InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRTM DEM

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1 Journal of Earth Science and Engineering (0) 47-5 D DAVID PUBLISHING InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM Huadong Hao, Guolin Liu, Xianlei Chen and Zhentan Cao. Metrology Verification and esting Institute, Zhoushan Institute of Calibration and esting for Qualitative and echnical Supervision, Zhoushan 360, China. Geomatics College, Shandong University of Science and echnology, Qingdao 6650, China Received: April 7, 0 / Accepted: April 5, 0 / Published: April 0, 0. Abstract: PU (phase unwrapping) is the ey step and important problem in DEM (digital elevation model) extraction and the measurement of surface deformation of InSAR (Interferometric synthetic aperture radar). he CKFPUA (conventional Kalman filter phase unwrapping algorithm) can obtain reliable results in the flat terrain areas, but it caused error transmission not maing the accurate inversion of surface deformation information in the steep terrain. Considering this situation, so it needs to introduce topographic information for guiding phase unwrapping. Here the 90 m resolution DEM data have been used and it is obtained by SRM (shuttle radar topography missio measured jointly by NASA (National Aeronautics and Space Administratio and NIMA (National Imaging Mapping Agency) of U.S. Department of Defense. his paper presents a SD-KFPUA (Kalman filter phase unwrapping algorithm) based on SRM DEM. With SRM DEM directing InSAR image to implement phase unwrapping, the speed and accuracy are improved. By analyzing with the conventional Kalman filter phase unwrapping algorithms, it is shown that the proposed method can achieve good results in particular to improve unwrapping accuracy in the low coherence region. Key words: InSAR, phase unwrapping, Kalman filter, topographic factors, SRM DEM.. Introduction Interferometric synthetic aperture radar (InSAR) is a great potential technology among the earth observation techniques. It has a wide application in terrain mapping, seismic deformation, volcanic activity, glacial drift, ground subsidence and landslides. PU (phase unwrapping) is the crucial procedure in InSAR data processing. In fact, SAR is placed on the surface by oblique irradiation, and there are inevitably the top and bottom of the displacement caused by undulating terrain, radar shadow and error generated in the original radar signal processing of spaceborne or airborne SAR. his can arouse that phase data is not continuous, giving rise to local phase error []. Corresponding author: Huadong Hao, master, engineer, main research fields: data processing of surveying, InSAR technology and its application, and large capacity measurement. gentlehhd@63.com. herefore, PU has been a difficult and hot issue for research scholars at home and abroad in InSAR area. Currently, the traditional PU algorithms are broadly divided into three categories. he first is path following method [-4], typically including Goldstein s branch cut method. he second is minimum norm algorithm [5-7], representatively comprising minimum Lp norm algorithm. he third is optimal estimation method, such as networ flow model [8], alman filter model [9-3]. Using alman filter model in Refs. [9-0], PU problem is transformed into state estimation issue, doing a direct effect on the real and imaginary part of complex interferogram, then to estimate true phase through the establishment of dynamic equation and observation equation of phase, achieving phase unwrapping and noise eliminating simultaneously. It is usually assumed that the state change is slow, the CKFPUA (conventional alman filter phase

2 48 InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM unwrapping algorithm) [] can get reliable and stable result in the condition that terrain is flat. However, the result is bad and duing the rapid state change, it caused error transmission in steep terrain. Ref. [] presents a CZ-KFPUA (Kalman filter phase unwrapping algorithm) based on topographic factors using CZ local frequency estimation, but accuracy is not high and computational efficiency is low. So this paper presents a SD-KFPUA (Kalman filter phase unwrapping algorithm) based on SRM DEM to improve the speed and precision of unwrapping. hen the feasibility and effectiveness of the proposed method are tested by using real InSAR data for experiment.. Kalman Filter Model of Phase Unwrapping Method. Observation Model he two noisy observations of the true interferometric phase consisted of the inphase and quadrature components of complex interferogram [9]. In order to write conveniently, it substitutes the n, m dependence by a -D dependence again, then it is written as Eq. (): z ( ) Re a ( ) cos( ( )) v y ( ) z ( ) sin( ( )) v Im a ( ) h ( x ( )) v ( ) ( ) ( ) () where, z() is complex interferogram, a() is observed interferogram amplitude, () is the true unambiguous phase, h() is nonlinearized mapping between y () and state vector (). he noise processes v ( ) and v( ) are assumed to be zero-mean white Gaussian noise with nown covariance depending on the coherence, i.e. Ev ( ) 0 () E v ( ) v( j) diag (, j) i R ( ) (, j) (, j) is Kronecer function, and (, j) 0, if j else. State-Space Model Based on opographic Information When interferometric phase is in the discrete-time case, terrain factors are considered on the impact to PU. he state-space model of existing KFPUA is modified as the improved model in follows: x ( ) Ax ( ) B w ( ) (3) Ew( ) 0 ; Ew( ) w( j) Q( ) (, j) (4) where, x() is the real phase at the point, A is system matrix of given state-space model, B is the input control matrix, is the input control variable related with topographic factors, w() is state noise, Q() is state noise covariance matrix. he improved state-space model is considered to introduce the input control variable related with topographic factors from SRM DEM of the measured area. Before the introduction, the blan area of SRM DEM is filled at first. After, it must mae registration with master SAR image, and simulate the phase, then use for the state-space model. 3. he Implementation Process of Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM he basic framewor of Kalman filter phase unwrapping algorithm based on SRM DEM is shown in Fig.. Firstly, it is started from the two SAR coregistrated images, and then calculated the interferogram, coherence map and measured interferometric phase simultaneously. he SRM DEM is coregistrated with SAR master image, then is used to simulate phase as the input control variable about topographic factors. Next, it needs to figure out the necessary parameters for KF. he observation noise variance used in observation model of KF can be calculated from interferometric intensity and coherence. he local phase frequency estimation and error variance (i.e., state noise variance) can be computed from the measured interferometric phase. Using these parameters, a very strong and effective state-space model has been established. -D Kalman

3 InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM 49 SAR Slave Image SAR Master Image SRM DEM wo SAR Coregistrated Images Coherence, Interferogram, Measured Phase Coregistrated DEM Simulated Phase by DEM Observation Noise Variance Error Variance Input Control Variable Related to opographic Factors Observation Model -D Kalman Filter State-Space Model Unwrapped Phase Fig. Kalman filter phase unwrapping based on SRM DEM. filter optimally combines the information from local frequency estimation and interferometric phase observations. he specific calculation process [3] is as follows: he first step according to Eqs. (5) and (6) calculates predictive value x ˆ, of state vector and the corresponding covariance matrix P, : x ˆ Axˆ,, B, (5) P, AP, A Q, (6) where, is the local frequency estimation,, Q, is the state noise covariance matrix. Here, the system matrix of state-space model A and input control matrix B are both taen as unit matrix. ˆx, and P are the initial, estimated phase and the corresponding covariance matrix respectively, which are selected based on experience value. he second step following the predictive value x ˆ, and covariance matrix P, from the first step, the state estimations xˆ and the corresponding covariance matrix P have been computed from Eqs. (7) and (8): ˆ ˆ x J r P, ( I J C ) P x (7) (8) where, J is the filter gain matrix, r is the residual, is linearized observation matrix. And C, J P C ( C P C R ) (9) r, y, C, xˆ, (0) d C, hx xˆ [ sin(ˆ x ),cos(ˆ x,, )] dx () Here, R is the observation noise covariance matrix, given by Eq. (). If is computed in Eq. (0), so r r r sign( r ) () hen, r in Eq. (7) is replaced by r in Eq. (), achieving effective control of residual, so error is limited in a very small range, preventing the spread of the error. In fact, the prediction estimate is calculated depending on two neighbours, only the following expected formula is to be corrected in SD-KFPUA. Using m as the azimuth index and n as the range

4 50 InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM index, then it gets: xˆ, ( mn, ) xˆ (, ) ˆ, m n x, ( m, n) m( m n ( m, n) P, ( mn, ) P, ( m P, ( m, n) 4 M wmq ( m MwmMwnQ( m, n) Mwn (3) where, M [0.5 0], M [0 0.5], ( m n wm wn and ( m, n ) are local frequency estimation in azimuth and range direction respectively. Q ( m, is the state noise covariance matrix, estimated from the normalized power spectral density of interferogram in a rectangular window centered around the pixel of interest [9]. m Based on the above two steps, the final unwrapping phase xˆ can be gained from the calculation of the second step by predictive value and its variance matrix P,. x ˆ, 4. Analysis of Experimental Results In order to test the effectiveness of proposed method, the real data has been used for validation. Adopting two ENVISA satellite SAR images acquired in Bam city of Iran in June 003 and January 004 to do interferometric processing, area with terrain fluctuating and detail richer is selected from flattened interferogram in Fig. a, and coherence map is in Fig. b (coherence coefficient is about 0. in blac part of the box area, (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. Real data phase images: (a) interferometric phase image; (b) coherence image; (c) phase simulated by SRM DEM; (d) the -D image of unwrapping result of CKFPUA; (e) the -D image of unwrapping result of CZ-KFPUA; (f) the -D image of unwrapping result of SD-KFPUA; (g) rewrapped phase of unwrapping result of CKFPUA; (h) rewrapped phase of unwrapping result of CZ-KFPUA; (i) rewrapped phase of unwrapping result of SD-KFPUA.

5 InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM 5 noise pollution is serious). he phase was simulated after registrated by using SRM DEM with the master image in Fig. c. Using CKFPUA in Ref. [], CZ-KFPUA in Ref. [] and the proposed method (SD-KFPUA) to unwrap respectively, the results are shown in Figs. d-f. he FE (frequency estimates) of three methods in the azimuth and range direction are shown in Figs. 3a-3f, respectively. 4. he Analysis Results of Frequency Estimation From FE result in Fig. 3, the result of CKFPUA is smooth and has error in the local area of serious noise pollution. here are many residuals in the results of azimuth direction (Fig. 3a) and range direction (Fig. 3d). For the result of CZ-KFPUA (Figs. 3b and 3e), there is a degree of improvement in smoothness, but still there are some residual. he proposed method has clear advantage in terms of smoothness and estimated accuracy (Figs. 3c and 3f). And serious noise pollution in the region (Fig. b in the box area) is relatively smooth. 4. he Analysis Results of Phase Unwrapping 4.. Qualitative Analysis Unwrapping result (Fig. d) of CKFPUA has clearly error transmission, losing part of the edge information. Although CZ-KFPUA (Fig. e) has valid estimate of the terrain, it still has error propagation. he proposed method is smooth and better reflected the original terrain in Fig. f. he serious noise area has been boxed in Fig. b, which has been successfully unwrapped. here is almost no error propagation, consistenting with phase simulated by SRM DEM in Fig. c. From rewrapped phase of unwrapping (Figs. g-i), filter effect of SD-KFPUA is better than CKFPUA and CZ-KFPUA, and the noise is almost completely removed. (a) (b) (c) (d) (e) (f) Fig. 3 FE (frequency estimatio of real data images: (a) FE in azimuth direction of CKFPUA; (b) FE in azimuth direction of CZ-KFPUA; (c) FE in azimuth direction of SD-KFPUA; (d) FE in range direction of CKFPUA; (e) FE in range direction of CZ-KFPUA; (f) FE in range direction of SD-KFPUA. able he computation time, number of discontinuities and value of three KF phase unwrapping methods. Phase unwrapping method Computation time (s) he number of discontinuities value (rad) CKFPUA CZ-KFPUA SD-KFPUA

6 5 InSAR Kalman Filter Phase Unwrapping Algorithm Based on SRM DEM 4.. Quantitative Analysis he proposed method is evaluated from three aspects of computation time, the number of discontinuities and value [4]. he time it cost is shorter, the efficiency it computed is higher; the discontinuities it have is fewer, the anti-phase distortion it performanced is better; the value is smaller, the quality of unwrapping is higher. As can be seen from able, it is clear that not only the computation time and value of SD-KFPUA are both less than OKFPUA and CZ-KFPUA, but also the number of discontinuities is relatively less. It is demonstrated that computation time is shorter, the anti-phase distortion performance is better and the unwrapping quality is relatively higher. 5. Conclusions Considering the problem that the conventional alman filter phase unwrapping algorithm has a greater error propagation in the steep terrain or larger slope, so the phase simulated by SRM DEM is introduced in state-space model of alman filter to estimate topographic factors on unwrapping. his paper presents a Kalman filter phase unwrapping algorithm based on SRM DEM that uses information of SRM DEM, guiding phase unwrapping to improve the computing speed and accuracy. he experimental results have shown that proposed algorithm (SD-KFPUA) has significantly improved the level in reliability, computational efficiency and the quality of unwrapping, particularly boosting the accuracy in the low coherence region. Acnowledgments he research is supported by the National Science Foundation of China ( ) and National 863 plans projects of China (009AAZ47). he authors would lie to express thans to ESA (European Space Agency) for providing ENVISA satellite data. References [] H. Zhang, C. Wang, Z. Liu, Spaceborne Synthetic Aperture Radar Interferometry, st ed., Science Press, Beijing, 00, pp (in Chinese) [] R.M. Goldstein, H.A. Zerber, C.L. Werner, Satellite radar interferometry: wo-dimensional phase unwrapping, Radio Science 3 (988) [3] W. Xu, I. Cumming, A region-growing algorithm for InSAR phase unwrapping, IEEE ransactions on Geoscience and Remote Sensing 37 (999) [4].J. Flynn, wo-dimensional phase unwrapping with minimum weighted discontinuity, Journal of the Optical Society of America A 4 (997) [5] D.C. Ghiglia. L.A. Romero, Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods, Journal of the Optical Society of America A (994) [6] M.D. Pritt, J.S. Shipman, Least-squares two-dimensional phase unwrapping using FFs, IEEE ransactions on Geoscience and Remote Sensing 3 (994) [7] M.D. Pritt, Phase unwrapping by means of multigrid techniques for interferometric SAR, IEEE ransactions on Geoscience and Remote Sensing 34 (996) [8] C.W Chen, Statistical-cost networ-flow approaches to two-dimensional phase unwrapping for radar interferometry, Ph.D. hesis, Stanford University, USA, 00. [9] R. Krämer, Auf Kalman-Filtern basierende Phasen- und Parameter estimation zur Lösung der Phasen vieldeutigeits problemati bei der Höhenmodellerstellung aus SAR-Interferogrammen, Ph.D. hesis, Universität-GH Siegen, Germany, 989. [0] O. Loffeld, H. Nies, S. Knedli, Y. Wang, Phase unwrapping for SAR interferometry A data fusion approach by Kalman filtering, IEEE ransactions on Geoscience and Remote Sensing 46 (008) [] G.L. Liu, H.D. Hao, Q.X. ao, Kalman filter phase unwrapping algorithm and comparison and analysis with other methods, Geomatics and Information Science of Wuhan University 35 (00) [] G.L. Liu, H.D. Hao, M. Yan, Phase unwrapping algorithm by using alman filter based on topographic factors, Acta Geodaetica et Cartographica Sinica 40 (0) [3] H.D. Hao, Study of Kalman filter applied in phase unwrapping of InSAR, Master hesis, Shandong University of Science and echnology, China, 00. (in Chinese) [4] D.C. Ghiglia, M.D. Pritt, wo-dimensional Phase Unwrapping: heory, Algorithms and Software, John Wiley & Sons Inc., New Yor, 998.