Fitting variogram models based on estimation errors and variances using genetic algorithms

Abstract

The hydraulic head is an important variable to determine the functioning of water in the subsoil; however, its spatial characterization is complicated due to the variability it presents in an aquifer. Measuring hydraulic head in piezometers or observation wells involves costs, so in some cases there is little data available. To obtain reliable configurations of the hydraulic head spatial distribution in an aquifer, interpolation methods that require few measurements have been used. Ordinary kriging is one of the most widely used spatial interpolation algorithms in geostatistics, which employs a theoretical variogram (circular, exponential, Gaussian, etc.). The variogram is a function whose parameters (nugget, sill and range) must be optimized because the accuracy of the estimation depends on them. As far as it has been reviewed in the literature, the adjustment of theoretical variograms has been carried out by means of genetic algorithms considering bi-objective functions where only the error in the adjustment of the variogram and the difference between the measured values and the estimated values by means of ordinary kriging are taken into account. In this paper we propose the adjustment using a new multiobjective function, where simultaneously the variogram adjustment, the accuracy of the interpolation result and the estimation error variances are considered. This nonlinear optimization problem contains three secondary objectives. The first is to obtain the best fit between the experimental variogram and the theoretical variogram function. Secondly, the aim is to minimize the difference between the measured values and the ordinary kriging estimates (measured with the mean square error) and thirdly that the error variances in the estimation are well represented by the selected model (using the standard mean square error). The tests of the proposed procedure were carried out with data measured in El Palmar aquifer located in the northern part of the state of Zacatecas, Mexico. The performance of this procedure was evaluated for different weights assigned to each of the secondary objectives. In the models where only the variogram adjustment is considered, the mean squared error and the standardized mean squared error turned out to be very large, it was also observed that when the estimation error variance is not taken into account in the objective function, the standardized mean squared error ranges from 20.94 to 56.41. It was observed that when the estimation error variance is incorporated in the objective function (even when its weight is small) the estimation errors are very close to the minimum obtained and that the variances are very reliable (with the standardized mean square error between 0.65 and 1.35).