CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 3, Number 3, Summer 1995

THREE-DIMENSIONAL INVERSION SCHEME FOR MISE-A-LA-MASSE PROSPECTING DATA

USING SIMULATED ANNEALING

D.J. MOSELEY, H. RASMUSSEN AND P. FORSYTH

1. Introduction. Electromagnetic methods are used extensively in the field of mineral exploration. These techniques involve generating a constant, sinusoidal or transient electromagnetic field and measuring the response caused by a buried conductive anomaly. The induced fields are then used to infer details of the conductivity structure of the subsurface. Electromagnetic techniques which utilize a constant current source (DC) are referred to as resistivity methods.

A common resistivity technique for surveying vein-type ore bodies is the mise-bla-masse prospecting method which was pioneered by Schlumberger in the 1920's [19].In this method, a constant current source is applied directly to the ore body, either through a drill hole which intersects the body or through an exposed outcropping. The current source is grounded at a large distance from the body to effectively isolate the source. Measurements of potential or potential gradient (electric field) are then made at numerous points on the surface to generate a 2-D response profile. Interpretation of the resultant surface fields represents a difficult inverse problem. The difficulty arises from the fact that the system is highly nonlinear, nonunique, and possesses a high degree of dimensionality. Discussion of the inverse problems associated with electromagnetic prospecting in geophysics can be found in [17].

Solution of the forward problem (i.e., knowing the conductive prop- erties of the medium and solving for the surface fields) is achievable numerically. Suitable numerical solution techniques include finite dif- ferences [7],finite elements [16]or integral equation methods [lo, 241. While these researchers consider full three dimensional models of the conductivity substructure, none attempt to solve the inverse problem; that is, recovering the conductivity structure through boundary mea- surements. Not only does the work presented here utilize a full 3D

Accepted by the editors on September 20, 1994. Copyright 01995 Rocky Mountain Mathematics Consortium

338 D.J. MOSELEY. H. RASMUSSEN AND P. FORSYTH

model but successful inversion using the surface fields is achieved.

A class of reconstruction methods phrase the solution to an inverse problem in terms of a multidimensional cost function [B, 181. The cost function in this case is based on fitting the measured surface field data in a least squares sense. Using the chosen numerical forward solver one could find an approximate solution to the inverse problem using a guess and improve method. The global minimum of the cost function now represents the best solution to the inverse problem. Finding the global minimum is a task ideally suited to the optimization technique, simulated annealing [I, 5, 6, 141. Inversions in geophysics have been attempted before using the simulated annealing technique. For example, in 1201, simulated annealing was used in the inversion of nonlinear seismic soundings for a 1D earth model.

This paper will formulate the mathematical model for the case of mise-8-la-masse prospecting. A numerical method using a finite differ- ence approximation is proposed. A least squares fit to the surface field is suggested as a suitable method to approximate the inverse prob- lem including geophysically motivated assumptions which regularize the problem. The value of the least squares fit is assigned to a cost function where the minimum value of the cost function is the best fit of the surface fields and hence the best approximation to the inverse prob- lem. The search for this best-fit is attempted with the use of simulated annealing. A proof of principle example using a synthetically generated data set is furnished to validate the technique. Further work includes an example which incorporates the dimensions from a real-life vein- type ore body. Extensions to the model which investigate the effects of random data noise and conductive overburden are also considered.

2. Formulation. The mise-8-la-masse prospecting method can be modelled by the electrostatic problem of an electrode buried in an inhomogeneous infinite half-space. The conservation of electric charge dictates that the current density J obeys

when no charge flux is contained. In a linear isotropic medium the relation between current density and electric field E is given by