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A Refined Strategy for Removing Composite Errors of SAR Interferogram Bing Xu, Zhi-wei Li, Qi-jie Wang, Mi Jiang, Jian-jun Zhu, and Xiao-li Ding
Abstract—In standard differential synthetic aperture radar interferometry, there could still be a residual tilt (orbital error) in the interferometric phase due to inaccurate baseline estimation. We demonstrated theoretically that the orbital errors were partially elevation dependent. On the basis of this, we introduced an elevation-dependent item to the conventional polynomial model to simulate, and therefore, compensate the orbital errors, as well as the small scale topographic and/or topography-related phase errors. Robust regression approach was suggested to determine the parameters of the proposed model. The model was validated with both synthetic and real ALOS PALSAR data of the Zhouqu, China mudslide. The synthetic test indicated that upon applying the refined model, the accuracies of phase measurements were improved by nearly two times, compared to those using conventional linear and quadratic models. The real data experiment indicated that after utilizing the refined model, the correlation between the interferogram and the digital elevation model of Zhouqu reduced to about 1/5 of those using linear and quadratic models. This demonstrates that the elevation-dependent phase components have been largely removed by the new model. More importantly, the interferogram corrected by the new model visibly disclosed the deformation area affected by the Zhouqu mudslide. Index Terms—Composite errors, elevation-dependent phase error, linear model, synthetic aperture radar interferometry (InSAR), quadratic model.
I. Introduction YNTHETIC aperture radar interferometry (InSAR) has been proven in the past few decades to be a very effective technology for mapping digital elevation model (DEM) and monitoring ground deformation caused by earthquakes, volcanoes, landsides, and others [1]. The accuracy of the InSAR measurement is, however, highly dependent on the quality of the interferometric phase. The dominant components in the interferometric phases are generally the variants due to the ellipsoid or reference surface (called flat-earth phase). Prior to further processing, these phases need to be removed using accurately estimated interferometric baseline [2]. However,
S
Manuscript received June 13, 2012; revised November 3, 2012; accepted January 18, 2013. Date of publication April 29, 2013; date of current version November 8, 2013. This work was supported in part by the National Natural Science Foundation of China under Grants 41222027 and 40974006, the National High-Tech. Research and Development Program of China under Grant 2012AA121301, the National Key Basic Research and Development Program of China under Grant 2012CB719903, and the Research Grants Council of the Hong Kong Special Administrative Region under Project PolyU 5154/10E. B. Xu, Z.-W. Li, Q.-J. Wang, and J.-J. Zhu are with the School of Geosciences and Info-Physics, Central South University, Changsha 410083, China (e-mail:
[email protected]). M. Jiang and X.-L. Ding are with the Department of Land Surveying and Geo-Informatics, Hong Kong Polytechnic University, Hung Hom, Hong Kong. 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/LGRS.2013.2250903
the trajectory of SAR satellite is difficult to be precisely modeled, as is the interferometric baseline. This makes the flat-earth phase removing incomplete, and there is still a residual ramp left in the interferometric phase. In addition, the inaccurate baseline would lead to incomplete topographic contribution removal for differential SAR interferometry (D-InSAR). These errors must be removed empirically for high precision deformation measurements. In order to eliminate these errors, different strategies in frequency and/or space domain have been proposed [2]–[5]. The frequency domain’s approaches are based on the assumption that the orbital error in the whole interferogram is linear. Under this assumption, the fast Fourier transform (FFT) [5]–[7] or wavelet multiresolution analysis [8] is used to construct and, subsequently, to remove the long-wavelength phase fringes. The space domain’s methods however can be divided into three categories. The first category uses external data, such as GPS measurement [3], to aid the orbital error removal. For example, Feng et al. [3] utilized GPS data to correct the PALSAR and ASAR coseismic deformation measurements of the 2011 Mw-9.0 Tohoku-Oki Earthquake. However, the applicability of this method is rather limited as synchronized GPS observations are not always conveniently available. The second category corrects the orbital error by tuning the trajectories of the satellite orbit [4]. This approach highly depends on the previous estimates of the orbital trajectories and is strongly dependent on the quality of the interferogram. Only highly coherent interferogram can achieve a good correction. The third category fits the orbital error with linear, quadratic, or higher order polynomials and makes the correction [5], [7], [8]. However, this method can only remove the long-wavelength orbital error, yet they are helpless for the elevation-dependent phase errors. In addition to the orbital errors, small-scale topographic errors and topographyrelated phase errors (e.g., vertical stratification atmospheric errors) generally exist in the interferometric phases and need to be corrected. We refer these errors collectively as composite errors in the following context. In this letter, a new strategy to remove the composite errors in interferometric phase is proposed based on the analysis of the dependence of orbital phase errors to elevation. Synthetic and real datasets are then used to validate the effectiveness of the proposed method. II. Review of Existing Models A. Linear Model The orbital phase error is mainly caused by the cross track and radial errors of satellite state vector [1]. For relatively
c 2013 IEEE 1545-598X
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 11, NO. 1, JANUARY 2014
small areas (e.g., < 30 × 30 km2 ), this error is generally modeled as a planar-phase ramp [1], [2] φorbit (x, y) = a0 + a1 x + a2 y
(1)
where x and y are the range and azimuth coordinates in the radar system, respectively. a0 , a1 , and a2 are the coefficients to be determined. In order to accurately estimate these coefficients, Shirzaei et al. [8] proposed an approach based on multiresolution wavelet analysis and robust regression (we call it wavelet-based approach hereinafter). However, the approach is easily affected by the errors of phase unwrapping and data gap interpolation, and suffers from the heavy computational burden of the multiresolution wavelet decomposition. Yet, according to the properties of linear models, a1 and a2 are the gradients of the planar-phase ramp along range and azimuth directions, respectively. They can be estimated by exploiting the relationship between phase gradient and instantaneous fringe frequency [9] fx = a1 /2π (2) fy = a2 /2π where fx and fy represent the fringe rates (or fringe frequency) of the orbital error in range and azimuth directions, respectively. We can, therefore, obtain a1 and a2 by performing FFT on the whole complex interferogram and retrieving the frequency of the fringes that correspond to orbital error (we call it the FFT-based approach hereinafter). The constant a0 is the initial phase and can be calculated by inverse FFT [6]. The FFT-based approach outperforms that of the wavelet-based one by less computational complexity and does not need phase unwrapping. However, we must note that when we need to mask out the area of poor coherence and consequently a sparse interferogram (i.e., interferogram with data gap) should be investigated, the Fourier transform can still be applied, but in a more sophisticated way as, for instance, suggested by [13]. B. Nonlinear Model In the previous section, we introduce the linear model for orbital phase error removal. However, in actuality, the orbital errors are seldom completely linear in the entire interferogram; a nonlinear model is, therefore, needed. Hanssen [2] and Feng et al. [3], therefore, suggested the quadratic polynomial model φorbit (x, y) = a0 + a1 x + a2 y + a3 xy + a4 x2 + a5 y2
(3)
where ai (i = 0, 1, . . . , 5) are the coefficients. In addition, Tobita et al. [10] modeled the orbital residual phases by a biquadratic surface, and Jiang et al. [7] discussed high-order polynomials for orbital error removal. The quadratic polynomial can fit a regular surface nicely, but it cannot catch the phase errors related to, e.g., elevation. The high-order polynomials can better represent the small scale errors than the quadratic model. However, with the rising of the order, serious oscillation [11] (known as Runge’s phenomenon) will occur, which would degrade the stabilities of the modeling and cause computational error. In addition, the high-order polynomials are still challenging to simulate and consequently to remove the elevation-dependent phase errors.
III. Model Development A. Elevation Dependence of Orbital Phase Error Generally, the interferometric phase can be divided into five parts [1], [2] ϕ = ϕflat + ϕtop + ϕdef + ϕatm + ϕnoi
(4)
namely the flat-earth, topographic, deformation, atmospheric, and noise phases. The flat-earth and the topographic term can respectively be denoted as [1] 4π Bn s φflat = − (5) λ R tan θ 4π Bn h φtop = − (6) λ R sin θ where λ represents the wavelength of the SAR system, R is the distance from the SAR sensor to the target, Bn is the perpendicular baseline, h and s are the ellipsoid height and slant range displacement of point target, and θ is the incidence angle. Neglecting the terms of deformation, atmospheric phases, and noise in (4), and taking the derivative with respect to baseline Bn , we get 4π s h dφ = − (7) + dBn . λ R tan θ R sin θ Equation (7) shows how baseline (orbital) errors will affect the flat-earth and topographic phases estimation and removal. The first and second terms in the square bracket denote the flatearth and topographic phase errors, respectively. It is clear that the effect is nonlinear, and most importantly, partially elevation dependent. If we wish to completely remove the phase errors due to the inaccurate baseline, the elevation-dependent nature must be considered. B. Model Development From the above analysis, we realized that the orbital errors are nonlinear and partially elevation dependent. As noted earlier, in addition to orbital errors, small-scale topographic (e.g., external DEM errors) and topography-related phase errors (e.g., atmospheric errors) also commonly exist in the interferometric phases. These composite errors cannot be modeled adequately by the linear model in (1). The quadratic or higher order polynomials can fit these errors better, but are still unable to account for the phase components that are almost linearly dependent on terrain elevation according to the second term in (7). With the above hypothesis and considering that high order polynomials may cause serious oscillation [11], we present a refined quadratic model by adding a term to account for the elevation φcom (x, y) = a0 + a1 x + a2 y + a3 xy + a4 x2 +a5 y2 + a6 h(x, y) + ε(x, y) (8) where φcom denotes the composite errors, h(x, y) is the elevation of point (x, y), ai (i = 0, 1, . . . , 6) are the coefficients to be determined, and ε is random phase error. Assuming that there are n sample points (outside of the deformation area) selected to estimate the coefficients of the model, we formulate the system into matrix form L = BX + ε
(9)
XU et al.: REFINED STRATEGY FOR REMOVING COMPOSITE ERRORS OF SAR INTERFEROGRAM
where L= X=
φ1com
φ2com
...
a0
a1
a2
a3
1 1 .. .
x1 x2 .. .
y1 y2 .. .
1 xn
yn
⎡ ⎢ ⎢ B=⎢ ⎣
φncom a4
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T T
a5
a6
x1 · y 1 x2 · y 2 .. .
x12 x22 .. .
y12 y22 .. .
h1 h2 .. .
xn · y n
xn2
yn2
hn
⎤ ⎥ ⎥ ⎥. ⎦
The observation equations are then ˆ − L. V = BX
(10)
A weighted least-squares (WLS) adjustment can be carried out to determine the coefficients
ˆ = BT PB −1 BT PL X (11)
Fig. 1. errors.
Synthetic dataset. (a) Simulated DEM. (b) Synthetic composite phase
where P is an n × n diagonal weight matrix, with Pii (i = 1, 2, . . . , n) defined as the interferometric coherences. C. Parameter Determination With Robust Regression Due to phase noise and/or the problem of under sampling, the observables L in (10) are sometimes unreliable, which can potentially degrade the quality of the solution. To decrease the effects of the unreliable observables, a robust regression approach [12] is suggested to adjust the weight P iteratively. The adjusting factor is defined as j
ωi =
1 j |vi |
+k
i = 1, 2, . . . , n; j = 1, 2, . . .
,
(12)
where j is the number of iterations, vi is the residual of the ith observable, k is a very small number (e.g., 10−7 ) that is used to avoid being divided by zero. The updated parameters determination procedures are as follows: j
Pii Pj ˆj X Vj
j−1
j−1
= Pii · ωi = diag P1j P2j . . .
−1 T j = BT P j B B P L ˆ j − L. = BX
Pnj
(13)
The initial diagonal weight matrix P 0 is defined by the interferometric coherences. The stopping criterions of the iteration are established either when iterations > 100 or until the following inequality is satisfied: j X − Xj−1 < δ (14) where δ is a predetermined small number (e.g., 10−7 ).
IV. Experiments A. Synthetic Test We first simulated a DEM, with a resolution of 30 × 30 m and size of 4250 × 3650 pixels based on the fractal theory [2]. The DEM [Fig. 1(a)] is used to simulate an interferogram with the PALSAR imaging geometry and a perpendicular baseline of 485 m (the same as that of the real data experiment in the next section). Random phase noise with a standard deviation (SD) of 0.52 rad is also simulated and added to
Fig. 2. Results of composite phase errors removal for synthetic interferogram. (a)–(c) Composite phase errors estimated from linear, quadratic, and new models, respectively. (d)–(f) Corrected interferograms by the linear, quadratic, and new models, respectively.
the interferogram. Subsequently, a 3.5 m baseline error and a zero mean normal error (with a SD of 16 m) are added to the baseline and the simulated DEM, respectively. The inaccurate baseline and DEM are used to remove the flatearth and the topographic phases from the interferogram. The resulting differential interferogram, mainly, orbital and topographic phase errors, are shown in Fig. 1(b). Although the linear component dominates the differential interferogram, the nonlinear components are, therefore, very obvious. The linear, quadratic, and new models will be tested to remove the residual phases in the differential interferogram. The linear model is determined by the FFT-based method through fringe frequency analysis. The average fringe frequencies of the whole differential interferogram are fx = −9.0785 × 10−4 and fy = −9.5372 × 10−5 in range and azimuth directions, respectively. Based on (2), we calculate the coefficients of the linear model, a1 = −5.7042 × 10−3 , a2 = −5.9924 × 10−4 and a0 = −1.6889 (calculated with inverse FFT; see [6] for details). For the quadratic and the new model, the unwrapped
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TABLE I Estimated Parameters for the Linear, Quadratic, and New Model (Synthetic Dataset) Parameter a0 a1 a2 a3 a4 a5 a6
Linear Model −1.6889 −5.7042 × 10−3 −5.9924 × 10−4 ——– ——– ——– ——–
Quadratic Model −2.5155 −6.3952 × 10−3 −7.7076 × 10−5 8.5988 × 10−8 1.3847 × 10−8 −1.5434 × 10−7 ——–
New Model 0.1193 −6.2440 × 10−3 −4.6727 × 10−5 −1.1442 × 10−8 1.6260 × 10−7 −1.1682 × 10−7 −1.6737 × 10−3
Fig. 4. Results of composite phase errors removal for the interferogram covering Zhouqu mudslide. (a)–(c) Composite errors estimated from linear, quadratic, and new models, respectively. (d)–(f) Corrected interferograms by linear, quadratic, and new models, respectively. Fig. 3. Results of standard two-pass differential interferometry. (a) Original differential interferogram. (b) Coherence map. (c) Radar-coded DEM.
phases, coherences, pixel locations, and/or heights at these sample points are then used to estimate the model parameters. The final parameters of the linear, the quadratic, and the new models are shown in Table I. The estimated parameters for the three models [(1), (3), and (8)] are then used to simulate and, therefore, to remove the composite phase errors in the differential interferogram in Fig. 1(b), respectively. The simulated phase errors and the corrected differential interferograms are shown in Fig. 2. For the convenience of comparison, the simulated phase errors are rewrapped to [−π, π]. From Fig. 2, we can clearly see that the new model reproduces the composite errors in Fig. 1(a) much better than the linear and the quadratic models do. Also, the corrected interferogram by the new model is flatter. While the linear and the quadratic models can remove the major part of the composite errors, there are still small-scale, elevationdependent errors remaining in the interferograms. For quantitative comparisons, the SDs of the corrected interferograms [Fig. 2(d)–(f)] are calculated as 0.64, 0.62, and 0.21 rad for the linear, quadratic, and new models, respectively. The new model can, therefore, remove the composite errors more thoroughly. B. Real Data Test Two ALOS PALSAR ascending fine beam double images (Track 472, Frame 660), related to the deadly mudslide occurred in Zhouqu County, China on August 8, 2010, are used to test the algorithms. The PALSAR data are processed using the method of standard two-pass differential interferometry [5]. The differential interferogram obtained is shown in Fig. 3(a). It is clear that it contains severe composite phase errors. Fig. 3(b) highlights the coherence of the interferogram. We can see that the interferogram maintains good coherence. Fig. 3(c) shows the radar coded DEM, where very large variations in topographic height could be found and potentially introduce elevation-dependent phase errors. Despite the severe mudslide in Zhouqu County, it only covered a relatively small area as shown in Fig. 3(a), enclosed
by the white ellipse. Correspondingly, the interferometric fringes should only be confined to this area if the composite phase errors were properly compensated. We tested the linear, quadratic, and new models to compensate the composite errors, which have filled the entire interferogram and almost completely obscured the deformation signals. The final parameters of the three models are shown in Table II. Fig. 4(a)–(c) highlights the modeled composite phase errors by the three models. Comparing these modeled phase errors, we can see that the linear model can only catch the linear fringes [Fig. 3(a)], while the quadratic model captures the large-scale nonlinear orbital errors. However, it is somewhat helpless for the local elevation-dependent errors, while the new model favorably reproduces the composite errors [Fig. 3(a)], particularly at small scales. The corrected interferograms by the three models are shown in Fig. 4(d)–(f), respectively. We can see that the long valley along Bailong River [denoted by A in Fig. 4(d)], an obvious phase error remains after applying the correction with the linear and the quadratic model [Fig. 4(d) and (e)], while these errors are eliminated almost completely by the new model. Furthermore, the upper- and middle-right part of Fig. 4(e) and (d) (as denoted by B) also show similar phase errors. Comparing with the radar-coded DEM in Fig. 3(c), we know that these errors, although inverse, are highly correlated with ground elevation. However, these errors are almost entirely removed in Fig. 4(f) because the ground elevation is being considered in the new model. The region C in Fig. 4(d) is the nonlinear part of the orbital errors that can be removed with the quadratic or the new models. Comparing Fig. 4(d), (e), and (f), we can see that the deformation area caused by mudslide [the red ellipse in Fig. 4(f)] is significantly better singled out in Fig. 4(f). This demonstrates the superiority of the new model, where the elevation information is additionally considered over the conventional linear and quadratic models. Nevertheless, some phase errors still remain as indicated in Fig. 4(f), denoted by D. These errors are possibly caused by DEM errors. To quantitatively evaluate the results, the correlations between the DEM and the three corrected interferograms are
XU et al.: REFINED STRATEGY FOR REMOVING COMPOSITE ERRORS OF SAR INTERFEROGRAM
TABLE II Estimated Parameters for the Linear, Quadratic, and New Model (Real Dataset) Parameter a0 a1 a2 a3 a4 a5 a6
Linear Model −1.7921 86708×10−3 5.9879×10−5 ——– ——– ——– ——–
Quadratic Model −2.0532 8.1022×10−3 1.3567×10−3 3.0093×10−8 1.7181×10−7 −4.6663×10−7 ——–
New Model 1.6904 7.9091×10−3 0.9507×10−3 1.8650×10−7 7.5536×10−8 −3.7054×10−7 −1. 4202×10−3
TABLE III Correlations Between the DEM and the Interferograms Corrected by the Linear, Quadratic, and New Model Model Applied Linear Quadratic New
Correlation Value 0.63 0.62 0.13
calculated. The results, as listed in Table III, are as high as 0.63 for the linear/quadratic models, indicating that there are still considerable elevation-dependent phases left in the corrected interferograms. For the new model, it is as low as 0.13, suggesting that the elevation-dependent phase components are largely removed. This again demonstrates that the new model outperforms the linear and quadratic ones in composite phase errors removal, especially when there are strong topographic variations. V. Discussion and Conclusion In this letter, we proposed a new model to simulate and thus remove composite phase errors. Robust regression approach was applied to determine the parameters of the new model. The proposed model, by taking the ground elevation into consideration, had not only good potential to remove the large scale, nonlinear orbital error, but also a remarkable advantage to deal with the small-scale errors that were related to ground elevation. Our approach was validated with both synthetic and real PALSAR data. The synthetic test showed that after applying the proposed model, the accuracies of phase measurements were improved nearly two times, compared with those using conventional linear and quadratic models. While the real data test indicates that the new model is better at reproducing the composite phase errors, especially those of elevation-dependent, with the correlation with DEM reduced to about one fifth, compared with the conventional models. More importantly, after applying the correction with our new model, the deformation area affected by the Zhouqu mudslide event is reasonably accurately captured. Our newly proposed procedure should be able to effectively eliminate both large- and small-scale phase errors during InSAR data analysis. This will benefit the studies of tectonic processes, such as earthquakes, volcanoes, and fault slip, which require the mapping of detailed deformation in space and time. Although the new model performs well, it should be noted that it also has limitations similar to the quadratic model. First, phase unwrapping remains as a prerequisite. Second, the deformation region should be excluded before the implementation
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of the robust regression. If the study area has long wavelength deformation signals, such as broad interseismic motions, the new model should be resolved with the help of external data. For example, when GPS measurements are available over the study area (e.g., the 2011 Mw-9.0 Tohoku-Oki Earthquake [3]), the deformation provided by GPS should be exploited to help resolve the model parameters. Last, we must bear in mind that when taking the elevation of pixels into consideration, it will remove not only the orbital phase errors related to the elevation, but also the atmospheric signal due to vertical stratification [2], and the contribution from external DEM errors in the context of long perpendicular baseline. After the correction with the new model, less phase errors will maintain in the differential interferogram. This significantly benefits the deformation monitoring. Acknowledgment The ALOS PALSAR data were provided by the JAXA under Project AO-582. The authors would like to thank two anonymous reviewers for their constructive comments, and Dr. G. C. Feng for his help on synthetic data simulation. References [1] A. Ferretti, A. Monti-Guarnieri, C. Prati, F. Rocca, and D. Massonnet, InSAR Principles: Guidelines for SAR Interferometry Processing and Interpretation, K. Fletcher, Ed. Noordwijk, The Netherlands: ESA Publications, 2007, p. 48. [2] R. F. Hanssen, Radar Interferometry, Data Interpretation and Error Analysis. Norwell, MA, USA: Kluwer, 2001. pp. 85–87. [3] G. C. Feng, X. L. Ding, Z. W. Li, J. Mi, L. Zhang, and M. Omura, “Calibration of an InSAR-derived coseimic deformation map associated with the 2011 Mw-9.0 Tohoku-Oki Earthquake,” IEEE Trans. Geosci. Remote Sens. Lett., vol. 9, no. 2, pp. 302–306, Mar. 2012. [4] A. O. Kohlhase, K. L. Feigl, and D. Massonnet, “Applying differential InSAR to orbital dynamics: A new approach for estimating ERS trajectories,” J. Geodesy, vol. 77, no. 9, pp. 493–502, Dec. 2003. [5] U. Wegmüller and C. L. Werner, “Gamma SAR processor and interferometry software,” in Proc. 3rd ERS Symp. Eur. Space Agency Spec. Publ. ESA SP-414, 1997, pp. 1686–1692. [6] J. Zhu, X. L. Ding, B. Xu, and Z. W. Li, “Refine method for removing the linear trend in interferogram based on fringe frequency test,” J. Geodesy Geodyn., vol. 31, no. 3, pp. 138–141, Jun. 2011. [7] M. Jiang, X. L. Ding, G. C. Feng, and Z. W. Li. (2011, Sep.). “Frequency and space domain orbit error corrections: A case study in Coseismic Deformations of 2011 Tohoku Earthquake,” in Proc. Fringe Workshop [Online]. Available: http://earth.eo.esa.int/workshops/ fringe2011/files/FRINGE2011 Abstracts Book.pdf [8] M. Shirzaei and T. R. Walter, “Estimating the effect of satellite orbital error using wavelet-based robust regression applied to insar deformation data,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4600–4605, Nov. 2011. [9] U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens., vol. 33, no. 3, pp. 579–589, May 1995. [10] M. Tobita, T. Nishimura, T. Kobayashi, K. X. Hao, and Y. Shindo, “Estimation of coseismic deformation and a fault model of the 2010 Yushu earthquake using PALSAR interferometry data,” Earth Planet. Sci. Lett., vol. 307, nos. 3–4, pp. 430–438, Jul. 2011. [11] C. J. Zarowski, An Introduction to Numerical Analysis for Electrical and Computer Engineers. Hoboken, NJ, USA: Wiley, 2004, pp. 251–252. [12] X. Z, Cui, Z. C. Yu, B. Z. Tao, D. J. Liu, Z. L. Yu, H. Y. Sun, and X. Z. Wang, Generalized Surveying Adjustment, 2nd ed. Wuhan, China: Wuhan Univ. Press, 2009, pp. 186–201. [13] V. Gradinaru, “Fourier transform on sparse grids: Code design and the time dependent Schrodinger equation,” Computing, vol. 80, no. 1, pp. 1–22, Feb. 2007.
