2.20.1.1 Microwave high resolution imaging

Active sensors make use of radars, typically installed on spacecraft, or on aircraft, or even on ground. They transmit a coherent (i.e., well controlled at the level of a single oscillation) signal and record the echoes scattered back to the sensor from the observed area. Accordingly, they are independent from any external illumination sources: this peculiarity, together with the fact that they work at wavelength that are, differently from optical and infrared sensors, almost immune to the presence of clouds and fog, provide the system with the possibility to operate day and night and also in adverse weather conditions.

Modern SAR sensors transmit signals whose bandwidth is of the order of tens/hundreds MHz, thus leading to spatial resolution along the range (typically called across track direction) of the order of meters or fraction of meter.

For comparable optical aperture and antenna size, the spatial resolution along the flight track of images acquired by microwave sensors should be several order of magnitude worse than the optical images. This drawback is however overcome by the possibility to synthesize a very large antenna (of the order of few kilometers) by moving a much smaller real antenna along a straight trajectory corresponding to the platform flight track. This possibility, first postulated by Wiley [2] with the Doppler beam sharpening concept, is a direct consequence of the coherent nature of the system.

The operation of synthesizing a large antenna is today carried out typically off-line, after the downlink to a ground station, via a digital processing operation, usually referred to as SAR focusing operation, that, coherently combine on a 2D domain the echoes received from the radar at different positions. The obtained images are characterized by a resolution along the direction of array synthesis (typically referred to as along track direction) of the order of the length of the physical antenna dimension, independently from the wavelength and the height of the platform. Depending on the operative mode (scan, strip, and spot-mode) and on the fact that the platform can be disturbed during its motion by turbulences as in the airborne case, this operation can be in some cases more problematic.

SAR images are complex entities where the intensity basically measures the energy backscattered by the ground targets toward the sensor, which depends on the geometric (shape, roughness, and slope) and on physical (conductivity and permittivity) properties of the observed scene.

SAR sensors provide information about the observed scene complementary to that provided by optical systems. SAR images are nowadays used in many areas of interest: In glaciology they are used for glaciers monitoring and snow mapping, in agriculture for crop classification and soil moisture monitoring, in forestry for biomass estimation, etc. They are also used in environment monitoring for the detection of oil spills, flooding, as well as to monitor the urban growth or moving targets.

The technique that has opened probably the widest range of application is SAR Interferometry [3,4].

As in any electromagnetic coherent system the phase information is related to the travelled path, that is the distance target-sensor (range). Radar measurements therefore embed the information of distances with extreme high accuracy, on the order of wavelength fraction: due to the randomness of the scattering mechanism this information can be however extracted only as a relative measurement between different images. SAR Interferometry (InSAR) is a technique that, by exploiting at least two SAR images acquired from slightly different angles, allows retrieving the topography of the observed scene. A single SAR image provides measurement of measure the backscattering scene property only on a 2D domain, i.e., by performing a projection onto the plane containing the flight direction and the radar line of sight. Similarly to the human eyes system, height sensitivity can be achieved by combining two images of the same area acquired from two slightly different positions. The key principle of SAR interferometry is the use of the phase difference between SAR images for the accurate measurement of the distances of a target from two sensors displaced in location in order to create a parallax.

The two images can be acquired simultaneously if two antenna are present at the same time on the platform (single-pass interferometry) or through different passes of the same antenna (repeat-pass interferometry). In the latter case changes of the scene backscattering properties and variations of the phase delay contribution from the atmosphere may strongly impair the accuracy of the results. The accuracy of the topography estimation depends on the component orthogonal to the line of sight of the antenna vector separation commonly referred to as (spatial) baseline: for this reason this technique is also referred to as across-track interferometry.

As an alternative to topographic mapping, when the two antennas are present on the same platform and are separated along the flight direction, they acquire images repeated with a revisit of a few milliseconds. This is the case of the along-track interferometry which allows monitoring fast movements of targets on the ground. Applications concern for example the estimations of ocean currents or moving detection and velocity estimation [5].

An interesting extension of across and along-track InSAR is the Differential SAR Interferometry (DInSAR): by exploiting phase difference of images acquired at times (epochs), separated typically by some days, it allows accurately monitoring slow displacements over the epoch sequence. Differential interferometric data can be acquired by radar observations separated in time either from a single radar on one platform (e.g., ERS-1, JERS-1, ENIVSAT, TerraSAR-X) or from multiple radars on different platforms provided the radar have similar radar operating a viewing parameters (e.g., Cosmo Skymed constellation). Since the precision of radar in estimating distance is in the order of fraction of wavelength, DInSAR can estimate movements with sub-centimetric accuracy using L-, C-, or X-Band radars.

Major applications of this technique regard the natural hazard and security area. Besides, by exploiting archives of past images multipass techniques are also extremely useful to provide a past monitoring.

Interferometry applications have dramatically increased the use of microwave remote sensing for the environment monitoring: This is also testified by the growing interest of the major international space agencies in the development and launch of spaceborne SAR sensors satellites. The twin satellites ERS-1 and ERS-2 [6] of the European Space Agency (ESA), operative since 1992 and 1996, respectively, each one with a revisiting time of 35 days each, were characterized by the possibility to acquire a pairs of tandem images, i.e., interferometric images separated by only one day. ERS sensors from nineties to the first decade of 2000 were the very first systems used for the operative demonstration and routinely application of interferometry. Their acquisitions have been deeply exploited for years to develop most of the interferometric processing algorithm currently used and to demonstrate the potentials of the application of SAR Interferometry in several natural risk areas. Recently, the Italian Cosmo Skymed [7] and German TerraSAR-X [8] missions improved the quality of SAR product by providing images with spatial resolution up to one meter. Together with its twin satellite TanDEM-X, TerraSAR-X is going to provide the most accurate Digital Elevation Model (DEM), i.e., the topography, of the Earth on a global scale with a relative accuracy of 2 m for slope lower than 20° and 4 m for higher slopes with a spatial grid of 12 m. From the other side, the Italian COSMO-Skymed mission [7] is, worldwide, the unique constellation of more than two SAR sensors exploited also for civilian applications. It is composed of four medium-size satellites, each one equipped with an X-band high-resolution SAR system, allowing acquiring images on the same area every 4 days on average, thus both reducing the effects of decorrelation and allowing a more frequent imaging which is useful for interferometric application to cases of emergency.

Polarimetry [9–16] and Polarimetric SAR interferometry [17,18] and are techniques that use multi-polarization channels to extract further information on the scattering mechanism. Polarimetric information allows separating different scattering mechanisms. Whereas SAR polarimetry is a technique that use single antenna data, Polarimetric SAR interferometry use data acquired by two antennas. The former has a wide use in the field of classification, the latter allows generating interferograms corresponding to different scattering mechanisms and has an area of application in the field of forest height retrieval and biomass estimation.

The advances in the SAR hardware has allowed to reach very high imaging resolutions at microwaves on the order of a meter and has in parallel stimulated the development of advanced processing techniques able to extract from the data the highest possible information content.

One of the most important and recent innovations in SAR processing is associated with the extension of the imaging process form a 2D domain to a multi-dimensional domain. The so-called SAR Tomography has been among the first examples giving to SAR the ability of reconstructing images of the backscattering property of the scene also along the direction (elevation) orthogonal to the two classical dimension (azimuth and range). The key aspect of this technique is the possibility to synthesize, similarly to the flight direction (azimuth) an array also along the “height” direction to sharpen and steer the beam in such a way to measure the backscattering characteristics of the scene along the elevation direction and hence to generate full 3D images.

SAR Tomography allows vertically profiling (3D imaging) the backscattering to detect targets which interfere in the same pixel of a single SAR image and even monitoring, with the extension of the imaging properties to the time direction (4D Imaging), their individual deformation. Beside the application to a distributed scenario such as forest where the scattering is distributed along the height, the tomographic technique also provide significant advances in the application to the imaging and monitoring of in areas characterized by an high density of scatterers, such as urban areas, opening the possibility to achieve dense imaging and monitoring of single buildings and individual structures from the space, for the first time comparable to what obtainable with in situ systems like laser scanner [19,20].

Polarimetric SAR tomography [21–24] takes benefits of both polarimetry and tomography: by accessing the multibaseline information on different polarization channels, it allows retrieving scattering profiles along the elevation direction associated with different scattering mechanisms such as single bounce, double bounce and volume scattering. This work concentrates on the development of SAR interferometry (including multipass Differential SAR Interferometry) and Tomography for 3D reconstruction and target deformation monitoring.

2.20.2.1 High resolution image formation

Among the several parameter characterizing an image, resolution plays certainly a major role. In the radar case the resolution along the range coordinate depends on the system bandwidth [25,26]. Large bandwidths are obtained, with simplified (i.e., low peak power) hardware, by transmitting long duration linear frequency modulated (chirp) pulses which are, after echo reception, compressed (typically on the ground) via correlation techniques: this operation is commonly referred to as range pulse-compression or range focusing (see Figure 20.1).

The transmitted chirp pulse has the following expression:

(20.1)

wherein rect[] is the window function, is the pulse duration, and is the Chirp rate . The correlation of the response of a target at range r with the transmitted pulse replica provides the expression of the range impulse response function (IRF), also known as range Point Spread Function (PRF):

(20.2)

with c being the light-speed and the bandwidth of the transmitted pulse. The (3 dB) range resolution is numerically given by [27]

(20.3)

Such a resolution value is also referred to as slant range resolution to highlight that it refers to the line-of-sight (LOS) direction. Scaling factors derived by standard trigonometry should be applied to achieve the resolution along the main scene direction, for instance along the direction corresponding to the projection of range onto the local cartographic reference system usually referred to as ground range resolution:

(20.4)

where is the so-called incident angle, defined as the angle between the radar LOS and the local normal to the surface at the point of the reflection on the ground (see Figure 20.2), and is the terrain slope. The ground range resolution is, of course, coarser than the slant range resolution.

Note that in the absence of terrain slope, the incidence angle is equal to the angle (known as look angle and defined with respect to the nadir direction) only when the Earth curvature can be neglected, i.e., as in the case of airborne sensors operating at low altitude.

In case of flat ground and if the SAR antenna beamwidth in the range direction is not too big, as usually occur for instance for new generation X-band SAR sensors, the ground resolution is almost constant along the footprint. Differently, in case of non-flat ground, as shown in Figure 20.3, the ground resolution can change significantly, as effect of topography, giving rise to the well known effects of foreshortening, layover, and shadowing. Under conditions of foreshortening different resolution cells can contain contribution of very different, in terms of dimension, ground areas (see Figure 20.3b). Layover is beyond the limit case just described: in this case, points more distant in the ground can appear closer to the SAR radar sensor, and are mapped erroneously in the SAR image (see Figure 20.3c). Such effect is very common in mountainous areas with steep slopes, and in urban areas [4]. The shadowing effect occurs when ground area are masked by reliefs, as in Figure 20.3d. In this last case, a slant range resolution cell does not map any ground area (the ground area BC in Figure 20.3d is not seen from the SAR antenna). The above presented distortion effects makes that SAR images of mountain and urban areas can look very different from optical images, as an effect of geometrical distortions.

In the azimuth direction the focusing operation is necessary to synthesize a long antenna with higher resolution capabilities, that is for achieving the beam sharpening.

With reference to Figure 20.4 where the system imaging geometry represented in the flight direction, the system “senses” the scene by transmitting pulses at regular time instants, regulated by the pulse repetition frequency. The echoes collected in each position may be coherently processed in such a way to synthesize (digitally) an antenna whose dimension is equal to the footprint (X) of the real antenna [25,28]:

(20.5)

where is the wavelength, L is the azimuth length of the real antenna, and r is the range of the target (range). Note that is the angular aperture of the real SAR antenna in the azimuth direction. The final resolution of the image, provided by the synthetic antenna is [25]:

(20.6)

where the resolution gain factor 2 in the first equality is associated with the capability of the array to transmit and receive the radiation from each position of the real antenna during the synthetic antenna formation.

The time interval in which the scatterer is illuminated is referred to as integration time: for a standard operating mode, as that illustrated in Figure 20.4 (referred to as stripmap mode), it is trivially given by the ratio between the real antenna footprint and the platform velocity (v).

A dual approach used for the computation of the azimuth resolution of the focused image is provided by the so called Doppler analysis which states that when either transmitters or receiver are subject to a uniform motion, the received radiation is subject to a frequency shift (called Doppler shift) equal , with being the velocity vector (in our case of the platform) and being the versor of the receiving radiation (in our case the direction locating the scatterer from the platform). It is therefore clear that, during the integration interval, the echo from the target sweeps an interval from to , which corresponds to a Doppler frequency interval from to . Accordingly, the Doppler bandwidth amounts to:

(20.7)

Such a bandwidth is able to provide pulses with time duration equals L/(2v), which corresponds a spatial extent of about L/2. Unfortunately, the relative motion between the sensor and the target introduces also linear (phase) distortions as well as motion through resolution cell. Therefore, to provide short duration pulses the phase distortions affecting the available bandwidth must be compensated at the azimuth focusing level with the use of filters which are intrinsically 2D and also space variant only (for rectilinear tracks) with the range. The SAR focusing topic is out of the scope of this work, readers can refer to [25,28].

2.20.2.2 Operational modes

The classical operational mode of a SAR system considers the antenna pointing with a fixed offset from the flight direction, non-necessary orthogonal (i.e., broadside) pointing: this is referred to as Stripmap mode to highlight the fact that the scene is illuminated along a strip. The stripmap imaging mode geometry is depicted in Figures 20.4 and 20.5a.

In this way the integration time for forming the image of a target is, as discussed in the previous section, limited by the ratio between the real antenna azimuth footprint dimension and the platform velocity. This poses a limitation on the maximum achievable resolution. Another limitation of this imaging mode is associated with the coverage of the imaged strip in the slant range direction, which is in this case provided by the ground range extent of the real antenna footprint [25]. Slant range coverage and imaging resolution can be traded-off by operating a beam steering during the acquisition. Beside the classical stripmap mode, the two most known operational mode are spotlight and scan mode.

In the Spotlight mode the antenna beam is steered backward with respect to the antenna flight direction in such a way to collect data from a fixed area on ground on a longer (compared with the classical stripmap mode) flight segment [26,29,30]. The spotlight imaging mode geometry is depicted in Figure 20.5b.

In particular with respect to the stripmap mode (Figure 20.5) where the beam orientation is fixed, in the spotlight mode what is fixed is the illuminated area: this spotlight configuration is usually referred to as staring spotlight mode. A configuration that allow obtaining illumination interval and hence resolutions between the stripmap and staring spotlight mode is the so called sliding spotlight in which the angular beam steering rate is reduced in such a way to allow the footprint to slide on the ground [31,32]. With respect to the staring spotlight, the resolution loss is compensated by an increase of the azimuth coverage.

A mode complementary to the spotlight is the ScanSAR mode [33] whose geometry is shown in Figure 20.6. In this case the antenna is steered in the range direction to increase the range coverage. During the aperture synthesis in azimuth, the beam is regularly steered in range to sweep among a fixed number (typically 2–4) of adjacent range subswaths. The sweep mechanism is carried out in such a way to avoid gaps along the azimuth direction over the subswaths, i.e., to avoid the presence of areas which are not illuminated in the azimuth direction. The data collected in an illumination sub-interval for a generic subswath is called burst. In the stripmap case each target is imaged from the whole antenna beam and therefore the radiometric accuracy is preserved, that is homogenous area are imaged in a (average) constant backscattering level. In the ScanSAR case, as the target is seen only from a, or from a few small portions of the azimuth beam during the burst acquisition, not only we have a reduction of the azimuth resolution (this is the price for the increase in the range cover), but also different areas can be seen by different portions of the azimuth antenna beam. The latter effect produces radiometric losses seen as stripes along the azimuth (scalloping): homogenous area are imaged in a variable backscattering level. A mitigation of the scalloping problem is achieved by the adopting a TOPS (literally reverse of SPOT) acquisition mode [34]. In this case, in addition to the range steering, a steering in azimuth, with a forward rotation (i.e., opposite to that of the SPOT mode), is carried out to allow the azimuth beam to run forward, faster that the platform, in such a way that almost all scatterers in azimuth are imaged by the highest possible beam portion.

The azimuth resolution for the different modes can be evaluated by referring to the Doppler bandwidth, evaluated in (20.7), which can be written as a product of the Doppler rate and the integration time :

(20.8)

Equation (20.8) follows directly from the fact that the signal collected along the azimuth direction (slow time) is with a good approximation a linear frequency modulated pulse; the associated Doppler rate equals:

(20.9)

In the Stripmap case the integration time is fixed by the real antenna beamwidth:

(20.10)

By substituting Eqs. (20.10) and (20.9) in Eq. (20.8), Eq. (20.7) is obtained. In the Scansar and Spotlight cases, due to the range or azimuth antenna sweep, the integration time is fixed to a value which is lower and higher than the limit in (20.10), respectively, in order to select the wanted azimuth resolution.

It is important to point out also that, while in the Stripmap case the spectral properties of the received signal are time invariant along the azimuth, in the case of Scansar and Spotlight acquisition, due to the antenna steering, the spectral properties show an azimuth space variance. For instance, in the spotlight mode, and particularly in the sliding spotlight configuration, due to the difference between the platform and footprint velocity, the angular view of the system to the scene is azimuth dependent and accordingly, the received Doppler bandwidth is progressively translated from positive to negative frequencies. All these aspects must be accounted during post processing, such as for instance image resampling and/or image filtering as in the case of SAR interferometry [35].

2.20.3.1 Across-track SAR interferometry for measuring the surface topography

The regular and controlled oscillation of coherent radiation used in SAR systems allows determining with high accuracy the variations of the propagation distance: such a basic property is the key principle of interferometric techniques.

A single SAR image provides measurement about the backscattering scene property along two directions: The target range (i.e., the distance of the target from the illumination track) and the position of the target along the track (the azimuth direction). Hence, no information is provided on the angle under which the target is imaged (look angle). Knowledge of the latter information completes the set of coordinates in a cylindrical reference system with the axis coincident with the track, thus allowing a full localization of the scatterers in 3D and therefore an estimation of topography.

Similar to the mechanism used in human visual system for the determination of the depth, SAR interferometry is a technique that exploits the parallax in the view of the scene to allows extending the capability of a single SAR system to the reconstruction of the scene elevation profile: as SAR is sensitive to distance whereas the visual system is sensitive to angles, the mechanism is indeed slightly different.

Figure 20.7 shows the geometry of a basic (two-antenna) interferometric acquisition in the plane orthogonal to the flight track: it is clear that by measuring the range of the target with a single (master) SAR system (say the blue line) it is not possible to uniquely localize the position of the scatterer because at the same range would be located all of the points distributed on a equi-range curve (the blue¹ one) in the elevation beam (dash line).

By using a second (slave) antenna that images the scene from a different look angle the system is able to measure also the range from a second location [48]: there is then only one point (the intersection of the two equi-range, i.e., blue and red lines in Figure 20.7) that obey to the distance measurements pair. The larger the separation between the two antennas, the sharper the crossing and hence the higher the height accuracy. As in the visual system the 3D sensitivity is given by the difference in the location of the object in the two images at the different eyes [37], the accuracy of the stereometric system in Figure 20.7 is related to the variation of the distance of the target from one to the other antenna (range difference). To provide sufficient accuracy in such a range variation, SAR interferometry uses the phase difference between the two SAR images: the path difference is hence measured to an accuracy which is a fraction of the wavelength (centimeters at microwaves).

Specifically, the terms that plays the key role in the determination of the height is the path difference . In particular, at large distances it can be shown that the (variation of) the path difference (with respect to a reference point for instance located on a plane) is [37,39]:

(20.11)

where (see Figure 20.7) is the variation of the look angle between the master and slave antenna, is the orthogonal baseline component, that is the component orthogonal to the (master) line of sight of the vector (baseline) connecting the two satellites, and is the incidence angle, that is the angle between the vertical direction related to the target and the direction of the incoming radiation (line of sight). It is however important to note that Eq. (20.11) represent a simplification of the true scenario which is useful to understand the key principles of across-track interferometry. In the reality, the height is referred to a geographic or cartographic reference system and determined by measuring and by knowing the orbital state vectors [49].

In SAR interferometry the path difference is measured to accuracy of the order of the wavelength by using the phase difference signal:

(20.12)

For application to topographic mapping the two interferometric images can, or better should be acquired at the same time by two antennas on the same platform (bistatic system). This is because changes in the scattering properties, as well as differences in the propagation phase delay through the atmosphere strongly impact the quality of the retrieved DEM. In the case of the Shuttle Radar Topography Mission (SRTM) in 2001 an extensible boom 60 m long was mounted on-board the Shuttle to separate the slave from the master antenna available in the fuselage [50]. The German TanDEM-X mission of 2010 is instead the first example of a bistatic system composed by two twin satellites (TerrSAR-X and Tandem-X) orbiting in a close (500 m one behind the other) formation [51].

2.20.3.2 Statistical characterization of across-track SAR interferometric signals

As mentioned before, an Across Track SAR interferometric (InSAR) system is used to reconstruct earth topography, providing high precision Digital Elevation Model (DEM) of Earth surface. The geometry of an InSAR system has already been shown in Figure 20.7, where two SAR systems look at the scene from two slightly different tracks. As already introduced, the distance b between the two SAR tracks is called baseline, its component orthogonal to the look direction is the orthogonal baseline, while its component parallel to the look direction (to the slant range) is the parallel baseline.

In order to understand how an InSAR system works, consider the distance between the first SAR antenna and a point target T on the ground, and the distance between the second SAR antenna and the same point target T, as shown in Figure 20.7, while and denote the angles at which the two antennas look at the point target on the ground (slightly different from each other).

Consider now the two complex (envelope of the) images and obtained processing raw data collected by the two SAR sensors, where (n,m) are the discrete coordinates corresponding to the continuous azimuth and range coordinates (x,r). Such images can be considered as random processes whose expression is [52]:

(20.13)

where is the complex envelope of the (deterministic) ground reflectivity function (which, in first approximation, can been assumed to be constant with the antenna position k  1, 2, since, as commented above, the view angles change only slightly from one position to another), are phase factors related to the different propagation paths between the two antenna positions and the point target, and

(20.14)

is the random process representing the multiplicative speckle noise at the kth antenna, typical of any coherent system, which is commonly assumed to be a complex Gaussian correlated process with zero mean and unit variance [53]. Of course, are also random processes.

As result of the SAR signal model given by (20.13) and (20.14), the phase of a SAR image pixel (n,m) is given by three main contributions:

Other phase terms related to geometrical uncertainties, to random propagation effects, or to the changes of the scattering mechanism between the two SAR image acquisitions (for instance, due to the time delay between the two acquisitions) can be also present in Eq. (20.13).

After a processing called image registration which aims to locate the response of a target in the azimuth range pixel in the two images same pixel [37,54], the two SAR images and are used to build the so-called multi-look SAR interferogram:

(20.15)

where arg( denotes the principal value of the phase, is the number of looks [26], and the explicit dependence on (n,m) has been understood (the same will be made in the following). Equation (20.15) represents, for homogeneous targets, the Maximum Likelihood Estimator (MLE) of the interferometric phase [39]. In the following, we will consider the single look case, with .

From (20.12) and (20.13), it is easy to show that the interferometric phases are related to the observed scene height profile through the well known mapping [3,53]:

(20.16)

where denotes the modulo operation and

(20.17)

is the decorrelation phase noise related to the phase differences of the speckle. In Eq. (20.16) it has been assumed, as commented before, that the scattering phase term , is constant in the two SAR images, so their differences vanishes.

The problem to be solved in across track InSAR consists of estimating the height values , starting from the measured (then, noisy) wrapped phases . Such problem is worldwide known as phase-unwrapping problem, as it amounts to find the unwrapped phase (not constrained to belong to the interval [0,2)) corresponding to the measured wrapped phase (constrained to belong to the interval [0,2)) [55]:

(20.18)

The unwrapped phase will be proportional to an estimate of the height h, according to the model given by Eq. (20.12):

(20.19)

Once that the phase unwrapping problem (20.18) has been solved, an estimate of the height profile can be obtained from Eq. (20.19).

Equation (20.19) shows how much sensitive is the unwrapped phase with the height. Considering a fixed geometry for the satellite (or airplane) carrying the SAR antennas ( and are fixed), it is easy to understand that the larger the orthogonal baseline and the larger the frequency, the more sensitive is the SAR interferometer. In other words, if we want to measure the height profile h, it could seem more convenient to use a larger baseline and a higher frequency, because for a given variance of the phase noise, the corresponding height variance (inaccuracy) decreases (see Eq. (20.19)). However, an increase of the baseline value may contributes to decorrelate the two speckle phase contributions ( and , thus increasing the interferometric phase noise (geometrical or spatial decorrelation) [56,57]. For a distributed scattering the correlation between the two speckle contributions decreases linearly with the increase of the baseline [37,39], for a point scatterer the decorrelation disappear because such targets are not affected by speckle. The difference between such two scattering mechanisms influences also the multipass interferometric processing chains (see Section 2.20.4). In any case an increase in the baseline impact also the degree of complexity of the phase unwrapping step described in the following sections.

Before describing in the following sub-section part the several methods that have been proposed in the scientific literature to solve the phase unwrapping problem [42,58–63], it is important to describe the random nature of the SAR complex signal and of the SAR phase terms.

Consider the random terms given by (20.14), representing the multiplicative speckle noise present in the SAR complex signals given by (20.13). Such random terms can be modeled as zero mean, mutually correlated Gaussian complex variables with unit variance, which assume uncorrelated values in adjacent pixels, and have real and imaginary parts mutually uncorrelated [52]. By understanding the dependence on range and azimuth discrete co-ordinates n and m, for the sake of simplicity of notation, we can consider the vector:

(20.20)

whose (real valued) elements and , with k  1,2, denoting the cosine and sine components of the speckle signal , are zero mean Gaussian random variables. The assumed statistical model implies that [52]:

Note that the independence is true if the speckle band-pass spectrum is Hermitian respect to the central frequency, assumption which can be considered always satisfied since the speckle vector is related to the complex envelope of a modulated real signal [64]. Moreover, from the assumption (b), it stems that .

In these assumptions, the probability density function of the vector is given by [65]:

(20.21)

where , and C is the covariance matrix given by:

(20.22)

where is the correlation coefficient of and , which by virtue of assumption (b) assume the same value of the correlation coefficient of and , given by:

(20.23)

where denotes expectation. Note that in Eq. (20.23) the terms present at the denominator are equal to one, as they represent the unit variance of the considered processes, so that their explicit presence should not be necessary. Nonetheless, we use such definition, as it is valid also in the case of not-normalized processes.

We note that, according to the assumptions (a) and (b), given by (20.23) is equal to the interferometric coherence usually employed in InSAR systems [57], defined starting from complex signals [3]:

(20.24)

Note that first result of Eq. (20.24) implies that the coherence of the (complex) speckle noise is real valued, due to the above assumptions (a) and (b). Note also that the module of the coherence of the complex received signals and is, in module, equal to the coherence of the (complex) speckle noise.

Starting from Eq. (20.21), it is possible to derive, by a change of variable from Cartesian to polar (from to , and following integration with respect to and , the pdf of the single-look speckle phase difference [53]:

(20.25)

where the dependence of phase difference and on (n,m) has been, as before, understood.

The coherence coefficient is influenced by all factors that cause differences between the two complex speckle images and . The larger these differences, the smaller the coherence coefficient’s value. A coherence reduction can be induced by actual physical changes occurring between the acquisition times of the two data sets (temporal decorrelation) and/or by changes of the ground reflectivity when it is seen from different angles (spatial decorrelation) [57]. Note that also the coherence is a function of the ground coordinate pair (n,m), so that it may change across the image.

The plot of the speckle phase difference pdf (20.25) for different values of the coherence coefficient is given in Figure 20.8. It has to be noted that the pdf become less picked by reducing the coherence value. The smaller the coherence value, the larger the variance of the speckle phase noise.

The joint pdf of the interferometric phases can be obtained from the joint pdf of the speckle noise phase given by Eq. (20.25) and plotted in Figure 20.8, exploiting the random variable transformation given by Eq. (20.16), which leads to:

(20.26)

where the dependence of phase difference , and on (n,m) has been again understood. The pdf’s (20.26) have the same shape of pdf of Eq. (20.25), but they are centered on the value h.

Such pdf family, that as it can be noted is parametrized by the height h, will be used as starting point of some of the phase unwrapping methods described in the following section.

The pdf (20.26) of the interferometric phase is strongly influenced by the coherence. Such parameter can be written as the product of four main contributions [3,39,57]:

(20.27)

where represents the influence of thermal noise in the receiver, represents the decorrelation effects due to the different SAR view acquisition angles, depending upon the spatial baseline, represents the decorrelation effects due to volume scattering mechanisms, and represents the so called temporal decorrelation effects [3,4].

The first factor in Eq. (20.27) can be computed starting from the circular Gaussian and independent nature of thermal noise and is given by:

(20.28)

where and are the signal to noise ratio on the two receiving SAR interferometric antennas [4,53,57].

The second factor in Eq. (20.27) is the so-called geometric coherence, also referred to as angular or baseline coherence; it is present for all scattering situations, it depends on the system parameters and on the overall observation geometry, including the different SAR view acquisition angles, and depending upon the spatial baseline.

Geometric coherence values can be easily computed, in case of flat terrain, and for a white scattering process, leading to [3]:

(20.29)

where is the orthogonal baseline and is the orthogonal critical baseline given by:

(20.30)

where all symbols in Eq. (20.30) have been previously introduced.

Geometric decorrelation effects, for flat terrain geometry, can be explained also from the so-called spectral shift effect [55]. This interpretation allows also to derive a filtering strategy of the interferometric channels aimed at mitigating such decorrelation. Such proper processing is also called “common band filtering” [55,66], because requires to process, to maximize the geometric coherence value, the common (overlapped) part of the spectrum of the two interferometric signals. The larger the baseline, the larger the terrain slope, the less the common part of the two spectra, and the larger the decorrelation effects. For a non-flat topography the approach in [67] can be adopted. For the ERS and ASAR-ENVISAT sensors, the critical baseline is of about 1100 m for , while for the last generations high resolution systems such as COSMO-Skymed and TerraSAR-X, this value is significantly enlarged. Therefore, for these new generation sensors, thanks to fact that the distribution of the baseline values is bounded by an “orbital tubes” significantly smaller than the critical baseline, such common band filtering is typically not required.

Similar effects can be induced also in the azimuth direction by the presence of variations of the azimuth antenna pointing [67]. This effect can be critical in some acquisition modes where antenna steering is present, such as in ScanSAR and Spotlight modes. Also in this case, the larger the acquisition geometric diversity, the less the spectra overlapping, and the larger the decorrelation effects.

The third factor in Eq. (20.27), is the volume coherence, it is due to volume scattering, and it is the effect induced by the scattering layer to increase the size projected range cell, and consequently, to decrease the correlation distance. As in the case of , it is dependent on the spatial baseline.

The last factor in Eq. (20.27) represents the so called temporal decorrelation [57] due to the instability of scattering mechanisms in the two different acquisition times, as the structure of the scatterer can change in the meantime between the two acquisitions. Such effect can be very important when the two SAR images are acquired at distance of several days or months in the case of vegetated or agricultural areas, or in presence of different climatic conditions.

2.20.3.3 The differential SAR interferometry technique for measuring displacements

Differential Interferometry (DInSAR) is a particular configuration of SAR interferometry. The reference geometry is the same of the classical InSAR case, but the target on the ground is allowed to move, displacing of say , between the two successive passes (see Figure 20.9).

In the following, for sake of simplicity, we indicate deterministic and stochastic terms all with non-capital symbols: the nature is specified whenever ambiguous. In this case, the interferometric phase is composed by three main factors:

(20.31)

where is the measured range variation that, in the far field observation approximation, is equal to the component of the displacement along the line of sight, is the phase contribution corresponding to the target height as in Eq. (20.12), is a stochastic term associated with the variation, between the two passes, of the wave propagation delay through the atmosphere, is the phase noise which in this case includes also the temporal decorrelation effects in addition to the decorrelation noise due to the speckle ( in Section 2.20.3.2). In the cases in which the topographic contribution is limited, that is if the baseline is negligible and/or an external DEM is available to compute and cancel out from Eq. (20.31), and in the case of a predominance of the deformation component and/or limited effects of atmosphere, displacements can be measured to accuracy which are on the order of the wavelength. By using this classical two passes DInSAR configuration in the past Scientists have been able to capture the surface deformation field generated by major earthquakes, or highlight deformation associated with volcanic activities.

The idea of mapping ground deformation via the interference of signals acquired by SAR systems was demonstrated for airborne systems in [43] and for the very first time using real data from the European Remote Sensing Satellite (ERS) in keystone experiments by [68], for ice-stream velocity measures in Antarctica, and by [69] for the co-seismic deformation field generated by the Landers earthquake (CA-USA). The Landers result received cover of Nature (vol. 364, 8 July 1993, Issue No. 6433) with a title “The image of an Earthquake” that translates the importance of the achievements and of the DInSAR technology with reference to application to seismic and to geo-hazards in general.

Today, with the availability of many SAR sensors with interferometric capabilities orbiting around the Earth, co-seismic, i.e., before and after main seismic events (see Figure 20.10), DInSAR data are almost analyzed routinely by scientists to study the displacements induced by known and unknown geological faults that are the causes of catastrophic events all over the world.

An example of measurement of DInSAR co-seismic displacement is reported to provide an idea of the powerfulness of this technology with reference to the 6.6 Mw Iran earthquake in 2003 that stroke the city of Bam. The image Figure 20.10, shows the co-seismic interferogram obtained by interferometric combination of the SCANSAR (Wide Swath Mode) acquisition of September 24th, 2003 and the STRIPMAP (Image Mode) acquisition of December 3rd, 2003: each color-cycle correspond to 1.55 cm in the line of sight. The coherence of the data and therefore the quality of the interferogram is very high due to the arid nature of the region: each color-cycle correspond to 2.8 cm displacement in the line of sight.

A key factor in such application is associated with the revisiting time. The retired satellites ERS-1, ERS-2, and ENVISAT of the Europen Space Agency have been the satellites on which repeat pass interferometry has been experimented for the very first time. These satellites were characterized in the normal operational situation by a strip width of approximately 100 Km EW and a revisiting time (i.e., the time necessary to repeat approximately the same orbit) of 35 days. The revisiting time poses a limitation to the minimum number of days necessary to generate an interferogram. The new generation of sensors operating at slightly lower orbits with respect to ERS and ENVISAT, such as TerraSAR-X and Tandem-X allows reducing the revisiting time to 11 days. The Italian COSMO-SkyMed (CSK) constellation is a constellation formed by four SAR satellites that acquires data for interferometric use, regardless of the specific satellite. This peculiarity allows CSK to provide the highest maximum revisit rate of an area of interest, that is one acquisition every 4 days (on average) for the whole constellation, instead of one acquisition every 16 days for a single satellite. CSK and TerraSAR-X provide spatial resolution (between and one order of magnitude better than the previous available C-Band satellite SAR data.

These systems operates in X-band and are characterized by higher spatial resolution with respect to the past ESA C-Band satellites; the counterpart of these advantages are however the reduced swath coverage which in the classical stripmap imaging mode reduces to 40 Km in the EW direction. An example of co-seismic interferogram obtained with a 8 days temporal baseline form COSMO-Skymed data is discussed in the following. On 6 April, 2009 the MW 6.3 L’Aquila earthquake occurred in the Central Apennines (Italy) causing extensive damage to the town of L’Aquila and killing 300 inhabitants. The event epicenter was located few kilometers southwest of the town of L’Aquila, the main shock nucleated at a depth of 9 km, was preceded by a preseismic sequence with the largest shock having a ML 4 magnitude, and was followed by a vigorous aftershock sequence. In Figure 20.11 it is shown the interferogram evaluated by the CSK pair of April 4th and April 12th: the activated fault is located NW-SE emerging to the right of the dense fringes area: each color-cycle correspond to 1.55 cm in the line of sight.

Since then, many experiments showed the potentiality of the technique in detecting deformation phenomena not only associated with earthquakes [70] but also in volcanic areas [71,72] and of glaciers [73].

However, in order to fully exploit the potentiality of the SAR technology in order to measure deformation with a centimetric/millimetric accuracy two or few images are typically not sufficient. At such accuracy level the presence of the atmospheric component and additional disturbing contributions such as orbital inaccuracies cannot be in fact neglected. For sake of simplicity we indicate still with the differential interferometric phase obtained after the subtraction of the contribution associated with the DEM (differential interferogram). We have therefore:

(20.32)

where is associated with orbital inaccuracies which affect the computation of the phase contribution associated with the topography (baseline error): as for Eq. (20.31), the noise term is associated with the noise contributions such as decorrelation of the response not only due to variation of the speckle contribution due to the angular imaging diversity (spatial baseline decorrelation), but also to changes of the backscattering response over the time (temporal decorrelation). Availability of on board GPS systems allows significantly mitigating the effects of orbital errors; for airborne systems which are subject to trajectory deviations due to turbulence the GPS must be integrated with accurate inertial navigation systems. Due to the DEM subtraction is now the residual target height, i.e., the height of the target with respect to the reference DEM. Accordingly, to have the possibility to achieve measurement of small deformation components, use of accurate external DEM as well as small baseline separations, is mandatory. In any case the atmospheric component plays a major role because it causes the presence of errors which are spatially correlated and therefore may be mixed with possible deformations.

The atmospheric contribution is typically separated in two components: a turbulent component which is associated with the air inhomogeneity and that causes a spatial variation of the Atmospheric Phase Delay (APD) and a stratified component which is associated with the vertical stratification of the atmosphere. Both these terms occur in the lower part of the atmosphere, the troposphere, whereas contribution in the upper part of the atmosphere mainly show contributions which are very low spatial variable and can be misinterpreted as orbital inaccuracies. The former is commonly referred to as wet component and is dependent on the relative humidity, the latter, also referred to as hydrostatic or improperly as dry component, is responsible of a contribution which is highly correlated with the topography and therefore almost negligible on quasi flat areas. A model that describes the statistical behavior of the turbulent component is due to Kolmogorov. In this case the turbulence is assumed spatially stationary and isotropic. The refractivity index, i.e., the excess in parts per million of the reflectivity index with respect to the vacuum, that provided the increase in the path difference due to the crossing of the atmosphere can be modeled in terms of the variogram, i.e., the variance of the difference of the refractivity contributions between two points. For separations below the order of a kilometer, the variance of the difference of the refractivity between two points small and decreasing with a power of 2/3 of the distance. A more thoughtful characterization and analysis of the tropospheric contribution can be found in [74].

The temporal correlation of the atmosphere is however typically low: this means that APD contributions over different epochs can be averaged together to diminish it contribution on the path difference. Therefore to measure small (up to millimeter per year) displacements and to handle the problem of monitoring at high resolution ground scatterers, techniques based on the use of several images acquired over the same scene have been developed. In fact, by exploiting a higher dimensional acquisition space, the “phase firms” of the different components such as DEM error, displacement, and APD can be deterministically or stochastically characterized and estimated directly from the received data. This topic is treated in more details in Section 2.20.5.

2.20.3.4 Phase unwrapping

Unwrapping aims at reconstructing an unrestricted (absolute²) signal starting from a measured wrapped signal version restricted a reference interval. In the case of interferometry, the phase is intrinsically wrapped in the () interval being extracted from complex values. Accordingly, we have:

(20.33)

where and are the restricted (wrapped) and unrestricted (absolute) phase. Phase unwrapping is a step necessary to reconstruct a phase signal which is an estimate of .

Figure 20.12 reports an example in a 1D case: phase unwrapping aims to estimate the absolute phase (shown in blue) starting from the measured restricted phase (shown in red).

It should also be noted that the term “absolute” phase in the interferometric context is typically used to refer to the phase corresponding to which has been corrected by an offset to account for the correct number of global cycles which are lost due to the wrapping operator and for the timing errors. In SAR interferometry such an offset is commonly evaluated after phase unwrapping by using one or more reference points with a known topography.

The problem of unwrapping is ambiguous as it admits in principle infinitely many solutions (the wrapped phase can be itself a possible absolute phase), and a reasonable solution can be obtained by imposing a certain degree of continuity: the absolute phase in Figure 20.12 is, among all possible absolute phase functions corresponding to the measured wrapped phase, the continuous one. Unfortunately, in real cases, in addition to the noise, the problem is further complicated by the presence of a finite sampling rate, see the measured red diamond samples in Figure 20.12 and the absolute black star samples corresponding to the absolute phase. Moreover, the actual solution could be locally not continuous.

In the following, some popular approaches to solve the phase unwrapping problem and to recover the height profile of the ground scene will be described.

2.20.4.1 Bayesian statistical solution to PhU problem

To solve the problem of the estimation of the ground elevation profile also Bayesian statistical techniques have been proposed [86–88]. In the framework of such techniques, in particular if a Maximum a Posteriori (MAP) estimation scheme is adopted, an a-priori joint pdf of the unknown height profile has to be introduced. It is based on the use of Markov Random Fields (MRF) as a-priori statistical term modeling pixels contextual statistical information in the 2D unknown height profile to be reconstructed. The MRF allows to describe the local spatial interaction between couples of pixels, through a set of model parameters (hyperparameters) which can be tuned following unsupervised procedures.

Consider a discrete (lexicographically ordered) two-dimensional (2D) points lattice , where is the number of pixels of the SAR image, and let the corresponding ground elevation values. Consider now an InSAR system, and let the wrapped phase values (single sample of a discrete interferogram) measured at the lattice point k and at nth interferogram. The wrapped phase values relative to the position k can be structured and ordered in the following way: let be the vector of the wrapped phases measured in k position at the N different interferograms, and be the vector collecting all available wrapped phase values. Then, h is the vector of the unknown height values, and the vector of the all available data (multi-interferogram).

The MAP estimation can be formulated as:

(20.60)

where is the a posteriori joint pdf of the unknown image, is the Bayesian likelihood function [89], and is the a-priori pdf of the unknown image.

The Bayesian likelihood function can be easily obtained by Eq. (20.59):

(20.61)

where the statistical independence of the interferograms in the different ground positions has been exploited. The Bayesian likelihood functions are formally equal to the likelihood function that can be obtained in the classical statistical case; in the Bayesian framework, differently from the classical one, the unknown image is seen as a sample of a random vector H.

The a-priori pdf is usually defined in such a way to express a-priori information about the unknown image, assigning high probability to particular pixels configurations. In the SAR interferometry case, being the unknown image representative of the ground elevation map of a geographic area, a strong contextual pixel information is very likely to be. In particular, it can be assumed that the unknown image can be modeled as a Markov Random Field (MRF) [90], a general image model able to represent contextual pixel information extending the 1-D Markov property to the 2-D case, whose corresponding joint pdf is given by a Gibbs distribution:

(20.62)

where is the so-called partition function (a constant factor needed to normalize the integral of the pdf to one), is the energy function, is the potential function between and is the neighbourhood system of kth pixel [91] (usually, the eight pixels around the kth one), is the hyperparameter vector, and are the hyperparameters. With such definition, the Gibbs-MRF model in Eq. (20.62), by means of the hyperparameters, well adapts to describe the image local nature, leading to a powerful and general model, well-suited to represent a very wide class of height profiles. The hyperparameter values are not know, of course, and they have to be estimated from the available interferometric data data.

The solution procedure essentially consists of two steps: ML estimation ( of the hyperparameter vector and MAP estimation ( of the actual realization of height profile process H.

This approach can work very well [87], even if the method is always limited by the coherence value and by the corresponding number of available independent acquisitions of the same scene. The lower are the coherence values over the entire image, the larger is the total number N of needed interferograms to obtain good quality reconstructions.

Performance of the Bayesian method can significantly outperform the one relevant to classic approaches [85,92] depending on the capability to estimate the a-priori model of the unknowns.

2.20.4.2 Graph cuts solution to PhU problem

Statistical approaches, especially the Bayesian ones, have proved to be effective dealing with noisy data and big discontinuities. However these algorithms can be time consuming and computationally heavy due to the a-priori model (estimation of hyperparamenters) and to the optimization step. It is possible to overcome these limits by introducing a fast and efficient (in term of global optimization) algorithm to unwrap the interferometric phase in the multichannel configuration [85].

To reduce the computational time needed to unwrap the multichannel interferometric phase, two aspects can be taken in consideration: first, a non-local a-priori energy function, the Total Variation (TV) model [93], and secondly an optimization algorithm based on graph cuts, Ishikawa algorithm [94], have been exploited.

The a-priori energy corresponding to the TV model can be written as follows:

(20.63)

Note that in this expression, is a scalar, making model in Eq. (20.63) a non local one, differently from the Bayesian model (20.62) presented in the previous section. This choice is done in order to make the algorithm faster. A non local a-priori model avoids the estimation of local hyperparameters , as only one parameter for the whole image has to be estimated.

Between the existing non local a-priori energy models, TV has been chosen, due to its main advantage: TV does not penalize discontinuities in the image while simultaneously not penalizing smooth functions either [93].

Given the TV a-priori energy model (20.63), the MAP estimation can be obtained from the minimization of the following function:

(20.64)

In order to minimize this energy function, graph-cut based optimization algorithms are used. Graph-cut optimization [95,96] is successful because the exact minimum or an approximate minimum with certain guaranties of quality can be found. Compared to the classical optimization algorithm, it provides comparable results with much less computational time and compared to the deterministic algorithm ICM [90], it avoids the risk of being trapped in local minima solution which can be far from the global one.

Let us, first, define a graph and a graph-cut problem. Suppose is a directed graph with non negative edge weights, where V is the set of vertexes and E the set of edges. This graph has two special vertexes (terminals) called the source s and the sink t. A s-t-cut C = {S,T} is defined as a partition of the vertexes into two disjoint sets S and T such that s  S and t  T. A cost of this cut is defined as the sum of weights of all edges that go from S to T. Figure 20.20a shows a simple graph made of four vertexes, two vertexes (x,y) plus the sink and the source (t,s). The edges are the links between the vertexes. An example of a cut is represented with the dashed line.

The minimum s-t-cut problem is to find a cut C with the smallest cost. This problem is exactly equivalent to its dual problem, which consists in computing the maximum flow from the source to the sink. Between the several algorithms proposed to solve the maximum flow problem, the one proposed in [97] turns to be the most adapted to computer vision problems.

To solve the MAP unwrapping problem, which means find the value of h that minimizes (20.64) the optimization procedure proposed by Ishikawa is implemented [94]. The interesting aspect of Ishikawa algorithm is that under some hypothesis on the energy to be minimized and if the graph is correctly constructed, the algorithm provides the global optimum of the considered energy.

Two hypothesis are at the base of Ishikawa algorithm: convexity of the a-priori energy and a linear order on the label set. Using the TV model the first hypothesis is satisfied. For the second hypothesis, in the case of phase unwrapping, the labels are the height of the image pixels. The heights are supposed to be represented as integers in the range {0,1,2,..,L−1}, where L is the size of the label sets. This condition satisfies the second hypothesis necessary for the Ishikawa graph construction.

Ishikawa method is based on computing a minimum cut in a particular graph. Ishikawa graph G = (V,E) contains nodes ( is the size of the image and L is the size of the label set) denoted by {}, plus two special nodes s and t. For each pixel k, we associate L nodes, that represent all the possible heights that the pixel k can take. The construction of Ishikawa graph, in case of a 1D image for legibility, is shown in Figure 20.20b.

Ishikawa graph contains three families of edges . is a set of directed edges called data edges (black arrows of Figure 20.20b). It represents the data energy term. is a set of directed edges called penalty edges (dotted arrows of Figure 20.20b). It ensures that only one data edge is in the minimum cut for each pixel k. Finally, is a set of constraint edges between all neighbor pixels (horizontal arrows in Figure 20.20b). It represents the a-priori energy term.

To better understand how to set the edges weight for the multichannel phase unwrapping problem let us consider Figure 20.20c. The vertex is the vertex identified by the pixel k and the label l. The cost of the edges in Ishikawa graph are reported in Eq. (20.65). For more details on the graph construction see [94].

(20.65)

Constructing the Ishikawa graph as shown and finding the cut with a minimum cost on this particular graph allows to find the exact (global) optimum solution for the multichannel phase unwrapping problem given by Eq. (20.64). The main disadvantage of Ishikawa method is related to the memory load. In fact, the algorithm stores the whole graph and then performs the cut. This can be a problem when the size of the images largely increase.

Before using Ishikawa algorithm, the hyperparameter has to estimated. The hyperparameter depends both on data energy and on a-priori energy term. A method to perform the hyperparameter parameter is the analysis of the so called L-curve. To find automatically the corner of the L-curve, the triangular method described in [98] can be used, providing good results in limited time.

2.20.5.1 Multipass phase unwrapping

As already stated in Section 2.20.4, Phase Unwrapping is necessary to extract the APD component and to estimate the non-linear deformation signal at either small or large scale: PhU is by far the most critical step of any A-DInSAR technique.

Most of PhU algorithms have been developed in the context of SAR Interferometry, and make use of the pixel-to-pixels spatial variations. One of the most used algorithm is the Minimum Cost Flow (MCF) PhU [60,79]: it is based on the use of triangulations defined on the sparse grid of useful pixels and the PhU is cast as the problem to minimize the L1 norm of a vector that corrects the measured phase differences subject to the zero-curl (i.e., non rotational) constraint, that is the constraint that forces to zero the circulation of the unwrapped phase differences on all the triangles. Very efficient MCF solvers are available in the framework of network-flow algorithms [60].

The MCF technique, typically exploited for the unwrapping of single interferograms, has been also extensively used in the multitemporal DInSAR context where commonly several interferograms must be unwrapped. More specifically, phase unwrapping of DInSAR data stacks is carried out by independently unwrapping each interferogram with the MCF optimization approach (single-step PhU).

The single-step PhU discards any relationship between the different interferograms, which are indeed all related to the same wanted signals; particularly it neglects the typical inherent redundant nature of the interferogram stack.

To overcome this limitation, the single-step PhU has been recently integrated with Model Based (MB) phase unwrapping algorithm that exploits the model in (20.67) also at the phase unwrapping stage. More specifically, starting from the measured interferograms, the MB PhU estimates the variations of the topography ( and deformation mean velocity ( over the set of spatial arcs defined on a grid of reliable pixels, i.e., with a coherence degree [118] which is above a threshold for a fixed percentage of the total number of interferograms:

(20.74)

where is the variation of the wrapped phase measured on the jth (out of spatial arc.

The set of all measured topography and deformation mean velocity deformation mean velocity, and , are then spatially integrated (generally in a Least-Square sense) over the network associated with the spatial arcs to retrieve the topography and deformation mean velocity at the pixel level. The phase signal corresponding to the signal component model is subtracted from each interferogram to ease the subsequent un-modeled unwrapping step, generally carried out with the application of the MCF algorithm applied independently to the each of the available interferograms [115,119].

Recently new algorithms has been proposed to improve the un-modeled PhU stage by exploiting the redundant nature of the interferograms [120–122].

A first contribution along this line is the two step PhU algorithm [122], where the use of a triangulation is in the spatial temporal baseline domain, see Figure 20.27, is used: each point represents an acquisition and arcs represents the interferograms. For each spatial arc, the interferometric measurements available in all the interferograms provide an estimate of the phase variations over the acquisition arcs in Figure 20.27. It is clear therefore that such phase variations must sum up to zero over closed circuits and therefore an MCF step can be therefore carried out in the acquisition domain to provide corrections of the phase variation over the selected spatial arc in each interferograms which are used for the subsequent MCF PhU implemented, as usual, in the spatial domain.

This two-step approach has the advantage to use the redundancy of the available interferograms, phase summation of interferograms over close loop should be zero, but imply a rigid scheme in the interferogram generation. The use of a triangulation does not allow fixing a limit to the baseline separation, as usually done in the SBAS approach, because the maximum baseline is defined by the triangulation scheme.

A solution to tackle problem is obtained by address the PhU problem in a generalized framework which makes use of the over-determined nature of the operator that relates the phase differences to the absolute phase values, [119]. More specifically, by referring to Figure 20.28 where a 3D representation of the stack of acquisition is shown, we let l and n indicating the pixel and the acquisition, respectively, and j the spatial arc. It is possible to define two operators implementing a differentiation along the acquisition (i.e., the interferogram generation) and along the space (spatial variation). A generic estimation of the absolute phase variation over the mth interferogram obtained by wrapping the difference of the interferogram phase values at the end point of a spatial arc can be seen as a result of a double differentiation (along the acquisitions and along the space) of the phase values:

(20.75)

where and are the indexes of the acquisitions of the mth interferogram on the jth spatial arc, and are the indexes of the acquisitions of the jth arc. Collecting all the measurements and unknowns for any spatial arc and interferogram in the vectors u and . Accordingly the following linear system can be written:

(20.76)

where is the matrix (typically with a number of rows larger than the number of columns due to the redundant selection of interferograms) that computes the differences at the right hand side of (20.75) along the acquisitions and space for all the interferograms (acquisition arcs) and spatial arcs; k is the vector of unknown 2 multiples. From (20.76) it is evident that errors in the measurement of u move the vector out of the range of the operator , and therefore the vector k must be selected as the vector that brings }. It is natural, as in the minimum cost flow approach to look for a correction vector k to be with integer values and with a convenient weighted minimum norm. An effective, and less computational demanding approach, is by exploiting the null-space matrix associated with , i.e., the matrix Z whose rows are the vectors of the null space of }, i.e., such that . By applying Z to the (20.76) we determine the a set of equations involving only the measurement u and the unknown vector k of 2 multiples. It is worth noting that in case triangulations are carried out in both the space and baseline domain Z is the matrix that evaluates the circulations over the elementary triangles. In conclusion, the vector k is determined by solving the following optimization problem:

(20.77)

where Z is the Left Null Space matrix associated with . The null-space approach is characterized by the desirable feature of being an approach in which the degree and the typology of redundancy of the measured interferometric phase variations (double differences) shall not follow any specific constraint, such as the triangulation scheme in the baseline domain used in [122] that imposes a critical constraint on the generation of interferograms. This characteristic allow to freely generate the set of interferograms.

2.20.6.1 Linear non-adaptive inversion for SAR tomography

A simple way to invert Eq. (20.81) and to recover the backscattering distribution is by applying the beamforming, i.e.,:

(20.82)

that is:

(20.83)

Note that the reconstruction of the backscattering sample is achieved via a filter which does not depend on the data (non-adaptive inversion).

Another possibility is offered by the Singular Value Decomposition (SVD) analysis of the operator in Eq. (20.79), i.e., by the SVD decomposition of the matrix A in the discrete case. In this latter case the assumption on the support can lead to a slight degree of super-resolution with respect to the Rayleigh resolution limit for elevation, given by:

(20.84)

where and are the maximum and minimum of {}. The corresponding height resolution is given by:

(20.85)

SVD decomposition provides the following fundamental equations pair:

(20.86)

(20.87)

where vector) and vector) are the left (data) and right (unknown) singular vectors of A and are the singular values. Note that we have assumed , i.e., a sampling of the elevation interval in a number of points which is larger than the number of acquisitions thus translating in a underdetermined characteristic of the matrix A.

Equations (20.86) and (20.87) represent the fundamental result of the SVD analysis: in particulars Eq. (20.86) states that in principle all the different vectors concur in the composition of the observed vector g; their contributions are however weighted by the singular values. Accordingly, low singular values indicates that the component along the corresponding data singular vector is attenuated by the imaging operator in the data formation and should be carefully treated when reconstructing the unknown via Eq. (20.87) because this data component may be overwhelmed by the unavoidable noise. As a result, typically the reconstruction in Eq. (20.87) is limited to the data and unknown singular vector pairs corresponding to significant singular values; the associated inversion scheme is referred to as Truncated SVD (TSVD). In the case in which the acquisitions are uniformly spaced with a separation of if the output sampling is chosen verifying the Nyquistic conditions, i.e., , i.e., N samples in the Nyquist interval with it can be shown that the operator in (20.81) is a Discrete Fourier Transform (DFT) matrix that is characterized by constant singular values. In such a case and therefore the direct and inverse operator in (20.86) and (20.87) becomes Hermitian conjugate pairs. In cases in which the singular values shows a decay which translate the presence of a redundancy in the acquired spectral samples that can be exploited to increase the resolution of the reconstruction below the Rayleigh limit in (20.84).

In other words, by progressively restricting the elevation ROI with respect to , a smooth decay in the singular values is observed. For a reduction factor , if the noise is small enough, the truncation can be arrested to a number of singular values which is larger than FN: in this condition a degree of super resolution is achieved [124].

In addition to the super-resolution degree, the SVD decomposition allows also to reduce the level of sidelobes with respect to a reconstruction carried out by simple Beamforming.

2.20.6.2 Linear adaptive inversion for SAR tomography

A more general expression for the evaluation of is given by:

(20.88)

where is the filter for the estimation of . Letting , that is the data covariance matrix, a solution obtained mutatis mutandis from the spectral estimation theory such that:

(20.89)

i.e., a solution that achieves the minimum output power, subject to unitary gain at the ‘frequency’ of interest (Capon filter), is provided by [128]:

(20.90)

Substituting (20.90) in (20.88) provides:

(20.91)

It is interesting to note that , white data spectrum, leads to . The advantage of the Capon filer is the achievement of higher super-resolution for line spectra (i.e., concentrated scatterers along s) compared to the SVD. However a disadvantage of the Capon filter is the need to estimate the data Covariance matrix. This estimation is carried out via spatial averaging, and operation known in the SAR processing as multi-look:

(20.92)

where g(l) is the data vector in the lth pixel located close to the pixels in which the tomographic processing aims to provide an estimate of the backscattering distribution.

Note that this operation inevitably leads to a loss of azimuth-range resolution. Note also that the reconstruction of the backscattering sample is in this case data dependent (adaptive inversion).

An example of reconstruction of a 3D SAR image on real data is provided in Figure 20.30: in this case the results are the first one obtained by using spaceborne SAR data, specifically data acquired by the ERS sensor [124]. In the first three columns the reconstructions obtained by SVD and Capon are reported. In particular six sections in the azimuth-elevation plane (i.e., at constant slant range) of the San Paolo stadium in Naples are shows: the locations of the constant range are indicated by the horizontal white line in the last column showing the classical azimuth-range representation. It is evident the capability of the tomographic technique to capture the 3D shape of the structure. It is also evident for the Capon approach that the increase in the height resolution is paid in terms of spatial (azimuth and/or range) resolution losses.

2.20.6.3 Compressive sensing inversion for SAR tomography

A recent approach to solve the SAR tomography problem is based on Compressive Sensing [123,126,129]. Compressive Sensing CS is a model-based framework for data acquisition and signal recovery based on the premise that a signal having a sparse representation in one basis, can be reconstructed from a small number of measurements collected in a second basis, that is incoherent with the first [130]. In our case sparseness requires a small number of stable targets in the same range-azimuth resolution cell [123].

In SAR Tomography, this approaches is particularly effective [123,131] because on data acquired high frequency systems (C- or X-band sensors), the scattering in the same range-azimuth cell along the elevation occurs typically on a few concentrated scattering points, as shown in Figure 20.29.

Considering the tomographic model (20.81), the matrix A in the context of CS is called the measurement matrix [130]. As already commented before, the inversion of Eq. (20.81) is equivalent to an inverse FFT operation, and would provide an estimate of the reflectivity function with a nominal 3-dB elevation resolution given by Eq. (20.84).

In practical cases, the orbits are usually not uniformly spaced and the number N of the available acquisitions is generally much lower than the number of unknown samples . As mentioned before in Section 7.1, truncated SVD (TSVD) can be used [125] for the inversion of (20.52). An alternative technique is based on Compressive Sampling [126]. It exploits the sparsity property of the unknown vector and allows obtaining very satisfactory reconstructions, even when a reduced number of acquisitions, almost randomly spaced within the overall elevation aperture, is available. Moreover, increased resolutions can be obtained adopting CS.

In the considered sparsity hypothesis, the sampled reflectivity function can be written as:

(20.93)

where is the sparsity matrix and is a sparse -dimensional vector. It has to be noted that in this case, by choosing sampling points to represent the unknown scattering distribution, and considering the circumstance that such function is sparse just in the domain of spatial samples (see also Figure 20.29, where only K = 3 scattering contributions are present in the whole scattering distribution along the elevation), the matrix becomes the identity matrix [123].

An estimate of the vector can be found solving the -norm minimization problem [130]:

(20.94)

The optimization problem in (20.94) is valid for the noiseless case. In the more realistic case, some noise is present in addition on the measurements:

(20.95)

with w a complex Gaussian vector with zero mean and uncorrelated elements. In this case, the solution can be found by solving the linear programming problem [132]:

(20.96)

where is a small positive number.

For discussing the super resolution issue, we can refer to the case of high signal to noise ratio, so that the maximum resolution that can be achieved depends only on the acquisition configuration and not on the noise level. The presence of an higher noise level can only degrade the evaluated performance.

In practical cases , so that Eq. (20.81) expresses an underdetermined system, which will admit a not-unique solution. Anyway, CS theory [130,132] ensures that if it is satisfied an incoherency property between A and , it is indeed possible to recover the K largest components of from the N measurements provided that the following inequality is satisfied:

(20.97)

where C is a small constant depending on the measurement and sparsity matrixes A and , which can be empirically estimated by numerical simulations, K denotes the number of non-zero coefficients of , and is the mutual coherence between the measurement basis A and the representation (sparsity) basis , defined in Ref. [132].

As far as the improvement or resolution, in Ref. [133] it has been shown that the elevation resolution (null-to-null) which can be obtained adopting a CS approach is given by:

(20.98)

where is the standard Fourier techniques elevation resolution and is the super-resolution factor (greater than one).

For a fixed number of acquisitions N, extension of illuminated scene in the elevation direction S, a number of scatterers K, and for a given standard resolution , there is an upper limit for the super-resolution factor given by [133]:

(20.99)

Combining Eqs. (20.98) and (20.99), a limit to the maximum resolution can be expressed as:

(20.100)

Equations (20.99) and (20.100) provide the super-resolution limits of the CS approach applied to SAR tomography.

Some experiments about the performance of the Compressive Sensing approach to solve the tomographic problem in terms of resolution capabilities, are presented in Ref. [123], with reference to simulated data using COSMO-Skymed system parameters (see Table 20.1), and to the real data acquired by the sensors ERS1–2, over the city of Naples, between 1998 and 2001.

The simulated observed scene is composed of two stable and coherent scatterers located in the same range-azimuth resolution cell at different elevations and responding with the same radar cross section.

In the simulation presented in Figure 20.31, it has been assumed that the SAR signals are acquired along multiple passes with a total orthogonal baseline span . The theoretical resolution in the elevation direction is given, according to Eq. (20.84), by . Assuming that the elevation extension of the ground scene is S = 200 m, the orbit spacing in the elevation direction corresponding to the Nyquist sampling rate is equal to 52 m [133].

Two scatterers at the elevation values =7 m and are considered, so that they cannot be resolved using standard FFT based techniques, since their distance is below the system configuration resolution. Moreover, orbits with orthogonal baselines −271 m, −238 m, −154 m, −68 m, 0 m, 64 m, 118 m, 209 m, 295 m, are considered, and the acquired signals have been corrupted with additive Gaussian noise, whose average power is 30 dB below the signal power.

The elevation profile obtained by the CS approach starting from the nine available acquisitions and with , corresponding to the upper limit for the resolution is shown in Figure 20.3 (black line) where it is compared with the TSVD reconstruction (blue line), this latter able to achieve conventional “Fourier” resolutions.

In the real data experiment, 15 passes, whose orthogonal baselines span a total baseline , are used. The resulting theoretical 3 dB elevation resolution is , which corresponds to a height resolution . The tomographic processing of the four azimuth-height constant range sections, indicated with four segments in the ortophoto image shown in Figure 20.32, are focused and shown in Figure 20.33 using CS (left) and TSVD (right). For the CS reconstruction, a super-resolution factor , corresponding to elevation and height resolutions of 8.87 m and 3.49 m, has been assumed. Also in this case, the sections of the San Paolo stadium in Naples in elevation (i.e., at constant slant range) shows, at different slant ranges, the capability of the technique to capture the 3D shape of the structure. A resolution improvement of the CS method with respect to the TSVD one is evident.

2.20.6.4 Performance comparison of SAR tomography methods

In the previous sections four SAR tomography approaches were described: Beamforming and TSVD, which attain an elevation resolution related and limited by the overall baseline span (see Eq. (20.84)), Capon, which allows super-resolution along the elevation direction, but requires the use of looks and therefore a loss of resolution, and Compressive Sensing based techniques, which exploits the sparsity of the scattering profile in elevation, and allows a strong improvement in elevation resolution.

The four method are compared with a data set simulating an observed scene composed by three stable and coherent scatterers, located in the same range-azimuth resolution cell at different elevations, and responding with the same radar cross section. More precisely the three scatterers are located at the height values , , and .

In the simulation presented in Figure 20.34, it has been assumed that the SAR signals are acquired along 30 multiple passes with a total orthogonal baseline span ( and therefore a mean baseline separation of 36.7 m: such a baseline distribution has been selected according to a real dataset of the Envisat satellite. According to Eq. (20.84), the theoretical resolution in the elevation direction is given by , corresponding to a height resolution . The three targets have been therefore located in two resolution cells (in eleation). A total of 15 independent looks have been generated by superimposing to the complex scatterer reflectivity a uniform phase independent from look to look. Moreover, the acquired signals have been corrupted with additive Gaussian noise, whose average power is 10 dB below the signal power.

The elevation interval 2a corresponding to the mean baseline is 645 m: this interval has been restricted with a reduction factor given by , thus leading to an investigated interval of 215 m.

The reconstruction of the height profile obtained by the CS, SVD and Beamforming tomographic approaches, starting from the 30 available acquisitions have been generated separately for all the available looks. The result for a particular look are shown in Figure 20.34a; all approaches are able to recover correctly the presence of three scatterer present in the profile and to resolve them from each other although the super-resolution capability of the CS based approach appear to be very evident. Due to the particular realization of the noise the scatterers separation is slightly overestimated and therefore the three targets are easily visible also from the Beamforming and SVD algorithms: the latter has slightly better performances on the resolution and on the sidelobe ratio.

Beamforming, SVD, and Capon were applied to the multilook data: the result are shown in Figure 20.34b. On one hand it is evident that on average the Beamforming and SVD are not able to resolve the scatterers, although the SVD provide generally better performances with respect to the Beamforming, particularly on the sidelobe ratio. On the other hand, it is evident that in this case the Capon provide better resolution capability with respect to the other two processing approaches, by resolving the presence of the three distinct scatterers which are also located at the three correct heights. In any case differently from the CS the super-resolution degree is paid by the necessity to have different looks. A more detailed analysis of the performances of the CS tomography in terms of resolution degree can be found in [129].