Earth Resources Laboratory :: Abstracts

Advanced Modeling and Inversion Techniques for Three-dimensional Geoelectrical Surveys

Weiqun Shi

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 1, 1998 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis develops an integrated methodology for high resolution geoelectrical surveys including a physically meaningful inversion method, an efficient inversion algorithm, a resolution and uncertainty analysis technique, and an effective data acquisition geometry. The methodology is applied to three important geoelectrical inverse problems: 3-D d.c. electrical resistivity, 3-D electrical induced polarization, and 3-D electrical self-potential.

The 3-D d.c. electrical resistivity inversion recovers the subsurface bulk resistivity distribution from static electrical potential measurements obtained on the surface of the earth or in boreholes. This is an ill-posed problem in the sense that a large number of solutions to the inverse problem exist due to incomplete and uncertain data. To reduce the ill-posedness, this thesis investigates inversion algorithms based on the Tikhonov regularization method which solves a minimization problem to find models that fit the data and also have minimum structure. Different smoothness constraints are investigated to obtain the minimum structure. A smoothness operator that employs the second-order spatial derivatives (the Laplacian) is found to be most effective in yielding a stable inversion solution and eliminating surface artifacts.

To implement the regularized inversion on a 3-D resistivity model, one faces a computational challenge owing to the nonlinear nature of the resistivity problem and the large number of model parameters and data which can possibly exist in a moderate 3-D model. This thesis develops an efficient numerical algorithm based on the nonlinear conjugate gradient method with pre-conditioning to minimize the objective functional and solve the inverse problem. Different pre-conditioners are investigated and their efficiency are compared. By using a pre-conditioner based on the approximate form of the Hessian matrix of the objective function, the nonlinear conjugate gradient method results in a tremendous time saving over the conventional Gauss-Newton approach.

The nonlinear regularized inversion methodology is then extended to solve the inverse problem of 3-D electrical Induced Polarization (IP). The subsurface complex resistivity distribution is reconstructed from the measurements of the amplitude and phase of the electrical potential in the frequency domain. Given a complex resistivity structure, the forward modeling which predicts the complex electrical potential distribution is solved by a bi-conjugate gradient method. Because the linear system of the equation for the forward modeling has a complex symmetric conductance matrix, the bi-conjugate gradient method is simplified to a special form which is comparable to the (real) conjugate gradient method that is used in the d.c. resistivity forward modeling. While in the IP inversion, the imaginary component of the complex resistivity is much smaller than the real part, the objective function is constructed in a complex form, and the minimization is solved directly in the complex domain using a bi-conjugate gradient method. This approach makes the inversion of 3-D Induced Polarization efficient because the computational cost is similar to that of the d.c. resistivity problem.

The inversion methodology is also extended to the inverse problem of 3-D electrical Self-Potential (SP), here the subsurface electrical current source distribution induced by underground mechanical and electrochemical activities is recovered. The SP inverse problem is inherently non-unique, in fact one can obtain a perfect data fit by appropriately adjusting the location, magnitude, and dimension of the electrical current source in many combinations. To reduce the nonuniqueness, the regularization constraints are justified and extended to a broader range of formulation including constraints on the resistivity structure and constraints on position, orientation, magnitude, or dimension of the SP source geometry.

Usually, the inversion reconstruction is evaluated in terms of how well the data are fit, but the suitability of the solution is better judged through uncertainty and resolution analysis. This thesis introduces an uncertainty and resolution analysis to quantify the variance and resolution length as a function of position for the geoelectrical inversion. It appeals to the Bayesian framework whereby both variance and resolution are inferred from the a posteriori covariance associated with the Tikhonov regularization method. The a posteriori covariance matrix is first calculated on an optimal nonlinear regularization solution by inverting the associated Hessian matrix or a Monte Carlo sampling method to give a local estimate of uncertainties about the optimal solution. Such resulted uncertainty does not posses an accurate measure for every model parameter. Therefore, the only uncertainties extracted are the ones associated with deterministically resolved model parameters. Then these uncertainties are calibrated from a sensitivity analysis. The uncertainty associated with the other model parameters are thus obtained. To measure the resolution power of the inversion technique, a Monte Carlo method is used to invert realizations of perturbed data and obtain the a posteriori model correlation. For computational efficiency the resolution is also analyzed through the Modulation Transfer Function, borrowed from the optical imaging community. The numerical analysis of synthetic data demonstrates that the method gives resolution and variance information that correlates well with the electrical current coverage and the character of the associated reconstruction.

In order to increase the accuracy of the geoelectrical imaging technique, a new survey geometry which employs a spatially varying source dipole is designed. This new survey geometry is investigated with sensitivity analysis and model correlation estimation, and it appears to be more effective than traditional pseudo-section acquisition geometry in cases where structure has an extended lateral variation.

Finally, our inversion techniques are successfully applied to various geoelectric field measurements. The first example applies the 3-D d.c. resistivity tomography to characterize subsurface soil properties in an effort to understand the transport mechanisms that have been involved in drinking water contamination in the Aberjona River of Woburn, Massachusetts. The inversion result correlates well with other geophysical results at the site (GPR sections and cone penetrometer logs) and extrapolates the sparse stratigraphic information into a full 3-D model of the study area. The second example uses the 3-D d.c. resistivity tomography to monitor changes in subsurface electrical resistivity caused by the movement of water and the conversion of water to steam in the Larderello-Valle Secolo geothermal field in Italy. Comparisons of the resistivity anomalies obtained from two surveys conducted in 1991 and 1993 indicate a correlation between the changes in resistivity and the water re-injection history. The results show that it is possible to evaluate and detect the re-injection of fluid through systematic observation of electrical resistivity at the site. The third example uses the 3-D d.c. resistivity tomography to map underground limestone caves in Barbados, West Indies. The inversion successfully identifies the known caves and a previously undiscovered cave. In the last example, the 3-D electrical Self-Potential tomography is used to investigate groundwater contamination associated with a jet-fuel leakage at Massachusetts Military Reservation in Cape Cod, Massachusetts. The inversion locates and describes the shape of the contaminant plume which matches well the plume geometry obtained from drill-hole samples.