Complex valued random fields, a natural generalization of real valued random fields, represent a powerful tool for modeling phenomena which evolve in time, spatial vectorial data in two dimensions and spatio-temporal vectorial data (i.e., a wind field). However, only a few efforts have been proposed in the literature to address some relevant aspects of spatial and spatio-temporal geostatistical analysis concerning complex valued random functions. For example, it is well known that covariance is, in general, a complex valued function; for this purpose, the main aim of this paper concerns the construction of parametric models for complex valued covariance functions, which are able to model wide families of complex valued random fields. Indeed, a previous class of complex covariance functions, which were built by simply translating an even spectral density function, was able to model just correlation structures characterized by damped oscillations because of the necessary presence of the cosine and sine functions in the real and imaginary part, respectively, of the complex covariance function. The wide class of covariance models, proposed in this paper, can be considered the basic building blocks which could be utilized in a spatial context, in several dimensional spaces, as well as in a spatio-temporal domain to model the correlation structure of complex valued random fields. Moreover, in order to provide a complete overview of the subject, a brief outline of the construction of the subset of real covariance functions has also been given; in particular, it has been shown how some parametric families of real valued covariance functions, whose spectral representation cannot provide an analytic solution, can be constructed using the formalism of differential equations.

复数值随机场是实数值随机场的自然概括，它代表了一种强大的工具，可以对随时间演变的现象，二维空间矢量数据和时空矢量数据（即，风场）。然而，在文献中仅提出了很少的努力来解决涉及复杂值随机函数的空间和时空地统计分析的一些相关方面。例如，众所周知，协方差通常是一个复数值函数。为此，本文的主要目的是为复数值协方差函数建立参数模型，该模型能够对复数值随机域的宽泛族建模。实际上，通过简单地平移偶数频谱密度函数而建立的上一类复杂协方差函数，由于在实部和虚部中必然存在余弦和正弦函数，因此能够对以阻尼振荡为特征的相关结构进行建模。 ， 分别，的协方差函数 本文提出的一类广泛的协方差模型可以被视为可在空间上下文，几个维度空间以及时空域中用来对复数值的相关结构进行建模的基本构件。随机字段。此外，为了提供对该主题的完整概述，还给出了实协方差函数子集的构造的简要概述。特别是，已经证明了如何使用微分方程的形式来构造一些实值协方差函数的参数族，其频谱表示不能提供解析解。可以认为是可以在空间上下文中，几个维空间以及时空域中用来对复数值随机域的相关结构进行建模的基本构件。此外，为了提供对该主题的完整概述，还给出了实协方差函数子集的构造的简要概述。特别是，已经证明了如何使用微分方程的形式来构造一些实值协方差函数的参数族，其频谱表示不能提供解析解。可以认为是可以在空间上下文中，几个维空间以及时空域中用来对复数值随机域的相关结构进行建模的基本构件。此外，为了提供对该主题的完整概述，还给出了实协方差函数子集的构造的简要概述。特别是，已经证明了如何使用微分方程的形式来构造一些实值协方差函数的参数族，其频谱表示不能提供解析解。还给出了实协方差函数子集构造的简要概述；特别是，已经证明了如何使用微分方程的形式来构造一些实值协方差函数的参数族，其频谱表示不能提供解析解。还给出了实协方差函数子集构造的简要概述；特别是，已经证明了如何使用微分方程的形式来构造一些实值协方差函数的参数族，其频谱表示不能提供解析解。