ALBERT

Notes: An accurate and efficient 3-D finite-difference forward algorithm for DC resistivity modelling is developed. The governing differential equations 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 grid cells. Variable grid spacing is incorporated. The algorithm does not limit the number and configuration of the sources, although all illustrative examples are computed using two current electrodes at the surface.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, the ordinary conjugate gradient method becomes applicable. It takes advantage of the matrix symmetry and, thus, is superior to the generalized methods. An efficient SSOR-preconditioner (SSOR symmetric successive overrelaxation) provides fast convergence by decreasing the spectral condition number of the matrix without using additional memory. Furthermore, a compact storage scheme reduces memory requirements and accelerates mathematical matrix operations.The performance of five different equation solvers is investigated in terms of cpu time. The preconditioned conjugate gradient method (CGPC) is shown to be the most efficient matrix solver and is able to solve large equation systems in moderate times (approximately 21/2 minutes on a DEC alpha workstation for a grid with 50 000 nodes, and 48 minutes for 200000 nodes). The importance of the tolerance value in the stopping criterion for the iteration process is pointed out. In order to investigate the accuracy, the numerical results are compared with analytical or other solutions for three different model classes, yielding maximum deviations of 3.5 per cent or much less for most of the computed values of the apparent resistivity.In conclusion, the presented algorithm provides a powerful and flexible tool for practical application in resistivity modelling.