Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients

Abstract

Four widely used algorithms for the computation of the Earth's gravitational potential and its first-, second- and third-order gradients are examined: the traditional increasing degree recursion in associated Legendre functions and its variant based on the Clenshaw summation, plus the methods of Pines and Cunningham--Metris, which are free from the singularities that distinguish the first two methods at the geographic poles. All four methods are reorganized with the lumped coefficients approach, which in the cases of Pines and Cunningham-Metris requires a complete revision of the algorithms. The characteristics of the four methods are studied and described, and numerical tests are performed to assess and compare their precision, accuracy, and efficiency. In general the performance levels of all four codes exhibit large improvements over previously published versions. From the point of view of numerical precision, away from the geographic poles Clenshaw and Legendre offer an overall better quality. Furthermore, Pines and Cunningham--Metris are affected by an intrinsic loss of precision at the equator and suffer from additional deterioration when the gravity gradients components are rotated into the East-North-Up topocentric reference system

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach

“…Later, in the context of synthetic [simulated] gravity field modelling, Holmes (2003) makes extensive use of firstorder Taylor expansions to continue functionals over short vertical distances, say ~100 m, from the ellipsoid to the geoid. Tóth (2005) and Keller and Sharifi (2005) use higher-order gradients in the context of satellite gradiometry, and Fantino and Casotto (2009) derived firstto third-order gradients of the gravitational potential. However, to the knowledge of the author, the use of higher-order gradients has not yet been systematically presented, investigated and applied for the accurate continuation of high-degree GGM functionals, from the ellipsoid to Earth's surface.…”

“…For the SHS expansions see, e.g., Wenzel (1985 p30f), Torge (2001), Holmes (2003, p16). Fantino and Casotto (2009) published the respective radial derivatives up to second-order, which we generalize here to arbitrary order k. …”

Evolution of the Selenopotential Model and Its Effects on the Propagation Accuracy of Orbits around the Moon

“…In addition, , are the normalized degree and order harmonic coefficients, respectively. The spherical-potential function shown in (1) can be rewritten with defined "lumped coefficients," as shown in [54][55][56] …”

“…More details regarding the force modeling method, for example, those related to implementation of the recursion algorithm and determination of the inertial acceleration due to the nonsphericity of the central body, expressed in body-fixed Cartesian coordinates, can be found in [54][55][56][57].…”

Problems and methods of calculating the Legendre functions of arbitrary degree and order

“…Since "absolute zero" destroys at the fi rst place the fnALFs which values are close to zero, there is a lag effect in its appearance, due to which H.-G. Wenzel has increased the maximum degree to M = 1900, but only for The method of S.A. Holmes and W.E. Featherstone (2002a) is widely used to compute the disturbing potential (Peng and Xia, 2004), the Bouguer and isostatic anomalies (Balmino et al, 2012), the spherical harmonic analysis and synthesis (Blais, 2008;Fantino and Casoto, 2009;Hirt, 2012), the gravitational potential of the topographic masses (Wang and Yang, 2013). GrafLab software (Bucha and Janák, 2013) for spherical harmonic synthesis contains three methods of fnALFs calculation, one of which is the method of S.A. Holmes and W.E.…”