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Optimized formulas for the gravitational field of a tesseroid

A Comment to this article was published on 13 May 2016

Abstract

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.

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Acknowledgments

The authors would like to thank Dr. Horst Holstein and two anonymous reviewers, as well as the handling editor and the Editor-in-Chief, for their valuable comments, which helped to improve the manuscript.

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Correspondence to Thomas Grombein.

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A comment to this article is available at http://dx.doi.org/10.1007/s00190-016-0907-8.

Appendix A1

Appendix A1

Taking the \(M^*_{23}\) component of the Marussi tensor as an example, the intermediate steps used for deriving Eqs. (29) and (30) in Sect. 4 are explicitly provided in the following:

Inserting the spherical derivatives of Eqs. (26) and (27) into the relationship for \(M^*_{23}\) in Eq. (23) results in

$$\begin{aligned} M^*_{23}&= \frac{1}{r\cos {\varphi }}\,\left\{ G\rho \int \!\!\int \!\!\int _{\varOmega ^*} \frac{r^{\prime }C_\lambda }{\ell ^3}\left[ \frac{3r\left( r^{\prime }\cos {\psi }-r \right) }{\ell ^2}+1 \right] {\mathrm{d}}\varOmega \right. \nonumber \\&\qquad \left. - \frac{1}{r} G\rho \int \!\!\int \!\!\int _{\varOmega ^*} \frac{ rr^{\prime }C_\lambda }{\ell ^3} {\mathrm{d}}\varOmega \right\} . \end{aligned}$$
(67)

As both volume integrals in Eq. (67) extend over the same domain \(\varOmega ^*\), they can be combined, yielding the more simplified expression

$$\begin{aligned} M^*_{23}&= \frac{G\rho }{r\cos {\varphi }} \int \!\!\int \!\!\int _{\varOmega ^*} \Biggl \{\frac{r^{\prime }C_\lambda }{\ell ^3}\!\left[ \!\frac{3r\left( r^{\prime }\cos {\psi }\!\!-\!\!r \right) }{\ell ^2}\!\!+\!\!1\! \right] \!\!-\!\!\frac{ r^{\prime }C_\lambda }{\ell ^3}\!\Biggr \} {\mathrm{d}}\varOmega \nonumber \\&= \frac{G\rho }{r\cos {\varphi }} \int \!\!\int \!\!\int _{\varOmega ^*} \frac{3rr^{\prime }C_\lambda \left( r^{\prime }\cos {\psi }-r \right) }{\ell ^5} {\mathrm{d}}\varOmega . \end{aligned}$$
(68)

Replacing \(C_\lambda \) by its definition given in Eq. (28) the final representation is derived

$$\begin{aligned} M^*_{23} = G\rho \int \!\!\int \!\!\int _{\varOmega ^*} \frac{3r^{\prime }\cos {\varphi ^{\prime }}\sin {\delta \lambda } \left( r^{\prime }\cos {\psi }-r \right) }{\ell ^5} {\mathrm{d}}\varOmega . \end{aligned}$$
(69)

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Grombein, T., Seitz, K. & Heck, B. Optimized formulas for the gravitational field of a tesseroid. J Geod 87, 645–660 (2013). https://doi.org/10.1007/s00190-013-0636-1

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Keywords

  • Forward modeling
  • Tesseroid
  • Gravitational field
  • Newton’s integral