# The relation between degree-2160 spectral models of Earth’s gravitational and topographic potential: a guide on global correlation measures and their dependency on approximation effects

## Abstract

Comparisons between high-degree models of the Earth’s topographic and gravitational potential may give insight into the quality and resolution of the source data sets, provide feedback on the modelling techniques and help to better understand the gravity field composition. Degree correlations (cross-correlation coefficients) or reduction rates (quantifying the amount of topographic signal contained in the gravitational potential) are indicators used in a number of contemporary studies. However, depending on the modelling techniques and underlying levels of approximation, the correlation at high degrees may vary significantly, as do the conclusions drawn. The present paper addresses this problem by attempting to provide a guide on global correlation measures with particular emphasis on approximation effects and variants of topographic potential modelling. We investigate and discuss the impact of different effects (e.g., truncation of series expansions of the topographic potential, mass compression, ellipsoidal versus spherical approximation, ellipsoidal harmonic coefficient versus spherical harmonic coefficient (SHC) representation) on correlation measures. Our study demonstrates that the correlation coefficients are realistic only when the model’s harmonic coefficients of a given degree are largely independent of the coefficients of other degrees, permitting degree-wise evaluations. This is the case, e.g., when both models are represented in terms of SHCs and spherical approximation (i.e. spherical arrangement of field-generating masses). Alternatively, a representation in ellipsoidal harmonics can be combined with ellipsoidal approximation. The usual ellipsoidal approximation level (i.e. ellipsoidal mass arrangement) is shown to bias correlation coefficients when SHCs are used. Importantly, gravity models from the International Centre for Global Earth Models (ICGEM) are inherently based on this approximation level. A transformation is presented that enables a transformation of ICGEM geopotential models from ellipsoidal to spherical approximation. The transformation is applied to generate a spherical transform of EGM2008 (sphEGM2008) that can meaningfully be correlated degree-wise with the topographic potential. We exploit this new technique and compare a number of models of topographic potential constituents (e.g., potential implied by land topography, ocean water masses) based on the Earth2014 global relief model and a mass-layer forward modelling technique with sphEGM2008. Different to previous findings, our results show very significant short-scale correlation between Earth’s gravitational potential and the potential generated by Earth’s land topography (correlation +0.92, and 60% of EGM2008 signals are delivered through the forward modelling). Our tests reveal that the potential generated by Earth’s oceans water masses is largely unrelated to the geopotential at short scales, suggesting that altimetry-derived gravity and/or bathymetric data sets are significantly underpowered at 5 arc-min scales. We further decompose the topographic potential into the Bouguer shell and terrain correction and show that they are responsible for about 20 and 25% of EGM2008 short-scale signals, respectively. As a general conclusion, the paper shows the importance of using compatible models in topographic/gravitational potential comparisons and recommends the use of SHCs together with spherical approximation or EHCs with ellipsoidal approximation in order to avoid biases in the correlation measures.

## Keywords

Correlation coefficient Degree correlation Reduction rate Gravitational potential Topographic potential Gravity forward modelling## Abbreviations

- SHC
spherical harmonic coefficient

- EHC
ellipsoidal harmonic coefficient

- SA
spherical approximation

- EA
ellipsoidal approximation

- STP
spherical topographic potential

- ETP
ellipsoidal topographic potential

- GGM
global gravity model

- CC
correlation coefficient

- RR
reduction rate

## Notes

### Acknowledgements

This study was supported by the German National Research Foundation (Grant Hi 1760/1) and the Institute for Advanced Study of TU Munich. We are grateful to all providers of data and models used in this study and to IAG’s ICGEM service for hosting some of the potential models used. Thanks go to Prof. Fernando Sansò and two anonymous reviewers for their comments on the manuscript. All models are available via the links provided in the document or upon request to allow replication of our results.

## References

- Ananda MP, Sjogren WL, Phillips RJ, Wimberly RN, Bills BG (1980) A low-order global gravity field of Venus and dynamical implications. J Geophys Res 85:8303–8318. doi: 10.1029/JA085iA13p08303 CrossRefGoogle Scholar
- Andersen OB (2010) The DTU10 global gravity field and mean sea surface. In: Improvements in the Arctic Presented at the 2nd gravity field symposium of IAG in Fairbanks, AlaskaGoogle Scholar
- Andersen OB, Knudsen P, Berry PAM (2010) The DNSC08GRA global marine gravity field from double retracked satellite altimetry. J Geod 84(3):191–199. doi: 10.1007/s00190-009-0355-9 CrossRefGoogle Scholar
- Balmino G, Vales N, Bonvalot S, Briais A (2012) Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J Geod 86(7):499–520. doi: 10.1007/s00190-011-0533-4 CrossRefGoogle Scholar
- Bamber JL, Griggs JA, Hurkmans RT et al (2013) A new bed elevation dataset for Greenland. Cryosphere 7:499–510CrossRefGoogle Scholar
- Becker JJ, Sandwell DT, Smith WHF et al (2009) Global bathymetry and elevation data at 30 arc seconds resolution: SRTM30_PLUS. Marine Geod 32(4):355–371. doi: 10.1080/01490410903297766 CrossRefGoogle Scholar
- Blakely RJ (1996) Potential theory in gravity and magnetic applications. Cambridge University Press, CambridgeGoogle Scholar
- Chambat F, Valette B (2005) Earth gravity up to second order in topography and density. Phys Earth Plan Int 151(1–2):89–106. doi: 10.1016/j.pepi.2005.01.002 CrossRefGoogle Scholar
- Claessens SJ (2005) New relations among associated Legendre functions and spherical harmonics. J Geod 79(6–7):398–406. doi: 10.1007/s00190-005-0483-9 CrossRefGoogle Scholar
- Claessens SJ (2006) Solutions to ellipsoidal boundary value problems for gravity field modelling. PhD thesis, Curtin University of Technology, Department of Spatial Sciences, Perth, AustraliaGoogle Scholar
- Claessens SJ, Featherstone WE (2008) The Meissl scheme for the geodetic ellipsoid. J Geod 82(8):513–522. doi: 10.1007/s00190-007-0200-y
- Claessens SJ, Hirt C (2013) Ellipsoidal topographic potential: new solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid. J Geophys Res 118(11):5991–6002. doi: 10.1002/2013JB010457 CrossRefGoogle Scholar
- Claessens SJ (2016) Spherical harmonic analysis of a harmonic function given on a spheroid. Geophys J Int 206(1):142–151. doi: 10.1093/gji/ggw126 CrossRefGoogle Scholar
- Colombo O (1981) Numerical methods for harmonic analysis on the sphere. Report 310, The Ohio State UniversityGoogle Scholar
- Forsberg R, Tscherning CC (1981) The use of height data in gravity field approximation by collocation. J Geophys Res 86(B9):7843–7854CrossRefGoogle Scholar
- Förste C, Bruinsma SL, Abrikosov O et al. (2015) EIGEN-6C4 The latest combined global gravity field model including GOCE data up to degree and order 2190 of GFZ Potsdam and GRGS Toulouse. doi: 10.5880/icgem.2015.1
- Fretwell P, Pritchard HD, Vaughan DG et al (2013) Bedmap2: improved ice bed, surface and thickness datasets for Antarctica. Cryosphere 7:375–393CrossRefGoogle Scholar
- Grombein T, Seitz K, Heck B (2016) The rock-water-ice topographic gravity field model RWI_TOPO_2015 and its comparison to a conventional rock-equivalent version. Surv Geophys 37(5):937–976. doi: 10.1007/s10712-016-9376-0 CrossRefGoogle Scholar
- Heiskanen WA, Moritz H (1967) Physical geodesy. WH Freeman, San FranciscoGoogle Scholar
- Hirt C, Reußner E, Rexer M, Kuhn M (2016) Topographic gravity modelling for global Bouguer maps to degree 2,160: validation of spectral and spatial domain forward modelling techniques at the 10 microgal level. J Geophys Res Solid Earth 121(9):6846–6862. doi: 10.1002/2016JB013249 CrossRefGoogle Scholar
- Hirt C, Rexer M (2015) Earth 2014: 1 arc-min shape, topography, bedrock and ice-sheet models–available as gridded data and degree-10,800 spherical harmonics. Int J Appl Earth Obs Geoinf 39:103–112. doi: 10.1016/j.jag.2015.03.001 CrossRefGoogle Scholar
- Hirt C (2014) GOCE’s view below the ice of Antarctica: satellite gravimetry confirms improvements in Bedmap2 bedrock knowledge. Geophys Res Lett 41(14):5021–5028. doi: 10.1002/2014GL060636 CrossRefGoogle Scholar
- Hirt C, Kuhn M (2014) A band-limited topographic mass distribution generates a full-spectrum gravity field–gravity forward modelling in the spectral and spatial domain revisited. J Geophys Res Solid Earth 119(4):3646–3661. doi: 10.1002/2013JB010900 CrossRefGoogle Scholar
- Hirt C, Kuhn M, Featherstone WE, Göttl F (2012) Topographic/isostatic evaluation of new-generation GOCE gravity field models. J Geophys Res Solid Earth 117:B05407. doi: 10.1029/2011JB008878 Google Scholar
- Hirt C, Kuhn M (2012) Evaluation of high-degree series expansions of the topographic potential to higher-order powers. J Geophys Res Solid Earth 117:B12407. doi: 10.1029/2012JB009492 Google Scholar
- Hirt C, Rexer M, Claessens SJ (2015) Topographic evaluation of fifth-generation GOCE gravity field models–globally and regionally. Newton’s Bull 5:163–186Google Scholar
- Hobson EW (1965) The theory of spherical and ellipsoidal harmonics. Chelsea Publishing Company, New York, p 500Google Scholar
- Holmes SA, Pavlis NK (2007) Some aspects of harmonic analysis of data gridded on the ellipsoid. In: Proceedings of the 1st international symposium of the international gravity field service, Gravity Field of the Earth, Istanbul, Turkey, J. Harita Dergisi, vol 73, pp 151–156 (General Command of Mapping), Ankara, TurkeyGoogle Scholar
- Hu X, Jekeli C (2015) A numerical comparison of spherical, spheroidal and ellipsoidal harmonic gravitational field models for small non-spherical bodies: examples for the Martian moons. J Geod 89(2):159–177. doi: 10.1007/s00190-014-0769-x CrossRefGoogle Scholar
- Jarvis A, Reuter HI, Nelson A, Guevara E (2008) Hole-filled SRTM for the globe v4.1. CGIAR-SXI SRTM 90m database at: http://srtm.csi.cgiar.org
- Jekeli C (1988) The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscr Geod 13:106–113Google Scholar
- Kaula WM (1992) Properties of the gravity fields of terrestrial planets. In: Colombo O (ed.) Proceedings of symposium 110 from gravity to Greenland: charting gravity with space and airborne instruments, Springer, New YorkGoogle Scholar
- Konopliv AS, Banerdt WB, Sjogren WL (1999) Venus gravity: 180th degree and order model. Icarus 139:3–18CrossRefGoogle Scholar
- Konopliv AS, Park RS, Yuan D-N et al (2013) The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission. J Geophys Res Planets 118(7):1415–1434. doi: 10.1002/jgre.20097 CrossRefGoogle Scholar
- Konopliv AS, Park RS, Folkner WM (2016) An improved JPL Mars gravity field and orientation from Mars orbiter and lander tracking data. Icarus 274:253–260. doi: 10.1016/j.icarus.2016.02.052 CrossRefGoogle Scholar
- Kuhn M, Featherstone WE, Kirby JF (2009) Complete spherical Bouguer gravity anomalies over Australia. Aust J Earth Sci 56(2):213–223. doi: 10.1080/08120090802547041 CrossRefGoogle Scholar
- Kuhn M, Hirt C (2016) Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET). J Geod 90(9):883–902. doi: 10.1007/s00190-016-0917-6 CrossRefGoogle Scholar
- Lambeck K (1976) Lateral density anomalies in the upper mantle. J Geophys Res 81(35):6333–6340. doi: 10.1029/JB081i035p06333 CrossRefGoogle Scholar
- Lee WHK, Kaula WM (1967) A spherical harmonic analysis of the Earth’s topography. J Geophys Res 72(2):753–758. doi: 10.1029/JZ072i002p00753 CrossRefGoogle Scholar
- Lemoine FG, Goossens S, Sabaka TJ et al (2014) GRGM900C: a degree-900 lunar gravity model from GRAIL primary and extended mission data. Geophys Res Lett 41(10):3382–3389. doi: 10.1002/2014GL060027 CrossRefGoogle Scholar
- Lowes FJ, Winch DE (2012) Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy. Geophys J Int 191:491–507. doi: 10.1111/j.1365-246X.2012.05590.x CrossRefGoogle Scholar
- Mazarico E, Lemoine FG, Han SC, Smith DE (2010) GLGM-3: a degree-150 lunar gravity model from the historical tracking data of NASA Moon orbiters. J Geophys Res 115:E05001. doi: 10.1029/2009JE003472 CrossRefGoogle Scholar
- McGovern PJ, Solomon SC, Smith DE et al (2002) Localized gravity/topography admittance and correlation spectra on Mars: implications, for regional and global evolution. J Geophys Res 107(12):5136. doi: 10.1029/2002JE001854
- Novák P (2010) High resolution constituents of the earth’s gravitational field. Surv Geophys 31:1. doi: 10.1007/s10712-009-9077-z CrossRefGoogle Scholar
- Pavlis N, Factor J, Holmes S (2007) Terrain-related gravimetric quantities computed for the next EGM. In: Dergisi H (ed) Proceedings of the 1st international symposium of the international gravity field service, vol 18, pp 318–323Google Scholar
- Pavlis N, Holmes S, Kenyon S, Factor J (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res Solid Earth 117:B04406. doi: 10.1029/2011JB008916 CrossRefGoogle Scholar
- Pavlis N, Holmes S, Kenyon S, Factor J (2013) Correction to the development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res Solid Earth 118(5):2633–2633. doi: 10.1002/jgrb.50167 CrossRefGoogle Scholar
- Phillips RJ, Lambeck K (1980) Gravity fields of the terrestrial planets: long-wavelength anomalies and tectonics. Rev Geophys Space Phys 18(1):27–76. doi: 10.1029/RG018i001p00027 CrossRefGoogle Scholar
- Rapp RH (1982) Degree variances of the Earth’s potential, topography and its isostatic compensation. Bull Geod 56:84–94. doi: 10.1007/BF02525594 CrossRefGoogle Scholar
- Rexer M, Hirt C (2015) Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000. J Geod 89(9):887–909. doi: 10.1007/s00190-015-0822-4 CrossRefGoogle Scholar
- Rexer M, Hirt C, Claessens SJ, Tenzer R (2016) Layer-based modelling of the Earth’s gravitational potential up to 10km-scale in spherical harmonics in spherical and ellipsoidal approximation. Surv Geophys 37(6):1035–1074. doi: 10.1007/s10712-016-9382-2 CrossRefGoogle Scholar
- Rummel R, Rapp R, Sünkel H, Tscherning C (1988) Comparisons of global topographic/isostatic models to the Earth’s observed gravity field. Report 310, Ohio State UniversityGoogle Scholar
- Sandwell DT, Smith WHF (2009) Global marine gravity from retracked Geosat and ERS-1 altimetry: Ridge segmentation versus spreading rate. J Geophys Res 114:B01411. doi: 10.1029/2008JB006008 CrossRefGoogle Scholar
- Sansò F, Sideris M (2013) Geoid determination. Lecture Notes in Earth Sciences, vol 110, Springer, Berlin, chap Harmonic Calculus and Global Gravity ModelsGoogle Scholar
- Sebera J, Bouman J, Bosch W (2012) On computing ellipsoidal harmonics using Jekeli’s renormalization. J Geod 86(9):713–726. doi: 10.1007/s00190-012-0549-4 CrossRefGoogle Scholar
- Simons M, Solomon SC, Hager BH (1997) Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus. Geophys J Int 131:24–44. doi: 10.1111/j.1365-246X.1997.tb00593.x
- Torge W, Müller J (2012) Geodesy, 4th edn. W. de Gruyter, BerlinCrossRefGoogle Scholar
- Tenzer R, Chen W, Tsoulis D et al (2015) Analysis of the refined CRUST1.0 crustal model and its gravity field. Surv Geophys 36(1):139–165. doi: 10.1007/s10712-014-9299-6 CrossRefGoogle Scholar
- Tenzer R, Abdalla A, Vajda P, Hamayun (2010) The spherical harmonic representation of the gravitational field quantities generated by the ice density contrast. Contrib Geophys Geod 40(3):207–223. doi: 10.2478/v10126-010-0009-1 Google Scholar
- Tsoulis D, Patlakis K (2013) A spectral assessment review of current satellite-only and combined Earth gravity models. Rev Geophys 51(2):186–243. doi: 10.1002/rog.20012 CrossRefGoogle Scholar
- Tscherning CC (1985) On the long-wavelength correlation between gravity and topography. In: Fifth international symposium geodesy and physics of the Earth, G.D.R. Magdeburg, 23–29 September 1984. Symposium Proceedings, edited by Kautzleben H, Veröffentlichungen des Zentralinstituts für Physik der Erde, 81(2), 134–142, Akademie der Wissenschaften der DDR, PotsdamGoogle Scholar
- Tziavos IN, Sideris MG (2013) Topographic reductions in gravity and geoid modeling. In: Lecture notes in Earth system sciences, vol 110, pp 337–400, Springer, BerlinGoogle Scholar
- Wang YM, Yang X (2013) On the spherical and spheroidal harmonic expansion of the gravitational potential of topographic masses. J Geod 87(10):909–921. doi: 10.1007/s00190-013-0654-z CrossRefGoogle Scholar
- Watts AB (2011) Isostasy. In: Gupta HK (ed) Encyclopedia of solid earth geophysics. Elsevier, Amsterdam, pp 647–662CrossRefGoogle Scholar
- Wieczorek MA, Simons FJ (2005) Localized spectral analysis on the sphere. Geophys J Int 162(3):655–675. doi: 10.1111/j.1365246X.2005.02687.x CrossRefGoogle Scholar
- Wieczorek M (2015) 10.05—gravity and topography of the terrestrial planets. In: Schubert G (ed) Treatise on geophysics, 2nd edn. Elsevier, Oxford, pp 153–193. doi: 10.1016/B978-0-444-53802-4.00169-X CrossRefGoogle Scholar
- Zuber MT, Smith DE, Watkins MM, Asmar SW, Konopliv AS, Lemoine FG, Melosh HJ, Neumann GA, Phillips RJ, Solomon SC, Wieczorek MA, Williams JG, Goossens SJ, Kruizinga G, Mazarico E, Park RS, Yuan DN (2012) Gravity field of the moon from the gravity recovery and interior laboratory (GRAIL) mission. Science 339(6120):668–671. doi: 10.1126/science.1231507 CrossRefGoogle Scholar