A 3-D FINITE-DIFFERENCE ALGORITHM FOR DC RESISTIVITY MODELING USING CONJUGATE-GRADIENT METHODS

An accurate and efficient 3-D finite-difference forward algorithm for DC resistivity modelling is developed. The governing differential equa tions of the resistivity problem are discretized using central finite differences that are derived by a second-order Taylor series expansion . Electrical conductivity values may be arbitrarily distributed within the half-space. Conductivities at the grid points are calculated by a volume-weighted arithmetic average from conductivities assigned to gr id cells. Variable grid spacing is incorporated. The algorithm does no t limit the number and configuration of the sources, although all illu strative examples are computed using two current electrodes at the sur face. In general, the linear set of equations resulting from this kind of discretization is non-symmetric and requires generalized numerical equation solvers. However, after symmetrizing the matrix equations, t he ordinary conjugate gradient method becomes applicable. It takes adv antage of the matrix symmetry and, thus, is superior to the generalize d methods. An efficient SSOR-preconditioner (SSOR: symmetric successiv e overrelaxation) provides fast convergence by decreasing the spectral condition number of the matrix without using additional memory. Furth ermore, a compact storage scheme reduces memory requirements and accel erates mathematical matrix operations. The performance of five differe nt equation solvers is investigated in terms of cpu time. The precondi tioned conjugate gradient method (CGPC) is shown to be the most effici ent matrix solver and is able to solve large equation systems in moder ate times (approximately 2 1/2 minutes on a DEC alpha workstation for a grid with 50 000 nodes, and 48 minutes for 200000 nodes). The import ance of the tolerance value in the stopping criterion for the iteratio n process is pointed out. In order to investigate the accuracy, the nu merical results are compared with analytical or other solutions for th ree different model classes, yielding maximum deviations of 3.5 per ce nt or much less for most of the computed values of the apparent resist ivity. In conclusion, the presented algorithm provides a powerful and flexible tool for practical application in resistivity modelling.