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In Geostatistics, primary interest often lies in the study of the spatial, or spatial-temporal, correlation of real-valued random fields, anyway complexvalued random field theory is surely a natural extension of the real domain. In such a case, it is useful to consider complex covariance functions which are composed of an even real part and an odd imaginary part. Generating complex covariance functions is not simple at all, but the procedure, developed in this paper, allows generating permissible covariance functions for complex-valued random fields in a straightforward way. In particular, by recalling the spectral representation of the covariance and translating the spectral density function by using a shifting factor, complex covariances are obtained. Some general aspects and properties of complex-valued random fields and their moments are pointed out and some examples are given.In Geostatistics, one of the main problems consists in finding a function which can be used as a covariance, in order to describe the spatial or spatial-temporal correlation of the phenomenon under study. Practitioners usually study the correlation of a random field in the real domain and fit known permissible covariance models to the experimental covariances. The even covariance function of a real-valued random field could represent a limitation in some geostatistical applications. In the multivariate context, for example, the Linear Model of Coregionalization (Journel and Huijbregts, 1981;Wackernagel, 1998), in spite of its computational semplicity, has the limitation to the simple and cross-covariance to be even functions; Grzebyk (1993) proposed an extension of the Linear Model of Coregionalization to the complex domain in order to allow the use of not-even cross-covariance functions. Anyway the major contribution given by Grzebyk in the complex theory was oriented to a multivariate context.

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