ASSESSMENT OF THE GOCE-BASED GLOBAL GRAVITY MODELS IN CANADA

Transcription

1 ASSESSMENT OF THE GOCE-BASED GLOBAL GRAVITY MODELS IN CANADA E. Sinem Ince 1, Michael G. Sideris 1, Jianliang Huang 2, and Marc Véronneau 2 1 Department of Geomatics Engineering, The University of Calgary, Calgary, Alberta 2 Geodetic Survey Division, Natural Resources Canada, Ottawa, Ontario The aim of this study is to test the first, second and third generation GOCE geoid solutions, obtained from the first 2, 8 and 18-month observations, respectively. These solutions are assessed over Canada and for two sub-regions (the Great Lakes and Rocky Mountains). The Canadian GPS/leveling-derived geoid heights are used as independent control values in the assessment of the GOCE geoid models. The study is conducted in two steps. First, the geoid models are computed from satellite-only models and truncated to different spherical harmonic degrees. These models are compared with the GPS/leveling geoid heights which are reduced to the same spectral band as the satellite models by EGM2008 predicted frequency components higher than the truncation degrees. The results suggest that the GOCE models show a full power of signal up to about spherical harmonic degree 180. Moreover, the second and third generation GOCE models (with the exception of the direct approach models) provide better agreement with the GPS/leveling-derived geoid undulations than the first generation models due to the longer observation period. The second step involves the combination of the two third generation GOCE models with terrestrial data. These models are tested against to the GPS/leveling-derived geoid undulations in full spectrum. EGM2008 global geopotential model and Canadian gravimetric geoid model CGG2005 are also included in the comparisons to measure improvement provided by the GOCE models. The GOCE-combined models yielded GPS/leveling results that are comparable with those obtained from EGM2008 and CGG2005 models. The best comparative results with the combined models give standard deviations of 4.8 cm, 6.0 cm and 12.2 cm for the Great Lakes, Rocky Mountains and Canada, respectively. These results indicate that the third generation GOCE models conform to the Canadian terrestrial gravity data from degrees 90 to 180. The new generation models show evident improvement over the first and second generation models. Cet article présente une étude qui a pour objectif de mettre à l essai les première, deuxième et troisième générations de solutions du géoïde GOCE, tirées respectivement de l observation des premiers 2 mois, 8 mois et 18 mois. Ces solutions sont évaluées à l échelle du Canada et dans deux sous-régions (les Grands Lacs et les Rocheuses). Les ondulations du géoïde canadiennes dérivées de mesures GPS et d observations de nivellement sont utilisées comme valeurs de contrôle indépendantes dans l évaluation des modèles du géoïde GOCE. L étude est menée en deux étapes. D abord, les modèles du géoïde sont calculés à partir des modèles élaborés d après les données des satellites seulement, puis sont tronqués à différents degrés d harmoniques sphériques. Ces modèles sont comparés aux ondulations du géoïde dérivées de mesures GPS et d observations de nivellement, qui sont réduites à la même bande spectrale que les modèles issus des données des satellites par composantes de fréquences prévues par EGM2008 et supérieures au degré de troncature. Les résultats laissent entendre que les modèles GOCE exhibent un signal à pleine puissance environ jusqu au degré d harmonique sphérique 180. En outre, les modèles GOCE de deuxième et de troisième générations (à l exception des modèles d approche directe) concordent mieux avec les ondulations du géoïde dérivées de mesures GPS et d observations de nivellement que les modèles de première génération, en raison de la période d observation plus longue. La deuxième étape consiste à combiner les deux modèles GOCE de troisième génération avec les données terrestres. Ces modèles sont comparés (selon un éventail complet) aux ondulations du géoïde dérivées de mesures GPS et d observations de nivellement. Le modèle géopotentiel mondial EGM2008 et le modèle gravimétrique du géoïde canadien CGG2005 sont également inclus dans les comparaisons pour mesurer l amélioration apportée par les modèles GOCE. Les modèles GOCE combinés ont donné des résultats dérivés de mesures GPS et d observations de nivellement comparables à ceux obtenus grâce aux modèles EGM2008 et CGG2005. Les meilleurs résultats comparatifs avec les modèles combinés donnent des écarts types de 4,8 cm, 6,0 cm et 12,2 cm pour les Grands Lacs, les Rocheuses et le Canada respectivement. Ces résultats indiquent que les modèles GOCE de troisième génération concordent avec les données canadiennes de gravité terrestres des degrés 90 à 180. Les modèles de nouvelle génération montrent une amélioration manifeste par rapport aux modèles de première et deuxième générations. E. Sinem Ince seince@yorku.ca Michael G. Sideris sideris@ucalgary.ca Jianliang Huang Marc Véronneau GEOMATICA Vol. 66, No. 2, 2012 pp. 125 to 140

2 Introduction The development of a geoid-based vertical datum in North America is a high priority and a collaborative initiative between Canada, USA, and Mexico. This approach is fundamentally different from the current Canadian and American vertical datums which are constructed from leveling observations. In Canada, the official vertical datum is the Canadian Geodetic Vertical Datum of 1928 (CGVD28) [Cannon 1929]. It is realized by spirit leveling which is constrained to the mean sea level at five Canadian tide gauges on the Pacific and Atlantic coasts. CGVD28 accessibility is restricted to some 80,000 benchmarks. Among them, only a small percentage (3%) of benchmarks has been occupied with GPS. Therefore, interpolation of the GPS/leveling geoid heights between the points cannot accurately provide a continuous vertical reference surface [Véronneau and Huang 2007]. In addition, systematic errors in the leveling network, local motion of the benchmarks, limited spatial coverage and accessibility, cost and time-consuming maintenance made the Geodetic Survey Division (GSD) of Natural Resources Canada rethink the height reference system in Canada [Véronneau et al. 2006]. Considering these factors, Canada is moving ahead with the implementation of a geoid-based vertical datum by 2013, which will eventually evolve into a North American datum after the USA and Mexico also adopt a geoid-based datum. The realization of the geoid model needs to be at the centimeter level accuracy as an accessible and practical tool to convert Global Navigation Satellite Systems (GNSS) obtained heights into physical heights [Huang et al. 2007]. The contribution of the satellite-based gravity technology is one critical component to the geoid-based vertical reference frame. Moreover, the current methodology developed for regional geoid determination is expected to achieve a cm-accurate geoid model everywhere except in the highest mountains [Vaníček and Martinec 1994]. Accordingly, the recently obtained satellite-based and terrestrial datasets are the sources that need to be investigated in our experiments to develop a cm-accurate geoid model as a vertical datum in Canada. The satellite-based gravity datasets acquired from the Gravity Recovery and Climate Experiment (GRACE) mission [Tapley and Reigber 2004] have already been included in the computation of the Canadian Gravimetric Geoid Model of 2005, CGG2005 [Véronneau and Huang 2007]. Previous studies suggest that the CGG2005 has weakness in the wavelength band between 100 and 450 km where the accumulation of the systematic errors can reach the decimeter level in a few regions [Véronneau and Huang 2007]. The latest gravity mission, Gravity Field and Steady-state Ocean Circulation Explorer (GOCE) [Drinkwater et al. 2003], is expected to contribute to the information within this wavelength interval and improve the accuracy of the existing Canadian geoid model. Recent studies by Pail et al. [2010b] and Gruber et al. [2011] show that the first generation of GOCE and GOCE&GRACE models have full powers of signals up to spherical harmonic degrees 150 to 170 in Canada. It needs to be investigated whether recent releases of GOCE models can outperform the first generation of GOCE models, and improve the geoid model in Canada. With the geoid-based approach, physical heights can be obtained anywhere where the geometric heights are known by using the following relation: H = h GNSS N grav, (1) where h GNSS is the ellipsoidal height obtained by GNSS measurements and N grav is the geoid undulation obtained from a gravimetric geoid model. In principle, the discrepancy between the GPS/leveling geoid undulations (h GNSS H) and gravimetric geoid undulations (N grav ) should be zero. However, gross, random, and systematic errors and datum differences involved in the three heights (h GNSS, H and N grav ) cause discrepancies between the gravimetric geoid undulations and the ones obtained from these two independent sources, GNSS and leveling measurements [Huang et al. 2007]. In this study, GOCE-only solutions determined from the first two-, eight- and eighteen-month observations, i.e. first, second and third generation models, are assessed in Canada and two subregions (the Great Lakes and Rocky Mountains). The global geopotential model EGM2008 [Pavlis et al. 2008], Canadian geoid model CGG2005 and Canadian GPS/leveling-derived geoid undulations are used in the evaluation of the GOCE-determined gravimetric geoid models. In section 2, we describe the procedure used in the development of satelliteonly geoid solutions for different expansions of spherical harmonic degrees. The methodology used in the combination of a GOCE or GOCE and GRACE combined satellite-only model with terrestrial gravity data for the determination of high-resolution regional geoid models is described in section 3. Section 4 investigates the GOCE-only geoid models against the GPS/leveling-derived geoid undulations in both absolute and relative senses. In section 5, the third generation GOCE satellite-only models are combined with terrestrial gravity data to

3 investigate possible improvements brought by GOCE. With the newest available satellite datasets, the accuracy of the geoid models is expected to be improved. Finally, conclusions and recommendations are given in section The Development of a Satellite-Only Geoid Model Global gravity field models are used in applications such as the determination of satellite orbits, inertial navigation, and development of geophysical and geodynamic models [Torge 2001]. In the 1950 s, it was only possible to compute Earth s geopotential models with a 2500-km spatial resolution from surface gravity data [Rapp 1997]. Today, a resolution of 9 km, corresponding to a maximum spherical harmonic degree 2190, has been realized by the Earth Gravitational Model of 2008 (EGM2008) built from satellite, altimetry-derived and terrestrial gravity data. In this section, we will review the determination of gravitational geoid model from satellite-only data. The satellite-only geoid models from GOCE-only or a combination of GRACE and GOCE are computed using National Geospatial-Intelligence Agency (NGA) s Harmonic_synth_v2 software package [NGA 2008]. The methodology given in Rapp [1997] is followed in the computation of the geoid undulations from satellite models. In this methodology, height anomalies are calculated first and converted into geoid heights by applying a correction that is function of Bouguer gravity anomaly and elevations, we use the file Zeta-to-N_to2160_egm2008, which is provided by NGA. Geoid undulation obtained by this method can be expressed as follows [Heiskanen and Moritz 1967]: N = r + g B (2) where ζ is the height anomaly computed at the point given on the Earth s surface, Δg BA is the Bouguer anomaly, H is the orthometric height measured at the point, and is the averaged normal gravity between P corresponding points on the ellipsoid and the telluroid. The height anomaly mentioned here can be expressed as: r = T, r H, (3) where T is the disturbing potential computed by the expansion:, n trc n sin, n = 2 T r = GM a n r r C nm cos m S nm sin m P nm m = 0 where GM is the geocentric gravitational constant of the Earth, r is the geocentric distance to point P, a is a chosen scaling radius, Cnm and Snm are the ful- ly normalized potential coefficients of degree n and order m, and n trc is the truncation degree of the gravity field model. The Bouguer gravity anomaly in equation (2) can be obtained from: Δg BA = Δg FA 2πρGH, (5) where Δg FA is the free-air gravity anomaly computed from the global model using the relationship of the disturbing potential [Rapp 1997], G is the gravitational constant, and ρ is the density of the crust assumed to be 2670 kg/m 3. One needs to note that for efficient computation, height anomalies are computed on the ellipsoid first and then corrected to obtain their values on the Earth s surface [see Rapp, 1997]. 3. Methodology for Computing the Combined Regional Geoid Model For the realization of geoid model, the gravity field can be generally decomposed into three bands. The long wavelength components (> 100 to 400 km) are obtained from a satellite-based global geopotential model. The medium wavelength components (10 km 100 to 400 km) are obtained from the regional terrestrial gravity observations. The short wavelength components (< 10 km) are recovered from the high-resolution topography data. The long wavelength components of the gravity field can be obtained from satellite-only solutions, but they do not provide any local details. On the other hand, the terrestrial data can provide local details, but they generally contain systematic errors (e.g., datum inaccuracy) which propagate biases in the long-wavelength components. An optimum regional geoid solution can be obtained by the combination of the satellite-only solutions with terrestrial data. The method used for the combination of satellite and terrestrial gravity data, namely removecompute-restore technique and the treatment of the datasets are given below Remove-Compute-Restore Technique In this study, the remove-compute-restore technique is applied to combine different datasets to 127 (4)

4 128 develop a high resolution regional geoid model [Rapp and Rummel 1975; Mainville et al. 1992; Sideris et al. 1992]. Also, the Helmert-Stokes Scheme [Vaníček and Martinec 1994] is followed to reduce the surface gravity anomalies to the geoid and to compute geoid heights from the gravity anomalies. The procedure applied in this study is summarized as following. 1. The gridded Helmert gravity anomalies are evaluated on the geoid (see section 3.2.). 2. The long-wavelength part of the gravity signal predicted from the geopotential model up to a chosen spherical harmonic degree (see section 3.3) is removed from the Helmert gravity anomalies. Removing the model-predicted gravity anomalies reduces the low-frequency part of the gravity signal (which is recovered from a global geopotential model in step 4) and the residual gravity anomalies can be expressed by: g res = g H g H GM, (6) where the gravity anomalies predicted from the geopotential model g H GM ed by can be express- g H GM r 2 n max a n n r n 1 C nm cos m S nm sin m P nm cos, n = 2 r,, = GM m = 0 + g DTE (7) where superscript H represents the quantities modified in order to specify Helmert space. The triplet (r, θ, λ) represents the geocentric spherical coordinates of the computation point, P nm are the fully normalized associated Legen- dre functions for degree n and order m, and Cnm and Snm represent the fully normalized spherical harmonic coefficients [see section 3.3, Heiskanen and Moritz 1967]. The second term δg DTE is the direct topographical effect that transforms the gravity anomaly into Helmert s space using the spherical harmonic model of Digital Elevation Model (DEM) [Vaníček et al. 1995]. 3. The residual co-geoid undulations are obtained from the residual gravity anomalies by applying a modified Stokes s integral: N res = R 4 where S M DB g res S M DB cos d, (8) is the modified degree-banded, Stokes function [Huang and Véronneau, 2010], ψ is the spherical distance between the computation point P and the running point, γ is normal gravity computed on the ellipsoid, Δg res is the residual local gravity anomaly and dω represents surface element. The standard Stokes s integration can be used to compute geoid undulations, but one would require a global coverage of gravity anomalies. Evidently, this is a challenging task to collect such a dataset that would be homogeneous, accurate and dense enough. Thus, the integration area is limited to a spherical cap around the computation point. As given above, the effect of the neglected area is obtained from the global models. This type of integration causes a truncation error that can be reduced by using a suitable modified Stokes kernel. The optimum combination of a global satellite model and regional terrestrial data can be performed by applying a modification of the Stokes s kernel. There are many modifications to the Stokes kernel based on different optimality criteria [Vaníček and Kleusberg 1987; Meissl 1971; Vaníček and Sjöberg 1991; Featherstone et al. 1998]. Before the gravity-dedicated satellite missions (CHAMP, GRACE and GOCE), the main reason for the kernel modification was to minimize the far-zone contribution of the Stokes s integral. This was mostly determined from satellite models, which were less accurate at the time [Huang and Véronneau 2010]. Today, gravity models obtained from the new satellite missions are more accurate than the terrestrial data as for the long-wavelength components of the gravity spectrum. Accordingly, the existing modification of the Stokes s kernel needs to be revised [Huang and Véronneau 2010]. In this paper, the degree-banded Stokes kernel formulation based on deterministic method is applied because the error information from global geopotential models (GGM) and terrestrial datasets is not considered in the geoid computation. The modified degree-banded Stokes kernel is expressed as: S DB = m TG 2n + 1 n = l + 1 n 1 P n cos, (9) where l is the maximum degree of the GGM used, m TG = π / Δ, and Δ is the sampling interval of the terrestrial gravity data. Accordingly, the spectral components higher than the data sampling frequency are removed by the modification [Huang and Véronneau 2005; 2006;

5 2010]. In this type of kernel, the geoid components of degree l + 1 to m TG are completely determined from the Stokes integration. The details can be found in Huang and Véronneau [2010] and Ince [2011]. 4. Geoid undulations obtained from the global geopotential model are computed in Helmert's Earth (N H GM) by equation (10) [Heiskanen and Moritz 1967] are restored to the residual cogeoid undulations: N H GM n max n = 2 = N 0 + GM r n a n r n C nm cos m + S nm sin m P nm =0 cos + N DTE, (10) where the first term on the right side represents the zero-degree term which enables the geoid heights to be referred to a specific equipotential surface W 0 and GM value. The second term is the spherical harmonic expansion for the geoid heights (Rapp, 1997). The third term is the direct topographical effect δn DTE on the geoid [Vaníček et al., 1995]. 5. As the Earth s potential changes due to the topographic reduction process, the potential of the geoid changes too. This potential change displaces the geoid to the co-geoid. Thus, to obtain the geoid, the indirect effect, Nind [Heiskanen and Moritz 1967] of the topography is added to the co-geoid values. The complete geoid undulation is represented by: N = N H GM + N res + N ind, (11) where N H GM is the model-predicted geoid undulation for the Helmert s Earth, N res is the residual co-geoid undulation obtained from the residual gravity anomalies and Nind is the indirect topography effect determined from the 0.75" x 0.75" gridded Canadian Digital Elevation Data Terrestrial Gravity Anomaly The gravity anomalies measured on the Earth s surface are reduced to the geoid. The topography above the geoid is removed and restored by applying the Helmert s second condensation method [Vaníček and Martinec 1994]. In this study, an intermediate Bouguer Earth is applied in the development of the Helmert gravity anomalies on the geoid. The computation procedure can be listed as: a. The gravity anomalies are measured on the Earth s surface. b. All the masses between the geoid and the observation points on the Earth s surface are removed to compute refined Bouguer anomalies on the surface of the Earth. c. The refined Bouguer anomalies are reduced to the geoid by downward continuation and spherical Bouguer anomalies on the geoid are created. d. The removed topography is then restored as a condensed layer on the geoid transforming the Bouguer anomalies to Helmert anomalies on the geoid. This procedure makes the downward continuation more stable (and smaller in magnitude) than the method where the Helmert gravity anomalies are evaluated on the Earth s surface first and then downward continued to the geoid, due to the smoothness of the refined Bouguer gravity anomaly [Huang and Véronneau 2005]. The spherical Bouguer anomaly on the Earth s surface created by the third step can be expressed as: g SRB = g FA + 2 r Hg BA + A t + g SITE + g a, (12) where Δg FA is free-air gravity anomaly, the second term is a correction for the separation between the geoid and the quasi-geoid, Δg BA is simple Bouguer gravity anomaly, the third term is the attraction of the topographical masses (Bouguer shell and terrain correction) on the gravity computed on the Earth s surface [Ellmann and Vaníček 2007], the fourth term is the secondary indirect topographical effect (SITE) on gravity which is reckoned on the Earth s surface and the last term is the direct atmospheric effect [Huang and Véronneau 2005]. In Huang and Véronneau [2005], the refined Bouguer anomalies are determined at each gravity station by using 1" x 1" gridded Digital Elevation Model. Terrain corrections are evaluated for only near-zone area within a radius of 50 km. Afterwards the refined Bouguer anomalies are interpolated in 40" x 40" grid by least-squares collocation and averaged 2" x 2" grid. Datasets over the oceans are obtained from satellite altimetry derived gravity data. After this step, the far-zone contribution (effect of the gravity outside the 50 km radius) is added to the refined Bouguer anomalies. Afterwards the refined Bouguer anomalies are downward continued to the geoid as mentioned in the third step. 129

6 130 h V GM Δg SRB (r g ) Δg FA + f DC, (13) where r g represents the geocentric radius of the point on the geoid and f DC represents the downward continuation. According to Huang and Véronneau [2005], the downward continuation effect on the geoid is smaller than 0.5 m in the Rocky Mountains region and is considered in our computations. Finally, Helmert gravity anomalies on the geoid can be expressed as: Δg H = Δg SRB (r g ) A ct, (14) where Δg H represents the Helmert gravity anomaly on the geoid and A ct is the condensed topographical effect Global Gravitational Model During the process of the remove-compute-restore technique applied to develop a combined regional geoid model, the long wavelength components of the gravity signal are removed from the terrain-reduced gravity anomalies by using gravity anomalies predicted from a global geopotential model up to a certain spherical harmonic degree n max. As the surface gravity anomalies are reduced to the geoid by using Helmert s second condensation method, the other components included in the regional geoid computations are required to be in the same model setting. Therefore the GGMs also need to be transformed to Helmert s space. The Helmertization process can be performed in two ways. The gravitational model can be corrected by taking the residual gravitational potential into account first and then modified potential coefficients representing the corrected gravitational potential can be used in the computations. The other way is to add the corrections to the gravity field functionals later which has been applied in this study. The direct topographical effect of the Helmert condensation to the gravitational potential at the geoid to the maximum M degree is expressed as follows: where M = 2G n n = 0 m = n H 2 nm = 1 4 n + 2 2n + 1 H p 2 H 2 nmy nm P, Y nm d, (15) (16) are the harmonic coefficients of the squared topography. The detailed derivation of the formulations can be found in Nahavandchi and Sjöberg [1998]. The similar studies (e.g. Helmert mass conservation or Helmert masscenter conservation) can be found in Vaníček et al. [1995]; Novak [2000], and Heck [2003]. The equation (15) and its first derivative are used to compute the direct topographical effects on gravity and the geoid in Equations (7) and (10). 4. Investigation of the Satellite-Only Geoid Models 4.1. Description of the Satellite- Only Models The investigation will evaluate the following models: eight GOCE-only models (DS01, DS02, DS03, TW01, TW02, TW03, SW01, and SW02), two GRACE and GOCE combined models (GOCO01S and GOCO02S) and one GRACE-only model (ITG2010s). The first generation GOCE models (01) are obtained from only a two-month observation period whereas the second and the third generation GOCE models (02 and 03) include longer observation periods of 8 and 18 months, respectively [ESA 2010; ICGEM 2010]. The DS models are calculated by applying the direct-solution approach [Bruinsma et al. 2010]. These models extend to spherical harmonic degree and order 240 for each generation [Bruinsma et al. 2010]. A second GOCE-only solution is based on the time-wise (TW) approach [Pail et al. 2010a]. TW models [Pail et al. 2010a] are developed up to spherical harmonic degree and order 224, 250 and 250 for the first, second and the third generation, respectively. These models are independent from any other gravity field data. No a-priori background model is applied and the model can be combined with any terrestrial data, satellite-only models or altimetry-derived data [ESA 2010; ICGEM 2010]. The last GOCE-only solution is developed based on the space-wise (SW) approach [Migliaccio et al. 2010]. SW-based models [Migliaccio et al. 2010] are created up to degree and order 210 and 240 for the first and second generation models, respectively [ESA 2010; ICGEM 2010]. The third generation SW-based model was not released yet at the time of the preparation of this paper. GOCO01S and GOCO02S [Pail et al. 2010b] are also satellite-only gravity field models developed based on different combination of GOCE, GRACE, CHAMP and SLR data. GOCO01S and GOCO02S are expanded up to spherical harmonic degree 224 and 250, respectively. According to Pail et al.

7 [2010b], comparisons done with GPS/leveling data have shown that GRACE is the most important dataset to determine the low to medium degrees whereas GOCE is a significant contributor from degree 100 and even more effective beyond degree 150 in GOCO01S. There is one GRACE-only model ITG2010s [Torsten et al. 2010] included in the assessments. It is an unconstrained static field model obtained from 7 years of GRACE data and expanded up to spherical harmonic degree 180. The description of the above mentioned models and datasets used in their development are given summarized in Table Assessment of the Absolute Agreement In this section, the GOCE-only models, the combined GOCE and GRACE combined satelliteonly models, and the GRACE-only model are tested from degree 90 to their respective maximum degrees in varying steps. This is conducted in order Table 1: Description of the satellite-only models included in the investigations. Model Max d/o Description and data used Data period Reference DS Direct solution, hybrid combined 2 months GOCE Bruinsma et al., background model EIGEN5C is applied DS Direct solution, a-priori model 8 months GOCE Bruinsma et al., ITG2010s is applied DS Direct solution, a-priori model 18 months GOCE Bruinsma et al., ITG2010s is applied TW Time-wise solution, developed from 2 months GOCE Pail et al., 2010a GOCE-only data. TW Time-wise solution, developed from 8 months GOCE Pail et al., 2010a GOCE-only data. TW Time-wise solution, developed from 18 months GOCE Pail et al., 2010a GOCE-only data. SW Space-wise solution, developed from 2 months GOCE Migliaccio et al., GOCE data and EGM SW Space-wise solution, developed from 8 months GOCE Migliaccio et al., GOCE data and EGM GOCO01S 224 GRACE and GOCE combined 7 years GRACE, Pail et al., 2010b satellite-only model. GOCE SGG, and 2 months SGG ITG2010s data are used, Kaula regularization is applied. GOCO02S 250 GRACE and GOCE combined 7 years GRACE Goiginger et al., satellite-only model. ITG2010s, 12 months SST, 2011 GOCE SST, GOCE SGG, CHAMP, 8 months SGG, and SLR data are used, Kaula 8 years CHAMP, regularization is applied. 5 years SLR ITG2010s 180 GRACE-only model. 7 years GRACE Mayer-Gürr et al., 2010 SST- Satellite-to-satellite tracking data from GOCE SGG- Satellite gravity gradiometry data from GOCE 131

8 to assess the performance of the GOCE solutions and any possible contribution to geoid modeling in Canada. The description of the comparison methodology is given in Gruber [2009 and 2011]. However, we do not apply a fitting to the GPS benchmarks or remove a mean value. Gravimetric geoids truncated at different spherical harmonic degrees (n trc = 90, 120, 150, 180, 190, 200, 210, 220, 224, 230, 240 in this paper) are compared with GPS/leveling-derived geoid undulations corrected to represent the same spectral band. This is achieved by calculating the EGM2008 contribution of geoid undulation from spherical harmonic degree n trc + 1 to 2190 and removing it from the GPS/leveling-derived geoid undulations. This procedure behaves as a low-pass filter and by this removal step both gravimetric and GPS/levelingderived geoid undulations are approximated in the same spectrum. It is a known fact that there are higher frequency components above the resolution of EGM2008 expanded up to spherical harmonic degree Gruber et al. [2011] suggest that higher frequency components of the gravity field (omission error) can be obtained by means of residual terrain model (RTM) technique. An example of German dataset is mentioned in Gruber et al. [2011] and given in Hirt [2011]. We will not include RTM based corrections in this study because the objective of this study is to investigate whether the GOCE models can improve long wavelength components of the existing models such as EGM2008. The omission error beyond degree 2190 affects all tests in the same way, thus it is not necessary here. Accordingly, the reader is reminded of considering the RTM corrections coming from the high frequency components (above spherical harmonic degree 2190) for the geoid modeling using EGM2008. Based on Jekeli s [2012] method, the RMS omission error of EGM2008 is estimated smaller than 1 cm globally, and smaller than 1.5 cm in a mountainous region of USA. In Canada, the GPS measurements on benchmarks were collected over three decades. Besides the epoch differences among the measurements, different GPS equipments, such as single and double frequency receivers, the length of the observations and observing procedures cause the precision of the ellipsoidal heights to vary [Véronneau and Huang 2007]. Their precisions range from millimeters to a few decimeters at the 95% confidence level. For the leveling data, it is also a well-known fact that there is an accumulation of systematic errors building up from the unique constraint at the fundamental benchmark at Rimouski, Québec and for the different epochs [Véronneau and Huang 2007]. The distribution of the benchmarks with the GPS observations 132 Figure 1: The distribution of the 2579 GPS/leveling benchmarks located across Canada and the differences between the geoid undulations derived from GPS/leveling and EGM2008 (degree 2190). The red rectangles highlight the two regional study areas: Rocky Mountains and Great Lakes.

9 used in this study is displayed in Figure 1. The 2579 GPS/leveling points are mostly located in the southern part of the country. The sub-regions, the Great Lakes and Rocky Mountains regions are outlined by the red rectangles in Figure 1, where 652 and 659 benchmark points are used in the comparisons, respectively. The discrepancies between the GPS/leveling-derived geoid undulations and the EGM2008 geoid heights (degree 2 to 2190) range from -90 to 10 cm. They are depicted by color-scaled points in Figure 1. The apparent east-west tilt is attributed to the accumulation of the systematic errors in the leveling [Huang et al. 2007]. The color-coded standard deviations of the misclosures between the GPS/leveling-derived geoid undulations and gravimetric geoid models from GOCE, GOCE and GRACE and GRACE up to different spherical harmonic degrees are depicted in Figures 2, 3 and 4 for Canada, the Great Lakes and the Rocky Mountains, respectively. These comparison results contain both the GPS/leveling error and the model commission error. The change with respect to the EGM2008 by using a satellite model is caused by the replacement of the EGM2008 spectral components with the counterpart components of the satellite models. In principle, a higher-accuracy satellite model reduces the EGM2008 commission error, thus leading to a better agreement in the GPS/leveling comparison, and vice versa. The change of the standard deviations from spherical harmonic degree 90 to maximums (210, 224, 240, and 250) in Figures 2 to 4 reflects the change of the satellite model commission error. Figures 2a, 3a and 4a indicate the improvement coming from the longer observation period of GOCE. The first, second and third generation models can be compared clearly in these figures. In Figures 2b, 3b and 4b the results of the recent models comparisons with GPS/leveling-derived geoid heights including GOCE latest models as well as EGM2008 are given. Using the EGM2008 as a baseline, relative quality of the satellite models with respect to EGM2008 can be assessed. Figure 2a shows comparisons of the first, second and third generation GOCE-only models in Canada evaluated on 2579 GPS benchmarks. In general, due to the hybrid background model applied in its development, DS01 has a better agreement with GPS/leveling-derived geoid undulations. Between the spherical harmonic degrees 150 and 200, the direct and time-wise solutions for the second and third generations are comparable and may provide more accurate solutions. In addition, the second and third generation models agree better with the GPS/leveling-derived geoid undulations than the Figure 2a: Standard deviations of h-h-n in meters as function of the spherical harmonic degree of the three generation of GOCE models (DS01, DS02, DS03, TW01, TW02, TW03, SW01 and SW02). The results are based on GPS/leveling data on 2579 benchmarks in Canada. Figure 2b: Standard deviations of h-h-n in meters as function of the spherical harmonic degree of the recent global gravity field models (DS03, TW03, GOCO01S, GOCO02S, ITG2010s and EGM2008). The results are based on GPS/leveling data on 2579 benchmarks in Canada. first generation models. For all models given here, the commission error increases after spherical harmonic degree around In Figure 2b, we compare DS03, TW03, GOCO01S, GOCO02S, ITG2010s and EGM2008. ITG2010s has the best agreement with GPS/leveling-derived geoid undulations for the interval band between spherical harmonic degrees 90 and , but higher than degree 150 the commission error increases steeply. DS03 has a better agreement at degree 150 than the other GOCE-based models. TW03 and GOCO02S are very similar because GOCO02S is developed based on GOCE datasets used in time-wise solution. Also, GOCO02S provides better agreement after degree 150 than GOCO01S because it includes longer period of GOCE data. 133

10 Figure 3a: Same as Figure 2a, but for the Great Lakes region performed on 652 benchmarks. Figure 3b: Same as Figure 2b, but for the Great Lakes region performed on 652 benchmarks. 134 Figure 3a shows comparisons of the same models for the Great Lakes region. In general, the second and the third generation GOCE models provide better agreement than the first generation models. However, again due to the background model used in the development of DS01, its agreement with GPS/leveling-derived geoid undulations is better in general except the degree interval between 150 and 180 where TW03 and DS03 have slightly better agreement. As it is the case for Canada, after degree the commission error of all the models increases steeply. In Figure 3b for the Great lakes, all models provide comparable results up to spherical harmonic degree around but after this point the latest models provide better agreement due to the longer GOCE observations periods included in their development. However, none of these models provide any indication of an improvement over EGM2008 in any interval of the spectrum. Figure 4a shows the results for the Rocky Mountains region. Again, all the models follow each other closely up to degree 150. The first generation time-wise and space-wise models diverge away from the other models after degree 150 due to the larger commission error in these two solutions. For the rest of the models, the analysis shows that an increasing commission error exists after degree 180. As mentioned for the other two figures, DS01 shows a better agreement due to the hybrid background model which it is referred to. Figure 4b shows comparisons of the latest models for the Rocky Mountains region. The behavior of the models follows the results of the other regions and none of the models show any improvement upon EGM2008. One needs to be aware of the fact that the scales of the figures are set differently to present the differences between the models and changes of the behavior of the models clearly. As expected in the Rocky Mountains, the standard deviation of the misclosures is larger than the standard deviation of the Great Lakes due to the rough topography. In general, the standard deviations follow a stable behavior up to degree 150, where they start showing a rapid increase. DS01 shows more stability again because of the use of the combined GRACE and terrestrial data background model EIGEN05-C [Förste et al. 2008]. The rest of the models are close to each other up to degree 180 from which their commission errors rapidly go up. As described above, GOCO01S and GOCO02S are also investigated to assess the possible improvement brought by GRACE to the GOCE measurements. Also, GRACE-only solution shows good agreement with the GPS/leveling-derived geoid undulations up to spherical harmonic degree about 150 for all the regions. This is a known fact that GRACE does not provide high-resolution solutions, but only for long-wavelength components of geopotential field. One can notice that none of these satellite models show a significant improvement over EGM2008 for any of the regions. The fact that the Canadian terrestrial gravity data have contributed to the EGM2008 from the spherical harmonic degree 90 to 2190 suggests that the GOCE models generally conform to the terrestrial data within the spectral band of degree 90 to 180, and deteriorate beyond degree Assessment of the Relative Agreement For the evaluation of the relative agreement of the gravimetric geoid models with GPS/leveling data, relative differences are computed based on the same spherical harmonic expansion degree, 240 of the GOCE-based models (DS03, TW03, SW02, and GOCO02S), and highest available degrees, 180 and 2190 for ITG2010s and EGM2008, respectively. The

11 results are plotted against baseline lengths. The baseline lengths are computed with an increment of 20 km among all GPS/leveling stations. Figures 5, 6 and 7 give the relative geoid undulation accuracy comparisons. In general, EGM2008 s relative accuracy is better than the satellite-only models in the three regions due to the contribution of the surface gravity data. The GOCE models show similar behavior with a systematic shift which may be due to the commission error coming from the models. Overall, the precision of EGM2008 is 0.07 to 1.12 ppm, which corresponds to 3.1 to 14.3 cm with an average of 9.5 cm relative agreement. GOCEonly model based on direct approach has precision of 0.1 to 3.31 ppm which corresponds to 8 to 30 cm in Canada. The relative error of the GOCE models shows a steady increase trend with decreasing baseline until km where a sharp increase starts. This disproportional increase likely indicates the fast deterioration of the GOCE models for baselines shorter than km. This number is inconsistent with the absolute assessment in which the GOCE models show stable performance until degree 180, equivalent to a baseline of about 117 km. This inconsistency is likely related to uneven and sparse spatial distribution of the GPS and leveling data, that results in low sampling counts for short baselines. For the Great Lakes area, EGM2008 has a precision of 0.03 to 1.08 ppm which corresponds to 3 to 8.5 cm. The results change to 0.08 to 2.93 ppm, corresponding to 8.9 to 24 cm relative agreement, with the direct approach based third generation GOCE-only model for the same region. Although the GOCE relative errors show similar behavior trend in the Great Lakes as in Canada, the increase is relatively slow compared to the Canada-wide results because of its flat topography. For the Rockies, due to the rough topography, the relative agreement is relatively worse than those for the Great Lakes and Canada. EGM2008 indicates results between 0.07 and 1.62 ppm corresponding to 3.1 to 14.6 cm relative agreement. The GOCE-only model DS03 indicates results from 0.15 to 4.45 ppm corresponding to 7.9 to 40.2 cm relative agreement. Evidently, for the Great Lakes area, the general relative agreement of the geoid models is better than it is for Canada and the Rockies due to the flat land features. Similar to the absolute accuracy case, factors such as rough topography, distribution of stations, noisy GPS and leveling data and long wavelength errors contribute to the large deviations and worse precision in the Rockies. One needs to note that the peaks of the satellite models in Figures 5-7 with the baselines of 80 to Figure 4a: Same as Fig. 2a, but for the Rocky Mountains performed on 659 benchmarks. Figure 4b: Same as Fig. 2b, but for the Rocky Mountains performed on 659 benchmarks. Figure 5: Relative undulation accuracy [ppm] as function of baseline length [km] for Canada from EGM2008, three GOCE solutions (DS03, TW03, and SW02), GOCE and GRACE model GOCO02S and GRACE-only solution ITG2010s. 135

12 100 km are likely due to the low number of sampling because of uneven and sparse spatial distribution of the GPS and leveling data. As observed in three of the figures, EGM2008 s superiority to other models is obvious. The better performance can be attributed directly to the inclusion of terrestrial gravity data in its development. 5. Investigation of the Combined Model from GOCE and Terrestrial Data Figure 6: Same as Fig. 5, but for the Great Lakes area. Figure 7: Same as Fig. 5, but for the Rocky Mountains. Figure 8: Geoid height differences between EGM2008 (expanded up to spherical harmonic degree 2190) and the DS03_180 combined solution. 136 In this section, the two third generation GOCEonly satellite models DS03 and TW03 are combined with the regional terrestrial data to analyze the possible improvement from the recent GOCE models. These two models are selected among others based on the results given in sections 4.2 and 4.3. To decide the cutting combination degree of the long wavelength components from DS03 and TW03, combined models of different spherical harmonic degree were tested. This process was done to determine the optimum combination for the satellite model and the terrestrial data. CGG2005, the Canadian gravimetric geoid model, used in the evaluation was developed by Natural Resources Canada. The GGM02-C combined GRACE model (degree and order 200) [UTEX CSR 2004] up to degree 90 defines its long wavelengths. In addition, it includes some 2.2 million gravity measurements obtained from different sources. Since the GRACE long-wavelength components have a relative precision of a few centimeters, the shortcomings occur due to the medium wavelength components along with the short wavelength which can cause decimeter level systematic errors in a few regions [Véronneau and Huang 2007]. In this study four different limits of degree and order by the Stokes kernel modification are used. The equivalent combination degrees are assigned as 90, 120, 150, and 180. The 8 final geoid models developed are named by the model name and the combination degree. For example, the final combined geoid model developed from DS03 at combination degree 90 is represented by DS03_90. The complementing gravity information is obtained from the 2 by 2 arc-minute gridded terrestrial Helmert gravity anomalies. Table 2 shows the GPS/leveling comparison results for EGM2008, CGG2005, and the 8 GOCE-combined models. It indicates the standard deviation of the misclosures whereas the standard deviations between paranthesis are given after applying a four-parameter corrector surface [Heiskanen and Moritz 1967].

13 Table 2: Comparison of EGM2008, CGG2005 and the combined GOCE models to the GPS/leveling data. The standard deviations in paranthesis are obtained after applying a 4-parameter corrector surface. Models Standard deviations (m) Canada Great Lakes Rocky Mountains EGM (0.079) (0.039) (0.063) CGG (0.086) (0.050) (0.064) DS03_ (0.084) (0.048) (0.065) DS03_ (0.079) (0.045) (0.063) DS03_ (0.076) (0.043) (0.060) DS03_ (0.074) (0.045) (0.058) TW03_ (0.085) (0.048) (0.065) TW03_ (0.080) (0.045) (0.063) TW03_ (0.076) (0.043) (0.061) TW03_ (0.075) (0.045) (0.058) According to the comparisons, the standard deviations of the combined models obtained from the modified kernel solutions range from 12.2 to 12.7 cm for Canada, 4.7 to 5.3 cm for the Great Lakes and 6.0 to 7.1 cm for the Rockies. The GOCE combined models are comparable with EGM2008 and CGG2005 in terms of their standard deviations of GPS/leveling comparisons. These results suggest that the recent GOCE models are spectrally consistent with the gravity field in Canada up to degree 180. They basically confirm the accuracy of the Canadian terrestrial gravity data for the spectral band between degree 90 and 180 for the most parts of Canada. The difference of the geoid models from EGM2008 expanded up to spherical harmonic degree 2190 and DS03_180 is depicted in Figure 8. In general, DS03_180 agrees well with EGM2008 except for some areas such as the north-east area of Canada, Yukon Territory, mountainous areas in the west, and western coastal regions and parts of Greenland where we see slight differences due to the possible contribution of GOCE or the differences of the terrestrial data included in DS03_180 with respect to EGM2008. In Huang and Véronneau [2010], different combinations of GOCE-based models and terrestrial gravity data are tested in Yukon Territory, British Columbia and Nunavut regions on 291 benchmark points. The results show that GOCE can contribute to the geoid model in the region close to cm level compared to EGM2008 [Huang and Véronneau 2010]. Also, new Canadian geoid model CGG2010 [Véronneau and Huang 2011] shows an improvement as large as centimetre or few centimetres over some regions such as the provinces of British Columbia and Alberta, Rocky Mountains, Yukon area compared to CGG2005 and EGM2008 due to the contribution of GOCE. For this study, it is to be further investigated if the differences shown in Figure 8 correspond to improvements coming from GOCE data too. 6. Conclusions and Recommendations In this study first three generation GOCE-only models developed by three different approaches are tested with respect to GPS/leveling-derived geoid undulations in Canada and two sub-regions, the Great Lakes and Rocky Mountains. Two generations of GOCE and GRACE combined models as well as the latest GRACE-only model is included in the investigations to measure their contributions and differences to GOCE-only models in the regions. The starting point of this study is the requirement of a precise geoid model to be used as a vertical reference surface in transforming the GPS 137

14 138 ellipsoidal heights into physical heights in Canada. A combined geoid model from the recent satellite models and Canadian terrestrial data was aimed to be developed in an optimal way by the help of the improvement achieved after the satellite gravity missions. The new combined regional gravimetric geoid models are developed from the third generation DS and TW GOCE models and Canadian terrestrial data for different combination degrees. These are tested by means of comparison with the GPS/leveling-derived geoid undulations and the latest global geopotential model EGM2008 and the Canadian geoid model CGG2005. The GOCE-only solutions expanded up to different spherical harmonic degrees are compared with the GPS/leveling-derived geoid heights for the same spectral band. The accuracy assessment shows that the GOCE-only models agree well with EGM2008 for the degrees lower than 180 in Canada, the Great Lakes area and as well as in the Rocky Mountains. It is shown that the GOCE models generally conform to the accuracy of terrestrial data within the spectral band of degree 90 to 180, and deteriorate beyond degree 180. Moreover, our comparisons confirm that the new generation models developed by using longer observation series provide more accurate models than the first two month observation based models. Assessment of the GOCE-only solutions in relative sense suggests that the GOCE commission error increases steadily with decreasing baseline until km where a sharp increase starts. This disproportional increase indicates inconsistency with the results in absolute sense, and implies likely insufficient sampling counts for short baselines. The combined models using the GOCE satellite-only models: DS03 and TW03 from degrees 90 to 180 and regional Helmert gravity anomalies created for Canada suggests 12.2 cm agreement at best with the GPS/leveling derived geoid undulations on 2579 benchmarks. This becomes 4.8 cm for the Great Lakes area and 6.0 cm for the Rockies. They suggest that the GOCE models are consistent with the Canadian terrestrial gravity data from degrees 90 to 180. In general, it is known that GOCE models do not provide accurate information for the lower degree components of the gravity field. Accordingly, GRACE-based models and/or data from other geodetic techniques (e.g., SLR) should be incorporated in the regional geoid solutions. However, the third generation of the GOCE-only models performs only slightly poorer in Canada than the latest GRACE model, ITG2010s, below degree 120 in terms of the comparisons against the GPS and leveling data in Canada. The best static geoid model for Canada can be developed by using the most recent and accurate satellite and terrestrial datasets available. The possible satellite model to be used in the combined regional geoid model can be a product of combination of GRACE, GOCE and other geodetic techniques, rather than GOCE-only model. Also, it is suggested to use another kind of independent datasets in the evaluation of the geoid models. It is known that many of the Canadian GPS/leveling benchmarks deteriorate rapidly, are not stable anymore, and the land movement due to the post-glacial rebound needs to be considered in the comparisons. Accordingly, other independent datasets not included in the development of the geoid model (e.g., airborne deflections of the vertical data) can provide additional information for the validation procedure of the existing and future Canadian geoid models. References Bruinsma, S.L., J.C. Marty, G. Balmino, R. Biancale, C. Förste, O. Abrikosov, and H. Neumayer GOCE gravity field recovery by means of the direct numerical method, presented at the ESA Living Planet Symposium, June 27-2 July, 2010, Bergen, Norway; See also: earth.esa.int/goce. Cannon, J. B Adjustments of the precise level net of Canada 1928, Publication No. 28, Geodetic Survey Division, Earth Sciences Sector, Natural Resources Canada, Ottawa, Canada. Drinkwater, M.R., R. Floberghagen, R. Haagmans, D. Muzi, and A. Popescu GOCE: ESA s first Earth Explorer Core mission, In Beutler, G.B., Drinkwater M., Rummel R., and von Steiger R. (Eds.), Earth Gravity Field from Space from Sensors to Earth Sciences, In the Space Sciences Series of ISSI, vol. 18, pp , Kluwer Academic Publishers, Dordrecht, Netherlands, ISBN: ESA ESA (European Space Agency)-Living Planet Programme, The Gravity Field and Steady- State Ocean Circulation Explorer (GOCE) mission website ( Ellmann, A. and P. Vaníček UNB application of Stokes-Helmert s approach to geoid computation, Journal of Geodynamics, vol. 43, pp Featherstone, W.E., J.D. Evans, and J.G. Olliver A Meissl-modified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations, Journal of Geodesy, 72(3), pp Förste, C., F. Flechtner, R. Schmidt, R. Stubenvoll, M. Rothacher, J. Kusche, H. Neumayer, R. Biancale. J-M. Lemoine, F. Barthelmes, S. Bruinsma, R. Koenig, and Ul. Meyer EIGEN-GL05C - A