A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation

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1 A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation HASAN YILDIZ * General Command of Mapping, Tip Fakultesi Cad., Dikimevi, Ankara, 06100, Turkey (hasan.yildiz@hgk.msb.gov.tr) Received: June 6, 2011; Revised: August 1, 2011; Accepted: September 9, 2011 ABSTRACT Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies ( ), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 Eötvös unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE with standard deviations of 11 mgal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately from GOCE and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However, using covariance function obtained from GOCE has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available, if error estimates are needed. Ke y wo r d s: GOCE vertical gravity gradient, covariance function, residual terrain modelling. * The manuscript solely reflects the personal views of the author and does not necessarily represent the views, positions, strategies or opinions of Turkish Armed Forces. Stud. Geophys. Geod., 56 (2012), , DOI: /s Inst. Geophys. AS CR, Prague

2 H. Yildiz 1. INTRODUCTION The GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite (Drinkwater et al., 2003) was successfully launched on 17 March GOCE is the first satellite mission that observes gradients of the Earth gravity field from space. The GOCE satellite observes gravity gradients with accelerometers over short baselines on a platform flying in drag-free mode. These gradient measurements are primarily used for the determination of high precision global gravity field models (Pail et al., 2011). The goal of GOCE global fields in terms of geoid is 1 2 cm. Wavelength is 100 km and longer. However, the use of densely distributed GOCE gradients may provide better information for the determination of regional gravity field and can be used to predict surface gravity anomalies in unsurveyed areas of the Earth or in mountains, where gravity usually is collected only along the roads in valleys (Tscherning, 2001) as well as to derive quasigeoid heights in regions where surface gravity measurements are not available. Tscherning et al. (1990) investigated different methods of regional gravity field modelling from satellite gravity gradiometer (SGG) data. They studied the Fourier, integral and least-squares collocation (LSC) methods and concluded that removing the long wavelength structure of the gravity field and the residual topography would decorrelate the data and thus reduce the error of the estimated quantities. Among these methods, LSC is the most general tool as it can take into account data located at different altitudes through the use of a spatial covariance function and provides the error of the predicted quantities. Arabelos and Tscherning (1990) studied recovery of the terrestrial gravity anomalies using LSC with simulated SGG data obtained from a high degree and order spherical harmonic expansion and concluded that gravity anomalies can be recovered with an error of 8.3 mgal and the geoid within 33 cm in a test area, surrounding the Alps. Arabelos and Tscherning (1995) further investigated the recovery from SGG data in the same test area in comparison with real surface gravity data considering the topographic information. They found a standard deviation of 22 mgal between the LSC predicted and the observed data when the low frequency part from OSU91A geopotential model to degree 50 are removed from the simulated SGG data. When the terrain effects computed from the residual terrain modelling (RTM) method (Forsberg, 1984) using the ETOPO5 at 5 spacing were also removed, a 17 mgal standard deviation was found, concluded as a smaller improvement than expected due to errors in ETOPO5 topographic values (Arabelos and Tscherning, 1995). Recently, Tscherning and Arabelos (2011) investigated the gravity anomaly recovery from real GOCE gradient data using LSC in different areas of the Earth, where gravity data have been used by Arabelos and Tscherning (2010) for the assessment of the Earth gravitational models. They used GOCE vertical gravity gradient anomalies ( ) and GOCE T xx gravity gradient anomalies in the Local North Oriented Frame (LNOF), where the x-axis points to north, the y-axis east and the z-axis radial outward. They removed the contribution of EGM96 (Lemoine et al., 1998) from degree 2 to 36 from the GOCE gradients. They concluded that the best results were obtained in areas with a smooth gravity field, where the difference between the LSC predicted and point surface gravity anomalies was in the order of 12 mgal. However, they found that LSC predictions obtained by removing the EGM96 from degree 2 to 36 from GOCE gradients were not 172 Stud. Geophys. Geod., 56 (2012)

3 Gravity field recovery from GOCE vertical gravity gradient better than the computations obtained from GOCE Earth geopotential models. Furthermore, Tscherning and Arabelos (2011) suggested that additional use of T xx provided a marginal improvement to the results obtained when only the were used. This is in agreement with the simulation studies (Arabelos and Tscherning, 1990, 1995). On the other hand, GOCE vertical gravity gradient observations are the least affected by the transformation from the gradiometer reference frame to the LNOF defining a local geographical coordinate frame. Therefore, only the GOCE vertical gravity gradient data are used in this study. The main motivation in the present study is to investigate whether using reduced GOCE, obtained by removing the long wavelength part from an Earth geopotential model (EGM) and the short wavelength part due to terrain effects, provide better predictions of the regional gravity field than those derived from GOCE EGMs in Auvergne test area. The performance of using covariance functions created separately from GOCE and terrestrial free air gravity anomalies for the regional gravity field recovery is also investigated. In Section 2, GOCE and ground truth data are described. Section 3 includes the methodology and numerical studies in Auvergne test area, and, finally, in Section 4 some conclusions are presented. 2. DATA 2.1. GOCE GOCE mission observes gravity gradients in the satellite sensor frame, the Gradiometer Reference Frame (GRF). However, gravity field quantities are derived by LSC in LNOF. Therefore, GOCE Level-2 calibrated vertical gravity gradients in the LNOF are used in this study, provided in monthly files with the product name EGG_TRF_2 (Gruber et al., 2010) by the GOCE Virtual Online Archive ( GOCE EGG_TRF_2 gradient data are obtained by a direct point-wise rotation of the GOCE gravity gradients from the GRF to the LNOF (Bouman, 2007). Four of the GOCE gravity gradients have high accuracy in the Measurement Bandwidth (MBW), whereas the other two (V xy and V yz ) are less accurate. A direct pointwise rotation without any additional processing would project the larger error of the less accurate gravity gradients into the accurate gravity gradients in the LNOF, which is undesirable (Gruber et al., 2010). Therefore, the GOCE V xy and V yz gradients are replaced by gravity gradients computed from a GOCE quick-look gravity field model (Gruber et al., 2010). Moreover, the increasing error for the long wavelengths of the GOCE gravity gradients may leak into the MBW due to the point-wise rotation (Bouman, 2007; Gruber et al., 2010). In order to prevent this leakage, all six gravity gradients (4 directly from GOCE and 2 from a GOCE quick-look) are high-pass filtered such that the signal in and above the MBW is kept and the gravity gradient signal below the MBW is replaced by model gravity gradients using a state-of-the-art a priori global gravity model which is expected to be accurate at long wavelengths (Gruber et al., 2010). Stud. Geophys. Geod., 56 (2012) 173

4 H. Yildiz Fig. 1. GOCE vertical gravity gradient anomalies ( ): a) without correction, b) after EGM2008 to degree and order 60 and with RTM effects removed. The unit is E.U. (Eötvös unit = 10 9 s 2 ). GOCE EGG_TRF_2 vertical gravity gradient data covering a period of 13 months are used. GOCE gradient data only of good quality are used taking into account flags for outliers (Gruber et al., 2010). All available gradient data are used, overlapping the test area by one degree both in latitude and longitude. A total of vertical gravity gradient data are used in an area with the borders of N and 2 W to 8 E. The data downloaded from GOCE Virtual Online Archive are the vertical gravity gradients of the Earth potential (V). The vertical gravity gradient anomalies ( ) corresponding to the anomalous potential (T) are obtained from T = V U, where U is the GRS80 normal potential. The scatter and the variability of the are shown in Fig. 1a. In order to compare the predicted regional gravity field quantities by LSC using GOCE data GOCE EGMs based on three different approaches: the direct approach (up to degree and order (d/o) 240), the space-wise approach (d/o = 240) and the time-wise approach (d/o = 250) (Pail et al., 2011) are used Ground Truth Data The best comparison data set for geoid determination methods in the mountains was distributed by IGN, France, on behalf of the International Geoid Service (IGeS), as a ground truth example for precise geoid determination methods (Duquenne, 2007). This data set consist of about gravity data points from the Bureau Gravimetrique International (Duquenne, 2007), covering a 6 8 area including most of France and a set of 75 GPS/leveling points in the central area, all with first order leveling connections, and a quoted GPS ellipsoidal height accuracy of 2 3 cm. (Fig. 2a). The 174 Stud. Geophys. Geod., 56 (2012)

5 Gravity field recovery from GOCE vertical gravity gradient Fig. 2. a) Distribution of GPS/leveling data (black triangles) with ETOPO1 elevation model (in m) in the background. b) Terrestrial free-air gravity anomalies (in mgal). elevations of the GPS/leveling points range from 206 to 1235 m, and the highest mountain in the central area is 1886 m; it is therefore a relatively moderate mountainous area. In total point free air gravity anomalies are selected from the Auvergne gravity dataset (Duquenne, 2007) at resolution in the area with the borders of N and 1 W to 7 E. The accuracy of gravity values was evaluated as mgal by Duquenne (2007). The free air gravity anomaly data having a standard deviation of 23.8 mgal are shown in Fig. 2b. 3. METHODOLOGY AND RESULTS The Least Squares Collocation (LSC) is a method where each observation, regardless of the data type, is treated as the value of a functional applied to the anomalous potential, and some optimal smooth least norm approximation is constructed in accordance with the observed functional values in a way that the approximation is a harmonic function (Forsberg and Tscherning, 1981). This approximation may subsequently be used for the estimation of all the gravity field components and their standard error needed for geodetic applications, such as geoid heights (Tscherning, 1982). LSC takes into account data located at different altitudes through the use of a spatial covariance function. The applied covariance model implies that the associated approximation to the anomalous gravity potential is harmonic down to the so called Bjerhammar sphere, with radius R B smaller than the mean Earth radius (R E ; the investigations of this study use spherical approximation). Stud. Geophys. Geod., 56 (2012) 175

6 H. Yildiz Empirical covariance functions are determined with the EMPCOV program of the GRAVSOFT package using a sampling interval of 2.5 and then fitted to a pre-selected model covariance function of the Tscherning-Rapp model (Tscherning and Rapp, 1974) using the COVFIT program (Knudsen, 1987) of the GRAVSOFT package (Forsberg and Tscherning, 2008). Gravity field quantities are determined by LSC, where the required auto and cross-covariance functions are computed by covariance propagation from the analytically modeled local covariance function: 2 i+ 1 N R 2 cov ( T, E P TQ) = a σi Pi ( cosψ ) i 2 rpq r = 2 i+ 1 R B A + Pi ( cos ψ ), i= N+ 1 rpq r ( i 1)( i 2)( i+ 4) (1) where P and Q are two points having a spherical distance ψ and r P, r Q, are the distances of two points from the origin, R E is the mean Earth radius, R B is the radius of Bjerhammar Fig. 3. RTM effects at GOCE vertical gravity gradient. The unit is E.U. 176 Stud. Geophys. Geod., 56 (2012)

7 Gravity field recovery from GOCE vertical gravity gradient 2 sphere and σ i the error degree-variance related to the EGM2008 (Pavlis et al., 2008) up to degree (N) 60. The covariance parameters a (scale parameter), A (constant parameter in units of (m/s) 4 ) and the Bjerhammar radius R B are determined using an iterative nonlinear adjustment (Knudsen, 1987). The data for the use in covariance function estimation, and the subsequent collocation step, is required to be smooth with small variance and have a good statistical distribution, in order to properly interpret the error-estimates (Tscherning et al., 1990). Therefore, the long and short wavelength structures of the gravity field are removed from the free air gravity anomalies, the GOCE and the quasi-geoid heights of the GPS/leveling data. In order to account for the data outside the area where data are used, to decorrelate the data and to permit the use of spherical approximation in LSC, the EGM2008 (Pavlis et al., 2008) up to degree 60 is subtracted, as the spherical harmonic coefficients of EGM2008 below degree 61 are mainly based on GRACE satellite data. Subsequently, the terrain effects are subtracted, computed by the residual terrain modeling (RTM) method (Forsberg, 1984) using ETOPO1 (Earth topography at 1 spacing) elevation model (Amante and Eakins, 2009) over the area bounded in latitude by 35 and 57, and longtitude by 9 and 15 with respect to a mean elevation surface of 1 resolution. ETOPO1 has the advantage that it contains elevation as well as bathymetric information which is neccessary to model the ocean mass effects on the gravity field quantities in southern and the western parts of the test area (Fig. 2a) located near the Mediterranean Sea and North Atlantic Ocean, respectively. In the RTM method, only local high-frequent topographic irregularities are taken into account by referring all elevations to a smooth mean elevation surface (Forsberg and Tscherning, 1981). Terrain effects on the GOCE vertical gravity gradient are shown in Fig. 3 which demonstrates that terrain effects on the GOCE vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 E.U. in the mountainous areas (1 E.U. = 10 9 s 2 or 10 9 Gal/cm). The statistics of the data before and after the removal of the EGM2008 and RTM effects are shown in Tables 1 3 for free air gravity anomalies, GOCE and quasi-geoid heights of the GPS/leveling points, respectively. Using the RTM reduction, it is expected that the reduced gravity anomalies have a standard deviation at least equal to the 50% of the standard deviation of the original data (Arabelos and Tscherning, 1995). In our case, the reduced point free air gravity anomalies have a standard deviation of 12.4 mgal which is 52% of the standard deviation of 23.8 mgal of unreduced values, in agreement with the expectation. The reduced GOCE have a standard deviation of E.U. corresponding to 22% of the standard deviation of the unreduced values (0.17 E.U.) (Fig. 1b). Similarly, the reduced quasi-geoid heights have a standard deviation of 0.46 m, corresponding to 31% of the standard deviation of the unreduced values (1.47 m). The covariance functions can be estimated either using GOCE or free air gravity anomalies, which both are assumed to be isotropic (Tscherning and Arabelos, 2011). We determine covariance functions separately from reduced terrestrial free air gravity anomalies (Fig. 4a) and reduced GOCE (Fig. 4b). The covariance function parameters are shown in Table 4. Using a covariance function from reduced resulted in gravity anomaly variance at the Earth s surface Stud. Geophys. Geod., 56 (2012) 177

8 H. Yildiz Table 1. Statistics of terrestrial point free-air gravity anomalies (Δg). The indices EGM and RTM refer to the removal of the EGM2008 and RTM effects, respectively. Number of points is Values is mgal. Δg Δg Δ gegm Δg ΔgEGM Δ grtm Mean Std.Dev Max Min Table 2. Statistics of GOCE vertical gravity gradient anomalies ( ). Number of points is Values in E.U. T EGM zz Tzz Tzz EGM RTM Tzz Tzz Tzz Mean Std.Dev Max Min Table 3. Statistics of quasi-geoid heights (ζ) at GPS/leveling points. Number of points is 75. Values in meters. ζ ζ ζegm ζ ζegm ζrtm Mean Std.Dev Max Min ( mgal 2 ) larger than that of the reduced terrestrial gravity anomalies ( mgal 2 ) by a factor of about 2. The covariance from have a smaller depth (~1 km) to the Bjerhammer sphere than that of the covariance function obtained from terrestrial gravity anomalies. These covariance function parameters in Table 4 are used as input for GEOCOL18 program of the GRAVSOFT package (Forsberg and Tscherning, 2008). The observation error of the reduced is set to 0.01 E.U. (Arabelos and Tscherning, 1995). The results are given in Table 5 in terms of the mean and standard deviation of the predictions and differences between the terrestrial and predicted point free air gravity anomalies. Table 5 also indicates the average value of the error estimate of the quantities calculated using LSC (Moritz, 1980) in addition to the mean quasi-geoid heights. 178 Stud. Geophys. Geod., 56 (2012)

9 Gravity field recovery from GOCE vertical gravity gradient Fig. 4. Empirical data (full line) and model (dotted line) regional covariance functions of: a) residual surface gravity anomalies, b) residual GOCE obtained after the removal of the contribution of EGM2008 to degree 60 and the RTM effect. Table 4. The fitted covariance function parameters determined by reduced terrestrial free air gravity anomaly data and reduced GOCE. R E is the mean radius of the Earth and R B is the radius of Bjerhammer sphere. Region Auvergne Description of Dataset Reduced terrestrial free air gravity anomaly ( Δg ΔgEGM Δ grtm ) Reduced GOCE ( EGM RTM Tzz Tzz Tzz ) Depth of the Bjerhammer Scale Factor Sphere (R E R B ) a [km] Variance at Earth s Surface [mgal 2 ] Table 5. Results of the prediction of point free air anomalies and quasi-geoid heights from GOCE vertical gravity gradient anomalies ( ). Input Data Free Air Gravity Anomaly [mgal] Mean St.Dev. Estimated Average Error Mean Quasi-Geoid [m] St.Dev. Using covariance function obtained from point free air gravity anomalies Estimated Average Error Predictions Differences Using covariance function obtained from vertical gravity gradient anomalies Predictions Differences Stud. Geophys. Geod., 56 (2012) 179

10 H. Yildiz Fig. 5. Differences between observed and predicted point free air gravity anomalies (in mgal), using covariance functions of: a) terrestrial gravity anomalies, b) GOCE. The use of different covariance function parameters resulted in the same standard deviation (11 mgal) between the LSC predicted and the observed surface gravity anomalies. The differences between the LSC predicted and observed surface gravity anomalies at gravity points are shown in Fig. 5a,b for covariance functions of surface gravity anomalies and GOCE, demonstrating that both covariance functions give almost identical predictions at individual gravity points. The LSC estimated average error (8.7 mgal) is smaller than the standard deviation between the LSC predicted and the observed gravity anomalies (11 mgal) if the covariance function of terrestrial gravity anomalies is used. A larger average error (14.1 mgal) is obtained if the covariance function of GOCE is used. The estimated error of the predicted free air gravity anomalies using two different covariance functions is shown in Fig. 6a,b, indicating that the pattern of error estimates from different covariance functions are similar. However, the error estimates obtained using the covariance function of GOCE are larger by a factor of about 1.5 than those obtained using the covariance function of using surface gravity anomalies. This makes sense because the variance of the covariance function at Earth s surface derived from the GOCE is two times the variance derived from the surface gravity anomalies (see Table 4, last column). Standard deviations of 18 cm and 21 cm are obtained between the LSC predicted and observed quasi-geoid heights at 75 GPS/leveling points, using covariance functions of terrestrial gravity anomalies and GOCE, respectively. The LSC estimated quasi-geoid average errors are in good agreement with the standard deviations between LSC predicted and observed quasi-geoid heights. 180 Stud. Geophys. Geod., 56 (2012)

11 Gravity field recovery from GOCE vertical gravity gradient Fig. 6. Errors of the predicted point free air gravity anomalies (in mgal), using covariance functions of: a) terrestrial gravity anomalies, b) GOCE. The standard deviation of the differences between the observed and the predicted free air gravity anomalies (11 mgal) is slightly worse than estimated by Arabelos and Tscherning (1990) (8.3 mgal) in which simulated data derived from a spherical harmonic expansion are used, but better than the estimates of Arabelos and Tscherning (1995) (17 mgal), obtained from the comparison of simulated SGG data constructed by real ground gravity and topographic information with surface free air gravity anomalies. We further compare the results of the prediction using LSC with the corresponding results of the computation of the gravity anomalies using second generation GOCE SH models based on the time-wise, the direct and the space-wise approaches (Pail et al., 2011) (Table 6). For the comparison with the unreduced ground truth data (Table 6), the contributions of the EGM2008 to degree and order 60 and the RTM effects are added back to the LSC predictions from GOCE. The standard deviations between the predictions from GOCE EGMs and the unreduced ground truth data are mgal and cm for free air gravity anomalies and the quasi-geoid heights, respectively (Table 6). The results show that both the surface gravity anomalies and the quasi-geoid heights can be predicted by LSC better than values obtained using GOCE EGMs. This demonstrates that the information beyond the maximal degree of the GOCE EGMs is present in the measured gradients. In order to further investigate this issue, the GOCE are compared with corresponding data generated from the GOCE EGMs using the SH coefficients up to the maximum degree and order of each model (Table 7). The minimum and maximum and the standard deviation of the differences between the GOCE and those generated from GOCE EGMs are 0.049, and E.U. This means that there is a signal included in the GOCE data which is not included in GOCE EGMs, Stud. Geophys. Geod., 56 (2012) 181

12 H. Yildiz Table 6. Comparison of the predictions from GOCE EGMs and the LSC predictions from GOCE with the unreduced ground truth data in the test area. For the comparison with unreduced ground truth data, the contributions of the EGM2008 to degree and order 60 and the RTM effects are added back to the LSC predictions from GOCE. GOCE Method Free Air Gravity Anomaly [mgal] Quasi Geoid Heights [m] Mean Max Min Std.Dev. Mean Max Min Std.Dev. Time-wise Direct Space-wise LSC* predictions from GOCE * LSC predictions are from covariance functions obtained using point free air gravity anomalies. Table 7. Statistics of the differences between GOCE vertical gravity gradient anomalies ( ) and those obtained by second generation GOCE EGMs up to their maximal degree and order. Values are in E.U. EGM _ GOCE _ TIM, EGM _ GOCE _ DIR and EGM _ GOCE _ SPW represent generated by GOCE EGMs based on the time-wise (up to degree and order (d/o) 250), direct (up to d/o 240) and space-wise (up to d/o 240) approaches, respectively. Number of points is Mean Max Min Std.Dev EGM _ GOCE _ TIM EGM _ GOCE _ DIR EGM _ GOCE _ SPW showing the presence of information in the measured gradients beyond the maximal degree of GOCE EGMs. 4. CONCLUSIONS Using GOCE Level-2 vertical gravity gradients at satellite altitude as input data, surface free air gravity anomalies and quasi-geoid heights in Auvergne test area are predicted using Least Squares Collocation. The primary contribution of this paper is to show the value of topography in improving the accuracy of the predictions from LSC using GOCE vertical gravity gradient data. Comparison with ground truth data shows that surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE with standard deviations of 11 mgal and 18 cm. This is better than the values obtained using GOCE EGMs which provide standard deviations of mgal and cm in terms of free air gravity anomalies and quasi-geoid heights. This demonstrates that high frequency information beyond the maximal degree of the GOCE EGMs is present in the 182 Stud. Geophys. Geod., 56 (2012)

13 Gravity field recovery from GOCE vertical gravity gradient gradient measurements of GOCE in this test area and may be even more important in areas with higher mountains or around deep trenches at the ocean. Test of the performance of using covariance functions, created separately from at altitude and surface free air gravity anomalies to predict terrestrial gravity anomalies, suggest that the covariance function does not affect the prediction. This result is in agreement with Sansó et al. (2000) who suggested that the dependence of the solution of collocation on the covariance function is not very critical where the density of measurement points is high. However, the predicted formal error estimates from LSC using a covariance function of GOCE are considerably larger (by a factor of 1.5) than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE should be used with caution to determine covariance functions where surface gravity anomalies are not available. The use of an erroneous covariance function may have consequences for the quality of the error estimates. Acknowledgements: The author wishes to thank Prof. H. Duquenne of IGN for the availability of the Auvergne dataset and European Space Agency for providing the GOCE data. The author is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) for a oneyear post-doctoral scholarship on a research about Precise Geoid Determination, Airborne and Satellite Gravity Field Modelling to be carried out at National Space Insitute, Technical University of Denmark. I am grateful to C.C. Tscherning and R. Forsberg for their help with the implementation of GRAVSOFT programs. The reviews by Associate Editor and two anonymous referees helped to improve the manuscript. References Amante C. and Eakins B.W., ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24, 19 pp., NOAA, Boulder, Colorado. Arabelos D. and Tscherning C.C., Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT. Bull. Geod., 64, Arabelos D. and Tscherning C.C., Regional recovery of the gravity field from satellite gradiometer and gravity vector data using collocation. J. Geophys. Res., 100(B11), Arabelos D. and Tscherning C.C., A comparison of recent Earth gravitational models with emphasis on their contribution in refining the gravity and geoid at continental or regional scale. J. Geodesy, 84, , DOI: /s z. Bouman J., Alternative Method for Rotation to TRF. GO-TN-HPF-GS Issue 1.0. ( FrameTransformation.pdf). Drinkwater M.R., Floberghagen R., Haagmans R., Muzi D. and Popescu A., GOCE: ESA s first Earth explorer core mission. In: Beutler G., Drinkwater M.R., Rummel R. and von Steiger R. (Eds.), Earth Gravity Field from Space - from Sensors to Earth Science. Space Science Series of ISSI, 17, , Kluwer, Dordrecht, The Netherlands. Duquenne H., A data set to test geoid computation methods. Proceedings of the 1st International Symposium of the International Gravity Field Service (IGFS), Istanbul, Turkey. Harita Dergisi, Special Issue 18, Forsberg R., A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modeling. Report No Department of Geodetic Science and Surveying, The Ohio State University, Colombus, OH. Stud. Geophys. Geod., 56 (2012) 183

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