IEEE Geoscience and Remote Sensing Magazine - March 2019 - 48

Parametric-Based MB PU Methods
Use InSAR PDF to Formulate an
ML or MAP Framework to Estimate
Absolute Phase

Multiple Input
Interferograms With
Different Normal
Baseline Lengths

MB PU Result
Use Unsupervised-Learning
Techniques to Estimate
Absolute Phase
Nonparametric-Based MB PU Methods

FIGURE 10. The processing flowcharts of the MB PU methods. PDF: probability density function.

According to the Chinese remainder theorem (CRT), we
know that C/2r in (10) can be uniquely obtained in the interval from 0 to lC 1 C 2 [79], [80]. Then, k i(s)(i = 1, 2) can
be solved from (10). We can see that some special combinations of B i (i = 1, 2) enable us to solve k i(s)(i = 1, 2) by
using just (9) [79], [81]. However, CRT is too sensitive to
measurement bias to use directly [81]. The measurement
bias could be caused by surface deformation, atmospheric
artifact, or phase noise. To overcome this issue, a variety of
MB PU methods are proposed.
Recent MB PU methods can be divided into two major
groups: parametric-based and nonparametric-based methods. The processing flowcharts of these two groups of MB PU
methods are summarized in Figure 10. From Figure 10, we
can see that the main ideas of these two groups of MB PU
methods both derive from machine learning. The methods
in the first group make use of the InSAR probability density
function to formulate a statistical framework using maximum likelihood (ML) [82]-[86] or maximum a posteriori
(MAP) criteria [87]-[91] to estimate the PU result.
Baselice et al. [92] provide a survey article of ML- and
MAP-based MB methods. A comparison of the phase estimation accuracy between ML- and MAP-based MB methods
was analyzed in [93]. In [94], the PU performance of the ML
MB method was evaluated by comparing it to results based
on a coarse-to-fine algorithm [95]. Equation (11), shown in
the box at the bottom of the page, is one common likelihood function used to design the parametric-based MB PU
method. In (11), c i (s) is the coherence value of the sth pixel
in the ith interferogram.

sampling rate, but it is based on repeated observations with
different InSAR system parameters. For simplicity, we will
use a dual-baseline (DB) InSAR system as an example to
introduce the principle of MB PU.
If there are two interferograms with different normal
baselines, we can combine the absolute phases of the two
interferograms using the relationship established from (1)
and (2) [78]:
B 2 ·({ 1(s) + 2k 1(s) r) = B 1 ·({ 2(s) + 2k 2(s)r)(flattened),
(9)
where B i (i = 1, 2) is the normal baseline length of the ith
interferogram, and k i (s)(i = 1, 2) and { i (s)(i = 1, 2) are
the ambiguity number and wrapped phase of the sth pixel
in the ith interferogram, respectively. In (9), the term flattened indicates that the flat-Earth phases of the wrapped
phases have been removed. Theoretically, we cannot solve
k 1(s) and k 2(s) simultaneously using only (9) (one equation
cannot solve two unknowns). However, because k 1(s) and
k 2(s) are integers, if we allow C = B 2 ·({ 1(s) + 2k 1(s) r),
then (9) can be transformed to
Z
]] C = B 2 ·k 1 (s) + B 2 { 1 (s)
2r
2r
[
B { (s) .
] C = B 1 ·k 2 (s) + 1 2
2r
\ 2r

(10)

Under this condition, B 1 and B 2 are real moduli, and
B 2 { 1(s) 2r and B 1 { 2(s) 2r are real remainders. If there is
a normalization factor l, then it makes C i = B i /l coprime.

p ({ i (s) | h (s)) = 21r ·

1 - c i (s) 2

1 - c i (s) 2 cos 2 c { i (s) -

4rB i
h (s) m
mr (s) sin (i)

J
N
4rB i
4rB i
c i (s) cos c { i (s) h (s) m cos -1 c -c i (s) cos c { i (s) h (s) mm O
K
m
i
m
i
r
(
s
)
sin
(
)
r
(
s
)
sin
(
)
O.
# K1 +
1/2
K
O
4rB i
1 - c i (s) 2 cos 2 c { i (s) h (s) m
K
O
mr (s) sin (i)
L
P
48

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