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The determination of soil moisture content is often based on the measurements of the ratio of
the vertically and horizontally polarized cross sections for large angles of incidence, where
the cross sections could be significantly different. Using the high frequency, physical optics
model of the earth's surface, this ratio depends primarily on the Fresnel reflection
coefficients for the two polarizations, while the impact of surface roughness factors out of the
cross section ratio. Thus for highly conducting moisture saturated soils, this ratio approaches
one. Using the low frequency, small height-small slope perturbation model of the earth's
surface, the vertically and horizontally polarized cross sections are critically dependent on
polarization for large angles of incidence, even for the perfectly conducting rough surfaces.
However using the standard perturbation model, the ratios of the cross sections are also
independent of the surface roughness. Applying the small perturbation approach to highly
conducting rough surfaces, the ratio of the horizontally to vertically polarized cross sections
approaches zero for grazing angles of incidence, for which the two cross sections differ
significantly. There is ample experimental evidence that neither the physical optics nor the
small perturbation models are adequate.
The standard hybrid two scale physical optics-perturbation approach depends critically upon
the decomposition of the composite surface into smaller and larger scale surfaces. The
smaller scale surface is restricted to small Rayleigh roughness parameters, proportional to the
mean square height, and the larger scale surface is restricted by the large radii of curvature
criteria.
Using a two scale full wave approach, the cross section are expressed as a weighted sum of a
physical optics cross section for the larger scale surface, reduced by a factor equal to the
square of the small scale surface characteristic function, and a cross section for the smaller
scale surface that is modulated by the slopes of the larger scale surface. A variation technique
is used to decompose the surface height spectral density function in a continuous, smooth
manner into spectral density functions for the larger and smaller scale surfaces. It is shown
that the corresponding polarization dependent rough surface cross sections are stationary
over a wide range of the variation parameters. The ratio of the cross sections are dependent
of the surface roughness, since the horizontally polarized cross sections are significantly
dependent on modulation by the slopes of the larger scale surfaces, for large angles of
incidence.
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