Sentinel-1 Radar Interferometry Applications

The short revisit time in the ERS Tandem experiment shows the data quality reachable when the temporal coherence is high. Sentinel 1 [5] will have a revisit time of 12 days. Using the 3 days repeat data from the ice phase of ERS1, we evaluated the improvements for distributed scatterers interferometry. Slow ground motion, visible in many successive images, allows the estimation of the subsidence rate from the interferogram stacks and not only from Persistent Scatterers. Interferograms at multiple spans can be optimally combined. The dispersion of the subsidence rate estimate obtained in one year using interferogram stacks compares favourably to a PS, if the number of pixels used is greater than say 100. We compare the Cramér Rao bound for the subsidence estimates for a Markov model of the temporal decorrelation to an approximated optimal linear estimate found from the covariances of the interferograms at different spans. The unbiased subsidence rate estimates are consistent. 1 Exponential decorrelation: the fit Many targets in a SAR image are not coherent over long temporal intervals, but nevertheless they can be exploited for motion estimation using "conventional" DInSAR techniques. Most approaches can be generally defined as interferogram stacks [2, 4]. We study target decorrelation for interferogram stacks, and provide a statistically consistent estimator, to be used mainly for the assessment of the ground motion accuracy. If we suppose that the time decorrelation mechanism is primarily due to the motion of the scatterers in the resolution cell [6] we can model this as a Brownian motion, or the sum of many successive independent and equally distributed motions. It is possible to substitute the variable describing motion in the line of sight with the variable describing the unwrapped phase because of the linear relation between the two. The decorrelation law is: O SV V V W W U U J J Bd n n T 4 ; 2 ; exp ; 2 0 ̧ 1 · ̈ © § A Brownian motion in the look direction could have a standard deviation in a day of 1Bd =1mm/¥(day). This corresponds, for a single scatterer, to a time-constant 2=40[days] in C-band. If the resolution cell contains many scatterers so that the observed reflectivity is the sum of elemental contributions, then the coherence shows the same exponential decay with time, provided that each element is affected by the same independent Brownian motion. An alternative Markov model making the assumption that the elemental scatterers in the resolution cell change at random but suddenly the reflectivity leads to the same exponential decorrelation. 1.1 Validation with real data The results here discussed are based on scenes from an ERS-1 Ice-Phase data set (Track 22, Frame 2763) acquired over central Italy. During this acquisition phase, the revisit interval was 3 days. The images were range over sampled 2:1 and co-registered. A portion of the scene was then selected (20×15km, range× azimuth). It is near the Fiumicino (Rome) airport and shows the last part of course of the Tevere river [6]. We studied the decorrelation dynamics for the time span of a few weeks. We worked with a reduced set of 17 images in the range ±250m (Bperp). Figure 1 Histogram of the log coherences (best 60%) versus span We applied a spectral shift filtering in the common band and spatially averaged on windows of 12×12 pixels (range over sampled 2:1). For each window, we L1 fitted an exponential decay with variable initial coherence 0 and time constant 2. The histograms of these two parameters are presented in [6]. The average time constant is about 40 days. In figure 1 we show the histogram of the 60% best fitting logcoherences as a function of the time span of the interferogram, after rescaling with respect to the time constant and the initial coherence. The histogram is centered on the line with slope -1, as in the exponential model. Implication for 12 day repeat pass Making use of the above model we can predict the coherence for a given time span and compare it to the 12 day coherence from that measured from the dataset. A typical value we would expect is 0.4-0.5. A change in the revisit time impacts on resolution (larger swaths imply coarser azimuth resolution) and number of available interferograms. These two effects broadly compensate: more interferograms means more samples in time, coarser resolution means less samples in space. However, for caution, we neglect the increase of interferograms and consider only spatial averaging to improve the interferogram quality. Moreover, the Sentinel-1 system shortens the revisit time to 12 days without reducing the final resolution, thanks to the augmented system bandwidth [5]. The big difference is in the temporal decorrelation for which we have to consider the combined effect of á and 2. The Cramér Rao bound for the phase variance gives the well known expression [1]: L 2 2 2 2 / 1 J J VI d with L being the number of independent samples averaged. With the exponential model of coherence, the expected dispersion of the interferometric phase for a wide variation of á from 0.3 to 0.7 increases 4-5dB moving from 12 to 35 days. Conversely, operating Sentinel-1 with a halved spatial resolution in order to bring the revisit time to 6 days, we lose 1-2dB as the increased temporal coherence does not compensate the resolution degradation. 2 The model for the distributed scatterers Starting from the results presented in the previous section, we introduce the model to be used for the distributed scatter. The input is a distributed target made by L independent samples, subject all to the same subsidence and decorrelation. The index n is the discrete time at which the acquisition has been made, and the scene decorrelation comes out from the AR(1) model, so far discussed, i.e. a white source un that feeds a single pole filter. For the GS1 case we will assume the pole equal to: 74 . 0 ; 0 U U J J m n nm The pole is modulated by the subsidence rate. White noise nn ,to be added, accounts for the other target decorrelation sources. The L pixels phases can be averaged to get a single "super-pixel" with an SNR increased by a factor L. Finally, the Atmospheric Phase Screen [3] due to the water vapor, adds to each image a white phase noise, an that is identical on all the L looks as long as their mutual distance is not greater than say 500m. In this term, we can include also the target elevation error contribution. 2.1.2 DInSAR Subsidence rate estimate The optimal estimation of the subsidence rate, v, is complicated by the presence of target decorrelation, additive noise (clutter and thermal) and multiplicative noise (APS). In particular, this last contribution makes the PDF of the observations nonGaussian. As a matter of fact, we have observed that in literature most of the "interferogram stacking" techniques are heuristic combinations of interferograms weighted according to a mixture of coherence and temporal baseline. We will derive a bound for the estimate of the subsidence rate and we will define a suitable DInSARbased estimate. We approach the problem here by approximating the APS phase screen as an additive noise. Moreover, we assume large SNR, that is L>>1, and relatively large scene coherence, say >0.5, that allows us a simple derivation of the ML estimate of the subsidence rate as well as its Cramér-Rao Bound. An exact ML estimate can be derived by a more refined statistical analysis. This has been done in [7], splitting the problem of estimating the subsidence rate in a two steps, where first the interferometric phases (including the APS) are retrieved as ML estimate from the complex images, and then the subsidence rate is retrieved from the phase series. This approach, that cascades two optimal estimates, is not overall optimal in strict statistic sense, but it is likely to be close. In the other approach [6], the model is linearized and the subsidence rate is estimated in one shot. The results achieved by the three different approaches have been compared: although the estimators are different, they fit very well one against the other, and get close to the value that we find in simulations. 2.2 Simplified spectral properties We consider a fixed pixel, so that APS and thermal+clutter merges in a single, white noise added to the take and we account later for the averaging over L samples. The contribution of temporal decorrelation to the interferogram n,m is related directly to the AR(1) parameters. The center frequency of the power spectrum depends on the subsidence rate. We can thus estimate v in the frequency domain. Moving to the DFT: ̧ 1 · ̈ © § ¦ N kn j z Z n N