IEEE Geoscience and Remote Sensing Magazine - December 2014 - 26

as compared to SNR cr is SNR cr - SNR +2 dB, to be traded
off with the increase of b.
However, the SNR is not only affected by thermal noise,
speckle noise can be even dominant. A rough approximation to assess the effect of speckle noise is by adding it to the
thermal noise as:
(24)

where NSR speckle = 1/3.63 (- 5.6 dB), and it is independent
on the bandwidth. Finally, the SNR is improved by incoherent averaging as N i .

1
0.9

-1000

0.8
0.7

-500
Delay (m)

NSR lR/cr = NSR D/cr + NSR speckle ,

Signal Component Covariance Matrix
-1500

0.6
0

0.5
0.4

500

0.3
0.2

1000

0.1

B. The RealisTic appRoach
In Sections V and VI.A a crude approximation to the effect
of noise was presented. A more precise estimation of the
delay is presented here using the Cramér-Rao bound (CRB)
including the covariance matrix (C) of the data:
1
v x2 $
,
/ C -k,1l sl ^t k - xhsl ^t l - xh

1500
-1500 -1000 -500 0
500
Delay (m)
(a)

Thermal Noise Component Covariance Matrix
-1500

(25)

1
0.9

-1000

k, l

0.8
0.7

-500
Delay (m)

where sl is the waveform's derivative. In the case of
AWGN, uncorrelated from sample to sample, (25) reduces
2
to v 2x $ v n2 /R l " sl(l) , , which is equivalent to (18), but in the
(discrete) time domain, instead of the frequency domain.
In reality, the ultimate achievable scatterometry or altimetry performance depends on the snap-shot SNR (no
incoherent averaging or N i = 1) and bandwidth, and on
the cross-correlation of the noise present in consecutive
lags Y c, i ^t n, x m, fdh and Y c, i ^t n, x m +1, fd h (fast time) as well.
Additionally, the GNSS-R observable (Eqs. (17) and (21))
was estimated as the average of the square of N i complex

1000 1500

0.6
0

0.5
0.4

500

0.3
0.2

1000

0.1
1500
-1500 -1000 -500 0
500
Delay (m)

1000 1500

(b)
Complete Covariance Matrix
1

-1500
+
K(x + x)

0.9
-1000

K(x)

0.8
0.7

Delay (m)

-500

0.6
0

0.5
0.4

500

0.3
0.2

1000

0.1
F+t

F0

1500
-1500 -1000 -500 0
500
Delay (m)

1000 1500

(c)
Figure 17. (a) Signal component covariance matrix (signal
Figure 16. Physical interpretation of the noise correlation be-

tween consecutive lags of the same waveform (top), and between
the same waveform lag in consecutive waveforms (bottom) [82].
26

statistics). (b) Thermal noise component covariance matrix (noise
statistics). (c) Complete covariance matrix (complex cross-correlation statistics, including both signal and noise terms). Simulation
parameters: h = 700 km, Tc = 1 ms, N i = 12.000 (from [131]).
ieee Geoscience and remote sensing magazine

december 2014



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