Effective multidimensional resistivity inversion using finite-element techniques

SUMMARY This paper describes the development of a multidimensional resistivity inversion method that is validated using two- and three-dimensional synthetic pole‐pole data. We use a finite-element basis to represent both the electric potentials of each source problem and the conductivities describing the model. A least-squares method is used to solve the inverse problem. Using a leastsquares method rather than a lower-order method such as non-linear conjugate gradients, has the advantage that quadratic terms in the functional to be optimized are treated implicitly allowing for a near minimum to be found after a single iteration in problems where quadratic terms dominate. Both the source problem for a potential field and the least-squares problem are solved using (linear) pre-conditioned conjugate gradients. Coupled with the use of parallel domain decomposition solution methods, this provides the numerical tools necessary for efficient inversion of multidimensional problems. Since the electrical inverse problem is ill-conditioned, special attention is given to the use of model-covariance matrices and data weighting to assist the inversion process to arrive at a physically plausible result. The model-covariance used allows for preferential model regularization in arbitrary directions and the application of spatially varying regularization. We demonstrate, using two previously published synthetic models, two methods of improving model resolution away from sources and receivers. The first method explores the possibilities of using depth-dependent and directionally varying smoothness constraints. The second method preferentially applies additional weights to data known to contain information concerning poorly resolved areas. In the given examples, both methods improve the inversion model and encourage the reconstruction algorithm to create model structure at depth.