The localization of high‐frequency sound‐wave propagation around ray trajectories, and the reflection and(or) diffraction of these local plane‐wave fields by boundaries, inhomogeneities, and(or) scattering centers has been combined via the geometrical theory of diffraction (GTD) into one of the most effective means of analyzing high‐frequency wave phenomena in complex deterministic environments. These constructs are here incorporated into a stochastic propagation and diffraction theory for statistical moments of the high‐frequency field when the propagation medium has weak random fluctuations superimposed upon an inhomogeneous background profile, subject to the assumption that the correlation length ln of the fluctuations is small compared with the scale of variation, but large compared with the local wavelength λ=2π/k=2πv/ω, in the fluctuation‐free background, with k being the local wavenumber, v the local wave speed, and ω the radian frequency. Canonical problems of deterministic GTD furnish the propagators and, in the presence of interfaces, boundaries, or other types of scatterers, the local reflection, refraction, and diffraction coefficients that relate incoming to outgoing wave fields.
The major analytical building blocks include propagators described in local coordinates centered on the curved GTD ray trajectories in the deterministic inhomogeneous background environment; multiscale expansions in these coordinates to chart and solve for the propagation properties of statistical measures of the parabolically formulated ray fields; Kirchhoff or physical optics (PO) approximations, generated by stochastic GTD incident fields, to establish initial conditions for fields reflected from, or transmitted across, extended smooth surfaces; and ‘‘point scatterer’’ solutions to establish GTD initial conditions for small scatterers and edges. In addition to the conventional second‐ and higher‐order coherence functions, there are introduced as appropriate statistical objects two‐point random functions and corresponding higher‐order functions which are useful in treating correlation of incident and of backward reflected or diffracted fields that traverse the same propagation volume. By this solution strategy, one gains access to a much larger class of high‐frequency problems in a random medium than at present. Expressions for the average field and higher moments have been obtained for forward propagation in a fluctuating medium with inhomogeneous and caustic‐forming background, for reflection and refraction due to a plane or smoothly curved interface in such a medium, and for diffraction due to a wedge and a small scatterer.