IEEE Geoscience and Remote Sensing Magazine - June 2020 - 55

1 - c H0 cos ^z 2 h
L $ SCNR ,

(23)

where z 2 is given in (13). Using (22), it can be deduced
that the standard deviation of any unbiased estimator for
the unknown radial velocity component is bound by the
L-sample CRLB 1/2 v v r:
m vp
Evt $ v v r = 4rb
2
r

1 - c H0 cos c

4rb
vm
m vp r
.
L $ SCNR

(24)

The CRLB for the radial velocity becomes smaller for
less clutter (smaller c H0) and/or for higher RFs (smaller m)
alongside larger target power (larger SCNR) and increased
averaging (larger L). Another critical parameter is the ratio
2b/ | v p |, describing the time it takes the SAR to travel the receiver channel baseline. For instance, for RDS-2, this delay is
approximately 7.5 m/7, 500 m/s = 1 ms and for TerraSAR-X
is 2.4 m/7, 600 m/s = 0.31 ms, while airborne C-band SARs
(AER) typically show values several times larger, for instance, 3 ms.
Figure 9 shows the CRLB 1/2 in (24) for a CNR = 7 dB,
SCNR = 12 dB, and SNR = SCNR ^1 + CNR h = 19.8 dB for
the three different systems, versus a varying true underlying
radial velocity in (a) for L = 1 and in (b) for L = 4, illustrating the beneficial effect of multilooking. The SCR is large
enough to assume a quasi-unbiased estimate (compare Figure 4). The potential optimum estimation accuracies vary
cyclically between two extreme values; this demonstrates
that reasonable estimation accuracies are achievable (at least
theoretically), even from space. The accuracy for airborne
systems is commonly on the order of factors better.
We note that the CRLB does not take into account the estimation error due to phase wrapping [46]. Then, it provides
accuracy values that can be close enough to the estimation
variance in only the unambiguous interval (- v a, v a), with v a
given by (11). Nonetheless, the cyclic behavior in Figure 9 is
related to phase wrapping, and the period of the oscillations
corresponds to the unambiguous velocity interval (- v a, v a).
We note that TerraSAR-X has the biggest unambiguous interval and that a larger unambiguous interval corresponds
to a smaller theoretical estimation accuracy. In agreement
with ambiguity value v a, the accuracy is the smallest one:
this effect is due to the interferometric pdf spreading, which
increases when velocity increases (see Figure 4).
In [20], the radial velocity estimation accuracy using
ATI multichannel configurations (either multifrequency
or multibaseline) in conjunction with classical statistical
estimation techniques was investigated. The multichannel
approach can overcome the limitations due to the presence
of ambiguous solutions, intrinsic of the single-interferogram configurations.
Recently, for maritime domain awareness purposes, an
analysis of vessel motion estimation was carried out using
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

6
# Looks L = 1
5
4
σv (m/s)

^ J -1 h11 =

RDS-2 SAR images of littoral
zones and the open ocean
THE INCLUSION OF CLUTTER
comparing the estimated raAND NOISE COMPROMISES
dial velocities of vessels with
THE ACCURACY OF THE
truth data from automated
identification systems. In FigESTIMATED ATI PHASE
ure 10, an example is shown
VALUE.
where the velocity of a commercial vessel is estimated as
the phase of the average of the
interferometric products of all pixels comprising the vessel. This average speed estimate is vt r = 6 m/s = 11.7 knots
and was used to compute the average displacement of each
pixel [see (12)].

3
2
1
0
-50 -40 -30 -20 -10 0 10 20 30
vr (m/s)

40

50

(a)
8
# Looks L = 4

7
6
σv (m/s)

inverse 3 × 3 Fisher information matrix J [44] was calculated in [26] and [45] as

5
4
3
2
1
0
-50 -40 -30 -20 -10 0 10 20 30
vr (m/s)

40

50

(b)
TerraSAR-X 2b/vp = 0.31 ms
RDS-2 2b/vp = 1 ms
Airborne 2b/vp = 3 ms

FIGURE 9. A CRLB 1/2 for three SAR systems versus radial velocity

for coherence 0.833 and SNR = 19.8 dB, with (a) L = 1 and
(b) L = 4.
55
IEEE Geoscience and Remote Sensing Magazine - June 2020

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