Statistical moments of the sound field propagating in a random, refractive medium near an impedance boundary

Propagation of a monochromatic sound field in a refractive and turbulent medium near an impedance boundary is considered. Starting from the parabolic equation for a moving medium and using the Markov approximation, a closed equation for the statistical moments of arbitrary order of the sound-pressure field is derived. Numerical methods for directly solving the first- and second-moment versions of this equation are formulated. The first-moment formulation is very similar to parabolic equations (PEs) that are now widely used to calculate sound fields for particular realizations of a random medium. The second-moment formulation involves a large, fringed tridiagonal matrix, which is solved using iterative refinement and Cholesky factorization. The solution is computationally intensive and currently restricted to low frequencies. As an example, the first and second moments are directly calculated for upwind and downwind propagation of a sound wave through a turbulent atmosphere. For these cases, predictions from the second-moment PE were statistically indistinguishable from the result of 40 random trials calculated with a standard Crank-Nicholson PE, although the second-moment PE yielded smoother results due to its ensemble-average nature.