3-Dimensional data interpretation of resistivity method by numerical coordinate transformation

摘要

Previous studies on the theory of optimal computational space indicated that the forward problem for the resistivity method could be replaced by the problem of numerical coordinate transformation. This conclusion implies that an efficient algorithm could be developed as long as the problems of the numerical coordinate transformation were solved. Clearly, the coordinate transformation should be reciprocated between the physical space and the computational space. To avoid using of the interpolation method, two direction transformation equations are applied. However, the introduced transformation equations cannot be easily solved either in the physical space or in the optimal computational space because the grid systems in both spaces are non-uniform. In fact, the grid coordinates are the unknowns to be solved. These facts suggest that, to describe our transformation equations, an "executing coordinate system" is needed. In this study, a third coordinate space, the intermediate space, is introduced. In this new space, the grid system holds a one-to-one coordinate transformation relation to those in the physical and computational spaces, and remains as a uniform grid system. Solving all the transformation equations in this space provides the coefficient matrices with the regular structure regardless of the topographies and resistivity distributions. In this algorithm, the initial grid in the computational space is assigned uniform or whatever expected and the initial grid in the physical space is created by the hyperbolic transform system according to the prescript surface grid in the physical space and the whole grid system in the computational space. To refine the grid systems in both spaces, elliptical systems are applied. Several numerical tests by this algorithm are carried out to show its application to the forward problems.