Extended Born approximation as a preconditioning operator for conjugate gradient forward and inverse methods

Abstract

Interest in cross-well tomography of the Earth electrical conductivity has increased because of improvements in field instrumentation, computing power, and method of interpretation. Cross-well electromagnetic logging is a technique to investigate the geological properties of the region between boreholes from the electromagnetic measurements made in these boreholes at one or more frequencies for a large combination of source and receiver locations. Recently, the authors have developed an inversion method to reconstruct a 3D cross-well conductivity distribution from the static/electrode logging measurements. The method used is based on the conjugate gradient (CG) method, and is called the contrast source inversion (CSI) method. Unlike the most nonlinear inversion methods, the CSI method does not require some artificial regularization techniques to deal with the problems of the non-uniqueness in inversion of data. It attempts to overcome this problem by recasting the problem as an optimization problem, in which it seeks not only the contrast sources but also the conductivity contrasts itself to minimize a cost functional. The cost functional consists of two terms. The first term is the error in matching measured field data with the field scattered by a body with particular conductivity and the second being the error in satisfying the integral equations for the field produced in the body by each excitation. This cost functional is minimized iteratively using conjugate gradient directions. In each iteration of CSI method, they first update the contrast source with a conjugate gradient step weighted so as to minimize the cost functional. Then, in the second step, the conductivity contrast is updated to minimize the error in the object equation using the updated contrast source also with a conjugate gradient step. In this way, they do not solve full forward problems in each iteration of our inversion algorithm. In fact the total computational effort of our inversion scheme is approximately the same as the computational effort of solving only two forward problems with linear CG schemes. In the present paper they have extended the CSI method to handle the full-vector complex 3D cross-well induction.