Geometrical theory of diffraction for high‐frequency coherence functions in a weakly random medium with inhomogeneous background profile

Abstract

The localization of high‐frequency wave propagation around ray trajectories and the reflection, refraction, and/or diffraction of these local plane wave fields by boundaries, inhomogeneities, and/or scattering centers have been combined via the geometrical theory of diffraction (GTD) into one of the most effective means for 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 in a weakly fluctuating medium with inhomogeneous background profile, provided 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 = 2πc/ω in the fluctuation‐free background, with k being the local wavenumber, c the local wave speed, and to the radian frequency. The major analytical building blocks for coherence functions and paired random functions (PRF) 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 solve for statistical measures of the parabolically formulated ray fields; Kirchhoff or physical optics (PO) approximations for fields reflected from extended smooth surfaces; and “point scatterer” solutions for small scatterers and edges. The PRFs are useful for correlating incident and backward fields traversing the same propagation volume. The theory is illustrated for forward propagation in a fluctuating medium with inhomogeneous and caustic forming background, for reflections 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. [Work supported by RADC and ONR.]