IEEE Geoscience and Remote Sensing Magazine - December 2014 - 18

illumination. This noise is multiplicative; i.e., it disappears
together with the transmitted signal in contrast to an additive thermal noise originating from the observed scene
and the receiver. Both the fading signal and the thermal
noise are assumed to be uncorrelated, stationary random
processes, which obey circular Gaussian statistics, and have
different correlation times and different variances, both
with zero means.
The analysis shows that the statistics of sample-averaged
Y 2 N obtained by averaging N statistically independent
samples can be described by the ensemble average of Y 2 ,
or a mean correlation power, and number N, similar to
what is derived for traditional monostatic radars [76]. The
residual standard deviation of the sample-averaged crosscorrelation power Y 2 N due to speckle noise reduces approximately as 1/ N .
It is important to know the correlation time x cor of the
fading signal being received because it determines the
choice of the coherent integration time for the matchedfilter processing. Choosing a very small coherent integration time Ti, will not allow buildup of the correlator output
to its full potential, whereas integrating for too long will
not be an improvement over incoherent summation of independent samples. Also, knowing x cor allows one to estimate number N as N . T/x cor . The correlation time x cor
can be extracted from complex auto-correlation function
B Y (x) = Y (t) Y ) (t + x) , or from its power spectrum [81],
[82]. A stochastic model for the waveform time series measurements was developed in [82], which was validated by
experimental data in [83].
For the case of fast-moving platforms such as aircraft, or
satellite, x cor can be related to the signal characteristic spatial scale, the correlation radius, t cor, through the translational velocity of the receiver's platform v rec : x cor = t cor /v rec .
The GNSS transmitter motion should also be accounted
for. For estimates of t cor one can use the van CittertZernike theorem from which it follows that for a spatially
incoherent source of size D, the scale t cor at distance L
from the source obeys the classical diffraction formula:
t cor . mL/D, where m is the wavelength of the signal's carrier, and D is the scale of the surface footprint associated
with the DDM [82].
C. EM SCattEring ModElS
The effect of the Earth's surface enters the bistatic radar
equation (11) through the normalized bistatic radar cross
section (BRCS), v 0, which depends on the directions of
incoming and outgoing EM waves, and on the properties
of the scattering medium. If the diffuse part of the scattering is weak so that the specular reflection prevails, then v 0
reduces to the value v coh discussed above. In the opposite
case, when v coh can be neglected, one must deal with calculating the diffuse bistatic cross section, v 0. The problem
of finding v 0, due to its complexity, cannot be solved in a
general form. If the L-band radiation cannot penetrate the
surface (such as for the ocean surface, or bare moist soil),
18

this significantly simplifies the problem. In this case, the
BRCS is driven mainly by the surface roughness and by the
impedance, or dielectric permittivity of the very top layer
of the medium. If radiation penetrates the scattering medium, it might involve volumetric scattering from inhomogeneities, or multiple reflections from the layers inside the
medium (as for ice, snowpack, or vegetation canopy). This
would significantly complicate the problem of finding the
BRCS. Even for the former case of pure surface scattering,
calculating the BRCS can be a challenge. This also pertains
to regimes of low-grazing angle scattering, or of multiple
scattering from a very rough surface.
Here, we limit ourselves to consideration of a much simpler, single-scattering regime. Also, many practical cases
allow various simplifying assumptions regarding the surface roughness, so it is possible to determine manageable
formulations. There are numerous theoretical models and
approaches of that sort in the literature (see, e.g., [84]). We
will mention here only the most popular ones. Among
them are the Kirchhoff approximation (KA), the Integral
Equation Method (IEM), and the Small Slope Approximation (SSA), accompanied by their variants and further approximations [79], [85]-[94]. The difference between them
lies in specific limitations applied to wavelength, geometry
and parameters describing roughness, so, as a result, some
models are more successful than others.
For example, one of the most widely used approaches is
the Geometric Optics limit of the Kirchhoff Approximation
(KA-GO). It estimates the Kirchhoff diffraction integral by
the stationary phase method. Physically, it means that the
EM field at the point of reception is determined by contributions from a multitude of specular points distributed over a
portion of the rough surface, whereas diffraction effects are
neglected. Frequently, this type of scattering is called quasispecular. The KA-GO works quite well for forward scattering
around the nominal specular direction of the linearly polarized and left-hand circularly polarized (LHCP) waves. The
difference between the GO and KA approximations in this
regime most likely exceeds the accuracy of the KA itself. The
GO approximation gives an incorrect prediction for BRCS
for out-of-plane scattering, and for the right-hand circular
polarization (RHCP). This is due to the fact that for the latter
cases, one needs to account for diffraction effects which are
neglected in the KA-GO approach.
In principle, the Kirchhoff Approximation accounts for
diffraction but only partially because it does not transition into the expression predicted by the small perturbation method for directions away from the nominal specular reflection [88]. This means that it cannot reproduce the
Bragg resonant scattering accurately. Let alone, it is still difficult to obtain an analytical solution using the KA without
further simplifying assumptions. If the surface slopes are
small the Physical Optics approximation of the KA can be
used [76]. Some other simplifying ideas are used in alternative formulations of the KA [95], [96]. Overall, the KA (as
well as both the GO and PO approximations) is limited to
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2014