Computations of secondary potential for 3D DC resistivity modelling using an incomplete Choleski conjugate-gradient method

An accurate and efficient 3D finite-difference (FD) forward algorithm for DC resistivity modelling is developed. In general, the most time-consuming part of FD calculation is to solve large sets of linear equations: Ax = b, where A is a large sparse band symmetric matrix. The direct method using complete Choleski decomposition is quite slow and requires much more computer storage. We have introduced a row-indexed sparse storage mode to store the coefficient matrix A and an incomplete Choleski conjugate-gradient (ICCG) method to solve the large linear systems. By taking advantage of the matrix symmetry and sparsity, the ICCG method converges much more quickly and requires much less computer storage. It takes approximately 15 s on a 533 MHz Pentium computer for a grid with 46 020 nodes, which is approximately 700 times faster than the direct method and 2.5 times faster than the symmetric successive over-relaxation (SSOR) conjugate-gradient method. Compared with 3D finite-element resistivity modelling with the improved ICCG solver, our algorithm is more efficient in terms of number of iterations and computer time. In addition, we solve for the secondary potential in 3D DC resistivity modelling by a simple manipulation of the FD equations. Two numerical examples of a two-layered model and a vertical contact show that the method can achieve much higher accuracy than solving for the total potential directly with the same grid nodes. In addition, a 3D cubic body is simulated, for which the dipole-dipole apparent resistivities agree well with the results obtained with the finite-element and integral-equation methods. In conclusion, the combination of several techniques provides a rapid and accurate 3D FD forward modelling method which is fundamental to 3D resistivity inversion.