An Overview of SAR Interferometry
1 - the two-ways travel path (sensor-target-sensor: hundreds of kilometers in the satellite case) that, divided by the used wavelength (a few centimeters), corresponds to millions of cycles;
2 - the interaction between the incident e.m. waves and the scatterers within the ground resolution cell;
3 - the phase shift induced by the processing system used to focus the image.
Therefore, the phase of a single SAR image is of no practical use. On the contrary, if two SAR images from slightly different viewing angles are considered (interferometric pair) their phase difference (interferometric fringes) can be usefully exploited to generate Digital Elevation Maps (DEMs), to monitor terrain changes and to improve the range resolution. The interferometric fringes image is derived as the phase of the SAR interferogram, that is the complex image formed by cross-multiplying the two SAR images. The relation between the interferometric fringes and ground elevation is usually explained by means of the monochromatic approach [24, 14]. It is based on the assumption that the RF bandwidth is so small (and this is the case of most satellite systems including SEASAT, ERS-1, JERS-1, ERS-2 and RADARSAT) to be negligible. Thus the system is considered monochromatic. However, if the finite bandwidth of the system is considered (wavenumber shift approach), a relative shift of the ground wavenumber spectra dependent on the baseline and the local slope is found. A few important consequences come out from this result [16, 5]. Using the simpler monochromatic approach we show the relationship between the relative terrain elevation and the interferometric fringes. Then we will determine the quality of the Digital Elevation Model derived from interferometry. Two sections will be dedicated to the description of other applications of SAR interferometry: small terrain motion detection and uses of the coherence for image segmentation. Symbols that will be used through the text and the main parameters values of the ERS-1 SAR system (that will be assumed as typical parameters of SAR from satellites) are shown in Table 1.Let us consider two complex SAR images taken from two slightly different viewing angles (see Figure 1), and . Even if non simultaneous acquisitions are considered, we shall suppose, for now, that the terrain backscatter did not change.
( is a constant). The interferometric phase of a single pixel is still of no practical use. The travel path difference is usually much greater than the wavelength (in most of the practical cases, the travel path difference from the satellite can be as large as a few hundred meters whereas the used wavelength is of the order of a few centimeters) and the measured phase shows an ambiguity of many cycles. On the other hand, passing from one pixel to its neighbor (only a few meters apart in the slant range direction), the variation of the travel path difference ( ) may be much smaller than and the variation of the interferometric phase is not ambiguous. Moreover, a simple relation between and the relative terrain elevation can be derived.
In Figure 1 we have indicated the position of the two SAR sensors ( and ) and their relative displacements parallel ( ) and normal ( ) to the slant range direction. We have also indicated the position of two point scatterers and their relative displacements parallel ( ) and normal ( ) to the slant range direction. Let us assume as a reference the positions of and with their relative distance . By changing the sensor and target position by and respectively the sensor-target distance becomes:
Since the distance between the two SAR sensors is generally much smaller than the sensor-target distance (a few hundred meters compared with 800 km, in the ERS-1 case), the following expression of the interferometric phase variation holds:
This result shows that if we know the relative displacement of the two orbits normal to the slant range direction , the distance and the value of the SAR wavelength , then the phase difference depends only on the value, i.e. the elevation difference between the points displayed in Figure 1, measured in the direction normal to the slant range axis. Thus the interferometric phase image represents a map of the relative terrain elevation with respect to the slant range direction. A linear term can then be subtracted from the interferometric phase so that the phase corresponding to the flat horizontal terrain is a constant. After some algebra, the version of equation 0.3 that refers to the relative elevation with respect to a flat terrain becomes:
where we have indicated with the altitude of ambiguity, i.e. the elevation change correspondent to a 2 phase shift.
Up to now it has been assumed that the phase difference of equation 0.4 would be directly obtained from the interferometric fringes. However such an assumption is in some way misleading. In fact, from the interferogram, the complex values can be determined but not the values themselves. What is measured is not the phase difference value , but its principal value , limited between and . The interferometrical fringes show typical discontinuities like those shown in Figure 2.
Two adjacent discontinuities separated by a constant elevation increment, corresponding to a phase drop, represent the height contours of the terrain elevation map. However, since the real phase values are ``wrapped'' around a interval, to obtain the correct map (and, thus, the correct label of each contour line), a map of the multiples of to be added to the ''wrapped'' phase should be found (phase unwrapping).
At the beginning of the studies on SAR interferometry (1988-89), the phase unwrapping problem was investigated assuming that only two SAR images were available. That was the actual situation at that time since the only few interferometric SAR images were from the SEASAT and the JPL single pass airborne systems. Many techniques of phase unwrapping of interferometric SAR images have been proposed [7, 6, 19]. A summary of these techniques is out of the scope of this paper.
More recently, however, a different strategy to solve the phase unwrapping problem has been considered. With the huge amount of SAR images from the European satellites ERS-1 and ERS-2 together with the availability of Digital Elevation Models (typically from the optical stereo SPOT) the basic assumption changed: all the available information should be exploited to get a DEM . Then, an available Digital Elevation Model (DEM) can be transformed into the SAR geometry and subtracted from the interferogram. Depending on the DEM accuracy and on the normal baseline the phase of the residue can be so small to show a single fringe with no need of phase unwrapping.3]. Theoretically it would be enough to have three interferograms with baselines that are prime with respect to each other to remove ambiguities (Chinese remainder theorem). In a practical case, where data are noisy and baselines random, the use of multiple interferograms increases significantly the elevation ambiguity level. Precise location of the flight paths together with a good estimation of the standard deviation of the phase noise allows the use of Maximum Likelihood (ML) and Maximum A Posteriori (MAP) techniques to estimate the height difference of each pixel with respect to a reference one, taking into account a priori information.
The conditional density function of the elevation for each interferogram is periodic with a different period (the altitude of ambiguity) dependent on the baseline. In each period the higher is the quality of the fringes (the coherence) the sharper is the histogram. The product of the conditional densities shows a neat peak whenever the coherence is not close to zero and the baseline errors are not too high. The sharper is the global peak, the higher is the reliability of the results i.e. the probability that the correct value of the height variation lies inside a given interval.
The benefits of the multibaseline approach are twofold. First, combining all the information it is possible to limit the impact of the noise. Besides, there is minor risk of aliasing with respect to conventional single interferogram phase unwrapping: working simultaneously on more interferograms, phase unwrapping is possible even if the phase is undersampled. Of course the higher the noise, the worse the reliability and the more likely the phase unwrapping errors.
As a conclusion, even if a fully automatic phase unwrapper that works in all conditions is not available yet, phase unwrapping does not appear to represent a serious problem in SAR interferometry anymore.
The elevation map derived from SAR interferometry lies on a plane where the reference axes correspond to the azimuth and slant range directions.
Such a coordinates system is different from the reference systems in the azimuth-ground range plane used in the usual elevation maps. Thus it is necessary to transfer the unwrapped phase from the slant range coordinates system to the ground range system; the obtained values must be interpolated and resampled in terms of uniform ground range cells.
From Figure 3 it is evident that the horizontal position of a backscatterer depends on both the slant range coordinate and the elevation. Through simple geometrical expressions the relation between these three parameters can be found. The ground range coordinate with respect to the initial point, indicated by y, is the sum of two components: the first is the horizontal displacement in the case of flat terrain, the second caused by a non-zero elevation drop.
Since the position of the points depends on its elevation, the correspondence between ground range and slant range is quite irregular. In fact, the well known foreshortening effect causes a compression of the areas with ascending slope and a spread of the descending areas. As a consequence the ground difference, corresponding to a constant slant range displacement, will be much larger in the case of ascending slopes. Furthermore, when layover effects occur several areas of the earth's surface can disappear from the SAR image.
In a ground range reference system the obtained elevation map will have a quite uneven sampling interval. Thus to obtain a regular sampled map the elevation values must be interpolated. For our purposes a linear interpolation is quite adequate. In fact, in flat or descending areas the interpolating points are fairly close, whereas with an ascending slope the foreshortening effect produces such a large slant range compression that the interpolating points lie much further away and no interpolator would operate correctly. The results of the rectification process, performed for the SAR image of the area near the Vesuvius is shown in Figure 4 .
13]. First, to unravel the layover areas, that, if not in shadow, may appear well behaved with the other orbit. Second, and equally relevant, the combination of the two views is only possible if the DEM is correct; hence, it is a powerful check of its quality, to remove blunders, reduce the elevation dispersion, etc.
If the SAR acquisitions are not simultaneous, the interferogram phases (fringes) are also affected by the possible terrain changes. Two types of temporal terrain evolution can be identified by means of multitemporal interferometric SAR images: decorrelation changes and small terrain motions; a very interesting report on the possibility of monitoring terrain changes can be found in .
Decorrelation changes. A random change of the position and physical properties of the scatterers within the resolution cell can be detected as low coherence of the images and will be dealt with in another section.
Small terrain motions. Local centimetric relative motions in slant range, generate large local phase shifts in the interferogram. The phase difference thus generated is governed by a mathematical relation completely different from that of the interferometric fringes (that, in order to avoid any confusion we shall indicate with ) described by equation 0.3. It is, in fact, proportional to the ratio between the relative motion along the slant range direction and the transmitted wavelength . Thus, if we have a non simultaneous interferometric SAR pair with a given baseline and a small terrain relative motion occurs between the two acquisitions, the following expression of the interferometric phase difference holds:
From equation 0.6, it is clear that the two terms and should be separated in order to recover the terrain relative motion. The simplest way to estimate small motions consists of choosing an image pair with a very small baseline (zero or a few meters would be sufficient in the case of ERS-1) so that the first term is much smaller than the second. An interesting example of such a solution is shown in Figure 5.
If a stereo SAR pair with a very small baseline is not available, the topography contribution to the interferometric phase ( ) must be subtracted from the fringes. It can be accomplished in two different ways. If a topographic map of the area of interest is available, it has to be transformed in the azimuth slant range coordinates and scaled proportionally to the baseline of the interferometric pair in order to have an estimate of the phase component . Then, it should be subtracted from the fringes (an impressive example of this technique has been shown by CNES  for the Landers earthquake). As an alternative, an additional SAR image can be exploited in order to have an interferometric SAR pair with no terrain changes . For the sake of simplicity, let us label the three SAR images with 1, 2 and 3 with no regard to their time consecution. Let us also assume that during the first two images no terrain changes occur. Thus, the fringes generated from the first two images will be proportional to the phase to be subtracted from the fringes generated either from the couple 1-3 or 2-3. From equation 0.3, it is clear that the proportionality coefficient is given by the ratio of the two baselines and (or ). However, since the proportionality holds on the phases and not on their principal values (apart from the special case of an integer ratio between the baselines), the fringes obtained from the couple 1-2 must be first unwrapped and then scaled. Other very important results on the use of this technique for studies of volcano deflation (Etna) or glacier motion can be found in [11, 8].25,20]. Multibaseline techniques can be usefully exploited to get a DEM that is less affected by artifacts by averaging the uncorrelated atmospheric contributions coming from the single interferograms . When the ML DEM is generated it is possible to get the phase difference with respect to each interferogram. These phase residues are proportional to atmospheric changes.
The phase variations (about one fringe peak to peak) with very low spatial frequency (more than one kilometer in both directions) visible in the figure are generated by atmosphere changes within the surveys. These effects appear to be the major limitation to the use of SAR interferometry as a technique for generating highly accurate Digital Elevation Models and for detecting small surface deformations. However, the interferometric phase depends on the relative elevation through a coefficient that is directly proportional to the baseline (see equation 0.3): the higher the baseline, the higher the phase variation correspondent to the same topography. On the other hand, the phase variations due to atmospheric changes are independent of the baseline. Thus, the higher is the baseline of the interferometric pair the smaller is the topographic error due to parasitic effects. Moreover, if many interferometric pairs of the same area are available, ''outliers'' can be identified and eliminated from the database. The remaining results can be combined to ''filter out'' the effects of atmospheric changes.
An example of outliers identification is shown in Figure 9.
It should be mentioned here that once an accurate DEM is available, also low coherence interferometric pairs (typically from the 35 days repeat cycle or its multiples) can be usefully exploited to get info on terrain deformations with low spatial frequency and/or atmospheric artifacts. Low-pass filtering the differential interferogram that contains low frequency signal plus white spectrum noise, allows us to extract useful information. An example is shown in Figure 10.
The quality of the interferometric phase depends on the amount of noise that, in general, comes from distinct sources [14, 18]: i- system noise; ii- terrain change (non simultaneous acquisitions); iii- images misregistration; iv- approximate and unequal focusing of the two passes; v- decorrelation due to the baseline (''geometric'' decorrelation). It is obvious that there is no way to avoid the first two sources of noise. On the other hand, as far as the last three sources are concerned, they can be taken under control. In other words, since in most cases the system noise is quite small compared with the usually sensed signals, and the processor noise is well under control if it is designed to be phase preserving , it can be seen that the fringes quality is degraded by scattering change in time and volumetric effects only. The coherence of two complex SAR images and , is defined as follows :
where E[.] means the expected value (that in practice will be approximated with a sampled average) and the complex conjugate. The absolute value of is a fundamental information on the exploitability of SAR interferograms. The signal (usable fringes) to noise ratio can be usefully expressed as a function of the coherence:
Thus, it is clear that every effort should be dedicated to avoid coherence loss during the interferogram generation process.
The statistical confidence of the estimated coherence (sampled coherence) and of the derived measurements, depends on the number of independent samples (n) that can be combined for the computation. As a first approximation, the standard deviation of the estimator is proportional to . Thus, whenever uniform areas (in the statistical sense) are identified, the sampled coherence can be computed as:
In fact, since the coherence is estimated from the combination of the phases of a few pixels at the very least, the topography effects on the interferometric phase proportional to the known terrain changes have to be removed from the result. Thus, in order to compensate this unwanted effect, the vectors at the numerator of equation 0.9 must be deskewed before summing. It is also clear that, in order to generate an interferogram, the pixels of the images gathered in the two different images must be registered accurately, so that the random variates corresponding to the reflectivity are properly aligned. A single pixel shift, if the focusing processor is a good one, is enough to practically zero the correlation. In the following we will not consider the effects due to misregistration and system noise, since they can be avoided with a good system or with a proper processing. The elevation error of maps generated by means of SAR interferometry will follow the value of as:
As an example, the coherence map of the area of Mt. Vesuvius in Italy observed from ERS-1 on August 27th and September 5th 1991 is shown in Figure 11.
A multi-interferogram approach can be usefully exploited to estimate the coherence using an ensemble average instead of a space one. When a good DEM is available with the same resolution of the SAR images, it is in fact possible to combine all the data to compute a multi-baseline coherence map of the area of interest on a fine spatial resolution: the increased number of freedom degrees (due to multiple interferograms) allows to get high resolution products (say 20 20 m).
First the topographic contribution on the phases of each interferogram is compensated for using the DEM. Then the phases are high-pass filtered to eliminate local distortions due to atmospheric effects. Finally the mean phase value is subtracted in each interferogram so that all the data can be considered phase aligned. The estimation is then straightforward:
where is the standard space average on the i-th interferogram (this time using a small estimation window).
The achieved coherence map highlights what remains unchanged during the time interval between the first and the last acquisition and could be exploited for image segmentation and classification; it gives a measure of SNR on a fine spatial resolution.
SAR coherence is an additional source of information with noticeable diagnostic power. In the following we shall enumerate some of the most relevant applications. In  it was first observed that forests, that appeared with variable reflectivity in the ERS - 1 detected images, appeared almost black in the coherence images: this effect is due to the scarce penetration of C band radiation in the vegetated canopy, so that small variations of the positions of leaves and smaller branches were enough to change the disposition of the scatterers and therefore practically annihilate coherence; likewise happens for water bodies, that appear always with negligible coherence. In [17, 23, 22] it was also observed that cultivated field changed their coherence after plowing, harvesting etc., so that it was possible to detect anthropogenic effects in multitemporal sequences of takes of the same area, by looking at sudden coherence losses. In general, the combination of multitemporal observation both of detected images and coherence allows a very good segmentation of agricultural areas; it is thus possible to identify cultures (potatoes are harvested in that month, whereas corn matures in that other .. ). Other authors  observe that from the phase of the interferometric takes the height of the trees and therefore the biomass can be estimated.
In this overview, we have seen that interferometry is speckle free, since its effect disappears from the differential phase. Further, we have seen that the fringes, given the short revisiting times of the TANDEM mission, may give a very good DEM, with a vertical resolution that could be in the tens of meters range, using the combination of several passes to combat atmospheric effects. Millimetric motion of large areas of the terrain or of corner reflectors have been measured with good reliability and therefore the possibility of using the system to measure subsidence, landslides, coseismic motion has been demonstrated. However, several conditions have to be met, the most important being the maintenance of some coherence of the scatterers during the entire experience. This is possible in the case of exposed rocks, but in other cases artificial reflectors may be needed to link together scattering structures that may change due to vegetation, floods, storms, etc. Coherence is an important clue, that combined with the more usual backscatter amplitude, leads to high quality images segmentation. The combination of multiple images will improve resolution.
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