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

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 gravity data, 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.
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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).
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Significance of the characteristic density
Characteristic density is a smooth function and
thus it is not a realistic solution of the
inverse problem. Characteristic density contains
the same amount of information as the
surface gravitation, but in another form the
information about the distribution of the sources
of gravitational field with depth is hidden in
the 2-dimensional surface gravitation, but it is
restored in the 3-dimensional characteristic
density. The extrema-conserving property of the
characteristic density implies that for each
domain where this density is positive (negative),
there has to exist an anomalous body with
positive (negative) difference density located
roughly in this domain. This shows that the
characteristic density is an important tool for
finding the realistic solutions of the inverse
problem.
56
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 into which the halfspace is
divided. For any multi-domain density
, we calculate the surface gravitation
generated by this density and then the
corresponding characteristic density
. Finally, we calculate the residual surface
gravitation
and the residual characteristic density
.
This quantity is identically zero if the density
is a solution of the inverse
problem.
57
Determination of the realistic solution
If the residual characteristic density
corresponding to the chosen multi-domain density
is nonzero, the latter density has to be
changed. This is done by changing the boundaries
of the domains the values of density in 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 of these cells
has its value of density. The cell is called a
boundary cell just if at least one of the
neighbouring cells has a different value of
density the other cells are called the interior
ones. For each boundary cell, if the residual
characteristic density in its centre is positive
(negative), the value of density of this cell is
changed to the nearest higher (lower) value from
among its neighbours (if such neighbour
exists). The result of these changes is the new
multi-domain density.
58
Zero model
The surface gravitation generated by any infinite
horizontal layer with constant density is a
constant function. The characteristic density
corresponding to the constant surface gravitation
is identically zero. This means that the
infinite horizontal layers with constant density
cannot be found if the only input is the surface
gravitation. Therefore, the number and parameters
of these layers have to be known in
advance. The multi-domain density representing
the layered calculation domain is called the zero
model. The zero model serves as a reference
model for any other models The calculation of
surface gravitation generated by any multi-domain
density has to use the difference density, which
is equal to the difference of the actual density
and the value of the density of the zero model
corresponding to the same depth.
59
Starting model
The calculation of shapes of individual domains
of the multi-domain density according to the
above description has to start from some simple
multi-domain density the latter is called the
starting model. The starting model is created
from the zero model by changing the value
of density 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
germ is created at the same position as this
extremum. The density value of each germ 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)
than the density of the zero model at this
depth. The suitable value of the difference
density of the germ is of the order of the value
of the characteristic density of this germ.
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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,
gt 32 km 3300 kg /
m³. The starting model had 1492 germs of
anomalous bodies with densities in the range 2140
3300 kg / m³.
61
Results
The calculations were performed on the Origin
2000 supercomputer of the Computing 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
iteration steps and it took 930.81 hours of CPU
time.
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Disadvantages of the described method
Harmonic inversion method using the
characteristic density cannot be
easily generalized for the case of the arbitrary
surface of Earth. This is because it is difficult
to find the exact formula for calculation of the
characteristic density in such a case. Another
disadvantage is that the formula for calculation
of the characteristic density contains the second
derivative of the surface data, what increases
the numerical errors. However, if we examine the
procedure for the determination of the
realistic solution, we see that there was nowhere
used the fact that the characteristic density is
a solution of the inverse problem. The
determination of the realistic solution was
enabled by the extrema-conserving property of the
characteristic density (of course, there was also
important the maximal smoothness of
the characteristic density and its linear
dependence on the surface gravitation).
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Advanced harmonic inversion method
Therefore, we can use for the determination of
the realistic solution any function having the
above mentioned properties. We shall call such a
function the information function (for the given
surface gravitation) this is because it gives us
the 3-dimensional information needed for the
calculation of realistic solutions. 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 having the
extrema-conserving property and depending
linearly on the surface gravitation. The
realistic solution is the same as before.
110
Information function
The information function (for the given surface
gravitation) has to satisfy the following
conditions
1. It is a maximally smooth function
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.
111
Formula for the quasigravitation
We choose the information function to have the
same dimension as the surface gravitation it
will be therefore called the quasigravitation. Th
e quasigravitation is a
triharmonic function (thus in
the condition 1) and is expressed by the formula
(3)
The quasigravitation is normalized such that for
a single point source the local extrema of the
surface gravitation and quasigravitation have the
same value.
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Multi-domain density
As in the previous case, the realistic solution
of the inverse problem can be represented 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
and the residual quasigravitation
.
This quantity is identically zero if the density
is a solution of the inverse
problem.
159
Determination of the realistic solution
If the residual quasigravitation corresponding to
the chosen multi-domain density is nonzero, this
density has to be changed. As in the previous
case, this is done by changing the boundaries of
the domains the values of density in these
domains remain unchanged. For each boundary
cell, if the residual quasigravitation in its
centre is positive (negative), the value of
density of this cell is changed to the
nearest higher (lower) value from among its
neighbours (if such neighbour exists). The
result of these changes is the new multi-domain
density.
160
Zero and starting models
The surface gravitation generated by any infinite
horizontal layer with constant density is a
constant function. The quasigravitation
corresponding to the constant surface gravitation
is identically zero. This means that the
infinite horizontal layers with constant density
cannot be found if the only input is the surface
gravitation. Therefore, the number and parameters
of these layers have to be known in
advance. Thus, as in the previous case, we have
to choose the zero model which serves as a
reference model for any other models and the
starting model by creating the germs of the
future domains. For any local extremum of the
original quasigravitation, a single germ is
created at the same position as this extremum.
The suitable value of the difference density of
the germ can be calculated from the value of
the quasigravitation of this germ.
161
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,
gt 32 km 3300 kg /
m³. The starting model had 793 germs of
anomalous bodies with densities in the range 2140
3300 kg / m³.
162
Results
The calculations were performed on the Origin
2000 supercomputer of the Computing 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
iteration steps and it took 1113.10 hours of CPU
time.
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