Improving CPT-InSAR Algorithm with Adaptive Coherent
Distributed Pixels Selection
Longkai Dong 1,2,3 , Chao Wang 1,2,3,* , Yixian Tang 1,2,3, Hong Zhang 1,2,3 and Lu Xu 1,2,3
Citation: Dong, L.; Wang, C.; Tang,
Y.; Zhang, H.; Xu, L. Improving
CPT-InSAR Algorithm with Adaptive
Coherent Distributed Pixels Selection.
Remote Sens. 2021,13, 4784. https://
Academic Editor: Zhong Lu
Received: 9 October 2021
Accepted: 22 November 2021
Published: 25 November 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional afﬁl-
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
1Key Laboratory of Digital Earth Science, Aerospace Information Research Institute, Chinese Academy of
Sciences, Beijing 100094, China; firstname.lastname@example.org (L.D.); email@example.com (Y.T.);
firstname.lastname@example.org (H.Z.); email@example.com (L.X.)
2International Research Center of Big Data for Sustainable Development Goals, Beijing 100094, China
3University of Chinese Academy of Sciences, Beijing 100049, China
*Correspondence: firstname.lastname@example.org; Tel.: +86-10-82178161
The Coherent Pixels Technique Interferometry Synthetic Aperture Radar (CPT-InSAR)
method of inverting surface deformation parameters by using high-quality measuring points pos-
sesses the ﬂaw inducing sparse measuring points in non-urban areas. In this paper, we propose the
Adaptive Coherent Distributed Pixel InSAR (ACDP-InSAR) method, which is an adaptive method
used to extract Distributed Scattering Pixel (DSP) based on statistically homogeneous pixel (SHP)
cluster tests and improves the phase quality of DSP through phase optimization, which cooperates
with Coherent Pixel (CP) for the retrieval of ground surface deformation parameters. For a region
with sparse CPs, DSPs and its SHPs are detected by double-layer windows in two steps, i.e., mul-
tilook windows and spatial ﬁltering windows, respectively. After counting the pixel number of
maximum SHP cluster (MSHPC) in the multilook window based on the Anderson–Darling (AD)
test and ﬁltering out unsuitable pixels, the candidate DSPs are selected. For the ﬁltering window,
the SHPs of MSHPC’ pixels within the new window, which is different compared with multilook
windows, were detected, and the SHPs of DSPs were obtained, which were used for coherent estima-
tion. In phase-linking, the results of Eigen decomposition-based Maximum likelihood estimator of
Interferometric phase (EMI) results are used as the initial values of the phase triangle algorithm (PTA)
for the purpose of phase estimation (hereafter called as PTA-EMI). The DSPs and estimated phase are
then combined with CPs in order to retrievesurface deformation parameters. The method was vali-
dated by two cases. The results show that the density of measuring points increased approximately
6–10 times compared with CPT-InSAR, and the quality of the interferometric phase signiﬁcantly
improved after phase optimization. It was demonstrated that the method is effective in increasing
measuring point density and improving phase quality, which increases signiﬁcantly the detectability
of the low coherence region. Compared with the Distributed Scatterer InSAR (DS-InSAR) technique,
ACDP-InSAR possesses faster processing speed at the cost of resolution loss, which is crucial for
Earth surface movement monitoring at large spatial scales.
interferometric synthetic aperture radar (InSAR); statistically homogeneous pixels (SHPs);
distributed scattering pixel (DSP); phase-linking (PL); deformation
Interferometry synthetic aperture radar (InSAR) has been widely used in Earth surface
movement monitoring [
]. In the past 20 years, the rapid development of multi-temporal
InSAR (MT-InSAR) has effectively suppressed the problems of spatial-temporal deco-
] and atmospheric phase delay of the differential interferometric synthetic
aperture radar (D-InSAR) technique [
]. The permanent scatterer InSAR (PS-InSAR) tech-
is used to process permanent scattering targets with a high signal-to-noise
ratio, and it is widely used in surface deformation monitoring at the original spatial resolu-
Remote Sens. 2021,13, 4784. https://doi.org/10.3390/rs13234784 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2021,13, 4784 2 of 19
tion. The Small Baseline Subset (SBAS) InSAR technique [
] suppresses the spatiotemporal
decoherence problem by selecting interferometric pairs with short spatiotemporal baselines.
The coherent pixels technique InSAR (CPT-InSAR) [
], which combines the ad-
vantages of PS-InSAR and SBAS-InSAR, was proposed in order to reduce the errors caused
by the small number of images and to improve the accuracy of estimation parameters. CPT-
InSAR is also able to retrieve linear and nonlinear components of movement from a set of
low resolution interferograms, estimating DEM errors and atmospheric artifacts at the same
time, and it is widely used in surface deformation monitoring [
]. However, when the
selected coherence threshold is high, CPT cannot select a sufﬁcient number of points in the
low coherence region, which will cause observation points to be disconnect [
]. When the
selected coherence threshold is low, the points with low phase quality could be selected as
the measuring points, deteriorating the accuracy of the inversed parameter [
]. In order to
improve measuring point density and monitoring accuracy of CPT, phase quality should
be improved along with the low coherence region, where they are not selected as coherent
pixels. Currently, there is no method for increasing the measuring point density of the
The PS-InSAR technique usually selects PS points with relatively stable scattering
characteristics in time and strong echo signals as observation objects, and PS includes
artiﬁcial buildings, lighthouses, exposed rocks and artiﬁcially deployed corner reﬂectors.
Similarly to CPT-InSAR, it is difﬁcult to obtain enough measurement points in non-urban
areas with low coherence. In order to tackle this problem, various methods have been
proposed for increasing the density of measuring points, such as Stable Point Interferometry
over Unurbanised Areas (SPINUA) [
], Persistent Scatterer Pair InSAR (PSP-InSAR) [
and Cousin PSs InSAR (CSP-InSAR) [
]. Additionally, fusion algorithms were proposed
in order to properly combine PS and distributed scatterer (DS) to increase the density
of measurement points [
]. Ferretti A. et al. introduced new approaches in 2011,
], in order to jointly process PS and DS, taking into account their different
statistical behaviors. SqueeSAR takes DS points as PS candidate points for space adaptive
ﬁltering and phase linking, which greatly increases the density of measuring points, and
it is considered second-generation PS-InSAR technology. Most methods for increasing
the density of measuring point are suitable for single-look processing, and there is no
systematic method for CPT-InSAR multilook processing.
The SqueenSAR method and its derived DS-InSAR method processing framework
are processed in the single look, and the processing of homogeneous pixel detection and
phase linking in DS-InSAR method is very time consuming and only practically applicable
for ground surface deformation monitoring in small areas [
]. In 2014 and 2017, the
European Space Agency (ESA) launched Sentinel-1A and Sentinel-1B satellites with a revisit
period of 6 days, and its coverage width of wide scan mode is 250 km and is open to
global users, which indicates that SAR technology enters into the era of Big Data [
]. It is
obvious that traditional DS-InSAR technology is not applicable in the processes of regularly
monitoring large spatial scale deformations. The processing idea of improving the density of
monitoring points in SqueeSAR is a good method. We learn from the processing framework
and ideas of SqueenSAR and propose a method suitable for CPT multilook processing.
The identiﬁcation of statistically homogeneous pixels (SHPs) and phase optimization
of DS includes two key steps in the DS-InSAR technique. The Kolmogorov–Smirnov (KS)
], Anderson–Darling (AD) test [
], Kullback–Leibler scatter test (KL) [
and Cramervon Mises (CM) test [
] are nonparametric tests that have been used for
the identiﬁcation of SHPs. Mi Jiang et al. proposed a fast SHPs method (FaSHPS) to
extract homogeneous points, which signiﬁcantly improves the speed of homogeneous
pixels detection [
]. Compared with other test methods, the AD statistic is powerful for
detecting changes in the shape parameter and changes within a scale family [
of estimating the phase sequence from all possible interferograms fall under phase linking
]. The difference between PL and SBAS lies in the full and partial utilization of
interferogram abundance [
]. In 2011, Ferretti et al. proposed PTA in order to optimize the
Remote Sens. 2021,13, 4784 3 of 19
DS target phase, resulting in a new direction of MT-InSAR [
]. Since then, several methods
for phase optimization were proposed, e.g., phase-decomposition-based persistent scatterer
InSAR (PD-PS InSAR) [
], component extraction and selection SAR (CAESAR) [
decomposition-based Maximum-likelihood-estimator of Interferometric phase (EMI) [
and sequential estimator [
]. Multipolarization data were also used in DS target detection
and phase optimization .
In order to overcome the limitation that the CPT-InSAR method only selects high-quality
coherence scatterers that result in low density of measuring points in non-urban and low
coherence areas, we propose a new method called ACDP-InSAR, which can adaptively
extract and process urban and non-urban measuring points by combining CPs with DSPs,
simultaneously improving the density of measuring points under multilook processing
based on the DS-InSAR processing flow. The conventional DS-InSAR method improves
the density of monitoring points in single-look data, while ACDP-InSAR designs a set of
processing flow in multilook suitable for CPT-InSAR. For low coherent regions that are
not selected for CPs, DSPs and its SHPs are obtained through a double-layer window. For
the first layer of multilook windows, the maximum statistically homogeneous pixel cluster
(MSHPC) within the window needs to be detected, and if the pixel number of clusters is
greater than a certain threshold, the window is considered as a candidate DSP. For the second
layer of the filter window, the SHPs of each pixel in the MSHPC within the window were
detected. By integrating homogeneous pixels, the SHP of the DSP is obtained. In order to
obtain the optimal phase for each time period, the coherence matrix is processed by using
the PTA-EMI method. The DSP and its optimized phase are subsequently processed together
with CPs in order to achieve the purpose of increasing the density of measuring points.
2.1. CPT-InSAR Method
When generating an interferogram by conjugate multiplication of two SAR images, its
phase variation between neighboring pixels be expressed as follows :
δφint =δφﬂat +δφtopo +δφmov +δφatm +δφnoise (1)
is the ﬂat earth component related to the range distance;
is the topo-
is the component due to the displacement of the terrain in the range
direction (line of sight (LOS)) between both SAR acquisitions;
is the phase related to
atmospheric artifacts; and
comprises degradation factors related with temporal and
spatial decorrelation and thermal noise. With respect to this,
has two contributions,
linear and nonlinear displacement, described as follows:
δφmov =δφlinear +δφnonlinear =4π
is the wavelength;
includes the velocity increments between neighboring
pixels; and T is the temporal baseline between both SAR acquisitions. Not all points satisfy
the linear model, and points with high phase quality need to be selected for the solution.
The CPT-InSAR method generates an average coherence map by estimating the average
coherence of all points in the multilook window from the entire interferogram stack, and
then it sets a threshold to select points with high coherence quality as candidate CPs. Then,
all the neighboring pixels of irregularly gridded data generating nonoverlapped triangles
are related by the Delaunay triangulation method. The linear components of movement
and DEM errors are estimated by maximizing the following model coherence function:
Remote Sens. 2021,13, 4784 4 of 19
where Nis the number of interferograms,
is the differential phase increment and
is the phase increment of the theoretical linear model. According to the coherence
threshold of the triangulation edge, the connection with low quality is rejected. The CPT
method used an approach based on the classical region growing algorithm for phase
]. The integration [
] starts from different seed points, which are chosen
from those presenting links with better model coherences, and calculates the absolute value
of velocity for each pixel by using the following equation.
Subsequently, the linear deformation rates and elevation errors at each measuring
point solved by integration are then used to calculate the nonlinear components of ground
surface deformation and atmospheric artifacts.
2.2. SHPCs Detection for DSP
Since the mathematical expectation of radar signals is not attainable in practice, under
the assumption of distributed targets, coherence is usually estimated by spatially averaging
radar echoes in a moving window :
where idenotes the i-th pixel in a coherence-estimate window (CEW), s
signals from coregistered InSAR images and
is a complex conjugate operator. For
each CEW, Lpixels are used to obtain a coherence estimate.
is the estimated value of
window coherence. The averaging in this method does not consider the homogeneity of
Backscattering amplitude statistics are a suitable method for adaptively grouping
and averaging pixels in order to preserve the phase signatures of natural structures in
the observed area [
]. The AD statistic is powerful at detecting changes in the shape
parameter and detecting changes within a scale family [
]. In this paper, the AD test
method is used to identify homogeneous pixels based on multitemporal amplitude data:
is the empirical cumulative distribution functions of the pooled distribution
obtained by combining the two independent datasets,
. . .
. The AD test method is used to test each pixel within a multilook
window in order to obtain the MSHPC within the multilook window, and the schematic
diagram of AD test in multilook window is shown in Figure 1. The pixel number of MSHPC
is used to determine whether the window is considered as a candidate DSP, and the steps
for selecting a candidate DSP are as follows.
(1) In the multilook windows (size is N*M), for image pixel F(pixel x
, pixel y
applying the two-sample AD test to amplitude data vectors, other pixels within the window
that are considered statistically homogeneous with pixel F are selected given a certain level
(2) The image pixels in a multilook window that, although selected by the AD test, are
not connected to P directly or through other SHPs are discarded.
Remote Sens. 2021,13, 4784 5 of 19
(3) Repeat steps (1) and (2) for the pixels within the
is a multilook
window), respectively, where i= 1, 2, 3
. . .
N and j= 1, 2, 3
. . .
M until all pixels in all
windows are detected, and the homogeneous cluster
Ω=[Ω1,Ω2,Ω3. . . . ΩN×M]
individual pixel is obtained.
(4) Judging the pixel number of MSHPC in each cluster, the window for which its
pixel number of MSHPC is greater than a certain threshold is the candidate DSP (and the
MSHPC are Ωmax).
Process of MSHPC acquisition. After testing all pixels in the window, the SHPCs, which
have the maximum pixel number of SHPs, include MSHPC.
The above is the method of the candidate DSPs selection. We consider the pixel as
a candidate DSP when the pixel number of MSHPC in the multilook window is greater
than the threshold. As shown in Figure 2, the yellow pixels in the red box region are the
SHPs of the MSHPC in the multilook window. It is worth mentioning that not only the AD
test method could be used in testing but also various methods for testing homogeneous
pixels can be used for testing. The phase quality of candidate DSP is poor compared to
the high coherence region. In order to improve the phase quality of DSP, spatial ﬁltering
is performed based on homogeneous pixels after determining DSP with reference to the
spatial ﬁltering technique of SqueenSAR. The steps of homogeneous pixel identiﬁcation
for SHPΩmax are as follows.
Example of the double-layer hypothesis testing window. The red window is the multilook
window, the yellow pixels are the SHPs of the multilook window and the green window is the spatial
ﬁltering window of a single SHP.
(1) The second homogeneous pixel test was performed for each SHP in
the window size was winA
winR. By using pixel P
) as the central pixel,
Remote Sens. 2021,13, 4784 6 of 19
the intensity vectors of pixel P
and other pixels q (q
as the ﬁlter
window) in the ﬁlter window are tested by using the AD test method to identify SHPs
(2) The image pixels in the ﬁlter window that, though selected by the AD test, are not
connected to Pmax directly nor through other SHPs are discarded.
of DSP and
are selected to prepare coherent
estimations for improving the target phase’s quality. Once the proper estimation window
has been deﬁned for each image pixel, by carefully selecting SHP families, amplitude data
can be despeckled, interferometric phase values can be ﬁltered and coherence values can
be estimated properly.
2.3. The Phase Optimization of DSPs
It should be noted that the same
will exist in different
. In the phase
estimation, in order to avoid the duplication of the same pixel, the
should be integrated, and the duplicate
should be removed to obtain
for coherent estimation. After identiﬁcation of the
, it is
possible to estimate the sample covariance matrix given by the following:
C(W) = Ehd(W)d(W)Hi≈1
where H indicates Hermitian conjugation, and
is the set of
used in the
sample estimate of the covariance matrix.
d(P)=[d1(P), d2(P), . . . dN(P)]H
is the complex
data vector of generic image pixel, and
is the complex reﬂectivity value of the i-th
image of the data-stack corresponding to pixel P.
d(W)=[d(P)1,d(P)2, . . . d(P)NC]
the complex data vector of the
used to estimate coherence, and NC is the
pixel number of
. The covariance matrix
is an N
N Hermitian positive
semideﬁnite matrix, with N being the number of SAR images. For phase optimization,
suppose that the coherence matrix of multilook windows can be expressed as follows [
Y is an N ×N symmetric real-value matrix for which its elements correspond to the
coherence values of all the interferograms.
is an N
N diagonal matrix containing the
values of the “true” phase values of multilook-windows W, related to the optical path of
the radar beam in each acquisition. The probability distribution function of the SHP can be
expressed as follows :
θ=[θ1,θ2, . . . , θN]T
is simply the vector of the phase values of N available images in
correspondence to the multilook windows. The estimated result
of maximum likelihood
(ML) is obtained by maximizing this probability distribution function or minimizing the
absolute value of its logarithm [
]. The optimal estimate of the N phase values is then
given by the following.
is an N
N diagonal matrix,
is an N-dimensional vector,
represents the Hadamard product. Therefore, it is necessary to use an
iterative method to ﬁnd the parameters corresponding to the minimization of
Remote Sens. 2021,13, 4784 7 of 19
in order to obtain the optimal phase sequence. However, iterative computation using the
BFGS method is extremely time consuming [
]. If an initial value close to the optimal
solution is provided, it will greatly increase the speed of iteration.
The EMI method differs from the PTA method by introducing additional degrees of
freedom for the estimation in maximizing the Wishart likelihood distribution [
the problem into a biobjective optimization problem. In order to improve computational
efﬁciency, the EMI method reduces the optimization problem to a maximum eigenvector
problem by solving the following equation.
This is the formulation of eigen decomposition of the Hadamard product
as the minimum eigenvalue and
as its corresponding eigenvector. We use
the phase sequence
solved by the EMI method as the initial value for the solution
in order to speed up phase optimization and to enhance the speed of the
solution while ensuring the accuracy of the solution. We refer to this method as PTA-
EMI. After the optimal phase has been estimated, the phase estimation results need to be
evaluated. We evaluate the results by using the temporal coherence estimation method
used in the PTA algorithm [29,50].
2.4. The Process of ACDP-InSAR
We have described the processing chain of the CPT algorithm, the SHP used to identify
the DSP coherence estimate based on the AD algorithm and the optimization of the DSP
phase by using the PTA-EMI algorithm. After the coregistration image is processed by the
above method, the standard CPT-InSAR processing chain can be used to reproduce the
parameters of CPs and DSPs. The ﬂow chart of ACDP-InSAR is shown in Figure 3and can
be described as follows.
(1) The time baseline and the space baseline are used to select the appropriate com-
bination of the interference. The average coherence of the interferometric combination
is calculated with an N
M window, and the CPs are selected by setting the coherence
(2) The operations of preprocessing the magnitude images of N images and then
ﬁnding the non-CP region and sparse-CP region SHPs based on the method outlined in
Section 2.2 are conducted to determine the DSP target and to record the SHP used to
estimate the DSP’s coherence matrix.
(3) The estimation of the coherence matrix of the DSP target using SHP and of the
optimal phase sequence based on the PTA-EMI is performed as described in Section 2.3.
(4) Based on the phase triangularity and the optimized phase, the phase dataset of
DSP under the above interference combination is calculated.
(5) The phase information of CP and DSP are jointly processed according to the CPT
processing work chain in order to invert the physical parameters of the measuring point.
The N phase sequences estimated using the PTA-EMI method have the following
represents the phase difference between time points iand j;
the phase values between time points iand j. In coherence estimation, complex data are
used, and topographic phases should be removed [
]. Compared with the traditional
CPT method, the ACDP-InSAR method does not develop a hybrid processing chain but
retrieves homogeneous pixels in the low coherence region based on the AD test method in
Remote Sens. 2021,13, 4784 8 of 19
order to obtain the coherence matrix and uses the PTA-EMI method to obtain the optimal
phase value for jointly processing CPs and DSPs.
Figure 3. Algorithm ﬂow chart of ACDP-InSAR.
3.1. Case 1: Mountainous Areas in Southwestern China
3.1.1. Study Area and Dataset
The proposed method is validated by 31 descending Sentinel-1 SAR images acquired
from 12 October 2018 to 19 October 2019 over mountainous areas in the southern region of
Bijie city, Guizhou Province, China (see Figure 4a). Due to rough terrain and vegetation
in the area, it is difﬁcult for the conventional CTP-InSAR algorithm. Sentinel-1 SAR is
C-band, with its wavelength of 5.6 cm. The nominal resolutions in azimuth and slant-range
directions are 5 and 20 m, respectively, corresponding to pixel dimensions of 2.3 and 13.9 m.
The SAR image of the study area contains 1600
8000 pixels, covering approximately
21 km ×21 km
in the azimuth and range directions. Figure 4b shows the TanDEM-X 90 m
Digital Elevation Model (DEM) downloaded from the German Aerospace Center (DLR)
EOC GEOSERVICE data portal. As observed from Figure 4d, the study area mainly in-
cludes impervious, evergreen broadleaved forest, shrubland, evergreen shrubland, rainfed
cropland, water bodies and buildings. The results of land use classiﬁcation are obtained
from the Data Sharing and Service Portal supported by the Big Earth Data Science Engineer-
ing Program (http://data.casearth.cn/sdo/detail/5d904b7a0887164a5c7fbfa0, accessed on
9 October 2021) , and the classiﬁcation information is shown in Table 1.
Remote Sens. 2021,13, 4784 9 of 19
) The location of Sentinel-1 data in China, and the black box represents range of Sentinel-1. (
location of the coverage of the SAR acquisitions superposed on the TanDEM, and the blue box represents the range of
Sentinel-1 data. (c) The optical image of study area, and (d) is the classiﬁcation result of land used in the study area.
Table 1. Classiﬁcation Information of Figure 4d.
Number Classiﬁcation System Color
1 Impervious (195,20,0)
2 Evergreen broadleaved forest (0,100,0)
3 Shrubland (150,100,0)
4 Evergreen shrubland (150,75,0)
5 Rainfed cropland (255,255,100)
6 Closed deciduous broadleaved forest (170,200,0)
7 Irrigated cropland (170,240,240)
8 Water body (0,70,200)
3.1.2. Data Processing and Result
We use TanDEM and precise orbit data to preprocess Sentinel-1 data and then gener-
ated the single look complex (SLC) data after coregistration. The date of the main image
used in the registration is 12 October 2018, and the amplitude image of the main image is
shown in Figure 5a. We follow the steps in Section 2.4 in order to process SLC data. When
calculating the average coherence coefﬁcient, the coherence of the interference pairs within
a certain baseline threshold and spatial threshold is averaged. The multilook ratio between
the azimuth and range is 4:20. The average coherence coefﬁcient of the study area is shown
in Figure 5b. According to statistics, the average coherence coefﬁcient of the entire study
area is 0.165. When selecting CPs, the coherence threshold is 0.5; that is, the pixels with
the average coherence coefﬁcient greater than 0.5 are selected as CPs. The results of the
traditional CPs selection method with a coherent threshold at 0.5, 0.375 and 0.25 are shown
in Figure 5c,d,e, respectively. In Figure 5c,d and Figure 5e, there are 329, 808 and 1872
coherent pixels, respectively. Although the number of CPs with low threshold is obviously
more than the number of CPs with high threshold, the poor phase quality of CPs with
low coherence threshold will affect the accuracy of deformation results. Therefore, the
coherence threshold of CPs used in this experimental area is 0.5 for deformation inversion.
By comparing the spatial relationship between the magnitude map, optical image, land
Remote Sens. 2021,13, 4784 10 of 19
used data and CP, it was observed that CPs are mainly distributed in the built-up areas of
the township and on bare rock. It should be noted that the coordinate system in Figure 5is
the SAR coordinate system, while the optical image coordinate system in Figure 4is the
WGS-84 coordinate system.
) Magnitude image of the master image where the date is 20181012. (
) Average coherence
coefﬁcient of an interference pair. (
) CPs’ spatial distribution with a coherent threshold as 0.5. (
spatial distribution with a coherent threshold as 0.375. (
) CPs’ spatial distribution with a coherent
threshold as 0.25. (
) DSPs’ spatial distribution. The white pixel (value as 1) represents the measuring
point of CPs and DSPs in (c–f).
DSP is extracted by using the method outlined in Section 2.2. First, the amplitude
dataset is corrected, and the amplitude value is normalized. The multilook window in the
DSP selection is 4:20, and the threshold value of the maximum pixel number of clusters
is selected as 48; that is, the window with the maximum pixel number of MSHPC greater
than 48 is selected as the candidate DSP. The dimensions of the window used to identify
is about four hectares, which corresponds in pixels to a window of 21
considering the testing time and the distribution of homogeneous statistical areas, this
detection range is reasonable in non-urban areas. The threshold of SHP rejection is 20 in
ﬁlter window. By ﬁxing 20 as the minimum number of SHP on which spatial ﬁltering can
be carried out, it is possible to preserve information of highly coherent targets [
the candidate DSPs were judged based on the pixel number of SHPs: The candidate DSPs
Remote Sens. 2021,13, 4784 11 of 19
below 48 were rejected, and the remaining candidate DSPs are the ﬁnal DSPs. The results
of DSP pixel selection are shown in Figure 5f, with 4499 measuring points. By comparing
the spatial relationships of the magnitude map, optical images and DSP, it was found that
DSP points were mainly distributed in agricultural ﬁelds and submerged vegetation areas.
In the experimental area, the number of DSPs was approximately 10 times of the number
of CPs with the coherent threshold as 0.5. It should be noted that the effect of monitoring
point density enhancement is more related to both topography and geomorphology, and
the effect has a better performance in non-urban areas.
In order to show the validity of the PTA-EMI method, we processed data not only
possessing DSP but also all pixels. After using PTA-EMI, we acquired N ﬁltered differential
interferograms. These N differential interferograms are the optimized phase after spatial
ﬁltering and temporal ﬁltering. We selected three interference phases before and after
ﬁltering for comparison, as shown in Figure 6. It can be observed that the quality of the
interference phase obtained by the proposed method has been greatly improved, which
proves the effectiveness of the algorithm. Figure 7shows the results of temporal coherence
calculated according to Equation (11), which takes values in the range of 0–1, with larger
values representing better temporal coherence and better temporal phase estimation results.
It can be observed from the Figure 7that the temporal coherence of the county and low
vegetation is better, while the temporal coherence of the broadleaf forest is worse.
) are the original phases of interference pairs 20191007–20191019, 20190913–20191019
and 20190209–20190317, respectively. (
) Phases 20191007–20191019, 20190913–20191019 and
20190209–20190317 after ACDP-InSAR processing.
Remote Sens. 2021,13, 4784 12 of 19
Figure 7. Time coherence of the study area in mountainous areas.
After optimizing the phase of the DSP pixels, we fuse the CPs and their phase with the
DSP pixels and its phase, and the merged pixels are named mixed coherent pixels (MCPs).
The phase of the CPs is the original phase and has not been processed by phase optimization.
According to the CPT method, the Delauney triangulation [
] of MCPs is constructed
and enters the phase, slope distance, incident angle and baseline parameters into the
triangulation. Finally, the deformation rate of the MCPs is obtained through integration.
The results of the deformation rate based on the CPT method and the deformation rate
based on the ACDP-InSAR method in the study area of case 1 are shown in Figure 8. By
comparing the two results, it can be observed that the number of measuring points has
increased signiﬁcantly, and the results of the deformation rate are more consistent, which
proves the effectiveness and reliability of the algorithm proposed in this paper.
) The surface deformation rate of the study area obtained by the CPT method. (
) The surface deformation rate
of the study area obtained by the method proposed in this paper.
3.2. Case 2: Shigatse M5.9 earthquake in Tibet, China
3.2.1. Study Area and Dataset
The proposed method is validated by 29 descending Sentinel-1 SAR images acquired
from 10 January 2020 to 21 December 2020 over the middle Himalayan Mountains (see
Figure 9c). The SAR image of the study area contains 2000
14000 pixels, covering
approximately 31 km
27 km in the azimuth and range directions. Figure 9a shows the
TanDEM-X 90m DEM of the study area, and Figure 9b shows the Sentinel-2 optical images
of the study area. The study area is located at the junction of the Indian plate and the
Eurasian plate in the Himalayan seismic zone. According to the website of the China
Earthquake Networks Center, an earthquake of magnitude 5.9 occurred in Tingri County,
Shigatse City, Tibet, on 20 March 2020, and the focal depth was 10.0 km. There have been
Remote Sens. 2021,13, 4784 13 of 19
134 earthquakes of magnitude three and above within 200 kilometers of the epicenter in
the past 9 years. The largest earthquake was a magnitude 7.5 earthquake that occurred in
Nepal on 12 May 2015.
) The location of Sentinel-1 data in China, and the black box represents range of Sentinel-1.
) Geographic location of the coverage of the SAR acquisitions superposed on the TanDEM, and the
blue box represents the range of Sentinel-1 data. (
) Optical image of the study area of Shigatse M5.9
earthquake in Tibet, China.
3.2.2. Data Processing and Result
The date of the main image used in the coregistration is 8 January 2020, and the
amplitude image of the main image is shown in Figure 10a. We follow the steps in
to process the SLC data of the study area in case 2. The multilook ratio between
azimuth and range is 4:20. The ground features in this area are mainly grassland and bare
rock, with high coherence. Therefore, in order to monitor points with exceptional quality,
the coherence threshold is set to 0.6.
Figure 10. (a) Magnitude image of the master image 20200108. (b) Average coherence coefﬁcient of the interference pair.
Remote Sens. 2021,13, 4784 14 of 19
DSP is extracted by the method outlined in Section 2.2. The amplitude dataset is
corrected after coregistration, and the amplitude value is normalized. Then, the multilook
window in the DSP selection is 4:20, and the threshold value of the maximum pixel number
of cluster is 48. Then, the homogeneous pixels of the
were tested, and the other
parameters were the same as those in case 1. The coherence matrix is estimated by SHP,
and the phase of DSPs is optimized by the PTA-EMI method. Case 1 has proven that the
effect of phase optimization is obvious, and it is no longer veriﬁed in this experiment.
The results of the deformation rate based on the CPT method and the deformation
rate based on the ACDP-InSAR method in the Shigatse M5.9 earthquake are shown in
By comparing both results, it can be observed that the number of measuring
point has increased signiﬁcantly, and the results of the deformation rate are more consistent.
The measuring point number of CPs is 14516 and the measuring point number of CPs and
DSPs is 95462. The increase in the density of monitoring points allows detecting surface
deformations more clearly. It can be observed from Figure 11 that the fault zone of the
earthquake is very obvious, and the range and absolute value of surface subsidence on the
east side are larger than the range and absolute value of surface uplift on the west side.
There is no monitoring point in the center of the deformation ﬁeld because the deformation
gradient is too large, which results in decoherence. From the phase of the interference pair
20200320–20200427, the maximum deformation can be roughly calculated to reach 300 mm.
The study area is a medium-high coherence area, but the conventional CPT-InSAR cannot
meet the demand for measuring point density for seismic deformation inversion.
) The surface deformation rate of the study area in case 2 obtained by the CPT-In- SAR method. (
) The surface
deformation rate of the study area in case 2 obtained by the method proposed in this paper.
It can be observed from the experimental results that the method proposed in this
paper will increase the density of monitoring points and improve phase quality. When
detecting SHPs in a double-layer window, the size of the window can be dynamically
adjusted. The size of the multilook window depends on the multilook number of the
average coherence coefﬁcient, which should be uniﬁed; otherwise, CP and DSP cannot be
combined. The size of the ﬁltering window will affect the ﬁlter’s result. A small window
will render the ﬁlter unapparent, but a window that is too large induce computational
burden and may lose details. In some scenes, the ﬁlter window is too large to retrieve more
SHPs, which is an invalid operation. Therefore, the corresponding window size should
be set according to different application scenarios. In order ensure the homogeneity of
multilook windows, a large threshold should be used.
In the era of SAR Big Data, the processing efﬁciency of algorithms is the basis of big
data processing. The PTA-EML method used in this paper has improved efﬁciency over
PTA phase optimization. It should be noted that the algorithm for identifying MSHPC
is too redundant when extracting DSPs. The mutuality of hypothesis testing can be used
Remote Sens. 2021,13, 4784 15 of 19
to enhance the speed of the MSHPC test. Mutuality of AD test means that the test result
between data group B and data group A when A as main body is the same as the test
result when B as the main body. It is not necessary to examine each pixel when retrieving
MSHPC in multilook windows, because the SHPCs of pixel A and the SHPC of A’s SHP are
practically the same. Therefore, it is only necessary to classify the pixels in the window by
their statistical characteristics, and the cluster with the largest pixel number is the MSHPC
of this window.
Phase optimization is a more time-consuming step [
], and the pixel number of DSPs
selected by ACDP-InSAR in mulitilook scenes is much smaller than the pixel number of DS
points selected by SqueenSAR in the single-look scenes. Therefore, the processing speed of
the proposed method is faster than that of the conventional SqueenSAR algorithm. Taking
case 1 as an example, we counted the time consumed by ACDP-InSAR and SqueenSAR
in SHP selection and phase-link steps. The statistical diagram of algorithm running time
is shown in Figure 12. The diagram shows, in Figure 12, that the processing speed of
ACDP-InSAR in SHP selection is lower than that of SqueenSAR, and the processing speed
of the phase link is higher than that of SqueenSAR. The overall processing speed of the
method proposed in this paper signiﬁcantly improved. On the other hand, the proposed
method has a cost of resolution reduction. The ACDP-InSAR method is more suitable for
high precision surface deformation monitoring in large spatial scales.
Cumulative processing time of the ACDP-InSAR and SqueenSAR over case 1 as an
assessment of computational efﬁciency.
In Section 3, we used two experiments to prove effectiveness in non-urban areas,
which has a good effect of increasing the number of points. Unfortunately, no in situ
measurement data could be applied to quantitatively verify the accuracy of the algorithm
in t experiment areas. In order to quantitatively evaluate the algorithm accuracy proposed
in this paper, the measurement data for Beijing, China, was used. The ﬁeld observation
points covering 40 different regions in Beijing, China, from 2017 to 2018 were collected.
However, the measurement period of the measured leveling data is one year, which cannot
be compared with the InSAR results in a long time series. We changed the leveling data
into annual average settlement and compared it with the annual average deformation rate
of ACDP-InSAR. Before comparison, the vertical deformation of leveling data was changed
into the deformation along the LOS direction according to satellite parameters. The area of
accuracy veriﬁcation and data comparison between leveling data and ACDP-InSAR result
are shown in Figures 13 and 14, respectively.
As observed from Figure 13c, the leveling results are practically consistent with the
results of ACDP-InSAR, and individual ﬁeld observation points are different. The speciﬁc
data values are compared, as shown in Figure 14. According to statistics, the root mean
square error (RMSE) between ACDP-InSAR results and leveling data is 5.45 mm/year,
which proves that ACDP-InSAR results are reliable.
Remote Sens. 2021,13, 4784 16 of 19
The experimental area for ACDP-InSAR accuracy veriﬁcation. (
) The location of the
experimental area, and blue box represents the range of leveling data. (
) Distribution map of ﬁeld
observation points. Each benchmark in the ﬁgure has a corresponding label, and the color of each
ﬁeld observation point represents the annual average deformation rate obtain from leveling data.
The corresponding color label is same as the color label in (
). The result of overlaying the level data
deformation rate above the ACDP-InSAR deformation rate.
Comparison between in situ leveling measurements and InSAR results estimated by the
proposed method over the ﬁeld observation points in Figure 13b.
The ACDP-InSAR method not only has the advantages of the CPT-InSAR method
but also has better density with respect to measuring points and detectability than those
of the latter method. Of course, the ACDP-InSAR method also has some limitations.
Firstly, ACDP-InSAR is an improvement of the CPT-InSAR method, which estimates
coherent values by multilook processing; thus, it will degrade resolution. Low resolution
Remote Sens. 2021,13, 4784 17 of 19
is not suitable for ﬁne deformation measurement, such as deformation monitoring of
dams, bridges and regional buildings. Secondly, both the ACDP-InSAR method and the
CPT-InSAR method solve the results by constructing a network. In large spatial scale
deformation monitoring, the process of removing edges with low coherence and causing a
set of CPs to be divided into several clusters is easy. Future investigations should establish
a multi-layer network to connect multiple clusters during large spatial scale data processing
in order to ensure the consistency of regional deformation monitoring.
In this paper, we proposed method ACDP-InSAR method in order to improve measur-
ing point density based on distributed scattering targets for solving the problem of sparse
measurement points in non-urban areas caused by CPT-InSAR. ACDP-InSAR identiﬁes
SHPs with two-layer windows based on the AD test. The ﬁrst multilook window selects
the DSP, and the second ﬁlter window determines the SHPCs of the DSP. The CP target is
selected on the coherence coefﬁcient map after multilooking, while the DSP needs to be
selected under the single-look scene. Therefore, the resolution of DSP should be consistent
with that of CP. The SHP of DSP is used to estimate the coherence matrix of DSP, the PTA-
EMI method is then used to optimize phase sequences, and a set of optimal phase sequence
with phase triangularity is obtained. We selected two regions as experimental areas located
in the mountainous region of Southwest China and around Shigatse on the Tibet Plateau.
The experimental results show that the density of monitoring points increases approx-
imately 5–10 times, and the phase quality is improved, verifying the effectiveness and
applicability of the algorithm. The ACDP-InSAR method can not only acquire monitoring
points in a low-coherence region but also increase the density of monitoring points in a
medium-coherence region. The root mean square error (RMSE) between ACDP-InSAR
results and leveling data is 5.45 mm/y, which proves that ACDP-InSAR results are reliable.
Compared with the conventional DS-InSAR technology, this method improves processing
speed at the cost of resolution loss and is suitable for Earth surface movement monitoring
in large spatial scales.
Conceptualization, L.D., C.W. and H.Z.; methodology, L.D. and Y.T.; software,
L.D. and Y.T.; validation, L.D., C.W. and L.X.; formal analysis, L.D. and H.Z.; investigation, L.D.;
resources, C.W., H.Z., L.X. and Y.T.; data curation, L.D.; writing—original draft preparation, L.D. and
L.X.; writing—review and editing, C.W., H.Z. and Y.T.; visualization, L.D. and L.X.; supervision, C.W.
and H.Z.; project administration, C.W.; funding acquisition, C.W. All authors have read and agreed
to the published version of the manuscript.
This research was funded by the Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDA19090126, and the National Natural Science Foundation of
China, Grant No. 41930110.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
Sentinel-1 data were provided by the European Space Agency (ESA)
and are available from the Alaska Satellite Facility (ASF) (https://vertex.daac.asf.alaska.edu, accessed
on 9 October 2021). TanDEM-X 90m DEM data are available at https://download.geoservice.dlr.de/
TDM90, accessed on 9 October 2021.
The authors would like to thank ESA and EU Copernicus Program for providing
Sentinel-1A SAR data and DLR for providing TanDEM-X 90m DEM data. Bo Zhang, Fan Wu, Wei
Duan and Jing Wang of Aerospace Information Research Institute of CAS are acknowledged for
Conﬂicts of Interest: The authors declare no conﬂict of interest.
Remote Sens. 2021,13, 4784 18 of 19
Samsonov, S.; d’Oreye, N.; Smets, B. Ground deformation associated with post-mining activity at the French-German border
revealed by novel InSAR time series method. Int. J. Appl. Earth Obs. 2013,23, 142–154. [CrossRef]
Lu, Z.; Dzurisin, D. InSAR Imaging of Aleutian Volcanoes[M]//InSAR Imaging of Aleutian Volcanoes; Springer: Berlin/Heidelberg,
Germany, 2014; pp. 87–345.
Wang, C.; Zhang, Z.; Zhang, H.; Wu, Q.; Zhang, B.; Tang, Y. Seasonal deformation features on Qinghai–Tibet railway observed
using time-series InSAR technique with high-resolution TerraSAR-X images. Remote Sens. Lett. 2017,8, 1–10. [CrossRef]
Wang, C.; Zhang, Z.; Zhang, H.; Zhang, B.; Tang, Y.; Wu, Q. Active Layer Thickness Retrieval of Qinghai–Tibet Permafrost Using
the TerraSAR-X InSAR Technique. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2018,11, 4403–4413. [CrossRef]
Wang, J.; Wang, C.; Zhang, H.; Tang, Y.; Zhang, X.; Zhang, Z. Small-Baseline Approach for Monitoring the Freezing and Thawing
Deformation of Permafrost on the Beiluhe Basin, Tibetan Plateau Using TerraSAR-X and Sentinel-1 Data. Sensors
Hartwig, M.E.; Paradella, W.R.; Mura, J.C. Detection and monitoring of surface motions in active open pit iron mine in the
Amazon Region, using persistent scatterer interferometry with terrasar-x satellite data. Remote Sens.
,5, 4719–4734. [CrossRef]
Zhao, C.; Lu, Z.; Zhang, Q. Time-series deformation monitoring over mining regions with SAR intensity -based offset measure-
ments. Remote Sens. Lett. 2013,4, 436–445. [CrossRef]
Lu, Z.; Dzurisin, D.; Biggs, J.; Wicks, C., Jr.; McNutt, S. Ground surface deformation patterns, magma supply, and magma storage
at Okmok volcano, Alaska, from InSAR analysis: 1. Intereruption deformation, 1997–2008. J. Geophys. Res. Solid Earth
Dong, L.; Wang, C.; Tang, Y.; Tang, F.; Zhang, H.; Wang, J.; Duan, W. Time Series InSAR Three-Dimensional Displacement
Inversion Model of Coal Mining Areas Based on Symmetrical Features of Mining Subsidence. Remote Sens.
Zebker, H.A.; Villasenor, J. Decorrelation in interferometric radar echoes. IEEE Trans. Geosci. Remote Sens.
,30, 959. [CrossRef]
Massonnet, D.; Feigl, K.L. Radar interferometry and its application to changes in the Earth
s surface. Rev. Geophys.
Ferretti, A.; Prati, C.; Rocca, F. Permanent scatterers in SAR interferometry. IEEE Trans. Geosci. Remote Sens.
Ferretti, A.; Prati, C.; Rocca, F. Nonlinear subsidence rate estimation using permanent scatterers in differential SAR interferometry.
IEEE Trans. Geosci. Remote Sens. 2000,38, 2202–2212. [CrossRef]
Ferretti, A.; Prati, C.; Rocca, F. Process for Radar Measurements of the Movement of City Areas and Landsliding Zones; International
Application Published Under the Patent Cooperation Treaty (PCT): Washington, DC, USA, 2000.
Berardino, P.; Fornaro, G.; Lanari, R.; Sansosti, E. A new algorithm for surface deformation monitoring based on small baseline
differential SAR interferograms. IEEE Trans. Geosci. Remote Sens. 2002,40, 2375–2383. [CrossRef]
Mora, O.; Mallorqui, J.J.; Broquetas, A. Linear and nonlinear terrain deformation maps from a reduced set of interferometric SAR
images. IEEE Trans. Geosci. Remote Sens. 2003,41, 2243–2253. [CrossRef]
Mallorquí, J.J.; Mora, O.; Blanco, P. Linear and non-linear long-term terrain deformation with DInSAR (CPT: Coherent Pixels
Technique). In Proceedings of the FRINGE 2003 Workshop ESA, Frascati, Italy, 1–5 December 2003; Volume 36, pp. 1–8.
Blanco, P.; Mallorqui, J.; Duque, S. Advances on DInSAR with ERS and ENVISAT data using the coherent pixels technique (CPT).
In Proceedings of the 2006 IEEE International Symposium on Geoscience and Remote Sensing, Denver, CO, USA, 31 July–4
August 2006; pp. 1898–1901.
Blanco-Sanchez, P.; Mallorquí, J.J.; Duque, S. The coherent pixels technique (CPT): An advanced DInSAR technique for nonlinear
deformation monitoring. In Earth Sciences and Mathematics; Birkhäuser: Basel, Switzerland, 2008; pp. 1167–1193.
Duque, S.; Mallorqui, J.J.; Blanco, P. Application of the coherent pixels technique (CPT) to urban monitoring. In Proceedings of
the 2007 Urban Remote Sensing Joint Event, Paris, France, 11–13 April 2007; pp. 1–7.
Navarro-Hernánde, M.I.; Tomás, R.; Lopez-Sanchez, J.M. Spatial Analysis of Land Subsidence in the San Luis Potosi Valley
Induced by Aquifer Overexploitation Using the Coherent Pixels Technique (CPT) and Sentinel-1 InSAR Observation. Remote Sens.
2020,12, 3822. [CrossRef]
Costantini, M.; Ferretti, A.; Minati, F. Analysis of surface deformations over the whole Italian territory by interferometric
processing of ERS, Envisat and COSMO-SkyMed radar data. Remote Sens. Environ. 2017,202, 250–275. [CrossRef]
Bovenga, F.; Nutricato, R.; Guerriero, A.R.L. SPINUA: A ﬂexible processing chain for ERS/ENVISAT long term interferometry. In
Proceedings of the Envisat & ERS Symposium, Salzburg, Austria, 6–10 September 2005; p. 572.
Costantini, M.; Falco, S.; Malvarosa, F. A new method for identiﬁcation and analysis of persistent scatterers in series of SAR
images. In Proceedings of the IGARSS 2008-2008 IEEE International Geoscience and Remote Sensing Symposium, Boston, MA,
USA, 7-11 July 2008; Volume 2, pp. II-449–II-452.
Devanthéry, N.; Crosetto, M.; Monserrat, O.; Cuevas-González, M.; Crippa, B. An approach to persistent scatterer interferometry.
Remote Sens. 2014,6, 6662–6679. [CrossRef]
Hooper, A. A multi-temporal InSAR method incorporating both persistent scatterer and small baseline approaches. Geophys. Res.
Lett. 2008,35. [CrossRef]
27. Rocca, F. Modeling interferogram stacks. IEEE Trans. Geosci. Remote Sens. 2007,45, 3289–3299. [CrossRef]
Remote Sens. 2021,13, 4784 19 of 19
Zebker, H.A.; Shanker, A.P. Geodetic Imaging with Time Series Persistent Scatterer InSAR; American Geophysical Union: San
Francisco, CA, USA, 2008; p. G51C-02.
Ferretti, A.; Fumagalli, A.; Novali, F.; Prati, C.; Rocca, F.; Rucci, A. A new algorithm for processing interferometric data-stacks:
SqueeSAR. IEEE Trans. Geosci. Remote Sens. 2011,49, 3460–3470. [CrossRef]
Ansari, H.; De Zan, F.; Bamler, R. Efﬁcient phase estimation for interferogram stacks. IEEE Trans. Geosci. Remote Sens.
Ansari, H.; De Zan, F.; Bamler, R. Sequential Estimator: Toward Efﬁcient InSAR Time Series Analysis. IEEE Trans. Geosci. Remote
Sens. 2017,55, 5637–5652. [CrossRef]
Wang, G.; Xu, B.; Li, Z. A phase optimization method for DS-InSAR Based on SKP decomposition from quad-polarized data.
IEEE Geosci. Remote Sens. Lett. 2021, 1–5. [CrossRef]
Lilliefors, H.W. On the kolmogorov-smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc.
Razali, N.M.; Wah, Y.B. Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. J. Stat.
Model. Anal. 2011,2, 21–33.
Parizzi, A.; Brcic, R. Adaptive InSAR stack multilooking exploiting amplitude statistics: A comparison between different
techniques and practical results. IEEE Geosci. Remote Sens. Lett. 2010,8, 441–445. [CrossRef]
Polzehl, J.; Spokoiny, V. Propagation-separation approach for local likelihood estimation. Probab. Theory Relat. Fields
Deledalle, C.A.; Denis, L.; Tupin, F. NL-InSAR: Nonlocal interferogram estimation. IEEE Trans. Geosci. Remote Sens.
Anderson, T.W. On the distribution of the two-sample Cramervon Mises criterion. Ann. Math. Stat.
Jiang, M.; Ding, X.; Hanssen, R.F.; Malhotra, R.; Chang, L. Fast statistically homogeneous pixel selection for covariance matrix
estimation for multitemporal InSAR. IEEE Trans. Geosci. Remote Sens. 2014,53, 1213–1224. [CrossRef]
Jiang, M.; Guarnieri, A.M. Distributed scatterer interferometry with the reﬁnement of spatiotemporal coherence. IEEE Trans.
Geosci. Remote Sens. 2020,58, 3977–3983. [CrossRef]
Cao, N.; Lee, H.; Jung, H.C. A phase-decomposition-based PSInSAR processing method. IEEE Trans. Geosci. Remote Sens.
Fornaro, G.; Verde, S.; Reale, D.; Pauciullo, A. CAESAR: An approach based on covariance matrix decomposition to improve
multibaseline–multitemporal interferometric SAR processing. IEEE Trans. Geosci. Remote Sens. 2014,53, 2050–2065. [CrossRef]
Liao, M.; Balz, T.; Rocca, F.; Li, D. Paradigm changes in Surface-Motion estimation from SAR: Lessons from 16 years of
Sino-European cooperation in the dragon program. IEEE Geosci. Remote Sens. Mag. 2020,8, 8–21. [CrossRef]
Xu, W.; Cumming, I. A region-growing algorithm for InSAR phase unwrapping. IEEE Trans. Geosci. Remote Sens.
Touzi, R.; Lopes, A.; Bruniquel, J. Coherence estimation for SAR imagery. IEEE Trans. Geosci. Remote Sens.
Guarnieri, A.M.; Tebaldini, S. On the exploitation of target statistics for SAR interferometry applications. IEEE Trans. Geosci.
Remote Sens. 2008,46, 3436–3443. [CrossRef]
Lv, X.; Yazıcı, B.; Zeghal, M. Joint-scatterer processing for time-series InSAR. IEEE Trans. Geosci. Remote Sens.
Samiei-Esfahany, S.; Martins, J.E.; Van, L.F. Phase estimation for distributed scatterers in InSAR stacks using integer least squares
estimation. IEEE Trans. Geosci. Remote Sens. 2016,54, 5671–5687. [CrossRef]
Goodman, N.R. Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math.
Stat. 1963,34, 152–177. [CrossRef]
Shanker, P.; Zebker, H. Persistent scatterer selection using maximum likelihood estimation. Geophys. Res. Lett.
Cao, N.; Lee, H.; Jung, H.C. Mathematical framework for phase-triangulation algorithms in distributed-scatterer interferometry.
IEEE Geosci. Remote Sens. Lett. 2015,12, 1838–1842.
Zhang, X.; Liu, L.; Chen, X.; Xie, S.; Gao, Y. Fine Land-Cover Mapping in China Using Landsat Datacube and an Operational
SPECLib-Based Approach. Remote Sens. 2019,11, 1056. [CrossRef]
53. Delaunay, B. Sur la sphere vide. Izv. Akad. Nauk SSSR 1934,7, 1–2.