Skip to main content
Log in

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

Journal of Geodesy Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  • 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 

  • Buttkus B (2000) Spectral analysis and filter theory in applied geophysics. Springer Nat. doi:10.1007/978-3-642-57016-2

    Google Scholar 

  • Chambers DP (2006) Evaluation of new grace time-variable gravity data over the ocean. Geophys Res Lett. doi:10.1029/2006GL027296

    Google Scholar 

  • Chen Q (2015) Analyzing and modeling environmental loading induced displacements with GPS and GRACE. Ph.D. thesis, Universität Stuttgart, http://elib.uni-stuttgart.de/handle/11682/3995

  • 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 

  • Devaraju B (2015) Understanding filtering on the sphere : experiences from filtering GRACE data. PhD thesis, Universität Stuttgart, http://elib.uni-stuttgart.de/opus/volltexte/2016/10379

  • 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 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Ilk KH (1983) Ein Beitrag zur Dynamik ausgedehnter Körper: gravitationswechselwirkung. Deutsche Geodaetische Kommission Bayer Akad Wiss 288

  • Kaula WM (1966) Theory of satellite geodesy: applications of satellites to geodesy. Blaisdell Publishing Company, Boston

    Google Scholar 

  • Kaula WM (1967) Theory of statistical analysis of data distributed over a sphere. Rev Geophys 5(1):83–107

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Klees R, Zapreeva EA, Winsemius HC, Savenije HHG (2007) The bias in GRACE estimates of continental water storage variations. Hydrol Earth Syst Sci 11:1227–1241

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Landerer FW, Swenson SC (2012) Accuracy of scaled GRACE terrestrial water storage estimates. Water Resour Res. doi:10.1029/2011WR011453

    Google Scholar 

  • Lorenz C (2009) Applying stochastic constraints on time-variable grace data. Diplomarbeit, Institute of Geodesy, University of Stuttgart, http://elib.uni-stuttgart.de/opus/volltexte/2009/4012/pdf/Lorenz.pdf

  • 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

    Article  Google Scholar 

  • Marmer HA (1928) On cotidal maps. Geograph Rev 18(1):129–143

    Article  Google Scholar 

  • Oppenheim AV, Lim JS (1981) The importance of phase in signals. Proc IEEE 69(5):529–541. doi:10.1109/PROC.1981.12022

    Article  Google Scholar 

  • 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 

  • Rummel R, Schwarz KP (1977) On the nonhomogeneity of the global covariance function. Bull Géod 51:93–103

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

  • 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 Technology

  • 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 

  • 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 

  • 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 

  • 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

    Article  Google Scholar 

  • Varshalovich DA, Moskalev AN, Khersonskii VK (1988) Quantum theory of angular momentum. World Scientific, Hackensack

    Book  Google Scholar 

  • Velicogna I, Wahr J (2006) Acceleration of Greenland ice mass loss in spring 2004. Nature 443:329–331. doi:10.1038/nature05168

    Article  Google Scholar 

  • 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 

  • 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):30,205–30,229

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Balaji Devaraju.

Appendices

Appendix 1: Relationship between complex and real spherical harmonic coefficients

Here, we provide formulas for converting the complex spherical harmonic coefficients to real and vice versa. Further, we will also derive equations for computing the polar spectrum directly from the real spherical harmonic coefficients. The spherical harmonic coefficients are in general disseminated in their real form, and hence these formulas will facilitate the polar spectrum computation.

Converting the complex spherical harmonics \(F_{lm}\) into real spherical harmonics \((C_{lm}, S_{lm})\) can be done by (Ilk 1983; Sneeuw 1991)

$$\begin{aligned} \left. \begin{array}{l} C_{lm} \\ S_{lm} \end{array} \right\}&\,\,{}={}\,\,\displaystyle {\frac{(-1)^m}{2 \, \sqrt{2 - \delta _{0,m}}} \, } \left\{ \begin{array}{r} \displaystyle {\left( F_{lm} + F_{lm}^{*}\right) } \\ {} \displaystyle {i \, \left( F_{lm} - F_{lm}^{*} \right) } \end{array} \right. \, . \end{aligned}$$
(16)

Similarly, the real spherical harmonic coefficients can be converted to their complex counterparts as

$$\begin{aligned} F_{lm}&\,\,{}={}\,\,\displaystyle {\frac{1}{\sqrt{2 - \delta _{0,m}}}} {\left\{ \begin{array}{ll} \displaystyle {(-1)^m \, \left( C_{lm} - iS_{lm}\right) } \, , &{} m > 0\\ {} C_{lm} \, , &{} m = 0 \\ {} \displaystyle {\left( C_{lm} + iS_{lm}\right) } \, , &{} m < 0 \end{array}\right. } \, . \end{aligned}$$
(17)

From these two equations, the amplitude and phase spectra of the field can be calculated directly from the real spherical harmonic coefficients as they are the ones that are commonly disseminated.

$$\begin{aligned} A_{lm}&\,\,{}={}\,\,\displaystyle {\sqrt{\frac{C_{lm}^2 + S_{lm}^2}{2 - \delta _{0,m}}}} \end{aligned}$$
(18)
$$\begin{aligned} \phi _{lm}&\,\,{}={}\,\,\displaystyle { {\left\{ \begin{array}{ll} \arctan \left( -\displaystyle {\frac{(-1)^m \, S_{lm}}{(-1)^m \, C_{lm}}}\right) \, , &{} m > 0 \\ {} \arctan \left( \displaystyle {\frac{0}{C_{lm}}}\right) \, , &{} m = 0 \\ {} \arctan \left( \displaystyle {\frac{S_{lm}}{C_{lm}}}\right) \, , &{} m < 0 \end{array}\right. } \, . } \end{aligned}$$
(19)

Appendix 2: Polar form of the static gravity field spectrum

Since very little is known about the nature of the gravity field phase spectrum, we computed the amplitude and phase spectra of the static gravity field EIGEN6c4 (Shako et al. 2014). It is a spherical harmonic model up to maximum degree and order 2190, and it uses all the available gravity field observables. The information that we seek here is that given a signal of the gravity field, we do not expect it to display order-wise correlation, which is purely an error associated with data derived from satellite missions. This is the case of grace. However, gravity field signal that is not affected by this order-dependent errors should have the phase spectrum and order-wise phase distribution as shown in Fig. 10, which is nearly circularly uniform.

Fig. 10
figure 10

Amplitude and phase spectra of the state-of-the-art spherical harmonic gravity field model—eigen6c4. The circular histograms of the order-wise phase distribution are shown in the bottom panel. The numbers in the circular histograms indicate spherical harmonic orders, and they are arbitrarily chosen. Nevertheless, they are representative of the overall order-wise phase distribution. Please note that for illustrative purposes the model is plotted up to a maximum degree and order of 180

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Devaraju, B., Sneeuw, N. The polar form of the spherical harmonic spectrum: implications for filtering grace data. J Geod 91, 1475–1489 (2017). https://doi.org/10.1007/s00190-017-1037-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00190-017-1037-7

Keywords

Navigation