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Engineering >> 2020, Volume 6, Issue 8 doi: 10.1016/j.eng.2020.05.008

A High-Resolution Earth’s Gravity Field Model SGG-UGM-2 from GOCE, GRACE, Satellite Altimetry, and EGM2008

a School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China
b Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University, Wuhan 430079, China
c School of Resources and Civil Engineering, Northeastern University, Shenyang 110004, China

Received:2019-05-16 Revised:2020-01-20 Accepted: 2020-05-26 Available online:2020-06-10

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This paper focuses on estimating a new high-resolution Earth’s gravity field model named SGG-UGM-2 from satellite gravimetry, satellite altimetry, and Earth Gravitational Model 2008 (EGM2008)-derived gravity data based on the theory of the ellipsoidal harmonic analysis and coefficient transformation (EHA-CT). We first derive the related formulas of the EHA-CT method, which is used for computing the spherical harmonic coefficients from grid area-mean and point gravity anomalies on the ellipsoid. The derived formulas are successfully evaluated based on numerical experiments. Then, based on the derived least-squares formulas of the EHA-CT method, we develop the new model SGG-UGM-2 up to degree 2190 and order 2159 by combining the observations of the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE), the normal equation of the Gravity Recovery and Climate Experiment (GRACE), marine gravity data derived from satellite altimetry data, and EGM2008-derived continental gravity data. The coefficients of degrees 251–2159 are estimated by solving the block-diagonal form normal equations of surface gravity anomalies (including the marine gravity data). The coefficients of degrees 2–250 are determined by combining the normal equations of satellite observations and surface gravity anomalies. The variance component estimation technique is used to estimate the relative weights of different observations. Finally, global positioning system (GPS)/leveling data in the mainland of China and the United States are used to validate SGG-UGM-2 together with other models, such as European improved gravity model of the earth by new techniques (EIGEN)-6C4, GECO, EGM2008, and SGG-UGM-1 (the predecessor of SGG-UGM-2). Compared to other models, the model SGG-UGM-2 shows a promising performance in the GPS/leveling validation. All GOCE-related models have similar performances both in the mainland of China and the United States, and better performances than that of EGM2008 in the mainland of China. Due to the contribution of GRACE data and the new marine gravity anomalies, SGG-UGM-2 is slightly better than SGG-UGM-1 both in the mainland of China and the United States.


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[1]  Hofmann-Wellenhof B, Moritz H. Physical geodesy. 2nd ed. Wien: Springer Science & Business Media; 2006. link1

[2]  Featherstone WE. GNSS-based heighting in Australia: current, emerging and future issues. J Spat Sci 2008;53(2):115–33. link1

[3]  Rummel R. Global unification of height systems and GOCE. In: Sideris MG, editor. Gravity, geoid and geodynamics 2000. Berlin: Springer; 2002. p. 13–20. link1

[4]  Knudsen P, Bingham R, Andersen O, Rio MH. A global mean dynamic topography and ocean circulation estimation using a preliminary GOCE gravity model. J Geod 2011;85(11):861–79. link1

[5]  McKenzie D, Yi W, Rummel R. Estimates of Te from GOCE data. Earth Planet Sci Lett 2014;399:116–27. link1

[6]  Reigber C, Lühr H, Schwintzer P. CHAMP mission status. Adv Space Res 2002;30 (2):129–34. link1

[7]  Tapley BD, Bettadpur S, Watkins M, Reigber C. The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 2004;31(9):L09607. link1

[8]  Drinkwater MR, Floberghagen R, Haagmans R, Muzi D, Popescu A. GOCE: ESA’s first Earth explorer core mission. In: Beutler G, Rummel R, Drinkwater MR, von Steiger R, editors. Earth gravity field from space—from sensors to Earth science. Dordrecht: Kluwer Academic Publishers; 2003. p. 419–32. link1

[9]  Reigber C, Jochmann H, Wünsch J, Neumayer KH, Schwintzer P. First insight into temporal gravity variability from CHAMP. In: Reigber C, Lühr H, Schwintzer P, editors. First CHAMP mission results for gravity, magnetic and atmospheric studies. New York: Springer; 2003. p. 128–33. link1

[10]  Tapley BD, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, et al. GGM02: an improved Earth gravity field model from GRACE. J Geod 2005;79(8):467–78. link1

[11]  Rummel R, Yi W, Stummer C. GOCE gravitational gradiometry. J Geod 2011;85 (11):777. link1

[12]  Förste C, Schmidt R, Stubenvoll R, Flechtner F, Meyer U, König R, et al. The GeoForschungsZentrum Potsdam/Groupe de Recherche de Gèodésie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGENGL04C. J Geod 2008;82(6):331–46. link1

[13]  Pail R, Goiginger H, Schuh WD, Höck E, Brockmann JM, Fecher T, et al. Combined satellite gravity field model GOCO01S derived from GOCE and GRACE. Geophys Res Lett 2010;37(20):L20314. link1

[14]  Mayer-Gürr T, Eicker A, Kurtenbach E, Ilk KH. ITG-GRACE: global static and temporal gravity field models from GRACE data. In: Flechtner FM, Gruber T, Güntner A, Mandea M, Rothacher M, Schöne T, editors. System Earth via geodetic-geophysical space techniques. Heidelberg: Springer; 2010. p. 159–68. link1

[15]  Jäggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L. AIUB-GRACE02S: status of GRACE gravity field recovery using the celestial mechanics approach. In: Kenyon S, Pacino MC, Marti U, editors. Geodesy for planet Earth. Heidelberg: Springer; 2012. p. 161–9. link1

[16]  Brockmann JM, Zehentner N, Höck E, Pail R, Loth I, Mayer-Gürr T, et al. EGM_TIM_RL05: an independent geoid with centimeter accuracy purely based on the GOCE mission. Geophys Res Lett 2014;41(22):8089–99. link1

[17]  Hirt C, Claessens SJ, Fecher T, Kuhn M, Pail R, Rexer M. New ultrahighresolution picture of Earth’s gravity field. Geophys Res Lett 2013;40 (16):4279–83. link1

[18]  Pavlis NK, Holmes SA, Kenyon SC, Factor JK. The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res 2012;117 (B4):B04406. link1

[19]  Förste C, Bruinsma SL, Abrikosov O, Lemoine JM, Marty JC, Flechtner F, Balmino G, Barthelmes F, Biancale R. EIGEN-6C4: the latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. 2014 [cited 2019 Mar 15]. GFZ Data Services. Available from: escidoc:1119897.

[20]  Gilardoni M, Reguzzoni M, Sampietro D. GECO: a global gravity model by locally combining GOCE data and EGM2008. Stud Geophys Geod 2016;60 (2):228–47. link1

[21]  Fecher T, Pail R, Gruber T. GOCO Consortium. GOCO05c: a new combined gravity field model based on full normal equations and regionally varying weighting. Surv Geophys 2017;38(3):571–90. link1

[22]  Pail R, Fecher T, Barnes D, Factor JF, Holmes SA, Gruber T, et al. Short note: the experimental geopotential model XGM2016. J Geod 2018;92(4):443–51. link1

[23]  Liang W, Xu X, Li J, Zhu G. The determination of an ultra-high gravity field model SGG-UGM-1 by combining EGM2008 gravity anomaly and GOCE observation data. Acta Geod Cartographica Sin 2018;47(4):425–34. Chinese. link1

[24]  Xu X, Zhao Y, Reubelt T, Tenzer R. A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models. Geod Geodyn 2017;8 (4):260–72. link1

[25]  Rapp RH, Wang YM, Pavlis NK. The Ohio state 1991 geopotential and sea surface topography harmonic coefficient models. Technical report. Columbus: The Ohio State University; 1991. link1

[26]  Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, et al. The development of the joint NASA GSFC and the national imagery and mapping agency (NIMA) geopotential model EGM96. Technical report. Washington, DC: NASA Goddard Space Flight Center; 1998. link1

[27]  Holmes SA, Pavlis NK. Some aspects of harmonic analysis of data gridded on the ellipsoid. In: Proceedings of the 1st International Symposium of the International Gravity Field Service; 2006 Aug 28–Sep 1; Istanbul, Turkey. p. 151–6. link1

[28]  Hotine M. Mathematical geodesy, ESSA, US. Washington, DC: US Department of Commerce; 1969. link1

[29]  Jekeli C. The downward continuation to the Earth’s surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomalies [dissertation]. Columbus: The Ohio State University; 1981. link1

[30]  Jekeli C. The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscr Geod 1988;13(2):106–13. link1

[31]  Gleason DM. Comparing ellipsoidal corrections to the transformation between the geopotential’s spherical and ellipsoidal spectrums. Manuscr Geod 1988;13 (2):114–29. link1

[32]  Sebera J, Bouman J, Bosch W. On computing ellipsoidal harmonics using Jekeli’s renormalization. J Geod 2012;86(9):713–26. link1

[33]  Rapp RH, Pavlis NK. The development and analysis of geopotential coefficient models to spherical harmonic degree 360. J Geophys Res 1990;95 (B13):21885–911. link1

[34]  Lu Y, Hsu HT, Jiang FZ. The regional geopotential model to degree and order 720 in China. In: Schwarz KP, editor. Geodesy beyond 2000. Heidelberg: Springer; 2000. p. 143–8. link1

[35]  Huang MT, Zhai GJ, Ouyang YZ, Lu XP, Xu GX, Wang KP. Analysis and computation of ultrahigh degree geopotential model. Acta Geod Cartographica Sin 2001;30(3):208–13. Chinese. link1

[36]  Driscoll JR, Healy DM. Computing Fourier transforms and convolutions on the 2-sphere. Adv Appl Math 1994;15(2):202–50. link1

[37]  Cruz JY. Ellipsoidal corrections to potential coefficients obtained from gravity anomaly data on the ellipsoid. Technical report. Columbus: The Ohio State University; 1986. link1

[38]  Sjöberg LE. Ellipsoidal corrections to order e2 of geopotential coefficients and Stokes’ formula. J Geod 2003;77(3–4):139–47. link1

[39]  Petrovskaya MS, Vershkov AN, Pavlis NK. New analytical and numerical approaches for geopotential modeling. J Geod 2001;75(12):661–72. link1

[40]  Claessens SJ. Solutions to ellipsoidal boundary value problems for gravity field modelling [dissertation]. Perth: Curtin University; 2006. link1

[41]  Claessens SJ. Spherical harmonic analysis of a harmonic function given on a spheroid. Geophys J Int 2016;206(1):142–51. link1

[42]  Claessens SJ, Hirt C. A surface spherical harmonic expansion of gravity anomalies on the ellipsoid. J Geod 2015;89(10):1035–48. link1

[43]  Rexer M, Hirt C, Pail R, Claessens S. Evaluation of the third- and fourthgeneration GOCE Earth gravity field models with Australian terrestrial gravity data in spherical harmonics. J Geod 2014;88(4):319–33. link1

[44]  Sandwell DT, Smith WHF. Marine gravity anomaly from Geosat and ERS1 satellite altimetry. J Geophys Res 1997;102(B5):10039–54. link1

[45]  Zhang S, Sandwell DT, Jin T, Li D. Inversion of marine gravity anomalies over southeastern China seas from multi-satellite altimeter vertical deflections. J Appl Geophys 2017;137:128–37. link1

[46]  Andersen OB, Knudsen P. Global marine gravity field from the ERS-1 and Geosat geodetic mission altimetry. J Geophys Res 1998;103(C4):8129–37. link1

[47]  Hwang C. Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea. J Geod 1998;72(5):304–12. link1

[48]  Kvas A, Behzadpour S, Ellmer M, Klinger B, Strasser S, Zehentner N, et al. ITSGGrace2018: overview and evaluation of a new GRACE-only gravity field time series. J Geophys Res 2019;124(8):9332–44. link1

[49]  Colombo OL. Numerical methods for harmonic analysis on the sphere. Technical report. Columbus: The Ohio State University; 1981. link1

[50]  Sneeuw N. Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective. Geophys J Int 1994;118 (3):707–16. link1

[51]  Hirt C, Featherstone WE, Claessens SJ. On the accurate numerical evaluation of geodetic convolution integrals. J Geod 2011;85(8):519–38. link1

[52]  Liang W, Li JC, Xu XY, Zhao YQ. Analysis of the impact on the gravity field determination from the data with the ununiform noise distribution using block-diagonal least squares method. Geod Geodyn 2016;7(3):194–201. link1

[53]  Koch KR, Kusche J. Regularization of geopotential determination from satellite data by variance components. J Geod 2002;76(5):259–68. link1

[54]  Meyer U, Jean Y, Kvas A, Dahle Ch, Lemoine JM, Jäggi A. Combination of GRACE monthly gravity fields on the normal equation level. J Geod 2019;93 (9):1645–58. link1

[55]  Jean Y, Meyer U, Jäggi A. Combination of GRACE monthly gravity field solutions from different processing strategies. J Geod 2018;92(11):1313–28. link1

[56]  Moritz H. Geodetic reference system 1980. Bull Géod 1980;54(3):395–405. link1

[57]  Chapman B, Jost G, van der Pas R. Using openMP: portable shared memory parallel programming. London: The MIT Press; 2008. link1

[58]  Gruber T, Rummel R, Abrikosov O, van Hees R, editors. GOCE level 2 product data handbook. Technical report. The European GOCE Gravity Consortium; 2010. link1

[59]  Schuh WD. Improved modelling of SGG-data sets by advanced filter strategies. Final report. Noordwijk: ESA; 2002. p. 113–81.

[60]  Fuchs MJ, Bouman J. Rotation of GOCE gravity gradients to local frames. Geophys J Int 2011;187(2):743–53. link1

[61]  Reubelt T, Austen G, Grafarend EW. Harmonic analysis of the Earth’s gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP. J Geod 2003;77(5– 6):257–78. link1

[62]  Baur O, Reubelt T, Weigelt M, Roth M, Sneeuw N. GOCE orbit analysis: longwavelength gravity field determination using the acceleration approach. Adv Space Res 2012;50(3):385–96. link1

[63]  IERS. SINEX format [Internet]. Frankfurt: IERS; 2005. Available from: https:// html.

[64]  Olgiati A, Balmino G, Sarrailh M, Green CM. Gravity anomalies from satellite altimetry: comparison between computation via geoid heights and via deflections of the vertical. Bull Geod 1995;69(4):252–60. link1

[65]  Holmes SA, Featherstone WE. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J Geod 2002;76(5):279–99. link1

[66]  Li JC, Jiang WP, Zou XC, Xu XY, Shen WB. Evaluation of recent GRACE and GOCE satellite gravity models and combined models using GPS/leveling and gravity data in China. In: Proceedings of the IAG Symposium GGHS2012; 2012 Oct 9– 12; Venice, Italy. Heidelberg: Springer; 2014. p. 67–74. link1

[67]  Milbert DG. Documentation for the GPS benchmark data set of 23-July-98. IGeS Bull 1998;8:29–42. link1

[68]  Chen JY. Tide correction should be scientifically defined in the geodetic data processing. Geomat Inf Sci Wuhan Univ 2003;28(6):633–5. Chinese. link1

[69]  He L, Chu YH, Xu XY, Zhang TX. Evaluation of the GRACE/GOCE global geopotential model on estimation of the geopotential value for the China vertical datum of 1985. Chin J Geophys 2019;62(6):2016–26. Chinese. link1

[70]  Webster R, Oliver MA. Geostatistics for environmental scientists. 2nd ed. Chichester: John Wiley & Sons Ltd.; 2007. link1

[71]  Bachmaier M, Backes M. Variogram or semivariogram? Variance or semivariance? Allan variance or introducing a new term? Math Geosci 2011;43(6):735–40. link1

[72]  Slobbe C, Klees R, Farahani HH, Huisman L, Alberts B, Voet P, et al. The impact of noise in a GRACE/GOCE global gravity model on a local. J Geophys Res Sol Ea. In press.

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