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Permanent Scatterers in SAR Interferometry
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 Permanent Scatterers in SAR Interferometry  Alessandro Ferretti, Claudio Prati, and Fabio Rocca  AbstractTemporal and geometrical decorrelation often prevents  SAR interferometry from being an operational tool for surface  deformation monitoring and topographic profile reconstruction.  Moreover, atmospheric disturbances can strongly compromise  the accuracy of the results. In this paper, we present a complete  procedure for the identification and exploitation of stable natural  reflectors or permanent scatterers (PSs) starting from long  temporal series of interferometric SAR images. When, as it often  happens, the dimension of the PS is smaller than the resolution cell,  the coherence is good even for interferograms with baselines larger  than the decorrelation one, and all the available images of the ESA  ERS data set can be successfully exploited. On these pixels, submeter  DEM accuracy and millimetric terrain motion detection can  be achieved, since atmospheric phase screen (APS) contributions  can be estimated and removed. Examples are then shown of small  motion measurements, DEM refinement, and APS estimation and  removal in the case of a sliding area in Ancona, Italy. ERS data  have been used.  Index TermsDifferential interferometry, digital elevation  model (DEM) reconstruction, geodetic measurements, radar data  filtering, synthetic aperture radar (SAR).  I. INTRODUCTION  REPEAT-pass satellite SAR interferometry (InSAR) is potentially  a unique tool for low cost precise digital elevation  models (DEM) generation and large-coverage surface deformation  monitoring [9], [12], [16], [17]. As is well known, the technique  involves interferometric phase comparison of SAR images  gathered at different times and with different baselines and  has the potential to provide DEMs with meter accuracy and terrain  deformations with millimetric accuracy [20]. In principle,  DEMs and deformation patterns can be estimated on a very  dense grid (4 20 m for ERS images) at low cost compared  with any other traditional method.  Limitations are essentially due to temporal and geometrical  decorrelation and atmospheric inhomogeneities. Temporal  decorrelation [10] makes InSAR measurements unfeasible over  vegetated areas and where the electromagnetic profiles and/or  the positions of the scatterers change with time within the  resolution cell. Geometrical decorrelation [10], [25] limits the  number of image pairs suitable for interferometric applications  and prevents one from fully exploiting the data set available.  Atmospheric inhomogeneities create an atmospheric phase  screen (APS) superimposed on each SAR image that can seriously  compromise accurate deformation monitoring. In fact, the  APS exhibits a low-wavenumber spectral behavior (according  to the atmospheric water vapor distribution in the troposphere  Manuscript received May 1, 1999; revised March 21, 2000. This work was  sponsored by ERA-ESRIN under Grant AO3 ESA.  The authors are with the Dipartimento di Elettronica e Informazione, Politecnico  di Milano, Milano, Italy (e-mail: [email protected]).  Publisher Item Identifier S 0196-2892(01)00991-3.  [11], [8], [26], [24]) and cannot be detected and estimated from  the coherence map associated with each interferogram [2].  The main goal of this paper is the identification of image  pixels, hereafter called permanent scatterers (PSs), coherent  over long time intervals [3], [4]. When, as it often happens,  the dimension of the PS is smaller than the resolution cell, the  coherence is good even for interferograms with baselines larger  than the decorrelation one [6], and all the available images of  the ERS data set can be successfully exploited for interferometric  applications. On these pixels, submeter DEM accuracy  and millimetric terrain motion detection can be achieved once  APS contributions have been estimated and removed. It will  be shown that even if no fringes can be seen generating single  interferograms, reliable elevation and velocity measurements  can be obtained on this subset of image pixels and can be used  as a naturalGPS network to monitor sliding areas [3] (as in  the case presented in Section IV), urban subsidence [4], seismic  faults, and volcanoes [7].  The use of sparsely populated phase data to estimate a geophysical  signal of interest has lately gained increasing attention  in differential SAR Interferometry (DInSAR) [21], [22], [34].  Here we use a multi-interferogram framework to identify highly  coherent targets, to overcome most of the difficulties related to  phase unwrapping and to better discern the different signals that  concur to the interferometric phase. The starting point is a set  of differential interferograms that use the same master acquisition.  The DEM used for differential interferograms generation  can be either a topographic profile estimated from the Tandem  pairs of the ERS data set, or an a priori DEM already available.  Its accuracy is not a real constraint (20 m is enough). In fact,  as already mentioned, DEM refinement (in correspondence to  the PS) is one of the products of the processing presented in the  following sections. Even though we consider a constant velocity  model for targets motion (i.e., we estimate just the local velocity  field of the area under study in correspondence of the PS grid),  this constraint can be relaxed using a more complex processing  [4] but using essentially the same framework. Moreover, a uniform  strain rate hypothesis is often used in geophysical modeling,  e.g., seismic faults motion, urban subsidence, lava compaction,  etc.  The aim of the paper is to make a step further toward an operational  use of DInSAR data for civil protection purposes and  a fully exploitation of the huge data set already acquired by the  ESA ERS satellites.  II. PHASE CHANGE IN REPEAT-PASS SAR INTERFEROMETRY  Although the theory of SAR interferometry has already been  presented in some detail in several papers [25], [9], [12], [15],  [6], in this section, we review the main results to establish notation  and to highlight the different physical signals that con-  01962892/01$10.00 © 2001 IEEE  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 9  tribute to the interferometric phase. For the sake of simplicity,  phase terms due to thermal noise, image misregistration, wrong  focusing parameters, etc. will be neglected [27]. The analysis  outlines the rationale behind a multi-image approach to DEM  generation and surface displacements estimation.  It is well known that a pixel in a SAR image changes its  phase due to: 1) satellite-scatterer relative position; 2) possible  temporal changes of the target; and 3) atmospheric variations  (APS). Considering 1 SAR images of the same area, the  phase of pixel ( and being azimuth and slant range  coordinates respectively) of the generic th focused SAR image  is thus the sum of different phase contributions  (1)  where is the satellite-target distance, is the scatterer reflectivity  phase, and is the atmospheric phase contribution.  Let us now consider one of the 1 images as the reference  masteracquisition . The phase difference of the generic  slaveimage with respect to the master one will be indicated  with  (2)  In repeat-pass interferometry, we can express as follows:  (3)  where is the range variation due to the different satellite  position, and is the possible target motion in the direction  of the satellite line-of-sight (LOS), occurring during the time interval  between the two acquisitions. The order of magnitude of  the first contribution is usually tens or hundreds of meters, while  the latter can be a millimetric surface deformation. The interferometric  phase is then a blend of several signals that  depends on the acquisition geometry (satellite positions and topography),  terrain motion, scattering changes (due to temporal  variations and/or baseline decorrelation), and atmospheric inhomogeneities  (4)  where we posed  (5)  A. DEM Estimation from Highly Coherent Interferograms  We shall now discuss the problem of DEM estimation from  the interferometric SAR phase referring to the ERS Tandem  case. Let us begin with a single interferometric pair formed by  a master and a slave image . Due to the short time interval  between the images (one day), we can usually neglect terrain  motion. The perpendicular baseline between the slave and  the master image is in general much smaller than the decorrelation  one (about 1200 m [6]), and it is a function of both range  and azimuth coordinate. The interferometric phase of (4) can  then be approximated as follows  (6)  In fact, we can split the geometrical term into two terms  related to the elevation and to the slant range position of the  scatterer as  (7)  where is the master sensor-target distance, and is the local  incidence angle with respect to the reference ellipsoid.  It is well known that the elevation derived from the  unwrapped phase will be affected by baseline errors  (smaller than 1 m in the case of ERS German precise orbits  [30] and ERS orbital data processed by Delft University, Delft,  The Netherlands [29]), decorrelation noise ( ), and APS  ( ) [2]. The following elevation error expression holds  (8)  where is the normal baseline error relative to image  Using a baseline greater than 200 m and precise orbits, the ratio  is usually less than 10 , so the first elevation  error contribution in (8) is usually less than a few meters.  On the contrary, the second term is a low order polynomial that  can generate large systematic errors on wide areas. This contribution  can be compensated either by using ground control points  [23] or a reference low resolution DEM [2].  As far as the last term in (8) is concerned, can be considered  a random phase fluctuation with a power dependent on  the local coherence (and it can be very high even in tandem interferograms  on areas with dense vegetation), whereas  is usually a low frequency phase distortion essentially due to the  space inhomogeneities of the atmospheric water vapor concentration.  The impact of these phase contributions depends on the  normal baseline. In general, even with the highest tandem baselines,  APS and noise can produce errors of tens of meters [2].  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. 10 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 1, JANUARY 2001  As recently proposed in [2] several tandem pairs can be exploited  to reduce the elevation errors caused by atmospheric  variations. Given tandem interferograms, DEMs of the  same area can be generated. Each DEM will be affected by the  elevation noise described by (8). These DEMs can be averaged  with weights dependent on 1) the normal baseline of each  interferogram; 2) the mean square value of the APS estimated  in different wavenumber bands; and 3) the estimated local coherence.  In [2], it is also shown that after optimum data combination,  the elevation error is concentrated at very low wavenumbers  due to residual atmospheric contributions and baseline errors  [see (8)]. Thus, a fusion of a multi-interferogram DEM (at high  wavenumbers) and a DEM obtained with different techniques  characterized by elevated accuracy at low wavenumbers (e.g.,  stereo SPOT) allows one to get the best elevation accuracy.  As a conclusion, estimated DEM accuracy (56 m) is now  limited by the largest available tandem baseline ( 500 m).  The difficulties related to phase unwrapping of high-baseline  interferograms can be overcome using multibaseline techniques  [1], [5].  III. USING LARGE TEMPORAL AND GEOMETRICAL BASELINES  Following the framework presented in the previous section,  we shall now discuss the use of large temporal and geometrical  baselines starting from the problem of accurate DEM reconstruction.  In fact, to further improve elevation accuracy with  respect to the multitandem approach, more images and larger  baselines should be exploited. However, it is well known that, as  the baseline increases toward the critical value, only scatterers  smaller than the resolution cell would allow reliable phase measurements  [10], [6]. Moreover, ERS tandem pairs have generally  small baselines and if larger ones are requested, larger time intervals  between the acquisitions should be accepted. The latter  condition (long temporal baselines) implies temporal stability  of the targets and reduces their population. The former condition  (pointwise scatterers) implies a further selection.  Similar considerations hold for differential applications for  surface deformation monitoring. In particular, if the area of interest  suffers from a secular time variation where the strain rate  is uniform, high temporal baselines should be exploited for accurate  estimation of the local velocity field in the direction of the  LOS. Again, interferometric pairs with high temporal baselines  and low geometrical baseline might be not available. Moreover,  more interferograms should be used to reduce the impact of the  APS on the estimated motion.  As a consequence, large baseline SAR interferometry can be  carried out, fully exploiting the ERS data set, only on a sparse  distribution of pointwise stable scatterers on the ground.  In the following, we shall consider 1 ERS SAR images  taken on a large time interval (e.g., five to six years in the data  set used in this work) and with baselines up to the decorrelation  one with respect to a reference acquisition selected as the  master image. A novel technique that allows us to identify  pointwise stable scatterers (PSs) and to accurately estimate their  elevation and LOS velocity is now presented.  A. General Formulation of the Problem  Let us indicate with the [ ] matrix of the interferometric  phases of pixels considered PS candidates (selection  of this subset of image pixel will be discussed later on).  The th row of contains the interferometric phases of image  with respect to the master image of pixels arbitrarily ordered  with column index  (9)  where  are constant phase values;  and contain the slope values of the  linear phase components, along the azimuth and  slant range direction, due to atmospheric phase  contributions and orbital fringes;  contains the normal baseline values (referred to  the master image). For large areas, cannot be considered  constant, and the array may become a matrix  . However, for simplicitys sake, we shall use the  previous simplified equation;  contains the elevation of each PS times  ;  contains the time interval between the slave  images and the master;  contains the slant range velocities of the PSs;  contains the residues that include atmospheric  effects different from constant and linear components in  azimuth and slant range, phase noise due to temporal and  baseline decorrelation, and the effects of possible nonuniform  pixel motion.  As formulated in (9), the problem would be linear if the unwrapped  values of matrix phase were available. We have  equations and 3 2 unknowns: .  Data are . Thus, in principle, (9) could be inverted  to get the local topography, the velocity field, and constant  and linear phase contributions. In practice, however, we  face a nonlinear system of equations (phase values are wrapped  modulo 2 ) to be solved by means of an iterative algorithm, and  an available DEM (possibly obtained using the tandem pairs of  the same data set) should be exploited to initialize the iterations.  Moreover, PS candidate selection does not allow one to identify  and exploit all the coherent targets in the area of interest. As  will be shown, the PS candidates are a good starting point to  solve the nonlinear problem at hand. In fact, most of the PSs are  actually identified after APS estimation and removal by means  of a time series analysis of their phase values.  B. Zero-Baseline Steering  Our first goal is to rephase all slave images with respect  to the master one in order to compensate for the geometric phase  contribution [(5)] as if they were  taken from the same master orbit (i.e., differential interferograms  are generated). Anyway, due to unavoidable orbit indeterminations  and DEM errors, zero-baseline steering cannot be  perfectly achieved.  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 11  The topographic phase componentcan be estimated  from the satellite state vectors and the available DEM as  (10)  where is the estimated normal baseline and is the  available topography (i.e., its accuracy is generally around 10  m). The phase error of the topographic component has the following  expression:  (11)  where is the DEM error. As already  discussed in the previous sections, the first term in (11) is very  small and will be neglected. On the contrary, for large baseline  , the phase component proportional to the DEM error can  be relevant.  As far as is concerned, possible orbit indeterminations  impact as follows:  (12)  If the area of interest is small (say 5 5 km) and precise orbits  are used, this phase contribution can be well approximated by a  linear phase component.  The estimated geometric phase contribution can then be subtracted  from the interferometric phase in order to get a first  estimate of the zero baseline steered interferometric phases  (13)  where and now take into account, apart from the APS,  the residual linear components of .  The nonlinear system 13 can be solved (and the unknowns  can then be estimated) provided that: 1) the  SNR is high enough (i.e., the selected pixels are only slightly  affected by decorrelation noise); 2) the constant velocity model  for target motion is valid; and 3) the APS can be approximated  as a phase ramp. The last condition can be fulfilled (as a first  order approximation) if the area of interest is small (say 5 5  km), while a uniform strain rate hypothesis is often used in geophysical  modeling. More complex methods should be adopted  when target motion in nonuniform [4]. The main problem is then  to properly select the PS candidates (i.e., the pixels).  C. PS Candidates Selection  In order to identify stable targets, the coherence maps associated  with the interferograms could be exploited. Correlation  thresholding would be the easiest approach. If a target exhibits a  coherence always greater than a suitable value, that would be selected  as a PS candidate. However, due to the high dispersion of  the baseline values and the limited accuracy of the DEM, several  coherence maps turn out to be useless. In fact, coherence computation  implies space averaging of the data inside a suitable estimation  window. If phase values are not properly compensated  for, the topographic (and possibly the target motion) contribution,  coherence, is underestimated. We can limit the analysis to  interferometric pairs with baseline smaller than 200300 m, but  the choice of the estimation window and the coherence threshold  to be used for PS identification is not trivial at all.  The problem can be stated as follows. On the one hand, PS  candidates (PSC) selection should be reliable (i.e., only a small  percentage of selected pixels should be affected by decorrelation  noise). On the other hand, the detection probability should  be as high as possible (so that most of the PS can be effectively  identified). The two variables to be optimized are the coherence  threshold and the dimension of the estimation window.  The larger the window dimension, the higher the estimator accuracy  (low false-alarm rate), but the lower the resolution (low  detection probability). In fact, using large estimation windows  (i.e., averaging the data over large areas), many stable targets  surrounded by noncoherent clutter are lost. Similar considerations  hold for coherence thresholding. Window dimension and  coherence threshold are then the result of a tradeoff between  false-alarm rate and detection probability, a classical detection  problem [19]. The result of this kind of approach is usually a set  of a few disconnected regions (not single pixels) where several  selected targets are actually affected by decorrelation noise.  Better results in terms of resolution can be achieved using  a different strategy. Since we suppose that many SAR images  ( 30) are available, we can analyze the time series of the amplitude  values of each pixel in the area of interest, looking for  stable scatterers. In fact, while phase stability can be assessed  only after estimation and removal of the different phase contributions,  absolute values are almost insensitive to most of the  phenomena that contribute to the phase values (APS, DEM errors,  terrain deformation, orbit indeterminations, etc.). Since we  are looking for targets slightly affected by geometrical and temporal  decorrelation, pixels exhibiting a very stablesequence  of amplitude values (in spite of the high temporal and geometrical  baseline dispersion) should be selected as PSC.  More precisely, let us focus on a PS characterized by a complex  reflectivity . Without loss of generality, we suppose  0 (i.e., is a positive real number). We then consider a complex  circular gaussian noise characterized by a power for both  real ( ) and imaginary components ( ). The distribution of  the amplitude values is given by the Rice distribution [18]  (14)  where is the modified Bessel function. The shape of the Rice  distribution depends on the SNR (i.e., the ratio ). For low  SNR, the Rice probability density function (PDF) tends to a  Rayleigh distribution, which only depends on the noise variance  , while at high SNR ( 4), approaches a Gauss  distribution. In fact, provided that , the following equation  holds:  (15)  since the modulus is primarily affected by the noise component  parallel to ). The phase dispersion ( ) can then be estimated  starting from the amplitude dispersion  (16)  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. 12 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 1, JANUARY 2001  Fig. 1. Numerical simulation results. Signal model: z = g + n (i =  1;  ; K + 1). The value of g was fixed to 1, while the noise standard  deviation ( ) was gradually incremented from 0.05 to 0.8. For each value of   , 5000 estimates of the dispersion index of the amplitude values (A =jzj)  were carried out. 34 data were supposed to be available (K = 33). The mean  values (solid line) and the dispersion (error bars) of the estimates D are  reported, together with the values of the phase standard deviation (dotted line).  Small values of D are good estimates of the phase dispersion.  where and are the mean and the standard deviation of the  amplitude values . The dispersion index is then a measure  of phase stability, at least for high SNR values. PSC can  then be selected computing the dispersion index of the amplitude  values relative to each pixel in the area of interest and considering  only those targets exhibiting values under a given  threshold (typically 0.25).  Although more rigorous and computationally intensive statistical  analyses can be adopted (e.g., ML estimation of the parameters  of the Rice distribution given the data), this very simple  approach ( thresholding) turns out to be enough for our purposes.  In fact, for PSC selection, we are interested in high SNR  values only (low ) and approximations 16 are valid. The processing  to be carried out is then fast and effective, at least for a  first selection of the targets. As will be shown, others PSs can be  identified after APS removal by means of a time series analysis  of the phase values.  The results of a numerical simulation reported in Fig. 1 (using  34 amplitude data, to be consistent with the data set used in the  experimental section) highlight potentials and limits of such an  approach. The value of was fixed to 1, while noise standard deviation  was gradually increased from 0.05 to 0.8. For each value  of , 5000 estimates of the dispersion index were carried  out. The mean values (solid line) and the dispersion (error bars)  of the estimates are reported, together with the values of the  phase standard deviation (dotted line). For low SNR, the dispersion  index tends to the value proper for the Rayleigh distribution  ( 0.5 [18]), while small values of are good  estimates of the phase dispersion.  It is important to point out that before statistical analysis of  the amplitude values, images must be radiometrically corrected  in order to make them comparable. Actually, since we are not interested  in the backscattering coefficient , it suffices to compensate  the amplitude data for a suitable calibration factor ,  depending on the sensor (ERS-1 or ERS-2), the acquisition date  and the processing center. Values for are provided by ESA  [13].  The advantages of this kind of approach are twofold: 1) fast  processing and 2) no resolution loss. Of course, the accuracy  is a function of the number of images available. A by product  is the incoherent average of the SAR data ( , multi-image  reflectivity map), where the impact of speckle noise is strongly  reduced and the spatial resolution of the image is preserved.  D. System Solution  After PSC selection, system 13 can be solved by means of  an iterative algorithm (Appendix A). Basically, DEM errors  and velocities are computed starting from small spatial  and temporal baselines and improving their accuracy as soon  as better estimation and removal of the linear phase terms  have been carried out. Since the system is highly nonlinear,  convergence is not guaranteed, depending on the following  factors:  1) space-time distribution of the acquisitions (which should  be as uniform as possible: spatial and/or temporal holes”  in the data set should be avoided);  2) reference DEM accuracy ( should generate small  phase contributions for low );  3) dimensions of the area of interest (APSs and orbital  fringes should be well approximated by linear phase  components);  4) target motion should be slow enough to avoid aliasing and  be well approximated by the constant velocity model. For  convergence, should generate small phase contributions  for low .  Results obtained with different test sites [3], [7] (one of which  will be presented in the experimental section of this paper) have  shown how constraints 1) and 2) are easily met using the ERS  data set and a reference DEM obtained using the multitandem  approach [2]. Larger areas and targets suffering nonuniform motion  can be monitored using a more complex processing [4].  E. Atmospheric Phase Screen and Ensemble Coherence  Estimation  As a result of the procedure described in the previous section,  we get a precise estimation of DEM errors and LOS velocities  of the PSC, together with constant and linear components  of the APSs ( and . If the constant velocity model  is valid, phase residues are due to atmospheric effects different  from a phase ramp and phase noise (mostly due to temporal and  baseline decorrelation)  (17)  (18)  with obvious symbol meaning. In order to filter out and  to estimate the atmospheric disturbances, we can take advanAuthorized  licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 13  tage of the strong correlation of the atmospheric components at  short distances, and we can smooth spatially the phase residues,  taking into account the power spectrum of [24]. Moreover,  once the APSs have been estimated on the sparse grid  (the PS candidates), we can interpolate them on the uniform  image grid. Both operations (filtering and resampling) can be  performed at the same time using kriging interpolation [35].  The mean value of the estimated atmospheric components in  the differential interferograms  (19)  is an estimation of the atmospheric phase contribution relative to  the master image. Its accuracy depends on the number of available  images, the density of PSs, and the reliability of . Once  has been computed, the APSs relative to  each single SAR acquisition can be easily obtained by subtraction.  From these estimated quantities, the phase of each slave  image can be modified as if it were taken from the master  orbital position in absence of terrain motion and atmospheric  effects. The new set of phases of the modified slave images  will be indicated as  (20)  where  contains the phases of the PSs as seen by  the slave images;  contains the phases of the PSs as seen by  the master image, compensated for ;  contains the estimated APS of each slave  image;  is the residues matrix.  Consider now the following expression:  (21)  It should be noted that the absolute value of ranges from 0  to 1 depending on the dispersion of the phases of the modified  slave images with respect to the master. In correspondence of a  PS, phase dispersion is low and gets close to 1. Thus,  can be regarded as an ensemble phase coherence.  Apart from temporal decorrelation, phase stability is a function  of the actual size of each scatterer within the imaged area.  The effective dimension of the targets with respect to the resolution  cell can be inferred analyzing the dispersion of the residual  phases  (22)  as a function of the baseline (Appendix B). As will be shown,  in correspondence of the PS grid, values are only slightly  affected by the normal baseline.  F. PS Identification by Means of Phase Stability Analysis  So far, LOS velocity and DEM errors have been estimated  for image pixels selected by means of the amplitude dispersion  index . However, due to the limits of the method for  PSC selection, a number of PSs could have been neglected. PS  identification can now be carried out by means of a time series  analysis of the phase values.  In fact, once the APSs have been estimated and resampled on  the uniform image grid, data can be compensated for this unwelcome  phase contribution. After APS estimation and removal, we  can finally compute DEM errors and target velocity on a  pixel-by-pixel basis: small phase residues with respect to the  model will show the presence of further PSs.  To this end, we can use a simple periodogram, albeit with an  irregular sampling of the two dimensions, baselines and time  (23)  where is the phase value of differential interferogram  relative to the generic image pixel after APS removal,  and . Basically, the unknowns  and are estimated maximizing the phase coherence of  each image pixel. The accuracy of the estimates depends on the  geometrical and temporal baseline distribution and the phase  stability of the target. For high SNR the following expressions  hold [33]:  (24)  (25)  where is the phase noise variance (supposed independent of  the acquisition), and and are the mean values of the geometrical  and temporal baselines. Considering 1 and the  baseline distribution of the ERS data set described in the experimental  section, we get 0.5 m and 0.5 mm/yr. In fact,  the expected elevation accuracy of the PS is, as usual, proportional  to the baseline that in this case can be even larger than the  decorrelation one. Referring to the actual ERS data base, baselines  as large as 1600 m can usually be found and exploited.  The experiment carried out on the urban area of Ancona, Italy,  described in the next section, shows that many PS are smaller  than the resolution cell since the average value of the phase  coherence does not decrease with the baseline as fast as it  should in case of distributed scatterers (Appendix B). Therefore,  a reasonable hypothesis to be confirmed by electromagnetic inversion  studies is that, in the urban case, the PS are railings,  corners (or equivalent) of buildings in reinforced concrete, etc.  visible to the radar. In this case, the 1600 m baseline can be coherently  exploited to get an elevation of ambiguity of about 5.5  m and thus elevation accuracies in the order of 50 cm, taking advantage  of all the available data. Similar considerations hold for  the estimated velocity field, but the accuracy will depend also  on the agreement to the constant-velocity model.  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. 14 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 1, JANUARY 2001  Fig. 2. Ancona, Italy: multi-image reflectivity map obtained using 34 ERS  SAR acquisitions. Images were interpolated by a factor of four in range  direction.  IV. EXPERIMENTAL RESULTS  The test area is a region about 5 4 km-wide in the Marche  region (Eastern part of Central Italy). The area of the town  of Ancona (Fig. 2) is of high geophysical interest because it  is known to be very unstable. The big Ancona landslide of  December 13, 1982 caused damages estimated at one billion  U.S. dollars. The area is now periodically monitored and it is  still subject to very slow terrain motion ( 1 cm/yr). Since  the area is strongly affected by temporal decorrelation (Fig. 3),  the analysis was carried out in a multi-image framework. 34  ERS SAR images gathered over the city (with a maximum  relative temporal baseline of more than five years and a  maximum relative normal baseline of more than 1600 m)  were co-registered on a unique master (ERS-2 orbit 13 460  taken on November 16, 1997). The local DEM was estimated  starting from six tandem pairs using the wavelet technique  described in [2]. After DEM compensation, 33 differential  interferograms were generated.  In order to select the PSC subset, a map of the amplitude dispersion  index was computed, starting from the 34 amplitude images  of the data set corrected for the different calibration factors  (Fig. 4). Analysis of this map shows interesting features. In  particular, it should be noted that sea pixels and vegetated areas  are characterized by high values ( 0.5), corresponding  to fully developed speckle statistics (Rayleigh distribution) or  strong changes of the backscattering coefficient (e.g., due to  Bragg reflections over the sea). On the other hand, several pixels  Fig. 3. Example of a differential interferogram over Ancona, Italy. Temporal  baseline is 70 days (images were acquired on September 9 and November 18,  1993). Estimated normal baseline is 57 m. The area of interest, affected by  terrain motion, corresponds to the low coherence area inside the white rectangle.  characterized by low values (black spots) can be identified.  Values as low as 0.11 have been estimated. About 500 targets  with 0.25 were selected as PSC.  On the PSC sparse grid, we carried out a joint estimation of  DEM errors (with respect to that obtained from the six tandem  pairs included in the data set), LOS velocities, and linear APS  contributions, as described in the previous section of this paper.  Solving the nonlinear system 13, by means of the iterative algorithm  described in Appendix B, turned out to be very fast, since  convergence was reached after a few iterations ( 10).  Atmospheric phase contributions was then estimated and resampled  on the uniform image grid by kriging interpolation  [35]. All differential interferograms were then compensated for  the estimated atmospheric contribution. In Fig. 5, an example of  APS relative to the August 1996 ERS2 acquisition is reported.  A joint estimation of both DEM errors and target velocity was  then carried out on a pixel-by-pixel basis. PSs are characterized  by low phase residues (high ensemble coherence values:  0.75). We finally obtained that about 1% of the image pixels can  be exploited for reliable phase measurements.  Fig. 6 shows the phase coherence map (23) of the data. Areas  suffering temporal decorrelation look black, whereas stable targets  are easily identified. Two considerations are in order. First,  PS density in urban areas can be very high, allowing very accurate  spatial sampling ( 100 PS/km [4]). Nevertheless, not  all the buildings can be monitored by means of this technique.  Next, a comparison between Figs. 6 and 4 shows that dispersion  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 15  Fig. 4. Amplitude dispersion index (D ) of the 34 ERS SAR images of the Ancona data set computed on a pixel-by-pixel basis. It should be noted that sea pixels  exhibit vey high relative dispersions (D > 0.5). On the contrary, black spots correspond to stable targets, characterized by low amplitude dispersions (D <  0.25). In the area under study, about 500 PS candidates were identified. Though values as low as 0.11 have been computed, the gray level scale has been limited  for visualization purposes.  Fig. 5. Estimated atmospheric phase screen (on land pixels) relative to the ERS2 SAR image acquired on August 18, 1996. Values are given in radians.  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. 16 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 1, JANUARY 2001  Fig. 6. Phase coherence map after APS removal and compensation for target velocity and elevation error.  Fig. 7. Ancona: perspective view of the harbor. The DEM was estimated combining six tandem interferograms. For visualization purposes, the vertical axis has  been magnified.  index thresholding is not enough for detecting all the PS and it  is worth carrying out all the processing steps.  After DEM correction, PSs location is known to a fraction of  meter depending on the local SNR. In Figs. 7 and 8, a comparison  between the DEM obtained combining six tandem pairs  and the improved DEM estimated using all the data is shown  (for visualization purposes the vertical scale has been magnified).  Of course, DEM was improved only where PSs were  identified. The multi-interferogram approach strongly reduces  the impact of phase noise and residual atmospheric effects both  on DEM errors and motion estimation. In Fig. 9, the map of  the PSs affected by linear motion is reported. As already mentioned,  the sliding area is subject to very slow terrain motion.  Nevertheless, it can be monitored with a high degree of accuracy.  An example of a time series of the differential phase  values corresponding to a target in the sliding area is reported  in Fig. 10. After APS removal, the accuracy of the velocity estimation  of a PS can be lower than 1 mm/yr depending on the  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 17  Fig. 8. Ancona: Three-dimensional (3-D) perspective view of the harbor. The local DEM was optimized on the PSs using all the 34 SAR images available. PSs  elevation error is now less than 1 m.  Fig. 9. Multi-image reflectivity map of the Ancona area: target affected by  significant linear motion have been highlighted. In the image, up-triangles”’  correspond to PSs with positive LOS velocity greater than 3 mm/yr, and  down-trianglescorrespond to PSs with LOS subsidence rate greater than  3 mm/yr. For the sake of clarity, stable PSs (about 1000 targets with phase  coherence greater than 0.9 and zero LOS velocity) have not been reported.  number of acquisitions, the PSs density (for APS estimation),  and temporal baselines dispersion. Comparison of the final velocity  field with ground truth (optical leveling) relative to previous  ground surveys over Ancona sliding area confirmed the  reliability of the results [31].  Fig. 10. Example of time series relative to a PS in the sliding area of Ancona.  Estimated velocity is 50.4 mm/yr (after APS removal).  Finally, in order to get an estimation of the PS size, the dispersion  of the residual phase values in correspondence with the  PS (22) of each slave image has been computed as a function of  the normal baseline. Results (Fig. 11) show a very weak dependance  of on the look angle. Fitting the data with equation  (36), the estimated PS size in range direction turned out to be  about 0.25 of the resolution cell. The decorrelation rate is lower  than what we measured for a rocky area on the Etna volcano in  Sicily [7], confirming that, especially in urban areas, most of the  PSs correspond to almost pointwise targets.  V. CONCLUSIONS  We have shown that in urban areas and in rocky terrain, PSs  exist that allow us to extract useful phase information on a sparse  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. 18 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 1, JANUARY 2001  Fig. 11. Ancona data set: coherence  versus absolute value of normal  baseline. The dashed line corresponds to (36) with  = 2 m. PSs in urban  areas exhibit almost no geometrical decorrelation due to their small dimensions  with respect to the resolution cell.  grid of targets even if the time lapse between the takes is many  years long. The spatial dimensions of the scatterers can be selected  to be small with respect to the range resolution so that  baselines longer than the critical one can be used. The PS density  was seen to be sufficient (at least in towns and on rocky  terrain [7]) to allow an estimation the atmospheric disturbance  (the APS) with a sufficient spatial resolution. APS removal can  then be performed, and a better estimation of both local topography  and terrain deformation can be carried out.  The use of all the data of the ERS data set relative to the area  of interest allows one to estimate long-term pixel motion with an  accuracy that was previously attainable using optical techniques  only. The data base of the PSs locations that can be created using  ERS data could be used with other platforms, partially compensating  low orbit stability that may produce excessive baselines.  Other applications might be the use of the pointwise character  of the PS to bridge the frequency change between ERS and ENVISAT  or maybe even the angle change with RADARSAT, etc.  The synergistic use of more platforms could in turn improve  the revisiting times, bringing the interferometric tool to an operational  stage even during seismic or volcanic crises. It will  be interesting to check whether the phase measurements on the  sparse grid of the PS could be used to improve the positioning  of the satellite. Sure enough, the PS density may prove to be too  low in vegetated areas so that artificial PS, namely, corner reflectors  (CRs), will have to be added in some locations. However,  first tests indicate that rather small CRs, with approximately a  1500 m cross section, should suffice.  Several questions remain to be studied, such as the following.  What is the distribution of PSs in different types of terrain? What  is the possibility of reducing the threshold coherence level to  extend their number? What is the physical nature of the PSs in  towns and on rocky terrain, etc.? What is the quality of the APS  estimates and their statistics?  The use of PS appears very promising in town subsidence  studies to analyze small and slow motion of buildings trying  to detect collapse precursors and for volcanic up-swelling  studies. Another application that should be feasible could be  the daily measurement of pre seismic motions on entire cities  using bistatic radar in connection with quasi-geostationary  illuminators [32].  APPENDIX A  ALGORITHM FOR LINEAR APS CONTRIBUTIONS ESTIMATION  Let us consider a relatively small area on the ground and let  us suppose that PS candidates have been selected and their  motion is almost completely described by a pixel-dependent but  time-constant velocity. The nonlinear system 13 can be solved  iteratively using the following algorithm:  1. Let be the iteration counter (starting from ) and let  Repeat until convergence:  (a) Update iteration counter and unknown vectors and  :  (b) If the following conditions are satisfied:  or (where is the maximum number of  iterations, and and are suitable  thresholds), exit the cycle.  (c) Compensate for phase contributions due to  and  (26)  (d) For each row of , estimate  using a periodogram  (27)  where  (28)  (e) Compensate the data for the estimated linear phase contributions  (29)  Authorized licensed use limited to: Politecnico di Milano. Downloaded on January 13, 2009 at 07:02 from IEEE Xplore. Restrictions apply. FERRETTI et al.: PERMANENT SCATTERERS IN SAR INTERFEROMETRY 19  (f) For each PSC (i.e., for each column of  ), estimate the residual velocity and  DEM error , weighting each datum with the  absolute value of  (30)  with  (31)  where , and .  Basically, DEM errors and velocities are computed  starting from small spatial and temporal baselines (the only ones  giving rise to high values of during the first iterations) and  improve their accuracy as soon as better estimation and removal  of the linear phase terms ( ) have been carried out.  Provided that conditions outlined in Section III are satisfied,  convergence is usually very fast ( 10 iterations with 500 PSC  and 30 images).  APPENDIX B  MEASURING THE PS DIMENSION  In this Appendix, we discuss how to estimate the electromagnetic  widthof each PS from its residual phase dispersion  as a function of the baseline after APS removal. To get simple  formulas, we shall make an approximation, namely, that we have  many independent scatterers in the resolution cell and that terrain  slopes (in azimuth and range) are constant within the resolution  cell. Let be the reflectivity modulus of a single scatterer  within the cell and its interferometric phase  (32)  where  scatterer slant range position;  local incidence angle;  normal baseline.  Then the following coherence expression holds:  (33)  If we approximate the ratio of the sums with the ratio of the expected  values, and if we accept the independence of reflectivity  , we get  (34)  Moreover, if is the probability density of the scatterer slant  range position, we can write  (35)  where is the Fourier transform of . In the case of a  uniform scatterers distribution in the interval 2 ( being  the slant range resolution cell), we get  (36)  The first zeros of are in or, using the expression  of in (32) in correspondence of the decorrelation baseline  m for flat terrain in the ERS case  On the other hand, if the scatterers are not uniformly distributed  along the resolution cell but we have scatterers with a dominant  backscattering coefficient concentrated in a smaller area, we get  a larger decorrelation baseline. As a limit, dominant pointwise  scatterers would be coherent with unlimited baselines (pointwise  targets). If we know where these PSs are located, then we  can illuminate them with any other SAR of comparable resolution.  On these pointwise PSs, we would get interferometry  notwithstanding rather unstable platforms, and this could lead  to improved revisit times.  ACKNOWLEDGMENT  The authors would like to thank Prof. A. Mazzotti, Universitá  di Milano, Milano, Italy, and the Municipality of Ancona, Italy,  for fruitful discussions and for providing us with the results  of previous ground surveys over the sliding area, and Prof. G.  Puglisi, Istituto Internazionale di Vulcanologia, Catania, Ita;y,  for helpful discussions about Etna and Valle del Bove. Finally,  they would like to thank Dr. G. Rigamonti and Dr. C. Poidomani  for their support in data processing. The continuous support of  the European Space Agency and namely, of Dr. L. Marelli, Dr.  M. Doherty, and Dr. B. Rosich, has been extremely useful.  REFERENCES  [1] A. Ferretti, A. Monti Guarnieri, C. Prati, and F. Rocca, Multi-baseline  interferometric techniques and applications,in Proc. FRINGE 96  Workshop, Zurich, Switzerland, 1996.  [2] A. Ferretti, C. Prati, and F. Rocca, Multibaseline InSAR DEM reconstruction:  The wavelet approach,IEEE Trans. Geosci. 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Zebker, Mapping small elevation  changes over large areas: differential radar interferometry,J.  Geophys. Res., vol. 94, no. B7, pp. 91839191, July 1989.  [13] H. Laur, P. Bally, P. Meadows, J. Sanchez, B. Schaettler, E. Lopinto, and  D. Esteban, Derivation of the backscattering coefficient sigma-nought  in ESA ERS SAR PRI,ESA Document No: ES-TN-RS-PM-HL09,  http://earth.esa.int/esc_intro.htm, Ispra, Italy, Sept. 1998.  [14] C. Prati, F. Rocca, A. Monti Guarnieri, and E. Damonti, Seismic migration  for SAR focusing: Interferometrical applications,IEEE Trans.  Geosci. Remote Sens., vol. 28, pp. 627640, July 1990.  [15] T. Dixon, Ed., SAR Interferometry and Surface Change Detection,”  Rep. Workshop, Boulder, CO, http://southport.jpl.nasa.gov/scienceapps/dixon/index.html,  Feb. 1994.  [16] D. Massonnet et al., The displacement field of the landers earthquake  mapped by radar interferometry,Nature, vol. 364, pp. 138142, July 8,  1993.  [17] D. Massonnet and A. 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Zebker,  Techniques and applications of SAR interferometry for ERS-1: Topographic  mapping, change detection, and slope measurement,in Proc.  First ERS-1 Symp., Cannes, France, Nov. 46 1992, pp. 205210.  [24] S. Williams, Y. Bock, and P. Pang, Integrated satellite interferometry:  Tropospheric noise, GPS estimates and implications for interferometric  synthetic aperture radar products,J. Geophys. Res., vol. 103, no. B11,  pp. 27 05127 067, 1998.  [25] E. Rodriguez and J. M. Martin, Theory and design of interferometric  synthetic aperture radars,Proc. Inst. Elect. Eng. F, vol. 139, pp.  147159, Apr. 1992.  [26] R. Hanssen, Assessment of the role of atmospheric heterogeneities in  ERS tandem SAR interferometry,DEOS Report 12 698.1, Delft Univ.,  Delft, The Netherlands, 1998.  [27] R. Bamler and D. Just, Phase statistics and decorrelation in SAR interferograms,”  in Proc. Int. Geosci. Remote Sensing Symp., Tokyo, Japan,  August 1821 1993, pp. 980984.  [28] C. Carnec, ““Interferometrie SAR differentielle,Application à la detection  des mouvements du terrain,Ph.D. dissertation, Univ. Paris 7,  Paris, France, 1996.  [29] R. Scharroo, P. N. A. M. Visser, and G. J. Mets, Precise orbit determination  and gravity field improvement for ERS satellites,J. Geophys.  Res., vol. 103, no. C4, pp. 81138127, 1998.  [30] C. Reigber, Y. Xia, H. Kaufmann, T. Timmen, J. Bodechtel, and M. Frei,  Impact of precise orbits on SAR interferometry,in Proc. FRINGE 96  Workshop, Zurich, Switzerland, 1996.  [31] A. Mazzotti, private communication.  [32] C. Prati, F. Rocca, D. Giancola, and A. Monti Guarnieri, Passive  geosynchronous SAR system reusing backscattered digital audio  broadcasting signals,IEEE Trans. Geosci. Remote Sensing, vol. 36,  pp. 19731976, Nov. 1998.  [33] D. C. Rife and R. R. Boorstyn, Single-tone parameter estimation from  discrete-time observations,IEEE Trans. Inform. Theory, vol. IT-20, pp.  591598, Sep. 1974.  [34] S. Usai and R. Klees, On the feasibility of long time scale INSAR,in  Proc. Int. Geosci. Remote Sensing Symp., Seattle, WA, July 610 1998,  pp. 24482450.  [35] H. Wackernagel, Multivariate Geostatistics, 2nd ed. Berlin, Germany:  Springer-Verlag, 1998.  Alessandro Ferretti was born in Milano, Italy,  on January 27, 1968. He received the Laurea”  and Ph.D. degrees in electrical engineering from  the Politecnico di Milano (POLIMI) in 1993 and  1997, respectively, writing his thesis on the use of  multibaseline SAR interferograms for more reliable  phase unwrapping algorithms, and the Master degree  in information technology from Politecnico (private  firms consortium) CEFRIEL, in 1993, working  on digital audio compression (psychoacoustic  applications to source coding algorithms).  In May 1994, he joined the POLIMI SAR group working on SAR interferometry  and digital elevation model reconstruction. In the summer of 1996, he was  with the Department of Geomatic Engineering (formerly Photogrammetry and  Surveying), University College London, London, U.K. His research interests  are in digital signal processing and remote sensing. He is Managing Director  of the company Tele-Rilevamento EuropaT.r.E.,a commercial spin-off of  Politecnico di Milano.  Claudio Prati was born in Milano, Italy, on March  20, 1958. He received the Laureadegree in electronic  engineering in 1983 and the Ph.D. degree in  1987, both from the Politecnico di Milano (POLIMI).  In 1987, he joined the Centro Studi Telecomunicazioni  Spaziali of the National Research Counci,  Milano. He was with Department of Geophysics  of Stanford University as a Visiting Scholar during  the Fall of 1987. Since 1991, he has been Associate  Professor of systems for remote sensing, POLIMI.  He has been the external responsible for the interferometric  experiments with the European Microwave Signature Laboratory,  Joint Research Center, Ispra, Italy. His main research interests are in digital  signal processing in noise suppression, emission tomography, and synthetic  aperture radar (SAR), where he has studied new focusing techniques and  interferometrical applications of SAR data. He has written more than 50 papers  on these topics.  Prof. Prati received the IEEE Geoscience and Remote Sensing Society 1989  Symposium Prize Paper award. He holds, with Prof. F. Rocca, a U.S. Patent  (5 332 999, July 26, 1994) on a Process for generating Synthetic Aperture  Radar Interferograms.”  Fabio Rocca received the Dottore in ingegneria elettronica  from the Politecnico di Milano (POLIMI), in  1962.  He has worked in the Department of Electronic Engineering,  POLIMI, where he is currently Professor  of Digital Signal Processing. His research activities  in seismics were dedicated to multichannel filtering,  interpolation of faulted surfaces, migration in the frequency  domain, dip moveout processing, nonlinear  deconvolution, offset and shot continuation, and recently,  the use of drill bit noise for while-drilling”  investigations. His nonseismic research activities have been dedicated to television  bandwidth compression, where he introduced and analyzed the technique  of motion compensation in emission tomography and in synthetic aperture radar  (SAR). From 1982 to 1983, he was President of the Osservatorio Geofisico Sperimentale,  Trieste, Italy, where he is now Coordinator of the Scientific Council.  He is a member of the Scientific Council of the Institut Français du Pétrole,  Paris, France.  Dr. Rocca is a Past President and Honorary Member of the European Association  of Exploration Geophysicists and an Honorary Member of the Society of  Exploration Geophysicists. He was awarded the Honeywell International Award  (HUSPI) for biomedical image processing applications in 1979, the Symposium  Prize Paper Award at the IGARSS 89 and the 1990 Schlumberger Award o

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