UBC Theses and Dissertations
Forward modeling and inversion of DC resistivity and MMR data McGillivray, Peter Robert
This thesis is a presentation of research that has addressed the forward and inverse problems for the DC resistivity and magnetometric resistivity (MMR) experiments. The emphasis has been on the development of practical numerical procedures for solving problems involving a large number of data and unknowns. The relative value, in terms of information content, of the different data sets arising from DC resistivity or MMR experiments has also been emphasized. For the purposes of this work, a two-dimensional conductivity structure was assumed, although the results can be extended to three-dimensions. The first part of this research focused on the development of numerical algorithms to accurately and efficiently solve the forward problem. A forward modeling algorithm based on the integrated finite difference discretization was first developed. The algorithm was designed to model a variety of different responses, including pole-pole, pole-dipole, dipole-dipole and MMR measurements for surface, cross-borehole and borehole-to-surface arrays. Complex conductivity structures and topography can also be specified. The high accuracy of the algorithm was demonstrated by comparing forward modeled results with analytic solutions for different conductivity models. The algorithm was then used in the development of a multi-grid procedure for iteratively computing DC resistivity responses. The multi-grid algorithm makes use of a sequence of grids of increasing fineness to accelerate convergence. Testing of the multi-grid algorithm demonstrated its value as a fast and accurate solver for the DC resistivity problem. The use of non-coextensive grids was also found to be useful for achieving higher resolution in the vicinity of singularities and rapid changes in the conductivity. The multi-grid approach was used in the development of a novel adaptive grid design procedure that iteratively refines an initial numerical grid. The application of this algorithm to the modeling of DC resistivity responses for different conductivity structures illustrated the usefulness of the approach, and demonstrated the need for assessing the accuracy of a numerically computed solution. The second part of this research focused on the calculation of the sensitivities of the modeled responses to changes in the model parameters — quantities that are essential to the solution of the non linear inverse problem. A study of the available techniques for numerically computing sensitivities was carried out to determine the most suitable approach. Based on this study, an adjoint formulation for the 2D resistivity and MMR problem was developed. A comparison of these numerically computed sensitivities to ones obtained by perturbing the model parameters verified the accuracy of the approach. The final part of this research focused on the solution of the inverse problem. The inversion of DC resistivity data has traditionally employed a coarse parameterization of the model to reduce the non-uniqueness of the problem. Although this can succeed in reducing the non-uniqueness, it can also lead to problems of poor stability and slow convergence. In this work, a strategy of using a fine parameterization of the model was adopted. The resulting non-uniqueness was reduced by requiring that the final solution minimize a global norm of the model. Computational problems were addressed using a re-parameterization based on a generalized subspace approach. The use of linearized information to avoid computing a large number of forward solutions was examined. The resulting generalized subspace algorithm was tested by inverting synthetic data sets generated for pole-pole, pole-dipole, dipole-dipole and cross-borehole electrode configurations. The success of these inversions demonstrated the stability and efficiency of the generalized subspace approach. Synthetic MMR data Sets were also inverted, both individually and in a joint inversion with polepole resistivity data. The results indicated that the additional information provided by the magnetic field data can help to better resolve the subsurface. An E-SCAN pole-pole field data Set was also inverted, and a solution that delineated two conductors was obtained. The solution was consistent with geological cross-Sections that were available for the study area.
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