On the Exploitation of Target Statistics for SAR Interferometry Applications

On the Exploitation of Target Statistics for SAR Interferometry Applications A. Monti Guarnieri, S. Tebaldini Politecnico di Milano Abstract This paper focuses on multi-image Synthetic Aperture Radar Interferometry (InSAR) in presence of distributed scatterers, paying particular attention to the role of target decorrelation in the estimation process. This phenomenon is accounted for by splitting the analysis into two steps. In the first step we estimate the interferometric phases from the data, while in the second step we use these phases to retrieve the physical parameters of interest, such as Line of Sight (LOS) displacement and residual topography. In both steps we make the hypothesis that target statistics are at least approximately known. This approach is suited both to derive the performances of InSAR with different decorrelation models and for providing an actual estimate of LOS motion and DEM. Results achieved from Monte- Carlo simulations and a set of repeated pass ENVISAT images are shown. I. INTRODUCTION The next generation of satellite-born SARs operating in P, L, C and X band, will provide new and undiscovered potentials for SAR interferometry (InSAR). Better orbital control and shorter revisiting times will allow to infer accurate information from repeated revisits over distributed targets, subjected to geometrical and temporal decorrelation. The lack of such systems in the past brought the literature to focus on the case of Permanent Scatterers (PS) that are, by definition, highly coherent and stable in the long term and for wide baseline spans [], []. Several approaches have been presented in literature to perform SAR interferometric analysis over scenes where the PS assumption may not be retained, such as forests, agricultural fields, soil or rock surfaces, or ice shelves. A number of these works share the idea to minimize the effect of target decorrelation by forming the interferograms from properly selected pairs, rather than with respect to a fixed reference image, as done in PS processing. Despite the good results achieved in the applications, however, there s no clear and formal assessment of the criteria which should drive the selection of the image pairs to be used. As a result, the processing is heuristically based on the exploitation of a set of interferograms taken with the shortest temporal and/or spatial baselines possible [], [5], [6]. A more sophisticated approach is the one exploiting the concept of Small Baseline Subsets (SBAS) [7], [8]. This technique accounts for spatial decorrelation phenomena by partitioning the data set into a number of subsets, each of which is constituted by images acquired from orbits close to each other. Then, the inversion of the parameters of interest Part of this work was presented at IGARSS7, Barcelona []. is carried out on the basis of a minimum norm criterion exploiting the phases of all the available interferograms in each subset. With respect to these works, the aim of this paper is to propose an approach that formally accounts for the impact of target decorrelation, in such a way as to drive the interferogram selection and the estimation of the parameters of interest basing on statistical criteria. The basic idea is to split the estimation process into two steps. In the first step, a maximum likelihood (ML) estimator is used that jointly exploits all the N (N )/ interferograms available with N acquisitions, in order to yield the best estimates of N phases, with one degree of freedom. Target decorrelation is accounted for by properly weighting each interferogram depending on the target statistics. In the following we will define the N estimated interferometric phases as Linked Phases (LPs), to remind that these terms are the result of the joint processing of all the N (N )/ interferograms. Once the first estimation step has yielded the estimates of the interferometric phases, the second step is required to separate the contributions of the Atmospheric Phase Screen (APS) and the decorrelation noise from the parameters of interest, such as the Line of Sight Deformation Field (LDF) and the topography. It will be shown hereinafter that this two-step approach is consistent with the best estimate of LDF parameters provided by the Hybrid Cramér Rao bound (HCRB) []. A work naturally related to this paper is the one by Rocca []. In a sense, also in that paper a two step approach is proposed, in that first N (N )/ interferograms are formed out of N acquisitions, and then the second order statistics of the interferograms are exploited to derive the optimal linear estimator of the parameters of interest, under the small phase approximation. The difference between the approach within this paper and the one by Rocca is that in this paper we deal with the problem of parameter estimation directly from the data, rather than from the interferograms, which constitutes a more rigorous treatment of the information carried by the data. Clearly, the two approaches converge asymptotically (i.e. large signal to noise ratio, large data space). The paper is organized as follows. Section II depicts the model of the SAR Single Look Complex (SLC) data to be exploited in the sections to follow. In section III the asymptotic From the point of view of the estimation theory, the theoretical justification for splitting the estimation process into two steps may be proved by invoking the Extended Invariance Principle [9].

properties of the estimates of the interferometric phases and phase related parameters, such as the LDF and the topography, are presented, according to the formalism introduced in []. Sections IV and V are devoted to depict the estimation of LPs from the SLC data and the subsequent estimation of the LDF parameters. Section VI shows the result obtained by processing a stack of SLC SAR images acquired by ENVISAT over a desert area in Nevada, US. Finaly, conclusions are drawn in section VII. II. SAR DATA MODEL We consider a data-set made by N focused SAR SLC images, y n, n being the index of the image (n =,,...N ). Under the hypothesis of distributed scattering, the probability density function (pdf) of the data may be regarded as being a zero-mean, multivariate circular normal distribution []. Therefore, the ensemble of the second order moments represents a sufficient statistics to infer information from the data. With reference to a particular location in the slant range - azimuth plane, the expression of the second order moment for the nm th interferometric pair is given by: where: E [y n y m] = γ nm σ n σ m exp (j (φ n φ m )) () y n represents a pixel in the n th SLC SAR image at the considered slant range - azimuth location; γ nm is the coherence of the nm th interfeometric pair; γ nn = for every n; σ n is proportional to the square root of the backscatter coefficient for the n th image; φ n is the interferometric phase for the n th acquisition. In the following we will assume, without any loss of generality, that the images are normalized such that σ n = for every n. The set of the coherences, γ nm, of each interferometric pair accounts for decorrelation sources like spatial and temporal, thermal noise, and other phenomena. A useful, and widely exploited, formalism is to account for all decorrelation sources in the following product form [], []: γ nm = γ (baseline) nm γ (temporal) nm γ (noise) nm () It is important to note that, from the statistical point of view within this paper, the physical nature of the mechanisms causing decorrelation is not relevant at all. Instead, what is relevant is only the overall coherence of the interferometric pairs, resulting from (). The role of the interferometric phases, φ n, in () is to account for the two way propagation. Such terms include both the phase contributions due to the physical quantities of interest, such as LOS displacement or local topography, and those due to the APSs. In formula: φ n = ϕ n (θ) + α n () where θ is the vector of the unknown parameters which describe the LDF and the scene topography to be estimated and {ϕ n (θ)} is a set of known functions of θ (like linear, exponential, periodic etc.). The term α n represents the APS at the time of the n th acquisition. Such term accounts for the variation in the optical path due to the stochastic fluctuation of the propagation delay, mainly caused by the wet refractivity of water vapors. In the case of interest, where SAR images are taken with repeat intervals of a few days and the geometric resolution is in the order of a few meters, the APS turns out to be highly correlated over space and uncorrelated from one acquisition to the other []. As a first approximation, the fluctuation of the propagation delay may then be modeled as a normal, zero mean stochastic process with variance σ α: α N (, σ αi ) () where α = [ ] T α... α N and I is the identity matrix. To express the covariance matrix of the SLC data, C, in a conveniently compact form, we proceed as follows. Let Ω denote a small neighborhood of pixels centered at the same slant range - azimuth location in each image, and let y n be the column vector corresponding to the pixels inside the Ω in the n th image. Under the assumption that the unknowns, θ, the APSs, α n, and the coherences, γ nm, are constant inside Ω, the nm th block of the data covariance matrix may be expressed as : C nm = E [ y n ym H ] (5) = γ nm exp (j (ϕ n (θ) ϕ m (θ) + α n α m )) I where the apex H stays for Hermitian transposition. It is straightforward, but tedious, to show that the determinant of C is neither affected by ϕ n nor α n. Therefore, the pdf of the data conditioned on the interferometric phases is: p (y θ, α) = const exp ( y H C y ) (6) where y = [ ] y H... yn H H [ ], C = E yy H, ϕ = [ ϕ (θ)... ϕ N (θ) ] T. It is important to point out the different nature of the vectors ϕ and α in this paper. The first is considered as a deterministic vector of functions of the unknowns θ, while the second is a vector of random variables with known prior distribution p (α). III. ASYMPTOTIC PROPERTIES OF THE ESTIMATES The model here shown leads to the use of the Hybrid Cramér-Rao Bound (HCRB), in order to assess the asymptotic structure of the covariance matrix of the estimates. The HCRB [5], [6] applies when some of the unknowns are deterministic and others are random, therefore it unifies the deterministic and Bayesian CRB in such a way as to bound at once the covariance matrix of the unbiased estimates of the deterministic parameters and the mean square errors on the estimates of the random variables [5]. In our contest, the HCRB applies as follows. With reference to (), let θ be an For sake of simplicity, we assume statistical independence between neighbouring pixels in the SLC images. A more detailed analysis is found in [].

unbiased estimate of the deterministic parameters θ and α an estimate of the random variables α, then the HCRB assures that, for every estimator, the following relation holds: ) ) T ) E y,α ) T ( α α) ( α α) T ( α α) ( α α) T J (7) where E y,α [...] denotes expectation with respect to the joint pdf of the data and the APSs and the inequality means that the difference between the left and the right member of (7) is a nonnegative definite matrix. The matrix J is called the hybrid information matrix. From the assumptions about the pdf of data and APSs it may be shown that []: [ ] Θ J = T XΘ Θ T X XΘ X + σ (8) α I The matrix Θ accounts for the particular parametrization chosen to model the LDF and the scene topography: {Θ} nm = ϕ n (θ) θ m (9) The matrix X represents the Fisher Information Matrix (FIM) associated to the LPs, i.e. to the estimates of the interferometric phases φ n. The FIM accounts for the loss of information about the interferometric phases due to target decorrelation. Let Γ be the matrix of the coherence of each interferometric pair, i.e. {Γ} nm = γ nm. Then, it may be shown [] that: X = L ( Γ Γ I ) () where indicates the Hadamard (i.e. entry-wise) product between two matrices and L is the number of (independent) looks inside Ω. It is important to note that the FIM, X, is always singular, since the phase of the master image is unaccessible. Finally, the matrix σ α I in the lower right block of J in (8) accounts for the a priori information about the APSs. The accuracy achievable on the estimate of θ provided by the HCRB in (7) can be expressed by the first block of the inverse of J in (8): E y,α [ ) ) T ] () ( ( ) ) lim Θ T (X + εi) + σ αi Θ ε It is easy to show that the limit exists finite iff Θ T XΘ is full rank. IV. PHASE LINKING The goal here is to estimate the set of the interferometric phases, φ n, comprehensive of the APS contribution, as indicated in (). The separation of the APS from the other phase components will be faced in the next section. One phase (say, n = ) is conventionally set to zero, in such a way that N Formulae( 8) and () are slightly different from those we presented in []. However, it may be shown that the two formulations are equivalent. phases are to be estimated. Rewriting (6) we get that the ML estimate of the interferometric phases is yielded by minimizing the function: f ( φ,..φ N ) = ξ H ( Γ I Ω ) ξ () where I Ω is the matrix of all the available interferograms [ averaged in the neighborhood Ω (L looks) and ξ H = exp (jφ )... exp ( ) ] jφ N. The minimization of f is achieved by iteratively minimizing with respect to each phase, which can be done quite efficiently in closed form: φ p (k) = N { Γ } nm {I Ω} nm exp (j ) φ (k ) n n p () where k is the iteration step. In our approach, the starting point of the iteration was assumed as the phase of the vector minimizing the quadratic form ξ H ( Γ I Ω ) ξ under the constraint ξ =. The phase estimates φ p in () at the end of the iteration represent the set of the LPs. This algorithm, which will be referred to as Phase Linking, has proven to be very effective in the case where the target statistics (i.e., Γ) is at least approximately known, getting close to the bound computed by (), even for highly decorrelated sources. As an example, Fig. () plots the variance of the estimates of the phases φ n under the constraint φ = or, equivalently, the variance of the estimates of the phase differences φ n φ, achieved by Monte-Carlo simulations with two different matrices Γ. To provide a comparison, we considered two other phase estimators. The trivial solution, consisting in reading the phase of the corresponding L-pixel averaged interferograms formed with respect to the first (n = ) image, is named PS-like. The estimator referred to as AR() is obtained by reading the phases of the interferograms formed by consecutive acquisitions (i.e. n and n ) and integrating the result. The name AR() was chosen for this estimator because it yields the global minimizer of () in the case where the sources decorrelate as an AR() process, namely γ nm = ρ n m, where ρ (, ). In literature, this solution has been applied to compensate for temporal decorrelation [], [5], [6]. Notice that, in the case where the target statistics is that of an AR() process, the Phase Linking algorithm defaults to this simple solution, see Fig. (, top). Finally, the HCRB variance of the estimate is obtained from () by posing Θ = [ I N ] T and σ α = 5. A. Phase unwrapping The algorithm so far discussed yields an estimate of the wrapped interferometric phases, therefore Phase Unwrapping (PU) should be performed prior to the estimating the parameters of interest. However, PU is not the main concern of the paper, nor the HCRB does account for phase aliasing. We will 5 This is formally a standard CRB, since the variance of the APSs is zeroed.

PS-like AR() Phase Linking HCRB decorrelation. It is the presence of this term which constitutes the peculiarity of model (5). Clearly, in the case where the phase noise due to target decorrelation noise is dominated by the APS noise, this model defaults to the standard model exploited in PS processing. Instead, the aim of this section is to design an estimator of θ from φ (the matrix Q in ()) to improve the estimate accuracy in the case where the phase noise due to target decorrelation is dominant. As for practical applications, this is the case where the LDF is to be estimated over distances smaller than spatial correlation length of the APS, which is typically about Km []. From (), by posing Θ = I and σ α =, we find that the covariance of υ is lower bounded by: Fig.. Variance of the phase estimates achieved by three different methods, compared to the HCRB bound. Top: variance of the phase estimates for an AR() decorrelation model. Bottom: variance of the phase estimates for a random decorrelation model. then assume the same approach as in conventional PS processing that is quite simple and well tested [], []. The idea is to retrieve a rough estimate of the phase contribution due to topography (and eventually to the linear part of the displacement field) directly from N wrapped interferograms computed with respect to a common master image, subtracting the result and unwrapping the residuals. The implementation in the case of distributed scatterers here discussed is straightforward, just by substituting the phases of the interferograms, corrupted by decorrelation noise, with the LPs, φ n. This approach is efficient, since only N estimated interferograms are required to estimate the topography, and provided good results in the real data experiment shown in section VI. V. PARAMETER ESTIMATION After an estimate of the unwrapped interferometric phases is available, the optimal estimation of the LDF and residual topography can be carried out by exploiting the phase model () and the asymptotical properties of the phase estimates which may be derived from (). Without loss of generality, we assume here a linear relation between ϕ and θ, i.e. ϕ = Θθ. The estimates are then obtained in a closed form as θ = Q φ () where Q is a matrix to be determined. After the analysis of section III, a proper model of the estimated interferometric phases is given as follows: φ = ϕ + α + υ = Θθ + α + υ (5) where υ represents the estimate error commited in the Phase Linking step or, in other words, the phase noise due to target C υυ lim ε (X + εi) (6) This bound is quite closely approached by maximumlikelihood estimators at sufficiently large signal-to-noise ratios, or when the data space is large. Under these conditions the estimate is also unbiased, thus E [υ] =, and normally distributed. In the InSAR case, these conditions are met when either the correlation coefficients, the number of images or the estimation window are large. Therefore, we consider (6) as a suitable approximation of the covariance matrix of the phase noise due to target decorrelation. Furthermore, notice that since the matrix X is not full rank the limit (6) doesn t exist finite. Nonetheless, we can exploit (6) for any finite value of ε in order to design an estimator of θ. If Θ T XΘ is full rank, then the estimator of θ from φ (the matrix Q in ()) is finite for any value of ε, therefore we can take the limit. Finally, accounting also for the APSs, we get the following expression for the covariance of the phase estimates: W ε = (X + εi) + σ αi (7) therefore, the best estimate (lowest variance) of θ is achieved by posing: ( Q = lim Θ T Wε Θ ) Θ T Wε (8) ε where Wε represents the set of weights which allows to fit the model accounting for target decorrelation and the APSs. As a result, we get: θ = Q φ = θ + Q (α + υ) To compute the covariance of the estimate, we write [ ) ) ] T E = QW ε Q T (9) ( ( ) ) = lim Θ T (X + εi) + σ ε αi Θ which is the same as (). The equivalence between () and (9) shows that the two step procedure herein described is asymptotically consistent with the HCRB, and thus may be regarded as an optimal solution at sufficiently large signal-tonoise ratios, or when the data space is large.

Δt = 79 days Δt = 9 days Δt = 9 days Δb = 9 m Δb = 8 m Δb = 5 m.8 slant range [Km].6.. Fig.. Scene coherence computed for three image pairs. The coherences have been computed by exploiting a 9 pixel window. The topographical contributions to phase have been compensated for by exploiting the estimated DEM. VI. AN EXPERIMENT ON REAL DATA We show here an example of application of the two step approach so far developed. We consider a dateset of 8 images acquired by ENVISAT 6 over a.5 Km (slant range - azimuth) area near Las Vegas, US. The scene is characterized by elevations up 6 meters and strong lay-over areas. The normal and temporal baseline spans are about meters and 9 days, respectively. The scene is supposed to exhibit a high temporal stability. Therefore, we expect both temporal decorrelation and LDF to be negligible. However, many image pairs are affected by a severe baseline decorrelation. Fig. () shows the interferometric coherence for three image pairs, computed after removing the topographical contributions to the phase. The first and the third panels (high normal baseline) are characterized by very low coherence values throughout the whole scene, but for areas in backslope, corresponding to the bottom right portion of each panel. These panels fully confirm the hypothesis that the scene is to be characterized as being constituted by distributed targets, affected by spatial decorrelation. On the other side, the high coherence values in the middle panel (low normal baseline, high temporal baseline) confirms the hypothesis of a high temporal stability. The aim of this section is to show the effectiveness of our approach by performing a pixel by pixel estimation of the local topography and the LDF, accounting for the target decorrelation affecting the data. There are two reasons why the choice of such a dataset is suited to this goal: an a priori information about target statistics, represented by the matrix Γ, is easily available by using an SRTM DEM; the absence of a relevant LDF in the imaged scene 6 The SAR sensor aboard ENVISAT operates in C-Band (λ = 5.6 cm) with a resolution of about 5 5 m (ground range - azimuth). represents the best condition to assess the accuracy of our technique. As a first step, we estimated the interferometric phases with respect to a reference image by applying the Phase Linking algorithm depicted in Section IV. In order to retrieve an estimate as local as possible, we averaged the oversampled interferograms over a 9 pixel estimation window, corresponding to approximately m in the ground range, azimuth plane. The Phase Linking algorithm has been implemented as shown by equations (), (), where the matrix Γ has been computed at every slant range - azimuth location as a linear combination between the sample estimate and the a priori information provided by the SRTM DEM. Then, we added all the image pairs (amplitude normalized) flattened by the LPs, and used the result as an index of the phase stability within each estimation window. In formula: γ = yn H y ( m y nm n y m exp j ( φm φ )) n () The precise topography has been estimated by plugging the phase stability map γ and the LPs φ n into a standard PS processors. More explicitly, we used the phase stability map γ as a figure of merit for sampling the phase estimates on a sparse grid of reliable points to be used for APS estimation. After removal of the APS, we could estimate the residual topography on the full grid by means of a Fourier Transform [], [], namely: { ( )) } q = arg max exp j ( φn k q z (n) q () n where q is the topographic error with respect to the SRTM DEM and k z (n) is the height to phase conversion factor for the n-th image [].

Fig.. Absolute height map in slant range - azimuth coordinates. Left: elevation map provided by the SRTM DEM. Right: estimated elevation map 5 5 Histogram Interferogram Phase Linked Phase - - - phase [rad] Fig. 5. Histograms of the phase residuals shown in the top and bottom left panels of Fig., corresponding to a normal baseline of 9 m. The resulting elevation map shows a remarkable improvement in the planimetric and altimetric resolution, see Fig. (). In order to test the DEM accuracy, we formed the interferograms for three different image pairs and compensated for the precise DEM and the APS, as shown in Fig. (, top row). Notice that the interferograms decorrelate as the baseline increases, but for the areas in backslope. In these areas, it is possible to appreciate that the phases are rather good, showing no relevant residual fringes. The effectiveness of the Phase Linking algorithm in compensating for spatial decorrelation phenomena is visible in Fig. (, bottom row), where the three panels represent the phases of the same three interferograms as in the top row obtained by computing the (wrapped) differences among the LPs: φ nm = φ n φ m. It may be noticed that the estimated phases exhibit the same fringe patterns as the original interferogram phases, but the phase noise is significantly reduced, whatever the slope. This is remarked in Fig. 5, where the histogram of the residual phases of the 9 m interferogram (continuous line) is compared to the histogram of the estimated phases of the same interferogram (dashed line). The width of the central peak may be assessed in about rad, corresponding to a standard deviation of the elevation of about m Finally, Fig. 6 reports the error with respect to the SRTM DEM as estimated by the approach depicted above (left) and by a conventional PS analysis (right). More precisely, the result in the right panel has been achieved by substituting the linked phases with the interferogram phases in (). Note that APS estimation and removal has been based in both cases on the linked phases, in such a way as to eliminate the problem of the PS candidate selection in the PS algorithm. The reason for the discrepancy in the results provided by the Phase Linking and the PS algorithms is that the data is affected by a severe spatial decorrelation, causing the Permanent Scatterer model to break down for a large portion of pixels. We then focused on the estimation of the LDF. A first analysis of the residual fringes (see Fig., middle panels) shows that - as expected - no relevant displacement occurred during the temporal span of 9 days we are considering. Therefore, we attribute the residual phases mostly to decorrelation noise and to the residual APSs. Thereafter, all the N estimated residual phases have been unwrapped, in order to estimate the LDF as depicted in section V. For sake of simplicity, we assumed a linear subsidence model for each pixel, that is Θ = π [ ] T t t t N () λ being λ the wavelength and t n the acquisition time of the n th image with respect to the reference image. The weights of the estimator (8) have been derived from the estimates of Γ, according to Sections III, V. As we pointed up in section V, the weighted estimator (8) is expected to prove its effectiveness over a standard fitting (in this case, a linear fitting) in the estimation of local scale displacements, for which the major source of phase noise is due to target decorrelation. To this aim, we selectively high-pass filtered the estimated phases along the slant range - azimuth plane, in such a way as to remove most of the APS contributions and deal only with local deformations. Figure (7) shows the histograms of the estimated LOS velocities obtained by the weighted estimator (8) and the standard linear fitting. As we expected, the scene does not show any relevant subsidence and the weighted estimator achieves a lower dispersion of the estimates than the standard linear fitting. The standard deviation of the estimates of the LOS velocity produced by the weighted estimator (8) may be quantified in about.5 mm/year, whereas the HCRB standard deviation for the estimate of the LOS velocity is.6 mm/year, basing on the average scene coherence. The reliability of the LOS velocity estimates has been assessed by computing the mean square error between the phase history and the fitted model at every slant range, azimuth location, see Fig. (8). It is worth noting that among the points exhibiting high reliability, few also exhibit a velocity value significantly higher that the estimate dispersion. VII. CONCLUSIONS A novel algorithm has been presented to estimate LOS subsidence and topography from SAR acquisitions in presence of a distributed scattering mechanism. The algorithm is able to optimally handle the target decorrelation, as it exploits a weighted joint combination of all the available interferograms.

Δt = 79 days Δb = 9 m Δt = 9 days Δb = 8 m Δt = 9 days Δb = 5 m slant range [Km] Interferogram Phases slant range [Km] Linked Phases Fig.. Top row: wrapped phases of three interferograms after subtracting the estimated topographical and APS contributions. Each panel has been filtered, in order yield the same spatial resolution as the estimated interferometric phases ( 9 pixel). Bottom row: wrapped phases of the same three interferograms obtained as the differences of the corresponding LPs, after subtracting the estimated topographical and APS contributions. x 8 6 Histogram standard linear fitting weighted linear fitting - - - LOS velocity [mm/year] LOS velocity D histogram - 5 Mean Square Error 6 Mean Square Error [rad ] 5.5.5.5 Fig. 7. Histograms of the estimates of the LOS velocity obtained by a standard linear fitting and the weighted estimator (8). The estimates have been to shown to be asymptotically unbiased and minimum variance. The concepts presented in this paper have been experimentally tested on an 8 image data-set spanning a temporal interval of about months and a total normal baseline of about m. As a result, a DEM of the scene has been produced with m spatial resolution and an elevation dispersion of about m. The dispersion of the LOS subsidence velocity estimate has been assessed to be about.5 mm/year. One critical issue of this approach, common to any ML estimation, is the need for a reliable estimate of the scene coherence for every interferometric pair, required to drive the algorithm. In the case where target decorrelation is mainly determined by the target spatial distribution, we have shown that a viable solution is to exploit the availability of a DEM phase [rad] - v = -.7 MSE =.8-6 - - Acquisition times [days] Fig. 8. Right: map of the Mean Square Errors. Top left: D histogram of LOS velocities estimated through weighted linear fitting and Mean Square Errors. Bottom left: phase history of a selected point (continuous line) and the correspondent fitted LDF model (dashed line). The location of this point is indicated by a red circle in the right panel. in order to provide an initial estimate of the coherences. The case where temporal decorrelation is dominant is clearly more critical, due to the intrinsic difficulty in foreseeing the temporal behavior of the targets. Solving this problem requires the exploitation of either a very large estimation window or, which would be better, of a proper physical modeling of temporal.5.5

Topography estimated from the linked phases Topography estimated according to the PS processing slant range [Km] - - - Fig. 6. Left: topography estimated from the linked phases. Right; topography estimated according to the PS processing. The color scale ranges from to meters. decorrelation, accounting for Brownian Motion, seasonality effects, and other phenomena. This point is left as an open problem for future researches. VIII. ACKNOWLEDGMENT The authors wish to thank the reviewers for helping improve the readability of this paper through their patient work. REFERENCES [] A.M. Guarnieri and S. Tebaldini, A new framework for multi-pass sar interferometry with distributed targets, Geoscience and Remote Sensing Symposium, 7. IGARSS 7. IEEE International, pp. 589 59, July 7. [] A Ferretti, C Prati, and F Rocca, Permanent scatterers in SAR interferometry, in International Geoscience and Remote Sensing Symposium, Hamburg, Germany, 8 June July 999, 999, pp.. [] Alessandro Ferretti, Claudio Prati, and Fabio Rocca, Nonlinear subsidence rate estimation using permanent scatterers in differential SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, vol. 8, no. 5, pp., Sept.. [] Ramon Hanssen, Dmitri Moisseev, and Steven Businger, Resolving the acquisition ambiguity for atmospheric monitoring in multi-pass radar interferometry, in International Geoscience and Remote Sensing Symposium, Toulouse, France, 5 July,, pp. cdrom, pages. [5] Y. Fialko, Interseismic strain accumulation and the earthquake potential on the southern san andreas fault system, Nature, pp. :968 97, June 6. [6] A. Hopper, H. Zebker and B. Kampes, A new method for measuring deformation on volcanoes and other natural terrains using insar persistent scatterers, Geophysical Research Letters, vol., pp. L6, Dec.. [7] Paolo Berardino, Gianfranco Fornaro, Riccardo Lanari, and Eugenio Sansosti, A new algorithm for surface deformation monitoring based on small baseline differential SAR interferograms, IEEE Transactions on Geoscience and Remote Sensing, vol., no., pp. 75 8,. [8] G. Fornaro R. Lanari M. Manunta M. Manzo E.Sansosti P. Berardino, F. Casu, A quantitative analysis of the sbas algorithm performance, in Proc. IEEE Geoscience and Remote Sensing Symposium (IGARSS ),, vol. 5, pp.. [9] A.L. Swindlehurst and P. Stoica, Maximum likelihood methods in radar array signal processing, Proceedings of the IEEE, vol. 86, no., pp., Feb 998. [] S. Guarnieri, A. M.; Tebaldini, Hybrid Cramér Rao Bounds for crustal displacement field estimators in SAR Interferometry, Signal Processing Letters, IEEE, vol., no., pp. 5, Dec. 7. [] F. Rocca, Modeling interferogram stacks, Geoscience and Remote Sensing, IEEE Transactions on, vol. 5, no., pp. 89 99, Oct. 7. [] Richard Bamler and Philipp Hartl, Synthetic aperture radar interferometry, Inverse Problems, vol., pp. R R5, 998. [] Howard A Zebker and John Villasenor, Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, vol., no. 5, pp. 95 959, sept 99. [] Ramon F Hanssen, Radar Interferometry: Data Interpretation and Error Analysis, Springer Verlag, Heidelberg, edition, 5. [5] Y. Rockah and P.M. Schultheiss, Array shape calibration using sources in unknown location, Part I: Far field sources, IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-5, pp. 86 99, Mar. 987. [6] H.L. Van Trees, Detection, Estimation and Modulation Theory, Part I, John Wiley and Sons, New York, NY, 968.