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Atmospheric and Oceanic Optics

2020 year, number 2

Spectral density of stationary random processes with a power structure function

V.A. Fedorov
V.E. Zuev Institute of Atmospheric Optics of Siberian Branch of the Russian Academy of Science, Tomsk, Russia
Keywords: стационарный случайный процесс, случайный процесс со стационарными приращениями, структурная функция, спектральная плотность, stationary random process, random process with stationary increments, structure function, spectral density


The conditions for a function to fall in the class of structure functions for stationary random processes are formulated. In the spatial domain, this corresponds to a homogeneous and isotropic scalar field. It is shown that the power function can be a structure function only if the power index is no larger than unity. The relation of spectral densities of stationary and random processes to stationary increments is demonstrated. The oscillating character of the behavior of spectral density of stationary processes is shown. Analytical equations are derived for their description with the analysis of accuracy characteristics. They are recommended for wide practical application.