Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

Abstract

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84 (3): [165][166][167][168][169][170][171][172][173][174][175][176][177][178] 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients C n,m and S n,m . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first-and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first-and second-order derivatives in terms of the geopotential coefficients. These equations are M. S. Petrovskaya (B) · A. N. Vershkov Central (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences, converted into recurrent relations from which the coefficients C n,m and S n,m are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.Keywords Terrestrial reference frame · Gravitational gradients of the first and second orders · Transformation of the gradients into geopotential models

Short guide to direct gravitational field modelling with Hotine’s equations

“…Detailed derivation of Hotine's equations can be found in Hotine (1969) or in the papers of Bettadpur (1995); Petrovskaya and Vershkov (2009). Recently, an introduction to the harmonic analysis with Hotine's equations is provided by Petrovskaya and Vershkov (2011), where the recurrences for computing the geopotential coefficients are derived. This paper is based on Cunningham (1970) instead of Hotine's analysis using a direct approach that we follow here.…”