Theory of GNSS Reflectometry - GNSS Remote Sensing: Theory, Methods and Applications - page 203

Fading events driven by this spectral component of the oscillation are the most
common, determining the shortest interval of time without a fading event: 1=f M .
Coherent integration times longer than 1=f M result in significantly diminished

power levels.
This is equivalent to the Cittert-Zernike theorem, for quasi-monochromatic
spatially incoherent source, Thompson et al. ( 1986 ). According to this theorem, the
illuminated areas act as uniformly illuminated radiating apertures, of diameter D.
Such apertures thus have a radiation patter, the first lobe of which becomes:

D

(8.56)

The signal is coherent within the lobe, so the longest time during which the receiver
can coherently integrate the signal is the time needed to cross the lobe. If L is the
width of the lobe at the receiver altitude h, this crossing time is coh D L= v

.h /= v , which is equivalent to the coherence time obtained from Eq. 8.55 .
Applying this formula separately in the along- and the across-track directions of
the GNSS-R receiver movement, the pattern appears as a set of extremely narrow
lobes across the Doppler stripe direction, determined by the distance between the
delay-Doppler cells (apertures). Along the craft's trajectory, the beam is limited
by the size of the Doppler stripe width, which in turn depends on the coherent
integration time i . Since the Doppler strip is much narrower than the distance
between spots, the angular size of the lobule in the along-track direction is much
larger. The time it takes the craft to cross the lobule is the time that the signal can be
coherently integrated. Beyond that interval the signal becomes incoherent and the
correlation with the replica drops off. An illustration of these ideas is presented in
Fig. 8.16 .
A more refined estimation of the coherence time for different delays along the
waveform, and taking into account the possibility of observations at any elevation
angle e, is presented in Zuffada et al. ( 2003 ):

r

2 v

h
2c sin e

coh D (8.57)

where c is the speed of light.
Elachi ( 1987 ) justifies that for a large number of individual scatterers within the
illuminated area, the total amplitude has a Rayleigh distribution, the total power an
exponential distribution, and the total phase a uniform distribution. The exponential
distribution of the total power indicates that it s standard deviation depends on the

mean power itself ( spec kl e D <P > N = p N ). This is the typical noise behavior
detected in GNSS-R observations. Fading/speckle has been identified as the major
source of random variability in the GNSS-R data. The most effective way to reduce
the speckle is by averaging in a non-coherent way the largest possible set of
observations, while increasing the signal-to-thermal noise of the sensor does not
reduce the relative standard deviation of the signal efficiently.