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FRINGE '96 Workshop: ERS SAR Interferometry, 30 September - 2 October 1996
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FRINGE 96

Multiple Images SAR Interferometry

Dominique Derauw Centre Spatial de liège, Avenue du Pré Aily, B-4031, Angleur, Belgium
CSLULG@vm1.ulg.ac.be or dderauw@ulg.ac.be
Jean Moxhet Laboratoire de Géomorphologie et Télédétection, Université de Liège, Sart-Tilman, B-4000 Liège, Belgium
Jean.Moxhet@physun.geo.ulg.ac.be
phypc9.geo.ulg.ac.be

Abstract

The interferometry SAR (InSAR) processor and the differential interferometry SAR (DInSAR) processor developed at CSL are described. We show that the use of multiple Tandem interferometric pairs allow to increase or decrease artificially the ambiguity altitude and, thus, to decrease or increase the fringe rate. In particular, this allows to generate DEM in regions having a highly energetic relief but at the cost of a lost of altimetric resolution.

Four images DInSAR is described and an exemple using two Tandem interferometric pairs is presented. This exemple shows that even Tandem interferograms may contain fringes due to optical path variations. The correct interpretation of such fringes can only be performed with the help of ancillary data. This exemple shows that interferograms used as topographical reference must be chosen with care. A second exemple of "four images SAR interferometry" is presented on the well-known Landers earthquake.

Keywords: SAR Differential Interferometry

Introduction

Interferometric radar techniques have been widely used to produce highly accurate digital elevation models (DEM's)[1-7] and, to a less extend, to measure displacement field or terrain motions in an observed scene. Mainly, two groups have demonstrated the ability to use differential interferometry for the study of seismic phenomena: Massonet & al. (CNES)[10] and Zebker & al. (JPL)[8,9] . Each of them has used a different approach to solve the problem. They both start from an interferogram containing elevation information (topographic fringes) and motion information (displacement fringes), but the DEM's they use to remove the topography come from different sources. In the case of the study of the Landers earthquake (28 June 1992), the JPL team uses a DEM produced by SAR interferometry with another pair of images and the CNES team uses a DEM coming from the U.S. Geological Survey (USGS).

SAR images and interferometric products derived therefrom cover very large areas over which geological structures can be studied at a regional scale. The interest of these products, in particular digital terrain models (DTMs) is to allow the caracterization and tri-dimensional analysis of these structures. Figure 1 shows the relationship between geological structures clearly visible in amplitude SAR images and the known structures widely documented in the literature. However, the image also shows sub-rectilinear features or lineaments that are not documented. This illustrates the wealth of informations contained in SAR data. The study of these structures is generally performed using "classical" methods (geologic, geomorphologic, seismic or geodesic), having point-like or linear character. On the other hand, differential interferometry over a sufficiently long period and with appropriate time-frequency can provide a synoptic vue of the displacements under concern. Using SAR differential interferometry, LGT ( Laboratoire de Géomorphologie et Télédétection, ULg; http://phypc9.geo.ulg.ac.be/geoinsti/lggq.html) hopes to explain global movements as well as dislocations and, as a result, to determine possible fracture zones.

Figure 1
Death Sea area: geological structures superimposed to the SAR image
One quadrant is corrected using the interferometric DTM.

However, interpretation based on SAR imagery and derived products is not yet accurate enough to allow a complete three-dimensional understanding of the lineaments. The Death Sea area was selected because of the absence of vegetation cover, its climate stability, its geologically favourable relief and an interesting tectonic context, related to the presence of a strike slip fault area and to the forming of pull-apart bassins.

In the following, we shall rediscuss the theory of SAR Interferometry and SAR differential interferometry. Next, we shall present the approach we use to combine two Tandem interferometric pairs. Interferogram combination allows to increase the ambiguity altitude to ease the phase unwrapping in high fringe density regions but to the detriment of the altimetric resolution. An example is given; we produced a DEM of the south-west side of the Death Sea using two Tandem pairs having a too small ambiguity altitude to be solved independently.

Finally, we shall present four-images DInSAR results obtained from two Tandem interferometric pairs covering the Death sea area. We show that even Tandem pairs may contain optical path variations and that interferogram used as topographic reference to perform DInSAR must be chosen with care.

Results obtained using four-images DInSAR to study the Landers eartquake are also presented.

SUMMARY OF THEORY

Digital Elevation Models

In this section, we derive the equations needed to generate DEM's from interferometric pairs of SAR images and we discuss the processing steps needed to combine two or more interferograms.

Geometry

In spaceborne SAR interferometry, two images of the same scene are acquired by a single sensor or by two different antennas passing above the area of interest along two successive orbits next to each other. Figure 1 represents the geometry here used for SAR interferometry.

From Figure 2, we can deduce the relationship which allows us to calculate the altitude of a point above the spherical Earth[11]:

(1)

To calculate the altitude, we need to determine the angle ( - at). This one is obtained from the triangle (z1 B z2):

(2)

Interferometry takes place in the calculation of z2:

(3)

where is the phase difference measured in the interferogram and kz0 is the Radar carrier wave number.

Figure 2
Viewing geometry in spaceborn SAR interferometry

Interferogram Combination

When generating a DEM from a single interferometric pair of SAR images, the processing steps are the following:

  • First, we must coregistrate one image (the slave one) with respect to the other (the master one). As a result of the coregistration, we get the best bilinear transform that must be applied to the slave image to make it superimposable to the master one[12] .
  • Secondly, the slave image must be interpolated according to this bilinear transform in order to genarate the interferogram.
  • Afterwards, the interferogram is unwrapped[13] .
  • Finally, from the unwrapped phase and with the use of relationships (1) to (3), we calculate the elevation of each point in the scene.

When combining several pairs, the procedure is slightly different mainly concerning the coregistration:

  • One image is chosen as master image and all the others are considered as slave images. Within each interferometric pair, we coregistrate the first image of each pair with respect to the master one using maximisation of the correlation coefficient of the modulus of both images as criterion. This generally allows to coregistrate the images with an accuracy of one pixel or better.
  • The first image of each interferometric pair is interpolated, using the bilinear transform obtained as a result of the coregistration.
  • The second image of each interferometric pair is coregistrated with respect to the first one using maximisation of local coherence on numerous anchor points as criterion. This allows to coregistrate the images with an accuracy up to a tenth of a pixel.
  • The second image of each interferometric pair is interpolated, using the bilinear transform obtained as a result of the coregistration to make it superimposable to the first one
  • Afterwards, each interferogram is generated. But now, they are all in the same Slant range-Azimuth geometry, i.e. the one of the master image.

Altitude retrieval

To calculate a DEM from two or more combined interferograms, we must use the first order developement of (1):

(4)

Where:

  • i is the index of the ith interferometric pair.
  • the index p means values relative to a flat Earth

For convenience, (4) may be expressed as:

(5)

Where is the flat Earth phase and ha is the ambiguity altitude.

If we remove the flat Earth phase from each interferogram and if we sum them, we get a combined interferogram with an ambiguity altitude given by:

(6)

It is thus possible to increase or decrease the ambiguity altitude by a convenient baseline combination. Moreover, since the phase noise from one interferogram to another may be considered as statistically independent, the summation of this term tends to zero with N. In particular, if we use two interferograms having each a good signal to noise ratio (SNR) to derive a combined one with a higher fringe rate, we will get an interferogram keeping a good SNR, even if this new ambiguity altitude corresponds to a single baseline wich theoretically generates a high decorrelation.

Increasing the ambiguity altitude, since it corresponds to a lowering of the fringe rate, may be useful to process areas having an highly energetic relief. On the contrary, lowering the ambiguity altitude and thus, increasing the fringe rate may be useful to get a better altimetric resolution on smooth reliefs.

Differential Interferometry

The interference pattern of two SAR images of the same region, acquired on two consecutivepasses depends on the topography.

If a movement occured between the two passes, or if there exists any local optical path variation between the two acquisitions, "displacement fringes" shall also be present.

To retreive only displacement fringes, one must substract the topographic component of the phase in the interferogram.

Differential SAR interferometry was first proposed by JPL[8] to measure small elevation changes or any other changes that induce optical path variations (called hereunder: "displacement"). They use three images to form two interferometric pairs; one containing only topographic fringes and used as a topographic reference, and the other containing topographic fringes and displacement fringes. But as we experienced it, it is often difficult to find such an interferometric triplet. Another solution is the one proposed by the CNES[10], where two SAR images are used to form an interferogram containing topographic fringes and displacement fringes and where topographic reference is taken from an other information source (map, DEM in ground range projection, ...). However, it may reveal difficult to obtain a good DEM for the region under concern.

Another solution consist in what we call here "four-images DInSAR". This method is a simple alternative to the three-images method used by JPL. It make use of two distinct interferometric pairs, one containing the displacements to be measured and another used as a topographic reference.

Three-Images Differential Interferometry

The priciple of SAR differential interferometry based on an image triplet is shematically described on Figure 3. It is supposed that no movements take place between the two first takes in order to generate a topographic reference. Displacements are supposed to occur between the second and the third image aquisitions. A small elevation change, or any optical path variation, induces an erroneous calculation of the view angle and, in turn, of the altitude h13, if the phase is interpreted as only due to the topography. From Figure 3, we deduce the relationship between the elevation angles , and , which are measured by "classical" interferometry (eq. 2) and the optical path variation :

(7)

Figure 3
Shematic representation of the "three-images Differential SAR interferometry" method

Remarks
  • The angle is obtained from the following relationships:
  • (8)
  • Equation 7 is directly applicable to both unwrapped interferograms.
  • h13 is the height that would be measured if displacement fringes were interpreted as topographical ones.

Four Images SAR Differential Interferometry

Four-images SAR interferometry makes use of two independent interferometric pairs: one containing a topographic phase as well as the displacement component that we want to measure, and a second interferometric pair which is supposed to contain only a topographic phase component in order to be used as reference.

Four images SAR interferometry is similar in its principle to the method developped by JPL since it uses a SAR interferogram to generate the phase component due to the topography. However it is often easier to find two coherent pairs than one coherent triplet.

All the images are coregistered with respect to a single one in order to generate a topographical reference interferogram which is directely in the same Slant Range - Azimuth geometry as the interferogram containing the displacement fringes. We thus obtain two interferograms; one containing the topography as well as the displacement information, and a second one containing only the topography:

The phase information in a SAR interferogram may be expressed as follow:

(9)
  • The first term corresponds to the "flat Earth" phase component.
  • The second term is the topographic phase component.
  • The third term is due to any local optical path variation.

Since the second interferogram is considered as a topographic reference, it is suposed that (10)

Remarks
  • Displacement may also be present in the interferometric pair considered as reference. In that case, the topographical reference is considered as corrupted and the measured result corresponds to the cumulated and weighted displacements that occured during each of the interferometric aquisitions:
  • (11)
  • It is preferable to choose || < 1 in order to lower the phase error in the transposed reference interferogram.
  • The former processing is also applicable to the classical three images DInSAR. In that case, the weihgting factor is simply [9]:
  • (12)

     

EXAMPLES

DIGITAL ELEVATION MODEL CALCULATION

Increasing of the ambiguity altitude

When a scene shows a highly energetic relief its often difficult to use interferograms with low ambiguity altitude because the fringe rate may be too high to allow phase unwrapping even if the signal to noise ratio is good. It thus may be useful to artificially increase this ambiguity altitude using two or more interferograms in order to lower the fringe rate in the resulting interferogram. This eases the phase unwrapping and allows to connect regions that would have been separated using a single interferogram, but at the cost of a loss of accuracy. An example is given hereafter; we used two Tandem interferograms of the "Death Sea" area, each one having an ambiguity altitude around 30 m. The images characteristics are summarized in Table 1.

Dates
E1 Orbit Ndeg.
E2 Orbit
Frame
Bx
By
ha
10-11 November 1995
22604
2931
621
375 m
-21 m
-25m
16-15 December 1995
23105
3432
621
-333 m
12 m
28m

Table 1

Because of the highly energetic relief in that area and the very small ambiguity altitudes of both the Tandem pairs, each interferogram exhibits a very high fringe rate (Figure 4). Residues are numerous even if the signal to noise ratio is good because of the too high fringe rate. The image of the residue connections shows a lot of independent zones. Finally, both of the Tandem interferograms reveals to be nearly impossible to be unwrapped suitably.

Figure 4
Tandem interferogram of the south-west side of the Death Sea.

.

Figure 5
Residue connections

Combining the two interfergrams allows us to generate a new one having an ambiguity altitude of 234m (Figure 6). This new interferogram shows fewer residues and can be entirely unwrapped with ease. As a result, we can generate a D.E.M. over the whole combined interferogram (Figure 7).

Figure 6
Combined interferogram (haeq = 234m) obtained from two Tandem interferograms
(ha1 = -25m, ha2 = 28m).

Figure 7
DTM issued from the combined interferogram.

FOUR IMAGES DINSAR EXAMPLE

The Landers Earthquake (June 28, 1992)

To validate our processor, we performed a study of the well-known Landers earthquake using four-images differential SAR interferometry. We thus used two interferometric pairs. The first one (figure 8) is the same as the one used by the CNES

Figure 8
First interferogram of the Landers area:
April 24, 1992 / August 7, 1992
Interferogram containing displacement fringes

(Click on the interferogram samples to get the full ones.)

Figure 9
Tandem interferogram of the Landers area:
January 7 and 8, 1996
Interferogram considered as the topographic reference

The processing steps are the following:

  • All the images are coregistered and interpolated with respect to the first one (April 24, 1992).
  • Both interferograms are generated in the same Slant range - Azimuth geometry.
  • The second interferogram is unwrapped and transposed in the viewing geometry of the first pair (i.e. multiplied by the ratio of the ambiguity altitudes).
  • The " transposed " interferogram is retrieved from the first one to generate the differential interferogram (Figure 10).

Figure 10
Differential interferogram of the Landers earthquake

The Death Sea Area

We performed four-images differential interferometry using the two Tandem pairs covering the Death Sea area. The relative position of the satellites are represented on figure 11. The image acquired the 10th of November 1995 was chosen as the master one. The three other images were coregistrated as already described. The two quadrants of the scene covering the Jordanian part, east of the Death Sea, showed a sufficiently smooth relief to allow phase unwrapping on large areas in the second interferogram (16-15/12/95) (Figure 12). This one was thus considered as a topographic reference and transposed in the geometry of the first Tandem interferogram in order to flaten it. It was expected to get a completely flat interferogram since no seismic event occured during both the Tandem acquisitions even if an earthquake occured between the 11 of November and the 15 of December.

Figure 11
Relative positions of ERS1 and ERS2 for the acquisition under concern.
(dx = -674m, dy = 31m)

Figure 12
Tandem interferogram used as topographic reference.

Due to a to high fringe rate in some areas, the reference interferogram is difficult to unwrap. As a result, the four-images differential interferogram is is made of numerous independent zones.

Figure 13
Four-images differential interferogram

The differential interferogram (Figure 13) is flattened in most of the areas even in regions which showed a very high fringe rate. But two residual fringes are clearly visible following the south-west to north-east direction.

An error in the calculation of the baselines is not sufficient to explain the presence of such remaining fringes. If those fringes were only due to a residual baseline component, fringes would have been also present elsewhere, particularily, on the summit located right to the east of the Lisan peninsula which shows a much more " energetic " relief.

It it thus supposed that these fringes contain information related to a local change in the optical path that occured in one or both of the Tandem acquisitions (eq. 11).

Determination of the Perturbed Image

Even if the baseline between the acquisition of the 10th of November and the one of the 16th of December is long, the coherence between these two takes is sufficiently preserved to generate an interferogram. This allowed us to generate two differential interferograms, each using one of the two Tandem pairs as topographic reference (Figure 14). We observe that the "displacement" fringes show a fringe rate twice higher when using the first Tandem pair (10-11/11/95) as a reference. If a reference interferogram is corrupted by "displacement" fringes, the corresponding phase shall appear twice in the differential interferogram; first from the interferogram covering the period 10/11/95 to 16/12/95 and secondly from the reference interferogram multiplied by the factor (eq. 11). Consequently, if the first Tandem pair is corrupted, the "displacement" fringes are approximately multiplied by 2 () in the first differential interferogram and must appear only once in the second one. From Figure 14, we can deduce that displacement fringes are mainly contained in the first Tandem pair and that the first image, the one of 10 November, is probably perturbed with respect to the others.

Figure 14
Three images differential interferogram of the Lisan peninsula
a) Tandem pair 10-11/11/95 taken as topographical reference
b) Tandem pair 16-15/12/95 taken as topographical reference

Fringes Interpretation

Two cases are possible. First, the Lisan peninsula and the north-east plateau underwent a one-day movement of approximately 3 cm between the 10th and the 11th of November. This hypothesis seems unprobable but is still under verification. Secondly, meteorological effects induced local optical path variations between the two takes. This hypothesis seems more probable because, even if there was no precipitation during the first Tandem aquisition, the cloud covering and the relative humidity were completely different for the two takes. The sky was nearly completely covered during the first acqisition and completely clear during the second (Figure 15).

Figure 15
Cloudiness and relative humidity during the first Tandem acqisition

(Sources: Meteorological Services, Ministery of Transport, State of Israel)

Moreover, the orientation of the residual fringes corresponds to the main orientation of the cloud front observed on 10th November.

Knowing that a variation of 15% of relative humidity may induce up to two fringes via optical variation[14], the fringes observed might correspond to a local variation of approximately 11%. This is comparable to the 14% measured by the Meteorological Services of the State of Israel.

Conclusions

It has been shown that combining interferograms allows to increase or decrease artificially the ambiguity altitude. Increasing the ambiguity altitude and, thus, decreasing the fringe rate may be usefull to unwrap interferograms covering hilly regions or having a too long baseline, but at the cost of a loss of accuracy.

When using a SAR interferometric pair as a topographic reference to perform differential interferometry, one must carefully verify the validity of this reference. If the reference interferogram contains fringes due to any optical path variation, the differential interferogram shall be completely corrupted. To assess the validity of an interferogram used as topographic reference, ancillary data (e.g., meteorological data as in the present study) are mandatory.

Acknowledgements

This work was supported by the Belgian Federal Offices for Scientific, Technical an Cultural Affairs (O.S.T.C.; http://www.belspo.be) under contract Nr. T3/12/32 of the TELSAT-III National Remote Sensing Program.
DD. wishes to thank Prof. H.A. Zebker and Dr. D. Massonnet for helpfull discussions.

References

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submitted to: J. Geophys. Res.- Solid Earth.

Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry