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Overview: Because the open channel through which we propagate electromagnetic radiation is often considered a turbulent medium, we present a brief review in this chapter of the main ideas associated with a random field, which in general is a function of a vector spatial variable R and time t. To begin, however, we start with the somewhat simpler concept of a random process and then present a parallel treatment for a random field. Fundamental in the study of random processes is the introduction of ensemble averages, which are used to formulate mean values, correlation functions, and covariance functions. The development of these statistics is greatly simplified for a stationary process, which means that all statistics only depend on time differences and not the specific time origin. From a practical point of view, however, we usually consider just the weaker condition of a stationary process in the wide sense, which demands only that the mean and covariance be invariant under translations in time. Whereas theoretical treatments of a random process ordinarily involve the ensemble average, measurements of various statistics of a random process make use of the long-time average. Nonetheless, if a random process is ergodic, then we can equate long-time-average statistics to ensemble averages. In addition to mean values, correlation functions, and covariance functions, we also introduce the notions of structure function and power spectral density. Structure functions, which involve averages of squared differences, are widely used in turbulence studies, particularly if the random process is not stationary but has stationary increments. The power spectral density is simply the Fourier transform of the covariance function and, consequently, contains the same information in a different form. Last, our treatment of random fields is virtually identical to that of random process but there are some subtle differences between the two. For example the notion of “statistical homogeneity” is the spatial counterpart of the temporal “stationarity”. It is also common to assume that a random field satisfies the additional property of isotropy, which means that the random field statistics depend only on the scalar distance between spatial points.
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