PHASE QUALITY OPTIMIZATION TECHNIQUES AND LIMITATIONS IN POLARIMETRIC DIFFERENTIAL SAR INTERFEROMETRY Rubén Iglesias, Dani Monells, Xavier Fabregas, Jordi J. Mallorqui, Albert Aguasca and Carlos Lopez-Martínez Universitat Politècnica de Catalunya, Departament de Teoria del Senyal i Comunicacions. D3 - Campus Nord, UPC, 08034, Barcelona, Spain. E-mail:
[email protected],
[email protected] ABSTRACT In this paper, the application of polarimetric optimization techniques for Differential SAR Interferometry (DInSAR) applications is studied. The objective of the proposed techniques is to increase the number of temporal coherent scatterers, improving thus the robustness of the DInSAR algorithms exploiting the polarimetric capabilities of data. The relationship between optimum coherences or amplitude dispersion maps, depending on the pixels selection method used, and the final DInSAR results is analyzed, using both orbital and Ground-Based SAR fully-polarimetric data. Moreover, the main advantages and drawbacks of each optimization method will be analyzed, especially when polarimetric stability does not apply. With the optimization techniques presented up to a twofold increase of the pixel candidates in the coherence case and up to a factor of seven in the amplitude dispersion case may be reached. 1.
INTRODUCTION
Differential SAR Interferometry (DInSAR) algorithms have been developed during the last decade, showing their feasibility and usefulness for the monitoring of deformation episodes in wide areas with millimeter precision. In this framework, any advanced Differential SAR Interferometry (DInSAR) technique is limited by the number and quality of reliable points within the area under study owning to decorrelation phenomena. In the literature, two main criteria for the pixels quality estimation are available: the coherence stability and the amplitude dispersion. In the first case, under the assumption of ergodicity and spatial homogeneity of the scattering process, the quality of the interferometric phase information is addressed through the coherence operator of each pair of data-sets [1][2]. In the second approach, the phase quality or phase dispersion along the whole stack of images is related with the amplitude dispersion index DA [3]. The higher the interferometric coherence, or accordingly the lower the amplitude dispersion index, the better the phase quality. Normally, the use of DInSAR techniques has been limited to the single-polarization case due to the lack of polarimetric SAR (PolSAR) data-sets. In this paper, the potential of polarimetric optimization methods is presented with the goal of enhancing the
density and quality of the deformation process retrieval. A comparison of the methods available in the literature, referred in this paper as Best [4][5], Equal Scattering Mechanism (ESM) [6] and Multi-Baseline Equal Scattering Mechanism (ESM-MB) [7], is presented for both, the coherence stability and the amplitude dispersion criteria accordingly to recent works that have already published [8][9][10]. Finally, in order to overcome some of the limitations of the previous methods, generally when polarimetric stationarity does not apply, the Sub-Optimum Scattering Mechanism (SOM) method [11] is proposed also for both pixel selection methods. 2.
TEST-SITE
Since the objective of the paper is evaluating the polarimetric optimization techniques from both, zerobaseline and orbital multi-baseline fully-polarimetric data-sets, two test-sited will be considered, one for each sensor. On the one hand, the Remote Sensing Laboratory (RSLab) of the Universitat Politècnica de Catalunya (UPC) carried out a one-year measuring campaign (October 2010 – November 2011) in the landslide of ‘El Forn de Canillo’, Andorra, using the UPC’s polarimetric GB-SAR (RISKSAR) [12],[4]. In this framework, a total of 10 daily data-sets have been collected with a temporal base line of approximately one month. The landslide of El Forn de Canillo constitutes one of the largest landslides of the Pyrenean area. It corresponds to a big landslide with a complex movement that took place in several instability events. Nowadays, the landslide is slowly moving, with some residual movement of the order of millimeters per year. In the North-West of the main landslide body there is a sector that is showing geological evidences of a present day activity in coincidence with heavy rainy periods [13]. On the other hand, 34 Fine Quad-Pol RADARSAT-2 acquisitions, from January 2010 to May 2012, over the metropolitan area of Barcelona, are used to evaluate the optimization methods when spatial baseline is present in data. Barcelona is affected by the construction of new underground infrastructures. A new underground line, that will connect the city with the airport, is generating subsidence in different urban areas of the city.
T
2. DINSAR OVERVIEW DInSAR is based on obtaining deformation and topographic error maps from a stack of differential interferograms. In this framework, a usual approach, such as the one applied in this paper, is the Coherent Pixels Technique (CPT) [1] [14] [15] which adjusts the data to a linear model. CPT has been widely used for monitoring subsidence episodes in different kind of scenarios and can use both, the coherence stability [1][2] and amplitude-based stability criteria [3] to perform the pixel selection step. Once the pixel selection is carried out, a triangulation of the pixel candidates is done; the objective is working with phase increments between pairs of pixels instead of working with absolute phases [16]. This step seeks to minimize the atmospheric artifacts taking advantage of its smooth behaviour in space. On the other hand, if a high density of pixel candidates is assumed, the phase increments are supposed to be lower than π radians. For this reason, no unwrapping process is needed. Going further, a linear model that accounts for the deformation and the residual topographic terms is defined. Data is then adjusted to the linear model though a minimization process of a test function addressed as Model Adjustment Function [14][15]. Finally, the absolute products of deformation rate and topographic error are calculated through an integration process, using one or multiple seeds with known behavior. 3.
POLARIMETRIC SAR INTERFEROMETRY
In this section the basic concepts of PolSAR are introduced in order to ease the understanding of the rest of the paper. The polarimetric information associated to each pixel of the scene could be represented by the scattering matrix Shv [17]
S hv
S hh S hv
S hv S vv
(1)
With the knowledge of the scattering matrix Shv in the linear polarization basis h , v , the scattering matrix in any elliptical orthogonal basis a , b defined by Sab could be derived by the following unitary transformation [17] [18]
refers to the vector transposition and the transformation matrix U2 can be expressed in function of the orientation and ellipticity angles , of the polarization ellipse by cos U2 sin
sin cos cos j sin
S ab
S ab U 2T S hv U 2 Sbb
0 e j0
(3)
where0 refers to the absolute phase. From an interferometric point of view 0 is irrelevant and for this reason it is normally set to zero. On the other hand, for i=(1,2) indicating two PolSAR acquisitions obtained at different times, the scattering vector ki, for each resolution element, is obtained as a vectorization of the scattering matrix Shv
ki
T 1 Shh,i Svv ,i , Shh,i Svv ,i , 2Shv ,i 2
(4)
From all possible basis, the Pauli’s basis is the most usual because it allows the direct interpretation of data in physical terms. The scattering vector ki could be projected onto an unitary vector obtaining a generic scattering coefficient Si w H k i for i=1,2. Hence, the i PolInSAR vector between two PolSAR acquisitions is defined by [19][20]
T
k k1T , k T2 .
(5)
Once the PolInSAR vector k is defined, under the assumption of spatial homogeneity and ergodicity, the 6×6 PolInSAR coherency complex matrix T6 could be defined to characterize the scatterering behavior from an interferometric point of view [19][20]
T T6 E kk H 11H Ω12
Ω12 T22
(6)
where H indicates the conjugate transpose operator, T11 and T22 refer to the coherency matrices of each PolSAR data-set and Ω12 is the polarimetric interferometric coherency matrix. At this stage the expression of the classical interferometric coherence can be generalized taking into account its polarimetric dependence
w1 , w2
S aa S ab
j sin e j0 cos 0
w1H Ω12 w2 w1H T11w1w2H T22 w2
(7)
(2) Since different projection vectors, w1 and w2, between
the acquisitions of the interferogram could lead to changes in the phase center of the scatterers, ensuring the same projection vector along the whole stack of interferograms is mandatory for PolDInSAR applications. Under this restriction, (7) can be rewritten as follows
w
w H Ω12 w
(8)
w H T11w w H T22 w
while the optimized interferometric phase is the one corresponding to the selected channel with the highest coherence value. For the multi-baseline case, this approach could be easily extended selecting the polarimetric channel that provides the highest value of temporally averaged mean coherence calculated along the whole set of interferograms. In this case, the method is referred as Best-MB. 4.2. Equal Scattering Mechanism. ESM
The objective of the polarimetric optimization processes presented in this paper is to find, for each pixel of the fully-polarimetric data-set of interferograms, either the transformation given by (2) or the projection vector w that provides a maximum coherence value from (8). Working with point-like scatterers, the amplitude dispersion index DA defined in [3] could be also generalized [8][9][10] DA w
A mA
1 H
w k
1 N
w N
H
i 1
ki wH k
2
(9)
This approach is based on finding the projection vector w that maximizes the generalized expression of the coherence seen in (8). Since the maximization problem cannot be solved analytically, the solution must be obtained using numerical methods. In this paper the solution presented in [6] is proposed. This approach makes use of an iterative solution in order to find the optimum projection vector w assuming that the two coherency matrices T11 and T22 are similar, or in other words, that polarimetric stability applies. Under this hypothesis, the estimated complex differential coherence is approximated by
where 1 wH k N
N
w i 1
H
ki
(10)
In this case the objective is, as stated in the coherence case, finding the polarimetric transformation or the projection vector that minimizes the generalized expression the DA defined in (9). 4.
ˆ w
COHERENCE STABILITY OPTIMIZATION
w H Ω12 w wH T w
(12)
where T
T11 T22 2
(13)
Notice how when polarimetric stability applies the condition ˆ is always accomplished. At the same
In this section the basis of the different polarimetric optimization methods are addressed in the frame of DInSAR applications to the coherence stability pixel selection approach. In order to ease the comprehension of the readers, the algorithms will be addressed first to the single-baseline case, which in fact applies to the zero-baseline GB-SAR data. Its extension to the multibaseline case, which applies to orbital sensors, will be briefly commented at the end of each subsection.
time, the phase is preserved if the polarimetric stability condition is fulfilled. For the multi-baseline configuration, the same projection w vector along the whole temporal stack of acquisitions have to be used, as it has been commented previously. The use of the ESM-MB method presented by Neumann et al. in [7], which in fact is an extension of the ESM method introduced by Colin et al. in [6], is proposed in this paper.
4.1. Best
4.3. Sub-Optimum Scattering Mechanism. SOM
This first approach is based on selecting the polarimetric channel that provides the highest coherence value for each pixel of the interferogram. This method is referred as Best In order to avoid changes in the phase centers, the polarization mechanism have to be the same for each pair of images that forms the interferogram. The High optimized coherence is given by
The optimization method showed in the previous subsection presents some restrictions since not all the points could be optimized for PolDInSAR applications. The algorithm only behaves correctly under the hypothesis of polarimetric stationary. When this hypothesis is not accomplished, the optimized differential phase may be affected by this difference and the optimization process may have no sense leading to erroneous optimized phase results. In this framework,
Best max hh , hv , vv
(11)
the symmetric revised Wishart dissimilarity measure can be used to evaluate the difference between the T matrices from both acquisitions [22]
d tr T11 T22 tr T22 T11 6 -1
-1
(14)
High Wishart distances will indicate large polarimetric changes between temporal acquisitions indicating thus when polarimetric stability does not applies. Fig.1 shows an example that illustrates these cases. The pixel analyzed corresponds to a stable area; hence, its differential phase is expected to be around zero after the optimization process. Since the matrices T11 and T22 are strongly different, this target is characterized by a high Wishart distance. Notice how the region of coherence, the area including all the possible complex coherence values given by the whole set of projection vectors, is spread in a large range of differential phases. For this reason, the optimization method leads to an erroneous phase moving away from the single polarimetric ones, which where around the zero value.
RC
HH HV VV Coh to each pair of w ESM optimal solution High phase variation
ESM solution
Fig.1. RC of ESM optimization in a pixel with a high Wishart distance.
In order to minimize this problem an alternative method, which is explained in [11], is proposed in this paper. As the ESM approach, it is based on preserving the same projection vector between acquisitions but now solving the coherence optimization problem in a closer way to a physical interpretation. The algorithm is based on sweeping all the possible combinations of ellipticity and orientation angles (ψ,χ) at the level of the scattering matrix Shv thorugh (2) defining thus all the polarization states of the propagating wave. Finally, the technique looks for the polarization basis transform that provides the highest value among all the co-polar and cross-polar coherence values:
SOM max aa , , ab , ,
(7)
where aa and ab reefers to the co-polar and cross-polar channels in the new (ψ,χ) polarization basis,
respectively. Since no statistical hypothesis is assumed about the scattering matrices, the solution corresponds to the highest coherence achievable with the constraint of maintaining the same projection vector for the entire possible basis. In fact, this method is a subspace of the ESM commented previously and, for this reason, it is referred as Suboptimum Scattering Mechanism (SOM). The main drawback of this technique is its computational load. Contrarily, this method leads to a significant reduction of the outliers produced when polarimetric stationary does not apply. Regarding the multi-baseline case (SOM-MB), the method can be again easily extended. All the possible combinations of orientation and ellipticity angles (ψ,χ) are swept but now maintaining the same transformation along the whole set of interferograms. The polarization basis transformation providing the highest temporally averaged mean coherence will be selected as the optimal one. 5.
AMPLITUDE STABILITY OPTIMIZATION
In this section the basis for the adaptation of the different polarimetric optimization methods, seen previously for the coherence stability case, are addressed to the amplitude-based pixel selection approach. Hence, the adaptation and the use of the three optimizations methods, Best, ESM and SOM, will be explained in the following. Recently, several works have been presented in this framework using the ESM method [8][9][10]. Notice that there are no distinctions in the mathematical formulation working either with zero-baseline or multi-baseline data since for this approach the spatial baseline plays no role in the estimation. 5.1. Best
The simplest way to reach an improvement in the amplitude dispersion coefficient is based on selecting the interferometric phases of the polarimetric channel that provides the minimum DA. As in the coherence case this method is referred to as Best. In this case, the optimized DA is given for each pixel by the following expression DA, Best min DA, hh , DA,hv , DA,vv
(15)
Consequently, the interferometric phases will be derived from the channel providing a less DA. 5.2. ESM
As in the coherence case, the Best approach does not completely exploit the polarimetric capabilities of data. The approach carries out a search over the whole polarimetric space with the objective of obtaining the
projection vector w that optimizes the generalized DA expression seen in (9). In order to solve this problem no analytical solution is yet found, hence, the optimization problem may be solved by brute force parametrizing the projection vector as follows cos w sin cos e j sin sin e j
0 2 0 0
(16)
5.3. SOM
The main drawback of the previous method is the computational cost since there is no analytical expression to obtain a mathematical relationship between the projection vector and the generalized expression of the DA. For this reason, the adaptation of the SOM approach is proposed. The basis are the same, it is based on sweeping all the possible orientation and ellipticity angles (ψ,χ) to achieve a scattering matrix in a new polarization basis providing a minimum DA value, among all the co-polar DA,aa and cross-polar DA,ab values ,
NUMBER OF PIXELS
hh 17553 (2.4%) hv 11638 (1.6%) vv 18280 (2.5%) Best 30345 (4.2%) SOM 45748 (6.4%) ESM 53095 (7.4%) Coherence Stability Pixel selection statistics for each method. Zero-Baseline GB-SAR PolSAR data. (%) is referred to the total number of pixels.
Fig.2 shows the retrieved deformation map from the daily-averaged acquisitions from October 2010 to November 2011 GB-SAR campaigns. Notice how while the global behaviour is identical for all the methods the density is increased in a spectacular way.
(17)
With this approach the computational load is drastically reduced since the solution now consists on exploring a 2D space corresponding to all possible orientation and ellipticity angles (ψ,χ). 6.
TABLE 1 NUMBER OF RELIABLE PIXELS SELECTED COHERENCE STABILITY PIXEL SELECTION METHOD
The solution consists in looking for the angles α, β, δ and γ that minimize the generalized DA over the whole polarimetric space. The optimized DA is directly calculated by (9) once the optimum projection vector wopt,ESM is found.
DA, SOM min DA, aa , , DA, ab ,
improvement in density as well in the quality of the retrieved deformation maps. This improvement in the number of pixel candidates and in their phase quality is a key issue in order to give a major robustness in the whole DInSAR processing, achieving up to a factor of three in the ESM case.
(a)
(b)
(c)
(d)
POLDInSAR results
6.1. Coherence Stability Optimization Results
Since the number of images is short for the GB-SAR case, and it is a natural environment in which is expected to find point-like scatters but also distributed targets, the coherence stability method has been considered more suited to perform a reliable the pixel selection. Hence, pixels presenting a coherence value above a threshold of 0.7 have been selected. A 9x9 multi-look window has been used. This coherence threshold selected corresponds to a phase standard deviation of 5º. Table 1 illustrates the advantages of using polarimetric optimization algorithms for DInSAR purposes. The usage of polarimetric capabilities lead to a large
0 cm/year
3 cm/year
Fig.2. Lineal velocity retrieved from the daily-averaged acquisitions from October 2010 to November 2011 GBSAR campaigns. Using the hh polarimetric channel (a), or Best (b), SOM (c) and ESM (d) approaches.
As it was expected results show that nowadays the main lobe of the landslide presents some residual movement (on the order of 1-1.5 cm per year). In the top-left side of the landslide exists a sector with higher activity that is reactivated coinciding with strong rainfall episodes with a motion rate of 2-2.5 cm per year. These movements perfectly matches with previous field measurements using geotechnical techniques [13], 6.2. Amplitude Stability Optimization Results
In this case the amplitude-based optimization methods have been applied to the RADARSAT-2 fullypolarimetric data-set that corresponds to the urban area of Barcelona. Moreover, the amplitude-based method is more suited in urban scenarios strengthening thus this choice. In this framework, a DA threshold of 0.25, which corresponds to a phase standard deviation of 15º, has been used.
(a)
TABLE 2 NUMBER OF RELIABLE PIXELS SELECTED AMPLITUDE DISPERSION PIXEL SELECTION METHOD
NUMBER OF PIXELS
hh 9398 (1.9%) hv 8522 (1.7%) vv 9927 (2.0%) Best 21721 (4.4%) SOM 40032 (8.1%) ESM 71702 (14.6%) DA Pixel selection statistics for each method. Multi-Baseline RADARSAT-2 PolSAR data. (%) is referred to the total number of pixels.
(b)
Fig.2. Lineal velocity retrieved from the RADARSAT-2 PolSAR data-set. Using the the hh (b) and ESM (b) DAbased optimization methods for the DInSAR processing.
7. The differences among the different methods are more noticeable for the amplitude-based pixel selection approach (see Table 2). Notice how ESM approach increases in a factor of seven the number of pixels with respect to the single-polarimetric case. Again, this increase in pixels’ density justifies the usage of PolSAR data in the DInSAR framework. Finally, Fig.2 shows the linear deformation map using ESM approach. The objective is demonstrating that the optimized phase information is reliable since the same deformation pattern is obtained compared with the single-polarimetric case. Despite the deformation maps achieved are almost identical in both cases an increase in the pixels’ density is evidenced, helping thus to the interpretation and characterization of results. The different deformation bowls observed in the figure match the path followed by a new underground tunnel construction in the city.
CONCLUSIONS
The different optimization algorithms have been tested with real fully-polarimetric data-sets for both, the coherence stability and amplitude-based pixel selection criteria. Results show a significant improvement in the number of pixel candidates selected with respect to the single-polarimetric case. Thanks to the polarimetric data it is possible the exploitation of a larger number of pixels compared with the single polarization case. This fact provides a major robustness in DInSAR algorithms. 8.
ACKNOWLEDGEMENTS
This research work received partial support from the Safeland project funded by the Commission of the European Communities (grant agreement 226479), from the Big Risk project (contract number BIA2008-06614), the project TEC2011-28201-C02-01 and the grant BES2009-015990 associated to the project TEC2008-06764C02-01, both funded by the Spanish MICINN and FEDER funds. The Radarsat-2 images were provided by MDA in the framework of the scientific project SOAREU 6779.
9.
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