Abstract
Statistical characteristics of geophysical fields and waves in random media often differ considerably from the behavior of their realizations. Practically, in each specific realization of the process one can observe features, that are completely absent in its statistical description.
In the simplest case, such features are described by the lognormal probability distribution. Our illustrations of this phenomenon include parametric stochastic resonance, dynamical and statistical energy localization for wavefields in randomly layered media, wave beam propagation in random parabolic waveguides, and diffusing tracers in random velocity fields.
Another example of this phenomenon is an appearance of certain singularities in the dynamics of individual realizations, accompanied by their absence in the statistical description. Such models are often reduced to boundary-value problems for the corresponding Fokker-Planck equations. Examples of such features of statistical solutions are provided and include a comparison of the mean exponential divergence of geometric-optical rays in a random medium with the almost sure existence of caustics on finite distances, and the phase fluctuations of plane waves in a randomly layered medium.
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Klyatskin, V.I., Woyczynski, W.A. (1997). Dynamical and Statistical Characteristics of Geophysical Fields and Waves and Related Boundary-Value Problems. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_10
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DOI: https://doi.org/10.1007/978-1-4613-8500-4_10
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