

54 Page: 53 of 173 P nm(cos ) Normalized associated Legendre polynomial of degree n and order m It is important to notice that due to the transformation of the coefficients the C00 coefficient not always is equal to zero. This has impact on the definition of the geoid computed from the global gravity field model. As in general the gravity potential is only harmonic outside the attracting masses one should compute the disturbing potential or the derived geoid at the Earth surface. According to the theory of Molodensky this implies that we are computing height anomalies instead of geoid heights over continental areas. For evaluating equation (3-60) the ellipsoidal co-latitude for a given point needs to be converted to a geocentric co-latitude taking into account the height of this point above the reference ellipsoid. REF N REF n n T(r,, ) GM a (r,, ) C cos m S sin m P cos r n0 r m0 max nm nm nm (3-61) with Normal gravity for reference ellipsoid at the Earth surface In order to compute height anomalies from a spherical harmonic series of the disturbing potential we have to know normal heights. Since not for all areas in the world normal heights are available (depending on the height system used in each country) they could be approximated by orthometric heights, which are also often used as height system. The difference between the normal and orthometric heights is equal to the difference between height anomalies and geoid heights (see (Heiskanen and Moritz 1967). It can reach up to a few decimeters or more in mountainous areas (e.g. 0.5 meters for Mont Blanc with an elevation of 4807 meters) Error Budgeting of Potential Differences from GPM s Quality assessment of height unification is based on propagation of error variances and co-variances of all required quantities. In the context of height unification with GOCE, error variances and covariances of GOCE-derived geoid heights (or heights anomalies) at or between datum points (or an adequate set of connection points) in different datum zones must be known. This information can be derived from the full variance-covariance matrix (VCM) of GOCE spherical harmonic coefficients as provided by ESA. Therefore software shall be developed for the propagation of GPM errors in order to assess the error budget of potential differences between arbitrary points on Earth. This shall be applied not only to GOCE, but also to GRACE and as far as possible, EGM08. For the full quality description, also estimation of the omission error shall be included. In the following, the related algorithms are described along with a description of the software requirements, necessary input data sets as well as availability and performance of existing software. Further development needs will be listed Algorithm for Propagation of GPM Errors to Geoid Heights Quality assessment of height unification is based on propagation of error variances and co-variances of all required quantities. Error variances and co-variances of GPM-derived geoid heights (or heights anomalies) at or between datum points (or an adequate set of connection points) in different datum zones can be derived from the full variance-covariance matrix (VCM) of the GPM. In the following the algorithms to be applied are specified. Height anomalies can be computed from a GPM spherical harmonic series according to equation (3-61). For a more general formulation of the error propagation problem, equation (3-61) is rewritten as follows using a single parameter for the constant and degree dependent factors. N max n n nm nm nm, (3-62) (r,, ) f C cos m S sin m P cos n0 m0

55 n Summary and Final Report Page: 54 of 173 REF REF where: GM a fn r r In order to propagate the full spherical harmonic series VCM to height anomaly difference variances and covariances between two points P and Q the following formula (after reordering of sums) has to be applied (see (Haagmans and Gelderen 1991). Nmax Nmax Nmax Nmax Cov(, ) f f Cov( C, C ) P (cos )P (cos ) cos m cos l P Q n k nm kl nm P kl Q P Q m0 l0 nm kl Nmax Nmax f f Cov( S, C ) P (cos )P (cos ) sin m cos l nm kl n k nm kl nm P kl Q P Q Nmax N max f f Cov( C, S ) P (cos )P (cos ) cos m sin l nm kl n k nm kl nm P kl Q P Q Nmax Nmax fn fk Cov( S nm, S kl) P nm(cos P)P kl(cos Q) sin mp sin lq nm kl where the factors depend on the coordinates of points P and Q. REF REF GM a f n (r P, P, P) rp P rp REF REF GM a f k (r Q, Q, Q) rq Q r Q n k (3-63) Introducing the abbreviations A mk, B mk, C mk and D mk for the four terms in parentheses we can write N N max max (3-64) Cov(, ) A cos m cos l B sin m cos l P Q ml P Q ml P Q m0 k0 Cml cos mp sin lq Dml sin mp sin lq Equations (3-63) and (3-64) allow to compute covariances between arbitrary points P and Q or point error variances for the case P = Q. Input data are error covariances of the potential coefficients which are available for the GOCE gravity field models, but usually not for other GPM s. Since the full error VCM of a GOCE GPM has an expected size of 8 (N max +1) 4 which corresponds to roughly 8GB N max =180, 13 GB for N max =200 and 32 GB for N max =250 respectively, error propagation is a computationally demanding task, which cannot simply be run on ordinary PCs in an efficient way. Therefore one has to either make use of super computers or computer clusters or one has to take into account certain approximations. Such approximations will be applied and it will be investigated if different levels of approximation lead to significantly different estimates of error covariances of height anomaly differences and ultimately of estimates of height datum offsets. It shall be noted here that error propagation of geoid heights instead of height anomalies is slightly less computationally demanding as the factors f n and f k are constant per latitude band. Differences between propagated height anomaly or geoid height variances and covariances should be marginal. Approximations: According to Sneeuw (2000), selection of a nominal circular orbit with constant inclination leads to a linear system of observation equation, where each individual spherical harmonic order m is decoupled from coefficients of all other orders. Therefore the normal equation matrix and the error VCM of the estimated potential coefficients have a block diagonal structure, provided that the matrices are ordered with respect to spherical harmonic order m (hereafter called m-order). Because the orbit of GOCE is almost circular and the inclination almost constant, one can expect, that the VCM is at least blockdiagonal dominant.

56 Page: 55 of 173 Therefore, as a first level of approximation, all co-variances between coefficients of different order m (m l) will be neglected. This results in an enormous reduction of computational load. The VCM is then split into individual blocks, each with a maximum size of (N max +1) 2. In addition, co-variances between sine and cosine coefficients are neglected. This leads to the following formulation: Cov(, Nmax Nmax Nmax ) f f Cov( C, C ) P (cos )P (cos ) cos m cos m P Q n k nm kl nm P kl Q P Q m0 nm km Nmax Nmax fn fk Cov( S nm, S kl) P nm(cos P)P kl(cos Q) sin mp sin mq nm km (3-65) or using the abbreviations A mk and D mk N max (3-66) Cov(, ) A cos m cos m D sin m sin m P Q m P Q m P Q m0 This level of approximation will be called the m-block approach. In a second level of approximation all error co-variances are neglected. This will be called the diagonal approach because only the main diagonal of the VCM is used. Accordingly, equation (3-65) further reduces to Cov(, ) f Cov( C, C ) P (cos )P (cos ) cos m cos m Nmax Nmax 2 P Q n nm nm nm P nm Q P Q m0 nm Nmax 2 fn Cov( S nm, S nm) P nm(cos P)P nm(cos Q) sin mp sin mq nm (3-67) Both approximations (m-block and diagonal) lead to a geographic structure with latitude dependent error variances. Longitude dependent features, which result from propagation of the full VCM disappear. Both approximations lead to error covariance functions which are neither fully isotropic nor homogenous. These properties result from a further simplification, where one does not even take into account the error variances of all individual coefficients, but considers error degree variances only, i.e. one fully neglects dependence from spherical harmonic order. The error degree variances are 2 2 computed from the error variances (C nm) and (S nm) of the potential coefficients according to (Heiskanen and Moritz 1967) n n nm nm m0 (C ) (S ) (3-68) The corresponding error covariance function which is isotropic and homogenous depends on the spherical distance between points P and Q only and can be derived from: n n nm nm m0 This approximation will be called the degree-variance approach Algorithm for Computation of the Omission Error (C ) (S ) (3-69) The error variance or co-variance of height anomalies or geoid heights takes into account errors of the GPM up to its maximum spectral resolution. This is not a measure for the deviation between the real geoid (which has infinite maximum degree) and the geoid computed from the GPM, because it does not take into account the signal contribution above the maximum degree of the GPM. The latter is called the omission error. Therefore full quality description takes into account both, the commission error (contribution of the VCM of the GPM) and the omission error above the resolution of the GPM. There are two methods to estimate the omission error, namely:

57 Page: 56 of 173 (1) Computation of the signal contribution from a ultra-high degree combination model like EGM2008 (Pavlis et al. 2012) eventually enhanced by topography derived gravity field estimates above the maximum degree and order of EGM2008 (Hirt, Gruber, and Featherstone 2011) (2) Estimation of the global average signal contribution from a degree variance model. Method (1) is useful because it accounts for spatial variations of the omission error. The omission error in terms of height anomalies is computed from equation (3-61) in the spectral band between the maximum degree of the GOCE GPM and EGM2008, e.g The signal in this bandwidth is not error free, but contains the commission error of EGM2008. There is a residual omission part, which is the contribution above the maximum degree of EGM2008. Since there is no model with higher degree available, the residual omission part has to be estimated either applying the RTM technique, which calculates the gravity field signal beyond a specific frequency band until a limit which depends on the spatial resolution of topography data, or from method (2). Method (2) makes use of degree variance which are defined according to (Heiskanen and Moritz 1967) as: n c C S 2 2 n nm nm m0 (3-70) The use of degree variances implies an isotropic and homogeneous averaging, i.e. the result reflects the global mean signal behaviour. Equation (3-70) is not applied explicitly because it requires knowledge of potential coefficients up to infinity. Instead, models are used which are fitted to data in the low and medium spectral band and are expected to reflect the signal behaviour for higher degrees as well. Degree variance model which are usually applied in the literature are Kaula s rule of thumb the model by Tscherning/Rapp or the model by Moritz/Jekeli. These models are given by the expression (see e.g., Rummel, 1997) c ( g) n c (Kaula) (3-71) n n 3 An 1 n 2n B s n2 with: A = mgal 2 B = 24 s = c n=2 = 7.7 mgal 2 (Tscherning/Rapp) (3-72) A n 1 A n 1 c ( g) s s 1 n2 2 n2 n 1 2 n B1 n B2 with: A 1 = mgal 2 A 2 = mgal 2 B 1 = 1 B 2 = 2 s 1 = s 2 = (Moritz/Jekeli) (3-73) Evaluation of these models yields omission error estimates for GOCE (above L max = 250) and EGM2008 (above L max = 2190) as provided in Table 3-10.

58 Page: 57 of 173 Table 3-10: Omission error from different degree variance models omission error Kaula Tscherning/Rapp Moritz/Jekeli GOCE 16 cm 34 cm 43 cm EGM cm 2 cm 1 cm Optimal Combination of Heterogeneous Data The offset of the local vertical datum with respect to the global datum is computed from the height misclosures l P h P N P H P. In a LSA problem, the (weighted) mean of the misclosures is in fact the offset of the level surface of the local vertical reference frame with respect to the level surface defined by the GGM. The height misclosure also includes other systematic errors in the three height components as well as random measurement errors. In the LSA model, eq. (3-74), the systematic component, Ax, describes accumulated levelling errors and is parameterized by a low degree bivariate polynomial or another simple mathematical surface. where: l l l T 1 2 n l = Ax + v, with E T vv C, (3-74) l is a vector of height misclosures at GNSS BMs of the vertical control T network, x [ x1x 2 x k ] is a vector of k unknown parameters; A is the coefficient matrix with vector rows a i [ a1a 2 ak], i 1,..., n and of full column rank, i.e., rank A k ; and T v [ v1v 2 v n ] is a vector of zero-mean residuals, i.e., Ev 0. For assumed disjunctive observations, h, H, and N, a block-diagonal covariance model is used. v C v C 0 0 h C 0 0 H 2 0 hq h 0 0 C N H Q 0 H N Q N (3-75) 2 where C j jq j is the (fully-populated or diagonal) positive-definite error VCM of the ellipsoidal (j=h), orthometric (j=h), or geoid (j=n) height with a variance factor 2 j and a positive definite cofactor matrixq (Fotopoulos, 2005). To estimate the three unknown variance factors, the Iterated j Minimum Norm Quadratic Unbiased Estimation (IMINQUE) algorithm is used. The effect of calibrated diagonal VCMs of the three height components on the computed datum offset is demonstrated by the result in Table The difference in the local datum offset value computed with non-calibrated VCMs is smaller than the omission error (Section ), but it is significant compared to its error. It should be noted that the use of diagonal VCMs generally provides an optimistic accuracy of the computed datum offset. Table 3-11: Computed offset of Nov07 datum using calibrated and non-calibrated error VCM Error W j o 2 /, m s 2 2 / 2, m s N j, cm W j o Non-calibrated ± ± 1.8 Calibrated ± ± 1.6

59 3.3.3 Estimation of Height System Offsets by the GBVP Solution of the GBVP Summary and Final Report Page: 58 of 173 The solution of the GBVP is based on two steps, the spherical harmonic synthesis of GPM according (see chapter 3.3.1, equation (3-60) and the integration of residual terrestrial gravity anomalies (after subtraction of the signal represented by the GPM). The latter integration for a point P has the form: R T(r,, ) St( ) g d (3-76) P P P PQ Q 4 Where R is the mean earth radius, St(ψ PQ ) is Stokes function and g are gravity anomalies. An efficient algorithm for evaluation of Stokes integral is based on a 1-dimensional FFT on the surface of the sphere (Haagmans et al., 1993). Thereby the convolution is carried out in the frequency domain. Therefore, the data matrix as well as the kernel function, are transformed to the spectral domain by means of FFT, followed by multiplication of the respective spectra and an inverse transformation back to the spatial domain. It is possible to implement various modifications of the Stokes function Least-Squares Estimation of Datum Offsets Least-squares estimation of datum offsets based on the solution of the GBVP was developed by (Reiner Rummel and Teunissen 1988). There exist L+1 non-overlapping height zones with 0,1,...L and the Earth s surface divided into E L. There is available a GPM up to high degree and order N max. This model is considered unbiased, i.e. unaffected of height system offsets. The missing residual part (omission part) will be computed by the solution of the GBVP using the available terrestrial global set of gravity anomalies g. The solution of the GBVP will be dealt with in spherical and constant radius approximation, which implies that the point P is located at a sphere and that geoid heights with respect to this sphere are regarded for further derivations. This level of approximation is clearly sufficient if a high d/o satellite gravity model is available. With these assumptions the disturbing potential at a station P in datum zone j can be computed according to: (GM) R 2. (3-77) j T (r P, P, P) St( PQ) gl Cl0 d R 4 R Q Thereby (GM) is the difference between the actual value of GM and the model value used, R is the mean earth radius and gl is a global set of gravity anomalies which is biased due to height offsets of all different datum zones l. C l0 is the potential difference between a chosen reference zone l=0 and datum zones l. Consequently the term (2 / R)C l0 is a bias correction term such that Stokes integration j is performed with unbiased anomalies. Applying Brun s equation (3-78), the biased geoid height N, i.e. the equipotential surface that passes through the datum point of datum zone j is given by: j N (r P, P, P) j T (r P, P, P) - W0 Cj0 j N (r P, P, P) W0 C j0 St PQ gl Cl0 d 0 R 4 R Q 0 (3-78) 1 (GM) R 2 (3-79) i.e. the geoid height corresponds to the unbiased geoid (computed from the unbiased disturbing potential by division through normal gravity 0 representing normal gravity on the sphere) which is

60 Page: 59 of 173 the shifted for the potential difference C j0 between datum zone j and a conventionally chosen reference datum and, in addition, by the offset W0 of this reference zone with respect to a global unbiased geoid. The first two terms are globally constant offsets, which cannot be separated in the adjustment procedure. They are combined to: The Stokes integral is separated according to: where the first term N 0 (GM) W0 R 0 0 R 2 R R 2 St g C d St g d St C d 4 R 4 4 R PQ l l0 PQ l PQ l0 0 Q 0 0 N Stokes P C L Stokes l0 NP St PQ d l1 2 0 l (3-80) (3-81) is the geoid at point P computed from the biased anomalies (this is the result of every real geoid computation where datum biases are not known beforehand) and the integration in the second term split into L datum zones such that the corresponding datum offset can be taken out of the integral. The last term actually is the negative of the indirect bias term, which is contained in the integration of biased anomalies. Introducing for the integration in the last term the abbreviation f we write: P l0 1 f St( )d P l0 PQ 2 (3-82) 0 l L j Stokes P P P P 0 j0 P l0 l0 l1 N (r,, ) N N N N f (3-83) Stokes where N j0 and N l0 are computed from C j0 / 0 and C l0 / 0, respectively. In reality N P is not computed from a global integration of gravity anomalies, but it is combined with a GPM such that integration can be limited to a certain spherical cap around the computation point. Then one may also take into account modifications of Stokes function, e.g. to ensure proper filtering of the long wavelength of the anomaly signal or to reduce the truncation error when limiting the integration radius. Using a GPM and the residual geoid we can write: N N N (3-84) Stokes GPM res P P P which leads to the final equation for solution of the GVBP: L j j GPM res P P P P P 0 j0 P P l0 l0 l1 N (r,, ) N N N N N N f (3-85) The geoid height on the left hand side can be observed, when ellipsoidal coordinates of levelling stations are determined from geodetic space techniques, i.e. it holds N = h - H (3-86) j j P P P where h P is the ellipsoidal height (e.g. from GNSS) and from a combination of levelling and gravimetry and referring to datum point Finally we end up with the observation equation L j GPM res P P P P P 0 j0 l0 l0 l=1 j H P is the orthometric height determined O j of datum zone j. h - H - N - N = N + N + N f (3-87)

61 Page: 60 of 173 All quantities on the left-hand side are measured or computed and are collected under the observation j vector y. The right hand side contains the unknowns, the globally constant offset P N 0 between the unbiased geoid and a chosen reference datum zone and relative offsets N of all datum zones with j0 respect to this reference zone. Separating the relative offset of the datum zone of observation point P from the summation, we find the observational model for least-squares estimation of datum offsets L j P P P 0 j0 j0 l0 l0 l=1,l j y = N f N + f N (3-88) Alternatively we can formulate the estimation in terms of geoid height differences between points P and Q (located in two different datum zones j and k): Δy = 1 + f N + f N f N + f N L M. (3-89) jk P P Q Q PQ j0 j0 l0 l0 k0 k0 m0 m0 l=1,l j m=1,m¹k In matrix notation the linear observation model can be written as y A x (3-90) with the observation vector y, the design matrix A and the unknown datum offsets x. The least-squares solution of this model is xˆ ( A' Q A) A' Q y, (3-91) yy yy where Q yy is the error VCM of the observations. The estimated error VCM of the unknowns is given by the inverse of the normal equation matrix Qˆˆ ( A' Q A ). (3-92) xx 1 1 yy The following example shows the structure of the design matrix. Let s assume we have 7 GNSS/levelling stations (A,B,C,D,E,F,G) available and a total of 4 different datum zones; station A is located in datum zone j=0, B and C in datum zone j=1 and D and E in datum zone j=2 and F in zone j=3. Let s further assume that we have observed absolute quantities in the sense of equation (3-88) and we chose datum zone 0 as reference datum zone. Then the unknowns are the constant global offset N 0 of zone 0 with respect to an unbiased geoid and the relative datum offsets N 10, N 20 and N30 of datum zones 1, 2 and 3 with respect to zone 0. The structure of the matrix is then given by Table Alternatively, observation of geoid height differences between observation stations results in a design matrix which contains differences of the elements given in Table 3-12 and evaluated at the respective stations/zones. In this case the first column disappears because only differences are observed and the absolute offset term N 0 cannot be determined.

62 Page: 61 of 173 Table 3-12: Structure of the design matrix for least-squares adjustment of height datum offsets for the station configuration as described N 0 N 10 N 20 N 30 j 0 y A A A A 1 f 10 f 20 f 30 j 1 y B B B B 1 1 f10 f 20 f 30 j 1 y C C C C 1 1 f10 f 20 f 30 j 2 y D D D D 1 f 10 1 f20 f 30 j 2 y E E E E 1 f 10 1 f20 f 30 j 3 y F F F F 1 f 10 f 20 1 f Combined Solution for the Geoid As described in sections and geoid heights and height anomalies can be derived by spherical harmonic synthesis from a global potential model or by global integration of gravity anomalies applying Stokes s equation. In practice Stokes s integral is limited to a certain inner zone Φ 0 around the computation point. This reduces the need to collect global sets of terrestrial gravity data and the effort in numerical integration. The omitted gravity signal from the outer zone Φ-Φ 0 is then taken care of by the global model. There exist two basic approaches for the combination of Stokes integration and a global model, namely (a) combination in the spectral domain and (b) combination in the spatial domain. Both approaches can be shown to provide identical results (Rapp and Rummel, 1976). Both approaches can be modified with the aim of choosing an optimal weighting of terrestrial data and the global model. The general structure of this weighting schema is shown in the sequel. In practical computations the weights need to be chosen according to the specific data situation considering all relevant constraints. Here we describe the general schema for weighting which can be applied to the relevant equations in the above sections. We will employ the formulas of section and in spherical and constant radius approximation. Then height anomalies can be computed through spherical harmonic synthesis from a GPM according to Nmax n GPM ( P, P) R Cnm cos mp Snm sin m P Pnm cos P n2 m0 (3-93) and through Stokes s integration from terrestrial gravity data according to R (, ) St( ) g d. (3-94) P P PQ Q Q 4 The GPM also allows computation of gravity anomalies according to with N n GPM g ( P, P) n 1 Cnm cos m P Snm sin m P Pnm cos P n2 m0 max Nmax n2 g, GPM n (3-95) n GPM n nm P nm P nm P m0 g (n 1) C cos m S sin m P cos. (3-96)

63 The isotropic Stokes kernel is given by the following spherical harmonic series Introducing the spectral weight function and PQ n PQ n2 n1 Summary and Final Report Page: 62 of 173 2n 1 St( ) P (cos ). (3-97) w n the kernel (3-97) can be split into the two components 2n 1 St ( ) w P (cos ) (3-98) 1 PQ n n PQ n2 n1 2n 1 St ( ) 1 w P (cos ). (3-99) 2 PQ n n PQ n2 n1 Correspondingly the height anomaly consists of two components which can be derived either by spherical harmonic synthesis or by means of Stokes s integration through R 2 R 2 (, ) w g 1 w g GPM GPM P P n n n n 2 n2 n 1 2 n2 n 1 R R St ( ) g d St ( ) g d PQ Q Q 2 PQ Q Q 1 2 The combination solution employing a GPM for 1 and Stokes s integration for 2 yields R 2 R (, ) w g St ( ) g d. GPM P P n n 2 PQ Q Q 2 n2 n 1 4 (3-100) (3-101) In practise the summation in the first term is limited to the maximum degree of the GPM and Stokes s integration is limited to the inner zone Φ 0. Separating the two terms in equation (3-101) accordingly yields Nmax R 2 GPM R 2 GPM P P n n n n 2 n2 n 1 2 nnmax 1 n 1 (, ) w g w g R R St ( ) g d St ( ) g d 4 4 (3-102) 2 PQ Q Q 2 PQ Q Q Thereby the term 12 is the contribution above the maximum degree of the employed GPM and the term 22 is the contribution of terrestrial gravity data outside the inner zone. Both terms represent truncation errors because they are neglected in practical geoid computation. The weights w n may be chosen such that the truncation errors are minimized. They may also be chosen such that error contributions from either the global model or the terrestrial data are minimized. Therefore the weighting schema depends on the specific application considering, the quality and distribution of available data and other possible constraints. The geodetic literature is full of deterministic or stochastic modifications of the Stokes function by introduction of weights of various type. Among others, examples can be found in Molodensky et al. (1962), Wong and Gore (1969), Meissl (1971), Jekeli (1981), Wenzel (1983), Sjöberg (1984), Heck and Grüninger (1988), Featherstone et al. (1998), Vanicek and Featherstone (1998), Featherstone (2003) and the references therein. All equations presented in sections to may be modified by introduction of weights according to the above schema..

64 Page: 63 of Preliminary Results Effect of the indirect Bias Term If GOCE would provide the global geoid without omission error, the problem of height datum unification could be solved easily. Unfortunately, this is not the case and the residual geoid height above the maximum degree L max of GOCE must be derived from terrestrial data. However, terrestrial gravity anomalies themselves are affected by datum offsets which distorts the solution of the GBVP. This is called the indirect bias term. It can in principle be determined from least-squares adjustment as shown by Xu (1992) or in an iterative manner as done by Amos and Featherstone (2009).We have carried out one and two-dimensional test computations to estimate, how large the magnitude of the indirect bias term is, in case geoid computation is based on the combination of a high resolution satellite gravity model (which is not affected by datum offsets) and residual terrestrial data (which are affected). The result of the 1-D simulation is shown in Figure There the indirect bias term is evaluated for different maximum degrees of the underlying GPM. For a GPM with resolution of D/O 200, the effect stays below the cm. This is supported by a 2D simulation of datum offsets inside Europe. We find that in case of GOCE, the indirect bias term stays below the level of 1 cm. This is below the expected commission error (about 2-3 cm) of GOCE at around spherical harmonic degree 200. Therefore we conclude that the indirect bias term can be neglected when GOCE is used for height system unification. In other words: one can fully exploit the high quality of GOCE for height system unification even when the indirect bias term is neglected.. Neglecting the indirect bias term simplifies the design matrix of the estimation algorithm. Recalling P the sample setup at the end of section , the indirect bias terms f l0 drop out and the structure of the new design matrix is simplified to the form given in Table Figure 3-20: Expected size of the indirect bias term for datum zones of different size. In all cases it is assumed that terrestrial data is combined with a GPM of different maximum D/O. The size of the bias corresponds to a 1m datum offset.

65 Page: 64 of 173 Figure 3-21: Datum offsets inside Europe (left, in cm) and the corresponding indirect bias term assuming combination of terrestrial data with a GOCE GPM up to D/O 200. Table 3-13: Simplified version of the design matrix shown in the previous table, here the indirect bias terms have been neglected N 0 N 10 N 20 N 30 j 0 y A j 1 y B j 1 y C j 2 y D j 2 y E j 3 y F Height Datum Connection by Regional Averaging of GNSS/levelling Data GNSS/levelling derived geoid heights or height anomalies are available for a number of areas like North America, Europe, Japan and Australia. Comparing this independent geoid information with a geoid computed from global models in principle delivers estimates of height system offsets between these regions because the local height systems are reflected in the GNSS/levelling geoid heights. This is well shown in Figure 3-4. The relation between GNSS/levelling derived geoid heights and those computed from the global models (in this case from a GOCE based model) can be written as follows: N N N N N (3-103) A GOCE res res do i A0 i i i The formula determines the geoid height at a point at the Earth s surface related to a vertical datum A, which is related to a mean geoid. The meaning of the individual components is as follows: A N i A0 Geoid height of point P i referring to the vertical datum A from GNSS/levelling N Offset of vertical datum A with respect to the mean geoid

66 Page: 65 of 173 GOCE N Geoid computed for point P i i from a global gravity field model (in this case a GOCE model) res N Residual geoid height (omission error) for point P i i res do N Impact of datum offsets in gravity anomalies on residual geoid for point P i i Rewriting equation (3-103) leads to the observation equation, which is applied for the investigations described further below. A GOCE res res do NA0 Ni ( Ni Ni N i ) (3-104) This observation equation says that by computing the differences between observed and computed geoid heights we can determine the offset for a specific datum from a global mean geoid for each point. Applying this formula to a set of points for a region, where we can assume that the same vertical datum has been used, and computing the mean one can get an estimate of the vertical datum offset for this region. Formula (3-104) then reads as: where: N A0 n i1 A GOCE res resdo Ni ( Ni Ni Ni ) n N A0 Mean regional offset of vertical datum A with respect to the mean geoid Number of GNSS levelling points available for this region n (3-105) Up to now nothing was said about the definition of the mean geoid, which is the reference of the vertical datum. At this point its definition has no impact on the results obtained by this procedure as long as it is kept the same for all computations. In order to eliminate the mean geoid one also can look for geoid differences and vertical datum differences for a regional data set. The formulas (3-104) and (3-105) to be applied in this case have to be rewritten for a point by point case and for a regional data set, respectively, as follows: B N j AB A B GOCE res resdo GOCE res resdo N ( Ni N j ) ( Ni Ni Ni ) ( N j N j N j ) n m A GOCE res resdo Ni ( Ni Ni Ni ) AB i1 j1 N B GOCE res resdo N j ( N j N j N j ) n Geoid height of point P j referring to the vertical datum B GOCE N j Geoid computed for point P j from a global gravity field model (in this case a GOCE model) res N Residual geoid height (omission error) for point P j j N Impact of datum offsets in gravity anomalies on residual geoid for point P j res do j m (3-106) (3-107) m Number of GNSS levelling points available for the second region AB N Mean offset of regional vertical datum A with respect to regional vertical datum B In order to preliminary assess the regional offsets between the various regional GNSS levelling data sets, which are available for this study the following computations have been done. The geoid from the global model (in this case for the 3 rd release pure GOCE model = TIM3) is computed for each GNSS levelling point coordinates (latitude, longitude and height) from the GOCE GOCE spherical harmonic series up to a specific degree and order (quantities N, N ). Here the series was solved up to degree and order 200, which is the target resolution of the GOCE models. The procedure applied here is described in detail in Gruber et al (2011). In addition the omission error ( res res N, N ) is computed for each point from the EGM2008 (Pavils et al, 2008) high resolution gravity i j i j

67 Page: 66 of 173 field model up to degree and order 2190 and in the case of the German GNSS levelling data set in addition from a digital terrain model using the RTM technique (Hirt et al, 2010). This is described in detail in Gruber et al (2011). For this preliminary study the impact of datum offsets in the gravity resdo resdo anomaly data sets used for the computation of the omission error (quantities N, N ) was not taken into consideration as the terrestrial data sets used for computing EGM2008 are not publicly available. In a separate analysis the relevance of this impact was investigated (c.f. previous paragraph). As a result of this analysis it can be concluded that its effect for the present analysis can be neglected as it is below the error level of the observed quantities (e.g. GNSS/levelling derived geoid heights). A B Geoid heights at the GNSS levelling points (quantities N, N ) are computed by subtracting the by spirit levelling observed physical heights (orthometric or normal heights) from the observed ellipsoidal GNSS heights. These geoid heights are independent from the global models and are dependent on the vertical datum used for determining the physical heights. It has to be mentioned that the available European data set was already harmonized by the IAG activity to determine a European Unified Vertical Network (EUVN). This implies that vertical datum offsets between the European countries should not be present in this data set. More details on the obtained results are shown further below. In order to assess the impact of the omission error on the derived vertical datum offsets in the following analysis two test cases were regarded. Global model geoid heights including the omission error and excluding it were computed and applied for datum offset estimation. In a first assessment the five available data sets (Australia, Canada, Europe, Japan and USA) were analysed regarding each of them as referring to one vertical datum (see also the comment above about the European data set). Applying equation (3-105) to each data set for the two test cases (with and without omission error) gives the results as shown in Figure i j i j Figure 3-22: Mean differences of GNSS Levelling and global model geoid heights for the five data sets under investigation taking the omission error into account or not. Coloured dots show differences per point when applying the omission error. Results show significant differences between the five data sets. Taking into account the omission error we get mean height system offsets (see yellow boxes in Figure 3-22) to a mean geoid, which is defined by the constants of the global GOCE model and the parameters of the ellipsoidal normal gravity potential. If we neglect the omission error and repeat these computations the results as shown in the blue boxes of Figure 3-22 are obtained. In summary one can identify that the difference between both

68 Page: 67 of 173 mean values (see red boxes in Figure 3-22) is between several millimeters (Europe) to more than a decimeter (Canada) per data set. Principally this difference is strongly dependent on the topography roughness of the area. In other words, the impact of the omission error on the mean geoid height differences and consequently the vertical datum offset is depending on the terrain roughness. This has to be taken into account in case more accurate results are needed. Applying equation (3-107) to these results deliver the following vertical datum offsets between the five test areas (see Table 3-14 and Table 3-15). Table 3-14: Mean vertical datum offsets between data sets including the omission error [m] (column = from, row = to) Europe Japan Australia Canada USA Europe Japan Australia Canada USA Table 3-15: Mean vertical datum offsets between data sets disregarding the omission error [m] (column = from, row = to) Europe Japan Australia Canada USA Europe Japan Australia Canada USA In order to further refine the results the European data set was investigated in more detail. Pointwise geoid height differences show systematic effects when looking to the map of individual differences (see Figure 3-23). Specifically, significant differences are visible between the results obtained for Great Britain and Italy. When looking to these results one should keep in mind that in principle this data set already represents a unified data set (EUVN), which should not have systematic differences of this level. For this reason a number of countries out of this data set were selected and a similar analysis as for the five large areas was performed. Results for the mean differences for the selected countries are shown in Table Regarding the mean values one can identify that there is a significant offset between Great Britain and all other countries (50 cm level). Similar, but on a somewhat lower level also the Italian heights seem to differ systematically (20 cm level). So both data sets might not be consistently transformed into the unified system. As a second observation one can identify that the mean values with and without taking account the omission error in most cases differ significantly for countries with a rough topography, i.e. high mountains (e.g. Austria, Norway), while for flat or hilly countries (Netherlands, Great Britain) the difference is almost negligible. There are exceptions like Switzerland with a rough terrain and Poland with a flat terrain, which need to be investigated in more detail. Applying equation (3-107) one can setup a similar table as for the large area data sets identifying the remaining vertical offsets between the data sets per country (see Table 3-17).

69 Country Summary and Final Report Page: 68 of 173 Mean with Omission Error [m] Mean w/o Omission Error [m] Austria (A) Belgium (B) Denmark (DK) Finland (FI) France (F) Germany (D) Great Britain (GB) Italy (I) Netherlands (NL) Norway (NO) Poland (PL) Spain (ES) Sweden (SW) Switzerland (CH) Figure 3-23: Geoid differences per point in Europe Table 3-16: Mean of geoid differences per country Table 3-17: Mean vertical datum offsets between data sets including the omission error [cm] (column = from, row = to) A B DK FI F D GB I NL NO PL ES SW CH A B DK FI F D GB I NL NO PL ES SW CH

70 Page: 69 of 173 Another attempt to identify the vertical datum offset between Germany and Great Britain was done by comparing mean GNSS levelling geoid height differences for both countries. This time not the EUVN extracted data per country were used, but data obtained from the national authorities (British Ordnance Survey and BKG). The Great Britain data set refers to the Newlyn reference tide gauge, while the German data set refers to the Amsterdam gauge. Figure 3-24 shows the geographical distribution of the geoid differences with and without taking into account the omission error. The mean values of these differences are given in Table As it can be identified the mean values hardly differ from each other for the two test cases with and without omission error even if the individual point differences strongly diverge. This supports the observation made above, that the omission error only plays a role in case of rough terrain. Figure 3-24: Geoid differences between the GOCE global model (d/o 200) and the national GNSS levelling geoid height data sets for Germany (left) and Great Britain (right) [m]. The top row shows the differences taking into account the omission error as described above and the bottom row the differences using only the information from the global model. Table 3-18: Mean of geoid differences for German and British national GNSS levelling data sets. Country Mean with Omission Error [m] Mean w/o Omission Error [m] Germany Great Britain

71 Page: 70 of 173 Computing the vertical offset between both countries (equation (3-107)) would provide the principle height offset between the Newlyn and the Amsterdam reference tide gauges. Taking the omission error into account one could argue that the the Newlyn reference tide gauge is 38 cm above the Amsterdam tide gauge (see Table 3-18). Comparing this to the difference obtained from Figure 3-3, which gives a value of +4 cm for the two reference tide gauges, we can identify a significant deviation of both results, which needs to be investigated in more detail. Synthetic Study on neglecting the Omission Error In the previous section, preliminary results have been shown from a height datum connection using real GNSS/levelling data and a GOCE GPM. In general there are two possible strategies for further improvement of such a connection: (1) Use, per datum zone, only some few selected GNSS/levelling stations with high accuracy. (2) Use as many GNSS/levelling points as possible; the quality requirements of each point is of minor concern, since one assumes random errors to vanish when computing the mean over all geoid heights observed by means of GNSS/levelling. res The residual geoid height N corresponds to the omission error of a GPM. It is often modelled as a stochastic quantity, i.e. it is assumed to behave like a random signal. Therefore one can assume the average of the residual geoid height to vanish, at least if the averaging area is large enough or the longest wavelength contained in the residual signal is short enough. Therefore it might seem reasonable to neglect the residual geoid height at all, if method (2) is chosen, i.e. if regional averages are formed. The advantage of such a procedure would of course be, that one would not have to estimate the residual geoid height at all, thus significantly lowering the computational burden. In the following we show results of simulations carried out to check the validity of this assumption. In order to check, whether the omission part can be neglected, a dense grid of residual geoid heights (above D/O 200) was derived from EGM2008. The grid is shown in Figure Its global mean is 0.09 mm. The colour-bar of this and the following plots is given in logarithmic scale in order to better visualize the pixel s different orders of magnitude. The basic idea is to compute area averages of different size of the residual geoid grid. If the area average is below the level of some centimetres, the omission part above the resolution of GOCE can be neglected because it does not significantly increase the total error budget (commission error of GOCE plus contribution of the other quantities, levelled heights and ellipsoidal heights). White and dark blue colours indicate that the omission error can be neglected, red/orange colours indicate, that it cannot be neglected. The procedure is not relevant for areas which are densely covered with terrestrial gravity stations of good quality, because there the residual geoid is known and can easily be taken into account. But for areas, with insufficient coverage, there is no high resolution geoid information available which makes the averaging procedure in principle attractive for datum connection. Of course, when assuming a situation without gravity and geoid information available, it is reasonable to assume, that there are also not many reliable GNSS/levelling stations available and it might be difficult to derive precise average values from a sparse station distribution. The simulation should give an idea on what the minimum size of the averaging area is. Therefore the dense grid of residual geoid heights is used to generate area averages over blocks of different size (1, 2, 5, 10, 20 ). In reality the situation is a bit different, because the average is not computed from a surface integral, but approximated by averaging inhomogenously distributed GNSS/levelling point values. Therefore realistic point distributions are also considered to check if the area average can be well represented by point averages. Figure 3-26 to Figure 3-30 show the block mean values of EGM2008 above degree 200 for different block sizes. It seems that for 2 x2 blocks, there are still many areas with averages on the level of some few centimetres. For 5 x5 blocks the picture is already changed, and there are hardly any such regions left. Because the signal properties vary around the globe, the minimum block size might vary

72 Page: 71 of 173 from region to region. But in general it seems, that the minimum block size is somewhere between 2 and 5, which corresponds to about 220 km to 550 km. Table 3-19 shows both, area and point averages of residual geoid heights for different countries. The difference is never larger than 2 cm, except for Switzerland, a small country (small integration/averaging average) with rough geoid signal. In summary it can be concluded, that by averaging over all GNSS/levelling points per datum zone, the height datum offsets can be determined from GOCE without considering the omission error at all. This conclusion holds at the level of some few centimetres if the area of the datum zone is large enough (extension of at least 2 to 5 ) and there are enough GNSS/levelling points available. However, the practical relevance as well as possible drawbacks of this approach must be investigated. Figure 3-25: Global grid of residual geoid heights from EGM2008 above degree 200. Figure 3-26: Global grid of residual geoid heights from EGM2008 above degree 200, averaged over 1 x1 blocks.

73 Page: 72 of 173 Figure 3-27: Global grid of residual geoid heights from EGM2008 above degree 200, averaged over 2 x2 blocks. Figure 3-28: Global grid of residual geoid heights from EGM2008 above degree 200, averaged over 5 x5 blocks. Figure 3-29: Global grid of residual geoid heights from EGM2008 above degree 200, averaged over 10 x10 blocks.

74 Page: 73 of 173 Figure 3-30: Global grid of residual geoid heights from EGM2008 above degree 200, averaged over 20 x20 blocks. Table 3-19: Comparison between area and point average of residual geoid heights (EGM2008 above D/O 200) Land Area average [cm] Point average [cm] Australia Austria Belgium Canada Denmark Europe Finland France Germany Great Britain Italy Netherlands Norway Poland Spain Sweden Switzerland Australia Austria Review of Height Systems in Europe Evaluation of National Vertical Reference Frames in Europe An overview about the European national height reference frames is given in the following. According to the RB section 3.3 the table includes all relevant quantities for the description of national height reference frames obtained by spirit leveling. Figure 1 provides a visual overview of height systems in Europe with given mean offsets to the European Vertical Reference Frame 2007 (EVRF2007). Analyzing the information displayed in Table 3-20 it becomes obvious that the initial data used for national height system establishments within Europe is inhomogeneous in many respects. Besides the differences in kind of heights, datum origin, permanent tidal treatment and measurement epoch the accuracy of the measurements itself also matters. Redundancy in geodetic networks is an indicator how the network allows the detection of gross errors. While a redundancy of zero means gross errors are not detectable, a redundancy of one means full

75 Page: 74 of 173 control against gross errors. In general, redundancy is low in spirit levelling networks and reaches values of 0.4 in maximum. Especially small countries with a low number of levelling loops such as Estonia and Lithuania show a low redundancy of less than 0.2. Table 3-20: Comparison of national height systems in Europe Country ID National identifier Datum Austria AT GHA Trieste Belgium BE DNG/TAW Ostend Bosnia and Hercegovina BA Trieste Kind of heights normal orthometric no gravity correction normal orthometric tidal system Epoch of measurements s 0 (nat.) s 0 (var. comp. est.) Redundancy mean tidal ,86 0,82 0,267 mean tidal ,25 1,24 0,383 mean tidal ,91 0,9 0,246 Bulgaria BG Baltic1982 Kronstadt normal mean tidal ,19 1,14 0,298 Croatia HR HRVD71 Trieste Czech Republic Denmark DK DVR90 normal orthometric mean tidal ,91 0,9 0,246 CS Bpv Kronstadt normal mean tidal ,92 1,16 0, Danish tide gauges normal non tidal ,87 0,91 0,245 Estonia EE BHS77 Kronstadt normal mean tidal ,27 1,3 0,164 Finland FI N2000 NAP normal zero tidal ,71 0,73 0,313 France FR IGN69 Marseille normal mean tidal ,08 2,02 0,275 Germany DE DHHN92 NAP normal mean tidal ,83 0,85 0,359 Great Britain GB ODN Newlyn Common epoch of adjustment normalorthometric mean tidal ,72 1,72 0,306 Hungary HU EOMA 1980 Kronstadt normal mean tidal ,46 0,47 0,261 Italy IT Genova 1942 Genova orthometric mean tidal ,69 1,75 0,329 Latvia LV Kronstadt normal mean tidal ,74 0,8 0,349 Lithuania LT NGVN Kronstadt normal mean tidal ,7 0,83 0,094 Netherlands NL NAP no gravity correction non tidal ,75 0,75 0,26 Norway NO NN2000 NAP normal zero tidal ,4 1.62/1.29 0,447 Poland PL Kronstadt 2006 Kronstadt normal non tidal ,87 0,88 0,39 Portugal PT RNGAP Cascais orthometric mean tidal ,21 2,09 0,348 Romania RO Constanta normal mean tidal ,84 1,75 0,333 Slovakia SK Bpv04 Kronstadt normal mean tidal ,74 1,55 0,227 Slovenia SI SI-NVN99 Trieste normal orthometric mean tidal ,91 0,9 0,246 Spain ES REDNAP Alicante orthometric mean tidal ,79 1,75 0,286 Sweden SE RH2000 NAP normal zero tidal ,99 1 0,237 Switzerland CH LHN95 Marseille orthometric mean tidal ,09 1,09 0,433

76 Page: 75 of 173 Figure 3-31 shows the RMS of the national leveling networks. While RMS is small for a number of countries (e.g. Hungary 0.46 mm, Lithuania 0.70 mm, Netherlands 0.75 mm, Germany 0.83 mm) it is larger than 2 mm for France (2.08 mm) and Portugal (2.21 mm). As a matter of fact, there are also tilts between national height reference frames and the EVRF2007. These are mainly caused i) by different conventions adopted for the national height systems (e.g. permanent tides, kind of physical heights), ii) by heterogeneous data sets (e.g. different epochs and network configurations), and, iii) by tensions in the leveling networks. Because of the existing tilts it might be not appropriate to model the differences between national height frames by a pure height datum offset. The additional small numbers given for some of the countries (e.g. UK, Portugal, Spain) in Figure 3-3 depict the presence of a relevant tilt. The systematic errors in the leveling observations, respectively in the national height reference frames, can be discovered only by comparison with independent observations, like GOCE or sea level data. Reliable estimates for offsets and tilts can be done only for larger areas. Figure 3-31: RMS of national leveling networks in Europe Evaluation of the European Vertical Reference Frame EVRF The European Vertical Reference System (EVRS) is related to the Earth gravity field on and outside the solid Earth. It is a geopotential reference system and Earth co-rotating. The EVRS is realized by the EVRF. For its realization geopotential values with respect to the reference potential and corresponding coordinates in a defined Terrestrial Reference System (TRS) are assigned to a set of physical markers at a specific epoch. The EVRS is defined as a kinematic height reference system and considers the following conventions ( The vertical datum is defined as the equipotential surface with a constant Earth gravity field potential on the Normaal Amsterdams Peil (NAP) level: W0 W0 const. E

77 Page: 76 of 173 The unit of length is meter (SI). The unit of time is second (SI). The scale is consistent with the TCG time coordinate for a geocentric local frame, in agreement with IAG and IUGG (1991) resolutions. This is obtained by appropriate relativistic modelling. The height components are given by differences W between the Earth gravity field P potential in a point P W and the potential at EVRS conventional zero level P W. The 0E potential difference W is also termed as geopotential number P c. P W c W W P P 0E P - - Normal heights are equivalent to geopotential numbers, provided that the reference gravity field is specified. The EVRS is a zero tidal system. This is in agreement with the IAG resolutions No. 9 and No. 16 adopted in Hamburg The recent realization of the EVRS is the EVRF2007 at reference epoch (Sacher et al. 2008). The EVRF2007 is computed by a re-adjustment of geopotential numbers of the United European Leveling Network (UELN). If possible, the observation data was reduced to epoch Especially, this is the case for the post glacial rebound area (PGR) of Fennoscandia. All heights were reduced to zero tidal system. The UELN adjustment for EVRF2007 was fitted to the previous EVRF2000 solution using a number of datum points. Figure 3-32 shows the leveling lines of the UELN within EVRF2007. The density of leveling lines differs significantly between national networks. Black dots in Figure 3-32 indicate connecting points between national networks across borders as a key element in height system unification based on leveling networks. While the national networks of Norway and Sweden as well as the national networks of central European countries are well connected, the connections of the national networks of France, Spain, Portugal and Italy and of south eastern European countries are poor. Due to the geographical situation, the connection between central Europe and Fennoscandia as well as between continental Europe and Great Britain is also rather poor. Figure 3-32: Leveling lines of the United European Leveling network (UELN). Black dots indicate connecting points between national networks at national borders.

78 Page: 77 of 173 In summary, following weaknesses of European height system unification by levelling can be identified: Different epochs of levelling observations in different countries Very long observation periods Very limited information about height changes with time (except PGR) Quality differences of levelling lines between neighbouring countries Differences in national standards for levelling Accuracy differences of national levelling networks Low redundancy of levelling networks Lack of independent information Comparison of Observed Geoid Heights with Global Geopotential Models Two data sets of observed height anomalies have been used for the comparison with global GPM (cf ): the German data set BKG_GNSS_NIV_geoid_points and the European data set EUVN- DA. The standards described in have been applied. The height anomaly at a point i ζ i hi H can be used for the evaluation of gravity field models by i ζ ζ ζ. GPM i i i The mean offset of all ζ can be interpreted as the offset of the national height system with respect to i the gravity field model GRS80. Their standard deviation describes the error budget of GPM, spirit levelling and GNSS observations. In order to evaluate the omission error of the GPM all analyses have been carried out in two different ways: i) the pure GPM without any omission part modelling have been used, ii) the omission part was modelled by EGM2008. Although not correct in strength, the GPM is composed with higher degrees of EGM2008 spherical harmonic coefficients. In a first analysis the GPMs have been compared to the German data set. This allows the evaluation the individual models among themselves and in comparison to ITG-GRACE2010s and EGM2008. Second, the European EUVN-DA dataset has been used. In this data set, the national physical heights have been unified by connecting national spirit leveling networks in a common adjustment (EVRF2007). Finally, this data set with the original national height reference has been used. For the evaluation of the GPMs against the German data set the spherical harmonic series were truncated in steps of d/o 5 starting with d/o 100 until the final nominal degree of the series expansion of the specific model. Thus, the omission error is caused by the part of shorter wavelengths which is omitted by the model. Figure 3-33 displays the evolution of the estimated mean offset with increasing model resolution. The results show, a common behaviour of the analyzed models until d/o 190 with a maximum variation of about 2 cm between the models. The ITG-GRACE2010s mean offset is systematically above the GOCE model estimates. The estimated mean offsets depend on the degree where the series expansion was cut. E.g. the mean offset of the TIM_R3 GOCE model cut at d/o 165 is 42.5 cm, at d/o cm and at d/o 190 almost 40.7 cm. The major reason for this uncertainty is the omission error which has not been considered and the limited spatial dimension of a country like Germany. It is to be expected, that the effect increases for smaller countries and decreases for larger countries. Nevertheless, the mean offset can be estimated with an accuracy of a view centimetre for Germany. Figure 3-34 shows the standard deviation of the mean offset estimate. It show, that the omission error decreases from about 70 cm at d/o 100 to about 30 cm at d/o 200. Furthermore, it is clearly visible, that several GOCE models show a better performance in the range of d/o 160 to d/o 210 than EGM2008. Now the omission part was modelled by composing the GPM with higher degrees of EGM2008 spherical harmonic coefficients. The GPMs are cut in steps of d/o 5 starting at d/o 100 until their nominal resolution. The missing part until d/o 2190 is filled up with EGM2008 coefficients. In this case, the mean offset estimate is much more stable than in the previous experiment without using EGM2008. The uncertainty of the estimate is in the range of 1 cm up to d/o 190 and fits well to the

79 Page: 78 of 173 EGM2008 value Figure The standard deviation of all GOCE models is in the same range until d/o 180. Nevertheless, the standard deviation of all investigated models is higher than the one computed for EGM2008 (Figure 3-36). Since there are improvements in Figure 3-34 this could be explained by the imperfect combination of GOCE GPMs and EGM2008. This analysis also indicates that beside their nominal resolution the resolution of the GOCE models is limited at d/o 180 to 190. The resolution of the GRACE model ITG-GRACE2010s is limited at d/o 150 to 160. Figure 3-33: Estimated mean offset between global gravity field models and observed terrestrial height anomalies from the German D924 data set. The omission error has not been modelled. The spherical harmonic series of the GPM were cut in steps of d/o 5 starting at d/o 100. Figure 3-34: Standard deviation of estimated mean offset between global gravity field models and observed terrestrial height anomalies derived from the German D924 data set. The omission error has not been modeled. The spherical harmonic series of the GPM were cut in steps of d/o 5 starting at d/o 100.

80 Page: 79 of 173 Figure 3-35: Estimated mean offset between global gravity field models and observed terrestrial height anomalies from the German D924 data set. The omission error is modeled by EGM2008. The spherical harmonic series of the GPM was cut in steps of d/o starting at d/o 100. Figure 3-36: Standard deviation of estimated mean offset between global gravity field models and observed terrestrial height anomalies from the German D924 data set. The omission error is modeled by EGM2008. The spherical harmonic series of the GPM was cut in steps of d/o starting at d/o 100.

81 Page: 80 of 173 Figure 3-37: Residuals of height anomalies of EUVN-DA data set with respect to GOCE time-wise model, release 2 d/o 250. The comparison of the unified EUVN-DA data set with a GOCE model (GOCE time-wise model, release 2 d/o 250) still shows inhomogeneities (Figure 3-37) especially in Great Britain, southern Italy, Spain and Portugal. These inhomogeneities clearly show the limits of height systems unifications in large spirit leveling networks and motivate the use of GPMs. For a thought experiment of unifying height systems in Europe on the base of a GPM, the model GOCO02S d/o 250 has been used. In a second computation the omission error was reduced by a model of composed spherical harmonic coefficients of GOCO02S until d/o 200 and EGM2008 between d/o 201 and Table 3-21 gives the mean values and standard deviations of for each country. The standard deviation comprises errors of the gravity field model including its omission error as well as systematic errors of the spirit leveling network and - to a smaller degree - errors of the GNSS ellipsoidal heights. Larger values of standard deviations for some countries such as Italy, Portugal, France and Spain are a result of the lower accuracy of their leveling networks. The estimated offset for Germany from EUVN-DA dataset is 38.6 cm and 38.9 cm for the GOCO02S and the GOCO02S+EGM2008 solutions respectively. This fits well to the estimates made for the German data set consisting of 924 points investigated above. There, an offset of about 38 cm was obtained (cf. Figure 3-35). The comparison of the estimated offsets related to the pure GOCO02S model and related to the fulfilled GOCO02S model gives a first estimate of the influence of the omission error. Since its influence is small and causes no offset change for a number of countries (e.g. Finland, France, Germany), a significant offset change up to 10 cm is found for others (e.g. Italy, Norway, Poland). Figure 3-38 shows all offsets estimated with respect to GOCO02S and Figure 3-39 with respect to GOCO02S fulfilled with EGM2008. In order to simplify comparisons to the estimated offsets from the UELN adjustment (Figure 3-3) the estimated offset were corrected by cm in Figure 3-38 and by cm in Figure Thus, the offset for Germany is 1 cm in both cases.

82 Page: 81 of 173 The comparisons of the estimated height system offsets based on GOCO02S and leveling adjustment shows a patchy result. A good agreement at a level of 2 to 3 cm is found for many pairs of countries, such as Germany/Poland, Germany/France, France/Spain, Switzerland/Austria, Slovakia/Hungary or Slovenia/Croatia. The agreement is also good within Scandinavia. On the other side, there is some disagreement, such as between Spain and Portugal or between Central Europe and Scandinavia. In the previous computations a pure height datum offset was estimated. As described in a slope between two national height reference frames may occur. Using a gravity field model is the only way to quantify such systematic effects in leveling networks. Significant slopes of up to 3 cm/100 km could be detected for France and Spain (Figure 3-40, Table 3-22). It could be shown, that national height systems can be unified based on a gravity field model. This requires physical heights obtained by spirit leveling in the national height systems and ellipsoidal heights at specific points. Connecting leveling lines across borders are often limited in occurrence and accuracy and are obsolete in this technique. Nevertheless, the accuracy of the estimated height offsets depends on the accuracy and spatial resolution of the gravity field model (omission error). Especially, the omission error is the crucial factor for small countries or networks with a less dense point distribution. In summary, three procedures of dealing with the omission error may be identified: 1. A satellite-only gravity field model is used and the omission error is ignored. According to this study, this procedure is sufficient for requirements in accuracy of about 10 cm. For large areas, such as continents, and well distributed point networks the accuracy may increase. 2. The satellite-only gravity field model is amended by a high resolution gravity field model, such as EGM2008. This procedure is sufficient for requirements in accuracy of about 2 to 3 cm. 3. The omission error of the satellite-only gravity field model is modelled from terrestrial observation data. In this case it can be expected that the accuracy can be increased up to 1 cm, depending on the quality and density of terrestrial gravity observations. Table 3-21: Mean offset and standard deviations obtained by a thought experiment, relating height anomalies computed from national physical heights and ellipsoidal heights to global gravity field models. Additionally, slopes have been estimated for the marked countries (c.f. Table 3-20) GOCO02S d/o 250 GOCO02S d/o EGM2008 d/o 2190 # points Offset [m] STD [m] Offset [m] STD [m] Austria Bulgaria Croatia Czech Denmark Finland France Germany Great Britain* Hungary Italy Lithuania Latvia Netherlands Norway Poland Portugal Romania Slovakia Slowenia Spain Sweden Switzerland All * The estimated offsets for Great Britain do not fit to the values reported by Fane (2011) and estimated within UELN adjustment. The reason could not be clarified.

83 Page: 82 of 173 Figure 3-38: Height system unification using EUVN-DA dataset across Europe (before adjustment). The national networks have been related to GOCO02S (d/o 250) satellite gravity model. All values are corrected by cm and given in [cm]. Figure 3-39: Height system unification using EUVN-DA dataset across Europe (before adjustment). The national networks have been related to GOCO02S (d/o 200) satellite gravity model. The omission error is modelled by EGM2008 d/o All values are corrected by cm and given in [cm].

84 Page: 83 of 173 [cm] [cm] [cm] Figure 3-40: Estimated slopes of national height systems with respect to GOCO02S d/o200 + EGM2008. Table 3-22: Estimated offsets and slops of national heights systems with respect to GOCO02S d/o200 + EGM2008. # points Offset [m] dh/dλ [m/100km] dh/dφ [m/100km] STD [m] France Germany Poland Spain In summary following statements could be carved out: 1. The height system unfication based on spirit leveling is limited in accuracy and reliability. Systematic effects and inconsistencies are detectable by help of GOCE GPMs. 2. The height system unfication based on GOCE GPMs is possible with an accuracy level of about 10 cm. This requires a sufficient number of GNSS-leveling data points. Our example show that this is also working for smaller countries. 3. Accuracy and reliability of height system unification based on GPMs can be increased by modeling of the omission part using a hig resolution GPM, such as EGM Since a composition of spherical harmonic coefficients of different models is not consistent in strength, a regional gravitiy field model would further increase the accuracy and reliabilty Review of Height Systems in North America Vertical Reference Frames in North America Table 3-23 contains a summary of the North American height systems. The North American Vertical Datum of 1988 (NAVD88) and the International Great Lakes Datum of 1985 (IGLD85) are international vertical reference frames for North America. NAVD88 was established in USA in The levelling networks of Canada, USA and Mexico were adjusted together with the height of one primary tidal benchmark held fixed (Zilkoski et al., 1992), i.e., NAVD88 and IGLD85 are constrained to the mean water level at Father Point/Rimouski, Quebec, Canada. Few other datum definitions were studied and evaluated, among them fix MSL at four tide gauges located at the corners of the network or fix the old NGVD29 heights at 18 benchmarks well distributed across the network. Additionally, two options for establishing the new vertical datum were under consideration: (1) the tidal epoch option which required that the MSL was held fixed at all primary tidal benchmarks and adopt the latest tidal epoch and (2) the minimally constraint adjustment option where the MSL at one tidal station is held fixed and all levelling data are adjusted to that level surface in order to maintain the integrity and avoid distortions of the vertical control network. In addition, the datum was shifted vertically to ensure minimum recompilation of the existing mapping products. IGLD85 is part of NAVD88. The only difference is that the heights of the benchmarks in NAVD88 are Helmert orthometric heights computed by scaling the geopotential numbers with the Helmert

85 Page: 84 of 173 approximation of the mean gravity along the plumb line (Heiskanen and Moritz, 1967, p.167) while the IGLD85 heights are dynamic computed by scaling the same geopotential numbers with the normal gravity value at 45 degrees latitude. The general adjustment of NAVD88 was completed in June It included improved existing levelling data and 81,500 km re-levelled first order levelling lines in order to reinforce the primary vertical control network (NGS, 1996). Rod scale and temperature, level collimation, astronomic (tidal), refraction and magnetic corrections were applied to the levelling data (Zilkovski et al., 1992). It was determined that the systematic errors have large local effects on the re-adjusted heights but a minimal continental effect (except for the magnetic correction). The use of true geopotential numbers instead of normal-orthometric geopotential numbers (as in the old reference frame NGVD29) had a small effect of 5-6 cm from coast to coast in USA but in the mountains this effect reached 50 cm. The levelling network was connected to 57 US primary tidal stations and 55 international water level stations along the Great Lakes shores. Connections between the US and Canadian networks were established at 28 benchmarks, and the US and Mexican vertical control networks were connected at 13 benchmarks. Geographical locations of BMs were scaled coordinates determined from digitized map products. Gravity values were interpolated from actual measured gravity data. Table 3-23: Summary of the characteristics of the North American official vertical reference frames USA Canada Mexico Great Lakes Reference frame NAVD88 CGVD28 NAVD88 IGLD85 3 Atlantic TGs, Father's Father's Datum definition Father's Point/Rimouski, 2 Pacific TGs Point/Rimouski, Point/Rimouski, (TGs constrains) Québec, Canada and Rimouski Québec, Canada Québec, Canada Kind of physical heights Permanent tidal convention Period of measurements Epoch of TG water level or network adjustment Helmert orthometric Astronomical correction (tide-free crust) Modernized archived (since 1887) 1.3 million km levelling data and 80,000 km re-levelled first order network to order 1 class II specifications Accuracy See Table 3-24 Normal orthometric No tidal correction (instantaneous crust) 2,300 km first order levelling ( ) Helmert orthometric No information No information mm error for 1 km levelled line Dynamic Astronomical correction (tidefree crust) (mean epoch 1985) See Table 3-24 See Table 3-24 Table 3-24: Statistics from the minimum-constraint NAVD88 LSA [adopted from Zilkoski et al., 1992] Adjustment Number of Standard error of unit Degrees of BMs weight freedom US/Mexican data 641, ,610 Canadian data only 67, ,161 Canadian-Mexican-U.S. data (final general adjustment) 708, ,939 Canada did not adopt NAVD88 because of unexplained large 1.5 m discrepancies likely due to accumulation of systematic errors from Atlantic to Pacific coasts (Véronneau and Héroux, 2006).

86 Page: 85 of 173 Approximately half of the first order BMs in Canada have NAVD88 orthometric heights. Canada s official vertical datum is the Canadian Geodetic Vertical Datum of 1928 (CGVD28) constrained to the mean sea level of five tide gauges on the Pacific and Atlantic coasts and the water gauge at Father Point/Rimouski on St. Lawrence River used to constrain NAVD88 (Véronneau, 2001). The heights are normal-orthometric, i.e., an approximation to the true orthometric heights, where the geopotential numbers were computed with mean normal gravity instead of the actual gravity. As a result, the CGVD28 heights are systematically lower than the true orthometric heights. In the Rocky Mountains, the deviations are up to several decimetres. No corrections for glacial isostatic adjustment and MSL rise were applied. Both CGVD28 and NAVD88 have big tilts with respect to recently computed regional geoid models. Across USA NAVD88 has a large northwest-southeast tilt with respect to a satellite-only GRACE geoid, and geoid height differences are within a meter range. In Canada, NAVD88 is tilted with respect to the gravimetric geoid model indicating that the mean sea level at the Pacific coast (near Vancouver) is higher than the mean sea level at the Atlantic coast (near Halifax) by 1.5 m while the difference in Rimouski is close to 0 m. Many regional distortions are found in CGVD28. For example, distortions range from -35 cm (Halifax, Nova Scotia) to 75 cm (Banff, Alberta) thus accounting for approximately a one-meter distortion nationwide. These are due to the fact that the vertical control network in Canada was established over the years in a piece-wise manner by combining observations after 1928 over several consecutive years and adjusting these observations locally. The vertical control network is characterized by a rapid rate of degradation due to destruction and loss of physical markers and limited maintenance. Furthermore, glacial isostatic adjustment, earthquakes, subsidence, frost heave, local instabilities and other sources of vertical crustal motion all contribute to the degradation of the accuracy of the official vertical reference frame at a regional and/or local level. The two external consultants to UoC for the ESA HSU project, the National Geodetic Survey (NGS) of the USA and the Geodetic Survey Division (GSD) of Canada, have decided that the levelling-based vertical reference frames NAVD88 and CGVD28 will be replaced by a common North American geoid-based, GNSS-accessible vertical datum by 2022 (Véronneau and Héroux, 2006; NGS, 2008; Smith et al., 2010). This modern vertical datum will comprise the vertical component of an accurate geocentric spatial reference system for North America. A new realization of IGLD integrated in NAVD is planned for At a workshop in October 2011, NGS and GSD agreed to define the new vertical reference frame by the local W 0 value determined through tide gauge averaging around North America. A subset of TGs along the coasts of North America that have been occupied by GNSS will be selected. Using a GGM, the W value of the local MSL will be computed at each tide gauge. Several critical issues will be evaluated: (1) criteria for selection of the TGs, (2) selection of the GGM that best represents W at the TGs, (3) evaluation of SSTop models. For example, the use of satellite-based GGMs (omission error at the TG locations) versus local geoid models (biases and errors in local gravity data) needs to be evaluated. Additionally, a model of the sea level change will be provided. For scientific purposes, GSD of Canada performs unofficial re-adjustments of the first order levelling network constrained to the mean water level at Father Point/Rimouski, Quebec, Canada. The newest unofficial vertical reference frame is labelled Nov07. Heights are true orthometric heights and are compatible with the gravimetric geoid model. The adjustment of the levelling data is realized in a stepwise procedure by constraining the older levelling measurements to the adjusted newer measurements. This procedure allows for reducing the magnitude of the regional systematic errors. The orthometric heights are classified in different levels with each level constrained to the upper level. Only level A heights have a realistic errors and are used in this study to assess the effect of the stochastic data information on the computed vertical datum offset Evaluation of GOCE-based Geoid Models in North America Evaluation of GGMs Performance Geoid models of max D/O were evaluated with the Nov07 and NAVD88 GNSS BMs data sets in Canada and the USA, respectively. Statistics of misclosures N GNSS/levelling N GGM are given in Table

87 Page: 86 of (Canada) and Table 3-26 (USA). All satellite GGMs perform similarly (in terms of standard deviation), but the time-wise tim_r3 performs slightly better than the rest of the models both in Canada and the USA. Table 3-25: Statistics of the geoid height misclosures in Canada, unit is meter Model Min Max Std Bias go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r eigen-6s goco02s go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r EGM Table 3-26: Statistics of the geoid height misclosures in the USA, unit is meter Models Min Max Std Bias go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r eigen-6s goco02s go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r EGM Evaluation of the GOCE GGMs Omission Error To evaluate the omission error of the GGMs, we extended each model to degree 2190 by means of EGM2008. According to the statistics in Table 3-27, the mean of the height misclosures in Canada is reduced by 9-11 cm, and the standard deviation is reduced by cm. The omission error for the USA GNSS BMs data set is smaller (Table 3-28). The mean is reduced by 2-3 cm, and the standard deviation is improved by 9-12 cm. Since the datum offset computed by GNSS BMs data is the (un)weighted mean of the height misclosures, the implication is that the omission error has a smaller effect on the NAVD88 offset than on the Nov07 offset. Table 3-27: Statistics of the geoid height misclosures in Canada (unit is meter) for the GGMs extended to degree 2190 Models Min Max Std Bias go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r eigen-6s goco02s go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r EGM

88 Page: 87 of 173 Table 3-28: Statistics of the geoid height misclosures in USA (unit is meter) for the GGMs extended to degree 2190 Models Min Max Std Bias go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r eigen-6s goco02s go_cons_gcf_2_dir_r go_cons_gcf_2_tim_r EGM Evaluation of GGMs Performance in Different Spectral Bands To evaluate the GGMs performance in different spectral bands, all models were truncated to a selected degree. EGM2008 spherical harmonic coefficients replaced the coefficients of each model up from the cut-off degree to the max D/O and extended the model to degree Plotting the standard deviation of the geoid height misclosures computed for each truncation degree allows one to assess the model error in different spectral bands relative to EGM2008. For Canada (Figure 3-41), all GGMs perform similarly to EGM2008 up to D/O 150 with a maximum difference in the standard deviation of 0.5 cm. For D/O larger than 180, the difference between each model and EGM2008 increases and reaches cm for eigen-6s and dir_r2 GGMs (max D/O of 240). In the spectral band D/O , both dir_r3 and tim_r3 are consistently better than the rest of the models in Canada. Similarly, tim_r3 is the best model for medium wavelengths in USA (Figure 3-42). Figure 3-41: Standard deviation of geoid height misclosures for Canada for different truncation D/O of GGMs

89 Page: 88 of 173 Figure 3-42: Standard deviation of geoid height misclosures for USA for different truncation D/O of GGMs Preliminary Computations of the North American Datum Offsets Basic Relationships Given a set of GNSS BMs points, the geopotential at the fundamental BM of the local height datum j, j j W o, and the separationn with respect to the global geopotential surface defined by Wo is computed by means of the relationships given in (Jekeli, 2002). For a point P with orthometric height j H P, GNSS ellipsoidal height h P and geopotential computed from a GGM at the point P on the geoid, i.e., T P', the offset of the local datum j is computed as where the geopotential difference is where j Q is the point at which N j j W o, (3-108) j j j T W P o Wo Wo Wo Uo Q h o P H ', (3-109) P j Q U j Q W P ', and Qo is the point on the ellipsoid along the normal at the point P on the physical surface. The targeted 1 cm accuracy of HSU allows eq. (3-109) to be simplified to j o j o o o o Q P j P W W W W U h H N. (3-110) o The offset of the height datums in North America is computed relative to the global equipotential 2 surface defined by the adopted by IAG W o m / s. Evaluation of the Effect of the GGM Truncation Degree The separation of the Nov07 fundamental level surface from the global level surface defined above was computed by eq. (3-108) at the 308 GNSS BMs using EGM2008 and varying the truncation D/O (Table 3-29). The table lists the estimated datum offset for several max D/O. For D/O = 180 (e.g., the P

90 Page: 89 of 173 max D/O of the GGM03S model), the datum offset is underestimated by 13.5 cm, for D/O = 250 (GOCE) by 12 cm, for D/O = 360 (e.g., GGM03C) by 5.4 cm and for D/O = 1400 (e.g., EIGEN-6C) by only 0.5 cm. This result suggests a spatial resolution of approximately 12 km half wavelength is sufficient to ensure that the omitted higher frequencies contribute less than 1 cm to the computed vertical datum offset value given the geographical distribution of the Canadian GNSS BMs. Table 3-29: Effect of the GGM truncation D/O on the computed datum offset Truncation D/O W j o 2 /, m s 2 2 / 2, m s N j, cm W j o ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.6 Computation of the Offset of the local Vertical Datums in Canada using GOCE The goco02s model of max D/O 250 and EGM2008 from degree 251 to 2190 were used. From the results given in Table 3-30 the following observations can be made: (a) N j, j = NAVD88, CGVD28 and Nov07, is a function of the geographical distribution of the GNSS BMs and TGs. The effect of the GOCE omission error on the computed datum offset is largest when GNSS TGs are used. This is because the effect of the higher frequencies of the regional geoid cancels out at the GNSS BMs over the landmasses. NAVD88 (b) The N value computed solely from the Canadian GNSS TGs data is much closer to the value computed with the USA GNSS BMs data in Table 3-31 than the value computed with the Canadian GNSS BMs. Because only one TG on the Pacific coast is used (the bulk of TGs is located on the Atlantic coast of Canada), the 1.5 m east-west tilt of NAVD88 does not affect the computed offset when using the GNSS TGs data set, which is not the case for the GNSS BMs data set. CGVD28 (c) The N value is the offset of the mean level surface of CGVD28. The value computed solely with the GNSS TGs differs from the value computed with the GNSS BMs by 21 cm. This is because the GNSS BMs data set may account for the large local and regional distortions of the CGVD28 level surface. Nov07 The N value computed with the GNSS TGs differs from the value computed with GNSS BMs by only 2 cm. This is because Nov07 has a much smaller coast-to-coast tilt than NAVD88 and much Nov07 smaller regional and local distortions than CGVD28. In fact the estimated N value aligns with the value computed with the USA data in Table 3-31 because NAVD88 and Nov07 datums are defined at the same primary tide gauge station.

91 Page: 90 of 173 Table 3-30: Computed offset of various vertical datums in Canada using GOCE (goco02s) and a combination of GOCE(goco02s) and EGM2008 Datum Data set # points GOCE GOCE+EGM2008 Difference N j,cm N j, cm d( N ), cm NAVD88 GNSS/levelling ± ± CGVD28 GNSS/levelling ± ± Nov07 GNSS/levelling ± ± NAVD88 Tide gauges ± ± CGVD28 Tide gauges ± ± Nov07 Tide gauges ± ± NAVD88 GNSS/lev+TG ± ± CGVD28 GNSS/lev+TG ± ± Nov07 GNSS/lev+TG ± ± Computation of the NAVD88 Offset NAVD88 Lastly, N is computed using the USA GNSS BMs data and the combined USA and Canada GNSS BMs data. Results in Table 3-31 show that the Canadian data have a little effect on the computed value. Note that the estimated errors are overly optimistic because no VCMs were available for the USA data. Furthermore, the addition of the higher frequencies (using EGM2008) improves the NAVD88 N value by only cm. Compare this to the Canadian case where the difference is 8.5 cm (the last column in Table 3-30). This is because the USA GNSS BMs points sample well the USA topography. Table 3-31: The NAVD88 local geopotential and offset computed using GOCE (goco02s) and a combination of GOCE (goco02s) and EGM2008 Territory W j o 2 /, m s 2 2 / 2, m s N j, cm W j o USA(goco02s) ± ± 0.3 North America (goco02s) ± ± 0.3 USA (goco02s+egm2008) ± ± 0.3 North America (goco02s+egm2008) ± ± 0.3 Comparison of the Computed Offsets of the North American Vertical Datums The graphics in Figure 3-43 of the computed offsets of the various vertical datums in Canada and USA using goco02s+egm2008 allows one to compare the position of the various level surfaces. The mean level surface of the outdated official Canadian vertical reference frame CGVD28 is positioned below the global level surface by some 21 cm. The Canadian data unrealistically suggest that NAVD88 is placed approximately 80 cm below the global datum. The much more realistic estimate with the USA data shows that NAVD88 is placed 48 cm below the global datum. Finally, the Canadian Nov07 level surface is close (by definition) to NAVD88 and is separated from the global level surface by 46 cm.

92 Page: 91 of 173 Figure 3-43: A graphical representation of the computed offsets of the various vertical datums in Canada and USA using a combination of the GOCE and EGM2008 geoids Conclusions (1) Release 3 GOCE-based direct and time-wise solutions are the best GGMs for medium frequencies in Canada and USA at present. These two models are recommended to be used for the North American height datum unification. (2) The omission error of the GOCE-based GGMs is significant in Canada and should be taken into account when using the GNSS BMs and TGs data for HSU. Though small, the omission error should also be taken into account when USA GNSS BMs are used if 1cm accuracy for HSU is aimed. (3) Given the distribution of the Canadian data points, wavelengths smaller than 12 km have a negligible effect on the computed vertical datum offset. (4) The higher frequencies of the local geoid should always be accounted for when tide gauges are used for HSU. (5) NAVD88 (USA) and Nov07 (Canada) vertical reference frames have similar definitions and consequently the two fundamental level surfaces differ by only 2 cm. CGVD28 (Canada) will be replaced by 2013 by a new geoid-based vertical reference frame. Therefore, it is recommended that the Nov07 height information is used DOT Determination This section refers to ocean levelling, the use of measurements and modelling to specify the Mean Dynamic Topography (MDT) of the ocean along a coastline in different ways, and thereby to obtain confidence in the ocean models and newly available geoid models. The subsequent aims are to use both sets of models in height system unification. For example, the ocean models can be used to provide estimates of MDT-difference between sections of coastline where different national datums apply, thereby providing a confident connection between datums. Meanwhile, validation of geoid models using coastal ocean information provides confidence in their use for height unification worldwide. There are two approaches to determination of the MDT at the coast. In the geodetic approach, ellipsoidal heights of mean sea level (MSL) at tide gauge stations obtained with the use of GNSS receivers geodetically-connected to the tide gauge zeros of the stations by means of conventional levelling, or ellipsoidal heights of sea level obtained from satellite radar altimetry, are compared to ellipsoidal heights from a geoid model. When gauge data are used, then clearly this exercise is

93 Page: 92 of 173 conducted exactly at the coast, whereas when altimetry data are employed, then the comparison is necessarily performed some tens of km offshore. The second approach is the ocean one. In some early versions of the ocean approach to determination of an MDT, sets of oceanographic and meteorological measurements were made (coastal sea level, ocean currents, temperatures and salinities and air pressures and winds) and analysed in the context of the known equations of motion in the ocean, so as to provide sets of sea surface gradient. Nowadays, it is more convenient and rigorous to make use of ocean numerical models in which the oceanographic data sets may have been assimilated. The result is a twodimensional field of the MDT which may be compared to those from the geodetic method. Our algorithms are better described as objectives : (1) To make a general comparison of the MDT profiles at locations along coastlines obtained from the two approaches and determination of the reasons for differences between them. (2) To make use of progressively more accurate and more complete geoid models within the geodetic approach. For example, MDT values at the tide gauges using any newer geoid model should ideally provide closer correspondence (root-mean-square) with MDT values from the ocean approach, recognising that a limiting factor in such a comparison is always likely to be due to omission errors in the geoid models (requiring the inclusion of additional terrestrial gravity information where available). Similarly, comparison between MDT derived from altimetry minus geoid should improve with a newer geoid model, given a correct treatment of both fields to ensure spectral consistency. (We assume that published MDT values derived this way by previous authors, which we use in our analysis, have ensured such consistency.) (3) To undertake both approaches along a number of coastlines thereby obtaining confidence in the use of each approach in different oceanographic regimes. (4) Where we settle upon a best ocean model for using in height unification along a coastline (or more likely an average of suitably performing ocean models) then the resulting MDT profile will be provided at full resolution (i.e. without any spectral filtering implied in the comparisons in (2) with estimates of uncertainties in the profile obtained from the ancillary surge and wave modelling described below. As stated, the extent to which the sets of MDT information at (or near) the coast are consistent between the two approaches provides assessments of the performance of the ocean and geoid models. In practice, a number of ocean models will be available which can be compared and one can thereby derive confidence in their use. The use of tide gauge (exactly coastal) and altimeter (near coastal) data in the geodetic approach provides validation of nearby ellipsoidal heights of sea level (subject to nearcoastal effects mentioned below), and a link to the larger-scale use of altimetry minus geoid for determination of the MDT and ocean circulation in the deep ocean. Different ocean models have different grid spacing, making it difficult to compare spectral content. There is no simple relationship between grid spacing and spectral content, as that also depends on the nature and strength of the numerical diffusion employed to avoid instabilities near to the grid scale, which also varies from model to model. Furthermore, not all models are completely global, and observational datasets additionally miss data close to the coast and in regions of frequent ice cover. In order to ensure fair comparisons among datasets, the models are first interpolated onto a standard 0.25 degree grid on the spatial domain common to all datasets. They are then smoothed with an isotropic Gaussian smoother (radius 150 km at half amplitude) to ensure similar spatial and spectral content in comparisons. This smoothing results in a loss of short wavelength information so, when examining the impact of new geoid data at the coast, the unsmoothed model data are used, and it is simply accepted that different models have different capabilities and hence different spectral content in the dynamic topographies. The implied values of coastal MDT derived from an ocean model may be affected in several ways. One is that, although the model will have contained forcing from winds and air pressures, most deep

94 Page: 93 of 173 ocean models employ daily or monthly versions of such forcing, rather than the hourly (or more frequent) forcing employed in barotropic storm surge models. The non-linearities in the equations will result in a net offset from a more rigorously modelled MDT; this effect can be estimated with the use of local surge models. Similarly, wave setup in bays, estuaries and harbours can result in an MDT measured at the coast differing from that implied by an ocean model; these effects can be investigated with wave models. Other factors include fresh water runoff from large rivers; this can be estimated with the use of three-dimensional shelf models. Another consideration relates to the spatial resolution of the ocean models and their extrapolation to the coast or to the off-shore locations of altimeter data. The finest resolutions tend to be employed in regional rather than global ocean models, and these are also the models which use the highestfrequency forcing as well as representation of contributions to the dynamics which are amplified in coastal regions, such as wave set-up and effects of river flows. Thus, while global models are best used to determine the large scale MDT variability, it is best where possible to supplement this information with fine scales taken from regional models, in order to extrapolate the global information to actual coastal locations. A natural place to begin merging global and regional models is at the continental shelf edge, where the steep topography acts to limit the coupling between deep and shallow dynamics, and tends to reduce the occurrence of short wavelength alongshore gradients in MDT. A consistent quasi-global patching of regional and global models has never been attempted by the community, most regional models having been developed for specific regional issues (e.g. water quality monitoring). The project which nearest meets our interests in this regard is the POLGCOMS project at the National Oceanography Centre which has developed models for several shelf seas but with a common ocean model. Data sets from this exercise around the world may well be useful for further investigation in the present project. However, in general, the utility of such patching, rather than for example the development of higher resolution ocean models in general which include shelf areas as far as possible, remains for further study. Consequently, the algorithms associated with ocean levelling can be seen to involve the use of a suite of models and data sets and assessments of consistency. There are then two benefits. One is that the best ocean model can then be used to relate datums between neighbouring (or even distant) countries with coastlines. The second is that the validation of geoid models at the coast provides confidence in their use across continents, thereby providing datum connections between countries without coastlines. Some details of the comparisons above may be mentioned. It is important that the data and model information used are for the same epoch. The coastal MSL can fluctuate by typically a decimetre from year to year and so a decade or more is needed to provide a MSL which is has centimetric stability as a datum. The tide gauge data should be corrected for inverse barometer air pressure effects and ellipsoidal heights (GNSS, altimeter and geoid) should be expressed in the same tide system (we have standardised on mean tide) and within the same reference frame (e.g. ITRF2005).

95 Page: 94 of Experiments and GOCE Impact Assessment 4.1 Overview and Approach for Study Experiments As already described in chapter height system unification is based on three baseline methods. These are: (1) Direct height connection inside a closed land area (continent) by geodetic (spirit) levelling in combination with gravimetry: the classical approach; it cannot connect islands separated by sea; GOCE could be used, however, to detect systematic errors in large levelling networks; (2) geodetic boundary value problem (GBVP) approach: a rigorous geodetic method of height unification; it is able to deal with the direct effect of height offsets and the indirect effect introduced by the use of gravity anomalies biased due to these offsets (3) ocean levelling: method suited to directly connect coastal tide gauges. All three approaches once more are summarized in Figure 4-1. In this figure there are three vertical datums each fixed at a different tide gauge (A, B and C) and separated either by an ocean or by a continent. For each tide gauge we assume that the equipotential surface going through this tide gauge defines the local vertical datum (datum A,B,C equipotential surface). The connection between ocean and land is done by linking the tide gauges to continental networks via spirit levelling, gravimetry and GNSS observations (see for example the right block in Figure 4-1). For the GNSS benchmarks one can determine ellipsoidal heights (h), physical heights above datum A, (H A ), and geoid heights for the equipotential surface representing vertical datum A, (N A ). In analogy this could be done for any other vertical datum equipotential surface as shown for the two continental blocks with vertical datum B and C. Having for example two tide gauges on both sides of a continent this could create a situation as it is shown in the middle of the continental block, where two different physical and geoid heights are realized just depending to which vertical datum the height refers (H B, N B, H C, N C ). Introducing the GOCE equipotential surface (globally), a GBVP equipotential surface (over continents) and the DOT (over oceans) opens the possibility to connect the different vertical datums. In order to do this one needs to determine the offsets of the different datum equipotential surfaces from them. An important issue to be investigated is the omission error of the GOCE equipotential surface, which is due to the limited sensitivity of the satellite to the high frequency gravity field signals. Its impact has to be quantified e.g. by estimating the high frequency signals separately from other sources (EGM2008, GBVP, RTM techniques). Over continents a GBVP equipotential surface from a combination of the global GOCE model and local gravity field observations can be determined, which contains quasi the full signal spectrum. Comparing a GNSS Levelling derived regional equipotential surface to the GOCE or GBVP equipotential surface delivers height offsets of datum related equipotential surfaces under the assumption of error free levelling. If this is not the case, levelling errors strongly influence the resulting offset estimates. Over the oceans, tide gauge height offsets can be determined by applying DOT estimates at these stations. The DOT is estimated from an ocean model or from an altimetric mean sea surface relative to the GOCE equipotential surface and therefore offers the possibility to quantify the offset for each tide gauge datum equipotential surface relative to the global GOCE derived equipotential surface. It shall be mentioned that also over the oceans the omission error plays a role and needs to be quantified.

96 Page: 95 of 173 Figure 4-1: Overview of methods for height system unification Taking the methods and relations as mentioned above into account one can formulate the overall study approach as shown in Figure 4-2. Starting point is the model level, where GPM s as well as ocean models are used in order to derive the required information (global equipotential surfaces, DOT). At the next level observations enter into the analyses. Here we distinguish between GNSS benchmarks, levelling networks and gravity field observations over the continents and tide gauges, altimetric mean sea surfaces and DOT over the oceans. It shall be mentioned that DOT is not a real observation, but either resulting from the ocean model or from a combination of altimetric mean sea surfaces with global gravity field models. Equipotential surfaces derived from the various sources (models, observations and combinations) represent the central level of the study layout. It is important to mention that for the determination of the equipotential surfaces from models and/or observations, data preprocessing and data conversions need to be taken into account in order to ensure consistency of the various surfaces for further analyses. The equipotential surfaces taken into account in this study are defined as follows: (a) local equipotential surfaces at GNSS Levelling points over land, (b) local equipotential surfaces at tide gauges at coasts, (c) oceanic equipotential surfaces, (d) regional/continental GBVP derived euipotential surfaces, and (e) global equipotential surfaces from GPM s. These surfaces can be compared and connected to each other by applying the methods as they are defined at the higher processing levels (experiments, connections). On the experiment level particular items to be investigated are specified in more detail. These are: (1) Impact of omission and propagated errors on equipotential surfaces. (2) Impact of height offset in gravity data on GBVP equipotential surfaces. (3) Compare GOCE and GBVP equipotential surfaces (also related to the omission error).

97 (4) Compare GOCE and GNSS Levelling derived equipotential surfaces. (5) Compare GOCE and oceanic equipotential surfaces. (6) Compare oceanic and tide gauge equipotential surfaces. (7) Compare tide gauge and GNSS Levelling equipotential surfaces. Summary and Final Report Page: 96 of 173 Figure 4-2: Height System Unification with GOCE Study Approach Experiments (1) and (2) can be regarded as basic investigations needed for most of the other follow-up experiments (3) to (7). The latter 5 experiments compare equipotential surfaces derived from two methods in order to identify potential synergies or problems. This is needed in order to make preparations for connecting the methods. A number of experiments already have been implemented and preliminary results are described in detail in chapter 3.3. Final results of the experiments are provided in chapter 4.2 of this document. The highest level in the study approach is the connection of the different experiments in order to estimate height system offsets between various domains, continents or regions. In order to reach this one has to connect the oceans to the continents via the tide gauges. This can be done based on tide gauges and GNSS Levelling equipotential surfaces or on tide gauges and GBVP equipotential surfaces. Tide gauges on two sides of the oceans are connected via the oceanic equipotential surface, while tide gauges on two sides of a continent are connected via GNSS Levelling or GBVP equipotential surfaces. As tide gauges are the connectors between all domains one needs to determine the offsets between all equipotential surfaces at these points. Ideally, if all connections are properly established one tide gauge is kept fixed and defines the worldwide height system. Refer to section 4.3 of this document. As not all elements nowadays are delivered with sufficient accuracy or with sufficient data coverage it is not expected that from this study a worldwide height system unification can be derived from the experiments and the established connections. Nevertheless, GOCE provides a significant better global

98 Page: 97 of 173 reference surface and it is expected that from the experiments performed within the study and from the attempts to connect the various equipotential surfaces one can identify the needs for homogenizing the various vertical datums currently in use. Therefore the final level of the study represents a scientific roadmap for height system unification with special emphasis on the contributions of GOCE as a new source of information (c.f. chapter 5). 4.2 Experiments for HSU with GOCE This chapter provides some detailed descriptions about experiments performed in the frame of vertical datum determination (compare also the experiment level in Figure 4-2). Chapters to first investigate the impact of height offsets in gravity data on derived geoid solutions, the amplitudes and structure of the omitted geoid signal in the GOCE gravity field solutions and finally an analysis about the properties of the propagated geoid errors of the GOCE global models. The results obtained from these initial analyses are required for all subsequent experiments and attempts to connect the various height system determination approaches. Chapters to describe the results for a number of experiments applying GOCE global models with different approaches. In chapter it is described that GOCE plays an important role for regional geoid computations applying the geodetic boundary value approach. Results for the two test areas Europe and North America diverge a little bit, but it is common to both that some improvements could be reached for the regional geoid solutions when using a GOCE satellite model as reference. Chapter provides the results for GNSS/Levelling derived equipotential surfaces as compared to the GOCE geoid. It could be concluded that the GOCE geoid provides a very good reference to identify distortions in national height datums as well as large scale levelling errors in the national networks. Chapters to describe the results obtained for ocean levelling by comparing the ocean model derived mean sea level to (1) an observed mean sea level at tide gauges referring to a specific national datum, (2) to an observed mean sea level from GNSS observations referring to a geoid model at the tide gauge stations, and (3) to an observed mean sea level from satellite altimetry off-shore the coastlines. From these ocean levelling analyses it can be concluded that all models are consistent to sub-dm level for the studied regions Height Offset Impact on GBVP Equipotential Surface The solution of the GBVP approach and its connection to global geopotential models is described in sections Thereby geoid heights or height anomalies are derived from a combination of a GPM and terrestrial gravity anomalies. According to equation (3-85) the long wavelength features of the geoid are taken care of by the GPM and the short scale features by terrestrial gravity data through evaluation of Stokes s integral. As outlined in section datum offsets in vertical reference frames lead to offsets in data sets of terrestrial gravity anomalies. Integration of such biased gravity anomalies g l generates long wavelength errors in the geoid. This is an indirect effect of height datum offsets and hence the corresponding last term in equation (3-85) is called the indirect bias term. Theoretically it needs to be included in the linear model for least-squares estimation of height datum offsets. This requires detailed knowledge of the boundaries of all globally existing datum zones and it complicates the adjustment considerably. It can however be shown that the effect stays below the level of one centimetre if a GPM with a resolution corresponding to GOCE gravity field models is applied and the Stokes s function is modified accordingly to filter out the contribution of the long wavelength structures of gl. For a deeper discussion of this issue and numerical studies on the size of the indirect bias term we refer to [P-2]. The indirect bias term in North America was evaluated on a 30 x30 grid with different residual Stokes kernels with n max = 70, 120, 150 and 200 related to the spectral resolution of existing satelliteonly GPMs. The adopted reference datum zone (datum zone 0) was the conventional equipotential surface defined by W 0 = m 2 /s 2 (Petit & Luzum, 2010). The -30 and -50 cm offset of CGVD28 in Canada and NAVD88 in the conterminous USA and Mexico, respectively, were used in the computations. The indirect bias term computed with the original Stokes kernel has its maximum value of 38 cm located over the USA. Consequently, the indirect bias term should be taken into

99 Page: 98 of 173 account in the least squares adjustment model and computational procedure of the height datum offsets. The indirect bias term computed with the residual kernels has its maximum values along the coastlines and country borders. These values are well below 1 cm for all n max for North America. Consequently, the indirect bias term can be neglected in North America if a global geopotential model of a maximum degree 70 or larger is used. The use of a residual Stokes s kernel reasoned out by the use of a GPM to define the long wavelengths of the geiod and gravity anomalies diminishes the significance of the biased local gravity anomalies in the datum offset computations. As a consequence, the computational procedure of the height datum offsets is simplified significantly (refer to [P-13].) This experiment was extended to include Europe and South America. The 20 offsets in Europe were obtained from the differences between EVRF2007 and national height systems. In order to have the offsets with respect to the reference datum zone defined by W 0 = m 2 /s 2, a constant value of cm was added to the original offsets. The 9 offsets for South America were obtained from Sanchez (IUGG, 2011) and transformed from the regional equipotential surface W 0 = m 2 /s 2 to the reference datum zone by adding 29 cm to all original offsets. Results in Table 4-1 show that for n max = 200 of the residual integration kernel, the indirect bias term for the combined grid - Europe, North America and South America - is below 1 cm and, consequently, can be neglected (refer to [P-2]). Table 4-1: Indirect bias term for different truncation degree of the residual Stokes kernel Kernel Residual Stokes kernel, degree Original Stokes Territory North America 38.4 cm < 1 cm <1 cm <1 cm <1 cm South America -55 cm 1 cm < 1 cm <1 cm <1 cm Europe 21.4 cm 2.9 cm 1.3 cm 1 cm < 1 cm Combined grid cm 2.9 cm 1.3 cm 1 cm < 1 cm

100 Page: 99 of 173 Figure 4-3: : Indirect bias term evaluated on a combined 30 x30 grid Table 4-2: Summary Height Offset Impact on GVBP Standards - W 0 from IERS CGVD28 offset in Canada - NAVD88 offset in USA - EVRF2007 offsets in Europe Equations (3-61) (3-85) (3-102) - IUGG, 2011 offsets for South America Main Conclusions - Indirect bias term can be neglected if a global geopotential model of maximum degree 200 or higher is used. - For all areas under investigation the indirect bias term is below 1 cm. - Computational procedure of height datum offsets by geodetic boundary value approach is simplified significantly.

101 4.2.2 Omission Error Impact on Equipotential Surface Summary and Final Report Page: 100 of 173 The omission error impact was investigated within a study on intercontinental height datum connection using GNSS-Levelling data. For this purpose the global GOCE geoid was compared to local geoid heights derived from GNSS-Levelling data with or without taking into consideration the omission error. The study was performed for specific regions, which could be a part of a country, a complete country or sometimes even an entire continent. Height offset estimates between the global and local geoids are computed by taking the mean value of these differences over the area under investigation. From the study results the following conclusions related to the impact of the omission error can be drawn. (1) It is important to identify how good the GOCE geoid represents the real geoid. In other words, the omission error representing the signal not observable by GOCE needs to be quantified. A full quantification of the omission error would require perfect knowledge of the geoid. As this information is in many cases not available, we make use of the EGM2008 global model, which incorporates terrestrial and altimetric gravity field information and has a resolution of about 8 km. In well observed areas one can assume that EGM2008 is a good representation of the high resolution geoid and can be used to estimate the omission error. For this reason, in this study a hybrid global geoid based on a pure GOCE model from degree 0 to 180 (TIM3) and EGM2008 from degree 181 to 2190 was used. (2) From the analyses performed for some areas, it can be concluded, that the omission error plays a significant role in case its mean value estimated at the GNSS-Levelling points does not vanish.. In extreme cases, like on small islands for which the resolution of the GOCE geoid is not sufficient, the omission error has a huge impact (more than 1 meter) and has to be taken into account. For some other areas the omission error sometimes is on the level of a few cm only. This is strongly dependent on the roughness of the terrain, i.e. the local structure of the omission error, and the distribution of the GNSS-Levelling points. In any case it is necessary to quantify the omission error beforehand. Details of this investigation can be found in [P-4]. The next experiment demonstrates the impact of the GPM omission error on the computed height datum offsets in North America using the EGM2008 model, which was truncated to different spherical harmonic degrees. Computed geoid heights from the truncated EGM2008 model and GNSS-Levelling (geometric) geoid heights in Canada and the USA were used to assess the impact of the omission error on the NAVD88 and Nov07 offsets. It was demonstrated that for the sparser GNSS-Levelling stations and much less regularly sampled landmasses in Canada, the gravity signal with half-wavelength of approximately 14 km will have a smaller than 1 cm contribution to the computed uncertainty of the Nov07 datum offset. Therefore, the GOCE geoid should be combined with local gravity information in Canada. On the other hand, the distribution of the US GNSS-Levelling stations allows one to use a GPM with a much lower resolution in the USA. A GOCE GPM would suffice the US NAVD88 offset computation without the need of using local gravity information if the required accuracy is few cm. Detailed investigations are presented in [P-1]. The hybrid geoid model used in the previous paragraph, i.e., TIM3 (degree 180) + EGM2008 (degree ), was used to compute the impact of the GOCE omission error on the North American height datums offsets: CGVD28, Nov07 and NAVD88 in Canada and NAVD88 in the USA. A set of GNSS-Levelling benchmarks from the first order levelling network in Canada was complemented by GNSS-tide gauge stations. The effect of the omission error was 13 cm when the GNSS-Levelling data points were used while it reached cm when the GNSS-tide gauge data points were utilized. This result shows that the impact of the GOCE omission error is of greater significance when tide gauges are used in height system unification. On the other hand, the much more regular and dense data coverage of the USA landmasses is the reason that the effect of the omission error of TIM3 on the NAVD88 offset is only 2 cm. Results of this study are given in [P-13]. A similar experiment for goco03s is given in [P-5].

102 Page: 101 of 173 For some large countries (e.g., Canada and Australia), the problem exists of connecting large islands to mainland when heights from the levelling network on the mainland cannot be transferred to the islands due to the large extent of the sea water bodies. An experiment similar to the above was designed to study the effect of the omission error of the goco03s GPM on the estimated datum offset on two main islands in Canada, i.e., Vancouver Island and Newfoundland. Results show that the GOCO03S omission error can contribute 31 cm to the offset of Vancouver Island that is characterized with a combination of rugged topography and a very small number of GNSS-Levelling data points (26). This result confirms the conclusions given in the previous paragraph. Details are given in [P-5]. Table 4-3: Summary: Omission Error Impact on EPS Standards - Omission error estimated from EGM2008 (refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-61) (3-43) (3-46) - Height system of levelled heights taken into consideration (refer to chapter ) Main Conclusions - Signal not observable by GOCE needs to be quantified. This is regarded as omission error. - EGM2008 is a good choice for quantification of the omission error in areas with good and high quality data coverage in EGM Omission error strongly depends on regional topography. - If omission error doesn t cancel out by mean value computation over a specific area, e.g. for GNSS/levelling points in this area, it needs to be estimated Impact of Propagated Error on Equipotential Surface Least-squares estimation of height datum offsets by the GBVP approach requires knowledge of the error characteristics of the related quantities. One component is the contribution of GOCE global gravity field models. Thereby the error variance-covariance matrix of geopotential coefficients needs to be propagated to the level of geoid heights or geoid height differences. Employing the full VCM of a GOCE geopotential model is numerically demanding due to the large size of the matrix. The corresponding equations are described in section along with two possible levels of approximation, the m-block and the diagonal approach. Since the error correlations between coefficients of different order m are considerably smaller in case of GOCE than correlations between coefficients of the same order, the m-block approach is assumed to be sufficient for error propagation. This is studied in [P-3] for a test case in Europe. The study shows that approximating the GOCE VCM by m-blocks leads to errors in the order of 1-2% or less. This is regarded sufficient for error assessment. Neglecting all correlation however yields errors in the order of 20-50% for the error estimates. Therefore restriction to the diagonal approach is not recommended. The m-block approach is much more accurate and numerically not very demanding. The even simpler diagonal approach over- or underestimates the size of error standard deviations by up to 50% depending on the geographic location. Table 4-4: : Summary: Impact of propagated Error on EPS Standards Equations - GOCE variance-covariance matrix (3-64) (3-66) Main Conclusions - m-block approximation of GOCE variance-covariance matrix is sufficient for error assessment. - Restriction to diagonal approach is not recommended.

103 Page: 102 of GOCE versus GBVP Equipotential Surface Experiment for Europe The gravity based approach is one of three approaches for the unification of height reference frames. At more than 1300 points, which are distributed irregularly over Europe, physical heights given in European national height reference frames and GNSS derived ellipsoidal heights have been available. Different gravity field models have been selected: the satellite-only gravity field model GOCO03S as a pure model up to d/o 250 and extendend in a specific way with EGM2008 or the European regional gravity field model EGG2008. Additionally, with GOCE TIM R3, a GOCE only model has been used in combination with EGG2008. The results based on the global satellite-only gravity field model GOCO03S up d/o 250, representing a spatial resolution of about 100 km (half wavelength), already gives satisfactory results for the offset estimates on an accuracy level of about 5 cm for the most cases. Due to its limited spatial resolution the model may cause larger differences up to 15 cm in areas with a rough gravity field, e.g. in mountainous areas. In general, the influence of the country size is rather small although this might not be true for very small areas below the spatial resolution of the gravity field model. GOCE has significantly increased the usability of satellite-only gravity field models for applications like height reference frame unification. This is of special value for areas with a lack of high quality terrestrial gravity data. For the verification of national height reference frames, a satellite-only gravity field model is not sufficient. High resolution models, such as EGM2008 or regional models, such as EGG2008, can be used to further improve the results. A widely used strategy extends the spherical harmonic coefficients of the satellite-only model by EGM2008 values. Doing so, inhomogeneities between the two models may decrease the accuracy of the combined model considerably. In a comparison to observed height anomalies at GNSS/leveling points in Germany, it could be shown, that this extended model even performs worse than the pure EGM2008. The performance of EGM2008 and EGG2008 is almost on the same level in Germany. The standard deviations are 2.80 cm and 2.63 cm for the EGG2008 and the EGM2008, respectively. Finally, EGG2008 was selected for the extension of GOCO03S and GOCE TIM R3 models. In a simple approach, based on Gaussian filtering, the global and the regional models were combined. The resulting combined models performed better when compared to the pure EGM2008 or EGG2008. The estimated standard deviation is 2.1 cm for the GOCE TIM R3 and 2.4 cm for the GOCO03S model. A further improvement could be expected, if spaceborne and terrestrial observations are combined in a joint analysis using a realistic stochastic model. The comparison of the results to the estimated offsets from the leveling approach gives an agreement of up to 5 cm for the most countries. High resolution gravity fields provide an independent validation method of national leveling data sets. Especially, they allow an inquiry of national leveling networks for systematic errors. The GOCE mission has significantly improved the capabilities of unifying height reference frames. Due to its global applicability, the gravimetric approach is the preferred method for the realization of a World Height System. Details of this investigation can be found in [P-11] Experiment for North America A regional geoid model can be obtained by means of the Stokes-Helmert solution (Novak, 2000) to the GVBP via an optimal combination of satellite-only global geopotential models with terrestrial gravity data and topographic information in remove-compute-restore computational procedures (e.g., Sideris et al., 1992). The optimal combination of a satellite GPM global satellite geopotential model and regional terrestrial data can be performed by applying a modification of the Stokes s kernel using different optimality criteria (e.g., Meissl, 1971; Vaníček and Kleusberg, 1987; Vaníček and Sjöberg,

104 Page: 103 of ; and Featherstone et al., 1998). Refer also to chapter for the theoretical derivations. The kernel modification was originally applied for minimizing the far-zone contribution of the Stokes s integral mostly determined from the less accurate satellite geopotential models before the dedicated gravity satellite missions. The improved accuracy of the long- and medium-wavelengths of the gravity spectrum in the CHAMP/GRACE/GOCE-based GPMs compared to the terrestrial gravity anomalies has required a revision of the kernel modification. Thus, the recent geoid models for Canada were computed using a degree-banded Stokes s kernel. The Stokes s integration contributes to the geoid height the signal coming from the frequencies in the band between a selected cut-off degree of the GPM and the degree that corresponds to the sampling frequency of the terrestrial gravity data. Evaluation of the GOCE and combined GRACE/GOCE GPMs against EGM2008 or a regional geoid model allows one to assess the quality of the satellite-only models for the purpose of height system unification. Usually, the evaluation is carried out using GNSS-Levelling stations, but components of the deflection of the vertical and terrestrial gravity anomalies can be used if available. The GOCE and combined GRACE/GOCE GPMs evaluation in Canada and USA using GNSS-Levelling data showed clearly the improvement from the first to the third generation of GPMs in terms of decreased standard deviation of the geoid height differences and increased signal content. In the spectral band degree 100 to 150, the GOCE and combined GRACE/GOCE GPMs perform slightly better than EGM2008 but after approximately degree 180 their performance deteriorates rapidly. Hence, the combination of the GOCE models with terrestrial gravity data in Canada does not lead to a significant overall improvement in the regional gravimetric geoid (Ince et al., 2012) over EGM2008. However, an improvement in some areas with rugged terrain in Western Canada could be expected. Details on the GOCE and GRACE/GOCE models evaluation are given in [P-1]. Table 4-5: Summary GOCE versus BBVP EPS Standards - Omission error estimated either from EGM2008 or from the European Gravimetric Geoid (EGG2008)(refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-64) (3-85) (3-102) - Height system conversions applied (refer to chapter ) Main Conclusions - European combined geoid based on GOCE performs better than the pure EGM2008 or the EGG2008 models. - In Canada the combination of GOCE with terrestrial data does not lead to a significant overall improvement in the regional gravimetric geoid, but in some areas with rugged terrain in Western Canada some impact became visible. - High resolution gravity fields based on GOCE allow an inquiry of national levelling networks for systematic errors GOCE versus GNSS Levelling Equipotential Surface In this experiment the GOCE equipotential surface is compared to GNSS-Levelling derived equipotential surfaces. For this purpose the global GOCE geoid is compared to local geoid heights derived from GNSS-Levelling data. This is done for specific regions, which could be a part of a country, a complete country or sometimes even an entire continent. Height offset estimates between the global and local geoids are computed by taking the mean value of these differences over the area under investigation. From the results obtained in this study one can draw the following conclusions. (1) GNSS-Levelling data are useful for this kind of analysis as they provide information about the height datum used for a specific area. The height datum is related to physical heights derived from spirit levelling and gravimetry at GNSS-Levelling points. Together with ellipsoidal heights, local geoid heights at these points can be determined and compared to geoid heights derived from GOCE.

105 Page: 104 of 173 (2) GOCE provides for the first time the opportunity to observe the global geoid with an accuracy of some few cm at a spatial resolution of 100 km independent of any terrestrial data. This was shown by analysing the signal degree variances of a number of GOCE based models. In order to identify the impact of GOCE it was decided to use a pure GOCE gravity field model for this study (TIM3 model) up to degree and order 180. (3) For intercontinental height datum connection it is important to select GNSS-Levelling data of good quality, as any error in the levelling networks completely maps into the offset estimates. This is nicely shown by an attempt to connect the European with the US height system as well as by connecting the US with the Japanese and Australian height systems. For this purpose US East and West coast states have been treated separately and offsets per state have been computed. It becomes obvious that there exists a relatively strong North-South trend for the offset estimates at both coasts. This trend can be linked to results shown by other investigations about the long wavelength error of the US levelling network model. It became obvious that intercontinental height datum connection strongly depends on the quality of the national levelling networks. (4) It was tried to quantify the impact of GOCE for intercontinental height datum connection by comparing newly obtained results from this study with those from an earlier study based on geometric heights on Doppler stations and an older global geoid solution. Results for vertical offsets between Australia, USA and Germany presented by Rapp (1994) and from this study differ by up to 10 cm. These differences can be addressed to improvements of the global gravity field model and the ellipsoidal heights. At this point it can not be stated, that these changes also can be regarded as improvements of the height offset estimates, as no reference values are available. (5) By comparing vertical offset estimates based on the full EGM2008 and the hybrid TIM3/EGM2008 models one can identify variations of several mm. As the comparisons have been done in well observed areas one can assume that the full EGM2008 solution already provides a high precision geoid and that GOCE can improve this geoid only marginally. On the other hand one may also conclude from the analysis of vertical datum offsets based on truncated gravity field models that the GOCE impact is at a level of up to 5 cm on height offset estimates. From both results it can be concluded that the quality of the omission error somehow dominates the accuracy of the datum offsets estimated by this method. Details of this investigation can be found in [P-4]. A similar experiment was carried out in North America with the following conclusions. (1) An important factor that has a significant impact on the accuracy of the GNSS-Levelling EPS is the large distortions of continental-size height datums such as the official Canadian datum CGVD28 and the international North American datum NAVD88. The latter is characterized with a large tilt, and the former has large, unknown regional distortions. The minimum constrained NAVD88 has a large tilt with respect to recently computed gravimetric geoid models due to accumulation of systematic errors in levelling data from coast-to-coast. Across USA, NAVD88 has a large northwest-southeast tilt. In Canada, NAVD88 indicates that the MSL at the Pacific coast is higher than the MSL at the Atlantic coast by 1.5 m while the oceanographic models of SST indicate that the Pacific Ocean near Vancouver is about 50 cm higher than the Atlantic Ocean in Halifax. The CGVD28 is an over-constrained datum to the mean sea level at five tide gauges on both the Atlantic and Pacific coasts. Therefore, the zero height level surface of CGVD28 is not an equipotential surface. In this case, GNSS-Levelling stations sample up to 1 m height datum distortions that are due to the fact that 1) the MSL constraints were not corrected for sea surface topography, 2) the heights were not corrected for crustal displacement of benchmarks over the decades of levelling measurements, 3) important systematic corrections to levelling observations were not taken into account including refraction, rod calibration, magnetic field and tidal corrections, and 4) the vertical control

106 Page: 105 of 173 network in Canada was established over the years in a piece-wise manner by combining observations after 1928 over several consecutive years and adjusting these observations locally. Because of these unknown datum distortions, the CGVD28 offset computed by averaging the offsets of the EPS at the GNSS-Levelling stations with respect to one conventional EPS or a GOCE-defined EPS is of no use for height datum unification. On the other hand, GNSS-surveyed tide gauges provide a more realistic estimate of the datum offset as (1) the distribution of the tidal benchmarks is not representative of the distortions within CGVD28 and (2) the datum offset is computed from fewer stations, the distortion of which is explained by the sea surface topography. More on this topic could be found in [P-1]. (2) An indicator of the quality of the GNSS-Levelling EPS is the standard deviation of the differences of the geometric GNSS-Levelling geoid and gravimetric geoid heights. The standard deviation contains the tilt and other more complex distortions of the height datum in addition to the random GNSS and levelling measurement and geoid modelling errors. For example, after filtering out the geoid height differences for the bias and tilt through a plane surface, the improvement in the standard deviation for NAVD88 was 25 cm in Canada and 14 cm in the USA. This shows that the systematic errors in the levelling data that accumulate from coast to coast have a large effect on the dispersion of the geoid height differences. More on this topic could be found in [P-1]. (3) An experiment was designed to assess the effect of the large east-west tilt (~ 150 cm) of NAVD88 and the smaller east-west tilt (~ 80 cm) of the Nov07 height datum in Canada on the estimated datum offset in a least-squares adjustment procedure. For this purpose, the offsets were computed with EGM2008 truncated at a different degree: 70, 90, 120, 150, 180, 200, 210, 360, 1400 and Each degree corresponds to the maximum spectral resolution of existing satellite and combined GPMs. The datum offsets were estimated by a least-squares fit of (1) a bias and (2) a plane (bias, north-south and east-west tilt). The differences between the computed offsets (biases) for each degree in the two cases show that the tilt affects significantly the computed NAVD88 offset in Canada through the correlation of the estimated parameters; the maximum difference is 4.6 cm for degree 180. The effect on the Nov07 offset in Canada is smaller and reaches 2.2 cm for degree 180. The effect of the tilt on the NAVD88 offset in USA is insignificant; the difference for each cut-off degree is below 0.5 cm. Three are the factors that are responsible for this significant effect of the NAVD88-Canada datum tilt: (1) the distribution of the GNSS-Levelling data points that do not constrain well the fitted plane while the opposite holds for the data points in the USA, (2) the omission error of the truncated EGM2008 at the GNSS-Levelling data points and (3) the GNSS and levelling measurement data errors. It should be noted that the effect of the datum tilt on the estimated offset computed with the full resolution EGM2008 is 3 cm for NAVD88-Canada and below 1 cm for Nov07. These results show that the systematic effects in the first order levelling network bias the estimated datum offset through the correlations between the estimated parameters in the least squares estimation procedure. Based on the discussion described in conclusions (1), (2) and (3), it is recommended that the geoid height differences are corrected for the known systematic effects before estimating the height datum offset. The tilt of the levelling datum with respect to the geoid should be determined by means of a GOCE-based geoid model and sea surface topography models. (4) In North America, both GOCE and GNSS-Levelling EPSs can be affected by the regional geodynamics in terms of mass change and crustal displacement due to the ongoing postglacial rebound. The effect of the mass change on the GOCE EPS would be approximately 2-3 mm as GOCE observations have been collected and processed since March The effect on the GNSS-Levelling EPS would be much more significant. Firstly, GNSS campaign and continuous measurements in the Canadian SuperNet GNSS network have been collected since 1994 while levelling measurements in the first-order control network were carried out for more than 80 years. Secondly, while the GNSS-determined ellipsoidal heights were corrected for the crustal motion in the last adjustment of SuperNet, such corrections were not applied to

107 Page: 106 of 173 the levelling measurements. Hence the combination of different measurement epochs and lack of corrections for crustal motion in the leveling data lead to incapability to properly quantify the effect of postglacial rebound on the GNSS-Levelling EPS. A somewhat simplified test showed that the crustal motion corrections in the GNSS ellipsoidal heights likely have a small effect of 1 cm on the computed height datum offset. Table 4-6: Summary GOCE versus GNSS/Levelling EPS Standards - Omission error estimated either from EGM2008 (refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-64) - Height system conversions applied (refer to chapter ) Main Conclusions - Large scale distortions of continental-size height datums (CGVD28, NAVD88) have significant impact on GNNS/Levelling derived EPS. - For intercontinental height datum connection it is important to select GNSS-Levelling data of good quality, as any error in the levelling networks completely maps into the offset estimates. - The amplitude of the omission error (mean over the selected GNSS/Levelling stations) and how good this can be determined somehow dominate the accuracy of the datum offsets estimated by this method. - In North America GOCE and GNSS/Levelling derived EPS are affected by regional geodynamics (PGR). This effect is more pronounced on the historic levelling measurements and orthometric heights, but corrections are not feasible. The vertical motion effect in the GNSS derived ellipsoidal heights needs to be considered as it could have an effect at a level of 1 cm on height datum offsets Oceanic and Tide Gauge/Datum Levelling In this experiment, the mean sea level (MSL) at a tide gauge in one coastal country, expressed as a height relative to a national datum, is compared to the MSL at a tide gauge in another country expressed to its own national datum. The use of ocean models enables MSL in the two countries to be related, hence a relationship can be established between the two datums. The important issue then is to decide how precise the ocean models are, most available models being deep ocean ones which omit consideration of coastal oceanographic processes. Our experiments have involved the use of a number of available ocean and coastal models. (The use of data from any geoid model does not enter into this experiment.) Table 4-7: Summary Oceanic and Tide Gauge Levelling Standards Equations - DOT Determination by Ocean Models (refer to chapter 3.3.6) Main Conclusions - Ocean models are consistent at the sub-decimetre level for the regions that were studied (North Atlantic coastlines and islands, North American Pacific coast and Mediterranean) Oceanic and Tide Gauge/GNSS-Geoid Levelling In this experiment, the MSL at the same tide gauges as in are expressed not in terms of nations datums, but first as ellipsoidal heights using GNSS measurements at their tide gauge benchmarks. The ellipsoidal heights can then be converted to heights above the geoid with the use of the latest, and most complete, geoid models. Such heights of MSL above the geoid should then be consistent with expectations of the Mean Dynamic Topography from the ocean models. (These are called the geodetic and ocean approaches to the determination of the mean dynamic topography (MDT) at the coast.) Such a consistency provides confidence in the use of both the ocean and geoid models, even extending confidence in the latter to terrestrial locations far from the coast, given that there is no reason to believe that the geoid models are intrinsically more precise at the coast. The consistency

108 Page: 107 of 173 between the geodetic and ocean approaches is demonstrated by Table 4-17and Table 4-18to be presented in chapter below. Table 4-8: Summary Oceanic and Tide Gauge/GNSS/Geoid Levelling Standards - DOT Determination by Ocean Models (refer to chapter 3.3.6) - Omission error estimated from EGM2008 (refer to chapter ) Equations (3-61) - Tide system conversion applied (refer to chapter ) Main Conclusions - New models of the geoid arising from the recent space gravity missions, and especially from the GOCE mission, are now accurate enough that one can derive the MDT along a coastline in terms of the ellipsoidal heights of MSL (measured at tide gauges) minus geoid height. - Resulting values of MDT are similar to those suggested by a number of ocean models Oceanic and Altimeter/Geoid Levelling In this experiment, the ellipsoidal heights of MSL are derived not from tide gauges and GNSS but from satellite altimetry obtained off-shore of the coast (typically a few 10s km). Otherwise, similar tests for consistency can be made as in Although in this experiment the MSL is not obtained exactly at the coast as for tide gauges, and as one really requires for datum connections and height unification, the continuity of altimeter information along a coastline provides a densification of information in such comparisons and a link to the general deep-ocean validation of new geoid models being performed by the oceanographic community. Details of the investigations described in 4.2.6, 4.2.7and may be found in [P-14]. We have concluded that models are consistent at the sub-decimetre level for the regions that we have studied (North Atlantic coastlines and islands, North American Pacific coast and Mediterranean) and has emphasised the importance of the model complete new geoid model (the Extended GOCO03S model). That level of consistency provides an estimate of the accuracy of using the ocean models to provide an MDT correction to the national datums of countries with coastlines, and thereby of achieving unification. It also provides a validation of geoid model accuracy for application to height system unification in general. Table 4-9: Summary Oceanic and Altimeter/Geoid Levelling Standards - DOT Determination by Ocean Models (refer to chapter 3.3.6) - Omission error estimated from EGM2008 (refer to chapter ) Equations (3-61) - Tide system conversion applied (refer to chapter ) Main Conclusions - New models of the geoid arising from the recent space gravity missions, and especially from the GOCE mission, are now accurate enough that one can derive the MDT along a coastline in terms of the ellipsoidal heights of MSL (measured from altimetry) minus geoid height. - Resulting values of MDT are similar to those suggested by a number of ocean models. 4.3 Experiments for Connection of HSU Approaches Connecting GBVP Solution and GNSS Levelling The geoid height as determined with the GBVP (see chapter , equation(3-85)) can be connected to observed geoid heights at GNSS levelling points for a specific datum zone. These geoid heights are determined by:

109 Page: 108 of 173 j J N (r P, P, P) h(r P, P, P) H (r P, P, P) N h H j j P P P (4-1) where h P is the ellipsoidal height (e.g. from GNSS) and from a combination of levelling and gravimetry and referring to datum point Finally we end up with the observation equation (compare (3-85)): L j GPM res P P P P P 0 j0 l0 l0 l1 j H P is the orthometric height determined O j of datum zone j. h H N N N N N f (4-2) All quantities on the left-hand side are measured or computed and are collected under the observation j vector y P. The right hand side contains the unknowns, the globally constant offset N 0 between the unbiased geoid and a chosen reference datum zone and relative offsets N j0 of all datum zones with respect to this reference zone. Separating the relative offset of the datum zone of observation point P from the summation, we find the observational model for least-squares estimation of datum offsets: L j P P P 0 j0 j0 l0 l0 l1,l j y N 1 f N N f (4-3) Alternatively we can formulate the estimation in terms of geoid height differences between points P and Q (located in two different datum zones j and k): y 1 f N N f 1 f N N f L M (4-4) jk P P Q Q PQ j0 j0 l0 l0 k0 k0 m0 m0 l1,l j m1,m k In matrix notation the linear observation model can be written as: y A x (4-5) with the observation vector y, the design matrix A and the unknown datum offsets x. The least-squares solution of this model is: ˆx (A'Q A) A'Q y (4-6) yy yy where Q yy is the error VCM of the observations. The estimated error VCM of the unknowns is given by the inverse of the normal equation matrix: Q (A'Q A) (4-7) xx ˆˆ 1 1 yy The following example shows the structure of the design matrix. Let s assume we have 7 GNSS/levelling stations (A,B,C,D,E,F,G) available and a total of 4 different datum zones; station A is located in datum zone j=0, B and C in datum zone j=1 and D and E in datum zone j=2 and F in zone j=3. Let s further assume that we have observed absolute quantities in the sense of equation (4-3) and we chose datum zone 0 as reference datum zone. Then the unknowns are the constant global offset N 0 of zone 0 with respect to an unbiased geoid and the relative datum offsets N 10, N 20 and N30 of datum zones 1, 2 and 3 with respect to zone 0. The structure of the matrix is then given by Table Alternatively, observation of geoid height differences between observation stations results in a design matrix which contains differences of the elements given in (4-4) and evaluated at the respective stations/zones. In this case the first column disappears because only differences are observed and the absolute offset term N 0 cannot be determined.

110 Page: 109 of 173 Table 4-10: Structure of the design matrix for least-squares adjustment of height datum offsets for the station configuration described in the text. j 0 y A 1 j1 y B 1 j1 y C 1 j 2 y D 1 j 2 y E 1 j 3 y F 1 N 0 N 10 N 20 N 30 A f 10 B 1 f 10 C 1 f 10 D f 10 E f 10 F f 10 A f 20 B f 20 C f 20 D 1 f 20 E 1 f 20 F f 20 A f 30 B f 30 C f 30 D f 30 E f 30 F 1 f 30 Some of the factors contributing to the accuracy of the computed offsets of the official height datums CGVD28 and NAVD88 in North America and the unofficial Canadian datum Nov07 were discussed in Chapter 4.2. Here we provide a summary of all of the factors we took into account in the offsets computations: (1) Tide system: GNSS ellipsoidal, orthometric and GOCE geoid heights are in a tide free system. (2) ITRF and epoch: the GNSS ellipsoidal heights used in the offsets computations are in ITRF2005, epoch These include the US GNSS ellipsoidal coordinates transformed from NAD83 (CORS96) to ITRF2005(2006.0). The Canadian GNSS-Levelling database consists of points in three different ITRF systems. Our tests showed that the combination of different ITRF has a 1-2 cm effect on the computed offsets. (3) Indirect bias term: Based on our investigations, the indirect bias term was omitted in North America. In this case, the datum offsets were computed as a (weighted) average of the geoid height differences in the left hand side of equation (4-2). (4) Data accuracy information: In the CGVD28 case, standard deviations of the normalorthometric heights were not available. In the Nov07 case, only the orthometric heights determined from the adjustment of levelling data after 1981 have meaningful error information. Our tests showed that the inconsistency of the data accuracy information is of less importance than the configuration of the network of GNSS-Levelling data points. Using properly weighted (rescaled by variance component estimation) diagonal VCMs for GNSS, orthometric and geoid heights could contribute up to 4 cm to the Nov07 offset. The difference in the datum offset when computed from a least-squares adjustment with fullypopulated and diagonal VCMs varies around 1 cm depending on the configuration of the network of GNSS benchmarks. The use of fully-populated VCMs is recommended to properly account for the correlations within each height group. (5) Systematic effects and regional distortions in the datums: a. For CGVD28, it is always a challenge to determine the mean datum offset because of the datum distortions. The precise determination of the mean offset requires a dense and well distributed network of GNSS/levelling stations, which is not the case in Canada. b. NAVD88 has a significant tilt in Canada, which can affect the computed datum offset by as much as 5 cm if GOCE GPM of D/O 180 is used. (6) GOCE GPM omission error: This is the main factor that contributes 8 cm to the uncertainty of the computed datum offsets in Canada if a GOCE GPM of D/O 250 is used and 13 cm for a GOCE GPM of D/O 180. The effect of the omission error significantly decreases to 2 cm for the NAVD88 offset in the USA. It should be noted that no

111 Page: 110 of 173 significant difference exists between the computed datum offsets with the EGM2008 geoid and the regional geoid models, CGG2010 in Canada and USGG2012 in the USA. This is because the EGM2008 model essentially uses the same terrestrial gravity data used in the development of the regional geoid models. (7) Configuration of the network of GNSS-Levelling points: This factor controls the effect of the GOCE GPM omission error on the estimated datum offsets. Two cases were investigated in North America: (i) use all available GNSS-Levelling data points and (ii) use selected points from the GNSS-Levelling database. In the Canadian case, we were able to down-sample the first-order levelling benchmarks from 1315 to 308 on the Canadian mainland while maintaining a 1 cm difference in the computed datum offset. Although we did not use an optimality criterion in the data point selection, such criteria can be employed. Relevant computational procedures were developed in Rangelova (2007). In the US case, we used all of the available GNSS-Levelling data points (including GNSS points with 5 th order ellipsoidal heights) in order to achieve more dense coverage of the US terrain. Results showed that even with this dense point distribution some local geoid signatures were not properly sampled. Nevertheless, the denser data coverage warranted that the mean of the GOCE GPM omission error would tend to vanish on the GNSS- Levelling data points and had a small effect of 2 cm (relative to EGM2008) on the computed NAVD88 offset even for a GOCE GPM of D/O 180. Table 4-11: Summary: Connecting GBVP Solution and GNSS Levelling Standards - Omission error estimated from EGM2008 (refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-61) (3-85) (3-102) - Height system conversions applied (refer to chapter ) Main Conclusions - Comparisons showed that the combination of different ITRF has a 1-2 cm effect on computed height system offsets. - Indirect bias term was omitted in North America. - Inconsistency of the data accuracy information is of less importance than the configuration of the network of GNSS-Levelling data points. - The GOCE omission error is the main factor that contributes 8 cm to the uncertainty of the computed datum offsets in Canada if a GOCE model of d/o 250 is used and 13 cm for a GOCE model of d/o 180. The effect of the omission error significantly decreases to 2 cm for the NAVD88 offset in the USA. - The configuration of the network of GNSS-Levelling points controls the effect of the GOCE GPM omission error on the estimated datum offsets Connecting Tide Gauges to GBVP Solutions A connection between mean sea level and the regional geoid was established at the GNSS-surveyed tide gauge stations on the Canadian Atlantic and Pacific coasts. The purpose of this study was the determination of the mean geopotential of the mean sea level for the Canadian coasts by averaging the geopotential computed at the tide gauge stations. The determined mean potential will define the zero height surface of the new Canadian and future North American height datums. Factors that contributed to the accuracy of the computed potential values are (1) the distribution and number of tide gauges with 19-year long recordings, (2) resolution, accuracy and interpolation errors at the tide gauge locations of the sea surface topography models, (3) different tide systems of the geoid model, GNSS ellipsoidal heights and sea surface topography, and (4) crustal motion corrections of the GNSS ellipsoidal heights. Finally, the omission error of the GOCE geopotential model is a source of significant error as the omitted higher frequencies of the geoid (less than ~110 km half-wavelength) contribute approximately 16 cm in terms of separation of the determined equipotential surface for the Pacific region and 9 cm for the Atlantic region. When regional gravimetric geoid models are utilized for the computation of the potential values at the tide gauge stations, results are consistent for all

112 Page: 111 of 173 investigated geoid models: the official high resolution gravimetric geoid for Canada, the unofficial GOCE-based geoid model developed by UoC/GSD and the hybrid GOCE (degree 180) + EGM2008 ( ) geoid. This shows that a significant improvement in the local geoid on EGM2008 cannot be expected in the Canadian coastal areas as the same gravity data information was used in the development of the regional geoid models and EGM2008. Details of this investigation are given in [P-6]. Table 4-12: Summary: Connecting Tide Gauges to GBVP Solutions Standards - Omission error estimated from EGM2008 (refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-61) (3-85) (3-102) Main Conclusions - Accuracy of potential values is dependent on the distribution and number of tide gauges with 19-year long recordings, on the interpolated sea surface topography at the tide gauges, on consistency of tide systems, and on the crustal motion correction. - Results for various regional gravimetric geoid models are consistent Connecting Tide Gauges to GNSS Levelling In order to connect tide gauges with available GNSS Levelling stations the following procedure was applied to the available data sets. (1) As starting point the list of tide gauge stations (coordinates) from the PSMSL was taken. (2) GNSS Levelling stations close to the tide gauge locations where identified. As a limit a distance corresponding to 1 degree was selected. This corresponds to the maximum reachable GOCE spatial resolution. For GNSS Levelling stations all available data sets were applied. (3) For each tide gauge station mean offsets of the GOCE geoid to the GNSS Levelling geoid were computed. Only the selected GNSS stations (see step 2) were taken into consideration. Offsets were estimated for the GOCE-TIM4 model (up to degree 180) with and without taking the omission error into account. The omission error was computed from the EGM2008 model (degree/order 181 to 2190) and from the RTM correction (above degree 2190). (4) The impact of the omission error was estimated for each tide gauge station separately. One can select tide gauges with small omission error surrounded by a sufficient number of GNSS Levelling stations to estimate the offsets of the height system to the GOCE geoid and to connect them with tide gauge recordings. Investigating the results obtained from the procedure described above one can identify individual tide gauges, where the omission error has minor impact (either due to the geoid signal characteristic in this area or due to the distribution of GNSS Levelling stations). Using these quasi omission error free geoid heights determined from GNSS Leveling one can estimate their offsets to the GOCE geoid. Such tide gauges can then further be used to connect them across the oceans. As an example for the procedure some results are shown in the subsequent figure.

113 Page: 112 of 173 Figure 4-4: Omission error (in [m])estimated from EGM2008 & RTM at GNSS Leveling stations close to tide gauge locations in Europe, Australia and along the US East coast (left column); Number of GNSS Leveling stations within 1 degree distance from tide gauge location used for computing the omission error (right column).

114 Page: 113 of 173 Table 4-13: : Summary: Connecting Tide Gauges to GNSS Levelling Standards - Omission error estimated from EGM2008 (refer to chapter ) - Tide system conversion applied (refer to chapter ) Equations (3-61) - Height system conversions applied (refer to chapter ) Main Conclusions - Tide gauge offsets to national height datums can be estimated at stations with minimum omission error impact. - Connection of tide gauges over large distances can be performed once they are corrected for their local offsets Connecting Tide Gauges by Ocean Levelling The connection of tide gauges by ocean levelling is described in detail in chapter [P-14]. From this study the following conclusions were drawn. (1) This study has demonstrated that the new models of the geoid arising from the recent space gravity missions, and especially from the GOCE mission, are now accurate enough that one can derive the MDT along a coastline in terms of the ellipsoidal heights of MSL (measured at tide gauges or obtained from altimetry) minus geoid height. Moreover, the resulting values of MDT are similar to those suggested by a number of ocean models. This exercise thereby provides a validation of both sets of models. (2) This work has some similarity to recent studies of MSL variations and datums along the North American Atlantic coast (Higginson, 2012) and around the coast of Australia (Featherstone and Filmer, 2012). In particular, it has resolved one of the longest-standing discussions in oceanography concerning the direction of the tilt of sea level along the American coast (Sturges, 1974). It is certain that, now that the geoid models are so good and will improve further, much more science will flow. (3) However, to return to the main reason for the present study, to assess the contribution of ocean levelling to worldwide height system unification, we believe that this is now possible with a typical uncertainty of better than a decimetre. However, this statement is subject to reservations concerning the limitations in the ocean models available for analysis, and to the fact that a global study remains to be made. Our use of data from North Atlantic coastlines and islands, the North American Pacific coast and the Mediterranean has demonstrated that our methods should be capable of being applied to any countries with at least one operational or historical tide gauge for which at least one of the benchmarks has been surveyed by GNSS. Table 4-14: Summary: Connecting Tide Gauges via Ocean Levelling Standards - DOT Determination by Ocean Models (refer to chapter 3.3.6) - Omission error estimated from EGM2008 (refer to chapter ) Equations (3-61) - Tide system conversion applied (refer to chapter ) Main Conclusions - New models of the geoid arising from the recent space gravity missions are now accurate enough that one can derive the MDT along a coastline. - Resulting values of MDT from the geodetic approach are similar to those suggested by a number of ocean models. - The study has resolved one of the longest-standing discussions in oceanography concerning the direction of the tilt of sea level along the American coast. - With ocean levelling global height system unification shall now be possible with a typical uncertainty of better than a decimetre.

115 4.4 GOCE Impact Assessment Summary and Final Report Page: 114 of 173 This chapter summarizes the impact of GOCE on height system unification compared with results obtained with pre-goce gravity fields. This assessment is based on the results obtained from the experiments described in the previous section Assessment of GOCE According to equation geoid heights (respectively height anomalies) are derived from the combination of a satellite-only global potential model (GPM) and terrestrial data, the latter taking care for the omission error of the satellite model. Thereby, GOCE provides superior accuracy for the medium wavelength band of the spectrum compared to earlier satellite-only models. This is depicted in Figure 4-5, which shows the error degree RMS (commission error) of the satellite-only models ITG- GRACE2010s (GRACE only), GOCO03S (GOCE-GRACE combination) and the high resolution combined model EGM2008 (GRACE plus terrestrial data). Figure 4-5: Commission error of different global potential models Since the medium wavelength band is represented best in GOCE based GPMs, one can expect, that GOCE leads to an improvement of the total error budget. In addition, higher spatial resolution of GOCE allows restricting the computation of residual geoid heights to increasingly smaller spherical caps around the area of interest. The latter affects the requirements for availability of terrestrial data. In order to give an estimate of the improvement brought by GOCE, the following experiment was conducted. The combination model EGM2008 is mainly used to provide the terrestrial information (according to Figure 4-5 this corresponds to the spectral band above spherical harmonic degree 100). For the long wavelength information either GOCO03s or ITG-GRACE2010s are used. The error budget is evaluated from the formal errors of the different models in the sense of commission errors. Thereby the long to medium wavelength are taken from the pure GRACE and GRACE/GOCE models, while the short wavelength errors are taken from EGM2008. Thereby, various spherical harmonic degrees are used for the transition from satellite data (GRACE or GOCE) to terrestrial data (EGM2008). The result is depicted in Figure 4-6.

116 Page: 115 of 173 Figure 4-6: Commission error for a combination of satellite only (GRACE or GOCO) and terrestrial (EGM2008) data as a function of the spherical harmonic degree of the transition between satellite and terrestrial data. The commission error of EGM2008 is at the level of about 8 cm in terms of geoid height. This value is dominated by the medium to short wavelengths of the EGM2008 signal. Therefore replacement of EGM2008 by the GRACE or GOCO models up to spherical harmonic degrees of about 70 does not improve the total error budget. Only after degree 70 the total error budget is decreasing. This means, that combination with GRACE or GOCE models improves the geoid solution. This behavior continuous until the commission error of the satellite-only models rises above the error level of the terrestrial data. Then adding more and more satellite-only data increases the total error budget. Figure 4-6 shows that this breaking point (minimum value) is reached at spherical harmonic degree 144 for the GRACE model and degree 168 for the GOCO model. For those degrees the commission error reaches about 5.8 cm for the GRACE model and about 5.1 cm for GOCO03s. This corresponds to an improvement of 30% respectively 38% for ITG-GRACE2010s and GOCO03s with respect to EGM2008. As the comparison is only based on the error standard deviations of the spherical harmonic coefficients of the different GPMs, the result corresponds to a global average. However, one has to consider that the improvement and the significant resolution (breaking point in Figure 4-6) of the satellite-only models strongly depend on the quality of the terrestrial data. However, the latter is quite inhomogeneous as is depicted in Figure 4-7. Figure 4-7: Geographic distribution of the commission error of EGM2008

117 Page: 116 of 173 In order to reflect the actual situation with, e.g., precise knowledge of the gravity field in Europe but poor knowledge in large parts of South America or Asia, the same experiment as shown in Figure 4-6 was repeated, but this time rescaling the EGM2008 error such that it reflects the quality of the terrestrial data in different geographic zones. Two extreme cases are discussed in the sequel. They are depicted in Figure 4-8 and Figure 4-9. In case of Figure 4-8 the error standard deviations of EGM2008 were rescaled such that the overall error budget amounts to about 1 cm. This reflects a situation with very dense coverage with high precision gravity values, allowing for the determination of a 1cm-geoid. This is much better than the global average of 8 cm shown in Figure 4-6. In contrast, Figure 4-9 shows a case, where the EGM2008 errors standard deviations are rescaled such that they reflect a situation with poor coverage and quality of terrestrial data, such that a geoid of only 50 cm precision can be determined. This is much worse than the global average and may reflect the situations, e.g., in some parts of South America, Africa or Asia. In case of the 1cm geoid the breaking point of GRACE is at spherical harmonic degree 112 and for GOCO at 120. This is a marginal improvement. Also the improvement in terms of the total error budget (with 33% and 37% being significant expressed in terms of relative numbers) is minor (3 to 4 mm expressed in terms of absolute values). In case of the 50cm-geoid, the effect of including GOCE data turns out to have a larger effect. The significant resolution is changed from degree 170 to 233 (GRACE vs. GOCO) and the error budget is decreased from 50 cm (EGM2008) to 37 cm (GRACE-ITGGRACE02s) and further down to only 30 cm (corresponding to a relative improvement of 26% and 41% for GRACE and GOCO, respectively). These numerical experiments show, that GOCE models can lead to very significant improvements of the geoid amounting to up to several decimeters even with respect to the best available GRACE models (one can expect even more with respect to earlier GPMs like EGM96). This however is restricted to areas with poor terrestrial gravity field data. In contrast, in areas with almost perfect knowledge of the gravity field ( cm geoid ) GOCE leads to only a very small improvement. This reflects the fact that the assessment of GOCE cannot be carried out by looking at GOCE only, but the data situation in different geographic regions needs to be considered. In general one can state, that the improvement brought by GOCE depends on the quality of terrestrial data. In the extreme case of perfect knowledge of the gravity field, GOCE will not contribute at all. Taking the formal error description of EGM2008 as a measure for the quality of terrestrial data, GOCO03s, on global average, leads to an improvement of about 10% in terms of geoid height error with respect to the currently best available GRACE model ITG-GRACE2010s and about 30% - 40% with respect to EGM2008 (employing an older GRACE model). Figure 4-8: Commission error for a combination of satellite only (GRACE or GOCO) and terrestrial (EGM2008) data. The formal errors of EGM2008 are rescaled such that they reflect terrestrial data allowing for a precise 1cmgeoid.

118 Page: 117 of 173 Figure 4-9: Commission error for a combination of satellite only (GRACE or GOCO) and terrestrial (EGM2008) data. The formal errors of EGM2008 are rescaled such that they reflect poor terrestrial data allowing for only an accuracy of 50 cm in terms of geoid heights Impact on HSU Methods GOCE Geoid to Check Levelling Example: Canada GNSS/levelling geoid residuals are computed for the set of approximately 2500 primary levelling benchmarks with normal-orthometric heights in Canada s official height datum CGVD28. The geoid heights are computed by means of TIM3 of maximum D/O 180 and the extended TIM3 up to degree 2190 and order 2159 by EGM2008. The residuals computed with both sets of geoid heights are denoted as I for TIM3 and II for TIM3+EGM2008, and their geographical distribution is given in Figure 4-10 a and b, respectively. Statistics of the residuals are given in Table The following observations can be made from the inspection of the figures and statistics: (1) There exists a predominantly east-west tilt of CGVD28 (Figure 4-10 a and b), which is due to constraining the datum to the mean sea level on the Atlantic and Pacific coasts. The tilt of the datum computed with the GNSS/levelling residuals II amounts for cm from Rimouski to the Pacific coast and cm from Rimouski to the Atlantic coast (see also [P-14]). In general, the mean sea level of the Canadian Pacific coast is 60 cm above the mean sea level of the Atlantic coast (Hayden et al., IAG, 2013). The north-south and east-west tilts in Table 4-15 were computed by fitting a plane to the GNSS/levelling geoid residuals I and II. Figure 4-11 c shows the differences between the longitudinal and latitudinal tilts computed with both data sets. The plane fitted to the TIM3+EGM2008 residuals is 20 cm above and 20 cm below the plane fitted to the TIM3 residuals on the west and east coasts, respectively. Thus, it can be conluded that the longitudinal tilt computed from the TIM3 residuals does not agree with the tilt computed by the oceanographic methods and is significantly smaller. The tilt computed with TIM3+EGM2008 residuals is subtracted from the residuals in Figure 4-10 a and b and the new residuals are given in Figure 4-10 c and d. It is easily seen that the positive residuals in Figure 4-10 d are located mostly in the regions with rugged terrain while the negative residuals are in low elevation areas in cental Canada, Great Lakes and Great Slave Lake in the northwest. (2) The omission error of the TIM3 geoid is significant as seen from the comparison of cases (c) and (d) in Table 4-14 and Figure 4-10 e: the standard deviation of residuals is reduced from 42.3 cm for TIM3 to 19 cm for TIM3+EGM2008, i.e., by 23.3 cm. (3) The estimate of the mean offset of CGVD28 with respect to the level surface defined by the geoid is largely affected by the TIM3 omission error (Table 4-14, cases (a) and (b)). It decreases by 20 cm for TIM3+EGM2008.

119 Page: 118 of 173 In summary, in order to study local biases and distortions in the CGVD28 height datum by means of the GOCE geoid, one should correct the geoid heights for the GOCE model omission error and correct the GNSS/levelling geoid residuals for the CGVD28 longitudinal tilt. We show in Figure 4-12 the distortion of CGVD28 in central and eastern Canada due to the vertical crustal motion of benchmarks as a result of the ongoing postglacial rebound. The TIM3+EGM2008 GNSS/levelling geoid residuals corrected for the datum tilt clearly show the areas of subsidence south and southeast of the line of zero vertical motion and the areas of uplift to the north. Table 4-15: Statistics of GNSS/levelling geoid residuals for the five cases in Figure 4-10 Residuals Min, cm Max, cm Mean, cm Std, cm (a) (b) (c) (d) (e) Figure 4-10: : GNSS/levelling geoids residuals: (a) computed with TIM3 d/o 180, (b) TIM3 + EGM2008, (c) and (d) same as (a) and (b) but with the longitudinal tilt removed, (e) omission error of TIM3

120 Page: 119 of 173 Table 4-16: Computed CGVD28 tilt and offset with respect to the geoid Residuals Mean Longitudinal tilt, Latitudinal tilt, cm/deg (CGVD28 offset), cm cm/deg TIM3 (I) TIM3 +EGM2008 (II) Figure 4-11: GNSS/levelling geoid residuals as a function of (a) latitude and (b) longitude; (c) differences between the computed longitudinal and latitudinal tilts from residuals II and I

121 Page: 120 of 173 Figure 4-12: TIM3+EGM2008 GNSS/levelling geoid residuals overlaid with postglacial rebound vertical crustal velocities in eastern and central Canada. Checking quality of national gravity data The Gravity database from Geoscience Data Repository of Natural Resources Canada that is used in the computations of the Canadian gravimetric geoid model is analyzed by means of the GOCE TIM3 free-air gravity anomalies expanded to degree and order 180. The database consists of approximately 700, 000 point gravity values (Figure 4-13) classified as follows: 0 = A depth of water or thickness of ice was present but was not measured 1 = Observation taken on land. 2 = Observation taken on the surface of an ocean or lake. 3 = Observation taken on the bottom of an ocean or lake. 4 = Observation taken on an ice cap. 5 = Observation taken on an ice-covered ocean or lake. 6 = Observation taken on the surface of the sea in a fiord. 7 = Observation taken on sea ice in a fiord. 8 = Observation taken on the bottom of the sea in a fiord. The gravity values refer to IGSN71, and the computed gravity anomalies refer to GRS80. Figure 4-13: Point gravity values clasified with respect to the surface on which gravity measurement were taken

122 Page: 121 of 173 The gravity data were analyzed by means of the following methodology: (1) Free-air gravity anomalies are computed from TIM3 (D/O 180) at the locations of the GDR gravity measurements on the topography surface on land and mean sea level at sea. Also, high frequency gravity signal from degree and order 181 to degree 2190 and order 2159 was computed by means of EGM2008. (2) The high frequency gravity signal is removed from the GDR gravity values in order to make the spectral content of the GDR gravity anomalies consistent with the spectral content of the TIM3 gravity anomalies. (3) Refined Bouguer anomalies are computed on the geoid from the reduced point GDR gravity values in step 2 using Eq.(3-21)) in (Heiskanen and Moritz, 1967). (4) Bouguer anomalies are cheched for consistency and merged on the geoid. Mean Bouguer anomalies are computed on a 15 x15 grid by averaging the point Bouguer anomalies in a 7.5 radius around the center of the grid cell. This size of the grid cell is the optimum one for the density of the GDR gravity measurements. (5) Mean free-air gravity anomalies are computed on the topography on a 15 x15 grid using the mean Bouguer anomalies on the geoid and mean elevations and topographic corrections computed in the same averaging procedure as in (4). Note that there is no difference between the mean gravity anomalies computed by averaging the reduced gravity anomalies in (2) and from the Bouguer anomalies in (4). The latter allows one to check the consistency of the land and sea gravity data along the coastlines. (6) Mean TIM3 free-air anomalies are computed on the same grid by applying the same avegaring procedure to the point TIM3 anomalies. (7) The mean GDR and TIM3 free-air anomalies are compared on topography. In the following we show an example for western Canada (Figure 4-14). Significant differences between the GDR and TIM3+EGM208 gravity anomalies exist in the Canadian Cordillera. These differences include not only the very high frequency gravity signal omitted in TIM3+EGM2008 but also the large differences between the CGVD28 heights of the GDR points and the DTM model used to compute the TIM3+EGM2008 anomalies. The DTM model is an improvement of its predecessor DTM2002 (based on digital elevation data) with SRTM data and ICESat ice elevations (not in the Canadian Arctic) and can be considered partly independent from CGVD28 on which the Canadian digital elevations are based. All other mountainous regions (these include the Laurentian Mountains and the Baffin Island mountains) show differences between the topographic information and, as a result, significant gravity anomaly differences. The GDR anomalies are positively biased in Foxe Basin and Foxe Channel due to flawed water depth and ice thickness information in the GDR databe. The bias found in the deep Lake Superior, is likely due to the existing very few gravity taverses in the lake and missing gravity information from the US part of the lake.

123 Page: 122 of 173 Figure 4-14: (a) Land GDR free-air anomalies and (b) gravity anomaly residuals in western Canada Figure 4-15: Difference between the mean free-air GDR and TIM3 gravity anomalies on a 15 x15 grid Example: Europe The national physical heights, ellipsoidal heights and a combined gravity field model have been used to compute offsets of the national height reference frames in Europe with respect to the common gravity field model obtained by a combination of a high resolution pre GOCE regional geoid model and a global gravity model including GOCE. Possible systematic effects in the levelling data were taken into account due to the estimation of a plane considering north-south and east-west tilts in addition to the offset. The procedure has been described in detail in [P-11]. The standard deviation of the residuals between the estimated plane and the observed height anomalies at GNSS/Levelling point locations has been computed for each country individually (Figure 4-16). Two different geoid models have been used for the analyses: a combined model from GOCO03S and EGG08 (Figure 4-16 in red)

124 Page: 123 of 173 and the pure EGG08 model (Figure 4-16 in blue). The overall error estimate incorporates errors of the levelling network, the GNSS solution and the geoid model. It allows an independent and more reliable quality assessment of all three observation types. For the most countries the combined model using GOCE reduces the estimated standard deviations in comparison to the pure EGG08 model. Assuming that the error budget of the geoid and the GNSS solution is constant over Europe in level of less than 1cm, differences in the standard deviations can be related to the different quality of the national levelling networks. The statistics shows that many national levelling networks have a pretty good quality (standard deviation below 3cm), which is an indication for the good quality of the levelling network as well as the quality of the geoid model including GOCE. Nevertheless, a number of countries show high standard deviations of more than 7cm, such as Italy, Spain, Portugal and Romania. Figure 4-16: Quality assessment of national leveling networks. In red: GOCO03S and EGG08 combined solution, in blue: pure EGG Characteristics of GNSS/Levelling Geoid Residuals Analysis of GNSS/Levelling Geoid Residuals in Europe For the detection of systematic effects misclosures h-h-n of GNSS/Levelling and geoid values have been computed in Europe. The obtained misclosures have been plotted against the longitude and latitude values of the corresponding GNSS/Levelling point locations (Figure 4-17) and their ellipsoidal height (Figure 4-18). The mean offset of approx. 30cm is caused due to the different reference levels of EVRF2007 and the geoid model. In the analyses no systematic effects are detectable. The points marked in blue are the results for Great Britain. The British levelling networks shows a significant north-south tilt, which can clearly be seen in the upper part of Figure The higher point scatter in the southern part is again an indication for the worse quality of the levelling networks in some southern European countries, which is also obvious in Figure An height dependent error in the European national levelling networks cannot be detected.

125 Page: 124 of 173 Figure 4-17: Misclosures h-h-n in Europe: EVRF2007 vs. a combined GOCO03S/EGG08 geoid model. Values for Great Britain in blue. Figure 4-18: Misclosures h-h-n in Europe: EVRF2007 vs. combined GOCO03S/EGG08 geoid model. Values for Great Britain in blue. Analysis of GNSS/Levelling Geoid Residuals in Canada The correlation of the GNSS/levelling residuals with elevation is clearly depicted after the tilt is removed from the residuals (Figure 4-11 c and d and Figure 4-19). This elevation dependence comes from the normal-orthometric heights being lower in the mountainous and high elevation regions than

126 Page: 125 of 173 the Helmert orthometric heights. The largest residuals are found in Pacific Cordillera on the west and Laurentian Mountains on the east. Correlation with topography is increased when the TIM3 omission error is corrected for and the scatter of the residuals is decreased. Approximate differences between the Helmert orthometric and normal-orthometric heights computed for the points of gravity measurements in the Pacific Cordillera given with their CGVD28 heights are shown in Figure Approximately 300 points have height differences between 50 and 80 cm. Differences between the NAVD88 Helmert heights and NGVD29 normal-orthometric heights in the mountains regions of USA reach 50 cm (Zilkovski et al., 1992). The correlation of the GNSS/geoid residuals with elevation has an effect on the residuals s mean and, respectively, on the CGVD28 offset. The mean of the TIM3 residuals (I) in Figure 4-21 gradually increases when the set of benchmarks is enlarged to include all points with elevation up to 1500 m and then levels off. Somewhat different behaviour is observed for the TIM3+EGM2008 residuals (II) for the small elevations. The mean decreases for all heights smaller than 300 m and then increases smoothly similar to the resuduals I case. Figure 4-19: Correlation of tilt and bias corrected GNSS/levelling geoid residuals II with elevation Figure 4-20: Differences between Helmert orthometric and normal-orthometric heights in the Pacific Cordillera; GNSS/levelling residuals corrected for the tilt of the datum are given with circles and (b) Correlation of differences (a) with elevation

127 Page: 126 of 173 Figure 4-21: Mean of the GNSS/levelling residuals as a function of cumulative number of points for increased elevation GNSS/levelling geoid residuals as a function of the baseline distance Another way to illustrate the effect of the GOCE omission error on the GNSS/levelling geoids residuals is to plot the mean absolute residual difference as a function of the baseline distance (1 deg increment). Even for small baseline distances (Figure 4-22, left plot), the mean difference for TIM3 GNSS/levelling residuals (I) is as large as 50 cm, and it correlates well with the geoid omission error. On the other hand, the mean difference computed with TIM3+EGM2008 (II) shows a linear increase with the baseline distance from as low as 5 cm for distances smaller than 100 km to 22 cm for distances between 800 and 900 km. The relative GNSS/levelling geoids residuals provide means to assess the realtive error of the TIM3 and TIM3+EGM2008 geoids in Canada in Figure 4-22 (right plot) with a baseline increment of 20 km.the total number of baselines is The error of TIM3 at a distance of 200 km is 3 ppm (60 cm) while the error of TIM3+EGM2008 is 0.8 ppm (16 cm). It should be noted that despite the long wavelength errors in CGDV28, this datum remains precise for small baseline distances. For Pacific Cordillera ( baselines), the error of TIM3 at 200 km increases to 74 cm and the error of TIM3+EGM2008 remains the same, i.e. 16 cm. For Eastern Canada ( baselines), the relative error of TIM3 is 44 cm and the TIM3+EGM2008 error is 14 cm.

128 Page: 127 of 173 Figure 4-22: (left) Mean absolute difference of GNSS/levelling residuals and (right) relative accuracy of the geoid as a function of baseline distance Ocean Levelling for Ocean Model Quality Assessment In our paper published in the Journal of Geodetic Science (refer to [P-14]), we have described the application of ocean levelling to worldwide height system unification. The study involved a comparison of geodetic and ocean approaches to determination of the mean dynamic topography (MDT) at the coast, from which confidence in the accuracy of state-of-the-art ocean and geoid models can be obtained. We conclude that models are consistent at the sub-decimetre level for the regions that we have studied (North Atlantic coastlines and islands, North American Pacific coast and Mediterranean). That level of consistency provides an estimate of the accuracy of using the ocean models to provide an MDT correction to the national datums of countries with coastlines, and thereby of achieving unification. It also provides a validation of geoid model accuracy for application to height system unification in general. We have shown how our methods can be applied worldwide, as long as the necessary data sets are available, and explain why such an extension of the present study is necessary if worldwide height system unification is to be realised. Table 4-17 provides an overview of our results, presenting statistics for offsets between the MDTs of the geodetic approach using tide gauge data (ellipsoidal height of MSL minus geoid height) and the MDT of the ocean approach at the same positions using the Liverpool-MIT ocean model. It has one line for each coastline using either the GOCO03S or Extended geoid models. Each line lists the number of stations in each coastline, the mean offset and the stdev of the differences between MDT values from the two approaches at each station. Table 4-18 provides further information which was not included in the paper. It shows the measured MDT (i.e. from tide gauge/gnss and geoid model), ocean model MDT and difference for the Atlantic and Pacific coasts of North America. This topic has a long history with classic papers published by Bowie (1929), Sturges (1974) and Fischer (1977) and recent work by Higginson (2012). Much of the fall in sea level on the Atlantic coast occurs going northward between Key West and Cape Canaveral (28 N) with a further fall until approximately 38 N whereafter sea level remains fairly constant. This contrasts with the Pacific coast where there is no significant slope. The consistency, at least in sign, between measurements and models on the Atlantic coast indicates that this classic problem in oceanography is gradually being understood. (Note that the ocean model MDT used here is that of the Liverpool implementation of the MIT ocean general circulation model and that other ocean model data sets are available; several of these other ocean models are discussed in [P-14].)

129 Page: 128 of 173 Table 4-17: Statistics of the differences between the MDT of the geodetic approach using tide gauge data (ellipsoidal height of MSL minus geoid height) and the MDT of the ocean approach at the same positions (using the Liverpool-MIT ocean model).units are millimetres Region Number of Stations Geoid Model Mean Offset Stdev Atlantic US/Canada 38 GOCO03S Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria Atlantic US/Canada 38 GOCO03S Extended Model Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria Table 4-18: Sea level gradients along the Atlantic and Pacific coastlines of North America using the geodetic approach (ellipsoidal height of MSL minus geoid height) and the ocean approach at the same positions (using the Liverpool-MIT ocean model).units are millimetres per degree of latitude Region Lat. Range Geoid or Ocean Model Sea Level Gradient Atlantic US/Canada All GOCO03S -19.1±8.9 Ditto GOCO03S Extended Model -8.8±1.4 Ditto L pool-mit Ocean Model -17.1±1.8 Atlantic US/Canada S of 38 N GOCO03S -87.2±21.9 Ditto GOCO03S Extended Model -12.5±4.8 Ditto L pool-mit Ocean Model -32.0±2.9 Atlantic US/Canada N of 38 N GOCO03S 12.4±18.0 Ditto GOCO03S Extended Model -6.0±3.0 Ditto L pool-mit Ocean Model 1.9±1.5 Pacific US/Canada All GOCO03S 8.9±19.6 Ditto GOCO03S Extended Model -1.2±4.1 Ditto L pool-mit Ocean Model 0.8±0.7 Update Using GOCE Release-4 Models We have most recently made use of Release-4 models provided by TUM, as an update to the tables above. The models include version of the time-wise TIM models to degree 160, 180, 200 and 220, each extended to degree 2190 using EGM2008. In addition there is the combined model EGU13 to degree 720 only provided by TUM and its extension to degree 2190 using EGM2008. The TIM models are GOCE-only models whereas the EGU13 includes other satellite and terrestrial/altimetric data. Table 4-19 shows the results.

130 Page: 129 of 173 Table 4-19: Statistics of the differences between the MDT of the geodetic approach using tide gauge data (ellipsoidal height of MSL minus geoid height) and the MDT of the ocean approach at the same positions (using the Liverpool-MIT ocean model).units are millimetres Region Number of Stations TIM4EGU_160 TIM4EGU_180 Mean Offset Stdev Mean Offset Stdev Atlantic US/Canada Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria Region Number of Stations TIM4EGU_200 TIM4EGU_220 Mean Offset Stdev Mean Offset Stdev Atlantic US/Canada Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria Region Number of Stations EGU13 (to deg 720) EGU13 (Extended) Mean Offset Stdev Mean Offset Stdev Atlantic US/Canada Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria We conclude that: The EGU13 model to degree 720 is definitely better than GOCO03S to 180, and is almost as good as any of the extended models. However, the EGU13 Extended is not particularly better than the other extended models, including the pre-goce EGM2008 model discussed below. All the extended models are similar but for the 4 TIM4 model results there are perhaps better stdev values for than for 220, indicating where the GOCE error contributions increase as might be anticipated. One wonders if progressive improvement in these stdev values is bottoming-out, given that they all use the same ocean model (L'pool/MIT) for comparison, with its own inaccuracies.

131 Page: 130 of Impact of GOCE Based Combined Models vs. pre-goce Models Tide Gauges We have repeated the analysis leading to the second half of Table 4-17 (which uses the Extended GOCO03S model where Extended means with GOCO03S itself to degree 180 and with EGM2008 at higher degree) but using the complete EGM2008 model instead. These results are shown in Table The results for the main North Atlantic coastlines where there are many tide gauges were almost identical to those for the Extended model, although there were significant differences for the Atlantic islands and particular Mediterranean stations. Altogether, this suggests that, while various studies have demonstrated undoubted improvement in ocean geoid modelling using GOCO03S, the accuracy of comparisons at point tide gauge locations depends greatly on the high frequency component which comes from in situ gravity information, which was already included to a great extent in EGM2008. (Some of the similarity of course comes from the fact that we used the same set of stations.) Table 4-20: As Table 4-17, but using EGM2008 instead of the Extended GOCO03S model. Units are millimetres Region Number of Stations Geoid Model Mean Offset Stdev Atlantic US/Canada 38 EGM Pacific US/Canada Gulf Coast USA European Atlantic and Atlantic Islands European Atlantic only Ponta Delgada, Azores Bermuda Marseille Alexandria Altimeter Data Studies Global and coastal dynamic topographies have been computed using the DTU10 mean sea surface and both EGM08 and Extended GOCO03S geoid models, on a ¼ degree grid (averages over ¼ degree grid squares). This simple averaging leaves considerable point-to-point noise in the dynamic topographies in both cases. In the case of the GOCO03S geoid it also introduces additional noise at wavelengths corresponding to degrees close to 180. Gaussian smoothing (radius of half amplitude = 100 km) suppresses much of this extra noise and produces a dynamic topography which, on visual inspection, appears slightly better than the EGM08 equivalent. Results are similar for the coastal dynamic topographies. After smoothing, the extended GOCO03S results are slightly smoother than the identically-smoothed EGM08 results, suggesting added information from GOCE. There are exceptions to this general picture. In regions with strong gradients of dynamic topography, smoothing is clearly removing signal as well as noise. There are a number of strong jets, particularly in the Southern Ocean, which appear sharper and more realistic when using the extended GOCO03S geoid than when using EGM08. An example is shown in Figure 4-23, which illustrates both noise and improved strong jets. The difficulty is that the signal to noise ratio is highly spatially variable, so in regions of strong signal it is better to smooth less than in regions of weak signal. This suggests that GOCE data are adding useful information, but that its effective exploitation requires an adaptive filter of the kind recently suggested by Bingham (2010).

132 Page: 131 of 173 Figure 4-23: Dynamic topographies compared with sea surface temperature contours in the Argentine Basin region Impact of GOCE for Regional Geoid Modelling Impact of GOCE Models on the regional Geoid Model of Canada As the evaluation of the GOCE GGMs in Canada showed, GOCE can have an impact and possibly lead to an improvement in the regional gravimetric geoids model in the spherical harmonic degree band (see section ). Figure 4-24 below illustrates the differences between the local and TIM3 free-air gravity anomalies for one of the regions in Canada, where improvement is expected. For the purpose of assessing the impact of GOCE on the regional gravimetric geoid, GOCE GGM and local gravity data are combined in a remove-restore procedure using Helmert s second condensation method. Figure 4-24 shows the computational and combination procedure. For the corresponding equations one can refer to [P-7]. Based on the performance of the various gravimetric geoid models using the GOCE go_cons_gcf_2_tim_r1 solution of maximum d/o 224 and different modification bands ( degree 90, 120, 150 and 180), it can be concluded that 1. Early GOCE releases do not lead to a noticeable improvement of the regional geoid over EGM2008 or the official geoid CGG2010. The agreement with the GNSS/levelling geoid is 11.5 to 12 cm for the whole Canada, 4.1 to 5 cm for the Great Lakes and 6.6 to 7.2 cm for the Pacific Cordillera (Ince, 2011). 2. In well surveyed areas (abundance of gravity data and good GNSS/levelling network coverage), e.g., the Great Lakes region, GOCE does not improve the EGM2008 geoid. The latter describes sufficiently well the local geoid. 3. GOCE does not visible improve the local geoid in the regions with rugged terrain such as the Pacific Cordillera. The improvement of the geoid in this region will come from the improvement of the short wavelength gravity field information cm differences between the GOCE-based regional gravimetric geoid models and EGM2008 are found in few areas: Yukon, Rockies, the Canadian Arctic, Greenland and the Atlantic coast of Canada. The geoid in Yukon, where the terrestrial gravity data coverage is considered to be insufficient, is likely improved by GOCE. The differences in the Atlantic may be due to improved ocean currents information from GOCE, but this needs to be investigated. 5. Release 4 GOCE GGMs can be used up to D/O 200 without worsening the gravimetric geoid model in Canada. These recent GGMs mostly confirm the Canadian terrestrial gravity data for the band from D/O 150 to 200. No significant improvement is observed in terms of comparison with the GNSS/levelling data that are located in regions with excellent gravity

133 Page: 132 of 173 data. Similar to the early releases of the GOCE GGMs, the local geoid model is updated mostly in regions with poor gravity data. Future regional gravimetric geoid models in Canada and North America should be developed based on the integration of GOCE and GRACE GGMs for improving the long wavelengths of the gravity field. The medium wavelengths from GOCE can be complemented by new altimetry and ship-borne gravity data in the oceans. Regional geoid modelling (the remove-restore procedure) Satellite GGM Gravity Data DEM Satellite GGM in Helmert space Spherical refined Bouguer anomalies on the Earth surface Direct topography effect (Bouguer plate & TC) Gravity anomalies g GGM ( n max ) Helmert gravity anomalies on the H geoid g Condensed topography effect Residual anomalies on the geoid g res g GGM g H GGM geoid heights N GGM ( n max ) Residual geoid heights using degree-banded Stokes kernel N res R g 4 S ( )cosd res m Indirect topography effect on the geoid ind N N Geoid height 0 N N GGM N res N ind Figure 4-24: Computational scheme of the regional geoid modeling procedure Impact of GOCE Models on the regional Geoid Model in Germany In [P-11] we describe a simple filter method for the combination of global and regional gravity field models. With this method GOCE models of the latest releases (GOCO03S, GOCE TIMEWISE R3 and GOCE TIMEWISE R4) satellite gravity field models have been combined with EGG08. Figure 4-25 shows the validation of gravity field models against a German dataset of GNSS/Levelling control points. A large filter width means a reduced influence of the GGM and a high influence of the regional model on the combined model. An optimum combination has been found for a Gauss filter, which was used in this application, with a width of approx. 400km. In comparison to the pure EGG08 and EGM2008 models, both models are from the pre-goce era, it can be seen that GOCE can clearly improve the regional geoid modelling even in an area with good terrestrial gravity data. The overall error budget could be decreased by about 25% using the GOCE gravity field models In addition a resolution improvement of the TIM R4 model in comparison to the TIM R3 and the GOCO model can

134 Page: 133 of 173 be detected which results in a shift of the curve to the left. It can be expected that the direct combination of GOCE models and regional terrestrial gravity data will further improve these results. Figure 4-25: Validation of pure, spherical harmonic combined and filter-combined geoid models versus GNSS/Leveling points (D954 dataset) in Germany Analytical Investigation on Expected Improvements of Regional Geoid Models in Europe from GOCE Global Potential Models The last section investigates the combination of GOCE models with regional data. The study makes use of the state-of-the-art regional quasigeoid model EGG2008 (Denker, 2013) which is available in the form of a gridded surface. The combination is performed by adding a low-pass filtered geoid surface from GOCE to the high-pass filtered EGG2008 surface. In the above section, a Gaussian kernel function is employed. In this section, we try to formulate the approach in a general way and derive analytically error estimates for a combined geoid. Filtering with the Gaussian kernel will be compared to results based on Wenzel s approach (Wenzel, 1981) for an optimal stochastic combination. Thereby realistic error covariance functions shall be used. Error covariance function for EGG2008: Denker estimates the error standard deviation of EGG2008 to vary between 3.1 cm (pessimistic) and 2.5 cm (optimistic) for those areas with good data coverage. For Germany, [P-11] derive the standard deviation of residuals between EGG2008 and GNSS-Levelling to be 2.8 cm. This value contains error contributions from the quasigeoid model, but also from levelling and GNSS heights. An error estimate of EGG2008 can be expected to be less than 2.8 cm in Germany. Because of the good data coverage in Germany as compared to other European countries, we will employ Denker s optimistic error estimate and employ the error covariance function published by Denker (2013) and shown in Figure 4-26.

135 Page: 134 of 173 Figure 4-26: Error covariance function of the regional quasigeoid model EGG2008 (after Denker, 2013) Error covariance function for GOCE: As shown in [P-3], the error covariance function of GOCE global potential models can be derived with sufficient accuracy (error less than 5%) by the m-block approach. Wenzel s method for optimal combination of satellite and terrestrial data, however, considers error degree variances only. As shown in [P-3] the m-block structure is latitude dependent. Using error degree variances in the weighting schema will therefore lead to overrating/underrating the weight of the GOCE model in dependence of geographic latitude. In the present study, the error degree variances were rescaled using the ratio between the error variance derived from the mbm-approach and the degree-variance approach (for both, see section ). Figure 4-27 shows several error covariance functions derived from GOCO03s using either the mbm-approach (gray and red line) or the simple degree-variance approach (green lines). The different gray lines represent the error characteristics in different azimuth directions. The red line is the isotropic average of those. The solid green line was derived from error degree variances only and clearly shows, that the error variance would be overestimated in this case (origin is located in central Europe at latitude 50 ). Using the degree variances directly in the weighting schema would lead to loading the regional data too heavily, i.e. the relatively good quality of GOCO03s would not be properly exploited. Using the numeric values of the variances of the red and the green curves one can rescale the error degree variances such that the error covariance function fit the regional error characteristics. This rescaled or locally adapted covariance function is represented by the dashed green line in Figure 4-27 which nicely resembles the average error model derived from the mbm-approach. When approximating the actual error characteristics by the rescaled degree variance model it is important to check, that the deviations of the covariances in different azimuths do not strongly deviate from their isotropic average. In the present case, the maximum deviation is about 10% of the variance, while the root mean square of all deviations over all spherical distances and azimuths is at the level of 2%. Based on these numbers, it is assumed, that the weighting schema can to a good approximation be derived from the locally rescaled degree variances. Scaling factors have been derived for the two GOCE models which were employed in [P-11] and for three different latitudes, namely a point in central Europe (φ=50 ) and for locations on the southern and northern border of EGG2008 at φ=35 and φ=65. The scaling factors are given in Table 4-21.

136 Page: 135 of 173 Figure 4-27: Error covariance function derived from GOCO03s using the mbm-approach for an origin in central Europe (λ=10, φ=50 ). The gray lines correspond to covariance functions in different azimuth directions while the red line represents an isotropic average over all azimuths. The green (solid and dashed) lines are derived from the degree-variance approach Table 4-21: Scaling coefficients for error degree variances of two GOCE models to resemble local error characteristics in Europe GOCO03s TIM-R3 φ= φ= φ= Combination of a GOCE global potential model and regional data: Section describes the general theory for combination of a global potential model and terrestrial data. Thereby terrestrial data is assumed to be given in the form of terrestrial gravity anomalies. In this section, the regional quasigeoid model EGG2008 represents the terrestrial data. EGG2008 is already a combination of a global potential model (EGM2008) and terrestrial gravity data in the surroundings of Europe. As described in [P-11] the combination of EGG2008 and GOCE can be performed by lowpass filtering a GOCE based geoid surface and adding the EGG2008 surface after filtering with the complementary high-pass filter. This procedure can be written as: Nmax GOCE EGG 2008 P P wn n wn n n2 n2 (, ) (1 ) (4-8) Thereby the degree dependent weights are given by w n, ζ GOCE is the GOCE based quasigeoid surface and ζ EGG2008 is the regional quasigeoid EGG2008. The upper summation limit of the second term expresses the fact, that the summation is performed to high degree numbers. In order to stay general we use infinity for the upper limit of the regional model, although we are aware of the fact that it is actually limited to a finite number given by the resolution of the geoid surface. Since weighting makes only sense in overlapping spectral bands it is also clear that for degrees above the maximum resolution of the GOCE model (250 for both GOCO03s and TIM-R3) it holds w n = 0, i.e, the full information is deduced from the regional model. Gaussian filtering: In [P-11] there is used a Gaussian filter to define the weights w n. They have used a 6-sigma value for defining the filter length and found best results for the combination of EGG2008 with TIM-R3 when

137 Page: 136 of 173 employing a filter with filter length of 380 km respectively 400 km for the combination with GOCO03s. Wenzel-type combination: This type of combination employs a Wiener-type filtering where the spectral weighting factors w n are derived from the error degree variances of the global and the terrestrial model according to w n ( EGG2008) ( 2008) ( GOCE) 2 n 2 2 n EGG n If the error degree variance of EGG2008 for a certain degree is much higher than the error degree variance of a GOCE model, the weight is roughly w n = 1, and the combination will be fully based on the GOCE model. The error degree variances for the GOCE models has been computed from the standard deviations of the coefficients and the values are scaled with the values provided in Table 4-21 for latitude φ=50. The error degree variances for EGG2008 have been derived by analysing the spatial error covariance function given shown in Figure Figure 4-28 shows the error degree variances for EGG2008 and the two GOCE models TIM-R3 and GOCO03s. For comparison also the two GRACE-based global models EGM2008 and EIGEN-5c are shown. The latter was used in the weighting schema of global and regional data during the production of EGG2008. (4-9) Figure 4-28: Error degree variances of different global potential models and derived from analysis of the regional error structure of EGG2008 Figure 4-29 shows the spectral weights of the Gaussian and Wenzel-type filtering for both TIM-R3 and GOCO03s. It is obvious, that due to the high quality of the GRACE information in GOCO03s, the combination is fully based on GOCO03s in the long wavelength. Only after degree 100 EGG2008 gets some weight. For degrees above 220, GOCO03s has hardly any influence and the full information is based on EGG2008. According to the degree variances in Figure 4-28 the quality of TIM-R3 is below GOCO03s for degrees below 170. Above that, the two models behave similar. This is also reflected in Figure 4-29 where the weights for both models are almost identical for degrees above around 170. Interestingly, the quality of TIM-R3 seems to be below the quality of EGG2008 in the whole spectra. Only around degree 100 they are almost equal. This results in weights which are always below 0.5; only around degree 100 are both sources of information weighted almost equally. For all other degrees

138 Page: 137 of 173 the combination is biased towards EGG2008. In consequence one should not expect that EGG2008 can be improved by combination with TIM-R3. This is in contradiction to [P-11], who find a standard deviation of the residuals between GNNS-Levelling and EGG2008 of 2.8 cm, and 2.1 cm for the combination of EGG2008 and TIM-R3. Interestingly the Gaussian weights roughly resemble the form of the curve for TIM-R3 above around degree 100. Due to the high quality of all global models (even TIM-R3) and EGG2008 in the degrees below , it can be assumed, that quality differences in this spectral range hardly affect the quality of the final geoid model, because the main error contribution comes from the degree band between and (see also Denker, 2013). Therefore it can be assumed that the significantly different weights in the band up to degree 100 (see Figure 4-29) hardly have an effect on the quality of the final quasi geoid. Figure 4-29: Spectral weights w n for the combination of GOCE and EGG2008 as derived from Gaussian filtering (dashed lines) or Wenzel s combination schema (solid lines). Finally Figure 4-30 shows cumulative geoid errors in the spectral band up to degree 250, which is the resolution of both GOCE models. It must be noted, however, that the errors do only slightly increase for higher degrees. The values at degree 2190 (which corresponds to the resolution of the global model EGM2008) are given in Table The table contains also values for a concatenated model consisting of one of the GOCE models up to degree 190 and the global model EGM2008 above degree 190. In addition, the table contains the standard deviations (numbers given in parenthesis) of residuals between the different quasigeoid models and GNSS-Levelling in Germany as provided in [P-11]. From Table 4-22 and Figure 4-30 the following conclusions can be drawn: (1) Combination of GOCE models with regional data clearly improves the results (2) Analytically derived errors for the Wenzel-type combination are better than Gaussian filtering (3) The combination of TIM-R3 and EGG2008 is worse than the original EGG2008, i.e. EGG2008 cannot be impoved by combination with TIM-R3. (4) The cumulative errors of the combination of GOCO03s and EGG2008 are lower than for the original EGG2008 in the degree band up to degree 196 (Gaussian filtering) respectively 2190 (Wenzel combination). This means, that only Wenzel s combination can improve EGG2008 over the full spectra. This is in contradiction with the numbers provided in [P-11], where both GOCE models, GOCO03s and TIM-R3 can improve EGG2008. (5) In contradiction to the numbers given in [P-11], where combinations with TIM-R3 always outperform those with GOCO03s, the analytic error estimates imply the opposite. Because GOCO03s basically consists of TIM-R3 combined with other satellite data (CHAMP, GRACE and satellite laser ranging to 5 different SLR-satellites) it seems reasonable, that the low

139 Page: 138 of 173 degree information in GOCO03s is considerably better than in TIM-R3, while the quality in the high degrees should be identical. This is also reflected by the formal error characteristics. Rülke et al. (see attachment) mention, that residuals to GNSS-Levelling from other data sets than the German one indeed imply GOCO03s to be superior to TIM-R3. At the moment this discrepancy (better performance between either GOCO03s or TIM-R3) between the observed residuals presented in P-11] and the analytic error estimates presented here cannot be explained. (6) In general, all observed residuals (from [P-11]) are smaller than the analytically derived error estimates. This may be surprising, because the observed residuals not only contain the errors of the quasigeoid models, but also errors of leveled and GNSS-derived heights. The observed residuals are on the level of 2-3 cm while the analytical cumulative geoid errors reach the 2cm level already somewhere (depending on the GOCE model and the combination approach employed) between spherical harmonic degrees 120 and 250. Therefore one may assume that the analytic error description of the GOCE models is too pessimistic in this spectral range. Figure 4-30: Table 4-22: Cumulative geoid errors derived from error degree variances for the original models (solid lines) TIM-R3, GOCO03s and EGG2008 as well as for the Gauss- and Wenzel-combined models (dashed lines). Cumulative geoid errors derived from formal error degree variances for different models and different combination of models. In parenthesis are given the residuals from GNNS-Levelling and the corresponding quasi-geoid model provided by [P-11] for Germany cumulative geoid error in cm at sh-degree EGM (2.6) TIM-R3(190) + EGM (4.2) GOCO03s(190) + EGM (4.7) TIM-R GOCO03s EGG (2.8) TIM-R3 + EGG2008 (Gauss) (2.1) GOCO03s + EGG2008 (Gauss) (2.4) TIM-R3 + EGG2008 (Wenzel) GOCO03 + EGG2008 (Wenzel)

140 4.4.3 Impact of Omission Error at Tide Gauges Summary and Final Report Page: 139 of 173 From the analyses shown in the previous chapters and as presented in several scientific publications resulting from this project ([P-1] to [P-14]), it was identified that the omission error could play a significant role for the estimation of height system offsets applying a GOCE global gravity field model. This is investigated in detail applying in the test areas of this project best local gravity information in order to determine local/regional geoids in the vicinity of tide gauges. This geoid information is compared to the results obtained from the global high resolution EGM2008 model, which currently is used for estimating the omitted signal in the GOCE models. Further-on the impact of using this local geoid information as compared to the GOCE/EGM2008 geoid on ocean levelling results is investigated for the tide gauges used in this study. The results of these investigations are reported in the subsequent chapters Local Geoid in North America at Tide Gauges and Impact of Height Offset A network of selected PSMSL tide gauges in Canada and USA is used in the computation of the CGVD28 and NAVD88 datum offsets. The ellipsoidal height of mean sea level (MSL) at each tide gauge is determined in ITRF2005 or 2008, and the height of MSL above the local datum is also known. The geoid height is computed by means of release 4 of the GOCE time-wise model, TIM4, (Pail et al. 2011) using maximum D/O 180, EGM2008 and GOCE TIM4 extended with EGM2008 above D/O 181. In addition, the geoid heights are computed from the recent gravimetric geoid models in the region: the national gravimetric geoids of Canada, CGG2010 (Huang and Véronneau, 2013) and the USA, USGG2012 ( both of which cover tide gauges in the neighbour country as well as the model UoC/GSD, which does not cover the USA. All datum offsets computations are in a conventional tide free system. Figure 4-31 shows the GNSS-Levelling residuals h-h-n computed with USGG2012, TIM4 (D/O 180) and TIM4 + EGM2008 for the Atlantic, Pacific and Gulf of Mexico US tide gauges and the Atlantic and Pacific Canadian tide gauges. Clearly visible in the GNSS-Levelling residuals is the slope of NAVD88 identified by means of TIM4 + EGM2008 and USGG2012, as well as the large omission error of TIM4. The slope of CGVD28 is also identified in the GNSS-Levelling residuals in the right plot. Pacific (17) Atlantic (7 TGs) Atlantic (28 TGs) NAVD88 Gulf (13) Pacific (5) CGVD28 Figure 4-31: Geoid height differences h-h-n computed with the USGG2012, TIM 4 and TIM 4+EGM2008 geoid heights at the USA tide gauges (left) and Canadian tide gauges (right). Huang, J., and M. Véronneau, Canadian gravimetric geoid model 2010, J. Geodesy, 87: GOCE Geoid Commission Error at the North American Tide Gauges (a) The GOCE geoid error was computed by means of the m-block approximation of the GOCE error VCM, eq. (3-66) and [P3]. The geoid error in Figure 4-32 varies from 1.9 cm at the northernmost North American tide gauges to 2.7 cm at the southernmost tide gauges. (b) A more substantial north-south variation of 1.3 cm is produced by the diagonal approach given by eq.(3-67).

141 Page: 140 of 173 (c) In areas where the GOCE geoid omission error (e.g., in coastal areas with flat terrain) can be disregarded, the achievable accuracy of the datum unification in North America is 2-3 cm and is determined from the commission geoid error. Figure 4-32: GOCE time-wise geoid height error at the North American tide gauges GOCE Geoid Omission Error (a) The GOCE TIM4 omission error is computed at each tide gauge location by means of EGM2008 from degree and order 181 to the maximum degree 2190 (Table 4-23). The omission error at the tide gauge locations can be as large as few decimetres, but it tends to cancel out when the mean of the omission error over the tide gauges is computed. (b) Another way to assess the omission error is to compute averages around each tide gauge station. The residual geoid height computed from EGM2008 above D/O 180 (the same is used in a.) is averaged within a spherical cap that is centered at each tide gauge and has a radius that increases from 0.1 to 2.0 degree. The standard deviation of this average omission error at the tide gauges in the five coastal regions is plotted in Figure 4-33 for each radius of the spherical cap. In the proximity of the tide gauges, the standard deviation is similar to the standard deviation in Table 4-23 for each region and it gradually decreases with the spherical distance to below 5 cm for all five regions at around 1.0 degree (~70 km for 50N), the halfwavelength defined by D/O 180. The mean omission error for each region is computed by averaging the average omission error over the tide gauges in the region. The graphs of the mean omission error are given in Figure 4-34 for the five coastal regions. Evidently, the rugged terrain in the Pacific coast of Canada sampled by a small number of tide gauges results in a large mean of the average omission error even for large radii of the spherical cap. For all other regions, the mean of the average omission error quickly decreases with the distance and from 0.4 to 1.0 degree varies within 2 cm. This variation is within 1 cm for distances larger than 1.0 degree. From the graphs in Figure 4-34, it can be determined that the tide gauges used for HSU should be at least 0.4 degree (~ 28 km for 50N) apart in order to ensure a smaller than 2 cm effect of the omission error on the mean datum offset. (c) The omission error of the GOCE TIM4 model D/O180 extended with EGM2008 to the maximum degree 2190 is computed by means of CGG2010 and USGG2012 (Figure 4-35) at the tide gauge locations. The omission error of the extended GOCE TIM4 geoid at the North American tide gauges is at the level of 3-6 cm. The agreement of the TIM4 geoid with CGG2010 is better at the Canadian tide gauges, and TIM4 agrees better with USGG2012 at the US tide gauges. The differences stem mostly from the different quality of the local gravity

142 Page: 141 of 173 information in the neighbour country used by the national agencies and differences in the geoid computational procedures. (d) The mean of the GOCE TIM4 omission error at the North American tide gauges evaluated by means of EGM2008 is small overall, but reaches -7 cm in the Pacific coastal areas of the USA. This is also confirmed by the mean of the omission error, which was averaged in a spherical cap around the tide gauges. Therefore, an improvement of the GOCE geoid heights is necessary in order to aim at a centimetre level of accuracy of the North American s height systems unification. The mean of the omission error of the GOCE TIM4 extended to the full resolution of EGM2008 is at the level of -3 cm to 1 cm (when evaluated with USGG2012), which indicates that the extended GOCE geoid height may still be improved with the use of local gravity and topography information. Table 4-23: Statistics of the GOCE TIM4 geoid omission error values at the North American tide gauges computed with EGM2008 Region Mean (cm) Std (cm) Min (cm) Max (cm) Atlantic Canada (7 TGs) Atlantic USA (28 TGs) Pacific Canada (5 TGs) Pacific USA (17 TGs) Gulf Mexico (13 TGs) Figure 4-33: Standard deviation of the average omission error at the tide gauges for the five coastal regions, in cm

143 Page: 142 of 173 Figure 4-34: Mean of the average omission error at the tide gauges for the five coastal regions, in cm Table 4-24: Statistics of the extended GOCE TIM4 geoid omission error values at the North American tide gauges computed with USGG2012 and CGG2010 (brackets) Region Mean (cm) Std (cm) Min (cm) Max (cm) Canada Atlantic (7 TGs) -3(0) 6(5) -11(-5) 8(6) US Atlantic (28 TGs) -3(2) 3(5) -8(-8) 4(15) Canada Pacific (5 TGs) 1(2) 6(5) -6(-2) 8(10) US Pacific (17 TGs) -3(-6) 4(6) -10(-16) 4(9) Gulf of Mexico (13 TGs) -2(0) 3(5) -7(-5) 4(9) Pacific (17) Atlantic (7 TGs) Atlantic (28 TGs) Gulf (13) Pacific (5) Figure 4-35: Geoid height differences of TIM4+ EGM2008 with the regional gravimetric geoid models USGG2012 and CGG2010 at the US and Canadian tide gauges. Datum Offsets Computed with the GOCE TIM4 Geoid and the Gravimetric Geoid With a known ellipsoidal height of the local mean sea level (MSL) h MSL at a GNSS-surveyed tide gauge station and a height of the local MSL H j MSL above the national datum j, eq. (4-2)for the datum offset N j0 can be written as

144 Page: 143 of 173 L j GOCE res MSL MSL MSL 0 MSL MSL j0 l0 l0 l1 h H (N N N ) N N f (4-10) The ellipsoidal height of MSL at each tide gauge is computed with respect to the reference ellipsoid by reducing the ellipsoidal height of the tide gauge benchmark with the height difference between the benchmark and the chart datum obtained by precise levelling and adding the measured water level from the chart datum. The mean datum offset is computed from a set of tide gauge stations by means of the least-squares solution given by eq. (4-5), (4-6) and (4-7). The indirect bias term be omitted based on the investigations in Chapter and in Table 4-1. L l1 N f MSL l0 l0 Two examples of the mean CGVD28 and NAVD88 datum offsets with respect to the level surface defined by W o = m 2 s -2, which will define the new North American height datum, are given in Table 4-25 and Table 4-26: (a) The mean CGVD28 offset is estimated by averaging 12 Canadian tide gauges on the Atlantic and Pacific coasts (Table 4-25). The difference in the mean offset computed by means of TIM4 and the high resolution GOCE-based gravimetric geoid models CGG2010, UoC/GSD and USGG2012 is less than 4 cm. The offset of CGVD28 at the individual tide gauge stations is shown in Figure Because CGVD28 was constrained to the local MSL at five Atlantic and Pacific tide gauges, the individual offset values show the east-west tilt of the datum, which is a result of the differences in the MDT (P-14] along both coasts. (b) The mean NAVD88 offset values in Table 4-26 are computed for the Pacific USA from averaging 17 tide gauges. The large offset values of 8 decimetres show the effect of error accumulation in NAVD88 from the datum origin in Rimouski across the continent. The difference in the mean offset computed with TIM4 and with the high resolution gravimetric geoid models is less than 4 cm. (c) with the configuration of the network of PSMSL North American tide gauges, the mean height datum offsets can be computed with an error of 10 cm or less provided that the GOCE omission error is taken care of. More on this in [P-10]. Table 4-25: Mean CGVD28 vertical datum offset from the level W o = m 2 s -2 (12 TGs) Geoid Model N j (cm) Diff N j (cm) TIM4 (d/o 180) TIM4 (180) + EGM2008 (2190) CGG2010 (22) UoC/GSD (22) USGG2012 (11) can Table 4-26: Mean NAVD88 (USA Pacific) datum offset from the level W o = m 2 s -2 (17 TGs) Geoid Model N j (cm) Diff N j (cm) TIM4 (d/o 180) TIM4 (180) + EGM2008 (2190) EGM2008 (2190) CGG2010 (22) USGG2012 (11)

145 Page: 144 of 173 Figure 4-36: CGVD28 offset values at the Atlantic and Pacific Canadian tide gauges and the mean dynamic topography from local oceanographic models Procedure for Computing the Local Geoid Height For the purpose of unifying the North American height datums by means of tide gauges, the local geoid heights should be computed by a unified procedure for all coastal areas by combining the most recent release of the GOCE GGM with the best local gravity data and topography/bathymetry models available. Such a combination can be performed by tailoring the GOCE GGM to the local gravity information and adding a correction for the RTM topography. The key steps of the GGM tailoring procedure are described in the following: (a) The input data are a GOCE GGM of D/O 200 and a 2 x2 grid of Faye gravity anomalies. The evaluation of the Release 4 GOCE GGMs with the North American GNSS-Levelling data shows that TIM4 and DIR4 can be used in North America up to D/O 200. (b) An initial GGM is developed from the GOCE GGM and a spherical harmonic analysis of the local gravity anomalies for the spherical harmonic coefficients above D/O 200. (c) The GGM tailoring is performed on a sphere with a radius that is an average of the geocentric radii of all gravity points. The local gravity anomalies are downward-continued from the topography to the sphere, and the GOCE gravity anomalies are computed on the sphere. (d) The spherical harmonic coefficients of the combined GGM are updated iteratively by computing corrections to the coefficients in each iteration step from a spherical harmonic analysis of the differences of the gravity anomalies computed from the updated GGM (at the previous iteration step) with the local gravity anomalies. (e) The spherical harmonic coefficients up to D/O 200 are defined by the GOCE GGM and remain fixed during the iterations. The higher than D/O 200 spherical harmonic coefficients are updated in each iteration from the local gravity information. (f) Geoid heights are computed by spherical harmonic synthesis of the coefficients of the final tailored GGM. As an example, local geoid heights are computed from a tailored GOCE DIR4 GGM of D/O 900 for Atlantic Canada. Table 4-27 shows the agreement of the tailored model with the GNSS-Levelling geoid and includes EGM2008 and the extended GOCE DIR 4 for comparison. The tailored GOCE DIR 4 has the same level of performance as EGM2008 and GOCE+EGM2008, which is expected as the local gravity data already contribute to EGM2008.

146 Table 4-27: Statistics of GNSS-Levelling geoid height differences in Atlantic Canada Summary and Final Report Page: 145 of 173 Model Min (cm) Max (cm) Mean (cm) Std (cm) EGM2008 (D/O 900) GOCE DIR4+EGM2008 (D/O 900) Tailored (D/O 900) The CGVD28 mean datum offset is computed in Atlantic Canada using the geoid heights computed at the tide gauges from the tailored GOCE DIR4 GGM and several other models (Table 4-28). The comparison of the mean offsets computed with models 2 and 5 shows that the gravity signal in EGM2008 above D/O 900 does not contribute to the datum offset. From the comparison of the mean offsets computed with models 2 and 4, it can be concluded that the local gravity information in the band from D/O 201 to 900 contributes less than 2 cm to the datum offset. From the comparison of the mean offsets computed with models 4, 7 and 8, it is obvious that, in addition to the differences in the long-wavelengths (CGG2010 uses GOCO01 while USGG2012 uses GOCO03), the procedure of combing the local gravity with the global GGM also has an effect on the computed datum offset. Table 4-28: The CGVD28 offset computed with the tide gauges in Atlantic Canada Geoid model Offset (cm) Diff Offset (cm) 1. EGM2008 (D/O 900) DIR4(200) + EGM2008 (D/O 900) Tailored (D/O 900) Tailored (D/O 900) + EGM2008 (D/O 2190) DIR4 (200)+EGM2008 (D/O 2190) EGM2008 (2190) CGG USGG In order to deliver the local geoid heights with the resolution and accuracy needed for HSU in the coastal areas of North America, the GGM tailoring procedure will be improved, as follows: (a) A tailored GGM of D/O 2160 would be sufficient in order to incorporate the new gravity information not included in EGM2008. The contribution of these high frequencies is not essential in the more flat Atlantic coast, but it is important in the rugged topography of the Pacific coast. (b) The contribution of the local gravity data in the band D/O 150 to 200 (see section ) will be modelled and evaluated. (c) The RTM (residual terrain model) contribution to the geoid, which accounts for the gravity frequencies above D/O 2160, will be computed for the Pacific coast. For this purpose the procedure given by Hirt et al. (2010) will be applied. (d) The local geoid model will be evaluated by means of the GNSS-Levelling data in the coastal areas and with the regional high resolution MDT models at the tide gauges Local Geoid in Germany at Tide Gauges and Impact of Height Offsets A set of five tide gauges at the German North Sea coast has been selected for the analysis of the omission error impact: The islands of Borkum (2 locations at Fischerbaje and Südstrand), Helgoland

147 Page: 146 of 173 and Sylt (tide gauge at Hörnum) as well as the tide gauge at Cuxhaven on the main land. The tide gauges and its locations are listed in Table A new local geoid for northern Germany and the adjacent marine area of the North Sea has been computed. For this new solution an updated database of marine gravity observations over the North Sea could be used. The new observation data was taken from the database of the Bureau Gravimétrique International (BGI). The data has been cleaned and homogenized using a cross over analysis. Figure 4-37 illustrates the improved data coverage over the North Sea. The largest differences of new German gravimetric geoid model BKG2011g_2013 in comparison to its predecessor, the gravimetric geoid model BKG2011g occur in the central North Sea. At the German mainland and the coastal areas the differences are small and do not exceed 2 to 3cm (Figure 4-38, left panel). The European Quasigeoid Model 2008 (EGG08, Denker 2013) is a high resolution geoid model for Europe. It has been adopted to the NAP reference level by adding a constant offset of 0.300m (Denker 2013), which has to be taken into account for model comparisons. In order to model the high frequency part of the global satellite only gravity field models, EGG08 has been incorporated by Gaussian filtering (cf. [P-11]). This has been done for the GGMs TIM_R4 and the GOCO03s models. Figure 4-38 (right panel) illustrates the differences between the combined TIM_R4+EGG08 and the pure EGG08 model. For the German coast the differences are small on the level of about 1cm. The combined TIM_R4+EGG08 model is suggested to be selected as the reference model in the area. In order to quantify the influence of the omission error at the German tide gauge locations a number of gravity field models have been compared: The satellite only gravity field models TIM_R4 and GOCO03s, the EGM2008, the EGG08 and the combined models TIM_R4+EGG08 and TIM_R4+BKG2011g_2013. The results are compiled in Table The differences of the satellite only GGMs with respect to the high resolution models may reach about 0.30m. This is related to the omission error of the satellite only models. Figure 4-39 illustrates the spatial pattern of the omission error of the TIM_R4 and the GOCO03s models. Although there are similarities in the spatial patterns between the two models, the omission errors differ up to 10cm at the tide gauge of Cuxhaven2 and up to 15cm at Helgoland. The analyses of the geoid heights at the tide gauge locations from the high resolution geoid models show a good agreement. With one exception at Hörnum (6cm between the TIM_R4+EGG08 and the TIM_R4+BKG2011g_2013) the differences do not exceed 3cm. Table 4-31 summaries the geoid heights of the different models at the German tide gauge locations. For comparisons the observed height anomaly is given for three tide gauges. Since the levelled heights in the German Height reference system DHHN92 do not refer to GRS80, a datum offset needs to be considered. This datum offset has been estimated using the TIM_R4+EGG08 model and 957 GNSS/Levelling points all over Germany and results in m.

148 Page: 147 of 173 Figure 4-37: Update of the gravity observations database in the North Sea. Left: Old observation coverage used for the BKG solution of the German Combined Geoid Right: Updated observation coverage. Table 4-29: Tide gauges at the German North Sea coast. Tide gauge Longitude [deg] Latitude [deg] Borkum Fischerbaje Borkum Südstrand ( ) Cuxhaven Helgoland ( ) Hörnum ( ) Figure 4-38: Left: Inclusion of the new observation data over the North Sea: Changes of the German geoid BKG2011g. Right: Differences in geoid heights of the combined geoid TIM_R4+EGG08 and the pure EGG08. The bias of 0.300m was reduced for the EGG08.

149 Page: 148 of 173 Figure 4-39: Differences of gravity field models at the German coast. Left: TIM_R4 model minus TIM_R4+EGG08 combined model. Right: GOCO03S model minus TIM_R4+EGG08 combined model. Table 4-30: Differences of the geoid heights of different geoid models at German Tide gauge locations. Borkum Fischerbaje [m] Borkum Südstrand [m] Cuxhaven 2 [m] Helgoland Binnenhafen [m] Hörnum [m] TIM_R4 minus EGM EGG TIM_R4+EGG TIM_R4+BKG2011g_ GOCO03S minus TIM_R EGM EGG TIM_R4+EGG TIM_R4+BKG2011g_ TIM_R4+EGG08 minus EGM EGG TIM_R4+BKG2011g_

150 Page: 149 of 173 Table 4-31: Geoid heights of different geoid models at tide gauge locations. All values refer to GRS80, zero tide system. (*) The EGG08 values have been reduced by 0.300m. (**) The observed height anomalies have been converted to GRS80, zero tide [m]. Borkum Fischerbaje [m] Borkum Südstrand [m] Cuxhaven 2 [m] Helgoland Binnenhafen [m] Hörnum [m] TIM_R GOCO03S EGM EGG08 (*) TIM_R4+EGG TIM_R4+BKG2011g_ Observed h-h(**) Local Geoid in Europe at Tide Gauges and Impact of Height Offsets A subset of European tide gauge stations of the PSMSL was used for a comparison of different geoid models. The geographic distribution of the stations is presented in Figure At all stations GNSS observations are available; therefore ellipsoidal heights of the mean sea level (MSL) observed at the tide gauges can be deduced. In addition, values of the mean sea surface (MSS) model DTU10 as well as values of the MIT-Liverpool mean dynamic topography (MDT) are extrapolated to the tide gauge coordinates. Besides MDT values derived from the dynamical model, MDT can also be deduced as difference between ellipsoidal heights h MSS/MSL of MSS or MSL and a geoid model N according to MDT h N (4-11) MSS / MSL Because in both cases (MSS or MSL) the sea surface height is based on observations, this type of MDT will be denoted observed MDT. Corresponding values at the tide gauge stations of Figure 4-40 are shown in Figure For this comparison a concatenated geoid model from GOCO03s (up to degree 180) and EGM2008 (above degree 180) has been employed. The offset between observed (red and blue circles) and modelled (green) MDT values is a consequence of the inherent datum choice implied by the ocean dynamic model and has no consequence for our results. The comparisons are used to investigate the consistency between oceanographic data and geoid information. This is in analogy to GNSS-leveling on land, where MDT takes the role of the physical height (orthometric, normal, etc.). It must be noted however, that both MDT from dynamic models as well as derived from observed MSS represent smooth surfaces with limited spatial resolution; only MSL observed at a tidegauge is comparable to the point values derived from GNSS-leveling. In addition, both are not available at the coast line, but values must be extrapolated to the tide-gauge locations. Therefore the quality of MSS and MDT at the tide-gauge will be limited to the level of several centimetres to decimeter. In this section, the quality of different geoid models shall be investigated. Therefore equation (4-11) will be written as: N h MDT (4-12) MSS / MSL

151 Page: 150 of 173 Figure 4-40: Tide gauge stations of the PSMSL which are used for the investigations Figure 4-41: MDT from MIT-Liverpool dynamic model (gree) compared to MDT computed by subtracting a GOCE geoid model from (blue) tide-gauge observations or (red) satellite altimetry (MSS model DTU10). Figure 4-42 shows the residuals between geoid heights derived from MSL-MDT and computed from (left) GOCO03s or (right) GOCO03s (up to d/o 180) and EGM2008 (above d/o 180). The left figure shows the omission error of GOCE. The right figure shows consistency between geoid information from global potential models and oceanographic data. In some cases the omission error is in the range of only a few centimetres, but in general it cannot be neglected, i.e. the satellite information provided by GOCE must be complemented by terrestrial data providing the short spatial signal components. Table 4-32 gives the standard deviations of the residuals between geoid heights and MSL-MDT respectively MSS-MDT.

152 Page: 151 of 173 Table 4-32: Standard deviation of residuals between geoid heights from the models indicated in the first column and geoid heights deduced from the differences between observed MSL or MSS and modeled MDT MSL-MDT [cm] MSS-MDT [cm] GOCO03s (180) GOCO03s (180) + EGM Figure 4-42: Difference between MSL observed at tide gauges and modeled MDT compared to a geoid model from (left) GOCO03s up to d/o 180 and (right) GOCO03s up to d/o 180 and EGM2008 above d/o 180. In order to assess the effect of GOCE on the geoid height at tide gauges, different sets of combined geoid models are employed. Again, residuals between the geoid height from the geoid model and corresponding values deduced from the differences between MSL (or MSS) and modelled MDT are computed. The corresponding standard deviations are given in Table According to the results in section GOCO03s is expected to improve regional geoid models. Therefore GOCO03s was employed to provide long to mid-wavelength information from GOCE. The short scale information is taken either from EGM2008 or from EGG2008. In all cases the filter-based combination using either Gaussian or Wenzel-type weighting was used. In case of EGM2008 also the concatenated model with sharp spectral cut at degree 180 was used. In order to compare GOCE models with pre-goce models, an additional combination of EGM2008 for the long to medium wavelength and EGG2008 for the short wavelength have been employed. Thereby the weighting of EGM2008 was identical to the combinations with GOCO03s. In general, the residuals employing MSL are much smaller (around 6 cm) than those employing MSS (around 14 cm). This expresses the fact, that MSL is a point value observed right at the tide gauge (with expected accuracy at the centimetre level) while MSS is derived from satellite altimetry which provides spatial averages over the ocean surface from which values at the tide gauge coordinates must be extrapolated; in addition, satellite altimetry standard products are provided for the open ocean and quality is decreasing towards the coast. All numbers also contain errors of the modelled MDT as well as errors from the different geoid models. The error budget of the left column (MSL-MDT) is expected to be dominated by MDT errors. The right column (MSS-MDT) by MSS errors. Because in both cases the geoid model is not the dominating error source, the numerical values of the standard deviations in Table 4-33 do not vary a lot. The smallest value is derived from the Wenzel-combination of GOCO03s and EGG2008. This is also the best combination according to the formal error estimate given in Table 4-22, where a cumulative error of 2.2 cm is given. Assuming the MSL-error to be 1 cm,

153 Page: 152 of 173 one can deduce an estimate for MDT at the tide gauges around 5.3 cm. Keeping this number fixed in all comparisons (all rows in the left column), variation of the standard deviation from 5.8 cm to 6.6 cm implies geoid errors with standard deviations between 2.2 cm and 3.9 cm, which is an improvement of 44%. The best results are achieved for the combinations of GOCO03s and EGG2008. The combinations with EGM2008 are worse. This implies that the regional geoid model EGG2008 is superior to the global potential model EGM2008 at these tide gauge stations. The GOCO03s combinations are also better than those employing long to medium information from EGM2008 (GRACE and terrestrial data instead of GOCE). This shows that GOCE improves the geoid in the medium wavelength. Using the rough estimate given above, improvements of regional GOCE based geoids of up to 44% seem reasonable at tide gauge stations. Table 4-33: Standard deviations of residuals between geoid heights from the model indicated in the first column and geoid heights deduced from the differences between observed MSL or MSS and modeled MDT. MSL-MDT [cm] MSS-MDT [cm] GOCO03s (180) + EGM GOCO03s/EGM2008 (Wenzel) GOCO03s/EGM2008 (Gauss) GOCO03s/EGG2008 (Wenzel) GOCO03s/EGG2008 (Gauss) EGM2008/EGG2008 (Wenzel) EGM2008/EGG2008 (Gauss) Impact of Local Geoid at Tide Gauges on Ocean Levelling As a test of the importance of omission errors in the geoid models used for determination of MDT values from tide gauge data so far, we have made use of the high spatial resolution modelling discussed in Sections , and CGG2010 is a gravimetric geoid model with a similar spatial resolution as EGM2008 (degree 2190) but with state-of-the-art regional improvements for North America (Huang and Véronneau, 2013). We have used this model as a comparison to others mentioned above, which have lower spatial resolution and have earlier representation of short spatial scales in the region. CGG2010 covers Canada and the Atlantic, Pacific and Gulf coasts of the USA. Values of mean offset and stdev for MDT values obtained using this model are shown in Table It can be seen that there is a marginal improvement in stdev for the Pacific and Gulf coasts using this model compared to the earlier GOCO03S Extended (all comparisons being again made using the Liverpool/MIT ocean model). However, for the Atlantic coast a stdev value similar to that from GOCO03S Extended was obtained. Similarly, we have also made use of a recent regional geoid model provide by NOAA in the USA. This is called USGG2012 and also covers Canada (although in the file we had available for study, 4 Canadian stations were not represented, hence the slightly different number of stations in Table 4-34 than for CGG2010). It can be seen that the performance of the recent Canadian and US models is equally good, reflecting no doubt the complementary improvements in regional geoid modelling that have been achieved in North America. However, there are some interesting differences between models that are not apparent by a simple computation of stdev. Figure 4-43 provides a comparison of geoid values at tide gauge sites along the US/Canada Atlantic coast, which is a coastline for which the meridional slope of sea level is of great

154 Page: 153 of 173 historical interest. Sea level unquestionably falls going north from Key West until 27 N, but an interesting question is whether there is another fall around Cape Hatteras, where the Gulf Stream no longer flows parallel to the coast but heads north-west towards Europe. The figure shows a big difference (order 10 cm) between CGG2010 and GOCO03S Extended at the southern end, where the use of CGG2010 would result in a larger apparent fall in MDT going north from Key West than would the use of GOCO03S Extended. CGG2010 is similar to EGU_13 (to 720 and Extended to 2190 using EGM2008) in this section of coast, while USGG2012 lies within the spread of the other models. In general, each of the four models shown has a similar latitudinal distribution with respect to GOCO03S Extended. Between N, CGG2010 and the others tend to be systematically larger than GOCO03S Extended by several cm which would contribute to a slightly larger fall in MDT at Hatteras when using GOCO03S Extended than when using CGG2010 or the others. These oceanographic aspects will be considered further in a separate study. We now turn to discussion of geoid values at German tide gauges (Section ). In [P-14], the larger scatter of MDT values at Netherlands/German/Danish/Norwegian stations using GOCO03S was found to be much reduced using the Extended GOCO03S model, indicating the importance of the degree terms in the geoid modelling (compare Figures 6(a) and (b) of that paper). We have since employed the EGU_13 (to 720 and Extended to 2190 using EGM2008) and TIM4EGU_180 (Extended to degree 2190 using EGM08) models, both computed at TUM, and also EGM08 itself, with similar closer grouping of MDT values along this coast as for GOCO03S Extended. In Table 4-30 (central section), it can be seen that the geoid values at German gauges are within a few centimetres for EGM2008 as for the state-of-the-art BKG model and for other models shown. Similarly, in Table 4-30 (bottom section), it can be seen that values for EGM2008 are within 2-3 cm of the other models shown, and also (not shown, private communication from BKG) with a version of TIM4EGU_180 (Extended) computed at BKG independently of the TUM version. The similarity at the 2-3 cm level of each model in Table 4-30 is consistent with our experience of a similar grouping of MDT values using different geoid models reported above. This level of consistency between geoid models is almost certainly within the range of variability of the MDT values themselves, given also the interannual variation in MSL at tide gauges together with GNSS positioning and levelling uncertainties. Altogether these exercises demonstrate the importance of high spatial resolution geoid models, but suggest that, unless there is further improvement in models in this region at the highest scales, there will be little impact on MDT studies such as those reported here with presently-available models. The work of Section has described further effectively the spatial consistency between values of MDT determined from tide gauge data and geoid models, and MDT from the ocean modelling, for coastlines around the NW European continental shelf. The improvements between Figures 6(a) and (b) in [P-14], discussed above, and the general conclusions arrived at above based in the information in Section for German sites, are confirmed by Section and Table These conclusions are two-fold: the improvement on using GOCO03S by employing one of the more complete geoid models is clear; but a limit has been reached (due to the inherent uncertainties in both ocean and geoid modelling) on deciding between more complete models. Further general improvements in observational data sets and ocean modelling may result in it being worthwhile to return to this issue in the future.

155 Page: 154 of 173 Table 4-34: Statistics of the differences between the MDT of the geodetic approach using tide gauge data (ellipsoidal height of MSL minus geoid height) and the MDT of the ocean approach at the same positions (using the Liverpool-MIT ocean model). Units are millimetres Region Number of Stations Geoid Model Mean Offset Stdev Atlantic US/Canada 38 CGG Pacific US/Canada Gulf Coast USA Atlantic US/Canada 35 USGG Pacific US/Canada Gulf Coast USA Figure 4-43: Comparison of geoid values at tide gauge sites along the US/Canada Atlantic coast

156 Page: 155 of Scientific Roadmap From the results achieved during this study a list of recommendations for the realization of globally consistent and accurate height systems is formulated. These recommendations are addressing essentially IAG and its bodies, because only they are in a position to initiate any such action. The recommendations have two objectives: They serve as (1) input to the discussion of the IAG-GGOS working group on Unified Height System and (2) information about the findings of this study to IAG and the scientific community. Therefore as a kind of final study result, this chapter provides a roadmap towards scientific applications. It is specifically elaborated on the situation for (1) a densely surveyed continent like Europe, (2) a large well surveyed continent like North America (3) a poorly surveyed area like a development country with little geodetic infrastructure and (4) across the ocean. The roadmap consists of two parts: a summary and a more detailed description. The summary should also serve as a self-contained, short version of the roadmap. Its structure and main elements are summarized in Figure 5-1. Figure 5-1: Roadmap structure and main elements. The geopotential and geoid improvements resulting from GOCE are the basis of a reassessment of global height systems. Shown are the essential tasks 5.1 Roadmap Summary Height System Unification is divided into three separate and independent parts: Global Height Unification (1) The objectives of global height unification are twofold: Objective One is the realization of accurate geopotential numbers C and as a measure of quality, their standard deviation σ(c) at a

157 Page: 156 of 173 selected set of stations (datum points of national height systems, geodetic fundamental stations (IERS), primary tide gauges (PSMSL) and primary clocks (IERS)). Objective Two is the determination of height off-sets between all existing regional/national height systems and one global height reference. (2) The primary geodetic method of height determination will be GNSS-levelling, i.e. the derivation of potential differences between points with known position given in one global earth fixed, geocentric reference frame. It requires (1) precise positioning (by GNSS, SLR, DORIS) in such a frame and (2) a best possible GRACE/GOCE geopotential model complemented with short scale geopotential information from terrestrial data. (3) Ultimately, the envisaged accuracy of height unification is about 10 cm 2 /s 2 (or 1cm). At the moment, in well surveyed regions, an accuracy of about 40 to 60 cm 2 /s 2 (or 4 to 6cm) is attainable. (4) In order to achieve this, the positioning part of GNSS-levelling should be better than 1ppb. (5) The geopotential numbers are derived from the best state-of-the-art combined Earth Gravity Model, based on GRACE and GOCE and terrestrial (and altimetric) gravity anomalies. Regional refinements may be required. (6) Geometry and gravity part must be consistent, i.e. refer to the same standards (ITRF, zero tide system, geodetic reference system). (7) Objective One can be realized by straight forward computation of geopotential numbers C, i.e. geopotential differences relative to an adopted height reference. No adjustment is required for this. (8) ObjectiveTwo, the unification of existing height systems is achieved by employing a leastsquares adjustment based on the GBVP-approach. In order to attain a non-singular solution, this requires for each included datum zone at least one geo-referenced station per zone, i.e. its ellipsoidal height h, and, in addition, the corresponding physical height H (geopotential number, normal height, orthometric height, etc.). (9) The height system definition and its realization may be denoted International Geodetic Height System/Frame, identified with the year of its definition/realization, e.g. IHSyy or IHFyy. Height information is provided to the user in parallel as geopotential number and normal height (or Helmert orthometric height). (10) Changes in geopotential numbers of consecutive realizations reflect (1) temporal changes of station heights, (2) improvements or changes of the applied geopotential (or geoid) model and (3) improvements of the adopted standards and methodology. (11) This recommendation will allow bringing all included stations into one and the same height datum. In sparsely surveyed regions of our planet the uncertainty of height off-sets may be at the level of 20 to 40cm (with extreme values up to 1m). In coastal regions, applying ocean levelling, these numbers may be improved. Ocean levelling is the combination of a best ocean topography model with either an altimetric mean sea surface or, at tide gauges, mean sea level as derived from a combination of tide gauge recording and GNSS positioning. (12) In the context of the GBVP approach, geopotential numbers C as well as the geopotential W itself are estimable quantities. (13) W 0 may be associated with (1) the geopotential value at mean sea level of a selected single tide gauge, (2) the average value of an ensemble of tide gauges or (3) the mean of the entire altimetric mean sea surface. Further study about the best operational selection is needed. (14) The classical geoid definition and realization is operational at the level of a decimetre. A geoid convention at the centimetre to decimetre level may be complex but is still feasible considering the distinction between geoid and quasigeoid, the influence of the atmosphere, topographic effects, and temporal variations (we refer to the well-known classical work by Moritz and by Mather). At the level of sub-centimetre accuracy the classical geoid definition is a challenge, mainly because of temporal changes of both sea level and geopotential due to atmospheric forcing, GIA, ice discharge, variations in continental hydrology and atmospheric and ocean loading.

158 5.1.2 Next Generation Regional/National Height Systems Summary and Final Report Page: 157 of 173 (1) Future height systems will be based on the method of GNSS-levelling. On large spatial scales it will replace traditional geodetic (spirit) levelling. The main reasons for this change are: (1) avoidance/elimination of large scale systematic distortions, (2) direct access to physical heights in the correct reference system by GNSS-techniques and (3) avoidance of the high costs of operation of spirit levelling. (2) For operational or legal reasons it may be advantageous, however, to maintain a national height system in its old (i.e. not unified) height reference (datum). Its connection with the globally unified system is established and available via the IGHFyy, see 1. (3) The technique of GNSS-levelling requires precise GNSS-positioning, a geopotential (or geoid) model of high accuracy and spatial resolution and the consistency of the two, i.e. of the positioning part and the geopotential part in terms of applied standards (ITRF, geodetic reference system, zero tide system etc.). (4) The geopotential model is assumed to be EGMnext, i.e. a combination of SLR tracking, GRACE and GOCE for the long to medium part of a spherical harmonic expansion and the combination with the best available global set of dense terrestrial and altimetric gravity anomalies. (5) The availability of high accuracy positioning at a set of permanent GNSS-stations is of great importance. These stations serve as reference for relative height determination and allow monitoring of local or regional height changes Well Surveyed Regions (a) (b) (c) (d) (e) In well surveyed regions the assumption is the availability of (1) a network of permanent, high quality GNSS stations, (2) records of past and present levelling campaigns and (3) EGMnext geopotential/geoid model as well as regional gravity and topographic data beyond the resolution of EGMnext. The positioning part is performed according to the standards of the IGS. The geopotential or geoid model is EGMnext, possibly refined with existing regional gravity data. Geodetic spirit levelling is very precise over short distances but subject to systematic distortions over larger distances. Therefore, one should consider taking advantage of the high quality of archived or recently measured geodetic leveling lines. In order to combine the strength of both methods, GNSS-leveling and geodetic leveling, an optimal combined adjustment procedure is to be designed. It is furthermore recommended, in good geodetic practice, to maintain and secure the real benchmarks, i.e. physical markers of a selected sub-set of points of the existing height framework. Geodetic markers provide a stable reference framework for monitoring, validation and independent determination of height differences Sparsely Surveyed Regions (a) (b) (c) (d) (e) Sparsely surveyed regions may be characterized by: missing GNSS infrastructure, low accuracy or lack of regional gravity data, lack of surveying and mapping infrastructure. EGMnext may be in error there by 20 to 40cm, in extreme cases even by 1 m, in terms of geoid height. Still, GNSS-levelling is a good start to a regionally consistent height determination with over short distances - rather high relative height accuracy. Again, ocean leveling may be employed for the establishment of a height reference along coastal lines and archipelagos. A master plan may help to assess the point-of-departure in terms of available data (positioning, levelling, gravity, topography and hydrographic data), the quality of the available data, the expected user community and its needs and the possible set-up and characteristics of the

159 Page: 158 of 173 (f) required regional height system. This calls for a joint action of geodesists (IAG, GGOS), oceanographers (IOC/IAPSO) involving also FIG and hydrographic agencies. IAG-GGOS may consider setting up a working group, advising and assisting countries in establishing modern and accurate height systems Ocean Levelling versus GNSS-Levelling (a) (b) (c) (d) (e) (f) A globally consistent and accurate height reference along coastlines and across ocean basins and straits is of high importance for ocean circulation studies and sea level research. At tide gauges the principle of GNSS-levelling can be applied for this purpose. It requires GNSS positioning at tide gauges, the connection of the GNSS marker to the tide gauge height reference and the use of a GRACE/GOCE geopotential or geoid model. This corresponds to a geodetic determination of mean dynamic ocean topography (MDT) at tide gauges. The same principle applied at open sea requires the combination of an altimetric mean sea surface with a geopotential/geoid model based on satellite gravimetry only (SLR, GRACE, GOCE). It is the altimetric method of determination of a geodetic MDT. Low pass filtering is required in both cases (tide gauge based and altimetric MDT) to make the high spatial resolution of the mean sea surface spectrally consistent with the lower resolution of the satellite-only geopotential/geoid model. The resulting MDT does not contain short wavelength ocean topography signal. It is a smoothed version of the actual MDT. In both cases, tide gauges and open sea, the same high consistency standards apply between the geometry and geopotential part (see under ). As shown by Woodworth et al. (2012), oceanic MDT models are a useful tool for validation of geodetic height systems along coastlines. In turn, geodetic MDT at tide gauges is useful as validation tool for ocean modelling and both, the altimetric MDT and MDT estimates at tide gauges are important input for ocean studies. 5.2 Detailed Roadmap Introduction With the new generation of satellite gravimetry missions and in particular with the missions GRACE and GOCE geodesy will have available global satellite-only geopotential models with an accuracy (commission error) of about 0.1 to 0.2 m 2 /s 2 at degree and order 200 of a spherical harmonic expansion. This corresponds to an accuracy of about 1 to 2 cm in terms of geoid heights and to a spatial half-wavelength resolution of about 100km. [In this document we will express height - differences as well as their computation in terms of geopotential differences. This is done to emphasize the fact that geopotential differences are the most physical way to express heights and height differences. Where convenient, we will give numbers also in terms of height and geoid height differences.] This is in accordance with the mission objective of the GOCE mission. Satellite-only models do not include the short-scale geopotential contribution. This missing part is referred to as omission part; it is of the order of 20 cm to 40 cm (RMS value) in terms of geoid heights, depending on the ruggedness of the terrain. Thus, for height determination satellite-only models have to be complemented with terrestrial data. In the well-surveyed regions of the world, such as e.g. North America, Europe, Australia, New Zealand and Japan, excellent data are available and there, after merging with a satellite-only geopotential model, the total (commission) error will amount from 0.4 to 0.6 m 2 /s 2 ; this corresponds to 4 cm to 6 cm in terms of geoid heights. In ocean regions the omission part may be derived from altimetric heights corrected for dynamic ocean topography. In other continental regions of the world, terrestrial data are missing, incomplete, inhomogeneous or inaccurate, resulting in higher commission errors. The expectation is that in the near future the best available terrestrial data sets will be re-assessed, complemented by new terrestrial data sets and optimally combined with one of the new generation satellite-only geopotential models, similar to the current available model EGM2008 (Pavlis et al., 2012). For convenience, let us refer to such a future model as EGMnext. EGMnext shall provide geopotential and/or geoid estimates, convenient,

160 Page: 159 of 173 anywhere and accompanied by a realistic quality measure. An alternative is a regional geoid determination, also optimally combining the best available satellite-only geopotential model with the terrestrial data of this region. The International Association of Geodesy (IAG) has set-up the Global Geodetic Observing System (GGOS), (Plag & Pearlman, 2009). One of its objectives is the standardization of height systems worldwide, compare e.g. (Ihde and Sánchez 2005, Sánchez, 2012). This activity is referred to as GGOS Theme 1 Unified Height System and was established in February One of the primary science objectives of GOCE is the unification of height systems and the improvement of GNSS-leveling, compare sec. 3.5 in (ESA, 1999). The Support To Science Element (STSE) of the European Space Agency (ESA) is a programmatic component of the Earth Observation Envelope Programme (EOEP). In the context of STSE, the present study (GOCE+ HUG) was initiated in order to demonstrate the feasibility of height system unification using the results of GOCE. The present Roadmap is resulting from this study. It is a set of recommendations for height system unification, the improvement and maintenance of state-of-the-art height systems and the role of these improvements for ocean applications. In view of these objectives it is also a contribution to the work of GGOS Theme 1. Before the background of the recent advancements in satellite gravimetry, the issue of the future role of GNSS-levelling and the improvement and unification of national and global height systems received new attention in the geodetic literature. Several of the associated aspects are discussed, for example, in the articles of the recent special issue on height systems, edited by M. Sideris and published in the fourth issue of the second volume of the Journal of Geodetic Science (see scientific project publications in section 1.3.) The classical method of measurement of height differences is geodetic (or spirit) leveling combined with gravimetry. In its essence it is the determination of geopotential differences between height benchmarks and it corresponds to the determination of geopotential numbers C (Heiskanen & Moritz, 1967). Physical heights are derived from them by introduction of a reference height value at a datum point and by choosing the appropriate type of height such as normal or orthometric height. It is common practice to relate the reference height to mean sea level at a chosen tide gauge. Since mean sea level is close to the geoid, its deviations being less than 2 meters, this choice leads to a rather consistent common global reference. Geodetic leveling has a long tradition; the first high precision, first order continental leveling networks were established in the 19thcentury. The method is very precise over short distances, but it is subject to an uncontrolled accumulation of systematic errors when applied on a continental scale. Furthermore the method is labor intensive and time consuming. These are good reasons to adopt a new, alternative approach. There is a clear trend of a complete or partial replacement of geodetic leveling by GNSS-leveling. GNSS-levelling is the new, alternative method of determining of height or geopotential differences between benchmarks. The ellipsoidal heights of the benchmarks are determined with best possible accuracy by geodetic positioning methods. This is primarily the use of GNSS or more generally of Global Navigation Satellite Systems (GNSS). Other techniques are Satellite Laser Ranging (SLR), the French DORIS system or VLBI. Potential differences or physical heights between the benchmarks are then obtained using a geopotential model based on GRACE and GOCE, either combined with regional terrestrial gravity data or extended to a high degree-and-order combined model such as EGMnext. Over short distances this approach is not quite as accurate as geodetic leveling. However, it delivers height information in a uniform system, fast, conveniently and at low cost. Only recently, because of the advancements in terms of quality and operational convenience of both, geodetic positioning as well as geopotential modeling, the method became competitive. The success of the satellite gravimetry missions GRACE and GOCE plays a pivotal role in this. For a recent discussion about the characteristics of the application of GNSS-leveling on a continental scale it is referred to (Featherstone et al., 2012).

161 5.2.2 Global Height Unification Summary and Final Report Page: 160 of 173 Global Height Unification, as proposed here has two objectives: Objective One is the computation of geopotential numbers C and the associated error standard deviations (STD) σ(c) at a selected set of globally distributed reference stations. These stations are (1) all height reference (datum) points of national height systems, (2) a selected set of tide gauges as identified by the PSMSL, (3) a selected set of geodetic observatories (fundamental stations) as identified by the IERS and, (4) after consultation with the IERS, clock reference locations of the bureaus of time standards, such as PTB in Germany or BIPM in Paris. Objective Two is the determination of the height off-sets Cl between all existing (and incorporated) height systems l and one global height reference. Objective One corresponds to the establishment (=realization) of a globally unified height system confined to a set of selected primary height reference points. An example is shown in Figure 5-2. After this exercise, the height differences between all points of this set are established. It is a list, a realization that is afterwards maintained and re-visited at regular intervals. IAG-GGOS, with its close collaboration with the IAG-services is the appropriate body to formulate a proposal with recommendations about the proper selection of stations and the operational and qualitative criteria to be followed. Figure 5-2: Demonstration of a height connection between Europe and USA using a GOCE geoid model combined with the omission part from EGM2008, from [P-4] Objective Two is the determination of the height differences between all incorporated height systems relative to one selected global height reference. The geopotential differences between all incorporated height systems are known then. If desired, they can be translated into any type of physical height offset (orthometric, normal, etc.). They are consistent and they are accompanied with a realistic quality measure. Objective Two - and this is an important quality of this recommendation provides the complete information about all existing height inconsistencies, it does not impose any specific action on the various national agencies. In other words, the global unification of height systems is de-coupled

162 Page: 161 of 173 from any re-definition, renewal, conversion or simply maintenance of existing continental and national height systems. Objective One can be met by directly computing geopotential differences between the selected set of stations. The assumption is that station coordinates are known to better than 1 ppb (1 ppb = 1 part per billion), which corresponds to height values of better than 6mm. With the coordinates known, the potential difference between point P and Q becomes: C PQ C C T U T with U the ellipsoidal normal potential and the anomalous potential T: T P U P0 Q0 P P Q Q (5-1) n1 nmax n ( GM) GM 0 a Pnm(cos P ) C nm cos mp Snm sin mp rp a n2 rp m0 In terms of heights, the same problem reads like the following expression, the well-known GNSSleveling -formula: whereby N P GM) W0 U a GM0 U0 (5-2) PQ P Q PQ PQ P Q P Q H H H h N h h N N (5-3) a nmax ( 0 n2 a r P n1 n m0 P nm P (cos ) C nm cos m S P nm sin m with H and N being orthometric and geoid height, respectively, or alternatively, normal height and quasi-geoid height and h the ellipsoidal height. The spherical harmonic coefficients C nm and S nm is the set given as EGMnext up to degree and order n max, ( GM ) is the uncertainty of our knowledge of GM and should currently be better than 1 ppb. If in some regions gravity data of high accuracy and spatial detail are available beyond those that went into and are represented by EGMnext, a regional improvement of the computation of the anomalous potential may be considered. Greatest care has to be taken to ensure the consistency of the normal potential U and the anomalous potential T in terms of the adopted geodetic reference field and choice of coordinates. Standards applied for point positioning need to be consistent with those applied to the computation of the geopotential. Following IAG recommendation, the permanent tide contribution is expressed in the zero-tide system. One may consider the estimation of a shift of origin between the geocentric coordinate system of the geopotential computation and ITRF, in which the point coordinates are expressed. Objective Two, i.e. the determination of the height off-sets between existing old and the new unified height system can be regarded as to be divided into two steps: (1) the actual determination of the height off-sets and (2) the diagnosis of the existing height system. Already long before GRACE and GOCE, the so-called GBVP-approach was proposed for height datum connection; see e.g. (Colombo, 1980, Rummel & Teunissen, 1988, Rapp & Balasubramania, 1992 and several others). The procedure is essentially a least-squares adjustment, based on a comparison of physical heights H derived from GNSS-leveling with those from geodetic (spirit) leveling at several points of each height datum zone with the height datum off-sets being the unknown parameters. The short scale geoid contribution is proposed to be determined from regional terrestrial gravity anomalies. Due to the unknown height datum off-sets in the gravity anomalies there is an indirect bias contribution, which makes the adjustment less straightforward. In ([P-2] and Gatti, Reguzzoni & Venuti, 2012) it was recently shown that due to the high accuracy and spatial resolution of the geopotential as derived from GRACE and GOCE the indirect bias term can be neglected. This makes the procedure much simpler and straightforward. The off-sets can be deduced directly from a comparison of geopotential differences P (5-4)

163 Page: 162 of 173 (GNSS-levelling versus geodetic leveling) at one point per datum zone. The inclusion of more than one GNSS-levelling -point per datum zone offers the additional possibility of detecting systematic distortions of the leveling network apart from the unknown height off-set parameters, see Figure 5-3. Figure 5-3: Long-wavelength errors of the US NAVD88 vertical datum (Wang et al., 2012) The linear least-squares adjustment model in terms of geopotential values at point P in datum zone l is: ~ P P ~ P ~ P o U T C W U C (5-5) l ~ 0 0 l0 where Cl0 is the unknown datum off-set between zone l and the global reference, P C ~ l is the observed P geopotential number (from geodetic spirit leveling); o ~ denotes the observed left-hand side of the adjustment, the residual and ~ indicating stochastic quantities. In terms of heights the adjustment model for a height at an arbitrary point P becomes: ~ P ~ P ~ P ~ P o h N H N N ~ (5-6) l 0 l0 with the unknown zero-order term N O and the height off-sets N lo. The result of the realization of the two objectives is a globally unified height system. In its essence it is a list of all incorporated stations with their geopotential number and, in addition, the height off-sets (also expressed in terms of geopotential) between all incorporated height reference systems and one common height reference. In analogy with the International Terrestrial Reference System (ITRS) and International Terrestrial Reference Frame (ITRF), one could refer to it as International Height System (IHS) and International Height Frame (IHF), the frame being the realization, i.e. the list of geopotential numbers and off-sets, and the system being the manual with a complete description of all involved standards, algorithms and definitions. The stochastic model is based on a best possible stochastic model of the three elements: point positioning, EGMnext (possibly complemented by regional data) and leveled geopotential differences. The GRACE/GOCE error variance-covariance matrix underlying EGMnext is a very large full matrix (VCM), containing all co-variances between the complete set of spherical harmonic coefficients. It could recently be demonstrated in [P-3] that the full VCM-matrix can be replaced by its dominant m- block structure, which implies a significant simplification.

164 Page: 163 of 173 The expected height accuracy is about 40 to 60 cm 2 /s 2 (or 4 to 6cm) in well surveyed regions with GOCE contributing only about 10 to 20 cm 2 /s 2 (or 1 to 2cm) to the error budget. In sparsely surveyed regions of our planet the uncertainty of height off-sets may be at an RMS level of 20 to 40cm with maximum values at the meter level. In coastal regions, applying ocean levelling, these numbers can be improved. Ocean levelling is the combination of a best ocean topography model with either an altimetric sea surface or, at tide gauges, mean sea level as derived from a combination of tide gauge recording and GNSS positioning. It is recommended to repeat the global height unification at regular intervals. Changes in geopotential numbers of consecutive realizations of the IGHF reflect (1) temporal changes of station heights, (2) improvements or changes of the applied geopotential (or geoid) model and (3) improvements of the adopted standards and methodology. The epoch of computation should be added to IHS and IHF, respectively, i.e. as IGHSyy and IGHFyy, respectively. An often discussed matter is the theoretical basis of height datum definition and of W 0. It is referred to e.g. Heiskanen & Moritz, 1967 (secs and 2-20), Rummel & Heck, 2001, Hipkin, 2001; Sacerdote & Sansò, In real world (physics) only potential differences are observable and not the magnitude of the potential itself (see the analogy with electrostatics). In geodesy the absolute potential W becomes estimable, because when solving the GBVP one has to introduce the regularity condition, which says that potential converges towards zero at infinity. With this boundary condition, the potential at points on the earth s surface and outside becomes estimable. Actually, this corresponds again with the determination of potential differences but with respect to a point at infinity. Under this assumption the absolute potential becomes an estimable quantity. Before this background, the actual datum choice is the regularity condition. The accuracy of this estimate depends solely on the accuracy of the positioning and that of the geopotential model EGMnext. The basic properties of datum parameters are discussed in (Baarda, 1973). Among all geopotential surfaces one is selected as global reference and denoted geoid. This corresponds to the datum choice! Its geopotential value is denoted W 0. The choice may be (1) the geopotential value at mean sea level of a chosen tide gauge, (2) the average value of an ensemble of tide gauges or (3) the mean of the entire altimetric mean sea surface. Each of these three choices is associated with a particular formalism of datum transformation. Further study about the best operational selection is needed. The classical geoid definition is operational at the level of a decimetre. A geoid convention at the centimetre to decimetre level may be complex but is still feasible considering the distinction between geoid and quasigeoid, the influence of the atmosphere, topographic effects, and temporal variations (we refer to the well-known classical work by Moritz and by Mather). At the level of sub-centimetre accuracy the classical geoid definition is a real challenge, mainly because of temporal changes of both sea level and geopotential due to atmospheric forcing, GIA, ice discharge, variations in continental hydrology and atmospheric and ocean loading Next Generation Regional/National Height Systems Future regional/national height systems will be based on the method of GNSS-levelling. It will replace traditional geodetic (spirit) levelling. Some countries, such as USA and Canada have already taken the decision to base their height systems on GNSS-leveling. There are three main reasons for this change: (1) Geodetic spirit leveling is subject to systematic distortions with, in addition, in terms of least-squares adjustment leveling loops offer only the minimal possible internal control. Thus, existing systematic errors remain largely undetected. GNSS-leveling is still less accurate over short distances but it is essentially free of systematic distortions. (2) With modern software tools, GNSSleveling offers instantaneous and direct access to geopotential values or physical heights wherever the geodetic GNSS-receiver is placed. (3) Levelling operation is very costly and time consuming while GNSS-leveling is highly efficient and fast.

165 Page: 164 of 173 It is well-known that the redundancy of geodetic leveling networks is small. Formal error models assume an increase of the error standard deviation with the square-root of the distance, such as the standard error σ in mm between fore and back leveling (with L distance in km): STD 2 L (5-7) For the German first order net the σ-requirement is 1mm for example. But this measure is overly optimistic when we think of the large systematic distortions of e.g. the levelling network of North America (Wang et al., 2012), again see Figure 5-3. Despite the profound change in paradigm proposed by switching to GNSS-levelling, i.e. the replacement of the century old method of geodetic levelling (for first and second order height networks), one should still consider to take advantage of the high quality of the available archived or recently measured geodetic levelling lines. In order to combine the strength of both methods, GNSSlevelling and classical geodetic levelling, an optimal combined adjustment procedure is to be designed. This is similar to the combination of GNSS and inertial navigation. A good choice would be to establish a primary framework of precise height values using all existing IGS stations and all existing (or planned) first and second order height benchmarks. In good geodetic practice it is furthermore recommended, to maintain and secure (many of) these primary points by real benchmarks, i.e. their physical markers. GNSS measurements in the field require a rather long occupation time. Again, with a network of permanent height reference stations available, relative GNSS height determination can be applied with much shorter occupation times. Low cost and low precision height values in a national system could be offered as an app together with any GNSSreceiver. As already pointed out, continental or national height systems are in principle not affected by the Global Height Unification, as described above. For operational or legal reasons it may be advantageous to keep a national height system in its old (i.e. not unified) height reference (datum). Its connection with the globally unified system is established and available via the IGHFyy, see above. As an example, the German NN-heights may be maintained and remain expressed in their present datum (Amsterdam). Nevertheless, their off-set to any other height system is available. The technique of GNSS-levelling requires three basic ingredients: (1) precise GNSS-positioning, (2) a geopotential (or geoid) model such as EGMnext of high accuracy and spatial resolution (possibly complemented by additional regional gravity/topographic/levelling data) and (3) the consistency of the two. The precision of positioning depends on the progress of geodetic satellite positioning techniques, in particular GNSS, but also that of SLR, DORIS and VLBI. In the framework of GGOS it is recommended to coordinate this part with the respective IAG-services. The geocentric coordinates system is the ITRS with its realization ITRF as provided at regular intervals by the IERS. The required precision of point positioning in a geocentric coordinate system must be better than 1 ppb relative the earth s radius, i.e. at the sub-centimeter level. The gravity geopotential model, denoted EGMnext, will be the currently best possible representation of the earth s gravity field. It needs to be adopted by IAG together with all accompanying standards and the algorithms of computation. Ideally, EGMnext should be the next generation of a combined geopotential field, the successor of EGM2008 (Pavlis, 2012). It should be based on SLR for the very low degree/orders, GRACE and GOCE for degree/orders up to about 220 and the best available altimetric and terrestrial gravity data. Also in view of property rights on data or political restrictions - the regional refinement of EGMnext may be an option to be considered. In the well surveyed regions of the world the accuracy is expected to be between 40 and 60 cm 2 /s 2 (corresponding to 4 to 6 cm geoid accuracy). Both, positioning and geopotential modeling have to follow the same geodetic standards (IERS standards, ITRF, zero permanent tide system, geodetic reference system (normal field)) and have to be consistent. Currently this is not the case. There are inconsistencies between on the one hand the

166 ) ATBD 23, - squares adjustment model for VD offset (Eq. 2 - A least Summary and Final Report Page: 165 of 173 ellipsoidal parameters of the IERS conventions 2003 (McCarthy and Petit, 2004) and conventions 2010 (Petit and Lutzum, 2010) and on the other hand those of the Geodetic Reference System 1980 (Moritz, 1979). Furthermore, no zero-tide convention is adopted for GRS80. There is now a clear need for IAG-conventions for global height determination that include the positioning as well as the geopotential (or geoid) component (ellipsoidal parameters, W 0 -value, zero-tide system etc.). It is referred to (Ihde, 2007, Ihde et al and Sánchez, 2012) The availability of high accuracy positioning at a set of permanent GNSS-stations is of great importance. These stations serve as reference for relative height determination and allow monitoring of local or regional height changes. Changes may be caused by (1) temporal changes of station heights, (2) improvements or changes of the applied geopotential (or geoid) model and (3) improvements of the adopted standards and methodology Well Surveyed Regions In well surveyed regions the assumption is the availability of (1) a network of permanent, high quality GNSS stations, (2) records of past and present levelling campaigns and (3) EGMnext geopotential/geoid model as well as regional gravity and topographic data beyond those already used for the determination of EGMnext. Different data sets and relations between position and gravity data in the height system unification is presented in Figure 5-4. Global Height Unification may identify systematic distortions in existing regional/national height systems. It would be of great value to better understand the cause of detected distortions. The renewed height systems will offer an improved data set for validation of future gravity field satellite missions. HSU of large, well surveyed areas by means of the GBVP approach Gravity field information GOCE EGM2008 Regional geoid free from VD distortions Local gravity and DEM GOCE omission part (residual geoid height) Path I HSU with GNSS BMs I.1 Data characteristics (a) Area coverage and omission error (b) Point distribution/density (c) Order of BMs & GNSS points (d) ITRF consistency I.2 VD distortions (a) Long- wavelength errors and tilts (b) Local biases (c) Crustal motion (d) Theoretical approximations of heights I.3 VCMs (a) Fully-populated or diagonal (b) Proper scale factors (c) GOCE commission error (d) Error of the GOCE omission part Path II HSU with GNSS TGs II.1 Data characteristics (a) Primary TG station or a set of TGs (b) Coastline sampling and location (c) GOCE Omission error (d) Ties with BMs (e) ITRF consistency II.2 SST models (a) Local& global SST models (b) Bias between SST models (c) Accuracy of interpolated SST II.3 MSL in local VD (a) 19-year records and data gaps (b) Crustal motion of BM (c) Long-term sea level change (d) VD tilts (e) Weighting based on SST quality Path III HSU with GNSS BMs and TGs III.1 Data characteristics (a) Distribution of GNSS BMs and link to TGs (b) GOCE omission error (c) Tide system and ITRF consistency III.2 VD distortions (a) Local/regional VD distortions at GNSS BMs (b) Cumulative levelling errors (c) SST errors at GNSS TGs III.3 Network of GNSS BM&TGs (a) GNSS BMs VCM (b) GNSS TGs relative weigting (c) GOCE commission error (d) Error of the GOCE omission art Figure 5-4: Schematic representation of height system unification for well surveyed areas Sparsely Surveyed Regions Before the background of global change, problems such as the fast grow of megacities, shortage of water and energy resources and lack of infrastructure, there is a rapidly increasing practical need for good and conveniently available height information.

167 Page: 166 of 173 Sparsely surveyed regions are characterized by missing GNSS infrastructure e.g. permanent GNSS stations, low accuracy or lack of regional gravity data, lack of surveying and mapping surveying infrastructure. From the comparisons of EGM2008 with GOCE-based models it is known that EGM2008 is rather accurate in well surveyed regions such as North America, Europe, Australia, New Zealand and Japan. It is much less accurate in regions such as parts of South America, Africa, Himalaya, parts of China and East Asia. A special case is Antarctica, where terrestrial and airborne gravity is collected only now. The accuracies found are in agreement with the quality of the data sources that were available for the determination of EGM2008. This situation will not change fundamentally in the near future. Thus, EGMnext may be in error in sparsely surveyed regions by 20 to 40cm, in terms of geoid height with maximum errors on the meter level. Still, GNSS-levelling is a good start to a regionally consistent height determination with rather high relative height accuracy. Furthermore, ocean leveling may be employed for the establishment of a height reference along coastal lines and archipelagos. A master plan may help to assess the point-of-departure in terms of an inventory of available data as well as their accuracy and completeness (positioning, levelling, gravity, topography and hydrographic data), the expected user community and its needs and the possible set-up and required characteristics of the required regional height system Ocean Levelling versus GNSS Levelling A globally consistent and accurate height reference along coastlines and across ocean basins and straits is of high importance for ocean circulation studies and sea level research, e.g. (ESA, 1999). Dynamic ocean topography, being the deviation of the actual (mean) sea surface from a geopotential surface, is the quantity at sea analogous to topographic heights on land. Mean dynamic ocean topography (MDT) is typically of the order of ±30cm with maximum values of 1 to 2 meters in areas of maximum ocean circulation. The determination of ocean topography by numerical ocean circulation models is referred to as ocean levelling. In analogy to GNSS-leveling on land, dynamic ocean topography can be recovered at sea and along coast lines by geodetic methods. Along coast lines, at tide gauges it is done by sea level recording connected to GNSS positioning and a GRACE/GOCE geopotential model. At sea it is derived from the combination of an altimetric mean sea surface and a GRACE/GOCE geopotential model. Both approaches offer a novel independent method of purely geodetic MDT-computation and they have various applications in oceanography. However, neither the oceanographic nor the geodetic method of MDT-determination is straightforward: (a) (b) (c) Oceanic and geodetic MDT have to be spectrally consistent, i.e. they have to contain the same level of detail. The spectral resolution of a numerical ocean circulation model is not well defined. Like on land, both, the tide gauge method and the altimetric method require the same high consistency between the geometry and geopotential part in terms of chosen coordinate frame, coordinate type, IERS standards and permanent tide model, compare (Hughes et al., 2008). The geometric mean sea level has to be spectrally consistent with the geopotential/geoid model. While mean sea level from altimetry or tide gauge measurements contain the full MDT and geoid signal, the spatial resolution of a combined GRACE/GOCE model is only about 100 km (degree/order 200).

168 (d) (e) (f) (g) Summary and Final Report Page: 167 of 173 EGMnext must not be applied at sea. It contains gravity material based on altimetry which requires already the application of an MDT model for data reduction. There are technical difficulties with processing satellite altimetry close to shorelines. General ocean circulation models do not apply along coast lines and have to be extrapolated from grid points to the coast. Currently for only a limited number of tide gauges accurate ellipsoidal heights, as derived by GNSS are available. This may be due to lack of GNSS measurements (permanent or campaigns), the missing connection (tie) from the GNSS marker to the tide gauge height reference, or the lack of communication between the science community (IAG, GGOS, sea level initiatives, etc.) and the respective tide gauge authorities. Nevertheless, as shown in [P-14], oceanic MDT models are a useful tool for validation of geodetic height systems along coastlines. In turn, geodetic MDT at tide gauges based on GNSS-levelling is useful as validation tool for ocean modeling and geodetic MDT as derived by satellite altimetry. Altimetric MDT represents important new input for global and regional ocean circulation studies. In a recent test, the historical controversy between geodesists ( sea level is increasing towards higher latitudes ) and oceanographers ( sea level decreasing towards higher latitudes ) about the correct sea level slope along the coast line of North America could be settled. The geodetic results from classical national scale spirit levelling nets could be shown to be wrong, see Figure 5-5. Still unknown is the origin of the systematic distortions. Similar distortions exist in other parts of the world. Figure 5-5: Sea level slope at tide gauges along the east coast of North America from classical geodetic leveling (USA in red, Canada in blue), from an ocean circulation model (black) and from GNSS-leveling (yellow); from: P.L. Woodworth, C.W. Hughes, R.J. Bingham and T. Gruber. Towards worldwide height system unification, Journal of Geodetic Science 2(4) DOI: /v