Mean kernels to improve gravimetric geoid determination based on modified Stokes's integration
Introduction
Many strategies used in gravity field modeling were developed at a time “when the goal for geoid determination was at a precision of a least one order of magnitude less than it is today, i.e., ∼10 cm or worse” (Sjöberg, 2005). For today's geoid and quasigeoid modeling at the centimeter level and better, it is therefore required to thoroughly assess and – if necessary – correct for approximations that are still inherent in the techniques used.
Regional geoid computations are often based on numerical integration of gravity anomalies using modified Stokes's integration where Stokes's integral is evaluated with some kernel modification (e.g., Featherstone, 2003, Ellmann, 2005). Inevitably, the practical evaluation of Stokes's integral is subject to approximations. This is because Stokes's integral is evaluated by numerical integration of gravity anomalies, given for small surface elements (aka blocks or cells) (cf. Heiskanen and Moritz, 1967; Torge, 2001). Not only are gravity anomalies required, but also values of the integral kernel which are “most” representative for the cells (Heiskanen and Moritz, 1967, Vaníček and Krakiwsky, 1986).
Sometimes, the integral kernel is computed at the center-of-the-cell, but this may be a coarse approximation in the neighborhood of the computation point, where the kernel changes nonlinearly across the cell (Strang van Hees, 1990). Rigorously, whole-of-the-cell means of the kernel (herein abbreviated to mean kernels) are required. These can be obtained through integration of the kernel over the cell (Vaníček and Krakiwsky, 1986). Using mean instead of center-of-cell kernels near the computation point may considerably reduce numerical integration errors (Hirt et al., in press).
For Stokes's integral and Stokes's kernel, Strang van Hees (1990) and de Min (1994) have developed approaches to compute estimates of mean kernels across the cell. Recently, Hirt et al. (in press) have combined these approaches, yielding a generalized computation procedure for mean kernels used in Stokes's integral and other geodetic convolution integrals, such as the integrals of Vening-Meinesz and Poisson (Torge, 2001) or Hotine (Hotine, 1969). However, the computation procedures and benefits of mean kernels are generally not or little addressed by scholars for modifications of Stokes's function, such as the deterministic modifications of Wong and Gore (1969), Heck and Grüninger (1987), Vaníček and Kleusberg (1987), and Featherstone et al. (1998) and the stochastic modifications of Wenzel (1982) and Sjöberg, 1984, Sjöberg, 1991. An exception is de Min (1996, p. 169) noting the use of cell mean values for the Wong and Gore (1969) kernel modification.
The aim of the present study is to demonstrate that the use of mean kernel estimates is an important issue for the accurate evaluation of Stokes's integral using modified integration kernels. Section 2 briefly summarizes the basic theory of geoid computation using Stokes's integral, modified Stokes's integration, and the Featherstone et al. (1998) kernel as an example for integral kernel modifications. Section 3 describes and compares one analytical and one numerical approach capable of providing whole-of-the-cell mean values of modified kernels. Results of closed-loop tests are presented in Section 4, revealing the benefits of mean kernels in practice. The Featherstone et al. (1998) kernel has been chosen to serve as an example; however, we consider our study representative for other kernel modifications used in practice (e.g., Featherstone, 2003, Ellmann, 2005). The present study is complementary to the paper by Hirt et al. (in press) that discusses the computation of mean kernels for geodetic convolution integrals in general and for the (unmodified) Stokes's integral in particular.
Section snippets
Stokes's integral
Stokes's integral (aka Stokes's formula) of 1849 allows computation of geoid heights N from gravity anomalies Δg which – at least theoretically – are required to be continously given for small cells of size dσ covering the whole of the Earth's surface σ (Torge, 2001):
Here R denotes the radius of the Earth and γ the normal gravity. The term S(ψ) is Stokes's function (aka Stokes's kernel), which is a function of the spherical distance ψ between the computation point P (where the
Computation of mean kernels
Whole-of-the-cell mean values of modified kernels can be obtained either through analytical or numerical integration. In the analytical integration approach, a planar approximation of the first term of Stokes's function is integrated over the area of the cell, serving as an aid in computing so-called kernel weighting factors (e.g., Strang van Hees, 1990, Featherstone and Olliver, 1997, Hirt et al.,). These are conversion factors allowing transformation from point to mean kernel values (of
Numerical closed-loop tests
Closed-loop tests were used to compare and evaluate the performance of FEO analytical/numerical mean and point kernels in RCR-based geoid computation. The recent high-resolution EGM2008 global gravity model (Pavlis et al., 2008) served as source to generate self-consistent pairs of gravity anomalies ΔgEGM2008 and geoid heights NEGM2008 using the state-of-the-art harmonic_synth spherical harmonic synthesis software (Holmes and Pavlis, 2008). Transformation of gravity anomalies ΔgEGM2008 to geoid
Conclusions and recommendations
The present study has investigated the computation of whole-of-the-cell means for the Featherstone–Evans–Olliver (1998) kernel, serving as an example of modified kernels used in gravimetric geoid determination based on Stokes's integration. We have described (i) the analytical approach to compute FEO mean kernels, adapting the concept of kernel weighting factors, and (ii) the numerical approach which is based on Gauss–Legendre quadrature. The comparison of both methods showed a satisfactory
Acknowledgments
The author thanks the Australian Research Council for funding through Discovery Project Grant DP0663020. Parts of the computations were performed using the Western Australian iVEC high-performance computational facility. The author is gratefull to Will Featherstone for his continuous support and provision of a basis version of the 1DFFT software. Sincere thanks go to Sten Claessens and Will Featherstone for productive discussions and to Robert Kingdon for helpful comments on the manuscript.
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