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Previous studies show that optimized arrays generated using the ‘Compare R’ method have significantly better resolution than conventional arrays. This method determines the optimum set of arrays by selecting those that give the maximum model resolution. The number of possible arrays (the comprehensive data set) increases with the fourth power of the number of electrodes. The optimization method faces practical limitations for 2-D survey lines with more than 60 electrodes where the number of possible arrays exceeds a million. Several techniques are proposed to reduce the calculation time for such survey lines. A single-precision version of the ‘Compare R’ algorithm using a new ranking function reduces the calculation time by two to eight times while providing results similar to the double-precision version. Recent improvements in computer GPU technology can reduce the calculation time by about seven times. The calculation time is reduced by half by using the fact that arrays that are symmetrical about the center of the line produce identical changes in the model resolution values. It is further reduced by more than thirty times by calculating the Sherman–Morrison update for all the possible two-electrode combinations, which are then used to calculate the model resolution values for the four-electrode arrays. The calculation time is reduced by more then ten times by using a subset of the comprehensive data set consisting of only symmetrical arrays. Tests with a synthetic model and field data set show that optimized arrays derived from this subset produce inversion models with differences of less than 10% from those derived using the full comprehensive data set. The optimized data sets produced models that are more accurate than the Wenner–Schlumberger array data sets in all the tests. [ABSTRACT FROM AUTHOR]

Abstract: Four different methods to automatically select an optimal set of array configurations that gives the maximum subsurface resolution with a limited number of measurements for 2D electrical imaging surveys were tested. The first (CR) method directly calculates the change in the model resolution for each new array added to the base data set, and uses this to select array configurations that gave the maximum model resolution. However this method is the slowest. The algorithm used by the CR method for calculating rank-one updates was optimized to reduce computational time by a factor of eighty. The sequence of calculations was modified to reduce the traffic between the computer main memory and the CPU registers. Further code optimizations were made to take advantage of the parallel processing capabilities of modern CPUs. The second (ETH) and third (BGS) methods use approximations based on the sensitivity values to estimate the change in the model resolution matrix. The ETH and BGS methods, respectively, use the first and second power of the sensitivity values to calculate approximations of the model resolution. Both methods are about an order of magnitude faster than the CR method. The results obtained by the BGS method are significantly better than the ETH method, and it approaches that of the CR method. The fourth method (BGS–CR) uses a combination of the BGS and CR methods. It produces results that are almost identical to the CR method but is several times faster. The different methods were tested using data from synthetic models and field surveys. The models obtained from the inversion of the data sets generated by the four different methods confirm that the models generated by the CR method have the best resolution, followed by the BGS–CR, BGS and ETH methods. [ABSTRACT FROM AUTHOR]

The smoothness-constrained least-squares method is widely used for two-dimensional (2D) and three-dimensional (3D) inversion of apparent resistivity data sets. The Gauss–Newton method that recalculates the Jacobian matrix of partial derivatives for all iterations is commonly used to solve the least-squares equation. The quasi-Newton method has also been used to reduce the computer time. In this method, the Jacobian matrix for a homogeneous earth model is used for the first iteration, and the Jacobian matrices for subsequent iterations are estimated by an updating technique. Since the Gauss–Newton method uses the exact partial derivatives, it should require fewer iterations to converge. However, for many data sets, the quasi-Newton method can be significantly faster than the Gauss–Newton method. The effectiveness of a third method that is a combination of the Gauss–Newton and quasi-Newton methods is also examined. In this combined inversion method, the partial derivatives are directly recalculated for the first two or three iterations, and then estimated by a quasi-Newton updating technique for the later iterations.The three different inversion methods are tested with a number of synthetic and field data sets. In areas with moderate (less than 10:1) subsurface resistivity contrasts, the inversion models obtained by the three methods are similar. In areas with large resistivity contrasts, the Gauss–Newton method gives significantly more accurate results than the quasi-Newton method. However, even for large resistivity contrasts, the differences in the models obtained by the Gauss–Newton method and the combined inversion method are small. As the combined inversion method is faster than the Gauss–Newton method, it represents a satisfactory compromise between speed and accuracy for many data sets. [Copyright &y& Elsevier]