Toward understanding the influence of the conductivity gradient on finite-difference methods for Maxwell's equations in 3D EM forward modeling

Previous research reported in the literature on the three-dimensional (3-D) finite difference (FD) methods to obtain discrete solutions of Maxwell's equations includes the staggered-grid and balance methods. The balance method 3-D algorithm exploits the conductivity gradient in order to make the FD formulation a seven-point scheme and the resulting matrix a banded septa block diagonal but not symmetric. On the other hand, the staggered grid algorithm is conductivity gradient free and results in a symmetric 13-diagonal banded matrix. The objective of this paper is to examine and understand better the influence of the conductivity gradient on the modeling accuracy when incorporated in the FD equations. We study the impact of the horizontal and vertical derivatives of the conductivity gradient on the accuracy of the electromagnetic (EM) simulation for two 3-D benchmark models. We use three various discretizations (fine, mildly coarse, and coarse) for each model. The forward modeling results of each discretization have been computed separately by the balance method and staggered grid method. We have found that the staggered grid method produces stable and accurate results for the fine, mildly coarse, as well as the coarse discretization. However, the balance method encountered some inaccuracy for the mildly coarse and coarse discretizations. This appears to be due to the presence of the conductivity derivatives in the 3-D modeling algorithm. Tables for the various discretizations used in earth and air are provided, and the corresponding modeling results are presented and discussed. The model studies presented here show that the thicknesses of the horizontal and vertical discretizations at the conductivity boundaries should be 1/25 and 1/100 skin depth to maintain accurate forward modeling results when the conductivity derivatives exist in the 3-D modeling algorithm.