Overview on the spectral combination of integral transformations

Geodetic boundary-value problems (BVPs) and their solutions are important tools for describing and modelling the Earth’s gravitational field. Many geodetic BVPs have been formulated based on gravitational observables measured by different sensors on the ground or moving platforms (i.e. aeroplanes, satellites). Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface integrals with Green’s kernel functions. When solving this problem for higher-order derivatives of the gravitational potential as boundary conditions, more than one solution is obtained. Solutions to gravimetric, gradiometric and gravitational curvature BVPs (Martinec 2003; Šprlák and Novák 2016), respectively, lead to two, three and four formulas. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.

In this contribution we present combinations of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green’s kernel functions by a spectral combination method. We investigate an optimal combination of different orders and directional derivatives of potential. The spectral combination method is used to combine terrestrial data with global geopotential models in order to calculate geoid/quasigeoid surface. We consider that the first-, second- and third-order directional derivatives are measured at the satellite altitude and we continue them downward to the Earth’s surface and convert them to the disturbing gravitational potential, gravity disturbances and gravity anomalies. The spectral combination method thus serves in our numerical procedures as the downward continuation technique. This requires to derive the corresponding spectral weights for the n-component estimator (n = 1, 2, … 9) and to provide a generalized formula for evaluation of spectral weights for an arbitrary N-component estimator. Properties of the corresponding combinations are investigated in both, spatial and spectral domains.