The influence of discretizing conductivity gradients in the 3-D finite difference EM forward modelling algorithms

Abstract

The objective of this paper is to seek a generalized understanding for the influence of incorporating the gradient of the model?s physical properties (e.g., conductivity, velocity) in the forward modelling numerical algorithm. In order to take a step towards that, we examine an example from Geophysics for solving 3-D Maxwell?s equations using finite difference (FD) methods. The 3-D FD methods to obtain discrete solutions of Maxwell's equations include 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. The staggered grid algorithm is free of conductivity gradient and results in a symmetric 13- diagonal banded matrix. The objective now is to examine and understand better the influence of the conductivity gradient incorporated in the FD equations on the accuracy of the electromagnetic (EM) modelling for two 3-D benchmark models. We use three various discretizations (fine, mildly coarse, and coarse) for each model. The modelling 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 accurate results for all the three discretizations investigated. However, the balance method encountered some inaccuracies for the mildly coarse and coarse discretizations. This appears to be due to the presence of the conductivity gradient in the 3-D modelling algorithm. The model studies also suggest that the thicknesses of the horizontal and vertical discretizations at the conductivity boundaries should be about 1/25 and 1/100 skin depth to maintain accurate modelling results when the conductivity derivatives exist in the 3-D modelling algorithm.