Automatic Variogram Modeling by Iterative Least Squares: Univariate and Multivariate Cases

In this paper, we propose a new methodology to automatically find a model that fits on an experimental variogram. Starting with a linear combination of some basic authorized structures (for instance, spherical and exponential), a numerical algorithm is used to compute the parameters, which minimize a distance between the model and the experimental variogram. The initial values are automatically chosen and the algorithm is iterative. After this first step, parameters with a negligible influence are discarded from the model and the more parsimonious model is estimated by using the numerical algorithm again. This process is iterated until no more parameters can be discarded. A procedure based on a profiled cost function is also developed in order to use the numerical algorithm for multivariate data sets (possibly with a lot of variables) modeled in the scope of a linear model of coregionalization. The efficiency of the method is illustrated on several examples (including variogram maps) and on two multivariate cases.