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Abstract:
Integral transformations represent a useful mathematical tool for gravitational field modelling. Basic assumption of integral transformations is the global data coverage. However, availability of high-resolution and accurate gravitational data is restricted. For this reason, we decompose the global integration into two effects: 1) the effect of the near zone calculated by numerical integration of data within a spherical cap, and 2) the effect of the far zone due to data beyond the spherical cap synthesised by external spherical harmonic series and EGM coefficients. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transforms used in geodesy, such as Poisson’s, Stokes’s, and Hotine’s. In this contribution, we present a complete and unified theory for the far-zone effects of integral transformations mutually relating quantities from the gravitational potential up to the third-order gravitational tensor components. These transformations include both isotropic and non-isotropic integral kernel functions. To understand mathematical aspects of the resulting external spherical harmonic series, we investigate behaviour of truncation error coefficients in the spectral domain in connection with the properties of the corresponding integral kernels in the spatial domain. We systematically quantify magnitudes of the far-zone effects with respect to the signal powers of the corresponding quantities. Preliminary numerical experiments indicate that formulas for evaluation of the far-zone effects are stable and reach less than a hundred percent of the total signal for smoothing integral formulas. For de-smoothing integral transformations, the magnitude of the far-zone effects may be several orders larger than the total signal.
