Empirical and quadrature approximation of acoustic field and array response probability density functions

Numerical methods are presented for approximating the probability density functions (pdf's) of acoustic fields and receiver-array responses induced by a given joint pdf of a set of acoustic environmental parameters. An approximation to the characteristic function of the random acoustic field (the inverse Fourier transform of the field pdf) is first obtained either by construction of the empirical characteristic function (ECF) from a random sample of the acoustic parameters, or by application of generalized Gaussian quadrature to approximate the integral defining the characteristic function. The Fourier transform is then applied to obtain an approximation of the pdf by a continuous function of the field variables. Application of both the ECF and generalized Gaussian quadrature is demonstrated in an example of a shallow-water ocean waveguide with two-dimensional uncertainty of sound speed and attenuation coefficient in the ocean bottom. Both approximations lead to a smoother estimate of the field pdf than that provided by a histogram, with generalized Gaussian quadrature providing a smoother estimate at the tails of the pdf. Potential applications to acoustic system performance quantification and to nonparametric acoustic signal processing are discussed.