Abstract
This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi:10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.
Highlights
The global gravity field is typically represented using spherical harmonics (SH)
Theoretical and numerical comparisons of Stokes’s formula and leastsquares collocation (LSC) were done by de Min (1995), while a theoretical comparison of LSC and spline kernel (SK) was discussed by Eicker (2008), both of which we review and present in a unified framework
We have reviewed the theoretical equivalence of Stokes’s formula, LSC, and SKs in the global case, as well as in regional applications, where Stokes integration is restricted to a spherical cap around the computation point, and no data outside this cap is considered
Summary
The global gravity field is typically represented using spherical harmonics (SH). in regional gravity modeling, one usually splits the gravity signal into a global long-wavelength part which is modeled using SH, and a regional short-wavelength part, which is modeled using a suitable regional method (Sansò and Sideris 2013). There is a vast amount of RBFs to choose from, as long as they represent harmonic kernel functions They are versatile in that their approximation characteristics and spatial distribution can be adjusted, making it possible to use them for all kinds of data sets and for combining different types of observations (e.g., Lieb et al 2016). Regional gravity field modeling with RBFs can be done using numerical integration We aim to show that regional geoid computation with RBFs is equivalent to Stokes’s formula and LSC, in theory and in practice. The breakdown of their equivalence in regional applications is shown, and the remedial modifications of LSC and SKs to restore their equivalence to Stokes’s formula are applied.
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