Testing of two variants of the harmonic inversion method on the

territory of the eastern part of Slovakia

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 anomalousbodies 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.

Abstract

These versions of harmonic inversion method are 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.

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.

Disadvantages of the described method

Harmonic inversion method using the characteristic density cannot be easilygeneralized for the case of the arbitrary surface of Earth. This is because it isdifficult to find the exact formula for calculation of the characteristic densityin such a case.

Another disadvantage is that the formula for calculation of the characteristicdensity contains the second derivative of the surface data, what increases thenumerical errors.

However, if we examine the procedure for the determination of the realisticsolution, we see that there was nowhere used the fact that the characteristicdensity is a solution of the inverse problem. The determination of the realisticsolution was enabled by the extrema-conserving property of the characteristicdensity (of course, there was also important the maximal smoothness of thecharacteristic density and its linear dependence on the surface gravitation).

Therefore, we can use for the determination of the realistic solution anyfunction having the above mentioned properties. We shall call such a functionthe information function (for the given surface gravitation); this is because itgives us the 3-dimensional information needed for the calculation of realisticsolutions.

The strategy for finding the solution of the inverse problem is thus as follows:

1. to find the information function;2. to find some realistic solution(s).

The information function is defined as the maximally smooth function havingthe extrema-conserving property and depending linearly on the surfacegravitation. The realistic solution is the same as before.

Advanced harmonic inversion method

The information function (for the given surface gravitation) has to satisfy 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),,( zyxfk k

),(),,(d),,(:0 yxazyyxxLSzyxfz

Information function

1. It is a maximally smooth function:

π2

0)sin,cos(d

π21

),,( uyuxauyxa

),,()(

d8),,(:00 2/522

32

uyxazuzu

uzyxqz u

(3)

Formula for the quasigravitation

),,( zyxq

We choose the information function to have the same dimension as thesurface gravitation; it will be therefore called the quasigravitation.

The quasigravitation is a triharmonic function (thus in thecondition 1) and is expressed by the formula

3k

The quasigravitation is normalized such that for a single point source thelocal extrema of the surface gravitation and quasigravitation have the samevalue.

Multi-domain density

),,( zyxm),( yxam

),,( zyxqm

),(),(),( yxayxayxa mr

and the residual quasigravitation

),,(),,(),,( zyxqzyxqzyxq mr .

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

),,( zyxm

As in the previous case, the realistic solution of the inverse problem can berepresented by a multi-domain density.

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

Determination of the realistic solution

If the residual quasigravitation corresponding to the chosen multi-domaindensity is nonzero, this density has to be changed.As in the previous case, this is done by changing the boundaries of thedomains; the values of density in these domains remain unchanged.

For each boundary cell, if the residual quasigravitation 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 and starting models

The surface gravitation generated by any infinite horizontal layer with constantdensity is a constant function. The quasigravitation 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.

Thus, as in the previous case, we have to choose the zero model which servesas a reference model for any other models and the starting model by creatingthe germs of the future domains.

For any local extremum of the original quasigravitation, a single germis created at the same position as this extremum. The suitable value of thedifference density of the germ can be calculated from the value of thequasigravitation 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 793 germs of anomalous bodies with densities in therange 2140 – 3300 kg / m³.

Results

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

The calculation of the quasigravitation took 2.08 hours of CPU time.

The calculation of the resulting multi-domain density consisted of 288 iterationsteps and it took 1113.10 hours of CPU time.