Physics-based distributions for randomly scattered signals and their Bayesian conjugate priors

Sound waves propagating through the atmosphere and ocean are randomly scattered by turbulence, internal waves, surface roughness, and other processes. In some limiting cases, probability density functions (pdfs) for the scattered signal variations can be derived, e.g., the log-normal pdf for weak scattering (in the Rytov approximation) and the exponential pdf for strong scattering. A variety of more general, usually empirically based, distributions are available which reduce to these limiting cases, such as the Rice, gamma, and generalized gamma. For situations involving multiple receivers, multivariate log-normal, Wishart, and matrix gamma pdfs may be employed. Parametric uncertainties and spatial/temporal variability in the scattering process can be addressed with a compound pdf formulation, which involves an additional distribution for the uncertain or variable parameters. From a Bayesian perspective, the scattering pdf corresponds to the likelihood function, the pdf for the uncertain parameters to the prior/posterior, and the compound pdf to the marginal likelihood. Many common scattering pdfs possess Bayesian conjugate priors, which lend themselves to simple updating equations and analytical solutions for the posteriors and marginal likelihoods. This presentation summarizes important pdfs for randomly scattered signals and their conjugate priors when available.