f Two-dimensional resistivity inversion using a new topographical correction method

In shallow geoelectric exploration, e.g. in problems occurring in environmental and engineering geophysics, the topographical effects can be larger in measured values than changes caused by resistivity inhomogenities to be found. In the simplest topographical correction method the forward problem is solved for homogeneous halfspace having actual topographical relief and the measured apparent resistivities are corrected before interpretation with coefficients calculated for homogeneous halfspace. This method is inaccurate if large resistivity variations occur near to the surface and it is problematic to represent the results of two-dimensional inversion (section prepared from cel-resistivities) taking into account the topographical data, too. Therefore it is very important to use real topography both in the direct and forward modeling but using conventional 2-D methods it is very time-consuming. To solve the reconstruction (inverse) problem by e.g. a Newton-type method a number of direct problems have to be solved. Therefore the choice of the applied solution method of the direct problem is of great importance. Here we used a numerical method adopted from the computational flow modeling which makes it possible to reduce the computational cost by a remarkable amount. The method is based on a non-equidistant, non-uniform but Cartesian cellsystem (computational) generated by the so-called quadtree algorithm. This algorithm came from the computational graphics and is a heart of the "unconstructed grid generation ". Special finite difference schemes have been defined on the above computational grid. To speed up the computations a simple but efficient multigrid technique has been developed in the quadtree context. It is shown that to solve the direct problem (2-D Poisson equation) the number of the necessary arithmetic operations is proportional to the first power of the number of the surface points only which results in a much more economie method compared with the traditional solvers.