FINITE‐DIFFERENCE RESISTIVITY MODELING FOR ARBITRARILY SHAPED TWO‐DIMENSIONAL STRUCTURES

Abstract

Resistivity surveying is commonly done by using a point‐source dipole. Consequently, a finite‐difference evaluation of apparent resistivity curves implies the use of three‐dimensional simulation models which necessitate prohibitive computer costs. However, if we assume variation of resistivity only in two dimensions and use a line‐source dipole for setting up the finite‐difference model of a given structure, the potential field can be evaluated easily. A discrete version of the resistivity problem in two dimensions, which takes into account nonuniform grid spacing, is presented as a system of self‐adjoint difference equations. Since the iterative solution of such a system does not require grid spacing to be less than a certain critical value, it was successfully used for the development of fast‐convergence finite‐difference models. By examining in detail the characteristics of the matrix associated with the evaluation of the potential field, it is demonstrated that the proposed modeling procedure will remain stable for all conceivable geometries and resistivity distributions. It was used for the investigation of certain models for which the corresponding results could also be computed analytically. A direct superposition of results obtained in the two cases shows that they are virtually identical. By making use of the reciprocity theorem, a computational short‐cut, which provides the evaluation of vertical sounding curves for a line‐source dipole in a single step, is put forward. Special problems related to the optimization of acceleration parameters as well as the estimation of the potential function along the subsurface boundaries of the model are discussed. It is concluded that by surrounding the model by a termination strip of very large effective width, either Neumann‐ or Dirichlet‐type boundary conditions can be used for simulating a semiinfinite medium without introducing signficant errors in the results.