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The stream function described is employed for the presentation of 2D
DC modeling results. The 2D model is understood as a 2D medium with linear
current electrodes, oriented along the inhomogeneities' strike direction.
In this case both the medium and the electric field depend on two space
coordinates only. Modeling becomes much easier than considering point
current electrodes, where the electrical field always is three-dimensional.
Meanwhile the actual results of such modeling are qualitatively equivalent
to 3D modeling with point electrodes, as long as the measurements are
conducted across the objects.
Stream-function used for Current-lines' CONSTRUCTION in 2-dimensional DC
modeling

Bobachev A.A., Modin I.N., Pervago E.V., Shevnin V.A.
Geological faculty, MSU, Moscow, 119899, Russia.
Fax: (7-095)-9394963, E-mail: Boba@geophys.geol.msu.ru
Internet site: www.geol.msu.ru/deps/geophys/rec_labe.htm
Report, presented at the 5th Meeting EEGS-ES in Budapest, 5-9 September
1999.


The classical modeling presentation is in apparent resistivity which
reflects an electric field distribution on the earth's surface. Quite often
the connection of measured anomalies with a geoelectrical model is rather
complex (fig. 1, A and C). The visualization of DC current lines simplifies
understanding of the electric field's structure. Current lines are used in
almost each textbook, but a practical technique for their construction is
usually not included.
[pic]Fig.1. Apparent resistivity for a gradient array (A), DC current lines
for central part of model (B) and the model with the current electrodes'
position (C).
The evident way for drawing current-lines is the step by step
continuation of a line from some point along the electric field direction.
The practical realization of such approach is not trivial. For a 2D field
it is possible to use the stream-function. This function is often used in
EM field modeling [flux function, Berdichevsky, 1984]. A contour map of the
stream-function corresponds to the stream-line distribution. Thus the
problem of current streamlines' construction is reduced to the calculation
of the stream-function in the research area. This can be achieved by
calculating secondary surface charges, which are determined at 2D modeling,
using Fredholm's integral equation of the second type relatively of
electric field [Escola, 1979].
The stream-function's (y) physical definition is the difference
between stream-function's values in two points in space, which is equal to
the electric current intersecting a curve connecting them:
[pic] (1)
where i is the vector of electric current density , n is the normal vector
to the AB curve, E is the electric field intensity and s(l) is the electric
conductivity in a point l. The value of the integral (1) does not depend on
the integration's path in regions without current sources. Therefore the
stream-function can be calculated by integration from any point. Assuming
that the stream-function is equal zero at point M, subsequent for any point
K we can write an equation:
[pic]. (2)
The electrical field is the sum of a primary field of current from the
current electrodes and a secondary field of charges from the modeled body's
surface (surface charges). If the density of surface charges is known from
2D modeling, the integral (2) can be calculated analytically. The
connection of stream-line distribution with apparent resistivity ([pic])
depends on the kind of array. For pole-dipole array with ideal MN dipole
the following relation is valid and exact:
[pic], (3)
[pic]
Fig. 2. The model with highly resistive object (rbody=1000 Wm, rmedium=100
Wm) for pole-pole array with buried current electrode. Apparent resistivity
on the surface (A), in the cross-section (B) and DC stream-lines (C).
[pic] and [pic] are the resistivity and current density in MN center, [pic]
is the current density for a uniform half-space. Therefore, if the upper
layer is homogenous ([pic]), then ra is proportional to the current
density. This approach clarifies the ra physical explanation as shown in
fig.1A. The current distribution around conductive object (fig.1B) results
in changes of current density on the surface and this finds it's expression
in ra anomalies. ra for pole-pole array is related to secondary charge
distribution. An expression can be derived for the potential value in point
M ([pic]): [pic], (4)
where [pic] is the potential of the primary electric field, [pic] is the
potential of the secondary electric field, defined by the surface charge
distribution. The surface charge density ([pic]) on the boundary of the
inhomogeneity is determined from the boundary condition (Strattom 1941,
p.163):
[pic]. (5)
Thus the surface charge density reveals a distribution of streamlines
(fig.2). The upper left corner of the object is less expressed in the
apparent resistivity than the upper right. Both corners will be expressed
clearer in the case of a conductive object.
The best way to visualize surface charge distributions, is the ra
cross-section for a pole-pole array (2B), corresponding to measurements
with a buried potential electrode. Maxima and minima of such cross-section
show a maximum density of positive and negative surface charges.
In our practice stream-line function is used both for educational tasks and
practical investigations. We can calculate on its base current lines,
potential lines and apparent resistivity isolines in vertical cross-
section. Like in formula 4 it is possible to separate potential, electric
field and apparent resistivity into normal and anomalous parts and draw
these lines distributions near anomalous object. Possibilities of this
approach are listed in the table below and at some examples shown at fig.3-
10.
[pic]


|[pic] |[pic] |
|3. Example of ra field's |4. Current lines' distribution near|
|distortions, caused with high |high resistive layer with |
|resistive object near measuring |conductive zone in three layered |
|electrodes. (rmedium=30, rbody=100 |model |
|Ohm.m, XA=100). A - ra -graph, B - | |
|current lines distribution in | |
|vertical cross-section. | |
|[pic] |[pic] |
|5. A single current electrode near |6. A single current electrode near |
|contact with conductive medium |thin conductive layer |
|[pic] |[pic] |
|7. ra graph as function r=AO (A), |8. 2D modeling from current source |
|current lines and apparent |near conductive object |
|resistivity distributions (B) for |(rMedium=100, rbody=15). A. Current|
|model with negative relief's form. |lines and ra vectors. B. Current |
|Field technologies are shown on the|lines of anomalous electric field |
|right. |outside of anomalous body and |
| |vectors of anomalous ra. |
|[pic] |[pic] |
|9. DC current and iso-potential |10. DC current and iso-potential |
|lines near inclined high resistive |lines from point current source |
|body |near high and low resistive |
| |cylinders |

For streamlines calculation DC_flow software has been developed.
Working version of DC_flow software you can find in Internet at our web-
site.

References
Berdichevsky M.N. and Zhdanov M.S. 1984, Advanced Theory Of Deep
Geomagnetic Sounding, ELSEVIER, Amsterdam.
Escola, L. 1979, Calculation of galvanic effects by means of the method of
sub-areas, Geophysical Prospecting 27, 616-627.
Stratton J.A. 1941, Electromagnetic Theory, McGraw-Hill, New York.