Effects of irregular terrain on waves-a stochastic approach
Yeh, K.C.; Lin, K.H.; Ying Wang
Antennas and Propagation, IEEE Transactions on
Volume 49, Issue 2, Feb 2001 Page(s):250 - 259
Digital Object Identifier 10.1109/8.914292
Summary: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
(1965) 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
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