IEEE Geoscience and Remote Sensing Magazine - December 2014 - 27

(26)

which can be understood as the sum of two terms: one corresponding to the signal, and the other one corresponding to the noise. Fig. 17 illustrates this for a spaceborne
cGNSS-R instrument. Interested readers are referred to a
simulation study for the cGNSS-R case in [130], a detailed
analytical study for the cGNSS-R case in [131], and for the
iGNSS-R case in [74].
As can be observed from Figs. 17(a) and (b), the noise
component is present in all the delays, while clearly this
is not the case for the signal term since it is dependent on
the backscattered signal. The covariance noise term follows
the shape of the auto-correlation function (ACF), and the
covariance signal term is dependent on the complex multiplication of the ACF at delays x 1 and x 2. This analysis is
fundamental to estimate the achievable SNR and the ultimate instrument performance, as well as to specify the
instrument in an optimum way in terms of bandwidth
[72], sampling frequency, width and central position of
the tracking window [129]. In addition, the correlation between consecutive lags is also related to the achievable data
compression that can be achieved, for example, using the
wavelet transform [118].
Fig. 18 shows consecutive waveforms (cGNSS-R) plotted vs. the correlation lag (x-axis) for up to 1.000 snapshots
(Ti = 1 ms, N i = 1.000). As can be observed, the amplitude
fluctuations are quite strong, and the correlation between
the noise in the same lag x in consecutive observables
("bin-to-bin" correlation) limits the effectiveness of the
incoherent averaging. The speed of these fluctuations depends on two factors: a) the properties of the surface under
observation (i.e., how fast it changes, if it changes at all,
for example the ice, the land, or the variable ocean surface
under different wind speeds), and b) the relative movement
between the transmitter and receiver (i.e., how fast the two
coronae move away one from the other).
This effect is better illustrated in Fig. 19, which shows
the progress of the waveform estimation using the PIR-A
airborne instrument (iGNSS-R) in a field experiment at the
Baltic sea on November 11th, 2011. The coherent integration
time is Ti = 1 ms, and the number of incoherent averages increases from 1 up to 10.000. As can be understood, longer
integration times are required to achieve a clear waveform
due to the higher noise of the interferometric processing.
The zoom shows that individual waveforms are formed by
december 2014

ieee Geoscience and remote sensing magazine

06
#10
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1000
800

;Y(t, x, fd = 0);2

C ^x 1, x 2h = Y c, i (t, x 1, fd = 0) $ Y c, i ) (t, x 2, fd = 0) ,

the scattering on a few facets only (in particular, the zoom
shows one around the specular reflection point), and how
the impulse response to that facet is simply the ACF squared
(Fig. 7a).
Fig. 20a shows the standard deviation of each correlation lag as a function of the incoherent integration time (as
in Fig. 19). It can be recognized that:
a) the standard deviation is higher where the waveform amplitude is higher (speckle noise or "multiplicative"
noise), and
b) the standard deviation does not reduce as the squared
root of N i, because the corresponding area to the lags associated to higher peaks is smaller and contains fewer scatterers. Before the leading edge, and in the tail, thermal noise
dominates and the SNR increases as the square root of N i
(Fig. 20b). This effect can be interpreted as an "effective"
number of incoherent averages N i, eff that depends on the

600
t (ms) 400

200

0

40

60

50

70

x (Range Bin Number)

Figure 18. One thousand consecutive waveforms plotted vs. the
fast time (correlation lag: x) and the slow time (t). [118, 131].

6
5
Waveform (a.u.)

cross-correlations of Y c ^t, x, fdh or Y i ^t, x, fdh, as the only
way to reduce speckle noise. However, the amount of reduction depends on the correlation between the noise in
the same lag x in consecutive observables Y c, i ^t n, x, fd h and
Y c, i ^t n +1, x, fd h (slow time). The physical interpretation is
presented in Fig. 16.
The analysis of the noise correlation between consecutive lags is performed using the covariance matrices defined as:

4
3
2
1
0

0

0.5

1

1.5
2
2.5
Delay (ns)

3

3.5

4

Figure 19. Sample interferometric waveforms obtained with
PIR-A instrument for Ti = 1 ms, and N i = 1 (red), 10 (purple),
100 (orange), 1.000 (green), and 10.000 (blue) waveforms [132].
Zoom for red plot ( N i = 1) around the main peak compares well
with Fig. 7a squared both in the relative amplitude of the side lobes
and the location at !1 P-chips of the main peak.

27



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