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.
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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.
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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)
Similarly, the real spherical harmonic coefficients can be converted to their complex counterparts as
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.
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.
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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
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DOI: https://doi.org/10.1007/s00190-017-1037-7