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 inhomogeneities 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 cell-resistivities) taking into account the topographical data, too. Therefore it is very important to use real topography both in the forward and inverse 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 cell-system (computational grid) 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 economic method compared with the traditional solvers. Grid shown in Figure 1 is generated automatically using only some control parameters for a halfspace having rough enough topography and two buried inhomogeneities. This is only an initial grid because by the help of multigrid technique using e.g. 10 levels every cell will be split up into 1024 small cells. On the earth-air interface and near the boundaries of inhomogeneities the size of cells is small already in the starting grid and going away from this regions, which are the most interesting in the description of the resistivity distribution, the size of cells increases gradually. Of course the presented grid will be expanded both on the left and right to fulfill by solution the prescribed boundary conditions. On the upper figure that grid can be seen which was generated automatically on the prescribed rectangle. However during the solution of forward problem those cells are not taken into account which are in full extent in the air so only the grid shown in the lower figure is used in the forward ( and inverse ) modeling. During inversion conductivities of the cells of the initial grid are determined from the apparent resistivities measured by a multi-electrode system using electrodes laid down equidistantly along real topography. It is important that the effect of topography is taken into account not by applying a topographical correction on measured data before inversion using a homogeneous halfspace having the actual topography but the real topography is used in every step of inversion.