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Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain


The isotropic Gaussian filter has been used extensively in Gravity Recovery and Climate Experiment (GRACE) temporal gravity field solutions, and is still being applied to GRACE Follow-On products to remove high-frequency errors and improve the estimation of mass transport events on the Earth’s surface. For such applications, the only known rigorous method to calculate the spherical harmonic coefficients of an isotropic Gaussian filter is by the use of a second-order recurrence relation. As an alternative, an approximate expression is also used frequently. In this paper, we provide some additional expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain. Specifically, we derive a new recurrence relation, a closed-form expression, expressions involving modified Bessel functions of the first kind, and a new approximate expression. We also examine and compare them from a computational viewpoint. The results of our numerical investigations indicate that the new recurrence relation and the closed-form expression are unstable in a way similar to the second-order recurrence relation that has been used so far. The expressions involving modified Bessel functions, and particularly the ones using exponentially scaled modified Bessel functions, provide a simple, elegant and stable way of calculating isotropic Gaussian filter coefficients, since routines for their stable evaluation are readily available in many programming languages. Alternatively, the new approximate expression can be used, which is also stable and offers better accuracy than previous approximations.

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We thank Roelof Rietbroek and an anonymous reviewer for their remarks on the improvement of our manuscript. Financial support for this work has been provided by a grant from Canada’s Natural Sciences and Engineering Research Council (NSERC) to the second author. GRACE-FO Level 2 CSR RL06 data are obtained from the Jet Propulsion Laboratory PO.DAAC service at A MATLAB function for the calculation of isotropic Gaussian filter coefficients using all the methods described in this study is available for download at

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Correspondence to Dimitrios Piretzidis or Michael G. Sideris.

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Piretzidis, D., Sideris, M.G. Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain. Stud Geophys Geod 66, 1–22 (2022).

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  • isotropic Gaussian filter
  • Bessel functions
  • spatial filtering