Height System Unification with GOCE

Height System Unification with GOCE North American Analyses and Main Results M.G. Sideris and E. Rangelova University of Calgary

Overview of the UoC contributions Tailored GOCE model Computing MSL at Canadian tide gauges Analysis of local gravity databases. Checking Canadian GDR gravity data with GOCE Indirect bias term investigations: North America & global Evaluation and impact of GOCE on the regional gravimetric geoid model. Relative accuracy of the GOCE geoid Analysis of NA height systems. Canadian first order levelling network check with GOCE Assessment of the impact of the GOCE omission and commission errors on the NA datum offsets and on the connection of large islands to mainland. Analysis of GNSS-Levelling EPS: long-wavelength and local distortions and geodynamics. Height datum offset computations with GOCE release 3 and 4 gravity models. Connecting MSL at Canadian tide gauges with the regional gravimetric geoid. Computing the mean potential of MSL for W o determination. Contributions to the HSU roadmap for well surveyed regions (including the dynamic height datum) and poorly surveyed areas. Height datum offset computations using tide gauges, GOCE and regional geoids models.

Outline Overview of height datums in North America Checking levelling datums with GOCE Checking gravity data base with GOCE The role of GOCE in the definition and realization of NAVRS Impact of GOCE on the regional geoid in North America Datum offset estimation by means of GOCE and GNSS-levelling data Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges A tailored GOCE model for HSU Dynamic height datum in North America Contributions to the scientific roadmap Conclusions

Overview of the North American vertical datums CGVD28 and NAVD88 official vertical datums in North America Levelling data collected over the time span of more than 8 decades CGVD28 constrained to the Atlantic and Pacific MSL Normal-orthometric heights An east-west slope of ~60 cm Poor absolute accuracy and local distortions Vertical crustal motion NAVD88 constrained to Rimouski Orthometric heights Very large NW-SE slope w.r.t. GOCE geoid Poor absolute accuracy IGLD85 is part of NAVD88 Dynamic heights Objectives Investigate the North American height datums unification by means of GOCE Using the GBVP approach GNSS-Levelling benchmarks GNSS-Surveyed tide gauges CGVD28 CONUS and Alaska NAVD88 Mexico NAVD88

Check of levelling datums with GOCE the CGVD28 case The CGVD28 coast-to-coast distortion w.r.t. the GOCE geoid Conclusion: The estimated mean offset and the coast-to-coast datum distortion is largely affected by the GOCE geoid omission error. Regional subsidence and uplift caused by postglacial rebound Benchmarks and tide gauges in Eastern Canada are subject to crustal motion, and the CGVD28 height of MSL can be in error of 20-30 cm. D = h ITRF2005 - H CGVD28 N GOCE&EGM2008 Findings: Geodynamic effects in the datum are revealed after correcting the geoid heights for the GOCE geoid omission error and the CGVD28 heights for the coast-to-coast datum distorion.

GPU Check of levelling datums with GOCE Canadian control network re-adjustments Spatial distribution of the Nov07 levelling GOCE geopotential differences at first order GNSS-levelling stations Conclusion: Tilts of few levelling lines in the first order vertical control network are observed only if the GOCE geoid model is corrected for the omission error.

Check of gravity data base with GOCE the Canadian case Geoscience Data Repository of NRCan Gravity Data 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. (1) GRS80 (2) Atmospheric correction (3) Second-order vertical gravity gradient (4) SST correction

Check of gravity data base with GOCE the Canadian case Mean -1.4 mgal Std 5.4 mgal Min -56.7 mgal Max 56.0 mgal

Check of gravity data base with GOCE the Canadian case Mean -1.6 mgal Std 7.4 mgal Min -56.7 mgal Max 56.0 mgal Mean -1.3 mgal Std 3.5 mgal Min -23.4 mgal Max 17.8 mgal Mean -0.6 mgal Std 2.9 mgal Min -43.4 mgal Max 14.9 mgal Mean -2.0 mgal Std 4.0 mgal Min -25.4 mgal Max 28.8 mgal

Check of gravity data base with GOCE the Canadian case Findings The GOCE GGM omission error should be accounted for in order to check the gravity anomay data base. In areas with good agreement between the GOCE GGM corrected for the omission error and the national gravity data, GOCE can be used for outlier detection. In mountains areas, GOCE cannot be used by itself to check the gravity data, but it can help identify biases due to flawed water depth and ice thickness information in remote areas.

The Role of GOCE in the Definition and Realization of the North American Vertical Reference System (NAVRS) Definition (GSD, Natural Resources Canada and NGS, NOAA) NAVRS will be an equipotential surface (W 0, m 2 s -2 ). Will be realized by a geoid model (N, m) A dynamic surface (N dot, mm/yr) Will comply with the adopted national, continental and international standards Requirements One geoid model will be realized for North America Accuracy 1-2 cm in coastal and flat areas 3-5 cm in mountainous regions Updates To be updated at certain time intervals A velocity model of the geoid will be produced and distributed Realization: The W 0 value is determined through tide gauge averaging around North America GSD and NGS signed an agreement on 16.04.2012 to realize and maintain a common vertical datum for USA and Canada Zero-height level defined by W 0 = 62636856.0 m 2 s -2

The Role of GOCE in the Definition and Realization of the North American Vertical Reference System (NAVRS) Determination of the W 0 value Obtain the geopotential at the tide gauge sites Long and medium wavelengths from a GOCE GGM with the highest possible spectral resolution) Omission error (may not average out at the tide gauge sites) A local high resolution geoid model is needed based on a GOCE GGM and local gravity and topography/bathymetry information Assess the quality and agreement of MDT models in coastal areas Computation of a common geoid model in North America by 2022 The CGVD2013 geoid-based datum (GSD, NRCan) is available for the users now (contains the newest GOCE gravity information) Link of STSE GOCE+ Theme 1 with NAVRS Compute high accuracy local datum offsets with respect to the regional W 0 level Improve gravity anomaly data and height information

The Role of GOCE in the Definition and Realization of the North American Vertical Reference System (NAVRS) Determination of the W 0 value by averaging the potential of mean sea level Obtain the potential at GNSS-surveyed tide gauge stations a connection between mean sea level and the GBVP solution (local geoid) Factors that contribute to the accuracy of the computed potential values distribution and number of tide gauges resolution, accuracy and interpolation errors of MDT models tidal systems of ellipsoidal heights, mean sea level and the geoid all computations are in a conventional tide-free system GOCE geoid omission error a significant error source the omitted higher frequencies of the geoid (less than ~110 km half-wavelength) contribute 16 cm in terms of separation of equipotential surfaces for the Pacific region (rugged topography) and 9 cm for the Atlantic region. References: Hayden, T., Rangelova, E., Sideris, M.G., Véronneau, M., 2012. Evaluation of W 0 in Canada using tide gauges and GOCE gravity field models. Journal of Geodetic Science, 2(4), 290-301, DOI: 10.2478/v10156-012-0003-9. Hayden T., Rangelova E., Sideris M.G., Véronneau, M., 2013. Contribution of tide gauges for the determination of W o in Canada, In: IAG Symposia Vol. 141, Gravity, Geoid and Height Systems (GGHS2012), Venice, October 2012, Springer (accepted).

The Role of GOCE in the Definition and Realization of the North American Vertical Reference System (NAVRS) W 0 values for Pacific, Atlantic, and Pacific + Atlantic tide gauges with 19 years of water level data with various GOCE GGMs expanded to degree 180 and 2190 (add 62636800.00 m 2 /s 2 ) GGM Model Pacific Atlantic Pacific + Atlantic goco03s n max :18 0 goco03s+egm2008 n max :180+181 to 2190 tim_r3 n max :180 tim_r3+egm2008 n max :180+181 to 2190 dir_r3 n max :180 dir_r3+egm2008 n max :180+181 to 2190 52.59 ± 2.13 58.30 ± 1.16 55.19 ± 1.39 54.18 ± 0.13 59.21 ± 0.16 56.47 ± 0.56 16 cm 52.84 ± 2.13 9 cm 58.56 ± 1.17 13 cm 55.44 ± 1.40 54.43 ± 0.12 59.48 ± 0.19 56.73 ± 0.56 52.78 ± 2.16 58.51 ± 1.20 55.39 ± 1.41 54.38 ± 0.19 59.43 ± 0.19 56.67 ± 0.56

Impact of GOCE on the regional geoid in North America Evaluation of the GOCE GGMs using GNSS-levelling data Canada Nov07 CONUS NAVD88 Mexico NAVD88 Alaska NAVD88 Conclusion: Better performance of RL-04 models compared to EGM2008 is observed in the spectral band from approximately degree/order 100 to 200.

Impact of GOCE on the regional geoid in North America Impact on the Canadian gravimetric geoid The early GOCE releases did not lead to a noticeable improvement of the regional geoid over EGM2008 or the official geoid CGG2010. The agreement with the GNSS/levelling geoid was 12 cm for the whole Canada, 4.1 cm to 5 cm for the Great Lakes and 6.6 cm to 7.2 cm for the Pacific Cordillera. 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. GOCE does not visibly 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. 10-25 cm differences (improvement?) 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. Reference: Ince, E.S., Sideris, M.G., Huang, J., Véronneau, M. (2012) Assessment of the GOCE-Based Global Gravity Models in Canada. Geomatica, 66, 125-140.

Data variances known only and uncorrelated h, H and N Datum offset estimation by means of GOCE Basic relationships for computing height datum offsets Rummel and Teunissen (1988): Multiple Vertical Datum Problem If the indirect bias term is negligible (less than 1 cm), the observation equation reduces to Mean datum offset, i.e., the offset of a mean equipotential surface from the geoid or a reference level defined by a conventional W o value dn j = h P - H P j -(N o + N PGOCE + N Pres ) i o P j P P o j P W R N N N N N res GOCE S 2 J i i Po i P P o j P P j f N N N N H h N res GOCE 1 ) ( an iterative solution NN GOCE NN res HH hh ll j Q Q Q Q Q l N A, 1 1 ˆ ˆ 1 1 1 ) (, ) ( ˆ A Q A Q l Q A A Q A N ll T N N ll T ll T j n i i n i i i j p p l N 1 1 / ˆ n i i o N p 1 2 2 ˆ / ˆ ˆ 1 2 2 2 2 ] ) ( ) ( ) ( ) [( i NN i NN i HH i hh i res GOCE p 1) /( ˆ ) ( ˆ 1 2 2 n p N l n i i i o

Datum offset estimation by means of GOCE Investigations on the indirect bias term in North America Procedure like in Gerlach and Rummel (Journal of Geodesy 2013) Mean offsets computed with DIR4 D/O 200 using GNSS/levelling points Reference level defined by W o = 62636856.0 m 2 s -2 A conventional tide-free system used Modified fftgeoid Fortran software by UofC Large offset of Alaska because of the NAVD88 errors that propagate from east to west 176 pts 1315 pts Country Datum Offset (cm) Canada CGVD28-28 USA NAVD88-48 18399 pts 744 pts Alaska NAVD88-148 Mexico NAVD88-6 Canada and USA GNSS heights in ITRF2005. Mexico unknown reference system.

Datum offset estimation by means of GOCE Indirect bias term computed with the Stokes s kernel Maximum effect of 43 cm located in Alaska The reference level is defined by W o = 62636856.0 m 2 s -2

Datum offset estimation by means of GOCE res Indirect bias term computed with a residual Stokes s kernel St ) St( Finding: the indirect bias term is below 1 cm everywhere for n max = 200. ( nmax ) St ( )

Datum offset estimation by means of GOCE Global investigation of the indirect bias term Mean offsets w.r.t. the reference level W o = 62636856.0 m 2 s -2 (IERS2010)

Datum offset estimation by means of GOCE res Indirect bias term computed with a truncated Stokes s kernel St ) St( ) Finding: the indirect bias term is below 1 cm everywhere for n max = 200. nmax ( St ( )

Datum offset estimation by means of GOCE Conclusions Indirect bias term can be neglected if a global geopotential model of maximum degree 200 or higher is used. Computational procedure of height datum offsets by geodetic boundary value approach is simplified significantly. Kernel Stokes s kernel, degree Original Stokes Territory 70 120 150 200 North America South America Europe Combined grid 38.4 cm < 1 cm <1 cm <1 cm <1 cm -55 cm 1 cm < 1 cm <1 cm <1 cm 21.4 cm 2.9 cm 1.3 cm 1 cm < 1 cm -63.4 cm 2.9 cm 1.3 cm 1 cm < 1 cm

Datum offset estimation by means of GOCE and GNSS-levelling data Canadian mainland (1315 pts) Vancouver Island (26 pts) Newfoundland (34 pts)

Datum offset estimation by means of GOCE and GNSS-levelling data Example 1: Effect of the GOCE geoid omission error Mean Nov07 offset for Canadian mainland (CML), Newfoundland (NFLD), and Vancouver (VAN) regions using the local gravimetric geoid and GOCE-based geoid Region CGG2010 goco03s n max =180 goco03s+egm2008 n max = 180 + 181 to 2190 CML(1315) -45.0 ± 0.3-59.2 ± 1.1-44.5 ± 0.4 NFLD (34) -44.1 ± 1.9-43.7 ± 5.0-33.5 ± 1.1 VAN (26) -38.9 ± 1.3-10.4 ± 14.8-41.3 ± 1.1 Conclusions The effect of the truncation degree of the GOCE model (omission error) on the datum offsets is significant in Canada, at the decimeter level. The omission error factor is especially important for regions with very few benchmarks, limited geographical coverage, and rugged terrain, such as independent levelling networks on islands, where the geoid model errors and the measurement errors of the GNSS/levelling heights may not average out. Hayden, T., Amjadiparvar, B., Rangelova, E., Sideris, M.G. (2012) Estimating Canadian vertical datum offsets using GNSS/Levelling benchmark information and GOCE global geopotential models. Journal of Geodetic Science, 2(4), 257-269, DOI: 10.2478/v10156-012-0008-4

Datum offset estimation by means of GOCE and GNSS-levelling data Example 2: Effect of the GOCE geoid commission error Mean Nov07 offset for CML, NFLD, and VAN regions using goco03s (n max = 180) and error information for the ellipsoidal, orthometric, and geoid heights Region δn j (cm) without error information δn j (cm) with error information CML -59.2 ± 1.1-63.2 ± 1.0 NFLD -43.7 ± 5.0-56.9 ± 2.5 VAN -10.4 ± 14.8 77.3 ± 8.9 Conclusions GOCE commission error information up to degree and order 180, in combination with the error estimates for the ellipsoidal and levelling heights, results in a 4 cm difference in the LVD offset with respect to the LVD offset estimated without any error information for the CML region. The effect is more pronounced for Newfoundland, where the difference is 13 cm. One of the reasons contributing to this difference can be explained by the fact that the geographical coverage of the NFD network is much smaller when compared to the geographic coverage of the CML network, and therefore it may be affected by the commission errors of the GGM wavelengths that exceed the size of the test area.

Observation equation Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges h TG H j TG Applied with the following observables: Height of mean sea level in the local datum GNSS ellipsoidal height of mean sea level Geoid height from GOCE and residual geoid height from local data Errors of the heights of MSL in the local datums may not be available even in well surveyed countries j ( N N N ) N N o TG GOCE TG res J i1 i f i

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges PSMSL tide gauges data set Canada 7 TG stations at the Atlantic Coast 5 TG stations at the Pacific Coast CGVD28 orthometric heights PSMSL tide gauges data set in USA 28 TG stations at the Atlantic Coast 17 TG stations at the Pacific Coast 13 TG stations at the Gulf Coast NAVD88 orthometric heights

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges GOCE omission error at tide gauges Differences h-h-n computed with the USGG2012, TIM Rl04 (d/o 180) and TIM Rl04+EGM2008 geoid heights at the USA tide gauges (left) and the Canadian tide gauges (right). Pacific (17) Atlantic (7 TGs) Atlantic (28 TGs) NAVD88 Gulf (13) Pacific (5) CGVD28 Findings: Clearly visible in the differences is the slope of the datums identified by means of TIM4 + EGM2008 and USGG2012, as well as the large omission error of TIM4 (several decimetres at individual tide gauges).

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges GOCE geoid omission error at tide gauges Contribution of EGM2008 from degree 181 to 2190 Region Mean (cm) Std (cm) Min (cm) Max (cm) Canada Atlantic 1 33-56 38 US Atlantic -3 37-84 72 Canada Pacific 6 40-36 64 US Pacific 7 33-49 74 Gulf of Mexico 3 26-62 28 Findings: The large GOCE geoid omission error tends to average out at the PSMSL network. The residual effect varies from region to region but remains within few centimetres. Conclusions: In Atlantic and Gulf of Mexico, a satellite GOCE GGM of degree/order 250 may be sufficient for computing datum offsets. Improvement is required in the Pacific region, but it will be mainly in the higher geoid frequencies.

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges GOCE omission error in the vicinity of the tide gauges (areal averages) Left: Standard deviation of the average TIM-Rl04 (d/o 180) omission error at the PSMSL tide gauges in different coastal regions, in cm Right: Mean of the average TIM-RL04 (d/o 180) omission error, in cm Findings: A small number of tide gauges combined with a rugged terrain as in the Pacific Canada results in a large mean omission error even for large radii of the spherical cap. Tide gauges used for HSU should be al least 0.4 degree (~ 28 km for 50N) apart in order to mimimize the effect of the omission error on the mean datum offset.

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges Example: Mean CGVD28 and NAVD88 offsets w.r.t. the W o = 62636856.0 m 2 s -2 level Geoid Model CGVD28 NAVD88 (Pacific coast) N j (cm) Diff N j (cm) N j (cm) Diff N j (cm) TIM4 (degree 180) -28 13 - -81 9 - TIM4 (180) + EGM2008 (2190) -25 10 3-74 5 7 Findings: CGG2010 (22) -24 10 4-80 5 1 USGG2012 (11) -26 10 2-77 5 4 Larger a-posteriori offset error with the TIM4 geoid (an increase of 3-4 cm) with non-weighted observations When the TIM4 geoid commission error is taken into account, the a-posteriori offset errors are 13 cm and 11 cm for CGVD28 and NAVD88(Pacific), respectively. Conclusion: 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 TIM4 omission error is taken care of.

Tailored GOCE geoid model for North America Objective: Develop a unified methodology to compute high resolution geoid heights based on GOCE Input: GOCE GGM and local gravity data g, GGM Earth ' s Surface observed Downward continuation Mean Sphere g Downwarded Initial Model C Initial nm, S Initial nm Residual data g g g Mean Sphere Downwarded Mean Sphere GGM g C S Mean Sphere GGM Initial nm Initial nm C S g New nm New nm Mean Sphere New GGM Spherical harmonic analysis C nm, S nm New model C New C Initial C, S New S Initial CS nm nm nm nm nm nm C C, S S Tailored New Tailored New nm nm nm nm Geoid undulations from tailored model + RTM correction Output: an ultra high degree (5400) local gravity field model.

Tailored GOCE geoid model for North America Tailoring of the DIR-Rl04 (D/O 200) model to the local gravity anomalies in the Atlantic region. Left: Model gravity anomalies, unit is mgal Right: 2 Faye gravity anomalies, unit is mgal

Tailored GOCE geoid model for North America Coverage All tide gauge stations of the northwest Atlantic plus the stations in the Gulf of Mexico. Input data Gridded Faye (2 ) gravity anomalies by GSD, Natural Resources DIR4 model up to D/O 200 EGM2008 up to D/O 2160 for the computation of the radial gravity anomaly gradient for the purpose of downward continuation to the geoid Initial model Obtained by spherical harmonic analysis of the gridded gravity anomalies Methodology Two MATLAB functions developed for ultra high degree global spherical harmonic analysis and synthesis Associated Legendre functions based on Fukushima (2012). Parallelized computations in MATLAB

Tailored GOCE geoid model for North America Coverage GNSS-levelling benchmarks for model validation and offsets computations 551 first-order points in Canada 15518 points in USA

Tailored model validation at GNSS-levelling stations Standard deviation of the geoid height differences (h - H - N) as function of the model cut-off degree at the USA GNSS-levelling stations in the test region *Tailored model developed to degree and order 5400 (2 ) **NAVD88 orthometric heights used in the model validation

Tailored model validation at GNSS-levelling stations Standard deviation of the geoid height differences (h - H - N) as function of the model cut-off degree at the Canadian GNSS-levelling stations in the test region *Tailored model developed to degree and order 5400 (2 ) **Nov07 orthometric heights at the first order levelling network used in the model validation

Tailored model validation at tide gauge stations Standard deviation of the geoid height differences (h - H - N) at tide gauge stations, in cm Region USGG2012 ExtGOCE Tailored (2160) Tailored (2700) Tailored (3600) Tailored (5400) Canada Atlantic 16.2 15.6 13.1 13.7 13.5 15.4 US Atlantic 17.7 17.2 16.7 16.7 16.7 16.6 Gulf of Mexico 13.7 12.8 12.7 12.7 12.7 12.7 Canada Pacific 9.9 14.6 27.1 27.4 27.5 27.5 US Pacific 20.4 19.8 35.7 35.7 35.7 35.7 Slightly better performance of the tailored model compared to USGG2012 in the US Atlantic and Gulf of Mexico Canada Atlantic has a small number of tide gauges; therefore, random errors may not average out -> larger variation of standard deviation Very large standard deviation for the Pacific region; shows the GOCE DIR Rl04 contribution *Tailored model developed to degree and order 5400 (2 ) **Nov07 and NAVD88 heights of the local MSL used in the validation ***USGG2012 has a resolution of 1

HSU with the tailored GOCE geoid Datum offset from GNSS-levelling stations in Canada Estimated datum offsets from EGM2008 and Tailored models are almost identical. Apparent 1 cm bias between the offsets from ExtGOCE and EGM2008 or Tailored model. 1 cm Conclusion: A GOCE-based geoid model of D/O 400 is sufficient for computing the datum offset (1 cm precision) with the GNSS-levelling stations over the Canadian Atlantic and Great Lakes.

HSU with the tailored GOCE geoid Datum offset from GNSS-levelling stations in USA Estimated datum offsets from EGM2008 and Tailored models are almost identical. Apparent 1 cm bias between the offsets from ExtGOCE and EGM2008 or Tailored model. 1 cm Conclusion: A GOCE geoid model of D/O 200 is sufficient for computing the datum offset (1 cm precision) with the GNSS-levelling stations over the US Atlantic US, Great Lakes and Gulf of Mexico.

Dynamic height datum in North America A postglacial rebound vertical rate surface from an optimal combination of GRACE and GNSS vertical velocities GRACE only GRACE + GNSS Rangelova, E., Fotopoulos, G., Sideris, M.G., 2009. On the use of iterative re-weighting least-squares and outlier detection for modelling rates of vertical displacement. J. Geod., Vol. 83: 523-535, doi: 10.1007/s00190-008-0261-6.

Dynamic height datum in North America Significance of the dynamic component of the height datum (the Canadian case) Calibration of Error Variance- Covariance Matrices through Variance Component Estimation Signal-to- Noise Ratio shows the significance of the time correction in a decade

A least-squares adjustment model for VD offset (Eq.2-23, ATBD) Contributions to the Scientific Roadmap 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

Contributions to the Scientific Roadmap HSU of poorly surveyed areas (UofC) I. Areas without established GNSS networks (e.g., isolated islands and remote areas, Antarctica) and tide gauges I.1 Geoid height (a) GOCE (b) EGM2008 solely on GRACE (Antarctica) (c) GOCE omission error can be assessed by SRTM, geology, topography-isostastic potential (d) GOCE & altimetry in coastal areas II. Areas with unreliable networks of GNSS BMs and local gravity II.1 Geoid height (a) GOCE plus GOCE omission error from EGM2008 (b) EGM2008 improved by GOCE for medium wavelenghts (c) GOCE & altimetry in coastal areas Required data for HSU may be missing or of poor quality EGM2008 may be of inferior quality no local gravity information Combination of GOCE and altimetry in the coastal areas to get the local potential. GOCE omission error from SRTM, geology and topography-isostatic potential I.2 GNSS height (a) Establish a network of GNSS points such that GOCE omission error cancels out (b) Establish one GNSS fundamental point I.3 Orthometric height (a) H= h-n GOCE (b) Main source of error: GOCE omission error (I.2.b) or mean of the omission error (1.2.a) II.2 SST at fundamental TG or network of TGs (a) Oceanographic models (b) Geodetic data (c) Tide gauges referenced in ITRF II.3 Local W j o and datum offset (a) N j = (W j o - W GOCE )/ (b) Main error source: GOCE omission error and SST model error Establish a vertical control by means of GNSS and the GOCE geoid At tide gauge stations, the potential of MSL can be termined from GOCE & the tide gauge records or GOCE & a global MDT model

Main conclusions The omission error of the GOCE Rl-04 models is the most important factor to account for in the HSU in North America shown for the GNSS-surveyed tide gauges in five coastal regions and the GNSSlevelling data in Canada, W o computations in Canada, and checking the levelling network the omission error averages out over the US GNSS-levelling points The omission error can be corrected for by a tailored GOCE gravity model uniform gravity model computation procedure; faster than similar available procedures new gravity data and/or GOCE models can be easily included in the local gravity model The effect of the height bias in gravity anomalies on the datum offsets can be kept below 1 cm over whole North America by using a GOCE GGM of degree/order 200. Rl05 models will most likely increase this degree, most significantly in Mexico, where the Rl-04 models show the largest improvement over EGM2008. The outcome of the studies performed in North America under STSF-GOCE+ HSU are important for NAVRS (improvement of local data sets/corrections for height biases, GOCE models evaluations, high resolution GOCE-based continental geoids models) Global HSU (proof of the GBVP concept in a region with abundance of geodetic data) GGOS Theme 1 A UNIFIED WORLD HEIGHT SYSTEM

List of publications Amjadiparvar, B., Rangelova, E. & Sideris, M. G. (2013) North American height datums and their offsets: Evaluation of the GOCEbased global geopotential models in Canada and the USA. J Appl Geodesy, 7(3), 191-203, DOI: 10.1515/jag-2012-0033 Amjadiparvar, B., Rangelova, E. V., Sideris, M. G. & Véronneau, M. (2013) North American height datums and their offsets: The effect of GOCE omission errors and systematic levelling effects. J Appl Geodesy, 7(1), 39 50, DOI: 10.1515/jag-2012-0034 Hayden, T., Amjadiparvar, B., Rangelova, E., Sideris, M.G. (2012) Estimating Canadian vertical datum offsets using GNSS/Levelling benchmark information and GOCE global geopotential models. Journal of Geodetic Science, 2(4), 257-269 DOI: 10.2478/v10156-012-0008-4 Hayden, T., Rangelova, E., Sideris, M.G., Véronneau, M. (2012) Evaluation of W0 in Canada using tide gauges and GOCE gravity field models. Journal of Geodetic Science, 2(4), 290-301, DOI: 10.2478/v10156-012-0003-9 Hayden T., Rangelova E., Sideris M.G., Véronneau, M. (2013) Contribution of tide gauges for the determination of Wo in Canada, In: IAG Symposia Vol. 141, Gravity, Geoid and Height Systems (GGHS2012), Venice, October 2012, Springer (accepted). Ince, E.S., Sideris, M.G., Huang, J., Véronneau, M. (2012) Assessment of the GOCE-Based Global Gravity Models in Canada. Geomatica, 66, 125-140. Rangelova, E., van der Wal W., Sideris, M. G. (2012) How Significant is the Dynamic Component of the North American Vertical Datum? Journal of Geodetic Science, 2(4), 281 289, DOI: 10.2478/v10156-012-0005-7 Rangelova, E., Sideris, M.G., Amjadiparvar, B., Hayden, T. (2013). Heights datum unification by means of the GBVP approach using tide gauges. VIII Hotine-Marussi Symposium, Rome, Italy, June 17-21, 2013 (in second review cycle). Sideris, M. G. (2012) Preface to the Special Issue of the Journal of Geodetic Sciences on Regional and Global Geoid-based Vertical Datums. Journal of Geodetic Science. 2(4), 246 246, DOI: 10.2478/jogs-2013-0006 Sideris, M.G., Rangelova, E., Amjadiparvar, B. (2013) First results on height systems unification in North America using GOCE. In: IAG Symposia Vol. 141, Gravity, Geoid and Height Systems (GGHS2012), Venice, October 2012, Springer (accepted).

MODT at the coast: UoC past and current related work Improved ocean tidal corrections for altimetry in east and west Canadian coastal regions based on the Webtide local tide model and analyses of residual SLA variability determined residual constituents larger than 6 months Improved monthly altimetry and tide gauge sea level series regional common mode corrections and proper noise modelling Determined absolute and relative sea level rates at both coasts MDT from altimetry data and local and GOCE-based geoid models (current work) for the North American Atlantic GOCE geoid models from the fourth and upcoming fifth releases CGG2013, USGG2012, and a tailored GOCE geoid model Developed software and vast experience in local geoid computations

Backup slides

Offset in cm Effect of the truncation degree of the GGM on the datum offsets 0 70 90 120 150 180 210 250 360 1400 2190-20 -40-60 Nov07 (Canada) NAVD88 (USA) NAVD88 (Canada) -80-100 -120 Spherical Harmonic Degree

Effect of the truncation degree of the GGM on the datum offsets Amjadiparvar, B., Rangelova, E. V., Sideris, M. G. & Véronneau, M. (2013) North American height datums and their offsets: The effect of GOCE omission errors and systematic levelling effects. J Appl Geodesy, 7(1), 39 50, DOI: 10.1515/jag-2012-0034

Estimated offsets with GNSS-levelling data Amjadiparvar, B., Rangelova, E. V., Sideris, M. G. & Véronneau, M. (2013) North American height datums and their offsets: The effect of GOCE omission errors and systematic levelling effects. J Appl Geodesy, 7(1), 39 50, DOI: 10.1515/jag-2012-0034

Estimated relative accuracy of the GOCE geoid Residuals I: D = h ITRF2005 - H CGVD28 N GOCE Residuals II: D = h ITRF2005 - H CGVD28 N GOCE&EGM2008 Canada (4393298 baselines) TIM3: 3 ppm (60 cm) @ 200 km TIM3+EGM2008: is 0.8 ppm (16 cm) @ 200 km Pacific Cordillera (723388 baselines ) TIM3: 74 cm @ 200 km TIM3+EGM2008: 16 cm @ 200 km Eastern Canada (349930 baselines) TIM3: 44 cm @ 200 km TIM3+EGM2008: 14 cm @ 200 km CGVD28 remains locally precise for GNSS-levelling purposes

Datum offset estimation by means of GOCE and GNSS-surveyed tide gauges

Comparison between the geodetic and oceanic approaches of HSU Pacific Canada Atlantic Canada 54.40.6 m 2 s -2 54.30.6 m 2 s -2 W o =62636856.0 m 2 s -2 10 10 cm 16 6 cm 16 6 cm TGs TGs Ocean TGs TGs Ocean TGs: H MSL above CGVD28 TGs: h MSL = N GOCE+EGM08 +MDT model 50 10 cm 31 5 cm 59 32 cm Ocean: h MSL = N GOCE+EGM08 +MDT model 59.00.5m 2 s -2 61.83.1m 2 s -2

W o estimates with various MDT models MDT Models Pacific W 0 (m 2 /s 2 ) Latitude: 30 N to 60 N Longitude: 150 W to 115 W Tide Gauges with CGG2010 Atlantic W 0 (m 2 /s 2 ) Latitude: 30 N to 60 N Longitude: 80 W to 50 W 62636854.17 ± 0.19 62636860.70 ± 0.18 Local: Foreman 62636854.25 ± 0.63 -- Local: Wright -- 62636857.28 ± 4.50 Local: Thompson & -- 62636857.11 ± 5.04 Demirov Maximenko 62636854.69 ± 1.03 62636859.61 ± 6.86 CLS 62636854.04 ± 0.65 62636859.99 ± 6.87 ECCO2 JPL 62636854.69 ± 0.29 62636860.84 ± 6.84 OCCAM12 62636854.24 ± 0.29 62636861.09 ± 6.75 GECCO 62636854.11 ± 0.24 62636861.08 ± 6.85 ECCO-godae 62636854.23 ± 0.37 62636861.77 ± 6.74 Liv Fine 62636854.15 ± 0.31 62636861.33 ±6.75 Liv Coarse 62636853.32 ± 0.35 62636861.23 ± 6.72 GOCE1 62636854.33 ± 0.45 62636859.79 ± 6.85 GOCE2 62636854.75 ± 0.82 62636859.73 ± 6.86 The MSL of the Pacific is approximately 20 cm above the global conventional reference surface adopted by GSD and NGS, while the MSL of the Atlantic is approximately 40 cm below this surface. The local MDT models of Thompson & Demirov and Wright yield a lower potential for the Atlantic when compared with the 10 other MDT models. Hayden T., Rangelova E., Sideris M.G., Véronneau, M. (2013) Contribution of tide gauges for the determination of Wo in Canada, In: IAG Symposia Vol. 141, Gravity, Geoid and Height Systems (GGHS2012), Venice, October 2012, Springer (accepted).

Gravimetric geoid computation 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 Ince, E.S., Sideris, M.G., Huang, J., Véronneau, M. (2012) GGM res ind N 0 N N N Assessment of the GOCE-Based Global Gravity Models in Canada. Geomatica, 66, 125-140

HSU with the tailored GOCE geoid GOCE omission error at the PSMSL tide gauges from the ExtGOCE (D/O 2160) and USGG2012 Region ExtGOCE-GOCE USGG2012-ExtGOCE GOCE omission Canada Atlantic 2.9 4.0 6.9 US Atlantic 4.6 2.0 6.6 Gulf of Mexico -1.7 0.9-0.8 Canada Pacific -13.4-3.0-16.4 US Pacific -14.8 3.6-10.9 GOCE omission error at the PSMSL tide gauges from the Tailored (D/O 2160) and USGG2012 Region Tailored-GOCE USGG2012-Tailored GOCE omission Canada Atlantic 6.5 0.4 6.9 US Atlantic 3.5 3.1 6.6 Gulf of Mexico -2.7 1.9-0.8 Canada Pacific 0.4-16.9-16.6 US Pacific 0.9-12.2-11.3 Conclusion: The omission error of the GOCE model does not average out over the PSMSL tide gauges. Given the configuration of the network of tide gauges, the tailored model accounts for the omission error of the GOCE model better than EGM2008 in the Canadian Atlantic. In the US Atlantic and Gulf of Mexico it performs similarly to EGM2008.

N max Global Spherical Harmonic Synthesis Global Spherical Harmonic Synthesis CPU time for Standard, Extended-Range and Parallelized Extended-Range computations Spacing Standard [seconds] Extended [seconds] 30 6 o 0.3 0.5 0.6 60 3 o 0.4 0.7 0.6 90 2 o 0.3 1.1 0.6 180 1 o 0.4 3.0 0.9 360 30 0.7 10.6 2.3 720 15 4.3 47.5 10.8 900 12 8.6 84.1 18.9 Parallelized Extended [seconds] 1800 6 58.1 507.4 121.6 2160 5 94.6 900.6 229.4 2700 4 179.3 1581.1 397.1 3600 3 409.6 3639.5 983.9 5400 2 1322.9 11578.8 3075.4

N max Global Spherical Harmonic Analysis Global Spherical Harmonic Analysis CPU time for Standard, Extended-Range and Parallelized Extended-Range computations Spacing Standard [seconds] Extended [seconds] 30 6 o 0.2 0.2 0.3 60 3 o 0.3 0.4 0.4 90 2 o 0.3 0.6 0.4 180 1 o 0.8 2.0 0.7 360 30 3.8 9.6 2.5 720 15 27.5 65.1 23.1 900 12 55.0 131.4 72.2 Parallelized Extended [seconds] 1800 6 920.3 1691.1 1341.0 2160 5 2172.1 3671.7 2764.2 2700 4 5866.9 9423.1 6468.2 3600 3 19733.7 28975.4 20090.2 5400 2 96716.9 144741.1 97782.5