IEEE Geoscience and Remote Sensing Magazine - December 2014 - 17

direction, while the unit surface is being illuminated by the
unit wave incoming in another direction.
The WAF in (11) can be approximated by the square
product of two functions: the triangularly shaped (in the
case of GPS L1 signals; see (2)) correlation function K (x)
and the sinc-shaped function S ( f ). The first term determines an equi-range annulus zone, and the second one
determines an equi-Doppler-frequency zone. The width of
K (x) is determined by two times the PRN chip length, 2x c,
and the width of S ^ f h is determined by two times the inverse of the coherent integration time, fDop = 2/Ti . For fixed
positions of the transmitter and the receiver, both WAF and
BRCS are functions of reference surface coordinates.
In applying this model to new GNSS signals with new
modulation schemes such as BOC, it would be necessary
to substitute the appropriate delay and Doppler responses
for the new signal characteristics. Some of the new characteristics to be found in future GNSS signals were discussed
previously in the Section "GNSS Signal Structure."
The normalized bistatic radar cross section (BRCS) v 0
describes the strength of the scattered signal originating
from a particular point on the rough surface and propagating in the direction of the receiver's antenna. So, in terms of
surface coordinates, it describes the glistening zone of the
rough surface. In the commonly used geometric-optics limit of the Kirchhoff approximation this term is represented
by the following expression [78], [79]:
v 0 = r 0 2 ^q/q zh4 P ^-q = /q zh .

(12)

Though this value is a function of the scattering vector,
qv, for fixed positions of the transmitter and the receiver
above a surface, this vector can be expressed as a function
of the coordinate tv in the mean surface plane. The value of
v 0 depends on the complex Fresnel coefficient 0 which in
turn depends on the signal polarization state, the complex
dielectric constant of the reflecting medium, f, and the local incidence angle.
Factor P ^ vs h in (12) is the probability density function
(PDF) of the slopes of the large-scale (larger than several
radio wavelengths) component of the sea surface which is
responsible for the quasi-specular scattering within the glistening zone. Usually, the most probable orientation of surface slopes is parallel to the mean plane, z = 0. Then, the
PDF has a maximum at s = 0, and the bistatic cross-section
v 0 has a maximum at qv = = 0, i.e., at the nominal specular
direction with respect to the mean surface.
There are several limitations associated with the bistatic radar equation in the form presented by (11). First, it is
limited to the case of completely diffuse surface scattering;
i.e., when the coherent specular component is absent, or
can be safely neglected. However, sometimes the coherent
component is noticeable, or even dominant in the scattering process. This can be found for forward scattering from
calm seas, lakes, relatively flat land, or sea ice characterized by the weak surface roughness (with heights much
december 2014

ieee Geoscience and remote sensing magazine

smaller than the signal wavelength). This situation can be
remedied by augmenting (11) with a term describing a coherent reflection. It can be constructed from a product of a
mirrored proxy of the direct signal cross-correlation power
Y0 ^x, f h 2, the absolute value squared of the Fresnel reflection coefficient, and the factor that takes into account the
loss of the spatial coherence due to the presence of some
relatively weak surface roughness:
Y ^x, f h 2

spec

= Y0 ^x, f h 2 0 2 exp ^-8r 2 v 2h cos 2 i/m 2h . (13)

This term will describe a sharp peak of the DDM centered at the delay and the frequency offset associated with
the nominal specular point on the surface. The same weak
surface roughness will be responsible for the rest of the
DDM forming a very shallow pedestal described by (11). In
most cases, it probably will not be detectible due to thermal
noise. The specular component can be incorporated into
the bistatic radar equation (placed under the integral) in the
form of a so-called coherent BRCS, v coh, added to a diffuse
BRCS, v 0 (see, for example, [80]). This additional BRCS includes a delta function over spatial coordinates multiplied
by the factor that accounts for factors such as a reflection
coefficient, antenna pattern and distance. Therefore, the
definition of v coh differs from the diffuse cross section, v 0,
definition as a characteristic of the scattering object alone.
The factorization of the WAF into a product of a delaydepended K function and frequency-dependent S function was used in (11). The limitations imposed by it are not
obvious. At least, they are of the same nature as in monostatic radar signal processing [77], and to our knowledge, in
practice these limitations play no significant role.
Frequently, the bistatic radar equation is used with a
BCRS, v 0 in the form of the geometric optics limit of the
Kirchhoff approximation (12) but it is not a necessary condition. Any other reasonable EM scattering model can be
used in (11) for v 0.
While (11) describes an ideal ensemble average
Y ^x, f h 2 , in practice we deal with an incoherent integration of Y ^x, f h 2 over some observation time T ; i.e.,
with averaging over a finite number of statistically independent samples. Such limited-sample averaging itself
contains residual noise, which might affect our ability to
accurately retrieve the ideal average waveform from the
measured one.
When a narrow-band GNSS signal is scattered from a
rough surface, the carrier phases arriving from individual
reflection points sum together in unpredictable ways at the
down-looking antenna, resulting in a completely random
received phase and amplitude at the receiver. This causes
the total received level of Y 2 to fluctuate over time due
to constructive and destructive interference between individual reflections.
This phenomenon is known in traditional remote sensing applications as fading, or speckle noise, and is unavoidable for a diffusely scattering surface and a coherent
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2014