Maximum Likelihood Multi-Baseline SAR Interferometry

Transcription

1 Maximum Likelihood Multi-Baseline SAR Interferometry G. Fornaro(*), A. Monti Guarnieri(**), A. Pauciullo(*), F. Rocca(**) (*) Istituto per il Rilevamento Elettromagnetico dell Ambiente (IREA), Consiglio Nazionale delle Ricerche (CNR), Via Diocleziano 38, 824 Napoli Italy, (**) Dipartimento di Elettronica e Informazione Politecnico di Milano Piazza Leonardo da Vinci 32, 233 Milano Italy, monti@elet.polimi.it. May 7, 25 Abstract We propose a technique to provide interferometry by combining multiple images of the same area. This technique exploits all the images jointly and performs an optimal spectral shift pre-processing to remove most of the decorrelation for distributed targets. It s applications are mainly for DEM generation at centimetric accuracy, and for differential interferometry. The major requirement is that targets are coherent over all the images: this may be the case of current multi-pass over desert areas, or better the case of images coming from future short revisit time systems (constellations, cart-wheel, geosynchronous SAR etc.). Introduction Interferometric SAR (InSAR) surveys have been exploited for terrain mapping and DEM generation since more than two decades. The enhanced performances that would have been achieved by combining multiple interferometric acquisitions have been studied in the early papers in SAR interferometry [], even though a large amount of literature has been built up since the early sixties for holographic applications. In papers [2], [3] a combination was proposed of many interferograms by exploiting techniques similar to the Chinese remainder theorem to reduce phase unwrapping problems. A Maximum Likelihood (ML) approach was proposed in [4] for solving the same problem, applied to Tandem ERS acquisition. In that case, the problem was made simple as only pairs of interferogram, assumed each one independent upon the other, were combined. The same technique was extended to the case of 3 images by exploiting jointly all the information in [5], or in the case of multi-frequency acquisitions in [6]. In all these papers, the model assumed for SAR reflectivity resembles a collection of point-scatterers. Then, in that case, a complete and assessed methodology has recently developed under the name of Permanent Scatterer [7]. On the other side, when the SAR reflectivity is modeled by a distributed target (speckle), these approaches becomes far from optimal, as they do not properly handle the frequency domain incorrelation of the sources. Indeed, it was shown in [8] that, a sensible increment in quality could be obtained in 2-pass interferometry if a proper prefiltering, tuned to the baseline and the terrain slope is exploited. In [9] and [] this spectral shift (SS) filtering was extended to the case of non-constant slopes, and in [] this approach was shown to be optimal in the MMSE sense, and extended to the combination of two images coming very different SAR modes (ScanSAR, SPOT etc.). The purpose of this paper is to extend such approach to a multi-channel environment, where multiple acquisitions are made available either with different frequencies or with different baselines. The purpose is to establish This work has been committed under ASI contract I/R/69/2 Multiple pass SAR Interferometry for very accurate Earth modeling 23-24

2 a theoretical framework based on a Maximum Likelihood estimate, in which the best estimate of the underlying geometry, linked to the terrain topography, is provided. Although we focus mainly on DEM generation, we will show that the approach is equally suited to retrieve motion fringes in multi-temporal acquisition, thus for mapping landslides, earthquakes etc. The major scope of the paper is to jointly exploit the information provided by distributed targets. Hence, the technique would give the best result with a set of coherent images. This would be the case of long temporal baselines in long term correlated areas (they should be non-vegetated in C or X band surveys), or using short temporal baseline. For such reasons the main applications will be framed in the next L band missions (ALOS, TERRASAR-L), or in the forthcoming constellations (Cartwheel etc.). 2 Multi-Channel model Let us refer to the multibaseline geometry in Fig., an extension of the conventional SAR Interferometry [2, 3] to the multi channel case. In the figure, the target P corresponds to a distributed scatterer, that is imaged using three different positions. Let us define the normal baseline B n as the distance between the master and the slave s track, measured normal to the slant range, azimuth plane. In the case shown in the picture, we could compute up to three interferograms, as the Hermitian products between two images, all of them coregistered in the same master reference. The contribute of the target P to the interferogram s phase would be proportional to the travel path difference between the two acquisitions ϕ = 4π λ (R s(p ) R M (P )) () where R S and R M are respectively the slave-target and the master-target travel path. Eventually we approximate the interferogram phase as a slow varying linear term[4]: θ φ(t) = f tan(θ α(t)) t (2) B n f R tan(θ α(t)) t (3) α(t) being the local slope (range-varying), R the closest approach, f the carrier frequency, and θ the look angle difference. The approximation in (3) holds for range intervals narrow enough to assume θ constant. In (3) the interferogram phase scales linearly both with the B n and with f, therefore we may get different interferograms either by exploiting multi-baseline or multi-frequency systems. In this paper we will refer to both cases as multi-channel, however we will address mostly the multi baseline case, that is much more common in spaceborne SARs. Let us assume to have N channels, including one master and (N ) slave images, and M samples (range bins) from each acquisition. Following reference [], we express each single acquisition as a filtered version of the wide-band reflectivity: y i (P (t)) = γ(t) exp (jφ(t)) + w i (t) (4) We limit our study to the D case, as the slant range, or fast-time, t, is the direction mostly affected by phase aliasing and baseline decorrelation due to the geometric deformations in the ground-to-range mapping [5]. In (4), γ is the source reflectivity, y i the focused signal of the i-th acquisition, and w i a noise source that cumulates all the decorrelation terms [6]. In order to proper process the data, we need to up-sample the received signal, so that the final range sampling would accommodate for all the possible spectral shifts, e.g. by computing (3) for the largest baseline. After up sampling each channel in 4, we end up in a discrete-time model that can be expressed by the following vector formulation [], represented schematically in Fig. 2: y i = F i Φ i γ + w i = H i γ + w i (5) where we assume bold notation for matrixes and vectors. In particular, we assume that the impulse response on each channel is of length L, and we to have N channels, and M samples out of each channel. We require then M+L- samples of the source. The vectors and matrixes involved in (5) are the following: 2

3 S B n B n2 Master S 2 P Figure : Multi-baseline Interferometric SAR: a master and two slave sensors (may be the same sensor following the same orbit) are shown. 3

4 y i is the column vector [M,] that corresponds to the complex SAR image, the data, coregistered in the reference of the Master: y i = [ y y M ] T where the suffix T stands for Hermitian matrix transposition, and we assume row and column indexes starting from. γ is the column vector [D,] that represents the source reflectivity. This is white over a large band. Φ i is a diagonal modulation matrix, [D,D] that express the topographic-dependent (or motion dependent) contributions: φ i ()... Φ i = φ i () φ i (D ) its element on the diagonal being just the sampled version of the topographic-dependent phase, φ(t) in 3. Notice that for different channel the phases simply scales according to the baseline. Henceforth, we will gather all the elements on the diagonal of Φ i on a vector of unknown parameters (to be estimated): { } φ = f R tan(θ α(t i )) t i (6) t i being the sampling times. φ is computed for a normalized Baseline (B n = ). The model further simplifies when we assume first that the terrain has constant slope. In that case there is only parameter to retrieve: φ(α). F i is the filter matrix [M,D] where D=M+L-, that is Toeplitz and contains the impulse response of the equivalent SAR end-to-end channel (that can be well approximated by an ideal bandpass), f L f L... f F i = f L f f L f H i is a matrix [M,D] H i = F i Φ i (7) that represents the channel to be estimated. Notice that the channel is now linear, but space-variant due to the modulation matrix Φ i, hence, H i is not block Toeplitz as usually assumed in literature; w i is the noise in the specific acquisition, it has the same size as y i. The model (5) frames in the Single-Input-Multiple-Output (SIMO) channel blind estimation, where the unknown channels, H i are to be retrieved from the outputs, y i, given only a few information on the inputs. Such model is widely studied in literature, as it is shared by many fields: estimates of Direction Of Arrivals (DOA), wireless cellular networks, tomography, etc. The reader could refer to paper [7] for a summary on blind deconvolution techniques for SIMO problems. Following the trend in literature, we provide a compact matrix formulation for the model (5), where the N input vectors, the N outputs and the N channel matrixes are stacked one upon the other: where: y = Hγ + w (8) y = [ y T... yi T... yn T w = [ w T... wi T... wn T ] T ] T 4

5 y and w are then column vector of size [NM,]. The channel matrix H in (8) is also a block matrix of size [NM,M+L-]: h H =... h i h N = φ ()f L φ (D )f M+L... φ ()f... h i h N w Φ F. γ Φ i F i. Φ N- F N- x w i x i w N- x N- y y i y N- Figure 2: Multi-Baseline Interferometric SAR system, modeled as a SIMO system. 3 ML estimate The straight and simplest derivation of the Maximum Likelihood estimate of the topographic dependent contributes, the vector φ, starts from the observation that the data, y, belong to a mutlivariate Normal process, therefore the Log-Likelihood maximization result in the following expression: ( )) φ ML = arg max log f (y; φ i bφ i = arg min ( y C y y ) (9) The inverse covariance matrix in (9) accounts for both the superimposed noise, w, and the so-called baseline or geometric decorrelation, that may arise when different spectral components of the same source, γ are imaged by the different channels. The covariance matrix C y, in (9) can be expressed as the following block notation: E [yy ] E [y y]... E [ ] y N yn E [yy ] E [yy ] E [yi y i]... C y = E [ yn y ] E [ yn y ] N () 5

6 Each block can be estimated basing on the model (8) and (7) (see Fig. 2): E [y i y i ] = F i φ i C γ φ i F i + C wi () E [y i y j ] = i j F i φ i C γ φ j F j + E [w i w j ] (2) = F i φ i C γ φ j F j (3) where C wi is the covariance matrix of the noise term in the i-th channel, and where we have introduced the definition: C γ = E [γγ ] The need of the covariance matrix inversion makes the ML estimator unfeasible, but for simple cases requiring one or two parameters to be retrieved and involving few images. The most interesting case could be the estimate of slopes basing on a very local scale (a few pixels) in a conventional two channel interferometric system [8]. 4 DML estimate The Deterministic Maximum Likelihood (DML) method approaches the same problem by including in the vector of parameters the samples of the source, γ [7], [9]. This choice provides a simpler solution, at the price of increasing the number of parameters to be estimated and potentially worsening the accuracy of the estimates. The problem is linear in both the channel and the sources, therefore we can cascade the optimizations with respect to γ and φ: ( ( { φdml, γ} = arg min min (y Hγ) C (y Hγ) ) ) (4) φ b bγ where C is the noise covariance matrix. Eventually we define the SIMO channel covariance: We first minimize with respect to the sources: R H = H C H = Φ F C FΦ (5) that leads to the solution: min bγ ( y C γh + H γ C y H γ C γh ) (6) γ DML = R H H C y = R H Φ F C y (7) where R H is the pseudo-inverse of the channel covariance. We do not need the explicit computation of γ DML if we are not interested, we just need to substitute its expression back into (4), to retrieve the estimate of φ: where P H is a projector in the norm of the channel H : φ DML = max (y P H y) (8) P H = C HR H H C (9) = C FΦR H Φ F C (2). The DML estimate φ DML appears as complex as the ML approach in (8), as one matrix inversion is required to compute R H at each iteration. However, a sub-optimal implementation is known as the Two-Step ML (TSML), introduced by Hua in [9]. This approach makes a first iteration by assuming that the channel covariance matrix is diagonal: R H = I. In that case the DML estimate would be ( φ DMLI = max y C FΦΦ F C y ) (2) bφ 6

7 this leads to an estimate of φ DMLI that is shown to be optimal as for SNR [9]. An approximate value of R H is computed by combining (5) with the first-step estimate φ DMLI in (2). Thereafter, the final estimate φ DML is retrieved from (2). In a similar way, in papers [] and [] a preliminary information of the geometry and the topography was used to derive the proper whitening operator. 5 Single baseline interferometry Let us compare the two estimates of the topographic-dependent phase in the simplest case, the single baseline interferometry. The model of the Interferometric SAR channel is represented in Fig. 3.a, whereas Fig. 3.b draws the linear estimate of the parameters φ. The linear estimate is justified by the Normal statistics of both sources and noise. We will however derive it, and show that, by a proper selection of the coefficients, the same scheme in Fig. 3.b can represent either the ML or the DML approach. In the figure, we have conventionally attributed half of the interferometric phase, Φ /2, to the master, and the other half to the slave. Furthermore, we changed the conventional SIMO model (see Fig. 2) by moving the noise contribution in front of the blocks F (that represents the SAR acquisition and focusing overall transfer function). This choice is closer to the actual SAR acquisition. However, the two models are both correct, provided that the proper noise covariance matrix is used. w Master Slave synthesis γ Φ F y H w 2 Slave G 2 y 2 y 2 Φ 2 F 2 y 2 G 2 y 2 y 2 H 2 (a) Δφ Master synthesis (b) Figure 3: Single baseline interferometry: (a) the acquisition model, and (b) the linear estimator of the interferometric phase. This estimator applies for both the ML and the DML technique. 5. DML estimate Let us first find a suitable approximation of the inverse of the noise covariance matrix involved in the DML estimate (4). Let us assume the same kernel for both channels, F = F = F that is usually the case in multi-pass interferometry. The model in Fig. 3.a allows us to express the noise covariance as the following Kronecker product: [ ] σ 2 C = σ 2 R F where σ 2 and σ 2 are the noise variances in the two channels, and R F expresses the channel cross-correlation. Eventually we take advantage of the inversion expression for Kronecker products: (A B) = A B 7

8 that leads to the inverse of the channel covariance matrix: [ ] C σ = σ R F [ σ = R F σ R F ] = [ ] C C Having expressed the inverse noise covariance as a block matrix, with two non null blocks, we may write the DML estimator, (8), (2), as follows: where the channel covariance matrix is: M = y C H R H H C y + y C H R H H C y (23) (22) + y C H R H H C y + y C H R H H C y (24) R H = H C H = σ H R F H + σ H R F H (25) 5.. TSML: Fist Step Let us now follow the TSML approach by providing a first estimate of φ by assuming R H = I. The two addends in (23), involve auto-correlations of each of outputs separately, therefore are blind to the interferometric phases and can be dropped. In fact the following expression holds for both i = and i = : y i H i H i y i = y i F i Φ /2 Φ /2 F i y i = y i F i F i y i that no longer depends in φ. The first step of the DML is then the maximization of the summation in (24): M = y C H H C y + y C H H C y (26) = 2 Re(y C H H C y ) (27) = 2 Re(s s ) (28) According to the scheme in Fig. 3.b, we interpret the DML (st step) as the maximization of the cross-energy of two signals, s and s obtained by applying the kernels: G = H C = σ Φ/2 F R F (29) G = H C = σ Φ /2 F R F (3) to the outputs y and y. We may assume that the channel kernels have no phase bias, i.e. F and F are Toeplitz, hence the products P = F R F and P = F R F will still be two Toeplitz matrixes, P and P. We can now derive the DML estimate form (27): φ DMLI = arg max (Re (yp ΦP y )) ( = arg max Re ) exp(jφ n )yp P y n (3) that is maximized by imposing: φ DMLI (n) = arg(y(n)y (n)) = arg I (32) I being the complex interferogram. Not surprisingly, the first step of the DML corresponds to the usual approach, that retrieves the phase, as the argument of the complex interferogram. Notice that this approach is 8

9 optimal as SNR. In the case of constant sloped terrain (3), would be the Fourier Transform of the complex interferogram, evaluated at the frequency φ n /(2π), and the maximization of the likelihood corresponds to the conventional periodogram-based frequency estimation TS-DML: Second Step The second step is performed by maximizing the likelihood: M = 2 Re(y C H R H H C y ) (33) where R H is the pseudo inverse of the channel autocorrelation (25) computed basing on the parameter φ DMLI from (32). Apart from this term, we would get the same result as in (32). The linear estimator in Fig. 3.b still holds, but we now need to modify (29,3) by including R /2 H to both terms. In order to have a reasonable idea of the role played by the term R H, let us assume () a constant sloped terrain, and (2) a stationary case. The system can now be studied as in terms of linear, time variant convolutions. The forward model in Fig. 3.a becomes: y (t) = {γ(t) exp( jωt) + w (t)} f (t) (34) y (t) = {γ(t) exp(jωt) + w (t)} f (t) (35) that is then sampled at a frequency consistent with the bandwidth of the on-board filters, f (t) and f (t). The frequency domain representation of the spectra Y (f) and Y (f) are shown in Fig. 4 (the two upper plots). Figure 4: Single baseline model in the frequency domain, for a constant sloped terrain. Notice the spectral shift, f, that depends on the local slope and the baseline. We have assumed ideal boxcar bandpass spectra, centered on the carrier frequency. Under this assumption, the autocorrelation of the two channels will have the same bandpass shape, then we will approximate the inverse noise covariance (22) as follows: [ C σ = F σ ] (36) 9

10 Eventually, we get from (25): R H = σ ( ) ( Φ /2 F F F F Φ /2) + σ ( ) ( Φ /2 F F F F Φ /2) (37) In the constant slope case, the inverse kernel R H will perform a whitening, e.g. the equalization in the frequency domain, shown in Fig. 4, last plot. Even if in this specific case, we do not expect any difference with the result achieved at the first step, this equalization would be effective in the Multi-Baseline (SIMO) environment, when many channels could contribute to the same spectrum. 5.2 ML estimate Let us now find a suitable approximation for the ML estimate in (9). We first introduce a matrix in the complementary space of the signal covariance: G = I C y, the ML is now the following maximization The inverse of the matrix G can be expressed as the block matrix: [ ] G G G = G G The ML estimate is then: φ ML = arg max ( y G y ) (38) Φ = arg max (y G y + y G y + y G y + y G y ) = arg max (y G y + 2 Re (y G y ) + y G y ) (39) The likelihood to be maximized is similar in topology (but non identical in weights) to the one in the DML estimate, (23). Instead of inverting the covariance matrix, we can directly derive the ML by finding the weights and G ij that maximizes (39). We observe that the solution is already known in the literature, in fact, in [] the linear model in Fig. 3.b: s = G y and s = G y (4) was exploited to minimize the squared error: [ min y G y 2 + y G y 2] (4) G,G [ ] y = max G G y + yg y + yg y yg G y + yg y + yg y that is formally equivalent to the likelihood (39). The minimization of (4) gives the result (see []) : G + SNR F Φ /2 F (F F ) (42) + SNR F Φ /2 F (43) G = + SNR F Φ /2 F (44) where SNR, SNR are the ratio between the noise power (in the signal bandwidth) and the signal power in each channel, and (42) is an approximation that gives almost the same results if the channels F, F have flat frequency response in their pass-band. 5.3 Deterministic and statistic likelihood

11 We have just shown that both DML and ML estimates are performed by means of linear kernels, whose expressions (29,3) and (43,44) appears quite similar. The most outstanding difference is due to the inclusion, in the ML case, of the weighting factor: ρ = + SNR + SNR that corresponds to the definition of the coherence of the interferometric pair [6]. The channel decorrelation is not handled in the DML approach, as this one regards the sources as set of parameters to be estimated rather than the realization of a statistical process. A qualitative comparison of the two techniques is provided in Fig. 5. In the figure, the upper plots show the squared absolute value of the Fourier Transform of the interferogram, that we have shown to be equivalent to the DML approach. ML DML Coherence Frequency Figure 5: Estimate of the fringe frequency made by () the DML technique (upper plot), (2) the ML technique (middle plot). All frequencies are normalized wrt 2f s. The correct frequency (.35) is retrieved in both cases, however, the ML technique, that exploits the coherence (lower plot) in its Likelihood, appears to be more robust. The ML estimator was applied following the idea shown in Fig. 3.b. In theory, a loop would be required where frequencies f are evaluated on a grid as fine as the one provided by the FFT (the DML approach). For each frequency, both the master and the slave are filtered according to the spectral shift filtering (43,44), to keep their common band contributions (see Fig.4). The likelihood is then estimated as the cross-energy of the filtered signals multiplied times the coherence. We can estimate the coherence by the sampled estimator: ρ bφ = s Φs (s s ) (s s ) where Φ is the modulation matrix evaluated at the current frequency, f. In practice, this loop can be made much faster by using a coarser grid for the spectral shift filtering, and refining the estimate by performing a FFT on the interferogram: I(n) = s (n) s (n). As an example, Fig. 5 (middle plot) shows the Fourier Transform of the interferogram after filtering the two images in the case of the correct spectra shift, e.g. f = f. The lower plot shows the coherence estimated at the various spectra shifts. A comparison of the plots in Fig. 5, motivates the following observations: both the FT of the interferogram and the spectral shift approach give the same results if the search is limited to a narrow interval of the correct value: f = f ± ɛ

12 the spectral shift approach is more robust due to the use of the coherence in the likelihood. We then notice that the inclusion of coherence in the likelihood allows to better discriminate which of the spectral peaks in Fig. 3 is the true one, whereas the coherence peak is too flat to provide an improvement in the fine estimate f. We once again remind that the weakness of the DML is due to the higher number of parameters to be estimated. We than expect that, at the highest SNR, and for large estimation windows, the two techniques would provide the same results, whereas different results would be achieved at the low SNR. A practical idea of this limit value is provided in Fig. 5. The figure shows the standard deviation of the estimate of f as a function of the coherence and for different window sizes, L, achieved by means of a Monte- Carlo simulation, with the DML approach. For high coherence/snr, all the estimates gives the same accuracy, once that they are scaled times the square root of the window length, L. However, for the low SNR, the performances drops and the estimated frequency is biased towards, a fact well known in literature [2]. In that case, the ML estimate would get significant improvements. σ f L=5 L = L=2 L=5 L= Coherence Figure 6: Standard deviation of the frequency estimate made by the DML approach, for different size of the estimation window, as a function of the coherence. Notice the threshold behavior for coherences ranging form.35 (L=5) to.95 (L=). 6 Multi-Baseline Slope Estimate The multi-baseline case is more complicated than the single baseline case. In fact, although we are still able to extend the DML approach, we have no closed form solution for the ML in (9) that would avoid the inversion of the covariance matrix. We than propose a straight extension of (26), by simply adding coherence-based weights to get close to the ML results. We need then to maximize the following figure of merit: φ = argmax N j= N i= ρ ij (φ)y i G ij y j (45) where the weights are computed basing on (29), and assuming the noiseless case (as we already accounts for noise with the coherence): G i = Φ /2 ij F i The diagonal matrixes Φ ij keeps the elements of φ in (6) properly scaled by the baseline of the pair i, j. The maximum is then found by iterating for all the values of the normalized frequency shift, f. Furthermore the co-channel terms (for i = j), are useless for the estimate of the phase term φ, and they can be dropped from the summation (45). This can be shown by writing the Fourier Transform of the SAR channel (34): Y i (f) = {Γ(f) δ(f f(α(t)))} F i (f) = Γ(f f) F i (f) (46) 2

13 where [ f(α(t) is the spectral shift, that depends on the slope. The power spectrum of the reflectivity, E Γ(f) 2], is white over a large bandwith, therefore the power spectrum of the output, E [ Y i (f) 2] carries no information on f. Nor we can retrieve any information from the phase of Y i (f), as the sources are Gaussian. Having dropped the co-channel contribution, (45) reduces to the summation of averaged interferograms (i j): ρ ij (φ)y i G ij y j Attention should be taken in order to exclude, at each value of the parameter f, the contributions that would be aliased. The constraint we have to impose is indeed more stringent, as we want to avoid biasing in the coherence estimate. Given a spectral shift f, the common bandwidth would be B c = B f, where B i is the SAR channel bandwidth. If we exploit patches on the ground of L pixels, sampled at a rate f s, the number of effectively independent samples would be: N i = L B c f s = L ( ) B n B max f f s R tan(θ α) and we need N i to avoid biasing of the coherence estimate ρ ij. Finally, we remark that in the straight extension from the single baseline, (42), to the MB case (45), we have not accounted for a spectral equalization like the one provided by the operator R H (see Fig. 4), however this topic could be a subject for future analysis. 6. Experimental results The proposed MB InSAR estimation algorithm has been tested first with synthetic and then with real data. The estimation algorithm has been applied with respect to the pixel-to-pixel phase differences in range, in order to remove any potential problem coming from tropospheric phase screen or other unknown phase offsets. We selected a 5 samples window length as a compromise between accuracy and resolution. The algorithms tested were based on the combination (45) in the two following implementations: - DML unweighed, - ML, including the spectral shift filtering and weights derived by the a-posteriori estimate of the coherence. In presenting the results we have also added a single interferogram image for comparison. 6.2 Results from simulations A set of 6 images was simulated by assuming the parameters of ENVISAT-ERS systems and the topography of Mt. Vesuvius. The baselines were chosen to cover a range of m with uniform sampling but taking care to avoid periodicity. The baseline vector is [ ] m with respect to the master. We did not add any further decorrelation, apart from the unavoidable baseline decorrelation. Having the perfect knowledge of the topography, we could perform an accurate evaluation of the estimator performances for different slopes. These results are shown in Fig. 7. Fig. 7.a maps the local phase gradient in range expressed in radians and measured on the theoretical unwrapped phase that would come from the 47 m baseline interferogram.the other graphics in the same figure show the natural logartihm of the counts of measured slopes with respect to the true slope, a sort of scatter-plot. The estimates are based on a single interferogram, in Fig. 7.(b), on the DML estimator, Fig. 7.(c) and on the proposed ML-derived estimator, Fig. 7.(d). The single interferogram is clearly limited in the trade-offs between altimetric resolution and phase aliasing. In the case shown we selected a small baseline ( m), nonetheless there is a sensible loss of the contribution coming from the higher slopes. We also notice the large variance, due to the fact that only two images were exploited instead of the five available. The DML estimate, in Fig. 7.(c) is almost useless, due to the strong biasing of the estimate, as discussed in section 5.3. On the other hand, the estimate achieved by the proposed technique, shown in Fig. 7.d, provides the higher accuracy for the low values of the gradient, due to the cooperation of all the images, and the capability to solve the highest gradient with no aliasing, due to the combinations with the lowest baselines. 3

14 φ e (a) φ 2 (b) φ e 3 2 φ e φ (c) φ (d) 4 6 Figure 7: (a) Phase gradient φ, along slant range, of a 5 m interferogram on Mt. Vesuvius. (b-d) Scatter diagram of φ estimated versus true value for different estimator. (b) single-baseline interferogram, (c) MB based on the unweighted combination, (d) MB achieved by both exploiting coherence weights and spectral shift filtering. 4

15 6.3 Real data processing The same technique has been exploited in a real ENVISAT dataset acquired over the area of Las Vegas. In particular we have considered the data relative to track 356, frame Acquisition parameters, in terms of perpendicular baseline and temporal separation, are collected in Table. Name Data Baseline Normal [m] Baseline Parallel [m] Delta Days M 25-Oct-22 S 29-Nov S2 3-Jan S3 7-Feb S4 27-Jun S5 -Aug Table : ENVISAT datasets, Las Vegas test site. The estimation algorithm has been applied, with respect to the pixel-to-pixel phase differences in range, over an image patch of approximately pixels 2 km (range, azimuth). An external DEM of the SRTM mission have been used with different purposes: - to remove aliasing in the single baseline interferogram, - to avoid threshold effects, and slope biasing in the DML approach, - to reduce the search range, hence speed up computations in the ML estimate. A smoothing window of 3 5 pixels in range and azimuth was used. The reference baseline used for referring phase gradient measures is 237 m. The slope measured on the reference SRTM DEM are reported in Fig. 8.a, together with the single baseline estimate, 8.b and the proposed ML combination, 8.c. Figure 8: Phase gradient, referred to a 237 m baseline image and estimated from: (left) an SRTM DEM, (center) a single baseline interferogram, (right) the proposed ML combination. 5

16 Here again we find the same features as in the simulated results, although the altimetric resolution is less evident due to the smaller baseline span. Notice the ML combination gets result similar to the SRTM DEM for both robustness with respect to ambiguity and the accuracy. 6.4 Acknowledgments The author would like to thank the Italian Space Agency for sponsoring of the work (ASI contract I/R/69/2 Multiple pass SAR Interferometry for very accurate Earth modeling). Thanks also to Dr. Ing. F. De Zan for the proficuous discussions on the statistic of the ML approach. 7 Conclusions In the paper we have studied two different Maximum Likelihood techniques to provide estimates of the terrain topography in a Multi-Baseline SAR interferometric case, namely the Deterministic ML and the statistic ML. The first approach includes the sources in the vector of parameters, and thus it is less robust than the second (particularly when few samples are available). However, its quite efficient and suited for applications when a preliminary knowledge of the slope exists. The second estimator, the ML approach, was derived as an extension of the MB case the exact ML estimated for conventional interferometry. A preliminary validation of this technique has demonstrated the capabilities of solving aliasing for the higher slopes, due to the use of the lower baselines, and to attain at the same time the high altimetric resolution that would come from the higher baselines. The major limit of the technique, the computational complexity, leaves open opportunities for further researches. References [] F. K. Li and R. M. Goldstein, Studies of multibaseline spaceborne interferometric synthetic aperture radars, IEEE Transactions on Geoscience and Remote Sensing, vol. 28, no., pp , Jan. 99. [2] D. Massonnet, H. Vadon, and M. Rossi, Reduction of the need for phase unwrapping in radar interferometry, IEEE Transactions on Geoscience and Remote Sensing, vol. 34, no. 2, pp , Mar [3] R. Lanari, G. Fornaro, D. Riccio, M. Migliaccio, K. P. Papathanassiou, J. ao R Moreira, M. Schwäbisch, L. Dutra, G. Puglisi, G. Franceschetti, and M. Coltelli, Generation of digital elevation models by using SIR-C/X-SAR multifrequency two-pass interferometry: The Etna case study, IEEE Transactions on Geoscience and Remote Sensing, vol. 34, no. 5, pp. 97 4, Sept [4] A. Ferretti, C. Prati, F. Rocca, and A. Monti Guarnieri, Multibaseline SAR interferometry for automatic DEM reconstruction, in Third ERS Symposium Space at the Service of our Environment, Florence, Italy, 7 2 March 997, ser. ESA SP-44, 997, pp [Online]. Available: [5] F. Lombardini, Optimal absolute phase retrieval in three-element SAR interferometry, Electronics Letters, vol. 34, no. 5, pp , July 998. [6] V. Pascazio and G. Schirinzi, Estimation of terrain elevation by multifrequency interferometric wide band SAR data, IEEE Signal Processing Lett., vol. 8, pp. 7 9, 2. [7] A. Ferretti, C. Prati, and F. Rocca, Permanent scatterers in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, vol. 39, no., pp. 8 2, Jan. 2. [8] F. Gatelli, A. Monti Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, The wavenumber shift in SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, vol. 32, no. 4, pp , July 994. [9] G. W. Davidson and R. Bamler, Multiresolution phase unwrapping for SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, vol. 37, no., pp , Jan

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