Families of complex-valued covariance models through integration

In geostatistics, the theory of complex-valued random fields is often used to provide an appropriate characterization of vector data with two components. In this context, constructing new classes of complex covariance models to be used in structural analysis and, then for stochastic interpolation or simulation, represents a focus of particular interest in the scientific community and in many areas of applied sciences, such as in electrical engineering, oceanography, or meteorology. In this article, after a review of the theoretical background of a random field in a complex domain, the construction of new classes of complex-valued covariance models is proposed. In particular, the complex-valued covariance models obtained by the convolution of the real component are generalized and wide new classes of models are generated through integration. These families include even non-integrable real and imaginary components of the resulting complex covariance models. It is also illustrated how to fit the real and imaginary components of the complex models together with the density function used in the integration. The procedure is clarified through a case study with oceanographic data.