f Stream-function used for current-lines' construction in 2-dimensional DC modeling

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. Modelling becomes much easier than considering point current electrodes, where the electrical field always is three-dimensional. Meanwhile the actual results of such modelling are qualitatively equivalent to 3D modelling with point electrodes, as long as the measurements are conducted across the objects. The classical modelling 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 techniques for their construction is usually not included. 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 make use of 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 by 2D modeling, using Fredholm's integral equation of the second type relatively of electric field [Escola, 1979]. The stream-function’s (ψ) physical definition is the difference between stream-function's values in two points in space is equal to the electric current intersecting a curve connecting them: