On frequency response of Stokes and Hotine-Koch integral transforms in calculation of height anomaly in the local area by means of SRBF

The problem of determining the height anomaly in a local area of the radius ψ₀ using gravity disturbances and gravity anomalies is discussed. The influence of the far zone, as usually, is approximately taken into account using the global gravity field model and the truncation coefficients Q_n (ψ₀) introduced by M.S. Molodensky [1]. The modification Q_n⁰(ψ₀) by O.M. Ostach [2] of these coefficients is described. They provide - in contrast to the original coefficients - the continuity of the used integral transform kernel Ker⁰ (ψ) in the whole its definition domain. As a consequence, the modified coefficients decrease faster compared to the original ones with an increase of the degree n (frequency). It reduces the error of the far zone influence. Coefficients are interpreted as Fourier coefficients of the outer part of the kernel when it is decomposed into the orthogonal system of nonnormalized Legendre polynomials. The relationship between Q_n (ψ₀) and Q_n⁰(ψ₀) is indicated. In the frequency domain, the expression for the truncated kernel ΔKer⁰ (ψ) of the integral transform used (Stokes or Hotine-Koch) differs from the corresponding full kernel by a multiplier, which is proposed to be called the frequency characteristic of the kernel truncation operator onto the inner zone of radius ψ₀.

In local modeling, when describing the details of the "useful signal", it is advisable to use approximation by means of spherical radial basis functions (SRBF) instead of traditional integration due to their good spatial localization [3, 4]. The procedure of constructing scaling functions and corresponding wavelets is briefly described. New scaling functions, based on the above-mentioned concept of frequency characteristic of the kernel truncation operator onto the inner zone of the radius ψ_o, are proposed. To prove the effectiveness of these scaling functions, numerical experiments were conducted. Both gravity anomalies Δg and disturbances δg were used as input data. The results of the calculations showed a high accuracy of recovering height anomalies from gravity anomalies. Besides, introduction of frequency characteristic of kernel truncation of corresponding integral transform onto the inner zone allows to cut off implicit influence of far zone. Known scaling functions that do not use this frequency characteristic lead, as experiments have shown, to biased results.

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