Second- and Third-Order Derivatives of the Somigliana-Pizzetti Reference Gravity Field
The computation of second- and third-order derivatives of the Somigliana-Pizzetti reference gravity field (reference gravity gradients and reference gravity field curvature values) is investigated. Closed expressions for these second- and third-order deri
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Abstract
The computation of second- and third-order derivatives of the Somigliana-Pizzetti reference gravity field (reference gravity gradients and reference gravity field curvature values) is investigated. Closed expressions for these second- and third-order derivatives are derived in spheroidal coordinates. Rigorous equations for the second-order derivatives in a local north-oriented frame are also given. It is shown that on the surface of the reference ellipsoid, these lengthy expressions can be reduced to simple elegant formulas, akin to Somigliana’s formula for the first-order derivative. Numerical results provide insight into the curvature of the reference plumb lines and spheropotential surfaces. It is shown that spheropotential surfaces up to 10,000 m in altitude differ from an oblate ellipsoid of revolution by less than 0.04 m. It is also shown that this fact can be utilised to approximate the reference gravity gradients through simple formulas. Keywords
Gravity field curvature Gravity gradients Reference potential
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Introduction
Terrestrial, airborne and space-borne observation of gravity gradients has been commonplace for many years (e.g. Völgyesi 2015, DiFrancesco et al. 2009, Rummel et al. 2011). The direct observation of rate of change of gravity gradients (gravity field curvature) has become an area of active research in recent years (e.g. Rosi et al. 2015). Gravity gradients are the second-order derivatives of the gravity potential, and the rates of change of gravity gradients are the third-order derivatives of the gravity potential. Since in geodesy the gravity potential is customarily separated into a reference potential and a disturbing potential, the computation of the second- and third-order derivatives of the reference potential is of interest. These reference poten-
tial derivatives can be computed through the spherical harmonic expansion of the reference potential (e.g. Petrovskaya and Vershkov 2010, Hamáˇcková et al. 2016). Manoussakis (2013) presents an exact method to compute the second-order derivatives of the reference potential on the surface of the reference ellipsoid, and approximately in the vicinity of the reference ellipsoid through a linear approximation. In this paper, closed expressions for the second- and thirdorder derivatives of the Somigliana-Pizzetti reference gravity field are derived in spheroidal coordinates. Rigorous, closed expressions for the second-order derivatives in a local northoriented reference frame are also provided. It is shown that on the surface of the reference ellipsoid, the lengthy expressions are reduced to simple elegant formulas. Numerical results show the approximation error in these formulas when applied at altitude, and provide insight into the curvature of the reference plumb lines and spheropotential surfaces.
S. Claessens () The Institute for Geoscience Research, School of Earth and Planetary Sciences, Curtin University, Perth, Australia e-mail: [email protected]
International Association of Geodesy Symposia, https://doi.o
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