IEEE Geoscience and Remote Sensing Magazine - December 2014 - 22

specular reflection is dominant, and therefore, the spatial
resolution of the bistatic GNSS-R radar will be limited to
the first Fresnel zone, l Fr = mR r . For aircraft altitudes of 5
to 10 km, l Fr is several tens of meters. For satellite altitudes
of 300 to 600 km, it grows to several hundred meters. That
would be a rather good spatial resolution for any radartype sensor.
Unfortunately, diffuse, quasi-specular scattering is the
more frequently occurring process when dealing with natural surfaces. In this case, scattered radio waves arrive from
the "glistening" zone, which is much larger than the first
Fresnel zone. With the help of the Woodward ambiguity
function (see (11)), which acts as a spatial filter, the scattered signal can be transformed into a delay-Doppler map.
Ultimately, pixels of that map would determine the spatial
resolution of the bistatic GNSS-R radar.
As seen from Fig. 9a, for large time lags, the equi-range
zones are getting increasingly closer to each other. So, the
pixels created by intersections of those distant equi-range
zones with the equi-Doppler lines might seem small. But
the power in those small pixels is low (and noisy), and additionally, they are affected by the ambiguity that would
significantly diminish the spatial resolution.
In practice, surface pixels which contribute the most to
the power of the corresponding DDM pixels have the largest size; so they are most suitable for measurements. They
occupy the area near the nominal specular point which
corresponds to the maximum in the DDM. The intersection of the first annulus zone with the first Doppler zone
creates the pixel with the best spatial resolution. The size
of the first annulus zone is proportional to aR r , where a
is the code chip length. The angle of incidence, of course,
is also contributing to a more accurate expression for the
annulus zone size. It can be found, e.g., in [96]. For 600-km
receiver altitude, C/A code, and 1-ms coherent integration
time (which determines the size of the Doppler zone), the
size of the pixel on the ground will be of the order of 20 to
30 km. For the P(Y), or M code, it will be approximately
10 times smaller along the equi-Doppler line. This size
would determine the best instantaneous spatial resolution
for this configuration.
However, because of the low signal-to-noise ratio, the
signal needs to be incoherently accumulated over some
time. During that time the receiver and the footprint asso-

ciated with the considered pixel will move over the surface
with a speed of about 2-5 km/s, thus making the resolution lower along the direction of motion. Therefore, finding a resulting spatial resolution would depend on several
geometrical, dynamical, and receiver's parameters, with a
particular choice of both the coherent and incoherent integration time.
V. GNSS-R RECEIVER DATA
ACQUISITION TECHNIQUES
As in navigation receivers, the most common GNSS reflectometers' architecture (conventional GNSS-R or "cGNSS-R"
in short) correlates coherently during Tc seconds (typically
+1 ms) the reflected signal s R (t) with a locally generated
replica of the transmitted signal a (t) (open C/A codes only)
after proper compensation of the Doppler frequency shift
fd or for a number of Doppler frequencies as sketched in
Fig. 11 [10]:
1
Y c ^t, x, fd h = T
c

FIGURE 11. Basic concept of a conventional GNSS-R instrument.

22

s R ^t lh a ) ^t l - x h e -j2r^Fc +fdht l dt l ,

(16)

Ni

1
Y c ^x, fd h 2 . N / Y c ^t n, x, fdh 2.
i
n =1

(17)

As explained in Section III, although the width of the
auto-correlation function is not critical for scatterometry
applications, for altimetry applications it determines the
best achievable time (range) resolution, which, under the
assumption of uncorrelated additive white Gaussian noise
(AWGN), is given by the Cramer-Rao bound (CRB) [117]:
v 2x $

1
,
SNR $ b 2

(18)

where SNR is the signal-to-noise ratio, and b is the socalled rms bandwidth, defined as:

b _

1 Tc
8 ( ) dt
Tc 0

t +TC

where t is the time when the integration starts. However,
since the reflected signal is of even weaker amplitude than
the direct one, that is, the signal-to-(thermal) noise ratio is
even poorer, and -more important- it usually suffers from
speckle noise, a large number of incoherent averages (N i)
are required in order to improve the signal-to-noise ratio
of Y c ^t, x, fdh:

2

Local Code

#t

#0

min " B, B IS ,

#0 B

f 2 S ( f) 2 df

S ( f) 2 df

.

(19)

In Eq. (19) B and B IS are the baseband bandwidths of
the receiver's filter and the transmitted signal according
to the Interface Specification documents [70], [73], and
S ( f) 2 is its spectrum. Actually, for low SNRs (1 5 dB)
the Ziv-Zakai bound [117] provides a better indication of
the magnitude of the estimation errors, which can be actually much larger than the CRB ones. Therefore, for the
same SNR, the larger the rms bandwidth (b), the better the
achievable range resolution.
ieee Geoscience and remote sensing magazine

december 2014



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