IEEE Geoscience and Remote Sensing Magazine - June 2020 - 52

where R is the distance from the sensor to the target. Considering typical values for R and v p in the case of spaceborne
systems, it may happen that a moving vehicle is located several kilometers off the road; this makes its identification
more difficult.
Moreover, when clutter plus noise power increases, the
ATI phase estimate is affected by noise, and it will deviate
from the nominal phase value, z = z 2 (see Figure 3), that
according to (3) is
z=

4rb v r
m vp

2r

(13)

.

To reduce noise, interferometric phase is typically averaged over L independent pixel pairs where L is commonly
called the number of looks [18]. This averaging increases the
detection performance if the same target is actually present
in every pixel of the group.
In the pixels without a target, (H 0), signals Z 1 and Z 2,
acquired by the two interferometric antennas, are zeromean Gaussian signals, and the pdf of the interferometric
phase fU (z H 0) was derived in [40] as
fU ^z H 0 h =

C ^L + 0.5 h^1 - c H0 2 hL c H0 cos z
+
2 rC ^L h^1 - c H0 2 cos 2 z hL + 0.5
^ 1 - c H0 2 hL
2
2
+
2 F1 ^ L, 1; 0.5; c H 0 cos z h, (14)
2r

where c H0 is the complex coherence between the two images and 2 F1 ($) denotes Gauss' hypergeometric function.
For single-look, that is L = 1, we can obtain a closed form:
1 - c H0 2
1
fU ^z H 0 h = 2r
1 - c H0 2 cos 2 z
c H0 cos z cos -1 6- c H0 cos z@
# )1 +
3,
61 - c H0 2 cos 2 z@1 2
z ! ^- r, r@.
(15)
The complex coherence can be derived using C H0 in Table 2
and is given by

1

SCR = 0 dB

SCR = −‫ ۻ‬dB

SCR = 6 dB

0.8
pdf

SCR = −6 dB

SCR = 20 dB

0.6

0.2
−1

−0.5

0

0.5
1
Phase (rad)

1.5

2

FIGURE 4. A phase pdf (17), including a Gaussian target with phase

z 0 = 1.3 rad and varying SCR.
52

1
1 .
1 + CNR

(16)

This equation has been derived from the assumption of
clutter fully correlated between the two acquisitions and
takes into account only the decorrelation due to thermal
noise. Regardless, the temporal decorrelation, due to a
change in the surface condition between the two observations (for example, the fast movements of scatterers in the
clutter), and the spatial decorrelation, due to a change in
sensor beam pointing, can also be taken into account when
including these effects in the thermal noise decorrelation
by considering a lower equivalent value of CNR in the definition of C H0 in Table 2.
When a moving target is present, the pdf of the interferometric phase z assumes different expressions depending
on the statistical model of the target.
INTERFEROMETRIC PHASE DISTRIBUTION FOR
DETERMINISTIC TARGET MODELS
In this case, signals Z 1 and Z 2, acquired by the two interferometric antennas, are still Gaussian but not zero mean,
and the marginalization of the pdf (10) with respect to the
phase cannot be expressed in closed form. The pdf of the
interferometric phase fU (z ; H 1) can be numerically computed via Monte Carlo simulation and depends on v r, CNR,
and SCR. In [20], it was shown that the pdf strongly depends
on SCR and weakly depends on CNR. Moreover, it assumes
a peak value in a position that is different from the nominal
interferometric phase value (13) and exhibits a variance that
gradually increases for increasing v r .
INTERFEROMETRIC PHASE DISTRIBUTION FOR
RANDOM TARGET MODELS
In the Gaussian assumption, signals Z 1 and Z 2, acquired by
the two interferometric antennas, are still zero-mean Gaussian signals, identical to the case when a target is absent. The
pdf of the interferometric phase fU (z ; H 1) can be expressed
in the closed form [20], [22] and, for instance, for L = 1 is
1 - c H1 2
1
fU ^z H 1 h = 2r
1 - c H1 2 cos 2 ^z - z 0 h
c H1 cos ^z - z 0 h cos -1 6- c H1 cos ^z - z 0 h@
# )1 +
3,
61 - c H1 2 cos 2 ^z - z 0 h@1 2
z ! ^- r, r@,
(17)
where c H1 is the complex coherence between the two images and z 0 is its phase. The complex coherence can be derived by C H1 in Table 2 and is given by

0.4

0

c H0 =

c H1 =

1 + SCR e jz 2
,
1
1 + CNR + SCR

(18)

where phase z 2 is the nominal phase difference and depends on the target velocity [see (13)]. Note that z 0 is different from the noise-free value z 2.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
IEEE Geoscience and Remote Sensing Magazine - June 2020

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