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A Curvature Based Method for Combining Multi-Temporal SAR Differential Interferometric Measurements

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A CURVATURE BASED METHOD FOR COMBINING MULTI-TEMPORAL SAR DIFFERENTIAL INTERFEROMETRIC MEASUREMENTS
Mario Costantini, Federico Minati, and Luca Pietranera
Telespazio S.p.A, Via Tiburtina, 965, 00156 Rome, Italy E-mail: mario_costantini@telespazio.it
ABSTRACT
Considering multiple interferograms is a common practice in SAR differential interferometry. In fact, multitemporal SAR acquisitions are used for monitoring terrain displacements over a long period of time. Moreover, the use of multiple data sets can help reducing atmospheric artifacts in SAR interferometric measurements. Given a series of SAR acquisitions, when all the interferograms between consecutive dates are computed, consecutive differential phases (after phase unwrapping) can be simply summed to obtain the differential phases correspondent to all the possible time intervals (including the total one). More generally, to this goal it is not necessary that all the computed interferograms are time-consecutive, but it suffices that the differential phases of each possible time interval can be obtained through a linear combination of the computed ones, i.e. by the solution of a determined linear system of equations. It is common in typical cases (e.g. with ERS data), that not all the interferograms necessary to the goal above can be computed, unless one accepts that only few pixels (corresponding to point-like scatterers) remain coherent. In fact, spatial and temporal baselines can be very large. So, an under-determined linear system of equations is in generally available. Previous works proposed to solve this system by singular value decomposition, i.e., by assuming that the solution (i.e. the terrain displacement) has minimum velocity. In this work, a different assumption is exploited in order to find a determined solution to the problem of combining SAR multi-temporal differential interferometric measurements. The proposed approach is based on the idea that the solution should have minimum curvature. Tests performed on simulated and ERS real data confirm the validity of the method.
1 INTRODUCTION
Differential synthetic aperture radar (SAR) interferometry is based on the concept of observing the same scene at two different times from different radar trajectories. Suppose that the imaged scene is not too different, from the electromagnetic point of view, in the two acquisitions, i.e. the scattering characteristics of the scene did not practically change in the occurred time and the view angles are nearly identical. In this case, the two SAR backscatterers are coherent, and their phase difference is a direct measure, modulo 2
π
(i.e. wrapped), of the difference of the electromagnetic paths traveled in the two cases. This phase difference, or interferometric phase, contains information on the possible small displacement of the points occurred in the time interval between the acquisitions [1], [2]. Extracting these information is a rather complex process, that is limited, among the other things, by the fact that the in real cases the view angles in the two acquisitions can differ more than desired, and the scattering scene itself can change with the time, which results in the fact that only few pixels remain coherent. In addition, different view angles cause a dependency of the measured phase on the terrain topography, which can be removed if a digital elevation model (DEM) of the observed scene is available. Another limitations is the need to solve the ambiguity of the measured phase (see for example [3], [4], and references therein): phase unwrapping can be impossible when the coherent pixels are not dense enough or the time occurred between the two acquisitions is too long. These considerations show the importance of minimizing the time between two interferometric acquisitions and the difference between the view angles, or, in other words, temporal and spatial baselines (the
interferometric baseline
is the difference between the points of the two radar trajectories from which each point of the scene is imaged). As an alternative, it is possible to concentrate only on the few pixels (corresponding to point-like scatterers stable with time) that remain in any case coherent, and discard the possible other information [5]. The simplest way to monitor terrain displacements over a long period of time is to use a series of SAR acquisitions and to compute the interferograms between image pairs with small temporal and spatial baseline. Suppose that a chain of interferograms between pairs with consecutive dates is computed. Then consecutive differential phases (after phase
____________________________________________________________ Proc. of FRINGE 2003 Workshop, Frascati, Italy,1 – 5 December 2003 (ESA SP-550, June 2004)
27_costant
unwrapping) can be simply summed to obtain the differential phases correspondent to all the possible time intervals [6]. More generally, to this goal it is not necessary that all the computed interferograms are time-consecutive, but it suffices that the differential phases of each possible time interval can be obtained through a linear combination of the computed ones, i.e. by the solution of a determined linear system of equations. It is common in typical cases (e.g. with ERS data), that not all the interferograms necessary to the goal above can be computed. In fact, spatial and temporal baselines can be rather large and in these cases only few pixels (corresponding to point-like scatterers) will remain coherent. So, an under-determined linear system of equations is in general available. Previous works proposed to solve this system by singular value decomposition, i.e., by assuming that the solution (i.e. the terrain displacement) has minimum velocity [7]. In this work, a different assumption is exploited in order to find a determined solution to the problem of combining SAR multi-temporal differential interferometric measurements. The proposed approach is based on the idea that the solution should have minimum curvature. Tests performed on simulated and ERS real data confirm the validity of the method. The proposed method is described in Section 2. In Section 3 the experimental results obtained on simulated and ERS real data are reported.
2 PROPOSED METHOD
Let N be the number of SAR acquisition and M the number of the computed differential interferograms. For each pixel, the vector of displacements
[ ]
10
,
−
=
N
x x x
l
and the vector of the differential measurements
[ ]
10
,
−
=
M
d d d
l
are linearly related:
[ ][ ] [ ]
d x A
=
. (1) The matrix
A
has as many rows as the differential measurements, and as many columns as the acquisitions. Given the differential measurement
i
between times
k
and
l
, the correspondent row
i
of the matrix is:
−==
otherwise 011
,,
l ik i
A A
. (2) Since spatial and temporal baselines can be rather large, often not all the interferograms can be computed, unless one accepts that only few pixels (corresponding to point-like scatterers) remain coherent. Therefore, the linear system (1) can be under-determined. Previous works proposed to solve the under-determined system by singular value decomposition, i.e. by assuming that the solution has minimum velocity [7]. In this work a different assumption is exploited to find a determined solution. The proposed approach assumes that the solution has minimum curvature (velocity variation in consecutive intervals). This can be done by adding to the system (1) further equations expressing the conservation of velocity in consecutive intervals, which makes the system over-determined:
[ ]
−−−=
−−−
0
d xV A
α
, (3) where the matrix
V
has N-1 rows and as many columns as the matrix
[ ]
A
, and can be defined as follows:
( )( ) ( )( )
−−=−+−=
−−=
+++−−−
otherwise 01111
11,11,11,
k k k k
k k k k k k
k k k k
t t V t t t t V
t t V
, (4)
or
( ) ( )( ) ( )
−−−=
=−−−=
−+−+
−++−
otherwise 01
1111,
,1111,
k k k k k k
k k k k k k k k
t t t t V
V t t t t V
(5) if to conditions corresponding to time intervals of different lengths are given different weights. The parameter
α
is the relative weight of the two sets of equations, those expressing velocity conservation and the srcinal ones related to the actual measurements. For small values of
α
, the regularization conditions added to the srcinal system allow to connect different times when the relative differential measurement is not available. In addition, for higher values of
α
, an effect of noise filtering can be obtained. Differential interferograms can be affected by errors on reference DEM used to remove topographic contribution. Let us denote with
[ ]
10
,
−
=
M
θ θ θ
l
,
[ ]
10
,
−
=
M
r r r
l
, and
10
,
−
⊥⊥⊥
=
M
B B B
l
, the view angle with respect to nadir, the SAR range (i.e. the distance SAR-target), and the baseline orthogonal to the view direction, respectively. For each pixel, the DEM error
h
∆
can be retrieved together with the displacement vector
x
by solving the following system:
−−−=
∆−−−
−−−−−−
⊥
00||sin4|
d h xV r B A
α ϑ λ π
. (6) The term added in system (6) with respect to (3) express the fact that the measured interferometric phase contains an additive term due to the DEM error.
3 EXPERIMENTAL RESULTS
Several tests of the proposed algorithm have been performed. In this section we present some results of the application of this method to simulated and real data. In Fig. 1 the results relative to four simulated cases are reported. Linear (a), quadratic (b, c) and sigmoidal (d) trends for the displacement evolution of a single point are considered. At the bottom of each plot the red and black solid lines represent the two disjoint sequences of differential measurements assumed available in the processing, and the green points identify the acquisition dates. For each case the minimum curvature solution (diamonds) is compared with the ideal solution (continuous line) and the minimum velocity solution (plus). It can be noted that the minimum curvature method allows reconstructing very well the ideal solution both for overlapping (a, b, d) and not overlapping (c) disjoint temporal sequences of available measurements. The real data used to test the algorithm consist on a series of ERS SAR images covering an area of Abruzzo region, Italy, around the city of Pescara. In Table 1 the list of the interferometric pairs used to perform the interferometric process is reported. The pairs were chosen in order to minimize spatial and temporal baselines. By integrating all the interferometric measurement, the displacement map relative to the period Aug. 1995 - Dec. 2000 shown in Fig. 2 was obtained. A displacement map practically identical to that of Fig. 2 was obtained, assuming that the differential measurements between 26/05/1998 and 22/12/1998 were not available, by applying the method proposed in this paper. To show some details of the result, we report in Fig. 3 the displacement evolution of the points highlighted in Fig. 2. The continuous lines in Fig. 3 represent the ideal solution obtained by integrating all interferograms (solution of determined system). Diamonds and plus symbols characterize the minimum curvature solution obtained without using the differential measurements between 26/05/1998 and 22/12/1998, with
α
= 0.1 and
α
= 0.5, respectively. At the bottom of each plot the two disjoint temporal sequences of measurements (black and red solid lines) used in the processing are shown. It can be noted that the ideal solution is reconstructed without appreciable error. As expected, a slight noise filtering effect start to be present for
α
= 0.5, and could be increased by using a higher value for the parameter
α
.
(a) (b) (c) (d)
Fig. 1: Plots of simulated displacement evolutions and reconstructions for a single point: (a) linear trend, (b, c) quadratic trend, (d) sigmoidal trend. Two disjoint, overlapping (a, b, d) and not overlapping (c) temporal sequences of differential measurements were assumed available in the processing (black and red solid line below the plots). The minimum curvature solution is practically identical to the ideal one.
Date
⊥
B
[m] Time [days] 15/08/1995-02/01/1996 -83 140 02/01/1996-21/05/1996 65 140 21/05/1996-25/02/1997 -114 280 25/02/1997-23/09/1997 -27 210 23/09/1997-10/02/1998 26 140 10/02/1998-26/05/1998 29 105 26/05/1998-22/12/1998 -150 210 22/12/1998-11/01/2000 -16 385 11/01/2000-26/12/2000 -30 350
Table 1: List of interferometric pairs (with relative spatial and temporal baselines) used to perform the interferometric processing on the test area. The interferograms were obtained from a series of ERS SAR acquisitions covering an area of Abruzzo, Italy (track 308 and the frame 2745). Fig. 2: Displacement map of Pescara, Italy, relative to the period Aug. 1995 – Dec. 2000, with superimposed 1:50000 cartography.
Point 1 Railway Point 2University Point 4 Harbour Point 3 Via Caracciolo Viale Vespucci

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