MLMATERN: A computer program for maximum likelihood inference with the spatial Matérn covariance model

The Matérn covariance scheme is of great importance in many geostatistical applications where the smoothness or differentiability of the random field that models a natural phenomenon is of interest. In addition to the range and nugget parameters, the flexibility of the Matérn model is provided by the so-called smoothness parameter which controls the degree of smoothness of the random field. It has been the usual practice in geostatistics to fit theoretical semivariograms like the spherical or exponential, thus implicitly assuming the smoothness parameter to be known, without questioning if there is any theoretical or empirical basis to justify such assumption. On the other hand, if only a small number of sparse experimental data are available, it is more critical to ask if the smoothness parameter can be identified with statistical reliability. Maximum likelihood estimation of spatial covariance parameters of the Matérn model has been used to address the previous questions. We have developed a general algorithm for estimating the parameters of a Matérn covariance (or semivariogram) scheme, where the model may be isotropic or anisotropic, the nugget variance can be included in the model if desired, and the uncertainty of the estimates is provided in terms of variance–covariance matrix (or standard error-coefficient of correlation matrix) as well as likelihood profiles for each parameter in the covariance model. It is assumed that the empirical data are a realization of a Gaussian process. Our program allows the presence of a polynomial trend of order zero (constant global mean), one (linear trend) or two (quadratic trend). The restricted maximum likelihood method has also been implemented in the program as an alternative to the standard maximum likelihood. Simulation results are given in order to investigate the sampling distribution of the parameters for small samples. Furthermore, a case study is provided to show a real practical example where the smoothness parameter needs to be estimated.