Journal of Geodesy

, Volume 91, Issue 12, pp 1475–1489 | Cite as

The polar form of the spherical harmonic spectrum: implications for filtering grace data

  • Balaji Devaraju
  • Nico Sneeuw
Original Article


Representing the spherical harmonic spectrum of a field on the sphere in terms of its amplitude and phase is termed as its polar form. In this study, we look at how the amplitude and phase are affected by linear low-pass filtering. The impact of filtering on amplitude is well understood, but that on phase has not been studied previously. Here, we demonstrate that a certain class of filters only affect the amplitude of the spherical harmonic spectrum and not the phase, but the others affect both the amplitude and phase. Further, we also demonstrate that the filtered phase helps in ascertaining the efficacy of decorrelation filters used in the grace community.


Polar form of spherical harmonics Amplitude and phase of spherical harmonic coefficients Filtering on the sphere Low-pass filtering Gravity Recovery and Climate Experiment (GRACE



This study was initiated within the German Research Foundation (dfg) funded project “Direct Water Balance” within the special priority programme spp1257 Mass transport and mass distribution in the system Earth. The study was completed within the framework of the dfg Sonderforschungsbereich (sfb) 1128 Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q). The authors would like to thank the dfg for the financial support given to the study through the two projects. We thank the editor, associate editor and two anonymous reviewers for their constructive review, which has helped us in improving the manuscript. We thank the grace data centers for making the level-2 data publicly available. All the figures in this document were prepared with the Generic Mapping Tools (gmt) software.


  1. Berens P (2009) CircStat: a MATLAB toolbox for circular statistics. J Stat Softw 31(1):1–21. doi: 10.18637/jss.v031.i10 Google Scholar
  2. Buttkus B (2000) Spectral analysis and filter theory in applied geophysics. Springer Nat. doi: 10.1007/978-3-642-57016-2 Google Scholar
  3. Chambers DP (2006) Evaluation of new grace time-variable gravity data over the ocean. Geophys Res Lett. doi: 10.1029/2006GL027296 Google Scholar
  4. Chen Q (2015) Analyzing and modeling environmental loading induced displacements with GPS and GRACE. Ph.D. thesis, Universität Stuttgart,
  5. Davis JL, Tamisiea ME, Elsegui P, Mitrovica JX, Hill EM (2008) A statistical filtering approach for gravity recovery and climate experiment (grace) gravity data. J Geophys Res Solid Earth. doi: 10.1029/2007JB005043 Google Scholar
  6. Devaraju B (2015) Understanding filtering on the sphere : experiences from filtering GRACE data. PhD thesis, Universität Stuttgart,
  7. Devaraju B, Sneeuw N (2016) On the spatial resolution of homogeneous isotropic filters on the sphere. In: Sneeuw N, Novák P, Crespi M, Sansò F (eds) VIII Hotine-Marussi symposium on mathematical geodesy: proceedings of the symposium in Rome. Springer, Cham, pp 67–73. doi: 10.1007/1345_2015_5 Google Scholar
  8. Duan X, Guo J, Shum C, van der Wal W (2009) On the postprocessing removal of correlated errors in GRACE temporal gravity field solutions. J Geod 83(11):1095–1106. doi: 10.1007/s00190-009-0327-0 CrossRefGoogle Scholar
  9. Han SC, Shum CK, Jekeli C, Kuo CY, Wilson C, Seo KW (2005) Non-isotropic filtering of GRACE temporal gravity for geophysical signal enhancement. Geophys J Int 163:18–25. doi: 10.1111/j.1365-246X.2005.02756 CrossRefGoogle Scholar
  10. Ilk KH (1983) Ein Beitrag zur Dynamik ausgedehnter Körper: gravitationswechselwirkung. Deutsche Geodaetische Kommission Bayer Akad Wiss 288Google Scholar
  11. Kaula WM (1966) Theory of satellite geodesy: applications of satellites to geodesy. Blaisdell Publishing Company, BostonGoogle Scholar
  12. Kaula WM (1967) Theory of statistical analysis of data distributed over a sphere. Rev Geophys 5(1):83–107CrossRefGoogle Scholar
  13. King M, Moore P, Clarke P, Lavalle D (2006) Choice of optimal averaging radii for temporal GRACE gravity solutions, a comparison with GPS and satellite altimetry. Geophys J Int 166(1):1–11. doi: 10.1111/j.1365-246X.2006.03017.x CrossRefGoogle Scholar
  14. Klees R, Zapreeva EA, Winsemius HC, Savenije HHG (2007) The bias in GRACE estimates of continental water storage variations. Hydrol Earth Syst Sci 11:1227–1241CrossRefGoogle Scholar
  15. Klees R, Revtova EA, Gunter BC, Ditmar P, Oudman E, Winsemius HC, Savenije HHG (2008) The design of an optimal filter for monthly GRACE gravity models. Geophys J Int 175:417–432. doi: 10.1111/j.1365-246X.2008.03922.x CrossRefGoogle Scholar
  16. Kusche J (2007) Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models. J Geod 81:733–749. doi: 10.1007/s00190-007-0143-3 CrossRefGoogle Scholar
  17. Kusche J, Schmidt R, Petrovic S, Rietbroek R (2009) Decorrelated GRACE time-variable gravity solutions by GFZ, and their validation using a hydrological model. J Geod 83(10):903–913. doi: 10.1007/s00190-009-0308-3 CrossRefGoogle Scholar
  18. Landerer FW, Swenson SC (2012) Accuracy of scaled GRACE terrestrial water storage estimates. Water Resour Res. doi: 10.1029/2011WR011453 Google Scholar
  19. Lorenz C (2009) Applying stochastic constraints on time-variable grace data. Diplomarbeit, Institute of Geodesy, University of Stuttgart,
  20. Lorenz C, Devaraju B, Tourian MJ, Sneeuw N, Riegger J, Kunstmann H (2014) Large-scale runoff from landmasses: a global assessment of the closure of the hydrological and atmospheric water balances. J Hydrometeorol 15(6):2111–2139. doi: 10.1175/JHM-D-13-0157.1 CrossRefGoogle Scholar
  21. Marmer HA (1928) On cotidal maps. Geograph Rev 18(1):129–143CrossRefGoogle Scholar
  22. Oppenheim AV, Lim JS (1981) The importance of phase in signals. Proc IEEE 69(5):529–541. doi: 10.1109/PROC.1981.12022 CrossRefGoogle Scholar
  23. Rangelova E, van der Wal W, Braun A, Sideris MG, Wu P (2007) Analysis of gravity recovery and climate experiment time-variable mass redistribution signals over North America by means of principal component analysis. J Geophys Res 112(F03):002. doi: 10.1029/2006JF000615 Google Scholar
  24. Rummel R, Schwarz KP (1977) On the nonhomogeneity of the global covariance function. Bull Géod 51:93–103CrossRefGoogle Scholar
  25. Sasgen I, Martinec Z, Fleming K (2006) Wiener optimal filtering of GRACE data. Stud Geophys Geod 50:499–508. doi: 10.1007/s11200-006-0031-y CrossRefGoogle Scholar
  26. Seo KW, Wilson CR, Chen J, Waliser DE (2008) GRACE’s spatial aliasing error. Geophys J Int 172:41–48. doi: 10.1111/j.1365-246X.2007.03611.x CrossRefGoogle Scholar
  27. Shako R, Förste C, Abrikosov O, Bruinsma S, Marty JC, Lemoine JM, Flechtner F, Neumayer H, Dahle C (2014) EIGEN-6C: a high-resolution global gravity combination model including GOCE data. Springer, Berlin, pp 155–161. doi: 10.1007/978-3-642-32135-1_20
  28. Sneeuw NJ (1991) Inclination functions: Group theoretical background and a recursive algorithm. Tech. Rep. 91.2, Mathematical and Physical Geodesy, Faculty of Geodetic Engineering, Delft University of TechnologyGoogle Scholar
  29. Sneeuw N, Lorenz C, Devaraju B, Tourian MJ, Riegger J, Kunstmann H, Bárdossy A (2014) Estimating runoff using hydro-geodetic approaches. Surv Geophys 4:1303–1318. doi: 10.1007/s10712-014-9300-4 Google Scholar
  30. Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE). J Geophys Res 107(B9):2193. doi: 10.1029/2001JB000576 Google Scholar
  31. Swenson S, Wahr J (2006) Post-processing removal of correlated errors in GRACE data. Geophys Res Lett 33(L08):402. doi: 10.1029/2005GL025285 Google Scholar
  32. Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins MM (2004) Grace measurements of mass variability in the earth system. Science 305(5683):503–505. doi: 10.1126/science.1099192 CrossRefGoogle Scholar
  33. Varshalovich DA, Moskalev AN, Khersonskii VK (1988) Quantum theory of angular momentum. World Scientific, HackensackCrossRefGoogle Scholar
  34. Velicogna I, Wahr J (2006) Acceleration of Greenland ice mass loss in spring 2004. Nature 443:329–331. doi: 10.1038/nature05168 CrossRefGoogle Scholar
  35. Vishwakarma BD, Devaraju B, Sneeuw N (2016) Minimizing the effects of filtering on catchment scale GRACE solutions. Water Resour Res. doi: 10.1002/2016WR018960 Google Scholar
  36. Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30,205–30,229CrossRefGoogle Scholar
  37. Werth S, Güntner A, Schmidt R, Kusche J (2009) Evaluation of GRACE filter tools from a hydrological perspective. Geophys J Int 179:1499–1515. doi: 10.1111/j.1365-246X.2009.04355.x CrossRefGoogle Scholar
  38. Wouters B, Schrama EJO (2007) Improved accuracy of GRACE gravity solutions through empirical orthogonal function filtering of spherical harmonics. Geophys Res Lett 34(L23):711. doi: 10.1029/2007GL032098 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of GeodesyLeibniz Universität HannoverHannoverGermany
  2. 2.Institute of GeodesyUniversity of StuttgartStuttgartGermany

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