A Novel Across-Track SAR Interferometry Simulator

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 3, MAY 1998

A Novel Across-Track SAR Interferometry Simulator Giorgio Franceschetti, Fellow, IEEE, Antonio Iodice, Student Member, IEEE, Maurizio Migliaccio, Member, IEEE, and Daniele Riccio, Member, IEEE

Abstract—A novel across-track interferometric synthetic aperture radar (SAR) raw signal simulator is presented. It is based on an electromagnetic backscattering model of the scene and an accurate description of the SAR system impulse response function. A set of meaningful examples are also presented. They show that the proposed simulator is structurally consistent and correctly simulates the decorrelation effect, both in the mean and in the distribution sense.

I. INTRODUCTION

S

YNTHETIC Aperture Radar Interferometry (IFSAR) is a powerful remote-sensing technique that performs global and continuous monitoring of the environment. With regard to the specific feature to be monitored, different operational modes must be employed [1], [2]. In particular, the capability of across-track interferometry to generate high-resolution Digital Elevation Models (DEM) of the scene under survey from a SAR image pair is well described in literature [1], [3]–[8]. In fact, the phase difference of the signals received at two spatially separated SAR antennas is directly related to the topography, and it is possible to generate DEM’s [1] whose quality is governed by the coherence of the image pair [3]–[8]. Within this framework, it is certainly very useful to benefit from an IFSAR raw signal simulator. In fact, simulation is helpful in the experimentation of processing procedures (focusing, registration, and phase unwrapping), system design, and mission planning. Furthermore, IFSAR simulation can be exploited to better understand the physics involved in SAR interferogram formation. In this paper, we present an across-track IFSAR raw signal simulator that is based on an appropriate electromagnetic model of the backscattered fields and on the correct inclusion of the SAR system transfer function (TF). The electromagnetic model is based on a two-scale composite model of the surface scattering; the surface is described by planar facets with known (average) electromagnetic parameters (mesoscale) with superimposed roughness (microscale). The TF is implemented by means of a two-dimensional (2-D) formulation derived from [9]. A key point is the correct inclusion of the correlation Manuscript received January 16, 1997; revised October 13, 1997. G. Franceschetti is with the Consiglio Nazionale delle Ricerche, IRECE, 80124 Napoli, Italy (e-mail: [email protected]) and the Universit`a di Napoli Federico II, Dipartimento di Ingegneria Elettronica, 80125 Napoli, Italy. A. Iodice and D. Riccio are with the Universit`a di Napoli Federico II, Dipartimento di Ingegneria Elettronica, 80125 Napoli, Italy (e-mail: [email protected]; [email protected]). M. Migliaccio is with the Istituto Universitario Navale, Istituto Teoria e Tecnica delle Onde Elettromagnetiche, 80133 Napoli, Italy (e-mail: [email protected]). Publisher Item Identifier S 0196-2892(98)02858-7.

of the two backscattered electromagnetic fields; in fact, height accuracy of the IFSAR DEM’s is strongly dependent on such a correlation [3]–[8]. Main sources of decorrelation include [4] system thermal noise, temporal changes of the scene, SAR interferometric processing errors, and spatial diversity. Decorrelation due to thermal noise can be easily modeled by an additive noise; decorrelation due to temporal changes is present only in the repeat-pass interferometry mode; and SAR interferometric processing sources of decorrelation can be minimized by accurate processing. Conversely, the baseline decorrelation, which is related to spatial diversity, is unavoidable [4] and intrinsic in any SAR interferometric system. It can be reduced only at the expense of the image resolution [5]. In this paper, most attention is devoted to the baseline decorrelation, therefore, single-pass mode (or repeat-pass mode over time-stationary scenes) is considered. Accordingly, modeling of the correlation of the two backscattered electromagnetic fields is here accomplished by means of an approach similar to that in [4]–[8]. Once the two raw signals have been simulated, it is possible to generate the two SAR images and then the corresponding interferogram. A complete set of meaningful examples are shown. Interferograms corresponding to relevant cases are illustrated to test the performance of the IFSAR simulator. In particular, cases relevant to reference scenes are examined in detail. These results show that the simulation code is effective and implements a realistic model of the considered problem. As a conclusion, the purpose of this paper is the presentation and discussion of a novel across-track SAR interferometry simulator. Focus is on the capability to correctly simulate the raw signal pairs of an extended scene, and no emphasis is posed on some other relevant problems, such as deviations from the reference platform path. These effects are more relevant to airborne rather than spaceborne SAR. Novelties of presented simulator are manifolds. First of all, to the best of our knowledge, it is the first time that all elements of simulation and interferometry are dealt with together (IFSAR simulator is the main topic, not a side one, of this paper). In fact, some papers that focused on IFSAR processing [10], [11] dealt with some simulations of interferometric fringes based only on the geometrical description of the scene under survey. Conversely, our simulator is able to generate the SAR raw signal pair, not simply the fringes, and this can be of great interest for testing processing procedures. Further, the simulation is based on an electromagnetic model of the scene, which allows us to evaluate the mean backscattered fields and their fading characteristics. This is a key point for

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efficient of the scene, and the SAR impulse response function is given by rect (2.2) (2.3) (2.4) is the normalized illumination function of the real where antenna over the scene, is the azimuth footprint, is the chirp duration, and is the chirp frequency rate. is the We explicitly note that, in (2.1), backscattering coefficient incorporating the propagation factor (2.5)

Fig. 1. IFSAR geometry.

implementation of a realistic simulator. In particular, modeling of the decorrelation effect is the main novelty, with respect to the noninterferometric simulator presented in [9]. Speckle over final images and the decorrelation phase noise over the fringes naturally come out from the electromagnetic model of the scene. This varies from what is usually accomplished in literature, where an unnatural noise superposition over the SAR images and fringes is applied. Finally, our simulator is modular and therefore amenable to future improvements and extensions. The paper is organized as follows. In Section II, the rationale of the simulation scheme is presented. In Section III, we describe the electromagnetic model used to evaluate the facets’ backscattering coefficients. In Section IV, the raw signal computation method is presented. In Section V the structure of the simulator is depicted. In Section VI, we show and discuss a meaningful set of examples, and finally, in Section VII, concluding remarks are reported. II. RATIONALE In this section, we briefly describe the rationale of the overall simulation scheme. The geometry of the problem is depicted in Fig. 1. The locations of the two interferometric (across-track mode) SAR antennas are . They both illuminate the same area and are separated by the baseline length are the distances from the generic point to the two antennas; and are the distances from the generic ground point to the two antenna flight paths. The SAR raw signal (after heterodyne, i.e., downconversion to baseband) can be written as [9]

(2.1) wherein subscripts 1 and 2 refer to the signal received by the antenna and , respectively, is the backscattering co-

where is the conventional backscattering coefficient. The propagation factors contain the information on the topography to be extracted by means of the IFSAR processing [1]; this factor is fundamental in an IFSAR simulator and cannot be neglected or incorporated into the stochastic microscopic phase term, as in a single SAR raw signal simulator [9]. and are by no means Note that, in general, identical, even for single-pass interferometry, due to variation of the signal backscattered by each element of the scene along the two look angles. Hence, and are only partially correlated (see also Section III), and the important conclusion is that an IFSAR simulation cannot be obtained by simply running a SAR simulator twice. Note, however, that, even if and were identical at any point of the scene, the final images would be only partially correlated because of the finite system resolution (see Section IV). Above considerations clearly show that the simulation scheme encompasses two major steps: generation of the two reflectivity maps and evaluation of the SAR raw signal integrals shown in (2.1). The first step is based on the SAR raw signal simulator presented in [9]. The geometry and height profile of the scene can be either prescribed analytically or provided by a DEM. This profile is then approximated by planar rough facets, large, with respect to the incident wavelength, but smaller than the resolution length. In other words, three scales are defined: the macroscale, the mesoscale, and the microscale. The macroscale is related to the SAR processor resolution; the mesoscale to the facet size, where the macroscopic characteristics of the scene under survey are assumed to be homogeneous; and the microscale is referred to the electromagnetic wavelength and therefore to scene features, such as the facet’s roughness, that cannot be resolved by the sensor but interact with the electromagnetic field. Each facet is characterized by its center position, the normal to the facet mean plane, the electromagnetic parameters (permittivity and conductivity ) of the underlying material, and by its microscopic roughness. Computation of the individual facet backscattering (at the

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Fig. 2. Facet backscattering: geometry of the problem.

two interferometric antennas) takes into account the two local angles of incidence, the polarization of the incident wave, the facet roughness, and any geometric distortion (i.e., layover, foreshortening, and shadowing). Electromagnetic backscattering is performed by using Kirchhoff approximation and physical optics (PO) solution [9]. We further note that correct correlation between the individual facet returns at the two antennas must be generated, as shown in detail in Section III. Once the two backscattering coefficient maps (one for each antenna) have been evaluated, the two raw signals are generated by properly summing the elementary returns, according to (2.1); this is the second step already cited. This task is accomplished by means of the 2-D fast Fourier transform (FFT) code described in [9], appropriately employed, as explained in Section IV.

III. ELECTROMAGNETIC MODELING In this section, we consider the two returns from a single facet and compute them at the interferometric antennas sites. Only surface scattering is modeled, while multiple or volume scattering is neglected. However, these features may be incorporated in the simulator, due to its modular nature. The backscattering coefficients of the elementary facet can be written as follows [9]:

If in the single-pass mode the second antenna is only receiving, (3.2) must be replaced by

(3.3) In (3.1)–(3.3), is the facet surface, and are the distances from the facet center to the two antenna flight paths, is a function of the average Fresnel reflection coefficients over the facet [9], and are the wavenumber vectors, and is the vector describing the generic point of the facet (see Fig. 2). Equations (3.1)–(3.3) assume a scalar backscattering problem, which is not the case for IFSAR applications. Accordingly, the function and, consequently, the backscattering coefficient become 2 2 matrices and . Computation of the matrix can be performed according to [9] and is fully implemented in our simulator. A random roughness is superimposed over the facet, and and are random variables due to the randomness of . The mean square values , , and the correlation can be analytically evaluated (under the PO approximation) if a statistical model of the roughness is and are assumed. These mean square values functions of the incidence angle and are strongly dependent on the facet roughness [12]; the correlation is function of the baseline as well. The correlation coefficient of and is defined as

(3.1)

(3.4)

(3.2) and can be analytically evaluated.

FRANCESCHETTI et al.: ACROSS-TRACK SAR INTERFEROMETRY SIMULATOR

Under the hypotheses of isotropic Gaussian surface and of microscopic roughness correlation length much smaller than the facet size, we obtain

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derived from (3.7) [7], [13], and [14]. For the first one we have

(3.5) by following an approach similar to that of [8]. In (3.5), and are surface correlation length and surface standard deviation, are the incidence angles over the facet. respectively, and Under the same hypotheses, we also obtain

rect

(3.9)

with (3.10) In (3.10), one we have

means the principal value. For the second

sinc sinc

(3.6)

where is the average antenna (facet center range), is the is the baseline component perpendicuaverage look angle, lar to the look direction, and are the facet dimensions, and and are the tilt angles of the mean plane along and directions, respectively, (see Fig. 2). If the second antenna is only receiving, the arguments of the exponential and sinc functions in (3.6) must be divided by two. depends on the facet dimenThe correlation coefficient sion and roughness on the incident wavelength and on the geometric parameters of the IFSAR system. Examination of (3.6) shows that in the Gaussian surface case is only slightly influenced by the facet roughness, at least when only surface scattering mechanism is involved [5], [6], [8]. Once the mean square values and the correlation coefand have been computed via (3.5) and ficient of (3.6), generation of these two random variables, with the proper self and joint probability density functions (pdf’s), is in order. In the same hypotheses leading to (3.5) and and can be modeled as complex (3.6), we have that circular Gaussian random variables, characterized by uniform phase and Rayleigh amplitude distribution. The interaction between the two random variables and is rather involved. However, their mean square values are practically coincident because the reirradiation diagram is rather broad (hence, the received powers at the interferometric antennas are equal). Under this hypothesis, the joint pdf is given by [13]

(3.7) wherein (3.8) The pdf of the (wrapped) phase difference and the conditional pdf of , given the value of

can be

(3.11) is the zero-order modified Bessel function of In (3.11) the first kind [18]. Accordingly, the IFSAR simulator first generates the phase and amplitude of and, then, from knowledge of their values, generates the correlated phase and amplitude of by using the pdf’s of (3.9) and (3.11) (the relation is used). Generation of a random variable distributed according to (3.9) is performed by means of a rejection method [19], while generation of a random variable distributed according (3.11) is straightforward because it is identical to the Rice one [14]. The wrapped phase difference distribution of (3.9) achieves , and therefore, represents its maximum value for the mode of the distribution. Since has a physical meaning (3.10), it is effective to view the phase difference of the facet returns as the sum of a deterministic component , due to the difference between the two ranges of the facet center, and , due to the finite (nonzero) a random component dimension of the rough facet. It turns out that the pdf of this random component is equal to (3.9), where rect is replaced by rect . Previous operations are iterated for all facets in the scene to be simulated, so that two reflectivity maps are generated. Returns from different facets are treated as statistically independent; this is consistent with the assumption that the microscopic roughness correlation length is much smaller than the facet size. Finally, a ground range to slant range transformation is performed to correctly include foreshortening, layover, and shadowing, as detailed in [9]. Mutual geometric distortions between the two raw signals, i.e., between the two corresponding images (i.e., misregistration effects) can be taken into account by projecting each reflectivity map on its own range plane and by using at least eight facets per resolution cell along the range direction. In this way, a relative shift of one-eighth of resolution cell can be appreciated. This is an important point because a misregistration of less than one-eighth of resolution cell has a negligible effect on the coherence, while a larger relative

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shift causes an appreciable decorrelation [7]. As a drawback, we have that use of eight facets per pixel is time and memory consuming. Since in the current application of our simulator we are not focused on testing image registration procedures, we directly project both reflectivity maps on the same (mean) slant range plane. Accordingly, the number of facets per pixel need not be increased1 and no registration procedure must be performed between the final complex images. Some comment on above procedures are now in order. First of all, we note that for usual values of the baseline length, moderate relief, and for usual roughness statistics, is closer to unity [see (3.6)]. the correlation coefficient in (3.6), tends to This would imply to set a Dirac function centered in tends to a Dirac function centered in , and all of the procedure simplifies. However, the presence of temporal decorrelation and volumetric scattering changes this simplified picture and significantly lower than unity. Accordgenerates values of ingly, from both a theoretical viewpoint and in view of future development of the simulator embodying volume scattering and/or repeat-pass mode, we implemented the most general does not formerly described procedure. Note that imply that the correlation coefficient of the final complex images and (3.12) is also unity. As a matter of fact, decorrelation is provided by the presence of more than one facet per resolution cell and by coupling of neighboring facets via the processing step (see Section IV). As a further remark, we note that an alternative way to perform the simulation consists of considering a deterministic (nonrandom) facet backscattering and of placing at least 4 4 facets per resolution cell with a random height displacement of the facet vertexes; the proper speckle statistics and coherence on the final complex images would be assured by the summation of elementary returns, according (2.1), and subsequent processing (see also Section IV). However, we and shown above prefer the method of generation of for the following reasons. 1) It takes into account more properly the effect of surface roughness (and possibly of temporal changes and volumetric scattering) on the baseline decorrelation [6], [8] by means of (3.6). 2) Simulation is time and memory saving because fewer facets per resolution cell may be chosen. 3) The same operation described above can be used if we want to simulate directly the interferograms. In fact, we can substitute in (3.9)–(3.11) , the correlation coefficient of the backscattering coefficient, with , the correlation coefficient of the resulting complex image amplitudes and (evaluated in [8]); then we can proceed to simulate directly the interferograms without passing through the raw signal simulations. This procedure can be sufficient for some applications. 1 The minimum number of facets per pixel must be established according to considerations detailed in Section IV.

IV. RAW SIGNAL COMPUTATION In this section, we briefly depict the logical scheme to be followed to evaluate the integral of (2.1), i.e., the simulated SAR raw signal. As a preliminary step, we recall some general problems that must be faced to perform an efficient and precise SAR raw signal simulation, even in the noninterferometric case. Evaluation of (2.1) is generally by no means straightforward. Computation of the integral in time domain is not efficient, and its evaluation in the transformed Fourier domain is desirable. However, this is made difficult by the presence of a spacevarying kernel. This situation is handled by an asymptotic evaluation of the (2-D) Fourier transform (FT) of (2.1) (see [9]) and subsequent use of a grid deformation in the Fouriertransformed domain [15] or of a modified kernel FT method [16]. In such a way, range migration, curvature effects, and variations of focus depth are automatically included. This procedure is efficient, is amenable of modular extension (for instance, inclusion of a squint angle [25]), and compares favorably with other available techniques. Coming to the specific aspects of the IFSAR simulator, we note that obviously the evaluation of (2.1) must be performed twice. Further, we have a fundamental additional requirement: oversampling of each reflectivity map by an amount depending on the baseline to correctly account for the cross correlation among the pixels of the two raw signals and consequently over the final interferogram. In other words, we need to put more than one facet per resolution cell; however, as explained in the following, we only need two facets per resolution cell. The remaining part of this section is completely devoted to the explanation of this requirement. Let us evaluate the correlation coefficient of the two complex images (after processing). To simplify the discussion, we assume independent scatterers uniformly distributed on a flat (tilted) surface; i.e., , where is the normalized radar cross section [4]–[7] of the scene. If we neglect processing errors, thermal noise, temporal changes, and misregistration, the two single-look complex (SLC) images are given by [6]

(4.1)

sinc sinc and wherein are the azimuth and slant range resolutions, respectively, and . The correlation coefficient of and can be analytically evaluated, as in [6] (4.2)

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dimensions2; and therefore, (4.2) is substituted by (4.6) sampled at rates which is the FT of the function and . If we consider again , we obtain the behavior plotted in Fig. 3(b); it is apparent that after simulation and processing we recover the correct correlation coefficient only if the baseline (once all other parameters are fixed) is short enough to have (a)

(4.7)

(b) Fig. 3. Plot of the correlation coefficient ji j as a function of ar : (a) x; y continuous variables; (b) x; y discrete variables.

wherein (4.3) and geometric symbols are defined in Section III. in the and domain, Equation (4.2) is the FT of shown in (4.4) at the bottom of the page. If we set the azimuth surface slope to zero, becomes equal to zero and we obtain the well-known linear decrease of the correlation coefficient as a function of the baseline depicted in Fig. 3(a). are discrete-space In the simulation case, however, and processes so that (4.1) must be substituted by

(4.5)

and where being the azimuth, ground range, and slant range facet

(i.e., there are at least two facets per If resolution cell along the range direction), we can recover the , i.e., for correct correlation coefficient for any all the values of the baseline length smaller than the critical baseline and for any value of . Even for the case of not equal to zero, previous considerations about the range [see oversampling still hold, due to the factorized form of (4.4)]. All of this can also be seen from the viewpoint of the wavenumber shift approach [5], if we note that the relative shift of the two reflectivity spectra becomes a circular shift in the case of discrete-space . Such considerations allow us to determine the facet dimension and therefore the rate at which is sampled along range. Since the pixel is smaller than the resolution, in our IFSAR simulator, two facets per pixel (along the range direction) are considered; the obtained raw signal is then decimated to obtain the proper SAR system pixel spacing. Let us now consider the azimuth direction. Following the same rationale previously employed we conclude that the correlation coefficient is correctly simulated provided that (4.8) Results similar to the range channel are reached. When (i.e., there are at least two facets per resolution cell along the azimuth direction), we can recover the correct , i.e., correlation coefficient for any value of for all values of the baseline length smaller than the critical baseline and for all values of . We note, however, that (4.8) is usually easily verified because is usually small, i.e., , due to the fact that is proportional to , which is expected not to be too large for natural relief. As a can be reasonably matter of fact, the condition relaxed. We experienced that a proper choice for can be related to the azimuth pixel spacing. This is also useful to easily achieve the desired output grid spacing. Within such a

1 1

1

2 The facet dimensions x and r are not necessarily related to the output pixel spacings x and r, dictated by the SAR system pulse repetition frequency (PRF) and sampling frequency fs . However, both are smaller than the resolution (in the SLC case).

1

for for

(4.4)

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Fig. 4. Simulator structure. TABLE I MAIN ERS-1 SYSTEM DATA USED IN

THE

SIMULATION RUNS

framework, for the azimuth direction, we only need one facet per pixel. To better illustrate this point, we consider the ERS-1 m mission (see Table I). We have a range resolution and an azimuth resolution m. A reasonable baseline is 100 m. If we set the range and azimuth facet sizes and equal to the pixel ones, i.e., PRF, we have that (4.7) is verified for any less than 13 and (4.8), in the worst case, , for any less than 65 . This shows that the critical condition is given by (4.7), and therefore, the safe condition must be employed. We note further that, if we let again, , for m (4.7) is not verified even for , whereas (4.8) is verified as soon as is less than 65 ( is set to 0). If the surface is not flat, these results are only approximate. However, they give us an indication on the choice of the facet size. As a conclusion, two facets per pixel and are sufficient under a very wide range of terrain slopes whenever surface scattering is in order. Note that former considerations are strictly valid when the aforementioned scattering model is in question and must be checked again for different scattering mechanisms. However, if a very simple volumetric scattering model (see [6]) is used, similar considerations can be developed and the number of facets per resolution cell remains practically unchanged. V. SIMULATION PROCEDURE In this section, we briefly outline the structure of the IFSAR simulator and the simulation procedure (see Fig. 4). Two main blocks (large dashed boxes) are depicted. The first one

is relevant to the electromagnetic modeling of the extended natural scene; i.e., it is concerned about generation of the two reflectivity maps (Section III). The second one is relevant to the SAR raw signal evaluation (Section IV). In order to generate the two backscattering coefficients of the generic facet, we need to insert system and environmental data. The system data include the electronic and geometrical system data, while the environmental data are needed to describe the extended scene under survey. This latter data set must encompass the large-scale (macroscale and mesoscale) description of the scene as well as its microscopic one. Within the large-scale description of the scene, we have the height map and the surface electromagnetic parameter maps. In the microscopic description of the scene, we have the microscopic roughness standard deviation and correlation length (see Section III). Once the input data set has been inserted, scene scanning can be accomplished for any equi-azimuth line, from near to far range, to use a ray-tracing recursive method to sort out the shadowed zones and include all geometric distortion effects [9]. In other words, input data points are provided in ground range coordinates, while the reflectivity data points must be evaluated with reference to slant range coordinates. For each scanned facet, the correlation coefficient , the function (3.5), and hence, the two backscattered powers are evaluated (see Section III). This set of operations is schematized by the first block of Fig. 4. We note further that, varying from what is accomplished in [9], a proper correlation between the two reflectivity maps must be correctly modeled and implemented. This is done by means of the model based on (3.5)–(3.11). As a matter of fact, the first backscattering coefficient is obtained by first generating a complex random variable whose amplitude is Rayleigh distributed in accordance to the before evaluated backscattered power and whose phase is uniformly distributed. This models the fading and therefore generates the speckle over the final simulated images. Afterwards, the round-trip interferometric phase is included. This is encompassed by the block of Fig. 4. Next, the second backscattering coefficient is evaluated by means of the correlation model expressed in (3.5)–(3.11). This block. Henceforth, the step is accomplished by the two reflectivity maps are generated and can be processed via

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(a)

(b)

(c)

(d)

Fig. 5. Simulated single-look interferograms of a flat rough scene with (a) ERS-1 system data. Near range is on the left.

TABLE II MAIN E-SAR SYSTEM DATA USED IN

B

= 50 m, (b)

B

= 100 m, (c)

B

= 200 m, and (d)

B

= 400 m.

VI. EXAMPLES THE

SIMULATION RUNS

an efficient 2-D FFT code to obtain the two SAR raw signals (Section IV). Some final information about the required computational time is appropriate. Reference is made to the IBM RISC 6000, running on an AIX 3.2 operating system, used to develop and test our program, which has been written in Fortran-77 programming language. Simulation of a 2048 2048 pixel ERS-1 raw signal pair requires about 13’ CPU time if two facets per pixel have been used. Normal task conditions were used.

In this section, we show some interferograms generated by processing simulated raw signal pairs and discuss the obtained results. To make use of practical parameters, we consider existing SAR systems: for a spaceborne SAR system, we consider ERS-1, and for an airborne one, we consider the DLR E-SAR. Main system parameters of ERS-1 and E-SAR are listed in Tables I and II, respectively. The case of a flat (possibly tilted) scene is widely examined in the literature [4]–[7]. In such a case, theoretical analytical expressions of the complex image-pair correlation coefficient [4]–[6] and of the phase difference pdf [7] are available. Therefore, we first simulate the interferograms of a flat scene using different values of the baseline length and measure the correlation coefficient3 and the phase difference distribution of these data to compare them with the theoretical ones. 3 It is interesting to note that estimation of the correlation coefficient of simulated interferometric pairs is very reliable because the nonstationarity [20], [22] of i1 i32 can be suppressed. As a matter of fact, we have an apriori knowledge of the height profile and can suppress the deterministic space-varying component of phase difference. Therefore, we can use very large windows (at least 16 16 pixels), thus, strongly reducing the estimator standard deviation and bias [23].

2

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(a)

(b)

(c)

(d)

Fig. 6. Theoretical and estimated phase difference pdf’s relevant to the interferograms of Fig. 5.

First of all, we simulated the flat scene by using the ERS-1 parameters (see Table I) for four baselines lengths: 50, 100, 200, and 400 m (the baseline angle has always been considered equal to 0 ). The considered scene is an electromagnetically homogeneous, flat, untilted surface; simulation has been performed with and without including the effect of the by (3.6) or setting microscopic roughness, i.e., evaluating equal to one. The roughness standard deviation was set equal to 1 m, that is, about 18 times the carrier wavelength of the ground range resolution. and about As a first check, we generated the two complex images and and performed a comparison among the measured and the theoretical ones evaluated by using (4.4). Results are collected in Table III. In the first two columns, the meaare collected, with (column 1) and without (column sured 2) taking into account the roughness; in the third column, are displayed. As anticipated in the theoretical values of Section III, the dependence on the roughness parameter is

hardly appreciable, except for very long baselines. We note the excellent agreement between the second and third column of Table III, i.e., between the theoretical estimate of and the experimental value obtained in the simulation in absence of roughness effect inclusion. This is a good mark for validation of the simulator because the theoretical model does not account for any roughness effect. Note that the discrepancy of the first and third columns is also a good mark because it shows that for large values of the baseline and for very rough surfaces the roughness cannot be neglected, as theoretically anticipated in [8]. These results show that the simulator can be fruitfully exploited to better understand the physics involved in the SAR interferogram formation. The single-look, slant range interferograms of the considered flat scene are shown in Fig. 5. The azimuth is vertical, the range is horizontal, and near range is on the left. All subsequent interferograms and images are formatted accordingly. The azimuth and slant-range pixel spacing are about 4 and

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TABLE III MEASURED AND THEORETICAL CORRELATION COEFFICIENTS ji j RELEVANT TO INTERFEROMETRIC PAIRS OF FIG. 5, WITH AND WITHOUT INCLUSION OF THE EFFECT OF SURFACE ROUGHNESS ON i . ROUGHNESS STANDARD DEVIATION  WAS SET TO 1 m

TABLE IV MEASURED AND THEORETICAL CORRELATION COEFFICIENTS ji j RELEVANT TO THE DIFFERENT FACES OF THE PYRAMID OF FIG. 8. THE FACE TILTED TOWARD THE SENSOR [LEFT PART OF FIG. 8(a)] IS REFERRED TO AS FACE A, THE ONE TILTED AWAY FROM THE SENSOR [RIGHT PART OF FIG. 8(a)] AS FACE B , AND THE FACES TILTED IN THE AZIMUTH DIRECTION (TOP AND BOTTOM) AS FACES C AND D

Fig. 7. Simulated single-look interferogram of a flat scene with B = 1:30 m. E-SAR system data. Near range is on the left.

8 m, respectively, and the overall extension is about 3.8 10 km. The measured phase difference distributions (phase histograms) and the theoretical pdf’s relevant to interferograms of Fig. 5 (after subtraction of the deterministic flat Earth pattern) are plotted in Fig. 6. Theoretical pdf’s are computed by using (3.6) is replaced by (3.12). (3.9), wherein As a conclusion, we can state the following. (1) Geometrical consistency is achieved. In particular, fringe spacing is in agreement with the foreseen one [5] (Fig. 5). (2) Phase noise due to decorrelation is apparent (Fig. 5). (3) Measured and theoretical correlation coefficients are in remarkable agreement (Table III). (4) Measured phase difference distributions fit very well with the theoretical phase difference pdf’s (Fig. 6). To further appreciate the first statement, we consider an airborne system: the E-SAR radar installed onboard a DLR DO-228 aircraft, and the same flat scene of the previous case. The baseline length is set to 130 cm, as in the real system [21]. The interferogram is formatted analogously to the ERS-1 case, except that in the E-SAR case we have an azimuth and slant range pixel spacing of about 27 and 150 cm, respectively, and an overall extension of about 424 1671 m, respectively. The effects of the variation of the look angle from near to

far range are appreciable. In particular, an increase of the fringe spacing from near to far range is apparent in Fig. 7. The measured correlation coefficient in the middle of the scene is , in agreement with the theory (4.4). In order to verify that the dependence of the correlation coefficient on the surface slope (both in range and azimuth directions) is correctly taken into account, we simulated the raw signal pair relative to a pyramid and processed it to obtain the corresponding interferogram. We used the ERS-1 system parameters and an horizontal baseline of 100 m. The surface slope was 10 in the range direction and 50 in the azimuth direction. In Fig. 8(a), a single-look amplitude image is shown: speckle noise and foreseen geometric distortion (foreshortening) are apparent. The corresponding correlation map is shown in Fig. 8(b), while the interferograms with and without flat earth pattern are depicted in Fig. 8(c) and (d), respectively. Theoretical and measured correlation coefficients on each face of the pyramid are reported in Table IV, where the face tilted toward the sensor, i.e., the near-range pyramid face, is referred to as face A, the one tilted away from the sensor, i.e., the far range one, as face B, and the faces tilted in the azimuth direction as faces C (top) and D (bottom). Again, a good agreement between theoretical and measured correlation coefficients is achieved. In particular, decorrelation is stronger on the side of the pyramid tilted toward the sensor (face A), as predicted by theory and as it happens for real SAR interferograms of mountainous areas (see, for instance, [17]). Theoretical and estimated phase difference pdf’s relevant to the different faces of the pyramid are also in remarkable agreement, as clearly shown in Fig. 9. As a summary of this set of examples, we can reasonably state that the proposed IFSAR simulator is structurally consis-

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(a)

(b)

(c)

(d)

Fig. 8. Simulated single-look amplitude image of a (rough) pyramid (a) corresponding correlation map (b) and interferograms without (c) and with (d) flat earth pattern subtraction. ERS-1 system data. Near range is on the left.

tent and correctly simulates the decorrelation effect, both in the mean sense and in the distribution sense.

VII. CONCLUDING REMARKS AND FUTURE DEVELOPMENTS A novel across-track raw signal IFSAR simulator has been presented and discussed. It includes an electromagnetic model of surface backscattering and a 2-D formulation of the SAR TF. The electromagnetic model is based on a facet model of the surface, and the backscattered fields are evaluated using the Kirchhoff approximation and the PO solution. The PO is also applied to analytically evaluate the correlation coefficient between the received fields at the two antennas in terms of the geometry of the problem and the facets’ characteristics. The presented 2-D formulation leads to an efficient simulation because computations are carried out in the Fourier domain. A set of meaningful examples relevant to some reference scenes, i.e., planes and pyramids, showed the potentiality of the simulator. Its effectiveness is assessed over these examples by comparing the correlation coefficient and the phase difference pdf measured on the interferograms obtained from the simulated raw signals with the corresponding theoretical behaviors reported in the literature. In particular, we showed

that geometrical consistency is achieved, and the baseline decorrelation effect is correctly simulated, both in the mean sense and in the distribution sense. The presented simulation code can be used with reference to an actual ground scenario. Subsequent comparison with real fringes is certainly desirable. However, we note that to perform such a task in a reliable manner is by no means straightforward due to many reasons. We underline the uncertainty of knowledge of the electromagnetic characteristics of the natural scene (i.e., permittivity and conductivity maps), of its topography (i.e., the scene DEM), and of the baseline length and orientation. Furthermore, the backscattering model employed in this work may not be sufficient, for instance, for vegetated areas. Finally, proper objective norms need to be introduced to measure the similarities between simulated and actual interferograms. With reference to the uncertainties over the input data, we call for high-resolution maps, but even in this favorable case, a proper interpolation scheme needs to be devised because these maps are generally referenced to a ground coordinate system by no means related to the SAR ground coordinate system (i.e., azimuth and ground range). A solution to this problem is provided by [24]. With reference to the scattering

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(c) Fig. 9. Theoretical and estimated phase difference pdf’s relevant to squared patches of the interferogram of Fig. 8 on (a) face A, (b) face B, and (c) face C.

model, our simulator is amenable of further extensions, due to its modular nature; volumetric scattering effects can be included as well as the influence of the atmosphere on propagation. However, the actual implementation of such effects depends on the availability of reliable and viable models. With reference to the definition of objective norms, we believe it is appropriate to compare appropriate moments particularly sensitive to phase signal characteristics. In any case, this is a sensitive estimation problem and must be properly considered [22], [23]. This subject is a matter for future developments. The simulator presented here is already a very useful tool for many applications and the basic groundwork for all of the presented ambitious goals.

REFERENCES [1] L. C. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE, vol. 62, pp. 763–768, June 1974. [2] R. M. Goldstein and H. A. Zebker, “Interferometric radar map of ocean currents,” Nature, vol. 328, pp. 707–709, Aug. 1987.

[3] F. K. Li and R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sensing, vol. 28, pp. 88–97, Jan. 1990. [4] H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sensing, vol. 30, pp. 950–959, Sept. 1992. [5] F. Gatelli, A. M. Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca, “The wavenumber shift in SAR interferometry,” IEEE Trans. Geosci. Remote Sensing, vol. 32, pp. 855–865, July 1994. [6] E. Rodriguez and J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” in Proc. Inst. Elect. Eng. F, 1992, vol. 139, pp. 147–159. [7] D. Just and R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt., vol. 33, pp. 4361–4368, 1994. [8] G. Franceschetti, A. Iodice, M. Migliaccio, and D. Riccio, “The effect of surface scattering on IFSAR baseline decorrelation,” J. Electron. Waves Applicat., vol. 11, pp. 353–370, Mar. 1997. [9] G. Franceschetti, M. Migliaccio, D. Riccio, and G. Schirinzi, “SARAS: A synthetic aperture radar (SAR) raw signal simulator,” IEEE Trans. Geosci. Remote Sensing, vol. 30, pp. 110–123, Jan. 1992. [10] D. Massonnet, M. Rossi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, and T. Rabaute, “The displacement field of the landers earthquake mapped by radar interferometry,” Nature, vol. 364, pp. 138–142, July 1993. [11] G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric phase

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[18] [19] [20] [21] [22] [23]

[24]

[25]

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unwrapping using green’s formulation,” IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 720–727, May 1996. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces. Norwood, MA: Artech House, 1987. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New York: IEEE, 1987. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. Berlin, Germany: Springer-Verlag, 1975, pp. 9–75. G. Franceschetti, R. Lanari, V. Pascazio, and G. Schirinzi, “WASAR: A wide angle SAR processor,” in Proc. Inst. Elect. Eng. F, 1992, vol. 139, pp. 107–114. G. Franceschetti, R. Lanari, and E. S. Marzouk, “Efficient and high precision space-variant processing of SAR data,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, pp. 227–237, Jan. 1995. R. Lanari, G. Fornaro, D. Riccio, M. Migliaccio, K. P. Papathanassiou, J. R. Moreira, M. Schw¨abisch, L. Dutra, G. Puglisi, G. Franceschetti, and M. Coltelli, “Generation of digital elevation models by using SIR-C/XSAR multi-frequency two-pass interferometry: The Etna case study,” IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 1097–1114, Sep. 1996. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1965. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. Cambridge, U.K.: Cambridge Univ. Press, 1987. J. O. Hagberg, L. M. H. Ulander, and J. Askne, “Repeat-pass SAR interferometry over forested terrain,” IEEE Trans. Geosci. Remote Sensing, vol. 33, pp. 331–340, Mar. 1995. R. Horn, “The DLR airborne SAR project E-SAR: Objectives and status,” in Proc. EUSAR’96, K¨onigswinter, Germany, pp. 53–58. M. Di Bisceglie, C. Galdi, and R. Lanari, “Statistical characterization of the phase process in interferometric SAR images,” in Proc. IGARSS’96, Lincoln, NE, pp. 1580–1583. R. Touzi and A. Lopes, “Statistics of the stokes parameters and of the complex coherence parameters in one-look and multilook speckle fields,” IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 519–531, Mar. 1996. G. Franceschetti, M. Migliaccio, and D. Riccio, “SAR raw signal simulation of actual ground sites described in terms of sparse input data,” IEEE Trans. Geosci. Remote Sensing, vol. 32, pp. 1160–1169, Sep. 1994. G. Franceschetti, R. Lanari, and E. S. Marzouk, “A new two-dimensional squint mode SAR processor,” IEEE Trans. Aerosp. Electron. Syst., vol. 2, pp. 854–863, Apr. 1996.

Giorgio Franceschetti (S’60–M’62–SM’73–F’88) was born and educated in Italy. He was appointed full Professor of Electromagnetic Wave Theory at the University of Napoli, “Federico II,” Napoli, Italy, after winning a national competition in 1968. In addition, he has been Visiting Professor at the University of Illinois, Urbana, from 1976 to 1977 and at the University of California, Los Angeles (UCLA), from 1980 to 1982. He was a Fulbright Scholar in 1973 and a Research Associate from 1981 to 1983 at Caltech, a Visiting Professor at National Somali University in 1984, an Adjunct Professor at UCLA from 1994 to 1995, and a Visiting Professor at the University of Santiago de Compostela, Spain, in 1995. He is currently an Adjunct Professor at UCLA. He is also a Scientific Consultant to several companies in the United States and Italy and Director of IRECE, (Istituto di Ricerca per 1’Elettromagnetismo ed i Componenti Elettronici), Napoli, a research institute of CNR, the Italian National Council of Research. He has published widely in the field of applied electromagnetics (reflector antennas, transient phenomena, shielding, nonlinear propagation and scattering) and, more recently, in the field of SAR processing and simulation. He is the author of about 100 papers published on journals of recognized standard and three books. He has lectured in several summer schools in China, Great Britain, Holland, Italy, Spain, Sweden, and the United States.

Antonio Iodice (S’97) was born in Napoli, Italy, on July 4, 1968. He received the electronic engineering degree from the University of Napoli, “Federico II,” Napoli, Italy, in 1993. He is currently pursuing the Ph.D. degree in electromagnetism from the University of Napoli. He received a grant in 1995 from CNR (Italian National Council of Research) to be spent at IRECE (Istituto di Ricerca per 1’Elettromagnetismo ed i Componenti Elettronici), Napoli, Italy, for research in the field of remote sensing. His main research interests are in the field of SAR remote sensing: electromagnetic modeling and SAR interferometry.

Maurizio Migliaccio (M’91) was born in Napoli, Italy, on February 22, 1962. He received the Laurea degree in electronic engineering from the University of Napoli, “Federico II,” Napoli, Italy, in 1987. He was with the Department of Electronic Engineering, University of Napoli, and the IRECE (Istituto di Ricerca per l’Elettromagnetismo ed i Componenti Elettronici), Napoli. He was a Guest Scientist at DLR, Germany, within the SIR-C/XSAR interferometry project in 1995. He is currently a Research Scientist at the Istituto Universitario Navale, Istituto Teoria e Tecnica delle Onde Elettromagnetiche, Napoli, Italy. His main research activities are in the fields of simulation and modeling of synthetic aperture radar signals relevant to terrestrial and oceanic scenes as well as in the application of the fractal geometry to electromagnetic scattering and remote sensing. Mr. Migliaccio is member of the IEEE Geoscience and Remote Sensing and Antenna and Propagation societies and a member of the AGU.

Daniele Riccio (M’91) was born in Naples, Italy, on April 13, 1962. He received the electronic engineering degree from the University of Napoli, “Federico II,” Napoli, Italy, in 1989. He won several fellowships from private and public companies (SIP, Selenia, CNR, CORISTA, CRATI) for research in the remote-sensing field. He was a Research Scientist at IRECE (Istituto di Ricerca per 1’Elettromagnetismo ed i Componenti Elettronici), an Institute of the Italian Council of Research (CNR), Napoli. In 1994 and 1995, he was a Guest Scientist at the High Frequency Institute, DLR, Germany. Since 1994, he has been a Research Scientist at the Department of Electronic Engineering, University of Napoli. His main research activities are in the fields of simulation and modeling of synthetic aperture radar signals relevant to terrestrial and oceanic scenes as well as in the application of the fractal geometry to electromagnetic scattering and remote sensing.