A Phase-Decomposition-Based PSInSAR Processing Method

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 54, NO. 2, FEBRUARY 2016

A Phase-Decomposition-Based PSInSAR Processing Method Ning Cao, Hyongki Lee, and Hahn Chul Jung Abstract—A phase-decomposition-based persistent scatterer (PS) InSAR (PD-PSInSAR) method is developed in this paper to improve coherence and spatial density of measurement points (MPs). In order to improve PS network density, a distributed scatterer (DS) has been utilized in some advanced PSInSAR process, such as SqueeSAR. In addition to the conventional DS that consists of independent small scatterers with a uniform scattering mechanism, processing the DSs dominated by two or more scattering mechanisms is a promising way to improve MP density. Estimating phases from DS with multiple scattering mechanisms is difficult for many DS algorithms because of the interference between different scattering mechanisms. Recently, a covariancematrix-decomposition-based method, which is named Component extrAction and sElection SAR (CAESAR), is proposed to extract different scattering components from the analysis of the covariance matrix. Instead of using a covariance matrix, the PD-PSInSAR in this study is developed to perform eigendecomposition on a coherence matrix, in order to estimate the phases corresponding to the different scattering mechanisms, and then to implement these estimated phases in a conventional PSInSAR process. The major benefit of using a coherence matrix rather than a covariance matrix is to compensate the amplitude unbalances among SAR images. A detailed study of comparison among SqueeSAR, CAESAR, and PD-PSInSAR is also performed in this study. It has been found that these three methods share very similar mathematic forms with different weight values. The PD-PSInSAR method is implemented to estimate land deformation over the greater Houston area using Envisat ASAR images, which verifies that the proposed method can detect more MPs and provide better coherences. Index Terms—Differential interferometric synthetic aperture radar (DInSAR), distributed scatterer (DS) interferometry, persistent scatterer (PS) interferometry (PSI), synthetic aperture radar (SAR).

I. I NTRODUCTION

D

IFFERENTIAL interferometric synthetic aperture radar (DInSAR) has already been proven to be a useful technique for measuring ground displacement [1], [2]. Among various multitemporal DInSAR techniques, the persistent (or permanent) scatterer InSAR (PSInSAR) technique has been widely used in a variety of cases, due to its high accuracy and resistance

Manuscript received September 18, 2014; revised February 14, 2015, July 13, 2015, and July 24, 2015; accepted August 21, 2015. Date of publication September 17, 2015; date of current version January 19, 2016. This work was supported by the National Center for Airborne Laser Mapping under Project NSF EAR-1043051. N. Cao and H. Lee are with the Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4003 USA, and also with the National Center for Airborne Laser Mapping, University of Houston, Houston, TX 77204-5059 USA (e-mail: [email protected]). H. C. Jung is with the Office of Applied Sciences, NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA, and also with Science Systems and Applications, Inc., Lanham, MD 20706 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2473818

to temporal and spatial decorrelations [3]–[5]. The basic idea of PSInSAR is to find and analyze the pointwise time-coherent persistent scatterer (PS) with long-time-span differential interferograms. One major drawback of the PSInSAR technique is the low spatial density of PSs, particularly over nonurban areas without man-made structures. Hence, several PSInSAR-related methods have been proposed to improve the number of measurement point (MP) by extracting information from distributed scatterers (DSs) [6]–[13]. Unlike dominant PSs, the DS pixel normally contains a coherent sum of individual small scatterers. The interference of these small scatterers causes the variation in the returned signal of the pixel, which gives rise to temporal and geometrical decorrelations. The conventional DS usually covers several pixels with similar statistically homogeneous behaviors. It is possible to get enough coherence by processing these DSs statistically [11]. By analyzing the interferograms with short time intervals and small geometric baselines, small baseline (SBAS)-related methods have been developed to reduce the decorrelation phenomena and, thus, can be applied to DS [6]–[8]. Basically, SBAS generates a deformation time series by a least squares (or singular value decomposition) method after unwrapping multilooked interferograms. The Stanford Method for Persistent Scatterers (StaMPS) utilizes the spatial and temporal correlations of the phase, combining with proper filtering and unwrapping methods, to extract the deformation signal at more MPs [9]. The basic idea of StaMPS is to isolate the phase component due to deformation in the PS pixel by removing estimates of the various nuisance phase components, which are estimated by the spatial and temporal filtering process. One common merit of the SBAS and StaMPS methods is that the deformation time series can be obtained without prior knowledge of the deformation model. Many other efforts have been made to improve the spatial density of MPs by jointly processing PSs and DSs [10]–[13]. PSInSAR and SBAS are combined in [10] to obtain more MPs and higher signal-to-noise ratio (SNR). SqueeSAR [11] applies a phase triangulation algorithm to get the best estimates of N − 1 phases associated with the deformation of DS from N (N − 1)/2 off-diagonal interferometric phases of a coherence matrix. Then, this optimized phase is used to perform conventional PSInSAR processing. A quasiPS (QPS) [12] technique uses the spatial coherence as a weight in the estimation process to extract information from partially coherent targets. The target height and displacement information are estimated by maximizing the value of the weighted temporal coherence. The improved PSInSAR approach in [13] uses homogeneous patches to estimate the local gradient of deformation velocity and the local gradients of the residual digital elevation model (DEM). Then, the deformation velocity is obtained by performing an integration process. A major

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CAO et al.: PHASE-DECOMPOSITION-BASED PSINSAR PROCESSING METHOD

Fig. 1. Four classes of pixels. (a) Random DS, which is useless for PSInSAR processing. (b) One dominant scatterer (known as conventional PS). (c) Conventional DS with one scattering mechanism. (d) DS consists of many small scatterers with two distinct scattering mechanisms.

disadvantage of this method is that the deformation pattern must be known a priori. The aforementioned DS interferometry approaches can only address a case of DS with many independent small scatterers sharing the same scattering mechanism, as shown in Fig. 1(c). In reality, however, there may exist a case that the DS is dominated by two or more different scattering mechanisms, as shown in Fig. 1(d). Exploring such DS with multiple scattering mechanisms (DS-MSM) is important, if the DS-MSM contains small scatterers with different deformation patterns or different elevations. Another case is the partially coherent DS, when the scattering characteristics of some of the scatterers in the DS are changed while others remain the same. Like the conventional DS, the DS-MSM mainly exists in rural areas. However, the DS-MSM can also occur over urban areas, particularly in coarse spatial resolution pixels, since it is likely to have several targets in one pixel, such as the mixture of man-made structures and vegetation. DS-MSM over urban areas can also occur from phase changes, due to the baseline span and their height difference, and the layover phenomenon [16]. In addition, a highly bright PS in an urban area can dominate its adjacent pixels beyond the pixel corresponding to its real location. As a result, these PSs would cover several pixels and act like DSs. Such multiple pixels of PS can also be introduced by an oversampling process. In PSInSAR processing, the oversampling is sometimes necessary, in order to improve the MP density [14], [15]. The oversampling of the pixels can cause pixels adjacent to the PS pixel to share the similar scattering properties with the PS. Therefore, one single PS pixel can span several adjacent pixels after oversampling. Extracting information from DS-MSM is not straightforward because of the destructive interference between different scattering mechanisms. Particularly for images with low spatial resolution, it is likely that DS-MSM can exist in a single pixel. Usually, average filtering (multilook process) is implemented in a DInSAR process to improve SNR. This average filtering process would decrease the resolution and deteriorate the interference [6], [7]. Even for high-resolution images,

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the interference between different scattering mechanisms could be significant because of the layover effect in urban areas. Starting from the SAR tomography principle, the Component extrAction and sElection SAR (CAESAR) algorithm based on covariance matrix decomposition was proposed in [16], to deal with the DS-MSM caused by the layover in urban areas. By applying eigendecomposition on the sampled covariance matrix, CAESAR can extract different scattering mechanisms from DS-MSM. For rural area, CAESAR estimates the dominant scattering mechanism to counteract the temporal and spatial decorrelation effects. Instead of the covariance matrix, the coherence matrix is utilized in this paper to extract and select different scattering components from DS-MSM. The major benefit of using the coherence matrix rather than the covariance matrix is to compensate the possible unbalance of the backscattered power among all the images [11]. Since we are more interested in using the phase properties of the DS to extract ground deformation over Houston in this study, the coherence matrix is considered to be an alternative choice to perform PSInSAR processing. In this paper, we named our method as a phase-decomposition-based PSInSAR method (PD-PSInSAR), which estimates phases from different scattering mechanisms using the coherence matrix. We then implement these estimated phases in the conventional PSInSAR process using StaMPS. As it will be shown, the proposed PD-PSInSAR method can be a complement to existing PSInSAR methods, such as CAESAR, to process DS-MSM. II. A LGORITHM A. Coherence Matrix of DS The DSs usually span several statistically homogeneous pixels (SHPs) and can be characterized using a covariance or coherence matrix [11] and [17]. In order to avoid the disturbance of the amplitude, the coherence matrix is utilized in this paper. Assuming a homogeneous patch Ω containing NP adjacent pixels with similar scattering properties, the coherence matrix for this homogeneous patch can be obtained by 1 H T = E[yyH ] ≈ yy (1) NP y∈Ω

where y = [y1 y2 y3 , . . . , yN ]T is the normalized complex observations of N SAR images, such that E[|yi |2 ] = 1. H stands for the conjugate transpose. The absolute values and the phase values of the elements of T are the estimated coherence value γ˜ and interferometric phase φ, respectively [11]. The coherence matrix can be rewritten as follows: ⎤ ⎡ ··· γ˜1,N ejφ1,N 1 γ˜1,2 ejφ1,2 ⎢ γ˜2,1 ejφ2,1 1 ··· γ˜2,N ejφ2,N ⎥ ⎥ ⎢ T=⎢ ⎥ .. .. .. .. ⎦ ⎣ . . . . γÑ,1 ejφN,1 = |T| ◦ Φ

γÑ,2 ejφN,2

...

1 (2)

where ◦ is the Hadamard product, and Φ is an N × N matrix with element ejφm,n indicating the interferometric phases between the mth acquisition and the nth acquisition. The operator

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| ∗ | represents the elementwise absolute value of matrix ∗. In this case, |T| represents an N × N matrix with element γ˜m,n . It is easy to see that φm,n = −φn,m . All the required interferograms for the PSInSAR process are contained in the coherence matrix. It should be noted that the N (N − 1)/2 interferometric phase values are not redundant due to the spatial filtering process. Thus, N (N − 1)/2, instead of N , interferometric phases are used in the following process.

L=

Det(Ti )Ni Det(Tj )Nj Ni +Nj Ni Ti +Nj Tj Det Ni +Nj

In order to process the homogeneous patches without averaging the coherent PS with the low-coherent neighboring pixels, SHPs should be identified from the start. Several statistical techniques have been used to detect SHP, such as the Kolmogorov– Smirnov test, the Kullback–Leibler divergence test, and the Anderson–Darling test [18]–[21]. SqueeSAR used the twosample Kolmogorov–Smirnov test [11], and the Anderson– Darling test was implemented in [13]. All these methods are based on evaluating the similarity of the amplitude distribution of two image pixels, which is reliable for urban areas since the returned signals tend to be strong and stable. However, for nonurban areas without man-made structures, these methods could be problematic when the echo amplitude of the scatterers vary spatially or temporally. Since our purpose is to utilize the phase properties, we applied a likelihood ratio test for equality of coherence matrices to examine if two adjacent pixels are SHP. The two pixels will be considered as SHP, when their corresponding sample coherence matrices have identical distribution. This coherence likelihood ratio test has already been successfully used in polarimetric SAR (PolSAR) image processing to detect polarimetrically homogeneous pixels [22]–[24]. In this paper, we extended this method to the PSInSAR process. Based on the central limit theorem, we made an assumption that the probability density function (pdf) of the normalized SAR data vector y follows a complex multivariate normal distribution with zero mean and a dispersion matrix Σ [2], [25]. The pdf function is given by f (y) =

(3)

where Det(·) denotes the determinant. Therefore, the scattering properties of a target can be represented by dispersion matrix Σ. The N × N coherence matrix T obtained from (1) is the maximum-likelihood (ML) estimator for Σ. It has been proven in [26] that the coherence matrix follows a complex Wishart distribution with NP degrees of freedom T WC (N, NP , Σ) defined by [24], [26]–[28]

(NP )N NP Det(T)NP −N exp −trace[NP Σ−1 T] p(T|Σ) = N π N (N −1)/2 Det(Σ)NP Π (Γ(NP − j + 1)) j=1

(4) where Γ() is the gamma function. If two pixels i and j follow the same distribution, their corresponding coherence matrices would accept the hypothesis Σi = Σj . A likelihood ratio test

(5)

where Ni and Nj represent the number of data vectors used to compute the coherence matrices Σi and Σj , respectively. A simple threshold Lth can be set to verify the hypothesis as follows: if L > Lth , we accept Σi = Σj .

B. Identification of Homogeneous Patches

1 exp{−yH Σ−1 y} π N Det(Σ)

1 exp −trace[Σ−1 yyH ] = N π Det(Σ)

has been proposed in [22]–[24] to test this hypothesis, and the likelihood ratio test statistic is obtained as follows:

(6)

Therefore, if the likelihood ratio test statistic of two pixels is larger than the threshold Lth , these two pixels are regarded as SHP. The threshold Lth can be set empirically or according to the fixed false-alarm probability. The relationship between the chosen threshold and the false-alarm probability has been analyzed in [22]–[24], and the threshold can be selected based on the desired false-alarm probability. The coherence matrix T estimated from (1) requires the homogeneous pixels to be known first, while the homogeneous pixel identification in (5) and (6) needs the coherence matrix to be known first. In order to solve this dilemma, we first select two windows with different sizes. First of all, a small window, inside which all the pixels can be assumed to be homogeneous, is used to calculate the initial T. For nonurban areas, the size of the small window can be selected to be the size of the multilook window. For urban areas, the small window can be set according to the size of the multiple pixel PSs. After the initial matrices T for all the pixels are calculated, a large window centered in an arbitrary pixel P is then used to identify the modified homogeneous pixels. The size of the large window should be consistent with the size of the expected DSs. By performing the aforementioned hypothesis test, all pixels within this large window satisfying (6) will be considered to be statistically homogeneous with pixel P . The selected pixels that are not directly connected to P or through other SHPs should be discarded. Similar to [11] and [13], a minimum patch size of 20 pixels has been used in this study to preserve stable PS. Finally, the homogeneous pixels are obtained, with which the modified coherence matrices T can be calculated. In this paper, the sizes of the small window and the large window are chosen to be 3 pixels × 15 pixels (i.e., 60 m × 60 m) and 5 pixels × 25 pixels (i.e., 100 m × 100 m) in ground range and azimuth, respectively. Again, it is assumed all the pixels in the small window share similar statistical properties, which is generally acceptable for nonurban areas. However, it is necessary to pay extra attention to urban areas, in order to avoid averaging the coherent PS with the adjacent low-coherent pixels. Therefore, in practice, we could implement amplitude-distribution-based methods, such as the Anderson–Darling test, over urban areas and the proposed likelihood ratio test over nonurban areas. C. Phase Estimation From Eigendecomposition Eigendecomposition-based methods have been widely exploited in PolSAR and PolInSAR image processing for target classification or detection [29] and [30]. In this paper, we implemented the eigendecomposition process to estimate the phase vectors for the DS-MSM from the coherence matrix. It should


be noted that the scattering mechanism in this study is not the same as the discussed polarimetric scattering mechanism in the PolSAR community. The PolSAR measures the SAR backscattered signal from a target in different combinations of transmit and receive polarizations: HH, HV, VH, and VV. The polarimetric scattering mechanism, such as surface scattering and doublebounce scattering, presents the characteristics (e.g., geometrical structure and reflectivity properties) of the target in different polarization states of the transmitting and receiving antennas. The exploited eigendecomposition-based methods in PolSAR aim at extracting the polarimetric scattering information by decomposing the target into different polarimetric scattering mechanisms, each of which is represented by a different eigenvector [29]. However, in this study, we implemented a single PolSAR signal and constructed the eigendecomposition to incorporate different interferometric phase information. Each eigenvector represents distinct interferometric phase properties on displacement and height difference from our identified DS-MSM. The coherence matrix T obtained from (1) is a positive semidefinite Hermitian matrix and can be written in the form of T = UΛU−1 = UΛUT

(7)

where Λ = diag(λ1 , λ2 , . . . , λN ) is a N × N diagonal matrix with nonnegative real eigenvalues, and U = [u1 , u2 , . . . , uN ] correspond to orthogonal eigenvectors. The eigenvalues are arranged in descending order, such as λ1 ≥ λ2 ≥, . . . , ≥ λN . T can be expanded into the sum of N scattering matrices, each of which represents an independent target. That is N N T= λi Ti = λi ui uH (8) i . i=1

i=1

The eigenvalues represent statistical weights for the scattering mechanisms. The contribution percentage of different mechanisms (ci ) can be calculated as follows: λi ci = N × 100 (%). (9)

λi

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ues (λn+1 , λn+2 , . . . , λN ). The corresponding n eigenvectors u1 , u2 , . . . , un represent n different scattering mechanisms. Consequently, the phase vectors of the DS-MSM can be obtained from the phase of the corresponding eigenvectors. The residual N − n eigenvalues and eigenvectors represent the decorrelation noise. Therefore, the coherence matrix can be decomposed into n dominant scattering mechanisms and the residual decorrelation noise. That is T=

n i=1

λi Ti +

N

λi Ti = Tsignal + Tnoise .

In reality, the number of dominant scattering mechanisms n is unknown. However, it is possible to estimate n from the analysis of the eigenvalues distribution. In this paper, we focus on studying the primary (T1 ) and secondary (T2 ) dominant scattering mechanisms to analyze the characteristics of the DS-MSM in relation to different land cover types. However, it should be noted that the secondary dominant scattering mechanisms could be merely a noise for the traditional DS, particularly over nonurban areas where decorrelation tends to be severe. Therefore, only the primary dominant scattering mechanism (represented by u1 ) is used to estimate land deformation. The phase of u1 was substituted for the original SAR interferometric phase for the DS. Unlike the phase estimated by the triangulation algorithm in SqueeSAR, the estimated PS phase in this study is free from the interference from the other n − 1 dominant scattering mechanisms. One important note about the eigendecomposition is that the obtained eigenvector is ambiguous, which means infinite eigenvectors exist for the same eigenvalue. In other words, if u is the eigenvector of matrix A with eigenvalue λ, we have Au = λu.

For a pure target, (1) becomes 1 H T = E[yy H ] ≈ yy = yyH . NP

(11)

y∈Ω

Comparing (10) and (11), we can see that the phase of the obtained eigenvector u1 is identical to the phase of the original data vector y. Therefore, the phase estimated from the eigenvector is correct for the ideal conventional DS. Implementing the eigendecomposition process would not degrade the phase information of the ideal conventional DS with one scattering mechanism. On the other hand, if all the eigenvalues are equal, this means that N orthogonal scattering mechanisms with equal amplitudes exist and the target is considered to be “random.” Assuming that there exist n(0 < n < N ) different dominant scattering mechanisms, we would get n larger nonzero eigenvalues (λ1 , λ2 , . . . , λn ) and N − n smaller eigenval-

(13)

For a nonzero complex constant α, we have A(αu) = α(Au) = α(λu) = λ(αu).

i=1

If only one eigenvalue is nonzero, T represents to a pure target (an ideal conventional DS with one scattering mechanism) and can be related to a single scattering matrix. That is H T =λ1 u1 uH λ1 u1 λ1 u1 . (10) 1 =

(12)

i=n+1

(14)

Consequently, αu is also the eigenvector of matrix A with eigenvalue λ. Therefore, an eigenvector multiplied by a nonzero complex constant is also an eigenvector. This would cause a constant offset between the real phase and the estimated phase. In order to solve this problem, we set the original interferometric phase of the master image to be zero. Thus, the offset can be calculated and removed by setting the estimated phase for the master image to be zero again. After the phases for all of the conventional DS and DS-MSM are estimated, we can implement the conventional PSInSAR procedure for DS and PS jointly to estimate the land deformation rates. The StaMPS [9] method is used in this paper because it can be applied to terrains, with or without man-made structures, and can obtain the deformation time series without prior knowledge of the deformation model, which is useful over areas that experience nonlinear deformation. StaMPS has been successfully applied in nonurban areas such as Volcán Alcedo [9]. In addition, we will also compare the conventional PSInSAR results using the StaMPS method with the results of SqueeSAR, CAESAR, and our PD-PSInSAR methods.

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III. D ISCUSSIONS This section describes the differences and similarities among SqueeSAR, CAESAR, and the proposed PD-PSInSAR methods. The basic procedures for these three methods are similar. The first step is to identify the homogeneous pixels and calculate the coherence or covariance matrix. The next step is estimating the optimal phases associated with the deformation of DS. SqueeSAR utilizes the phase triangulation algorithm to estimate N − 1 phases from N (N − 1)/2 interferometric phase values of the coherence matrix [11]. The CAESAR and our proposed PD-PSInSAR estimate phases with different scattering mechanisms from the covariance matrix and the coherence matrix, respectively. Finally, the estimated phases associated with DS are processed jointly with PS using the traditional PSInSAR procedure. Here, we mainly focus on analyzing how these three methods estimate phases associated with the conventional DS or DS-MSM. In order to compare the SqueeSAR and the proposed PDPSInSAR method, we have first shown in Appendix A that the phase triangulation algorithm in SqueeSAR can also be obtained by analyzing the statistical properties of the coherence matrix as the proposed PD-PSInSAR method. The phase triangulation algorithm in SqueeSAR intends to obtain the best phase estimation from the coherence matrix T, assuming that only one scattering mechanism exists. However, the proposed PD-PSInSAR considers the case that coherence matrix T contains multiple scattering mechanisms (MSMs). Hence, SqueeSAR is favorable for the conventional DS with one scattering mechanism, while PD-PSInSAR is suitable for the DS-MSM. Instead of using the coherence matrix, CAESAR focuses on extracting different scattering mechanisms from the covariance matrix, which is defined as follows: 1 H xx (15) C = E[xxH ] ≈ NP x∈Ω

T

where x = [x1 x2 x3 , . . . , xN ] is the vector containing the nonnormalized complex observations of N SAR images. The connection between x and the y in (1) is yi = xi / E[|xi |2 ]. Therefore, the relation between T and C can be shown as [16] C = T ◦ (σσ H )

(16)

T

the vector containing the data where σ = [σ1 , σ2 , . . . , σN ] is standard deviations, with σi = E[|xi |2 ]. CAESAR implements the eigendecomposition on covariance matrix C to get the eigenvectors associated with different scattering mechanisms [16] as follows: C=

N

λi vi viH

(17)

i=1

where the eigenvalues λi are arranged in descending order as λ1 ≥ λ2 ≥, . . . , ≥ λN . vi are the orthogonal eigenvectors associated with the eigenvalues λi . The connection between SqueeSAR and CAESAR has been analyzed in [16] by rewriting the expression of eigenvector v1 in the following form:

subject to v1H v1 = 1. (18) v1 = arg max v1H Cv1 v1

Similar to CAESAR, the eigenvector u1 from PD-PSInSAR can also be written by maximizing the quadratic form uH 1 Tu1 , as shown in Appendix B. In order to perform the comparisons among SqueeSAR, CAESAR, and PD-PSInSAR, it is necessary to rewrite (18) in the form follows: of(31) as T θCAESAR = arg max sum C ◦ |v1 ||v1 |T ◦ (ΛΛH ) θ N N cm,n vm,n cos(φm,n − θm,n ) = arg max θ

m=1 n>m

(19) where cm,n and vm,n are the elements of matrix |C| and T |v1 v1 | at row m and column n, respectively. We can obtain from (16) that cm,n = σm σn γ˜m,n . Consequently, (19) can be rewritten as follows: N N σm σn γ˜m,n vm,n cos(φm,n −θm,n) . θCAESAR = arg max θ

m=1n>m

(20) Therefore, by comparing (20), (25), and (31), we can see that SqueeSAR, CAESAR, and PD-PSInSAR share very similar mathematic forms, but with different weight values. Comparing with (31), it is easy to see from (20) that CAESAR can be affected by the unbalance of the amplitudes among SAR images due to the term σm σn . If the unbalance of the amplitudes can be ignored, CAESAR and PD-PSInSAR should obtain similar results. For the multitemporal InSAR processing, we also need to examine its computational requirements. As mentioned earlier, the fundamental difference among SqueeSAR, CAESAR, and PD-PSInSAR is how to estimate the optimal phases for a DS. Therefore, in this study, we only analyze the computational requirements to extract the optimal phases for a DS using these three methods. Although SqueeSAR, CAESAR, and PD-PSInSAR share similar mathematic forms, as shown in (20), (25), and (31), the computational requirements for CAESAR and PD-PSInSAR are much less than that for SqueeSAR. SqueeSAR requires an iterative optimization process, while CAESAR and PD-PSInSAR can be implemented by the basic eigendecomposition procedure. In general, the computational complexity of eigendecomposition is O(N 3 ). Many sophisticated and efficient algorithms have been proposed to perform eigendecomposition on a Hermitian matrix, such as QR and divide-and-conquer eigenvalue algorithms [31]. Moreover, SqueeSAR usually needs to insert a damping factor to avoid small negative or null eigenvalues, which requires the estimate of the eigenvalues in the beginning [11]. Therefore, eigendecomposition is also performed in SqueeSAR. Using 25 Envisat SAR images over the Houston area with a 3.5-GHz processor, the CPU time for the computation of extracting the phases from 1000 DSs using SqueeSAR was about 10 s, while it took only 0.5 s using CAESAR or PD-PSInSAR. The comparison among these three methods is summarized in Table I. IV. R ESULTS To evaluate our PD-PSInSAR method, we first perform a simulation study in Section IV-A and a real data analysis to estimate land subsidence over Houston in Section IV-B.


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TABLE I C ONNECTION AND D IFFERENCE OF S QUEE SAR, CAESAR, AND PD-PSI N SAR

A. Simulated Results and Analysis In this experiment session, we use a DS with two different scattering mechanisms to demonstrate the properties of a DS-MSM. Considering 30 SAR images, Fig. 2(a) shows the simulated phases for the random DS, which is useless due to complete spatial and temporal decorrelations. Fig. 2(b) and (c) shows the simulated phases of two elementary scatterers (named DS1 and DS2, corresponding to primary and secondary dominant scattering mechanisms, respectively) with increasing phase trend and decreasing phase trend, respectively. It should be noted that it is more likely to have two scattering mechanisms sharing similar displacement since they are spatially close to each other in reality. The reason we used two scattering mechanisms, with both increasing and decreasing phase trends, here is merely to highlight the effect of the interference between different scattering mechanisms. The decorrelation of the scatterers are simulated by adding small random scatterers, which is the reason the simulated phases of these two scatterers fluctuate around the ideal phase trend. The small random scatterers are simulated by noise signals with uniformly distributed amplitude from 0 to 1 and uniformly distributed phase from −π to π. The amplitudes of DS1 and DS2 are around 120 and 65 times of the average amplitude of the noise, respectively. This is why the simulated phase of the DS1 is more close to the ideal phase trend than DS2. The phases of a DS with two scattering mechanisms are simulated by creating a pixel containing many DS1 and DS2 scatterers. In order to demonstrate the phase decomposition properties of PD-PSInSAR, our simulation experiments are conducted for two cases. For the first case, the simulated signal of a conventional DS with one primary scattering mechanism is obtained by creating a pixel containing 20 DS1 scatterers. For the second case, the simulated phase of a DS-MSM with two scattering mechanisms is obtained by creating a pixel formed by 20 DS1 scatterers and 20 DS2 scatterers. For the first case, the contributions of different scattering mechanisms are shown in Fig. 3(a). It is noted that one prominent value exists with a contribution percentage more than 25%. This indicates that only one scattering mechanism exists in

this pixel, which is consistent with the simulated signal. The original simulated phase from this DS, the coherently averaged phase, and the phases estimated by SqueeSAR, CAESAR, and PD-PSInSAR are shown in Fig. 3(b). The averaged phase reveals the general pattern of the increasing phase trend, which demonstrates that the multilook process of a homogeneous patch can reduce noise and improve the quality of the phase. We can see that the phases estimated by SqueeSAR, CAESAR, and PD-PSInSAR are similar to the averaged phase. The comparison among the phases estimated by SqueeSAR, CAESAR, and PD-PSInSAR with the ideal phase trend is shown in Fig. 3(c). It can be seen that all these three methods can obtain reasonable phase estimations of the conventional DS. However, it should be noted that the phase estimated by CAESAR tends to be noisier than SqueeSAR and PD-PSInSAR, which is caused by the amplitude unbalance. Since SqueeSAR and PD-PSInSAR are based on the coherence matrix, such amplitude unbalance is mitigated. On the other hand, consistent offsets between the estimated phase and the ideal phase trend are also observed. Since we are interested in the phase change, the consistent offsets will not affect the estimation of the deformation rates. In addition to scatterer DS1, scatterer DS2 is also added as the secondary scattering mechanism in the pixel to simulate the DS-MSM consists of two scattering mechanisms [corresponding to the DS-MSM in Fig. 1(d)]. The estimated phase of this DS-MSM is shown in Fig. 4. It is shown in Fig. 4(a) that two major scattering mechanisms exist in the pixel. The phases of DS1 and DS2 corresponding to the two scattering mechanisms are estimated, using PD-PSInSAR, by extracting the phases of u1 and u2 . Similarly, the phases corresponding to DS1 and DS2 are also estimated by CAESAR, by extracting the phases of v1 and v2 . The coherently averaged phase in Fig. 4(b) shows distortion due to the interference between the two scattering mechanisms. Fig. 4(c) shows that the estimated phase by SqueeSAR is degraded, while CAESAR and PDPSInSAR can robustly get proper estimation. The estimated phases for DS2 by CAESAR and PD-PSInSAR are shown in Fig. 4(d), which fits the ideal phase trend. Therefore, we can see that both CAESAR and PD-PSInSAR can effectively estimate the phases from the DS-MSM with two scattering mechanisms.

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Fig. 2. Simulated phases for different scatterers. (a) Random DS. (b) One elementary scatterer (DS1) with increasing phase trend. (c) One elementary scatterer (DS2) with decreasing phase trend.

For a DS-MSM with more than two scattering mechanisms, it is expected to obtain a similar conclusion. After all, the aforementioned simulations demonstrate that all SqueeSAR, CAESAR, and PD-PSInSAR can effectively process a conventional DS with one scattering mechanism. Both CAESAR and PD-PSInSAR perform better than SqueeSAR for DS-MSM. The performance of CAESAR could be affected by the amplitude unbalance of SAR images, while SqueeSAR and PD-PSInSAR are more resistant to the disturbance of the amplitudes. B. Real Data Analysis: Land Deformation Over Houston The Houston metropolitan area in Texas, USA, has been experiencing significant land subsidence due to ground water

Fig. 3. Simulation of a conventional DS with one scattering mechanism formed by DS1 scatterers. (a) Contribution percentage ci of different scattering mechanisms. We can see that only the primary scattering is prominent. (b) Original simulated phase of the DS and three estimated phases. Average indicates the coherently averaged phase. (c) Comparison of the phases obtained from SqueeSAR, CAESAR, and PD-PSInSAR with the ideal phase trend.

pumping [32] and [33]. The proposed PD-PSInSAR technique is implemented, to estimate the land deformation over Houston, using 25 Envisat ASAR data spanning from July 2004 to June 2010, as summarized in Table II. The SAR image acquired on 2008/04/01 is used as the master image, and the phase of which is set to be zero. The averaged SAR amplitude image and the National Land Cover Database 2006 (NLCD2006) of the study area are shown in Fig. 5. The amplitude image is compressed in the azimuth direction for better visualization. As shown in Fig. 5(b), the land cover of this area can be divided into 15 categories, according to NLCD2006 data. In this paper, we regard the four developed categories as the urban area and the other eleven


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Fig. 4. Simulation of DS-MSM with two scattering mechanisms formed by DS1 scatterers and DS2 scatterers. (a) Contribution percentage ci of different scattering mechanisms. We can see that two scattering mechanisms are prominent. (b) Original simulated phase of the DS-MSM and four estimated phases. (c) Comparison of the SqueeSAR and DS1 phases obtained from PD-PSInSAR and CAESAR with the ideal increasing phase trend. (d) Comparison of the DS2 phases obtained from PD-PSInSAR and CAESAR with the ideal decreasing phase trend. TABLE II S UMMARY OF E NVISAT ASAR S CENES U SED IN T HIS S TUDY

categories as the nonurban area. The intensity of the developed area represents the cover of the impervious surfaces, which can be regarded as man-made structures. Hence, the density of man-made structures increases from “Open Space” to “High

Intensity,” as shown in Fig. 5(b). In additional PD-PSInSAR, conventional PSInSAR (StaMPS), SqueeSAR, and CAESAR are also applied for comparison. The result of SqueeSAR is obtained by implementing SqueeSAR-estimated phase with the triangulation algorithm with the StaMPS method. CAESAR is implemented by applying the StaMPS method to the phase estimated from (17). All the parameters for PD-PSInSAR, CAESAR, SqueeSAR, and conventional StaMPS PSInSAR procedures are identical. The threshold Lth is set to be 0.001, to detect homogeneous pixels in this paper. After performing eigendecomposition, the contribution of different scattering mechanisms can be obtained from (9), as shown in Fig. 6. In addition to the primary and the secondary contribution c2 , the rest are contribution c1 summarized as N i=3 ci to represent higher order contributions. Comparing with the land cover map in Fig. 5(b), it is shown in Fig. 6 that the primary contribution value c1 is relatively higher in urban areas than in nonurban areas such as forests. This is because more stable and coherent man-made targets are likely to exist in urban areas. It is shown in Fig. 6(b) that c2 tends to be large in urban areas, particularly for the downtown area in the bottom right of the image, which is caused by the layover effect from high density of man-made structures. Without dominant scatterers, nonurban areas tend to be noisier and have higher order of scattering mechanisms, as shown in Fig. 6(c). It should be noted that large c2 in urban areas does not necessarily mean that the interference between the primary and secondary dominant scattering mechanisms is severe in

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Fig. 5. (a) SAR amplitude image of the study area. (b) NLCD 2006 land cover data. Four developed categories are regarded as urban area, while other eleven categories are jointly treated as nonurban area. Houston’s downtown area is in the bottom right of the image. Two large nonurban areas exist in the bottom-left and top-right part of the study area.

Fig. 6. Contributions of different scattering mechanisms. (a) Primary contri N bution value c1 . (b) Secondary contribution value c2 . (c) c . (d) c2 /c1 . i=3 i

urban areas since c1 can be much larger than c2 . In Fig. 6(d), c2 /c1 is used to indicate the interference between the primary and secondary dominant scattering mechanisms. Small c2 /c1 means that the effect of the secondary scattering mechanism on the primary scattering mechanism is small, while high c2 /c1 indicates that the interference between two scattering mechanisms is significant. Overall, c2 /c1 is higher in nonurban areas than urban areas; thus, it indicates stronger interference between DS1 and DS2 in nonurban areas. In order to study the effect of the density of man-made structures, 30 000 samples are randomly chosen from “High Intensity,” “Medium Intensity,” “Low Intensity,” “Open Space,” and

“Non-urban” areas, respectively. Histograms of c1 , c2 , N i=3 ci , and c2 /c1 for these five different land cover types are generated

using these samples and shown in Fig. 7. It is clearly shown in Fig. 7(a) that c1 increases as the density of man-made targets increases. The increase of c1 indicates that more dominant stable scatterers (similar to PSs) such as man-made targets exist. This is consistent with the well-known fact that PSs mainly exist in urban areas. Furthermore, it is shown in Fig. 7(b) that c2 increases as the man-made target density increases. It indicates that the secondary dominant scattering mechanism tends to be stronger, when more man-made targets can represent more PSs. Interestingly, the histograms of c2 are less distinguishable from five land cover types, as compared to those of c1 . In Fig. 7(c)

and (d), N i=3 ci and c2 /c1 increase as the density of manmade targets decreases. Because of less stable scatterers from lower density of man-made targets, the interference between the primary and secondary scattering mechanisms deteriorates. In this study area, the interference becomes weaker in urban areas because the primary dominant scattering mechanism tends to be much stronger than the secondary dominant scattering mechanism. The obtained line-of-sight (LOS) land deformation velocities from conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR are shown in Fig. 8. CAESAR and PD-PSInSAR use the phases estimated from u1 and v1 , respectively. In other words, the LOS velocities from PD-PSInSAR in Fig. 8 are estimated using the primary dominant scattering mechanism only. In this paper, we do not intend to utilize the secondary or higher dominant scattering mechanisms. It can be seen that the deformation patterns of conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR are generally similar. Only the top-left corner of the image shows slightly different deformation, which is caused by the unwrapping error. For MPs with coherence higher than 0.8, the velocity differences between these four methods are less than 1 mm/year. It indicates that PD-PSInSAR can perform as well as the other three methods do for highly coherent MPs. The northwestern part of Houston shows significant subsidence with LOS velocity close to 2 cm/year. The obtained deformation results are consistent with previous studies about the Houston subsidence [32]–[35]. PD-PSInSAR obtained 725 013 MPs, while conventional PSInSAR, SqueeSAR, and CAESAR detected 371 015,


Fig. 7. Histograms of c1 , c2 ,

N

c, i=3 i

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and c2 /c1 for five different land cover types.

Fig. 8. LOS velocity obtained using (a) conventional PSInSAR, (b) SqueeSAR, (c) CAESAR and (d) PD-PSInSAR. Background is SRTM DEM. The red x mark denotes the reference point, which has negligible deformation according to the continuous geographic positioning system observations at TMCC station (G. Wang, personal communication, 2014). Deformation time series over the two points denoted by A and B are selected for further analysis.

448 925, and 682 147 MPs, respectively. The average MP density obtained by conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR are 160, 253, 294, and 312 MP/km2 , respectively. The MP densities detected from different land cover types are summarized in Table III. Since SqueeSAR jointly processes PS and DS, it is expected that more MPs are detected by SqueeSAR than conventional PSInSAR in both urban and nonurban areas. In Table III, CAESAR and PD-PSInSAR show more significant increase in the MP density, as compared to SqueeSAR and conventional PSInSAR. This indicates that eigendecomposition-based methods perform well to mitigate the interference of the secondary and higher order scattering mechanisms in the primary scatterer. However, since PD-PSInSAR is robust to the disturbance of the amplitude, PD-PSInSAR detects slightly more MPs than CAESAR. In addition, we can see that the MP density decreases as the density of man-made structures decreases for all these four methods. It supports the fact that the decorrelation phenomenon increases when less man-made structures exist. Furthermore, it can be seen that the percentage increases from SqueeSAR to CAESAR or PD-PSInSAR become more distinct as the density of manmade structures decreases. This is because the interference between different dominant scattering mechanisms becomes more significant when less man-made structures exist. Therefore, the improved performance of PD-PSInSAR becomes more obvious in nonurban areas. Since the results of the CAESAR are quite

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TABLE III MP D ENSITY D ETECTED BY C ONVENTIONAL PSI N SAR, S QUEE SAR, CAESAR, AND PD-PSI N SAR IN D IFFERENT L AND A REAS

Fig. 9. Histograms of c2 /c1 . (a) MPs jointly detected by conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR in common. (b) MPs detected by SqueeSAR rather than conventional PSInSAR. (c) MPs only detected by CAESAR rather than conventional PSInSAR or SqueeSAR. (d) MPs only detected by PD-PSInSAR rather than conventional PSInSAR or SqueeSAR. Statistically, the extra MPs detected by CAESAR or PD-PSInSAR have much higher c2 /c1 , which indicates that they have MSMs. As compared with CAESAR, the histograms of PD-PSInSAR in (d) skew more significantly to the right.

close to PD-PSInSAR, such pattern becomes unclear on the MP increase from CAESAR to PD-PSInSAR. In order to further demonstrate that it is the interference between different scattering mechanisms that causes the increase of detected MPs in PD-PSInSAR, histograms of c2 /c1 are generated. As shown in Fig. 9(a) and (b), most of the MPs detected by conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR in common have c2 /c1 smaller than 0.6 (slightly right skewed), while most of the extra MPs detected by SqueeSAR show c2 /c1 around 0.7 (slightly left skewed). The reason that Fig. 9(b) shows larger c2 /c1 value is because

the conventional DS is processed in SqueeSAR. It is shown in Fig. 9(c) and (d) that most of the extra MPs detected by CAESAR and PD-PSInSAR have c2 /c1 that tend to be larger than 0.7 (significantly left skewed), which is caused by the existence of DS-MSM. As compared with CAESAR, the histogram of c2 /c1 from PD-PSInSAR skews more significantly to the right. These histograms demonstrate that the interference of different scattering mechanisms lead to less number of MPs identifiable from conventional PSInSAR and SqueeSAR. Fig. 10 shows the coherence of conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR. The coherence value


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Fig. 10. Coherence maps from (a) conventional PSInSAR, (b) SqueeSAR, (c) CAESAR, and (d) PD-PSInSAR.

is measured using γx parameter in StaMPS, which is a coherencelike parameter and measures the variation of the phase residuals [9]. That is N √ 1 u γx = exp −1 ϕx,i − ϕ˜x,i − Δϕˆθ,x,i (21) N i=1

where N is the number of interferograms. x and i denote the xth pixel and the ith interferogram, respectively. ϕx,i , ϕ˜x,i , and Δϕûθ,x,i are the initial measured phase, the spatially correlated phase, and the spatially uncorrelated phase due to look angle error, respectively. ϕ˜x,i , which is obtained by implementing the bandpass filter in the frequency domain, is mainly caused by the spatially correlated part of LOS deformation, look angle error, atmospheric delay between passes, and orbit inaccuracies [9]. After estimating Δϕûθ,x,i using the least squares estimation, the residual phases can be obtained by subtracting ϕ˜x,i and Δϕûθ,x,i from ϕx,i . Therefore, γx measures the variation of these residual phases. It is also known that γx is a measure of the phase noise level and performs similar to coherence [36]. Again, here, γx of PD-PSInSAR represents the coherence of the primary dominant scattering mechanism. The means and standard deviations of the coherences (γx ) obtained from conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR are 0.71 ± 0.14, 0.79 ± 0.12, 0.76 ± 0.14, and 0.84 ± 0.11, respectively. In Fig. 10(a) and (b), it can be observed that SqueeSAR can obtain slightly better coherence than conventional PSInSAR. This is because the phase triangulation algorithm in SqueeSAR processing can reduce the noise. It is also obviously shown in Fig. 10 that PD-PSInSAR can obtain the highest coherence for both urban areas and nonurban areas. This is because PD-PSInSAR mitigates the interference of the secondary (and other higher order) scattering mechanisms from the primary scattering mechanism. However, we also see that the coherence over nonurban areas is not as good as urban areas. Although CAESAR is also intended to mitigate the interference

Fig. 11. MP distribution in the 2-D c1 –coherence plane and c2 /c1 –coherence plane. (a) and (e) Conventional PSInSAR. (b) and (f) SqueeSAR. (c) and (g) CAESAR. (d) and (h) PD-PSInSAR.

between different scattering mechanisms, the coherence is not as good as PD-PSInSAR. It may be because of the amplitude fluctuations. Since SqueeSAR is also based on the coherence matrix rather than the covariance matrix, the coherence of SqueeSAR is slightly better than CAESAR in this study. As mentioned earlier, the value of the primary contribution c1 reflects how close a pixel is to a pure target. Thus, the larger the c1 , the better coherence is expected. Fig. 11(a)–(d) shows the relation between the coherence and the primary contribution value c1 . It can be seen that the coherence increases as c1 increases for conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR. However, PD-PSInSAR shows the highest coherence among these methods. As described earlier, the value of c2 /c1 indicates the interference between the primary and secondary dominant scattering mechanisms. Therefore, larger c2 /c1 means that stronger interference exists, and thus, worse coherence is expected. In Fig. 11(e)–(h), it can be observed that the coherence decreases as c2 /c1 increases. However, the coherence of PD-PSInSAR decreases much slower than conventional PSInSAR, SqueeSAR,

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Fig. 12. Time series of DS A with one scattering mechanism. (a) Contribution percentage ci of different scattering mechanisms. We can see that only the primary scattering is prominent. (b) Coherence matrix. (c) Original real data phase of DS A and three estimated phases. (d) Deformation time series obtained from conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR. The dashed lines are their fitted linear trends.

and CAESAR. This again demonstrates that the interference is mitigated by PD-PSInSAR. Finally, we repeat the assessment of improvements of the PD-PSInSAR method, using the two DSs at A and B labeled in Fig. 8 chosen to be examples of DSs with one scattering mechanism and two scattering mechanisms, respectively. As will be shown in Figs. 12 and 13, we obtained similar results to the previous simulation study in Section IV-A. First of all, it is shown in Fig. 12 that the coherence is very good for the conventional DS with one dominant scattering mechanism. In Fig. 12(b), the coherence matrix shows that the first three SAR images do not have good coherence as other images, which is caused by the temporal decorrelation since only three images exist for the time span from 2004 to 2006. In Fig. 12(c), we observe that the phases estimated by PD-PSInSAR are similar to the averaged phase and the phase estimated from SqueeSAR and CAESAR. The deformation time series in Fig. 12(d) confirms that conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR can obtain similar results for conventional DS with high coherence. It is noted that CAESAR obtains slightly different deformation

time series with the other three methods. Again, it may be caused by the unbalance of the amplitudes of the SAR images. On the other hand, pixel B located around the boundary of two areas with different deformation trends is selected as a DS-MSM. It is shown in Fig. 13(a) that DS B contains two major scattering mechanisms (i.e., DS B is a DS-MSM). In Fig. 13(b), we observe that the coherence for DS B is much worse than DS A. Fig. 13(c) and (e) shows the case of the interference between different scattering mechanisms with distinctly different phases for DS1 and DS2. The deformation time series obtained by conventional PSInSAR, SqueeSAR, CAESAR, and PD-PSInSAR are shown in Fig. 13(d) and (f). The estimated linear LOS velocities from conventional PSInSAR and SqueeSAR are −6.58 ±1.32 mm/year and −7.78 ±1.27 mm/year, whereas CAESAR estimated −6.36 ± 1.19 mm/year and −7.39 ± 1.71 mm/year on DS1 and DS2, respectively. The difference of the linear velocities between DS1 and DS2 from CAESAR is within the uncertainties of the estimates. It indicates that DS1 and DS2 cannot be distinguished. In Fig. 13(f), the time series of DS2 from PD-PSInSAR looks much noisier than DS1, suggesting that the secondary dominant scatterers have less


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Fig. 13. Time series of DS B with two scattering mechanisms. (a) Contribution percentage ci of different scattering mechanisms. We can see that two dominant scattering mechanisms exist. (b) Coherence matrix. (c) and (e) Original real data phase of DS B and four estimated phases. (d) Deformation time series obtained from conventional PSInSAR, SqueeSAR, and CAESAR. (f) Deformation time series obtained from conventional PSInSAR, SqueeSAR, and PD-PSInSAR. The dashed lines are their fitted linear trends.

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energy with stronger decorrelation. The estimated linear LOS velocities are −6.23 ± 1.05 mm/year on DS1 and −9.23 ± 1.45 mm/year on DS2 from PD-PSInSAR. For both CAESAR and PD-PSInSAR, we can see that DS2 shows higher subsidence rate than DS1. In Fig. 13(d), due to the interference between DS1 and DS2, the time series from conventional PSInSAR and SqueeSAR tend to be noisy for years 2004–2006. It can be seen that the time series of DS1 obtained by CAESAR is also slightly affected by DS2 for the same period. On the other hand, the time series of DS1 obtained by PD-PSInSAR tends to be smoother, which indicates that the effect of the DS2 is mitigated. Therefore, we can see that PD-PSInSAR better mitigates the interference between DS1 and DS2.

tra caution is needed with regard to the orthogonality from eigendecomposition-based methods. In Fig. 13, the DS-MSM example is carefully chosen in urban areas based on the similar analysis of the distribution of the eigenvalues on the simulated DS-MSM in Fig. 4. This DS-MSM is considered to contain two dominant scattering mechanisms, which can be induced by the elevation difference and deformation difference between the scatterers. The effect of the elevation difference on the deformation time series is eliminated by filtering out the DEM error phase components for DS1 and DS2, independently, in the standard PSInSAR procedure. The difference of deformation time series between DS1 and DS2 is most likely supported by the fact that the selected DS-MSM locates in the boundary of two areas with different deformation trends.

C. Caveats Here, we discuss three potential issues in our proposed PDPSInSAR method. The first issue is the temporal effect of soil moisture on our interferometric phases in rural areas. It has been known that soil moisture changes between two radar acquisitions can influence SAR interferometric phase and coherence [37]–[41]. The soil moisture change can be characterized by nonlinear, diurnal, and seasonal variations. In this paper, we calculated average linear velocities over six years; thus, the phase term from soil moisture is mostly regarded as residual coherence in the PSInSAR procedure. When the temporal effect of soil moisture is severe, the corresponding areas cannot be selected as MPs due to low coherence. The second issue is the number and source of MSMs. As introduced in Section I, the DS-MSM addresses a case of DS containing multiple independent scatterers with different scattering mechanisms. For instance, MSMs can be caused by phase changes due to the baseline span and their height difference. In addition, the DS-MSM can be induced by many other factors such as land deformations, different vegetation properties, and soil moisture variation. It should be noted that the deformation patterns are generally spatially correlated, although limited areas may show different deformation patterns such as uplift and subsidence. Theoretically speaking, the different physical, chemical, and geometrical properties of the small scatterers within few pixels can lead to the DS-MSM. Without in situ data, however, it is difficult to determine the source of MSMs, particularly with coarse SAR image resolution. Therefore, although it is possible to determine the number of MSMs from the eigenvalue distribution, as shown earlier, we do not intend to investigate the source of MSMs. The third issue is related to the orthogonality of our DS-MSM eigenvectors. Since both the coherence matrix and the covariance matrix discussed in this study are positive semidefinite Hermitian matrices, the eigenvectors estimated from either CAESAR or PD-PSInSAR are orthogonal. While the scattering mechanism in PolSAR relates to different PolSAR signals through the eigendecomposition, our scattering mechanism incorporates different interferometric phase information about displacement and elevation difference from the DS-MSM. When the two scattering mechanisms are located at elevation differences larger than the Rayleigh resolution, the scattering contributions can be regarded as quasi-orthogonal, as demonstrated in [16] and [42]. However, for multiple scatterers with elevation differences smaller than the Rayleigh resolution, ex-

V. C ONCLUSION The modified PSInSAR method has been developed to improve the number and coherence of MP, by removing the influence of secondary (or higher order) dominant scattering mechanisms on the primary one. Instead of the covariance matrix, we used the coherence matrix in order to remove the unbalance of the SAR amplitudes. Under the assumption that the SAR data vector follows a complex multivariate normal distribution with zero mean, the statistical properties of the scattering mechanism can be represented by the coherence matrix. Consequently, a likelihood ratio test of coherence matrix was used to detect SHPs. Then, the phases corresponding to different scattering mechanisms are obtained by implementing eigendecomposition on the coherence matrix. Both the simulated results and the real data analyses over the Houston area verified the performance of the proposed method. It has been demonstrated that our PD-PSInSAR method has the advantage of providing higher MP density with better coherence by mitigating the interference between different scattering mechanisms, particularly over nonurban areas. For future study, we will focus on determining the source of dominant scattering mechanisms and extracting more information from the higher order scattering components of the DS-MSM. A PPENDIX A This appendix intends to show how the phase triangulation algorithm in SqueeSAR can be derived by analyzing the statistical properties of the coherence matrix. Assuming that the SAR data vector y follows the zero-mean complex Gaussian distribution, the coherence matrix will follow a complex Wishart distribution, as shown in (4). The Σ of a generic pixel can be expressed using true” coherence values and true” phase values as follows: Σ = ΘΥΘH where ⎡ jθ e 1 0 ⎢ 0 ejθ2 ⎢ Θ =⎢ . .. ⎣ .. . 0 0

··· ··· .. .

0 0 .. .

. . . ejθN

(22)

⎤

⎡

⎥ ⎥ ⎥ ⎦

⎢ γ2,1 ⎢ Υ =⎢ . ⎣ .. γN,1

1

⎤ γ1,2 · · · γ1,N 1 · · · γ2,N ⎥ ⎥ .. . . . ⎥. . .. ⎦ . γN,2 . . . 1

θ = [θ1 , θ2 , . . . , θN ]T is the optimal phase of the DS that needs to be estimated from the filtered N (N − 1)/2 phases.


Without loss of generality, one phase value of θ can be set to zero. Therefore, only N − 1 phase values need to be estimated. Given T obtained from (1), the ML estimation of Σ can be obtained by maximizing the absolute value of the logarithm of the pdf in (4) as follows: ˆ = arg max {ln [p(T|Σ)]} Σ Σ

= arg max −trace NP Σ−1 T − NP ln (Det(Σ)) Σ

= arg max −trace(Σ−1 T) − ln (Det(Σ)) Σ

= arg max −trace(ΘΥ−1 ΘH T) − ln (Det(Υ)) . (23) Υ,Θ

To estimate Θ, the true coherence Υ is required. In this paper, similar to [11], we also use |T| to estimate Υ. Therefore, (23) becomes

θ PTA = arg max −trace Θ|T|−1 ΘH T − ln (Det (|T|)) θ

= arg max −trace Θ|T|−1 ΘH T θ

= arg max ΛH −|T|−1 ◦ T Λ θ

= arg min ΛH |T|−1 ◦ T Λ (24) θ

where Λ = [ejθ1 ejθ2 ··· ejθN ]T . It is easy to see that (24) is the ML estimation of θ, which is identical with the phase triangulation algorithm (PTA) in SqueeSAR [11]. For further analysis, (24) can be rewritten as

θPTA = arg max ΛH −|T|−1 ◦ T Λ θ T = arg max sum −|T|−1 ◦ |T| ◦ Φ ◦ (ΛΛH ) θ N N γm,n cos(φm,n − θm,n ) = arg max N + 2 θ

= arg max θ

m=1 n>m N N

γm,n cos(φm,n

− θm,n )

(25)

where sum(∗) is the sum of all the elements of the matrix ∗. is the element of matrix −|T|−1 ◦ |T| θm,n = θm − θn . γm,n at row m and column n.

The basic mathematic meaning of eigendecomposition of the coherence matrix is given in this appendix. In the principal component analysis sense, the eigenvector with the largest eigenvalue is the direction” along which the variance of T is maximized. Assume that the vector u1 represents the variance-maximum direction,” which will be shown to be a generalized eigenvector. The normalized variance of T along direction u1 is given by (26)

Equation (26) can be rewritten as H λuH 1 u1 = u1 Tu1 .

∂λ H H u u1 + λuH 1 = u1 T. ∂u1 1

(28)

Setting ∂λ/∂u1 = 0, (28) becomes Tu1 = λu1 .

(29)

Therefore, it can be noted that u1 is the corresponding eigenvector of λ. In PD-PSInSAR processing, the phase corresponding to the primary scattering mechanism is extracted from the phase of u1 . Therefore, u1 can be rewritten as u1 = |u1 | ◦ Λ. For simplicity, u1 is normalized as uH 1 u1 = 1. The PD-PSInSAR phase estimator can then be expressed as follows: H u1 Tu1 θ PD−PSInSAR = arg max Hu θ 1 1

u H (30) = arg max u1 Tu1 . θ

Equation (30) can also be written as θPD−PSInSAR

T = arg max sum T ◦ |u1 ||u1 |T ◦ (ΛΛH ) θ N N = arg max γ˜m,n um,n cos(φm,n − θm,n ) (31) θ

m=1 n>m

um,n

where is the element of matrix |u1 ||u1 |T at row m and column n. It can be seen that not only the phase but also the weight is variable to get the largest critical values. It should be noted that (30) and (31) are used to estimate the phase of the primary scattering mechanism in the DS-MSM. The phase associated with the secondary (or higher order) scattering mechanism can be estimated in a similar way.

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable suggestions and comments.

R EFERENCES

A PPENDIX B

uH 1 Tu1 . uH 1 u1

Maximizing λ requires ∂λ/∂u1 = 0. Taking the differential with respect to u1 on both sides of (27), we obtain

m=1 n>m

λ=

1089

(27)

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Ning Cao received the B.S. and M.S. degrees in electronic engineering from the Harbin Institute of Technology, Harbin, China, in 2011 and 2013, respectively. He is currently working toward the Ph.D. degree in geosensing systems engineering and sciences with the Department of Civil and Environmental Engineering, University of Houston, Houston, TX, USA. He is also with the National Center for Airborne Laser Mapping (NCALM), University of Houston. His main research interests focus on multitemporal interferometric SAR techniques and airborne SAR imaging.

Hyongki Lee received the B.S. and M.S. degrees in civil engineering from Yonsei University, Seoul, Korea, in 2000 and 2002, respectively, and the Ph.D. degree in geodetic science from The Ohio State University, Columbus, OH, USA, in 2008. He is currently an Assistant Professor with the Department of Civil and Environmental Engineering and the National Center for Airborne Laser Mapping, University of Houston, Houston, TX, USA. Dr. Lee is the recipient of the NASA New Investigator Award in Earth Science.

Hahn Chul Jung received the B.S. and M.S. degrees in geology and remote sensing from Yonsei University, Seoul, Korea, in 1998 and 2003, respectively, and the Ph.D. degree in earth sciences from The Ohio State University, Columbus, OH, USA, in 2011. He is currently a Lead Research Scientist with the NASA Goddard Space Flight Center, Greenbelt, MD, USA, and also with Science Systems and Applications, Inc., Lanham, MD, USA. His research interests include radar interferometry, hydrological and hydraulic modeling, wetlands, floods, and water balance.