Three-dimensional DC resistivity forward modeling and inversion by finite-element method.

Abstract

DC resistivity inversion is a method for determining underground geoelectrical structures from discrete measurements of electric potential made on the surface or within a borehole. In this dissertation, a fully three-dimensional (3-0) resistivity inversion algorithm has been developed. Based on a finite-element forward solution of Laplace's equation, the program estimates several thousand unknowns in a rectangular grid by the linearized least-squares method. In the first Chapter, the main 3-D forward modeling techniques were investigated. These techniques include boundary condition implementation, secondary field solution and matrix inversion. Among the various kinds of mixed boundary conditions, the terminal-impedance method is particularly well suited for 3-D resistivity modeling. Its implementation is simple, but eliminates the mesh-edge influence effectively. The advantage of calculating the secondary fields instead of the total fields is that a coarse mesh may be used to achieve the same accuracy, which turns out to be particularly beneficial for 3-D modeling. Compared with other relaxation methods to solve the linear system iteratively, the incomplete Cholesky conjugate gradient (lCCG) algorithm is superior in convergence rate. However, to guarantee a stable solution, this method also requires more regular elements. To make the program capable of overcoming non-uniqueness and handling large numbers of parameters, the sensitivity matrix construction and three constraining conditions are discussed in Chapter two. In 3-D DC resistivity inversion, computing the sensitivity matrix is an enormous task even when using reciprocity. This is because the total number of forward calculations used to construct the sensitivity matrix for one iteration of the inversion is on the order of the number of observed data. By applying the conjugate-gradient method to solve the least-squares system, our program only needs to calculate the product of the sensitivity matrix, or its transpose, with an arbitrary vector, which requires only two forward runs for each source point. The different constraining conditions were tested by several synthetic models. Although each method can give a unique solution, we found that in our case, the smoothest solution method will reduce the data error better than the other two methods, the damped method and the stochastic method. A number of simple but geophysically significant structures are also modeled to test the program. These include a single isolated block anomaly, three connected blocks representing a dipping fault and a multi-layer model. Data were simulated by both integral-equation and finite-element approximations. The main features of most resistivity models were identifiable in the inversion result. As an example of a 3-D inversion program application, a field data set was processed in Chapter three. The effects of some important parameters used in the program were discussed. The results were compared with a 2-D solution and the known geological structures around that area.