Comparison of four least-squares inversion schemes for studying equivalence in one-dimensional resistivity interpretation

Abstract

Abstract The problem of equivalence in dc resistivity inversion is well known. The ability to invert resistivity data successfully depends on the uniqueness of the model as well as the robustness of the inversion algorithm. To study the problems of model uniqueness and resolution, theoretical data are inverted using variations of a nonlinear least-squares inversion. It is only through model studies such as this one, where the true solutions are known, that realistic and meaningful comparisons of inversion methods can be undertaken. The data are inverted using three schemes of fixed-layer thickness where only the resistivity varies, and the results are compared to the variable parameter inversion where both the layer resistivities and thicknesses are allowed to vary. The purpose of fixing the layer thicknesses is to reduce the number of parameters solved for during the inversion process. By doing this, nonuniqueness may be reduced. The fixed-layer thickness schemes are uniform thickness, geometrical progression of thickness, and logarithmic progression of thickness. By applying each inversion scheme to different models, the layer thickness that minimizes the data rms error for various numbers of layers is determined. The curve of data rms error versus model rms error consists of three general regions: unique, nonunique, and no resolution. A good inversion routine simultaneously minimizes the data rms and model rms errors. The variable parameter scheme is best at simultaneously minimizing the data rms and model rms errors for models that can be resolved through the inversion process. The optimum number of layers in the model can be determined by using a modified F-test.