IEEE Geoscience and Remote Sensing Magazine - December 2014 - 19

the cases of large correlation scales and average curvature
radii of the surface compared to the radiation wavelength.
Also, it works only for incidence well away from grazing.
The early attempt to unify the KA and the small perturbation method, so that both quasi-specular reflections
and Bragg scattering are described simultaneously, have
been undertaken by the classical two-scale, or composite
model [79], [97]. More rigorously, this was done in a set
of "advanced" theories which include, among others, the
integral equation method [85], the small-slope approximation method [86]-[89], and the reduced local curvature approximation of third order and its versions [90]-[94]. The
two-scale method is heuristic by nature. It requires an arbitrary spectral splitting parameter which divides the surface
elevation spectrum into two parts: small-scale and largescale spectral components of roughness. The small-scale
component is responsible for the Bragg scattering, whereas
the large scales provide a quasi-specular contribution. The
dividing wavenumber is established from the comparisons
of the model predictions with the measurements [98].
At the same time, the small-scale approximation and
similar approaches (like IEM and RLCA3) do not require
the dividing wavenumber because it employs the entire
elevation spectrum without splitting it on large-scale and
small-scale components. It takes into account the abovementioned diffraction effects. However, similar to the KAGO approach, it requires the ocean roughness to have small
slopes (1 0.2-0.4).
Since the SSA is a more accurate approximation, it can
be used to assess the validity of the KA-GO approximation [99]. There are two approximations of the SSA, the
SSA of the 1st order and the more accurate approximation,
the SSA of the 2nd order. Practice shows that the SSA of
the 1st order, or the SSA1 suffices for calculations of the
LHCP BRCS of the L-band signal in the forward-scattering regime. For calculations of the RHCP BRCS and for a
wide-angle scattering regime, the more accurate SSA2 is required. The SSA1 gives the expression for the BRCS v 0 in
the form of a 2D surface integral similar to that obtained
in the Kirchhoff approximation but with a more accurate
pre-integral factor [87], [88]; generally, the integral cannot be evaluated by the stationary phase method. Some
of the above-discussed theoretical models were employed
for modeling of the L-band polarimetric bistatic scattering
from sea and land surfaces (see, e.g., [100]-[104]).
For the cases when all the above-mentioned analytical methods are not satisfactory, direct numerical simulations are needed. However, for bistatic geometry and
wide-spectrum roughness, they might be prohibitively
time consuming.
D. Surface MoDelS
In most of the models mentioned above, when calculating
v 0, one needs to make an assumption about the probability
distribution of the surface elevations or slopes. Typically,
bivariate Gaussian, or normal, distribution is used in the
december 2014

ieee Geoscience and remote sensing magazine

literature for describing the statistics of ocean and land
surfaces. The advantage of a Gaussian distribution is that it
is fully expressed through the second-order statistical moments of the random field of surface elevations or slopes.
In the case of the KA-GO, the BRCS is explicitly expressed
through the PDF of surface slopes which, in turn, depends
on slope variances along orthogonal axes. The variance of
slopes, or mean-square slope (MSS) can be derived from
the correlation function B h ^tvh of the surface elevations, or
equivalently, from its spectrum W ^lv h (which is a Fourier
transform of B h ^tvh ) by integrating it over wavenumbers, l,
which are smaller than a dividing wavenumber, l )):
v 2x, y = s 2x, y =

## l 2x,y W^lv hd 2 l,

(14)

l # l)

Models such as the IEM, or the SSA employ the full
surface spectrum in their formulations directly without retreating to the variances of elevations, or slopes and, therefore, without using the dividing wavenumber.
To describe ocean surface wave spectra, semi-empirical
models are frequently used that were designed to explain
field observations, particularly, microwave radar data (see,
e.g., [105]). Others suggest obtaining sea-surface spectra by
solving a wave action balance equation (e.g., [106]). These
models describe wind-driven waves in deep water under
diverse wave age (often called 'fetch') conditions. For the
wind-driven waves, it is convenient to introduce the MSS
along wind direction, v 2u, and across it, v 2c . They can be calculated using (14). Under well-developed conditions (i.e.
the waves and wind have reached equilibrium), the statistical distribution of surface slopes would connect the wind
speed and direction. The stronger the wind, the larger the
MSS of slopes. The direction along which the corresponding mean-square slope is maximal indicates the up/down
wind direction.
There are situations when wind and waves are not in
equilibrium. Such conditions exist in hurricanes where wind
and wave propagation directions are not aligned. To our
knowledge, there are no analytical spectral models describing hurricane waves, but there are various numerical coupled
ocean-atmosphere models for wave fields in hurricanes, (e.g.,
WAVEWATCH III [107]). However, they are mostly concerned
with energy-bearing spatial frequencies at the peak of the
wave spectrum, leaving aside smaller scales that contribute
significantly to the total MSS of surface waves.
There are some indications that the actual PDF of ocean
wave slopes L-band filtered according to (14) does not exactly follow a Gaussian curve at their tails [108]. Accounting for non-Gaussian features in the PDF of surface slopes
would require knowledge of third and fourth statistical
moments of slopes, skewness, and peakedness, respectively. Unfortunately, in many cases those parameters are not
readily available.
The Gaussian statistics is often used to describe the
rough soil surface in soil-surface-scattering studies. It is
also rather common for these studies to use an exponential
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2014