The inverse problem of Schlumberger resistivity sounding measurements by ridge regression

Abstract

When the number of soil layers exceeds two, the determination of the resistivity and depth (ρi,hi) for layer # i from a resistivity sounding becomes more difficult. The proposed method is based on the identification of the theoretical resistivity transform ‘T’ for n-layer from the expression of the voltage distribution around an electrode in a multi-layer soil. Vertical soundings in the field measure a series of apparent resistivity values at a constant increment Δx of the logarithm of electrode spacing. We obtain the filter coefficient series. And thus, the resistivity transform can be easily derived from the experimental data by the application of linear electric filter theory. The problem of direct interpretation of apparent resistivity curves from horizontally layered earth models is solved by using the ridge regression estimator. This paper presents a methodology for estimation of soil parameters from Schlumberger measurements. The procedure has been applied using field test data and has been shown to provide a satisfactory soil model.