Multigrid methods for forward and inverse resistivity problems in geophysics

In forward resistivity modelling, conventional multigrid solvers have difficulty in representing the physics of an arbitrary distribution of electrical conductivity on very coarse grids. In general, traditional coarsening methods for discrete conductivity models cannot represent, at a coarse grid level, the effective structural conductivity anisotropy, which results from fine structure in the model. These coarse grid limitations are particularly damaging to multigrid solvers when a coarse cell is obliged to represent fine structure containing high conductivity contrasts. I have developed a variation of the usual resistive-network representation of the 2D continuum which avoids these problems, and have compared it to the traditional resistive-network currently used. In this variation, the usual five-point Laplacian stencil used on a finite-difference grid was replaced with a nine-point stencil, and the conductivity scalar was replaced with a 6 parameter conductivity parameterization.; In addition to resistivity forward modelling, I examined two methods for solving the inverse resistivity problem. In the first method, both forward and inverse solutions are obtained simultaneously by relaxing the conductivity model parameters in tandem with the potentials. Implementing such an inverse solver in a multigrid method could, in principle, obtain a solution to the inverse problem in time that is linearly proportional to the number of model parameters. However, this approach suffers from convergence problems which were not resolved.; The second approach to the inverse problem uses a multilevel method in which approximate solutions for both the forward and inverse problems are obtained on a coarse grid and gradually refined until a desired fine grid is reached. The inverse portion of this algorithm utilizes the standard iterative damped least-squares method that is solved with a bi-conjugate gradient method. Results are presented for three standard geophysical models and compared with the results obtained from a standard single grid inversion method. In addition to a substantial speed-up over the single grid inversion method, the multilevel inversion method distributes the model over a greater depth range, without coercion by weight functions. This gives more realistic solutions to the inverse problem than the unweighted single grid inversion method which tends to concentrate the solutions near the surface.