UBC Theses and Dissertations
Inversion of three-dimensional direct current resistivity data Li, Yaoguo
A direct current (d.c.) resistivity experiment investigates subsurface geo-electrical structures by measuring the electric field set up by introducing current into the earth. Information about geo-electrical structures is extracted by inverting the observed data to generate an image of the conductivity or to construct a conductivity model. The goal of this thesis is to develop efficient inversion techniques for the interpretation of three-dimensional (3d) d.c. resistivity data. The study assumes data consisting of pole-pole potentials measured over a regular grid on the surface for many current locations. The Born approximation is employed to linearize the inverse problem. The source of the electric field measured in the d.c. resistivity is the accumulated electric charges. Different aspects of the charge accumulation are reviewed, enlarged with new insights and presented in a unified notation. This provides the basis for understanding the fundamentals of d.c. resistivity experiments. Two algorithms are developed to image simple 2d conductivities. The first constructs a structural image by combining the charge density images obtained by inverting multiple sets of common current potentials. The second constructs a conductivity image directly. Processing and displaying the apparent conductivity, and constructing equivalent sources from secondary potentials are studied as the means of imaging. Assuming a multiplicative perturbation to a uniform half-space, the potential anomaly of pole-pole arrays is expressed as a depth integral of the logarithmic perturbation convolved with a kernel function in the horizontal directions. Applying the Fourier transform decomposes the data equation for a 3d problem into a set of id equations. A rapid approximate 3d inversion is developed based upon this decomposition by solving a sequence of id inversions in the wavenumber domain. The approximate 3d inversion is used to construct iterative inversion algorithms using the AIM (Approximate Inverse Mapping) formalism. The approximate inversion and an exact forward mapping are used to update the model successively so that the final result reproduces the observed data. The AIM inversion is applied to analyse a set of field data.
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