Effects of irregular terrain on waves - A stochastic approach

This paper treats wave propagation over an irregular terrain using a statistical approach. Starting with the Helmholtz wave equation and the associated boundary conditions, the problem is transformed into a modified parabolic equation after taking two important steps: 1) the forward scatter approximation that transforms the problem into an initial value problem and 2) a coordinate transformation, under which the irregular boundary becomes a plane. In step 2), the terrain second derivative enters into the modified parabolic equation. By its location in the equation, the terrain second derivative can be viewed as playing the role of a fluctuating refractive index, which is responsible for focusing and defocusing the energies. Using the propagator approach, the solution to the modified parabolic equation is expressed as a Feynman's path integral. The analytic expressions for the first two moments of the propagator have been derived. The expected energy density from a Gaussian aperture antenna has been analytically obtained. These expressions show the interplay of three physical phenomena: scattering from the irregular terrain; Fresnel phase interference; and radiation from an aperture. In particular, the obtained expected energy density expression shows contributions from three sources: self energy of the direct ray; self energy of the reflected ray; and cross energy arising from their interference. When sufficiently strong scattering from random terrain may decorrelate the reflected ray from the direct ray so that the total energy density becomes the algebraic sum of two self energies. Formulas have been obtained to show the behavior of the energy density function.