Applying an Expansion of the Second Derivatives of the Earth’s Gravitational Potential in Modified Spherical Harmonics for Constructing and Estimating Its Models

Abstract

Second-order derivatives of the Earth’s potential in a local north-oriented coordinate system are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. Based on these relations, recurrent procedures are developed for estimating the geopotential coefficients from the spectrum of each derivative and, conversely, for simulating the spectrum from a known geopotential model. The very simple structure of the expressions for the derivatives is convenient for estimating the coefficients of the geopotential by the least squares method at a certain step of processing satellite gradiometry data. Since the new series are orthogonal, the method with a quadrature formula can be applied, which helps avoid aliasing errors caused by the truncation of the series. The spectral coefficients of the derivatives are estimated using the derived relations for different models on an average orbital sphere of the GOCE satellite and at other altitudes above the Earth’s surface.