IEEE Geoscience and Remote Sensing Magazine - June 2020 - 50

zn =

4rb n v r
m vp

2r

,

(3)

n = 1, f, N,

with m as the operating wavelength, v p the flying platform
velocity, and < $ >2r the "modulo-2r" operation.
Due to the phase wrapping effect, different target radial velocity values v r are mapped into the same wrapped
phase value so that a maximum value of such velocity can
be unambiguously estimated from the interferometric
phases. This corresponds to the value that produces interferometric phases equal to r. From (3) we have
m vp
m vp
v aN = mcm c 4b , ....., 4b m,
2
N

(4)

where mcm denotes the minimum common multiple. Then,
by properly choosing the baselines, a large value of v aN can
be obtained.
To derive a statistical model of the received signal, two
statistical hypotheses have to be distinguished:

Without a moving target, the random vector Z is circularly
Gaussian with zero mean and covariance matrix C H0. Its
probability density function (pdf) can be written as

the vector Z T can be written in matrix form:
Z T = WA.

(9)

fZ ^z H 1 h =

1
exp " - ^z - n hH C -H11 ^z - n h,,
r N C H1

(10)

(5)

with
V
f c C 1NW
f c C 2NW
W + v 2W I N,
1 fW
f 1 W
X

(6)

where ) denotes conjugation; I N is the N-dimensional identity matrix; C C and C W are the clutter and
noise covariance matrices, respectively; and clutterto-noise ratio (CNR) = v 2C /v 2W and c Cnm are the clutter coherence values given by
c Cnm =

T

where n is the mean vector and C H1 is the covariance matrix.

1
exp " - z H C H-10 z ,,
r N C H0

R
S 1 c C 12
S c C) 12 1
C H0 = C C + C W = v 2C S
Sf f
Sc C) 1N c C) 2N
T

W = diag " { ,; { = 6e jz 1, f, e jz N@ ; A = 6A 1, f, A N@T , (8)

As far as the model in the presence of the target is concerned, two models are considered for vector A: a deterministic model and a random model. In both cases, the pdf
of Z is Gaussian, assuming the form

H 0: the moving target is absent Z T = 0;
H 1: the moving target is present Z T ! 0.

fZ ^z H 0 h =

with E [$] denoting the expectation operator. Note that,
for circularly symmetric zero mean Gaussian clutter, the
coefficients c Cnm for ground-moving target applications
can be considered real valued and commonly assume a
value equal to one, because the clutter signal from the
stationary ground is acquired by the SAR antennas with
the same view angle and a time delay on the order of milliseconds. This real-valued assumption is not valid in the
case of nonstationary clutter, for instance, for moving
ocean surface.
By introducing the diagonal phase matrix W and vector A ,

E 6Z Cn Z )Cm@
E 6Z Cn Z )Cm@
=
,
2
2
v C2
E 6 Z Cn @ E 6 Z Cm @

(7)

SIGNAL DISTRIBUTION FOR DETERMINISTIC
TARGET MODELS
This model is applicable to the situation of a target of simple physical shape (such as a corner reflector) whose radar
cross section (RCS) does not change in the small amount
of time required for the SAR antenna to cover the baseline
length and, in the absence of other information, to its a
priori statistical distribution. In this hypothesis, it can be
assumed that the target contribution takes the same (complex) value A = A e jz A in all of the acquired images, which
can be considered a deterministic unknown parameter influencing the SCR and statistical distribution of the data.
Under this assumption, the random vector Z is a circularly
symmetric Gaussian complex vector with mean vector n
and covariance matrix C H1, assuming the expressions listed
in Table 2 [27]. In this case, SCR = A 2 /v c2.

TABLE 2. SECOND-ORDER STATISTICS OF THE ATI SIGNAL FOR GAUSSIAN FULLY CORRELATED
CLUTTER (cCHK = 1) AND DIFFERENT TARGET MODELS.
DETERMINISTIC TARGET MODEL

RANDOM TARGET MODEL

Z T = A e jz A {

0

Covariance matrix C H1 = E 6^Z - nh^Z - nh @

C H0

v 2C SCR {{ H + C H0

Inverse of the covariance matrix C -H11

C -H10 =

Unknown model parameters

z A, SCR = A

Mean vector n = E 6Z@

H

50

1 c
CNR 2
CNR I N - 1 + N CNR 11 T m
v C2
2

v 2C, v r

C -H10 -

v 2C SCR
H
C -H10 { ^C -H10 {h
1 + tr ^v C2 SCR {{ H C -H10 h

SCR = v 2A v 2C, v r

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2020
IEEE Geoscience and Remote Sensing Magazine - June 2020

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