Feasibility Analysis of the Use of Binary Genetic Algorithms as Importance Samplers Application to a 1-D DC Resistivity Inverse Problem

Abstract

DC resistivity inverse problems are ill-posed. For the Vertical Electrical Sounding method the acceptable solutions lie in very narrow elongated-shape regions in the model space. To characterize this ensemble of solutions is a central question. In a Bayesian framework this issue is solved adopting as solution the so-called model posterior probability distribution. However, due to the nonlinearity of the problem, this distribution is not explicitly known, or it is difficult to calculate. Therefore, algorithms that efficiently sample the model space according to it (importance sampling) are very desirable. The main goal of this paper is to numerically explore the performance of binary genetic algorithms as posterior importance sampling strategies. Their behavior will be firstly analyzed using 2D synthetic posterior test functions bearing the relevant properties of the real geo-electrical inverse problem. The conclusions will be again checked through the histogram reconstruction of parameters in a synthetic VES case, and eventually, in a real, higher dimensional, sea-water coastal intrusion problem, by comparing the results with those obtained with a theoretically correct Metropolis-Hasting importance sampler (simulated annealing without cooling). Percentile curves are introduced as an effective tool for risk assessment. We show that binary genetic algorithms perform well under very general assumptions. When the roulette wheel is the selection method used, mutation is over 10%, and the algorithm does not incorporate elitism. The results do not depend on the values of the remaining tuning parameters. Finally, to improve the efficiency of the sampling strategy, we introduce a binary genetic algorithm with oriented search space. This is done with the help of linearization of the forward operator and singular value decomposition around the maximum posterior estimate. It is shown, also, that the logarithmic model parameterization is adequate for this task.