Abstract

1. Testing of two variants of the harmonic inversion method on the territory of the eastern part of Slovakia

2. Abstract The aim of this contribution is to compare two variants of the harmonic inversion method on the territory of the eastern part of Slovakia. The older variant uses in the determination of the position and shape of anomalous bodies the characteristic density, the new one uses the quasigravitation. Both the characteristic density and the quasigravitation are smooth functions obtained from the surface gravitational field by a linear integral transformation. Both functions restore the 3-dimensional distribution of sources of gravitational field that is hidden in the surface gravitational field. The comparison of these two variants is accompanied by numerous figures.

3. Introduction These versions of harmonic inversion method are suitable for the case of planar Earth surface. This means that it was not accounted for: 1. the ellipsoidal shape of the Earth; 2. the topography. In order to avoid the problem in the point 2, the original gravimetric data were continued downwards to the zero height above the sea level by the method of Xia J., Sprowl D.R., 1991: Correction of topographic distortion in gravitydata, Geophysics, 56, 537-541.

4. Inverse gravimetric problem Density Surface gravitation (1) Input: Output:

5. Harmonic inversion method The inverse problem of gravimetry has infinitely many solutions. In order to obtain a reasonable solution(s), the following strategy was proposed: 1. to find the simplest possible solution; 2. to find some realistic solution(s). The simplest solution is defined as the maximally smooth density generating the given surface gravitation and having the extrema-conserving property; this density is a linear functional of the surface gravitation. The realistic solution is defined as a partially constant density; in other words, the calculation domain is divided in several subdomains and in each of these subdomains the density is a constant.

6. Characteristic density The simplest solution described above is called the characteristic density (of the given surface gravitation); thus it satisfies the following conditions: 1. It is the maximally smooth density generating the given surface gravitation: for the smallest possible 2. It is a linear integral transformation of the surface gravitation: 3. For the gravitational field of a point source, it has its main extremum at the point source.

7. Formula for the characteristic density These conditions define uniquely the characteristic density; it will be denoted . In the condition 1 we have , thus the characteristic density is a tetraharmonic function. Formula for this density from the condition 2 reads (2) Details can be found in: Pohánka V., 2001: Application of the harmonic inversion method to the Kolárovo gravity anomaly, Contr. Geophys. Inst. SAS, 31, 603-620.

8. Input data Input was represented by 71821 points (coordinates x, y, gravitation a). The data were interpolated and extrapolated into a regular net of points in the rectangle 300 × 240 km with the step 0.5 km and the centre at 48º49'20" N, 21º16'20"E (totally 289081 points). The calculation domain was chosen as the rectangular prism whose upper boundary was the rectangle 200 × 140 km with the same centre as above and whose lower boundary was at the depth 50 km; the step in the depth was again 0.5 km (totally 401 × 281 × 100 = 11268100 points).