An accurate and robust multigrid algorithm for 2D forward resistivity modelling

We present an adaptation of the full multigrid algorithm in DC resistivity modelling in an effort to increase its accuracy. There is a great difficulty with conventional multigrid solvers in representing the physics of an arbitrary distribution of electrical conductivity on a very coarse grid. In general, conventional rectangular finite-difference or 5-point approximations of Poisson's equation cannot represent, at a coarse grid level, the effective anisotropy on a coarse scale which results from fine structure in the model. An exception to this generalization occurs when the principal axes of structural anisotropy are aligned with the coordinate axis. Additional and similarly generated problems arise when a coarse cell is obliged to represent fine structure containing very high conductivity contrasts. We have developed an adaptation of the usual resistive-network representation of the continuum, which avoids some of these problems, and have compared it with the traditional resistive network currently used. The network adaptation consists of replacing the usual 5-point Laplacian operator stencil used on the finite-difference grid with a 9-point stencil, and the conductivity scalar with a 6-parameter conductivity parametrization. This parametrization permits representation of arbitrarily orientated anisotropy as well as more complex behaviour related to high conductivity contrasts. The importance of multigrid solvers does not lie in their speed at forward modelling (which is comparable with other methods), but rather in their potential for inverse modelling. Inverse solvers which proceed by refinement of an initially very coarse solution can, in principle, take time only linearly proportional to the number of gridpoints in the final desired model.

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