THE PERMANENT scatterers (PSs) technique is a multiinterferogram

Transcription

1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 9, SEPTEMBER Validating the SAR Wavenumber Shift Priniple With the ERS Envisat PS Coherent Combination Daniele Perissin, Claudio Prati, Marus E. Engdahl, and Y. L. Desnos Abstrat Continuity of the European Remote Sensing Satellite Syntheti Aperture Radar ERS SAR) arhive by means of Envisat Advaned SAR ASAR) data aquired from Marh 00 has introdued the problem of the oherent ombination of images oming from sensors with slightly different frequenies. The spetral shift priniple states that in ase of extended distributed targets, the frequeny shift is equivalent to a hange of looking angle. In this paper, the same priniple is exploited to analyze the behavior of permanent satterers PSs) with an extension that is smaller than the ground resolution ell. The onditions under whih the PSs identified by ERS an be ontinued by Envisat are then theoretially determined and experimentally validated. Moreover, this analysis shows that aquisitions haraterized by different frequenies an be used to identify the slant-range position of satterers with high subell auray tens of entimeters). From the proessing side, a very preise images oregistration step is required to get the results desribed in this paper. Index Terms Interferometry, syntheti aperture radar SAR). I. INTRODUCTION THE PERMANENT satterers PSs) tehnique is a multiinterferogram algorithm for differential interferometri syntheti aperture radar DInSAR) analyses developed in the late 1990 s to overome the limits of onventional syntheti aperture radar SAR) interferometry approahes, namely phase deorrelation and atmospheri effets [1] [3]. The ore purpose of PS analysis is to reonstrut the terrain topography and to monitor deformation phenomena by exploiting long series of SAR data aquired under different geometries and time lags. The auray of the height estimate depends on the normal baseline dispersion of the aquisitions [4]. The wider the range of the baselines is, the better the digital elevation model DEM) estimate is. High values of normal baseline are also a soure of deorrelation in orrespondene of distributed satterers [5]; thus, only almost-point-like satterers are identified beause they keep on being oherently imaged. After the failure of the ERS- gyrosope, updating ERS PS measurements by means of the Advaned SAR ASAR) data beame a major issue. Apart from the enhaned ASAR features in terms of aquisition modes and polarimetri apabilities, the main differene between ERS and Envisat ERS-like aquisitions is the arrier frequeny 5.3 GHz for ERS-1/ and GHz for Envisat). The 31-MHz frequeny shift has a major impat on any interferometri appliation involving Manusript reeived Otober 17, 005; revised February 3, 006. D. Perissin and C. Prati are with the Dipartimento di Elettronia e Informazione, Politenio di Milano, 0133 Milano, Italy daniele.perissin@ polimi.it). M. E. Engdahl and Y. L. Desnos are with the European Spae Ageny, European Spae Researh Institute, Frasati, Italy. Digital Objet Identifier /TGRS Fig. 1. Radar aquisition geometry of the Master image. r axis: slant-range diretion; y axis: flight diretion azimuth); and x axis: normal to the ry plane ross slant-range diretion). 1) First point target, loated in the axes origin. ) Seond point target with oordinates r, x, and y. ERS Envisat data sets beause the bandwidth of both systems is less than 18 MHz. ERS Envisat interferograms usually turn out to be ompletely deorrelated, at least for typial baseline values < 1300 m), beause no ommon band is available for distributed targets [5]. Point targets, on the ontrary, are expeted to remain oherent, and their ERS Envisat phase histories an be oherently stithed. Moreover, the arrier frequeny variation an also be exploited to preisely loate the satterer in slant range [6] [8]. II. THEORETICAL FRAMEWORK First, the phase stithing theory is onsidered for point-like targets. Then, the theory will be extended to distributed targets introduing the progressive deorrelation proess indued by the satterers extension. A. Point-Target Analysis Let us onsider a pair of oregistered SAR images whih will be identified as Master and Slave) and the orresponding interferogram. The data are aquired with a ommon nominal geometry by two radar systems operating at two slightly different frequenies. Let B n be the normal baseline assoiated to a ross-trak angle θ) and f DC the Doppler entroid DC) frequeny differene assoiated to an along-trak angle ψ) between the Master image, whih was aquired at frequeny f 0 f, and the Slave, whih was aquired at frequeny f 0. In Fig. 1, two nearby point targets 1 and are shown. The /$ IEEE

2 344 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 9, SEPTEMBER 006 The interferometri phase an then be expressed as φ 1 = φ M,1 φ S,1 = 4π f 0 f) r 4π f 0 r + 4π f 0 x θ + 4π f 0δy ψ = 4π f r + 4π f 0 x θ + 4π f 0δy ψ. 4) In.4), besides the two well-known interferometri terms, a new ontribution arises. It depends on the frequeny differene between Master and Slave and on the slant-range satterer position. Equation.4) highlights the possibility of loating in the 3-D spae a target by means of SAR interferometry. Equation.4) an be rewritten as a funtion of the normal baseline B n and of the DC frequeny differene f DC. Defining R 0 the sensor distane from the origin, the inidene angle differene is θ = B n /R 0. Starting from [10], we have the following relation between the squint angle ψ and the DC frequeny f DC Fig.. Master aquisition geometry setions. a) and b) xr plane. ), d) yr plane. In b) and d), the Slave slant-range diretion is reported, forming two angles with the Master r axis: b) θ in xr plane and d) ψ in yr plane. In ) and d), δy is the subpixel azimuth position of target, assuming that target 1 is loated at the enter of the sampling ell. following orthogonal three-dimensional 3-D) referene system entered at the first point target is then defined: 1) r axis: slant-range diretion of the Master image; ) y axis: flight diretion azimuth) of the Master image; 3) x axis: normal to the ry plane ross slant-range diretion). In this referene system, the seond target has oordinates r, x, and y as shown in Fig. 1. The following general form of the interferometri phase between the two point targets 1 and ) holds φ 1 = φ M,1 φ S,1 1) where φ M,1 and φ S,1 are the phase differenes between the two point targets as seen by the Master and Slave images, respetively. Under the hypothesis of no ground motion and almost no atmospheri phase ontribution lose targets [9]), the phase differene φ M,1 only depends on the slant-range distane between the two point targets, whih is given as follows: φ M,1 = 4π f 0 f) r ) where is the speed of light. The omputation of phase differene φ S,1 has the same form of φ M,1 but for a different frequeny f 0 instead of f 0 f) and a slightly different slant-range displaement [ r x θ δy ψ instead of r as shown in Fig. b) and d)], whih is given as follows: φ S,1 = 4π f 0 r 4π f 0 x θ 4π f 0δy ψ. 3) Here, δy is the subpixel azimuth position of target, assuming target 1 is loated at the enter of the sampling ell. f DC = v f 0 sin ψ 5) where v is the sensor veloity, and ψ = ψ 0 + ψ is the Slave squint angle. Computing the differene with respet to the Master aquisition haraterized by a squint angle ψ 0 whose typial value is very lose to 0), we obtain f DC = v f 0 ψ os ψ 0 v f 0 ψ 6) and introduing the azimuth sampling step δ az and the pulse repetition frequeny PRF, we have ψ = Equation.4) an thus be rewritten as δ az PRFf 0 f DC. 7) φ 1 = 4π f r + 4πf 0B n x +π f DC δy. 8) R 0 PRF δ az In the framework of a multiinterferogram analysis, like the PS tehnique, the three terms in 8) an be separated by exploiting their different behaviors in the parameter spae f, B n, f DC, and the three oordinates r, x, and δy an be estimated. In ase of an ERS Envisat data set, only two different frequenies are present and thus a single value for f. Consequently, the first term in 8) is different from point to point but onstant in all ERS Envisat interferograms. This term shall be referred to as the loation phase sreen LPS) in the following. Given the 31-MHz frequeny shift, the LPS hanges by almost two yles aross a single slant-range resolution ell giving an LPS ambiguity of about 5 m), thus, it is extremely sensitive to the target slant-range position within the resolution ell. The variane of the slant-range position estimated from the LPS exploiting N ERS images and M Envisat images an be

3 PERISSIN et al.: VALIDATING THE SAR WAVENUMBER SHIFT PRINCIPLE WITH ERS ENVISAT PS 345 expressed from the first term in 8) as a funtion of the phase dispersion in the different aquisitions and is given by σ r = ) ) σ φers 4π f N + σ φ Envisat. 9) M Aording to 9), with N =60and M =10, a PS with multiinterferogram oherene γ =0.8 γ = e σ φ / [3]) an be positioned with unertainty of about 0 m. The LPS an be onneted to the slant-range satterer position estimated through the peak of the radar return amplitudes. A PS with a given radar ross setion RCS an be positioned in slant range, from N arhived ERS SAR images, with the following unertainty see the Appendix): σ r 3N σ0 A RCS 10) where σ 0 is the baksattering oeffiient of the lutter surrounding the PS, A is the area of the ERS resolution ell, and is the ERS slant-range resolution. In ase of urban areas and ERS typial values RCS = 500 m, A = 15 m, σ 0 =0 db, N =60, and =9 m), we get σ r 30 m, whih is omparable with what was obtained from the LPS. B. Impat of the Physial Extension of the Target Let us onsider the same interferogram desribed at the beginning of the previous setion. Whereas pointwise targets are seen oherently by definition in both images, a distributed satterer may lose oherene when observed under a different aquisition geometry. To quantify the phenomenon, we derive the expression of the target oherene as a funtion of the frequeny shift f, the inidene angle differene θ, and the squint angle variation ψ between the two images. The oherene is obtained as the absolute value of the normalized ross-orrelation oeffiient between the Master image S M and the Slave S S. The oherene has the usual well-known form [11], but in this ase, it depends also on a frequeny hange, i.e., E [S M SS γ f, θ, ψ) = ] E [ SM ] E [ S S ]. 11) The distributed target is modeled under the first-order Born approximation as the sum of a high number of elementary point targets with no mutual interation on an absorbing bakground. The omplex refletivity value s of the elementary point satterers is desribed as independent identially distributed zeromean omplex Gaussian variate that is fully defined by its variane σ. For the sake of simpliity, we analyze a single distributed target formed by many elementary point satterers uniformly distributed into a ubi box with sides L r, L x, and L y in slant range, ross slant range, and azimuth, respetively Fig. 3). Fig. 3. Distributed satterer model in the Master aquisition geometry. The target is formed by many elementary point targets uniformly distributed into a ubi box with sides L r, L x,andl y in slant range, ross slant range, and azimuth, respetively. The pixel omplex value of the Master image S M in orrespondene to the distributed target an be approximated as the sum of the elementary refletivities, whih is expressed as S M Lr Lr Lx Lx sr, x, y)e j 4π f 0 f)r M drdxdy 1) where r M = R 0 + r is the distane between the sensor and the satterers, whih is loalized by the oordinates r, x, and y. Here, the onboard filters both in time/slant range and azimuth) weights have been assumed onstant along the resolution ell to keep mathematis as simple as possible. Suh an approximation does not hange so muh the final result beause we assume that the distributed target extension is smaller than the resolution ell. Similarly, for the Slave, we have S S Lr Lr Lx Lx sr, x, y)e j 4π f 0r S drdxdy 13) where the sensor-targets distane r S = R 0 + r x θ y ψ takes into aount the different geometries. The expeted value of the ross-orrelation oeffiient an be alulated, observing that all ross-produts between different satterers are zero for the refletivity independeny, by E[S M SS] E Lr Lx Lr Lx Lr Lx Lr Lx =σ e j 4π fr 0 Lr Lx E sr, x, y)e j 4π f 0 f)r M drdxdy... Lr Lx s r,x,y )e j 4π f 0r S dr dx dy e j 4π r f+f 0x θ+f 0 y ψ) drdxdy. 14)

4 346 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 9, SEPTEMBER 006 Realling the Fourier transform of a retangle e jπξτ dτ = sinπ ξ) π ξ 15) and taking the normalized absolute value of 14), the oherene expression 11) beomes γ f, θ, ψ) sin πl r f πl r f ) ) ) f sin πl 0 θ f x sin πl 0 ψ y f πl 0 θ f x πl 0 ψ y. 16) Exploiting the relations found in the previous setion, we an write the oherene as a funtion of frequeny shift f, normal baseline B n, and DC frequeny differene f DC, i.e., Fig. 4. ERS Envisat interferogram on a rural area inluding the rivers Reveillon and Yerres near Paris haraterized by a normal baseline of 1500 m and a temporal baseline of 30 min. High oherene is found even in vegetated areas of the interferogram. γ f,b n, f DC ) ) f sin πl r πl r f f sin πl 0 B n x R 0 ) πl x f 0 B n R 0 sin πl y f DC δ az PRF πl y f DC δ az PRF ). 17) Equation 17) sums up in a simple form the omplex phenomenon of geometrial deorrelation in the 3-D spae. A pointwise target L r = L x = L y =0)is not affeted by geometrial deorrelation γ =1)even for high values of f, B n, and f DC. A target with finite dimensions on the ontrary is oherently observed only under a limited range of view angles. If the view angles overome suh limits, the target progressively loses oherene. On the other hand, given the distribution of aquisition parameters f, B n, and f DC ), a target must be smaller than a ritial threshold to be oherently seen by the sensor. The theoretial maximum size of a satterer an be easily obtained by alulating the values of L r, L x, and L y, making the argument of the ardinal sines equal to 1 γ =0). For ERS typial values entral frequeny f 0 =5.3 GHz and losest approah distane R km), we obtain numbers well known in SAR interferometry index means ritial) L x = R 0 f 0 B n 0 m with B n 1300 m 18) L y = δ azprf 4 m with f DC =1. 19) f DC PRF For the ERS Envisat oherent ombination, the 31 MHz of frequeny gap leads to L r = f 5 m with f =31MHz. 0) If ERS stable satterers were perfetly pointwise, we should expet to observe them oherently also under the ERS Envisat frequeny shift 100% of survival rate). For targets distributed in a volume, loss of oherene, and onsequently, lower ERS Envisat PS s survival rate is expeted. Fig. 5. Surfae sattering model in the Master aquisition geometry. a) Target of extensions L y and L w lies on the yw plane; the β angle is between the w axis and x axis. b) Angle β an be expressed as a funtion of the inidene angle θ and the target inlination with respet to the ground α: β = θ α. Fig. 4 shows an ERS Envisat interferogram on a rural area inluding the rivers Reveillon and Yerres near Paris haraterized by a normal baseline of 1500 m. It is worth noting that the ERS Envisat interferogram in Fig. 4 shows high oherene even in vegetated areas. This example suggests that the main sattering mehanism in the C-band annot be volumetri. In fat, as stated in 17), a volume target distribution would have generated muh lower oherene. The solution has to be found in a different sattering mehanism. In the following subsetion, we analyze the impat of a arrier frequeny shift on distributed surfae targets. C. Wavenumber Shift Priniple The ase of flat surfae sattering is a degeneration of volumetri sattering and deserves a separate analysis. The assumptions for the target model are the same as desribed for the ubi ase at the beginning of the previous subsetion, but the elementary satterers are uniformly distributed on a planar surfae of extensions L y and L w as shown in Fig. 5a). The target lies on the yw plane, where the w axis is rotated of an angle α from the horizontal [Fig. 5b)]. The expression of the oherene 11) as a funtion of the aquisition geometry an be omputed as previously done, taking into aount the surfae target distribution. The pixel omplex value of Master and Slave images are obtained as the

5 PERISSIN et al.: VALIDATING THE SAR WAVENUMBER SHIFT PRINCIPLE WITH ERS ENVISAT PS 347 double integral of the omplex refletivity along the diretions w and y, i.e., S M S S Lw Lw Lw Lw sw, y)e j 4π f 0 f)r 0 +r) dwdy 1) sw, y)e j 4π f 0R 0 +r x θ y ψ) dwdy. ) The r and x oordinates have to be expressed as a funtion of the oordinate w. Posing β = θ α Fig. 5), we have r = w sin β and x = w os β. Thus, the ross-orrelation oeffiient between the Master image S M and the Slave S S beomes E [S M SS] σ e j 4π fr 0 Lw E e j 4π w sin β f+f 0w os β θ+f 0 y ψ) dwdy. 3) Lw Following the same omputational steps desribed previously, we obtain the following expression of the oherene of a surfae satterer: [ sin πlw γ f,b n, f DC ) πl w f 0 B n R 0 f 0 B n R 0 sin )] os β f sin β ) os β f sin β ) f πl DC y δ az PRF. 4) πl y f DC δ az PRF Equation 4) is the produt between a ardinal sine that depends on the target azimuth dimension L y already seen in the volume sattering ase) and a new ardinal sine relative to the seond dimension L w. The argument of the new ardinal sine in 4) shows that there is a partiular baseline that ompensates for the frequeny shift the spetral shift priniple [5]): f 0 B n /R 0 )osβ f sin β =0. In this ase, the satterer extension does not introdue any deorrelation. The baseline B n that ompensates for a frequeny shift f is a funtion of the target slope α and inidene angle θ, i.e., B n = f f 0 R 0 tanθ α). 5) In ase of the ERS Envisat frequeny gap, the ompensation baseline for flat horizontal terrain is about km, whih is very lose to the baseline value of the interferogram in Fig. 4. Thus, surfae sattering is at the origin of the high interferogram oherene in Fig. 4. Equation 4) is an alternative formulation of the wavenumber shift priniple [5] where the target size has been left variable. Fig. 6. PS example. Top) The PS loation in Milan is identified with a white star on an airborne optial image. Bottom) PS time series. Eah marker of the series refers to a displaement measure derived from ERS-1 triangles), ERS- dots), and Envisat stars) at the date indiated on the absissa. Two adjaent replia ±8 mm) are plotted to eliminate ambiguities. III. EXPERIMENTAL RESULTS To assess the theoretial framework disussed in the previous setions, we arried out a PS analysis on a SAR data set inluding 8 ERS and 1 Envisat images aquired over Milan, Italy Trak 08, Frame 691). The test area overs about 400 km. The ERS image aquired on January 14, 1997 is seleted as the master sene, and all the slave aquisitions ERS and Envisat) are resampled on the ommon master grid, taking into aount the different PRFs and sampling frequenies in the range diretion. The PS proessing follows the desription reported in [1], i.e., adopting a two-step strategy. First, PS andidates PSCs) are seleted by means of the amplitude stability index [1]. For eah PSC, four phase terms dependent on the relative elevation, the Doppler differenes, the frequeny shift, and a possible onstant relative veloity are jointly estimated. Phase residuals should then depend only on atmospheri effets, noise, and possible phase omponents due to the nonuniform motion of the targets. The atmospheri phase sreen APS) of eah interferogram is estimated from the phase residuals exploiting its statistial behavior [1] and is then removed from the data. Finally, PSs are deteted on a pixel-by-pixel basis [1]. The multiinteferogram oherene [3] of eah PS is omputed on the ERS and Envisat data sets separately to estimate their survival rate when the radar operating frequeny hanges. In Fig. 6, an example of time series is shown, together with the georeferened position of the satterer on an aerial photo. A. ERS Envisat Continuity In Table I, the main results of the oherene analysis and PS survival rate are reported. The number of PSs having ERS

6 348 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 9, SEPTEMBER 006 TABLE I SURVIVAL RATE FOR DIFFERENT COHERENCES.FOR EACH THRESHOLD VALUE FIRST COLUMN), NUMBER OF POINTS WITH ERS COHERENCE N =8IMAGES)GREATER THAN THE THRESHOLD SECOND COLUMN) AND NUMBER OF POINTS WITH BOTH ERS AND ENVISAT M =1 IMAGES)COHERENCE GREATER THAN THE THRESHOLD THIRD COLUMN)ARE GIVEN.FOURTH COLUMN:PERCENT RATIO Fig. 8. Density of points of the satter plot between the ERS Envisat LPS in radians) y axis) and the PS position in meters) x axis) extrated from the peak of the refletivity map. For visualization purposes, the satter plot has been repliated both on the vertial π) and horizontal axes 5 m). Graysale indiates the data density. Fig. 7. ERS triangles) and Envisat dots) PS oherene histograms of the analyzed points. oherene estimated on 8 ERS images) greater than the value indiated in the first olumn is reported in the seond olumn. The third olumn reports the number of PSs showing oherene greater than that indiated in the first olumn on both ERS 8 images) and Envisat 1 images) time series. The PS survival rate is shown in the last olumn of Table I as the perent ratio between the seond and third olumns, and it tells the perentage of PSs that maintain the oherene over a ertain threshold passing from ERS to Envisat. More than 60% of PSs with ERS oherene greater than 0.7 have the same stability in Envisat data, whereas by inreasing the threshold to 0.95, the perentage beomes about 40%. The auray of the estimate of the multiinterferogram oherene is inversely proportional to the number of images [3]. The Envisat oherene is thus less aurate than the one alulated in ERS aquisitions. Notwithstanding, the PS survival rate an be onsidered reliable, beause it is estimated as the ratio of a high number of points. A more detailed analysis of the PS s survival rate is shown in Fig. 7. Here, the ERS oherene histogram triangles) of PSs showing oherene greater than 0.7 is ompared with the Envisat oherene histogram dots) of the same PSs. In Fig. 8, we show the density of points of the satter plot between the estimated LPS in radians y axis) and the slantrange position of the analyzed PSs expressed in meters x axis). The PS slant-range position has been obtained by extrating the peak of the refletivity map i.e., the inoherent time average of the images; see the Appendix) in orrespondene of the PS. As seen in the previous setion [see the first term of 8)], the LPS depends linearly on the satterer slant-range loation with a π phase ambiguity orresponding to 5 m of slant-range ambiguity. For visualization purposes, the satter plot has been repliated both on the vertial π) and horizontal 5 m) axes. The good orrelation found dispersion about 1 m) is a proof of the loation apability of the PS tehnique. B. Validation of the Wavenumber Shift Priniple The results on the survival of PSs when passing from ERS to Envisat data an be used for validating the wavenumber shift priniple. As seen in Setion II-C, the spetral shift priniple states an equivalene between a arrier frequeny slight hange and a variation of the normal baseline i.e., the off-nadir angle). This equivalene leads to the expression of the normal baseline that ompensates for the ERS Envisat frequeny gap f = 31 MHz 5). The ompensation baseline for flat horizontal terrain is about km note that, here, the baseline is derived with respet to the master aquisition, and therefore, it has a sign and an be negative). This means that at the ompensation baseline, Envisat observes a distributed satterer as it would have been observed by ERS at zero baseline. In this framework, when the ERS orbit is assumed as a referene, it is useful to introdue an equivalent Envisat baseline, whih is shifted by km with respet to its atual value, aounting for the ERS Envisat frequeny shift. The amplitude of the radar returns of the ERS images as a funtion of the view angle normal baseline) orresponds to a portion of the baksattering radiation pattern of the radar target at 5.3 GHz the ERS operating frequeny). Following the spetral shift priniple, the amplitude of the radar returns of the Envisat images should then orrespond to a part of the PS radiation pattern shifted by km on the ERS baseline

7 PERISSIN et al.: VALIDATING THE SAR WAVENUMBER SHIFT PRINCIPLE WITH ERS ENVISAT PS 349 Fig. 9. Amplitudes versus baseline in meters). ERS-1 triangles), ERS- dots), and Envisat stars) data relative to a PS. Envisat data are shifted by km, whih ompensates for the frequeny shift. Dashed line) Cardinal sine fitting the data. Envisat values fall lose to zero. Fig. 10. Amplitudes versus baseline in meters). ERS-1 triangles), ERS- dots), and Envisat stars) data relative to a PS. Envisat data are shifted by km, whih ompensates for the frequeny shift. Dashed line) Cardinal sine fitting the data. Envisat values fall around the maximum. axis. In Figs. 9 and 10, two examples of the PS amplitudes as a funtion of the baseline are shown. In ase of the Envisat data, the equivalent baseline shifted by km) is used. It should be noted how the ERS PS radiation pattern is onsistently ontinued by the Envisat measurements. The portion of the radiation pattern illuminated by Envisat an be onneted to the PS multiinterferogram oherene in the Envisat data. In fat, whenever the PS radiation pattern seen by Envisat is lose to zero as in Fig. 9), the Envisat PS phase oherene is low. On the other hand, if Envisat looks around a loal maximum as in Fig. 10), Envisat data show high oherene. In Fig. 11, we show this behavior on the whole set of PSs analyzed. Assuming, for the sake of simpliity, that the PS an be modeled as a mirror, its radiation pattern as a funtion of the looking angle an be approximated with a ardinal sine. The Fig. 11. Envisat oherene dependeny on the target geometry width and pointing). x axis) Cross slant-range satterer width in meters). y axis) Cross slant-range pointing in meters) amplitudes mainlobe peak position along baseline). Graysale: Envisat oherene mean value. The data plotted are ERS PSs γ ERS > 0.7). White line: theoretial 3.1) PS geometry width and pointing) that generates a first zero of the sattering pattern in km. width of the ardinal sine is therefore inversely proportional to the spatial extension of the PS, whereas the portion of the radiation pattern illuminated by the radar gives information on its orientation. In Fig. 11, the x axis represents the target ross slant-range width L x whih is inversely proportional to the ardinal sine mainlobe width) estimated from the ERS data set. The y axis is the target orientation i.e., the ardinal sine mainlobe position on the baseline axis, expressed in meters) again estimated from the ERS data set. The zero baseline orresponds to the ERS Master aquisition, whih is hosen for minimizing the dispersion of the baselines. The graysale in Fig. 11 shows the average oherene estimated from Envisat data only on the PS set identified by the ERS data with γ ERS > 0.7. Blak areas in Fig. 11 zero oherene) show a lak of ERS PSs with suh width and pointing. The atual baselines of the Envisat images luster around the same ERS zero referene baseline. However, for a simpler analysis of this key figure, it is useful to imagine that they luster around the equivalent referene baseline loated at km, aounting for the ERS Envisat frequeny shift. Point satterers have a onstant radiation pattern, and the ardinal sine model assigns a zero width to them. They do not need a partiular orientation to be seen by the satellite. In Fig. 11, they appear uniformly distributed along the whole range of baselines, and they also have high oherene in Envisat data despite the frequeny shift. When the estimated ross slant-range width inreases, the target must be pointed toward the master ERS aquisition to be identified as an ERS PS. This theoretially predited behavior for distributed targets is onfirmed by the experimental data in Fig. 11, where ERS PS distribution shrinks around the zero baseline for higher values of the ross slant-range width. Moreover, a darker low Envisat oherene) urved area is learly visible in Fig. 11 going from the upper left to the middle right of the graysale image. This interesting feature deserves a detailed analysis beause it represents the experimental validation of spetral shift priniple applied to SAR missions with

8 350 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 9, SEPTEMBER 006 Fig. 1. Sattering pattern examples of targets with a geometry width and pointing) that generates a first zero in km. Theoretial width and pointing of sattering patterns with a zero in km lie on the blak dashed line 6). slightly different arrier frequenies. Let us onsider a lass of PSs whose width L x estimated from the ERS data set is, say, 8 m. The Envisat average oherene of this lass of PSs is represented by the gray level of the small square pixels aligned along a olumn in Fig. 11. The vertial oordinate shows the estimated pointing angle whih is expressed as the normal baseline with respet to the referene ERS orbit) of the strongest target baksattering i.e., the angular orientation of peak of the ardinal sine baksattering pattern). As already pointed out, the atual Envisat orbits luster along the same ERS referene orbit namely the baseline 0 in Fig. 11); however, they should be shifted by km to take into aount the frequeny shift with respet to ERS. Thus, high Envisat oherene is expeted from targets with their strongest baksattering loated around km, whereas low oherene is expeted from targets whose baksattering radiation pattern shows a zero at km.the first zero of the baksattering pattern is obtained by B n,zero = R ) f 0 L x Getting bak to our example, for an L x =8mwidth target, the first zero is at about 3000 m. In our theoretial model, a low Envisat oherene value is thus expeted at those pixels loated along the 8-m-wide olumn in Fig. 11 about 3000 m above km. The same an be done for every target width as visualized in Fig. 1, thus identifying the urve white in Fig. 11 and blak dashed in Fig. 1) where low Envisat oherene is expeted. The experimental data of Figs. 11 and 1 learly validate the theoretial model. In Fig. 13, the histogram of the estimated ross slant-range widths of the analyzed PSs γ ERS > 0.7) is reported. The amplitude analysis shows that less than 0% of PSs an be onsidered as point satterers the estimated ross slant-range width is less than 50 m). The other urban radar targets are more omplex objets that behave as distributed, even if with widths that do not fill the resolution ell. Their Envisat oherene an thus be predited in first approximation by looking at the portion of their refletivity at the Envisat frequeny. The study on the eletromagneti harateristis of SAR urban Fig. 13. Histogram of the estimated ross slant-range widths of the analyzed PSs γ ERS > 0.7). targets deserves muh more analysis, whih will be overed in another paper. IV. CONCLUSION In this paper, we have presented the theory of oherent stithing of PS phase histories deteted by ERS and Envisat haraterized by slightly different arrier frequenies. The results obtained over Milan ombining 8 ERS and 1 Envisat images with the PS tehnique validate the theory and demonstrate that the ESA ERS C-band arhive an be ontinued with the new Envisat aquisitions. Envisat ASAR an be usefully exploited to ontinue the displaement time series of more than 60% of ERS SAR PSs, despite the frequeny gap between the two radars. It has been shown that a single PS data set an be formed using ERS and Envisat images with a ommon ERS master image. Moreover, the joint analysis of ERS and Envisat surveys adds new insight on the physial nature of the PS. APPENDIX The PS slant-range position an be extrated from the peak of the radar return amplitudes. The problem an be simplified as the estimate of the peak of a ardinal sine in the presene of additive white Gaussian noise. For the sake of simpliity, we take a normalized ardinal sine entered around the origin as Sr) = πr sin πr + n A1) where is the system range resolution and n is the white Gaussian noise with variane σ n. The peak is identified by the zero of the first derivative. Exploiting the Taylor series expansion of seond order, the ardinal sine around the origin is approximated as sin πr πr 1 πr 6ρ rg A)

9 PERISSIN et al.: VALIDATING THE SAR WAVENUMBER SHIFT PRINCIPLE WITH ERS ENVISAT PS 351 and thus S r) π r 3ρ + n. A3) rg The solution is found posing S r) =0, i.e., r = 3 π n. A4) We are interested in the dispersion of the loation estimate defined by σ r = 3ρ rg π ) E [ n ) ] A5) and, thus, we need to alulate the power of the noise derivative as follows: E[n ) ]= = 1 ρrg 1 ρrg 1 ρrg 1 ρrg S n k r )dk r σ n jπk r dk r = π σn 3ρ. A6) rg Substituting A6) in A5), we obtain σr = 3 π σ n. A7) Beause we took a normalized radar response, the noise power an be expressed as the noise-to-signal ratio defined by σn = σ 0A A8) RCS where σ 0 is the baksattering oeffiient of the lutter surrounding the PS, A is the area of the ERS resolution ell, and RCS is the PS radar ross setion. Exploiting N arhived SAR images, the slant-range satterer position unertainty is then given as follows: σ r 3N σ0 A RCS. ACKNOWLEDGMENT A9) The authors would like to thank ESA for the Envisat and ERS data provided under ESRIN Contrat 16564/0/I-LG) as well as the whole Tele-Rilevamento Europa staff for proessing ERS and Envisat images. REFERENCES [1] A. Ferretti, C. Prati, and F. Roa, Permanent satterers in SAR interferometry, IEEE Trans. Geosi. Remote Sens., vol. 39, no. 1, pp. 8 0, Jan [], Non-linear subsidene rate estimation using permanent satterers in differential SAR interferometry, IEEE Trans. Geosi. Remote Sens., vol. 38, no. 5, pp. 0 1, Sep [3] C. Colesanti, A. Ferretti, F. Novali, C. Prati, and F. Roa, SAR monitoring of progressive and seasonal ground deformation using the permanent satterers tehnique, IEEE Trans. Geosi. Remote Sens., vol. 41, no. 7, pp , Jul [4] A. Ferretti, C. Prati, and F. Roa, Requirements for a spae mission for DInSAR and PS analysis based on past and present missions, in Pro. IGARSS, 004, vol. 3, pp [5] F. Gatelli et al., The wavenumber shift in SAR interferometry, IEEE Trans. Geosi. Remote Sens., vol. 3, no. 4, pp , Jul [6] C. Colesanti, A. Ferretti, D. Perissin, C. Prati, and F. Roa, ERS Envisat permanent satterers interferometry, in Pro. IGARSS, 004, vol., pp [7] A. Ferretti, D. Perissin, C. Prati, and F. Roa, ERS Envisat permanent satterers, in Pro. IGARSS, 004, vol., pp [8] M. Arrigoni, C. Colesanti, A. Ferretti, D. Perissin, C. Prati, and F. Roa, Identifiation of the loation phase sreen of ERS Envisat permanent satterers, in Pro. FRINGE, Frasati, Italy, De. 1 5, 003. ESA SP-550, Jan [9] R. F. Hanssen, Radar Interferometry. Data Interpretation and Error Analysis. Dordreht, The Netherlands: Kluwer, 001. [10] G. Shreider, Ed., SARGeooding: Data and Systems. Karlsruhe, Germany: Wihmann, [11] H. A. Zebker and J. Villasenor, Deorrelation in interferometri radar ehoes, IEEE Trans. Geosi. Remote Sens., vol. 30, no. 5, pp , Sep [1] S. Williams, Y. Bok, and P. Pang, Integrated satellite interferometry: Tropospheri noise, GPS estimates and impliations for interferometri syntheti aperture radar produts, J. Geophys. Res., vol. 103, no. B11, pp , Daniele Perissin was born in Milan, Italy, in He reeived the master s degree in teleommuniations engineering and the Ph.D. degree in information tehnology from the Politenio di Milano, Milan, in 00 and 006, respetively. He is urrently working in the researh group of Prof. F. Roa and Prof. C. Prati at the Politenio di Milano. His main researh interest is in the field of SAR. Sine 00, he has been working on the permanent satterers tehnique. He is the holder of a patent on the use of urban dihedral refletors for ombining multisensor SAR data. Claudio Prati was born in Milan, Italy, on Marh 0, He reeived the laurea degree in eletroni engineering and the Ph.D. degree from the Politenio di Milano, Milan, in 1983 and 1987, respetively. He is urrently a Full Professor of teleommuniations in the Dipartimento di Elettronia e Informazione, Politenio di Milano. He is the holder of four patents on SAR image proessing. He has published more than 100 papers on SAR data proessing and interferometry. He is a Co-Founder of Tele- Rilevamento Europa, whih is a remote sensing spinoff ompany of POLIMI. Prof. Prati was awarded two prizes from the IEEE Geosiene and Remote Sensing Soiety IGARSS 89 and IGARSS 99). Marus E. Engdahl was born in Finland. He reeived the M.S. degree in tehnial physis from the Helsinki University of Tehnology HUT), Espoo, Finland, in He is an alumnus of the International Spae University, Strasbourg, Frane. From 1996 to 003, he was a Researh Sientist with the Laboratory of Spae Tehnology, HUT. Sine 003, he has been an EO Appliations Engineer in ESA-ESRIN, Frasati, Italy. In ESA, he has been responsible for oordinating R&D projets that deal with the observation of land and oean surfaes with SAR, the development of SAR-related software toolboxes, and the organization of international workshops. His main researh interests are in the land appliations of SAR interferometry. Mr. Engdahl was the main organizer of the Fringe 005 Workshop held in Frasati, Italy. Y. L. Desnos, photograph and biography not available at the time of publiation.