MODELING INTERFEROGRAM STACKS FOR SENTINEL - 1 Fabio Rocca (1) (1) Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy, Email: rocca@elet.polimi.it ABSTRACT The dispersion of the optimal estimate of a subsidence motion is determined, and compared with that of Permanent Scatterers, when the targets are progressively decorrelating due to Brownian motion. Approximately, with N consecutive images and revisit time T, the interferogram stack results are equivalent to PS's, if the pixel count in the window grows with N²T, independently of the frequency. In the case of Sentinel 1, it is possible to expect that, in lightly vegetated areas and in one year time, less than 100 pixels will create an interferogram stack as efficient as a PS 1. Interferogram stacks approximation holds. 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 [8]: with τ=2/σ². For instance for a Brownian motion in the look direction with a standard deviation in a day of, with say the phase deviation would be: Hence: Many targets in a SAR image are not coherent over long temporal intervals, but they can be exploited nevertheless for motion estimation using "conventional" DInSAR techniques. Despite the widely developed literature on differential interferometry, starting from the very first InSAR references, there is a substantial lack of estimates of the decorrelation of distributed targets, after the paper by Zebker, and of optimal techniques to provide an estimate of the motion field [1, 2, 5]. Most approaches can be generally defined as "interferogram stacks", and new and appealing methodologies appear that could be classified in this category [4]. I establish a model for target decorrelation for interferogram stacks, and provide a statistically consistent estimator, to be used mainly for the assessment of the ground motion accuracy. Modeling the decorrelation Exponential model: Brownian motion I suppose that the time decorrelation mechanism is primarily due to the motion of the scatterers in the resolution cell [8]. I model this motion as a Brownian motion, or the sum of many successive independent and equally distributed motions so that the normal 1 This work was partly carried out within ESA contract 19142/05/NL/CB. corresponding, for a single scatterer, to a time-constant τ=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. Exponential model: Markov This alternative model makes the assumption that the elemental scatterers in the resolution cell change suddenly their reflectivity. Supposing the change rate constant and the process without memory other than the state, then the differential variation of the unchanged population is proportional to the current unchanged population. A more complex model could consider different populations characterized by different time constants. In the case of only two populations with very different τ (e.g. with τ 0 very small compared to the acquisition time scale and the other time constant τ 1 ) then the coherence can be approximated with The term p 1 in the last equation can range from 1 to 0 and represents the fraction of the scatterers that didn't suffer from a "quick decorrelation" mechanism. The impact of decorrelation is the same as that of thermal noise, i.e. a multiplication times: Proc. Envisat Symposium 2007, Montreux, Switzerland 23 27 April 2007 (ESA SP-636, July 2007)
Then I account for all the multiplicative terms with a single constant γ₀ and eventually the model writes simply: where T is the interval between the takes, and λ is the wave length. The interferogram is obtained by cross multiplying two images at times k 1 T, k 2 T, and then 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, from the end of December 1993 to April 1994 (26 scenes). During this acquisition phase, the revisit time interval was 3 days, while all the other orbital parameters remained basically unchanged. The images were focused and over sampled by a factor 2 (range only) and co-registered on the master's common grid (image taken March 5, 1994). Then a portion of the entire scene was selected (20 15km, range azimuth). It's near the Fiumicino (Rome) airport and shows the last part of course of the Tevere river. The data set interest stays with the reduced revisit time, 3 days. We can study the decorrelation dynamics in the time span of a few weeks, a task otherwise impossible with the usual 35 days data sets. Even though the maximum baseline span is about 800m we chose to work with a reduced set of 17 images in the range ±250m. This measure is an attempt to reduce the impact of geometric decorrelation. At the same time we applied a spectral shift filtering in the range band common to all images. Approximately half the original bandwidth was retained by this step, as one is limited by the worst case (500m). Decorrelation dynamics Estimates have been made by spatial averaging on windows of 12 12 pixels (range over sampled 2:1) with no overlap, to make every measure independent. The histograms of the short term coherence and of the time constant τ, with and without spectral shift filtering are shown in Figures 1, 2. The peak in the histogram is at about 40-50 days. As expected, the coherence is increased if using the common band filtered images, because a source of decorrelation is eliminated. Linear Estimates of Ground Motion The covariance matrix of the interferograms Here I will deal with the approximate estimation of the progressive interferometric phase, namely the subsidence velocity correspondent to an additional phase shift φ identical from one pass to the next. I have Figure 1: Histogram of the time constant, in days. Figure 2: Histogram of γ 0 averaging over L pixels. The removal of the scatterer phases is obtained by multiplying one image times the conjugate of the other. Consistently with the thinner orbital tubes of the forthcoming systems [6], I suppose that the baseline is kept to zero, and thus I will not consider geometrical decorrelation. This impacts on the covariance of the decorrelation noise, too, now only dependent on temporal decorrelation. As the subsidence induced phase shifts could create geometrical decorrelation, if changing with range, I have assumed to have uniform subsidence in all the pixels to be considered. The value of L will found to be consistent with the hypotheses that all pixels stay within the correlation radius of the APS, say within 800m, and therefore are subject not only to the same subsidence as said before, but also to the same atmospheric phase. The interferogram value is: where is the received signal considered as the sum of the temporally decorrelating signal γx plus noise n. The expected value of the interferogram is:
where the temporal component of the coherence, can be expressed on the basis of the exponential decay as and ρ is real, smaller than 1. For short, I will call span the temporal baseline of an interferogram i.e. the time interval between the two takes, while calling lag the delay between the time centers of two interferograms with the same or with different span. Further, is the variance of the complex value of the noiseless received signal. The actual coherence will be lower than that due to longer term decorrelation, due to the additive noise n (say, the instantaneous decorrelation term). Using the exponential Markov model, the temporal component of the coherence decreases exponentially with the span. Indicating with ρ the coherence at span T (apart the noise), and considering the phase shifts due both to the progressive subsidence and to the APS, and therefore adding to the subsidence phase shifts the phase shifts due to the APS in the two takes, I get: For small phase shifts I can linearize with respect to the APS and subsidence terms: It is correct to contend that even if the subsidence could be low, the APS won't. In effect, we expect, the ms value of the APS to be of the order of one radian square [3]. The dispersion of the two way additional travel path due to local random variations of the refractivity of the atmosphere is The interferograms covariance, for any given atmospheric and subsidence phase shift is, using the gaussianity of the data and the Gaussian moment factoring theorem: Optimal linear estimates It is possible to check that the entries of the correlation matrix of the imaginary parts of the interferograms, without the additional phase shifts due to APS or ground motion, are: The optimal estimate of φ is a linear combination of the interferograms weighted with the weights vector represented in fig. 3 and obtained with the usual prediction techniques, imposing no bias. Another derivation of these results has been obtained using an extension of the Cramér Rao bound [7]. Figure 3: Interferogram weights vector The weight vector is represented here as a function of the two indexes correspondent to the two takes; for this figure, the total number of takes is 30 (one year work for Sentinel 1) and thus the matrix is 30 30. However, the matrix is zero on and below the main diagonal as the auto interferograms are irrelevant and not weighted (points on the main diagonal) and each interferogram is considered only once. In the case of no instantaneous decorrelation, and no APS, only the interferograms at span 1 (those on the first sub diagonal) would be used, as remarked in the discussion that will follow. In order to have a better understanding of the interplay of the variables, it is useful to study interferogram stacks, or to move to the frequency domain. Cross spectra of interferogram sequences We have shown that the complex covariance of two interferograms depends only on their mutual lag and on their span difference. Now, we Fourier transform the lag axis, and call the transform variable ψ. This is made in order to study the low frequencies (ψ ~ 0) correspondent to interferogram stacks, i.e. low pass data in the lag domain. The elements of the cross spectral matrix of the imaginary parts of the interferograms, at zero frequency, depend only on the difference and the sum of their spans (indicated as 2p, 2q, as the lags are transformed out) and are:
The cross spectra of the atmospheric perturbations at very low frequencies (zero for ψ=0) are: seen in the last section. To avoid (in)significant figures, then, if T is the repeat time in weeks: Hence, the cross spectral matrix of the APS contribution at low frequencies is a dyad increasing with ψ². Finally the signal vector has components: that in the Fourier domain become: i.e. the same dyad as the atmospheric contribution but the frequency behavior is delta like instead of ψ² like as for the atmosphere. The estimate of the interferometric phase φ in the presence of this colored noise can be carried by averaging over N samples of the interferograms, i.e. windowing the spectrum. I approximate this windowing with an ideal filtering in the band: If the decorrelation time is much shorter than the integration time NT, then in the band of the filter the cross spectrum of the decorrelation can be considered as a constant. Then, indicating with a bar the effect of filtering (stacking), the covariance matrix of the filtered (stacked) atmospheric components is: and using the matrix inversion lemma, I have: Discussion This result is reasonable in that the dispersion of the estimate of the subsidence rate decreases with N³ in the case of a PS. In the case of L distributed scatterers decorrelating in M revisits, we combine N interferograms, M spans, and L pixels, and thus the dispersion may well decrease with NML. Then, to make the two behaviors equivalent one needs L increasing with. This model, very crude and overestimating L PS as the atmospheric contribution is too small, still captures the interesting behavior of L PS versus frequency, as it will be seen later in the section on the extension of the model to different frequencies. In fact, as the product does not change with λ, frequency will minimally impact on L PS. Further, it is easy to check that with high SNR only one span is used for the estimate, while the others are redundant as the unique source of noise is decorrelation. This corresponds to say that the inverse of a Toeplitz exponential matrix is tri diagonal, or that with first order Markov processes, the memory to use for estimation is the shortest possible. With lower SNR, more spans are used. Anyway, with the expected Sentinel 1 spatial resolution of 5 20m, we expect well more than 50 independent APS measurements per km², not bad at all [6]. This would allow a good estimate of the APS, its reduction, and therefore the justification of the assumptions made, entering the PS regime and yielding a further reduction of the dispersion of the velocity estimate. Extension to Different Frequencies The optimal weights depend only on ρ, SNR. It is possible to notice from this last equation that the distributed scatterers act as a PS, i.e. the APS and not the decorrelation is the main cause for dispersion of the subsidence rate estimate, if the number of looks is greater than: One advantage of the model that has been considered is the possibility of its extension to different carrier wavelengths λ. The decorrelation at span 1 and the ms atmospheric phase shift become according to : Calling M the number of revisits during the decorrelation time constant, it results approximately: Then, the still rather complex formula previously shown can be approximated with the following very simple one, even frequency independent, as it will be As said, M is the decorrelation time constant measured in terms of revisit times. In figure 4, I show in ordinates the number of pixels yielding an estimate of the subsidence with a standard deviation of 4, 5, 6 mm/year for the center frequencies in abscissas, in the case of 30 acquisitions in one year. In figure 5, I show the standard deviation of the estimate of the subsidence
for various center frequencies, according to the model, again in the case of 30 acquisitions in one year, for a number of pixels ranging from 10 to 150. I see from this figures that say L=100 is close to be enough to ensure that, with 12 days repeat and 30 revisits, the average coherence is enough to enter the PS regime, making the atmospheric effect to prevail, while there is no appreciable change with carrier frequency. Indeed, higher carrier frequencies behave worse, but practically big changes are not to be expected until 8-10 GHz. On the very low frequency side, the increment of the dispersion is due to the limited SNR, and also as expected affects both PS as well as interferogram stacks. With longer observation times, there would be a shift of the optimum towards lower frequencies, while keeping the variation of the dispersion of the estimate rather small. [7] S. Tebaldini and A. Monti Guarnieri, Cramér Rao lower bound for parametric phase estimation in multi - pass radar interferometry, Personal Communication, Final report, ESA contract No. 19142/05/NL/CB, Analysis of ambiguity noise in the Sentinel-1 Interferometric Wideswath Mode. [8] H. A Zebker and J. Villasenor. Decorrelation in interferometric radar echoes. IEEE Transactions on Geoscience and Remote Sensing, 30(5):950--959, September 1992. Conclusion An evaluation of the subsidence rate error budget has been carried out for DInSAR interferometry, using the 3 days revisit interval data over Rome taken by ERS - 1. The results of modeling temporal decorrelation with a Brownian motion depend upon the number N of images used and the revisit interval T. Assuming a short term target coherence of 0.6 and averaging measures over L=100 independent looks the dispersion of the velocity estimate is lower than 4-4.5mm/year for a 12 days revisit time. For a wide band of frequencies including C band, the number of looks needed to make distributed scatterers as accurate as a PS is in the order of 0.1N²T [weeks], approximately. Figure 4: L PS as a function of frequency for different values (4, 5, 6 mm/year) of the standard deviation of the subsidence estimate and 30 acquisitions in one year. The lower the deviation, the higher is L PS. References [1] R. Bamler and P. Hartl. Synthetic aperture radar interferometry. Inverse Problems, 14, R1 --R54, 1998. [2] Y. Fialko. Interseismic strain accumulation and the earthquake potential on the southern San Andreas fault system. Nature, 441:968--971, June 2006. [3] R. Hanssen. Radar Interferometry: Data Interpretation and Error Analysis. Kluwer Academic Publishers, Dordrecht, 2001. [4] A. Hooper, H. Zebker, P. Segall, and B. Kampes, A new method for measuring deformation on volcanoes and other natural terrains using InSAR persistent scatterers, Geophys. Res. Letters, 31, L23611, doi:10.1029/2004gl021737, 2004 [5] P. Rosen, S. Hensley, I. R Joughin, Fuk K Li, Soren Madsen, Ernesto Rodríguez, and Richard Goldstein. Synthetic aperture radar interferometry. Proceedings of the IEEE, 88(3):333--382, March 2000. [6] Mission Requirement Document, Sentinel - 1 http://esamultimedia.esa.int/docs/gmes/gmes_sent 1_MRD_1-4_approved_version.pdf Figure 5: Standard deviation of the subsidence estimate for 30 acquisitions in one year, different center frequencies, and different values of L PS, the number of pixels in the window. The lower the deviation, the higher L PS. The behavior of a PS is also indicated. The sources of noise are thermal, atmosphere, and decorrelation (for non PS targets).