IEEE Geoscience and Remote Sensing Magazine - June 2020 - 66

methods is included to illustrate their potential, pointing
out challenges and potential future directions.
INTERFEROMETRIC DATA MODEL
From the view of statistical signal processing, phase denoising is performed to retrieve the true value from noisy observations, such as an interferometric pair of SAR images.
The statistical model of the interferometric data and the
noise model of the interferometric phase constitute the basic foundation of InSAR technology. To describe the existing
methods, it is necessary to introduce the signal model.
STATISTICAL MODEL
Let an interferometric pair of single-look complex SAR images be defined in a vector form as
k = 6z 1 z 2@T ,

(1)

Probability-Density Function

where z 1 and z 2 are the obtained SAR images of two channels and T denotes the transpose operator. Usually, k can
be assumed to follow a bivariate complex Gaussian distribution with zero means, and its probability-density function (PDF) is expressed as [25]

0.6
0.4

-3

-2

-1
0
1
Interferometric Phase
(a)

2

2

3

Look Number Is 1
Look Number Is 2
Look Number Is 4
Look Number Is 8

1.5

1

0.5

0

0

0.2

0.4
0.6
Coherence
(b)

0.8

1

FIGURE 1. The statistics of the interferometric phase. (a) The
probability-density function (L = 1). (b) The standard deviation of
the noisy phase.

66

1
exp 6- k H / -1 k@,
r 2 det ^/ h

(2)

where H is the transpose and conjugate operator and R is a
2 # 2 covariance matrix of k, which is given by
t $ R1 $ R2
R1
G,
/ = E ^kk H h = = *
t $ R1 $ R2
R2

(3)

where R 1 = E ^ z 1 2 h and R 2 = E ^ z 2 2 h are the underlying reflectivities and t is the correlation coefficient between z 1
and z 2, which can be further decomposed as [7], [26]
E ^ z 1 $ z )2 h
E ^ z 1 $ z )2h
=
= c $ e jz x
2
2
E^ z 1 h $ E^ z 2 h
R1 $ R2
E ^ z 1 $ z )2 h
c= t =
1 60, 1@,
R1 $ R2
t=

(4)

where c is the so-called coherence with a real value and
z x is the actual/noise-free interferometric phase. In (4),
I z = z 1 $ z 2) is defined as an interferogram by the conjugate
multiplication between two SAR images.
From (2) and (3), the InSAR observations [that is, a 1 = z 1 ,
a 2 = z 2 , and z z = + ^ z 1 $ z )2h, with + ^ $ h a phase operator,
z z ! ^- r, r h, and modulus 2r] can be statistically modeled
under the constraint of R = R 1 = R 2, and the PDF is expressed as [27]
p(a 1, a 2, z zu R, c, z x) =

a 1 + a 2 - 2a 1 a 2 b
2a 1 a 2
n,
expd rR 2(1- c 2)
R(1 - c 2)
(5)
2

2

where b = c cos^z z - z x h . Then, the phase distribution
can be obtained by integrating a 1 and a 2 out, which is expressed as [28]

0.2
0

Standard Deviation of Phase Rad

Coherence Is 0
Coherence Is 0.2
Coherence Is 0.4
Coherence Is 0.6
Coherence Is 0.8

0.8

p^k /h =

p^z zh =

1
1 - c 2 f b ` 2 r + arcsin ^bhj
1 p
$
.
+
2r
1 - b2
^ 1 - b 2 h3/2

(6)

The curve of p ^z z - z x h versus different values of coherence c is plotted in Figure 1(a). As c increases, the distribution of z z is much more compact, with a smaller noise
turbulence, showing a higher quality for the interferometric phase. All of the noise sources mentioned earlier can
be treated as decorrelation issues, reflecting the decrease
in coherence. As a result, lower coherence means the presence of stronger noise, while higher coherence indicates
a lower noise level. It is well known that the phase-noise
level of the SAR interferogram is spatially variant, which
is highly related to the reflectivity and tomography of the
observed scenes. Figure 2(a) and (b) shows the interferometric phase and estimated coherence through shuttleimaging radar-C/X-SAR data of Mount Etna [2], [13], used
to illustrate this phenomenon.
With InSAR development, the distribution of the interferometric phase, z z, has been further investigated. Several
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
IEEE Geoscience and Remote Sensing Magazine - June 2020

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