Automatic variogram model fitting of a variogram map based on the Fourier integral method

Abstract

One of the cornerstones of geostatistics is the variogram. Fitting a variogram model to observed data is certainly one of the routine tasks in daily geostatistics practice. The central role of the variogram and the difficulty of performing variogram modeling manually call for an automatic, or at least a computer-assisted, method. Fitting a parametric curve to observational points in an XY chart is far from new and the GSLib software package has included a module (called VARFIT) to perform automatic variogram fitting in 1D since its debut many years ago. Nevertheless, extending its objective function to fitting to variogram map (2D) resulted in a method with poor performance. Hence, this work proposes a computational method to fit variogram models to variogram maps with a different objective function based on the principles of the Fourier Integral Method (FIM). A map is synthesized from the theoretical variogram model using the principles of FIM. The metric of the objective function is the sum of squares of differences between each pixel value of the synthetic map and the input map. The genetic algorithm uses this metric to find the variogram parameters of a model with good fitness. Since the genetic algorithm has a stochastic component, the best fit is not certain. Hence, several runs with different random number generator seeds are performed. The parameters of all fitted models are collected and cluster analysis in parameter space is done with the DBSCAN algorithm. The centers of the clusters are the parameters of the sought variogram model. Open-source software was written and made available, in which the proposed objective function was tested alongside the 2D version of the one in VARFIT. Results showed better effectiveness of the proposed method in fitting variogram models for the variogram map case.