IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 5, NO. 2, APRIL 2008
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Simultaneous Least Squares Adjustment of Multiframe Velocities Derived From Interferometric and Speckle-Tracking Methods Hongxing Liu, Member, IEEE, Zhiyuan Zhao, Jaehyung Yu, and Ken Jezek, Associate Member, IEEE
Abstract—This letter extends the simultaneous least squares adjustment method to the calibration and merging of velocity measurements, which are obtained by applying interferometric and speckle-tracking methods to multiple frames of repeat-pass interferometric synthetic aperture radar (InSAR) data. The underlying observation equations have been derived for velocity control points and tie points. By exploiting a set of tie points, individual frames within a strip or a block can be correlated with each other, and the parameters of calibration models for all individual frames can be simultaneously determined. With Radarsat InSAR data over the Recovery Glacier, Antarctica, we demonstrate that our simultaneous least squares adjustment method has effectively removed velocity discontinuities between adjacent frames and greatly reduced the velocity-control-point requirement. Index Terms—Interferometric SAR (InSAR), least squares adjustment, phase unwrapping, speckle tracking, surface velocity.
I. I NTRODUCTION
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ANY glaciological and environmental applications require velocity measurements over a large drainage basin, and multiple interferometric synthetic aperture radar (InSAR) image frames from the same and/or adjacent orbits need to be processed to achieve full ground coverage [1]. The conventional frame-by-frame processing approach often causes velocity discrepancies and artificial discontinuities between adjacent frames, which may mislead numerical analysis in glaciological and geophysical applications. In addition, because it is often difficult to find a sufficient number of velocity control points, the frame-by-frame processing approach may not be able to produce absolute velocity information for some image frames. To tackle the problems associated with the frame-byframe processing approach, a least squares adjustment approach has been proposed in [1] for making a strip or block adjustment of velocity measurements derived solely from the speckletracking method. Manuscript received September 19, 2007; revised September 25, 2007. This work was supported by the National Science Foundation under Grant 0126149. H. Liu is with the Department of Geography, Texas A&M University, College Station, TX 77843 USA (e-mail:
[email protected]). Z. Zhao is with Vexcel Canada Inc., Nepean, ON K2E 7M6, Canada (e-mail:
[email protected]). J. Yu is with the Department of Physics and Geosciences, Texas A&M University—Kingsville, Kingsville, TX 78363 USA (e-mail: jaehyung.
[email protected]). K. Jezek is with the Byrd Polar Research Center, The Ohio State University, Columbus, OH 43210 USA (e-mail:
[email protected]). 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.2008.915936
Previous studies [2]–[5] have suggested that, whenever possible, the range-motion measurements derived by the conventional interferometric method should be fused with the azimuth-motion measurements derived by the specklematching method to achieve the best possible measurement accuracy. In this letter, we extend the simultaneous least squares adjustment method to the situation, in which conventional interferometric and speckle-tracking methods are combined to process multiple frames of repeat-pass InSAR data within a strip or a block. In the following sections, we will derive the underlying observation equations for velocity control points and tie points, formulate the simultaneous least squares adjustment problem, and present the application results with the Radarsat InSAR data over the Recovery Glacier, Antarctica. II. F USION OF C ONVENTIONAL I NTERFEROMETRIC AND S PECKLE -T RACKING M EASUREMENTS The conventional interferometric method [6]–[9] and the speckle-tracking method [2]–[5], [10] are two alternative approaches for processing repeat-pass interferometric data for surface-velocity measurements. Each method has its advantages and limitations [3]–[5]. The conventional interferometric method is based on the differential-phase measurements and is able to measure the surface displacements at a fraction of the radar wavelength. The range-velocity measurements derived from the phase-unwrapping-based interferometric method have an intrinsically higher resolution and superior accuracy compared with the speckle-tracking method [3]–[5]. However, the surface displacements in azimuth direction cannot be derived from the conventional interferometric method. The speckletracking method can be employed to measure surface displacements in both range and azimuth directions. However, its measurement accuracy in the range direction is much worse than that in the azimuth direction. When phase unwrapping is possible, the surface-motion measurements in the range direction from the interferometric method should be coupled with the surface-motion measurements in the azimuth direction from the speckle-tracking method to maximize the overall velocity accuracy, particularly for the areas with a relatively slow surface motion and for InSAR data with a short temporal baseline [3], [4]. The interferometric method measures the surface displacements in the radar line-of-sight (LOS) direction [6]–[9] by unwrapping the interferogram of the repeat-pass SAR data pair. The phase measurement from the unwrapped interferogram
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includes contributions from baseline, topography, and surfacemotion effects. The baseline effect, namely, the flat-terrain phase ramp, can be removed with knowledge of SAR imaging geometry. The topography effect can be further removed by using a digital elevation model. After the removal of the baseline and topography effects, the phase values in the interferogram are proportional to surface displacements in the LOS direction. To obtain the absolute surface motion measurements, an unknown constant phase, associated with the arbitrary selection of initial seed point for phase unwrapping, needs to be solved [3], [5] Φm = Φ − Φ0
(1)
where Φm is the surface-motion-induced phase in the LOS direction, Φ is the unwrapped phase after the removal of baseline and topography effects, and Φ0 is the unknown constant phase to be solved. Under the assumption that the ice flow vector is parallel to the ice surface, the radial LOS velocity can be projected into horizontal surface velocities in the range direction [3], [5]
Vr =
λ Φm 4πT sin(β + αr )
(2)
where Vr is the surface motion velocity in the range direction, λ is the wavelength of radar signal, T is the time interval between two acquisitions, β is the incidence angle, and αr is the surface slope angle in the range direction. Based on a correlation matching of speckle pattern, the speckle-tracking method measures pixel offsets in both azimuth and range directions. The azimuth offset δa detected from crosscorrelation matching includes nonmotion component contributed by the imaging geometry (orbit-crossing angle). The geometry-induced terms in the azimuth offset can be modeled and removed using a linear equation [3], [5] da = δa − (b0 + b1 x + b2 y)
(3)
where da is the surface displacement in the azimuth direction, which is measured in pixels; x and y are the range and azimuth coordinates of the slant-range image; and b0 , b1 , and b2 are coefficients for accounting for the geometry term in the azimuth direction. The velocity component in the azimuth (Va ) direction can then be calculated from the following [3], [10]:
Va =
da · Sa T cos(αa )
(4)
where αa is the terrain slope in the azimuth direction and Sa is the pixel size in the azimuth direction. The range direction velocity (Vr ) in (2) and the azimuthdirection velocity (Va ) in (4) can be combined to produce 2-D surface motion speed and direction.
Fig. 1.
Velocity control points and tie points in multiple frames of a block.
III. U NIFIED S IMULTANEOUS L EAST S QUARES A DJUSTMENT A. Need for Strip or Block Adjustment Multiple frames of SAR data can be acquired to form a strip or a block for the study area of interest. A strip consists of a sequence of overlapping image frames that are acquired from the same orbit, whereas a block comprises a number of overlapping strips from different orbits (Fig. 1). InSAR data were conventionally processed on a frame-byframe basis. This often leads to inconsistency and discontinuity in the surface motion speed and direction between neighboring frames when velocity measurements from individual image frames are merged [1]. Because a certain number of velocity control points are required for each individual frame, the independent processing may preclude the possibility of deriving absolute velocity information for areas where no sufficient control points can be identified [1]. Instead of independently processing a single InSAR image frame at a time, the least squares adjustment method numerically stitches multiple individual frames in a strip or a block together through a set of tie points and computes the calibration-model parameters for all frames at the same time. As shown in Fig. 1, velocity control points and tie points can be identified and deployed within the strip or block. The benefits of the simultaneous strip or block adjustment are twofold: to reduce the number of required control points to a minimum and to produce a consistent and seamless velocity mosaic. B. Observation Equations for Velocity Control Points Velocity control points play an important role in calibrating the unknown parameters. To calculate range and azimuth displacements with (1) and (3), we need to calibrate four unknown parameters, including Φ0 , b0 , b1 , and b2 . Once the four parameters are determined, the 2-D surface displacements and velocity field can be calculated. A velocity control point is a feature with a known geographic position and velocity. Namely, its range and azimuth coordinates and its surface displacements in both range and azimuth
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directions (Dr , Da ) during the orbit repeat cycle are known. In practice, features at rock outcrops are often selected as stationary velocity control points, and their surface displacements are set to zero, namely, Dr = 0 and Da = 0. Nonstationary velocity control points can be acquired through in situ GPS measurements, but they are often unavailable. Observation equations for a velocity control point can be derived from (1) and (3) as 4πDr λ b0 + b1 x + b2 y = δa − Da . Φ0 = Φ −
(5) (6)
Each velocity control point gives rise to two observation equations. The accurate determination of the unknown parameters requires a least squares calibration with redundant observations. The number and the spatial distribution of the velocity control points influence the estimates of the unknown parameters and, hence, the accuracy of the surface velocities. Three noncollinear velocity control points are the minimum requirement for calibrating the four unknown parameters in (5) and (6). More control points will increase the redundancy of observation equations and therefore reduce the overall estimate errors for unknown parameters. Ideally, velocity control points should be evenly distributed in both the range and azimuth directions. The well-distributed control points make the least squares solution for the parameters more stable and reliable. C. Observation Equations for Tie Points A tie point refers to the same ground feature that can be recognized in two overlapping images. Tie points can be identified visually or through an automated image-matching technique. For a tie point, its velocity can be derived from two different images. Logically, the velocity for a tie point derived from one image frame should be exactly the same as that from the other image. This condition is used as a constraint to formulate observation equations for tie points. As shown in Fig. 1, tie points are located in the overlap area between adjacent images. We can write two equations for calculating the range and azimuth surface displacements of a tie point based on the first image frame (i) as follows: Φm = Φi − Φi0 da = δai − bi0 + bi1 xi + bi2 y i
(7) (8)
where Φ is the measured phase from image frame i, xi and y i are the range and azimuth coordinates of the image frame i, δai is the measured azimuth offset of the tie point from the image frame i, and Φi0 , bi0 , bi1 , and bi2 are the four unknown parameters for the image frame i. Similarly, we can write two other equations for the same tie point based on the second image frame (i + 1) as follows: Φm = Φi+1 − Φi+1 0 i+1 i+1 i+1 i+1 da = δa − b0 + bi+1 + bi+1 1 x 2 y
(9) (10)
where Φi+1 is the measured phase from image frame i + 1, xi+1 and y i+1 are the range and azimuth coordinates of the image frame i + 1, δai+1 is the measured azimuth offset of i+1 the tie point from the image frame i + 1, and Φi+1 0 , b0 , i+1 i+1 b1 , and b2 are the four unknown parameters for the image frame i + 1. By subtracting (9) from (7) and then (10) from (8), we obtain two observation equations for a tie point Φi0 − Φi+1 = Φi − Φi+1 0
(11)
i+1 i+1 bi0 + bi1 xi + bi2 y i − bi+1 − bi+1 − bi+1 = δai − δai+1 . 0 1 x 2 y
(12) D. Unified Simultaneous Least Squares Adjustment Observation equations for velocity control points and velocity tie points within a strip or a block can be integrated to perform a unified least squares calibration of unknown parameters for all image frames. The basic assumption for the least squares method is that the observation errors associated with displacement measurements of control points and tie points are random, unbiased, and normally distributed [11]. The observation (measurement) errors for individual control points are assumed to be independent and equal for solving our least squares adjustment problem. The least squares adjustment method estimates the most probable values (maximumlikelihood estimates) for the unknown parameters in the model by minimizing the sum of squared deviation between measured and estimated surface displacements of velocity control points and tie points [11]. Assume that we have n velocity control points for image frame i, m velocity control points for image frame i + 1, and p velocity tie points between image frame i and image frame i + 1. We can form 2(n + m) observation equations for velocity control points from (5) and (6) and 2p observation equations for velocity tie points from (11) and (12). There are 2(n + m + p) observation equations in total. The least squares solution for the unknown parameters can be obtained by solving normal equations or using singular value decomposition method [5], [11], [12]. For a strip or a block with k image frames, there are 4k unknown parameters to be solved in the adjustment. Therefore, the required minimum number of linearly independent observation equations is 4k + 1. The simultaneous least squares adjustment is made so that the velocities of tie points fit together as well as possible and that the residual discrepancies at the velocity control points are as small as possible. IV. A PPLICATION R ESULTS The simultaneous least squares adjustment method proposed earlier has been applied to the processing of Radarsat-1 InSAR data over the upstream reach of the Recovery Glacier, Antarctica (Fig. 2). During the first Antarctic Mapping Mission [13], this area was imaged twice by the Radarsat-1 SAR sensor on September 23 and October 27 in 1997 during repeat orbits. We use two adjacent frames of interferometric data pairs to demonstrate the effectiveness of our adjustment method.
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Fig. 3. Velocity field derived by using the frame-by-frame method. (a) Frame 5544. (b) Frame 5527.
Fig. 2. Locations of InSAR frames 5527 and 5544 over the Recovery Glacier, Antarctica. TABLE I COMPARISON OF MODEL PARAMETERS OF FRAMES 5527 AND 5544 CALIBRATED BY THE FRAME-BY-FRAME METHOD AND THE SIMULTANEOUS ADJUSTMENT METHOD
Because the upstream Recovery Glacier has a relatively slow-moving surface with a low snow-accumulation rate, the two InSAR frames preserve sufficient coherence to enable phase-unwrapping operation in the interferometric method. A sequence of interferometric processing steps has been conducted on the single-look complex images of InSAR pairs to derive precise range-motion measurements. The branch-cut method of Goldstein et al. [6], [7] was used to unwrap the differential phase. The Ohio State University digital elevation model [14] is used to remove the topographical effect. By using the speckle-tracking method, we compute azimuth offsets for both frames 5527 and 5544 at a subpixel accuracy through a cross-correlation-matching algorithm. First, we perform an independent frame-by-frame least squares calibration of the unwrapped phases and azimuth offsets to determine parameters for each frame. For frame 5527, we identified five stationary velocity control points. Following (5) and (6), we obtained ten observation equations. The parameter values of Φ0 , b0 , b1 , and b2 calibrated for frame 5527 by the single-frame least squares adjustment are shown in Table I. Similarly, we calibrated the unknown parameters Φ0 , b0 , b1 , and b2 for frame 5544 using 13 stationary velocity control points (Table I). We computed the velocity field for each frame by using (2) and (4). The results are shown in Fig. 3. For the 10-km-wide overlap area, the velocity differences between the two frames have a mean value of 5.42 m/year and a standard deviation of 5.7 m/year. The velocity difference is caused by the fact that velocity control points for each frame have a different quality and spatial configuration.
Fig. 4. Velocity mosaic of frames 5527 and 5544 derived by coupling the phase-unwrapping method and the speckle-tracking method. (a) Processed by the frame-by-frame method. (b) Processed by the simultaneous least squares adjustment with tie points (color version is posted online).
To perform a simultaneous strip adjustment, 22 tie points were identified in the overlap area. Observation equations for each tie point were created following (11) and (12). By integrating tie points and stationary velocity control points from both frames, we obtained 80 observation equations. The unknown parameters Φ0 , b0 , b1 , and b2 for frames 5527 and 5544 were simultaneously determined over the course of the least squares adjustment. Table I lists the calibration results for these parameters. After strip calibration, the mean of the velocity difference for the overlap area has been reduced from 5.42 to 0.501 m/year, and the standard deviation of velocity difference has been also reduced from 5.7 to 4.11 m/year. The seam line in Fig. 4(a) has been avoided, and the velocity discontinuity along the cross-section A in Fig. 5(a) has disappeared. The resulting seamless velocity mosaic [Fig. 4(b)] and the continuous velocity profile [Fig. 5(b)] suggest that the simultaneous adjustment with tie points minimizes the discrepancy of velocity measurements between adjacent frames and enhances the consistency of velocity measurements within the strip. The observation equations contributed by tie points also free us from the stringent need of velocity control points (stationary
LIU et al.: SIMULTANEOUS ADJUSTMENT OF MULTIFRAME VELOCITIES
Fig. 5. Velocity profiles along the cross-section A shown in Fig. 4. (a) Profile from the frame-by-frame method. (b) Profile from the simultaneous least squares adjustment.
TABLE II MODEL PARAMETERS CALIBRATED BY THE SIMULTANEOUS ADJUSTMENT METHOD WITH OR WITHOUT USING ITS OWN VELOCITY CONTROL POINTS
or nonstationary) for each frame. To demonstrate this desirable property, we drop all the velocity control points of frame 5527. Namely, we pretend that no velocity control points can be identified for frame 5527. We only employ the velocity control points from frame 5544 and the tie points to perform the simultaneous least squares adjustment of calibration model parameters for both frames 5527 and 5544. The calibrated model parameters are shown in Table II. The mean of the velocity differences between using and without using velocity control points of frame 5527 is only 4.1 m/year. Compared with the stationary velocity control points of frame 5527, the root-meansquare error (rmse) is 3.73 m/year. Similarly, we also perform the simultaneous adjustment with the velocity control points of frame 5527 and the tie points while withholding the velocity control points of frame 5544. The mean of the velocity differences for frame 5544 is 7.3 m/year when comparing the results with and without using the velocity control points of frame 5544. Compared with the stationary velocity control points of frame 5544, the rmse is 8.48 m/year. This experiment result suggests that the use of tie points in the least squares adjustment makes it possible to calibrate unknown parameters for those frames without velocity control through distant velocity control points in other image frames. V. C ONCLUSION InSAR data are usually provided by data suppliers in the format of individual frames. To obtain velocity estimates over
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a large area, multiple InSAR image frames were previously processed on a frame-by-frame basis. We developed a simultaneous least squares adjustment method for calibrating and merging surface-displacement measurements, which are obtained by applying the interferometric and speckle-tracking methods to multiple InSAR data pairs within a strip or a block. Mathematical derivation and application results show that our simultaneous least squares adjustment method can achieve consistent and smooth transition of velocity measurements between adjacent frames and considerable reduction of the stringent requirements for velocity control points. For those frames that are devoid of velocity control points, absolute velocity measurements can still be derived with this method based on constraints imposed by distant velocity control points in other frames. R EFERENCES [1] H. Liu, J. Yu, Z. Zhao, and K. C. Jezek, “Calibrating and mosaicking surface velocity measurements from interferometric SAR data with a simultaneous least-squares adjustment approach,” Int. J. Remote Sens., vol. 28, no. 6, pp. 1217–1230, 2007. [2] A. L. Gray, N. Short, K. E Matter, and K. C. Jezek, “Velocities and ice flux of the Filchner Ice Shelf and its tributaries determined from speckle tracking interferometry,” Can. J. Remote Sens., vol. 27, no. 3, pp. 193– 206, 2001. [3] Z. Zhao, “Surface velocities of the East Antarctic ice streams from Radarsat-1 interferometric synthetic aperture radar data,” Ph.D. dissertation, Ohio State Univ., Columbus, OH, 200. [4] I. Joughin, “Ice-sheet velocity mapping: A combined interferometric and speckle-tracking approach,” Ann. Glaciol., vol. 34, no. 1, pp. 195–201, Jan. 2002. [5] H. Liu, Z. Zhao, and K. C. Jezek, “Synergistic fusion of interferometric and speckle tracking methods for deriving surface velocity from interferometric SAR data,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 1, pp. 102–106, Jan. 2007. [6] R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci., vol. 23, no. 4, pp. 713–720, 1988. [7] R. M Goldstein, H. Engelhardt, B. Kamb, and R. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: Application to an Antarctic ice stream,” Science, vol. 262, no. 5139, pp. 1525–1530, Dec. 1993. [8] R. Kwok and M. Fahnestock, “Ice sheet motion and topography from radar interferometry,” IEEE Trans. Geosci. Remote Sens., vol. 34, no. 1, pp. 189–199, Jan. 1996. [9] I. Joughin, R. Kwok, and M. Fahnestock, “Estimation of ice-sheet motion using satellite radar interferometry: Method and error analysis with application to Humboldt Glacier, Greenland,” J. Glaciol., vol. 42, no. 142, pp. 564–575, 1996. [10] A. L. Gray, K. E. Matter, P. W. Vachon, R. Bindschadler, R. Forster, and J. P. Crawford, “InSAR results from the RAMP data: Estimation of glacier motion using a simple registration procedure,” in Proc. IGARSS, Seattle, WA, Jul. 1998, vol. 3, pp. 1638–1640. [11] P. R. Wolf and C. D. Ghilani, Adjustment Computations: Statistics and Least Squares in Surveying and GIS. New York: Wiley, 1997. [12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge, MA: Cambridge Univ. Press, 1992. [13] K. C. Jezek, “RADARSAT-1 Antarctic mapping project: Change detection and surface velocity campaign,” Ann. Glaciol., vol. 42, no. 142, pp. 564– 575, 2002. [14] H. Liu, K. C. Jezek, and B. Li, “Development of Antarctic digital elevation model by integrating cartographic and remotely sensed data: A GIS-based approach,” J. Geophys. Res., vol. 104, no. B10, pp. 23 199–23 213, 1999.
