Intrinsic and Generalized Random Fields

An intrinsic random field can be described as a random field with stationary increments. In a generalized random field, the realizations are too rough to be ordinary functions. The spectral representation for the covariance function of a stationary random field can be extended to cover random fields that are intrinsic or generalized or both. This chapter focuses on an intrinsic random field of order k = 0. It illustrates some typical behavior in isotropic semivariograms. The class of function-indexed random fields is wider than the class of site-indexed random fields. Such random fields are called generalized random fields. A fundamental property that can be possessed by a random field is self-similarity. Many of the core models in the theory of Gaussian random fields possess the property of self-similarity. The chapter focuses on the simulation methods for stationary Gaussian random fields.