Inversion of two-dimensional resistivity and induced-polarization data

Abstract A fast ridge regression inversion technique has been devised for the interpretation of simple two-dimensional resistivity and induced-polarization data. The program will determine the rectangular source under a single layer of overburden which best fits the observed data.Several advantages are derived from using the ridge regression method; they include convergence from very poor initial guesses, stability in the presence of high-frequency geologic noise, readily obtained estimates of parameter statistics, and the ability for simultaneous inversion of multiple data sets. Unfortunately, each ridge regression inversion requires a great many forward problem evaluations; thus in order to achieve speed and reasonable cost, it is essential to reduce the calculation time for the forward problem to an absolute minimum. One method of achieving this is to store in the computer a data bank containing solutions for the entire range of expected parameter combinations. The forward problem then reduces to numerical interpolation between these precalculated data sets.For compilation of the data bank of forward solutions, two main numerical methods were investigated: the finite element and transmission-surface algorithms. Although these algorithms are conceptually quite different, the resulting matrix equations are very similar. The efficiency of either method depends mainly on the scheme chosen for solving the resultant large system of linear equations.Once the data bank has been created, it is possible to obtain inverse solutions for less cost than the computation of one finite element or transmission-surface forward problem. Tests on theoretical data and field data show the inversion technique to be reasonably accurate, stable, and fast.The statistics estimated by the inversion program provide additional useful information on the uncertainty in the parameters of the derived model and on high correlations between parameters. The most highly correlated parameters are, as might be anticipated, the resistivity and the width of thin conductive bodies. Two practical methods for carrying out inversion in spite of highly correlated parameters are, preferably, to add extra data sets which provide more information on some of the parameters or, alternatively, to fix some of the parameters at geologically reasonable values and invert to a more restricted model.