2-D resistivity inversion using spline parameterization and simulated annealing

Abstract

Successful inversion of geophysical data depends on prior information, proper choice of inversion scheme, and on effective parameterization of the model space such that the model representation is appropriate and efficient. Inversion of resistivity data has long been recognized as a nonlinear or quasi‐linear problem. Traditionally, 2-D resistivity inversion has been performed by trial and error methods and with linear and iterative linear methods. The linear and iterative linear methods are limited because of the requirement of good prior knowledge of the subsurface. Unlike linear and iterative linear methods, most nonlinear inversion schemes do not depend strongly on the starting solution, but prior information helps to reduce the computational cost and to obtain geologically meaningful results. In the present study, we have applied a nonlinear optimization scheme called very fast simulated annealing (VFSA) in the inversion of 2-D dipole‐dipole resistivity data to image the subsurface. Unlike Metropolis simulated annealing (SA) in which each new model is drawn from a uniform distribution, VFSA draws a model from a Cauchy‐like distribution, which is also a function of a control parameter called temperature. The advantage of using such a scheme is that at high temperatures, the algorithm allows for searches far beyond the current position, while at low temperatures, it looks for improvement in the close vicinity of the current model. We have used the mean square error between the synthetics and original data as the error function to be minimized. The synthetic response for 2-D models was obtained by finite‐difference modeling, and cubic splines were used to parameterize the model space to get smooth images of the subsurface and to reduce computational cost. VFSA was used to estimate the conductivity at each spline node location. The inversion was applied to various synthetic data to study the influence of the starting solution and the location of the spline nodes. Finally, we applied it to real data collected over a disseminated sulfide zone at Safford, Arizona, and compared the results with those obtained from a linearized inversion and from a model based on geologic and well‐log data. The VFSA results are in good agreement with the previously published results.