IEEE Geoscience and Remote Sensing Magazine - June 2020 - 53

Figure 4 illustrates the behavior of the phase pdf for increasing SCR. The peak of the pdf migrates toward the target phase value z 0, exhibiting a symmetrical behavior with
respect to z 0. When the SCR tends to infinity, the phase
pdf becomes identical to the original clutter pdf except for
the shift toward the target phase. In addition, in this case,
the dependence on CNR is weak, and, for increasing v r, the
pdf variance gradually increases [20]. This behavior implies
that velocity estimation is more accurate for small values of
velocity and less accurate when velocity increases, as shown
by the Cramér-Rao lower bound (CRLB) in the "Velocity
Estimation From Phase-Only Data" section.
In case of heterogeneous terrain, i.e., rough sea surfaces or urban areas, the Gaussian assumption is no longer
valid [18], and the clutter model must be adapted. Aside
from the introduction of heuristically motivated, longertailed distributions like the Weibull or log-gamma, the
product or compound model established itself as the prevailing model of choice because it is physically meaningful and flexible.
The basic idea is that the underlying Gaussian fluctuations are modulated by an independent texture random
variable T, that is,

(20)

k=1

where d in general can be a complex value.
This modification effectively decreases the width of the
clutter phase pdf in the hypothesis H 0, thereby pushing the
detection threshold away from ! r toward smaller values,
thereby permitting an increased probability of detection
PD. An example is shown in Figure 5, in which the original
ATI phase is included as special case d = 0. The addition of
the constant d in (20) also affects the pdf of the interferometric phase in the hypothesis H 1, producing a shift in the
peak and a slight decrease in the pdf width, as presented
in [42].
JUNE 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

-h

fUl ^zl H 0 h dzl,

(21)

and the corresponding probability of detection PD when a
target is present through
r

PD =

-h

fUl ^zl H 1h dzl.

(22)

Both integrals cannot be analytically solved and must
be numerically computed. Evaluating the PD for varying

9
d=0
d = 0.5
d=1

8
7

(19)

6
pdf

Note that the interferometric clutter phase is invariant
against such multiplicative modulation because the texture random variable cancels out when the phase is computed. Consequently, it was shown that the ATI phase is a
robust detector with respect to heterogeneity [41]. Nevertheless, the biggest weakness of using solely the phase to
detect moving targets is the restricted support of the phase
between - r and r, which, in many cases, does not permit the selection of a practically relevant small probability
of false alarm. In other words, for a diminishing PFA, the
threshold approaches the limits ! r, and any meaningful
detection becomes impossible.
To mitigate this problem, a simple but powerful extension of the ATI phase (called BATI) was introduced
[42], namely, adding a constant term prior to the phase
computation
Ul = arg d / Z 1 ^ k h Z )2 ^ k h + d n,

PFA =

5
4
3
2
1
0

-0.6

-0.4

-0.2
0
0.2
Phase (rad)

0.4

0.6

FIGURE 5. A BATI phase pdf for L = 2, c H 0 = 0.98, and varying d.

1
0.9
Probability of Detection

ZlC = TZ C.

Obviously, the additive constant d is not a priori known,
and a suitable value must be predetermined. However, it
was found via extensive simulations and real data that precise knowledge is not critical [42] (see also Figure 6).
The detection threshold h is computed via inversion of
the integral of the interferometric phase for an operatorgiven constant false alarm rate PFA via

0.8
0.7
d=0
d = 0.1
d = 0.3
d=2

0.6
0.5
0.4
0.3
0.2
0.1
0

-3

-2.5

-2
-1.5
-1
Phase (rad)

-0.5

FIGURE 6. The PD of BATI versus a target phase for L = 1,

c H 0 = 0.98, SCR = 3 dB, and varying constant d. The PFA was
chosen as 10 -4 .
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IEEE Geoscience and Remote Sensing Magazine - June 2020

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