Testing of the harmonic inversion method on the territory

of the eastern part of Slovakia

An improved version of the harmonic inversion method was tested on the territory of the eastern part of Slovakia. The improvement with respect to the initial version consists in the possibility to calculate position and shape of many anomalous bodies at once. The calculation is performed in two steps: first, from the surface gravitational field the characteristic density is obtained; second, the germs of anomalous bodies are placed in the local extrema of characteristic density and subsequently the true shapes of these bodies are determined iteratively. The evolution of the shapes of the anomalous bodies from the original germs to their final form was shown to be steady. The results of calculation are presented in numerous figures.

Abstract

This version of harmonic inversion method is suitable for the case of planarEarth 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 werecontinued downwards to the zero height above the sea level by the method ofXia J., Sprowl D.R., 1991: Correction of topographic distortion in gravity data,Geophysics, 56, 537-541.

Introduction

Inverse gravimetric problem

:0z 0),,( zyx

),,(),,(d),( zyxzyyxxGVyxa

2/3222 )(κ),,(

zyxz

zyxG

),( yxa

),,( zyx

Density

Surface gravitation

Input:

Output:

(1)

Harmonic inversion method

The inverse problem of gravimetry has infinitely many solutions. In order toobtain 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 generatingthe 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 thesesubdomains the density is a constant.

The simplest solution described above is called the characteristic density (of the given surface gravitation); thus it satisfies the following conditions:

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.

0),,( zyxk k

),(),,(d),,(:0 yxazyyxxKSzyxz

Characteristic density

1. It is the maximally smooth density generating the given surface gravitation:

π2

0)sin,cos(d

π21

),,( uyuxauyxa

),,(1

)(d

πκ20

),,(:00 2/722

44

uyxauzu

zuuzyxz uu

4k

(2)

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 isa tetraharmonic function. Formula for this density from the condition 2 reads

),,( zyx

Details can be found in:Pohánka V., 2001: Application of the harmonic inversion method to theKolárovo gravity anomaly, Contr. Geophys. Inst. SAS, 31, 603-620.

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 therectangle 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 upperboundary was the rectangle 200 × 140 km with the same centre as above andwhose lower boundary was at the depth 50 km; the step in the depth was again0.5 km (totally 401 × 281 × 100 = 11268100 points).

Characteristic density is a smooth function and thus it is not a realistic solutionof the inverse problem.

Characteristic density contains the same amount of information as the surfacegravitation, but in another form:the information about the distribution of the sources of gravitational field withdepth is hidden in the 2-dimensional surface gravitation, but it is restored inthe 3-dimensional characteristic density.

The extrema-conserving property of the characteristic density implies that foreach domain where this density is positive (negative), there has to exist ananomalous body with positive (negative) difference density located roughlyin this domain.

This shows that the characteristic density is an important tool for finding therealistic solutions of the inverse problem.

Significance of the characteristic density

Multi-domain density

The realistic solution of the inverse problem can be represented by a multi-domain density; this is the density that is constant in each of the domains intowhich the halfspace is divided.

For any multi-domain density , we calculate the surface gravitation generated by this density and then the corresponding characteristicdensity .Finally, we calculate the residual surface gravitation

0z

),,( zyxm),( yxam

),,( zyxm

),(),(),( yxayxayxa mr

and the residual characteristic density

),,(),,(),,( zyxzyxzyx mr .

This quantity is identically zero if the density is a solution of theinverse problem.

),,( zyxm

Determination of the realistic solution

If the residual characteristic density corresponding to the chosen multi-domaindensity is nonzero, the latter density has to be changed.This is done by changing the boundaries of the domains; the values of densityin these domains remain unchanged.

The changing of boundaries of particular domains is performed as follows:The whole calculation domain is divided into elementary cubic cells; each ofthese cells has its value of density. The cell is called a boundary cell just ifat least one of the neighbouring cells has a different value of density; the othercells are called the interior ones.

For each boundary cell, if the residual characteristic density in its centre ispositive (negative), the value of density of this cell is changed to the nearesthigher (lower) value from among its neighbours (if such neighbour exists).

The result of these changes is the new multi-domain density.

Zero model

The surface gravitation generated by any infinite horizontal layer with constantdensity is a constant function. The characteristic density corresponding to theconstant surface gravitation is identically zero.

This means that the infinite horizontal layers with constant density cannot befound if the only input is the surface gravitation. Therefore, the number andparameters of these layers have to be known in advance.

The multi-domain density representing the layered calculation domain is calledthe zero model.

The zero model serves as a reference model for any other models:The calculation of surface gravitation generated by any multi-domain densityhas to use the difference density, which is equal to the difference of the actualdensity and the value of the density of the zero model corresponding to thesame depth.

Starting model

The calculation of shapes of individual domains of the multi-domain densityaccording to the above description has to start from some simple multi-domaindensity; the latter is called the starting model.

The starting model is created from the zero model by changing the value ofdensity in some number of individual cells; these cells are called the germs(of the future domains to be created from these cells in the calculation process).

For any local extremum of the original characteristic density, a single germis created at the same position as this extremum. The density value of eachgerm is a free parameter and has to be entered; for the positive (negative)value of the extremum, the density of the germ has to be greater (lower) thanthe density of the zero model at this depth.

The suitable value of the difference density of the germ is of the order of thevalue of the characteristic density of this germ.

Calculation

The calculation domain was divided into 401 × 281 × 100 = 11268100 cells.

The layers of the zero model were defined as follows:

for the depth 0 - 3 km the density is 2680 kg / m³, 3 - 6 km 2700 kg / m³, 6 - 9 km 2720 kg / m³, 9 - 12 km 2740 kg / m³, 12 - 15 km 2760 kg / m³, 15 - 18 km 2780 kg / m³, 18 - 32 km 3000 kg / m³, > 32 km 3300 kg / m³.

The starting model had 1492 germs of anomalous bodies with densities in therange 2140 – 3300 kg / m³.

zero model

starting model

iteration 72

iteration 128

iteration 146

iteration 165

iteration 184

iteration 203

iteration 221

iteration 238

iteration 256

iteration 268

iteration 296

iteration 320

iteration 335

iteration 352

iteration 376

iteration 384

Results

The calculations were performed on the Origin 2000 supercomputer of theComputing Centre of the Slovak Academy of Sciences.

The calculation of the characteristic density took 2.08 hours of CPU time.

The calculation of the resulting multi-domain density consisted of 384 iterationsteps and it took 930.81 hours of CPU time.