1-D inversion of resistivity and induced polarization data for the least number of layers

Abstract

An automatic inverse method has been developed for generating layered earth models from electrical sounding data. The models have the minimum number of layers required to fit a resistivity sounding curve or a combined resistivity and induced polarization sounding. The ground is modeled using a very large number of thin layers to accommodate arbitrary variations. The properties of the layers are optimized using as a constraint the L 1 norm of the vertical derivative of the resistivity distribution. The use of linear programming leads to piecewise smooth distributions that simulate traditional models made up of a few uniform layers. The process considers from the simplest model of a uniform halfspace to models of many layers, without fixing a priori the number of discontinuities. The solution is sought by iterating a new linear approximation, similar to the classical process of linearization, except that a reference model is not present in either the data vector or the unknown function. For induced polarization soundings, the problem is linear and the solution is obtained in a single iteration, provided an adequate resistivity model is available. The performance of the method is illustrated using numerical experiments and published deep resistivity data from Australia. The method also is applied to combined resistivity and induced polarization soundings from a local groundwater prospect in México.

Alternative theory for signal contribution sections and depth of investigation characteristics in electrical prospecting

“…However, it is independent of resistivity for homogeneous media; therefore, the equation can be the basis of first‐order approximations or imaging techniques as developed by Pérez‐Flores, Méndez‐Delgado, and Gómez‐Treviño (). Updating the kernel and solving again for resistivity leads to iterative algorithms similar to those developed for small perturbations (Esparza and Gómez‐Treviño ).…”

Mapping of thin sandy aquifers by using high resolution reflection seismics and 2-D electrical tomography

“…After automatic interpretation of each sounding, the interpreted data is merged to form a quasi 2-D section. Many 1-D inversion techniques for interpretation electrical sounding have been published (Oldenburg, 1978;Constable et al, 1987, Zohdy, 1989Esparza and Gomez-Trevino, 1997). This technique is useful in stratified earth models, meaning basin conditions with slight or gradual lateral resistivity variations, and gradual changes of layer thickness.…”

Nonsmooth Nonconvex Optimization for Low-Frequency Geosounding Inversion

“…ψ is a strictly convex and decreasing function such that φ(t) = inf w (wt 2 + ψ(w)), and for the case of potential function in (11), ψ(w) = wβ 2 − 2 √ wβ + 1. The half quadratic procedure is based on alternating minimizations over m and b. Charbonnier et al proved convergence …”