The cgeostat Software for Analyzing Complex-Valued Random Fields

Abstract

Given a vectorial data set in two dimensions, a representation on a complex domain is often convenient. This representation is rarely considered in geostatistics, although interesting applications can be found in environmental sciences and meteorology (e.g., for wind fields). In such a case, some computational difficulties are related to the lack of software for estimating and modeling a complex covariance function, for predicting complex variables as well as for representing the output results. In this paper, the new Fortran software cgeostat for geostatistical analysis of complex-valued random fields is presented and an application is demonstrated.

New spatio-temporal complex covariance functions for vectorial data through positive mixtures

“…Indeed, the complex covariance depends on the direct covariances (in the real part) and on the cross covariances (in the imaginary part); however, it is not necessary to model separately the two direct covariances and the two cross covariances, but just the sum of the former and the difference of the latter. Further details on this aspect can be found in Wackernagel (2003) andDe Iaco andPosa (2016). It is worth highlighting that these perks are even more appreciable when these data have a spatio-temporal structure.…”

“…Under a computational point of view, it can be convenient to estimate the real and imaginary parts of the marginals, they can be employed to separately estimate the spatial shifting factor and the temporal one, by following the same idea given in De Iaco and Posa (2016) andDe Iaco (2017). A discussion on the effect of the shifting factor can be found in De Iaco et al (2013b).…”

Families of complex‐valued covariance models through integration

“…Thus, the reliability of the model has been evaluated with respect to a progressive reduction of the available data. Computational aspects have been handled by implementing the new model in the “ckriging” routine of the package cgeostat (De Iaco, 2017).…”

“…Nevertheless, the works of Yaglom (1987) and Lajaunie and Béjaoui (1991) have represented one of the first theoretical evidences regarding the complex framework and the methodology to study vector data in a domain of one (temporal) or two (spatial) dimensions as well as the results in Grzebyk (1993), Wackernagel (2003), andDe Iaco et al (2003) have contributed to the discussion on modeling and interpolating a complex variable. Further advances have enriched the parametric covariance families of complex-valued random fields (Posa, 2020(Posa, , 2021 and have focused on applications and computational tools concerning structural analysis and interpolation of vectorial data in two dimensions (De Iaco, 2017;De Iaco et al, 2013). In particular, this last article has introduced a specific package, called cgeostat, which can be used to deal with the various steps of the study of phenomena with a complex representation, from estimation and modeling of the real and imaginary parts of the complex covariance function to the complex kriging based on the use of a class of complex covariance models obtained, from the Bochner characterization, by translating an even spectral density function through a shifting factor.…”