International Journal of Geosciences, 2012, 3, 329-332
http://dx.doi.org/10.4236/ijg.2012.32035 Published Online May 2012 (http://www.SciRP.org/journal/ijg)
Polar 3D Transformation of the Full Gradient of
Gennady Prostolupov, Michail Tarantin
Laboratory of Geopotential Fields, Perm, Russia
Received December 21, 2011; revised February 9, 2012; accepted March 11, 2012
The method of 3D polar transformation of full gravity potential gradient vectors is based on the geometric properties of
the crossing points of complete gradient of the potential to localize the source region that causes the observed anomaly.
The cross-points—poles—are defined for rectangular polygons of different sizes where the full gradient vector is de-
fined at every vertex. The polygon size range could be specified. The set of poles, positive and negative, is then repre-
sented on the 3D chart in the form of clusters of dots or cubes and can be considered as a model image of the sources,
intended for visual analysis and further interpretation.
Keywords: Gravity; Anomaly; Interpretation; Model; Vector; Full Gradient; 3D Chart
Developing new methods of data express interpretation,
that give the first approximation for quantity analysis and
geological interpretation, is of actual type. Presented me-
thod of a polar transformation, as Euler deconvolution 
or tensor deconvolution , is just a transformation of
obtained data and is used primary for visual analysis.
Attention to such kind of methods is due to its low cost
in computing. The most close to such methods are that
based on a theory of special points. But the methods
mentioned above are much more productive but less
strong with theory.
The method described in this paper uses the geometrical
properties of the full vectors of potential gradient in
three-dimensional space: they concentrate in the positive
mass direction and disperse when mass is negative. The
crossing point of vector-lines lies in nearby of pertur-
bation mass and coincides with that mass if it is of point-
type. For brief these points of line crossing are called by
the authors as poles, because they became a special pro-
perty—sign of the attached mass (positive or negative).
All three gradients could be found from the vertical
one using the method of source-point approximation .
For survey data the construction of point masses below
every measure point is build. The gravity effect of such
construction should be equal to the measured data with
the given accuracy. When having such a system of masses
any type of derivatives could be found. For example, the
first order derivatives could be found by (1).
When having the gravity effect of a sphere the cross-
point of any lines of full gradient vectors coincides with
the center of that sphere. But the real field is additive and
complex and it is practically impossible to find 3 vectors
of attractive force that cross in one point of a realm.
Nevertheless we had introduced a following algorithm to
get the coordinates of poles.
Let’s consider determination of a cross-point coordi-
nates of a single mass in a plane by one pair of vectors
(Figure 1). Let us know the attractive force vectors in
two points A and B. Obviously both of them would look
in the direction of the attractor and exactly in its center.
So we could determine the mass center position of the
body by two coordinates XM and ZM by (2).
Having 2 vectors we can define 5 types of poles po-
ssible. When both of the vectors concentrate in the lower
half-space we got a positive pole, when both of them
concentrate in the upper part of the half-space we got a
negative pole. When vectors disperse to lower half-space
or disperse to upper half-space we got weakly negative or
weakly positive type respectively. The last type is neu-
opyright © 2012 SciRes. IJG
G. PROSTOLUPOV, M. TARANTIN
Figure 1. For cross-point coordinates definition.
For all calculations in three-dimensional space the 4-
vertex polygons are used. Every vertex is placed in the
measured point or in the grid point, if the measured data
had been gridded and processed. All further calculations
proceed only with polygons having all vectors directed in
one half-space at the same time. When all vectors point
up we got a negative or weakly positive pole, when all
vectors point down we got a positive or weakly negative
pole, depending on convergence of vectors. None of other
combinations is used at the time. Having 4 vertexes we
can define 6 pairs of vectors (4 at sides and 2 at diagonals)
to find 6 triplets of pole coordinates. The final coordinates
of the pole for the polygon are the mean-averaged of all
got for it. The range of polygon size variations could be
set by operator. The algorithm uses all possible polygons
within data grid.
As a result we got a 3D chart of poles P++ and P−−.
After some filtering with the chosen algorithm (for ex-
ample, one from ) the chart is divided into set of clu-
sters. These clusters in turn give us positions and appro-
ximate shapes of the sources of anomalies. The deve-
loped program based on this algorithm allows field-
source bodies to be a sphere, combination of blocks or a
star-body with an equivalent volume and mass. Having a
full force vector the attractive mass is calculated as
mgrG , where G is the gravitational constant, g is
the full force vector, r is a distance from the grid point to
the pole point. The mass of a cluster is an average mass
of all parts in it: for sphere and star-body—of all poles
and for combination of blocks—of all blocks. The ano-
malous density of a field-source could be determined by
its mass and volume.
As a theoretical example let’s consider results of two
point source field processing. These sources have differ-
rent masses and lay on different depths: 0.5 and 0.75 km
(Figure 2). Determination of the position of sources
gives results that are close to real. The discrepancies of
coordinates and mass lay in 2 per cent boundary. Results
for 5 point-sources show the same quality of localization.
As follows from the theory of a method  the most
appropriate type of fields to process is a point-source field.
But the real observed field are too complicated and do not
have such a morphology. So let’s consider a non-point
model—the field of a material rod. Such type of model
could be used in working with mines and pits, gas pipe-
lines or any lengthy objects. Figure 3 shows the results of
three rods field processing. It is seen that the red point
sources are combined in lengthy clusters of rod type.
Comparing with strong defined poles P++ and P−− the
weakly defined poles are non-obvious or cryptic for sour-
ces. During the processing of a real field it is impossible
to determine the type (positive or negative) of its sources.
We can not distinguish the portions of softening of con-
solidated ground from the consolidation of soft soils. So
the use of weakly determined poles could be informative.
If, for example, we consider a source field defined by a
sine function then it could be caused either positive
masses or negative. Using only strong defined poles gives
us only one type of sources while weak defined poles
gives both types (positive and negative). Weak defined
poles for rod field case is shown in Figure 3 with blue.
The next example is the field of a lengthy L-rod. Such
Figure 2. Model gravity field of two point mass (left) and the result of mass positions determination (right).
Copyright © 2012 SciRes. IJG
G. PROSTOLUPOV, M. TARANTIN 331
kind of source could model an ore body. As shown in
Figure 4 the result cluster shape is in good agreement
with the source model, but the left part of it is a bit lower.
The tests show that the best result for complex fields
could be gained when using the combination of divided
and processed apart fields. These parted results are to be
combined in one diagram like one in Figure 5. That fi-
gure shows the result of 2 km deep reef trap field pro-
cessing. The trap and its envelope could be located on the
The geometrical property of an attraction potential gra-
dient to point to the force sources allows building models
of equivalent mass distribution. The difference of the
shown above diagrams from the transformed field dia-
gram or up- and down-continuous is as follows. Here we
have a 3D view of equivalent mass source chart with
definite properties (position, size and mass) in spite of 2D
field being interpolated in 3D. These diagrams could be
Figure 3. 2D field shape for 3 rod s an d the result of this field processin g .
Figure 4. L-rod field shape and the result of its p roces sing.
Copyright © 2012 SciRes. IJG
G. PROSTOLUPOV, M. TARANTIN
Figure 5. The real field and the result 3D chart. Contour marks the known oil reservoir.
used within quality and quantity interpretation of geo-
 C. Zhang, M. F. Mushayandebvu, A. B. Reid, J. D. Fair-
head and M. E. Odegard, “Euler Deconvolution of Gravity
Tensor Gradient Data,” Geophysics, 2000, Vol. 65, No. 2,
pp. 512-520. doi:10.1190/1.1444745
 V. O. Mikhailov and M. Diament, “Some Aspects of
Interpretation of Tensor Gradiometry Data,” Izvestiya,
Physics of the Solid Earth, Vol. 42, No. 12, 2006, pp.
 A. S. Dolgal, “Geopotential Fields Approximation for
Practical Tasks Using Equivalent Sources,” Geophysical
Journal, Vol. 21, No. 4, 1999, pp. 71-80.
 A. D. Gvishiani, S. M. Agayan, Sh. R. Bogoutdinov and
A. A. Solovyov, “Discrete Mathematical Analysis and
Applications Geology and Geophysics,” Bulletin of Kam-
chatka Regional Association “Educational-Scientific Cen-
ter”, Earth Sciences, Vol. 16, No. 2, 2010, pp. 109-125
 G. V. Prostolupov and M. V. Tarantin, “Attractive Poten-
tial Full Gradient Vectors Transformation,” Proceedings
of 38th Uspensky Simposium on Theoretical and Practi-
cal Aspects of Geologycal Interpretation of Geophysical
Fields, Perm, 2011, pp. 245-248
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