We have developed a new method for the approximate mapping of subsurface resistivity based on a geostatistical approach. We have used the characteristic points (CPs) of a resistivity sounding, which are their inflection and extreme (maximum and minimum) points. The methodology consists of five stages: (1) smoothing the geoelectric soundings to assure stability in the presence of measurement errors, (2) identifying the CPs in the smoothed versions of the geoelectric soundings, (3) obtaining subsurface sparse point resistivity estimates for the subsurface resistivity from the CPs, using empirical relationships between electrode spacing and penetration depth, (4) calculating semivariograms using the subsurface sparse point resistivity estimates, and fitting them with semivariogram models, and, finally, (5) estimating the subsurface resistivity distribution by kriging interpolation of the sparse point estimates. No explicit assumption is made about the true subsurface resistivity and, as result, the method can be applied to models of any dimension, given that the empirical rule for the penetration depth is approximately valid. Computationally, the method is very fast because no numerical modeling (either direct or inverse) is demanded and the most intensive computer operation is just a kriging interpolation. Given its independence of the model dimension and stability in the presence of measurement errors, the method can be readily implemented as a fast automatic method of interpretation. As result, the best scenario to apply the method is the initial treatment of large data sets. The methodology was developed, with synthetic and field data, using direct current (DC) resistivity soundings obtained with the Schlumberger array. The proposed method might be generalized for other DC-resistivity arrays and electromagnetic techniques based on resistivity soundings.

We have developed an inversion approach based on particle swarm optimization (PSO) for 2D direct current resistivity data. Once the PSO algorithm is implemented, cost-function changes on constraints and misfit criteria can be easily performed and an approximate invariance of the Lagrange multipliers is obtained if each term of the cost function is properly normalized. More importantly, the interpreter can divide the mesh into partitions and apply a different constraint or combination of constraints in each partition. We explore several restrictions on the log-resistivity variation, either in single or in joint versions, including spatial continuity constraints in the [Formula: see text]-and [Formula: see text]-norms, total variation, sparsity constraints using the discrete cosine transform and Daubechies bases, and minimum moment of inertia constraints to impose concentration of resistive or conductive materials along target axes. In the latter case, the earth surface might be used also as a target axis. In addition, the final state of the PSO algorithm, once a stopping condition has been reached, contains not only the best solution but also a cluster of good suboptimal solutions that can be used for uncertainty analysis. As a result, the interpreter has the flexibility to perform an interpretation process based on a feedback trial-and-error inversion approach, in a similar manner that he/she has when using a friendly forward modeling software. The interpreter can then drive the tentative solutions obtained along the inversion process to incorporate their conceptions about the geologic environment, besides appraising misfit and stability of the solutions. We evaluate synthetic and field data examples. In the first case, we determine how the interpreter can drive the inversion process in a karst environment to image dissolution features. In the second case, we explore a similar situation aiming to optimize borehole locations in fracture zones in crystalline rocks.

One-dimensional forward modeling in direct current (DC) resistivity is actually computationally inexpensive, allowing the use of global optimization methods (GOMs) to solve 1.5D inverse problems with flexibility in constraint incorporation. GOMs can support computational environments for quantitative interpretation in which the comparison of solutions incorporating different constraints is a way to infer characteristics of the actual subsurface resistivity distribution. To this end, the chosen GOM must be robust to changes in the cost function and also be computationally efficient. The performance of the classic versions of the simulated annealing (SA), genetic algorithm (GA), and particle swarm optimization (PSO) methods for solving the 1.5D DC resistivity inverse problem is here compared using synthetic and field data. The main results are as follows: (1) All methods reproduce synthetic models quite well, (2) PSO and GA are comparatively more robust to changes in the cost function than SA, (3) PSO first and GA second present the best computational performances, requiring less forwarding modeling than SA, and (4) GA gives higher performance than PSO and SA with respect to the final attained value of the cost function and its standard deviation. From our experience, to put them into effective operation, the methods can be classified from easy to difficult in the order PSO, GA, and SA as a consequence of robustness to changes in the cost function and of the underlying simplicity of the associated equations. To exemplify a quantitative interpretation using GOMs, we compare solutions with least-absolute and least-squares norms of the discrepancies derived from the lateral continuity constraints of the log-resistivity and layer depth as a manner of detecting faults. GOMs additionally provide the important benefit of furnishing not only the best solution but also a set of suboptimal quasisolutions from which uncertainty analyses can be performed.

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