Corner reflectors designed for radar are targets that reflect the wave emitted by the satellite back to the antenna. The combined reflections on the 3 mirors redirect the waves in their original direction and it can be demonstrated by simple math. We can decompose the propagation vector of the wave k in the 3-dimentionnal basis make by normal to the 3 mirors, n1, n2 and n3 : k=a1.n1+a2.n2+a3.n3 . The optical law of reflection that we can use in radar says that the reflection on the miror 1 will change the sign of the composante n1 of the vector k: k’=-a1.n1+a2.n2+a3.n3 . After the reflection on the second miror, the composante n2 will change sign: k”=-a1.n1-a2.n2+a3.n3 and after the reflection on the last miror, k”‘=-a1.n1-a2.n2-a3.n3 = -k . In conclusion, after the three reflections the wave that leaves the corner reflector go in the opposite direction that the wave that entered in the corner reflector and thus go back to the antenna.

For our experience on the Eiffel Tower, we put 2 corner reflectors above the restaurant “Jules Vernes”, located at the second level on a platform which we know the altitude. The corner have been on the Eiffel Tower for nearly two month, during the new TerraSAR-X and TanDEM-X acquisitions over the city. We are now waiting for the delivery of the images. Then, when we locate them on the images, they  serve as reference for the height measurment that will allow us to have a precise deformation measurment.

The first corner and its view

The second corner and its view.

SAR sensors are echo-localization systems. They measure the time between the emission of an electromagnetic pulse and its reception after that it have been backscattered by targets. This time is firstly converted in distance and then in pixels. The coordinates of a target in the image correspond to the number of resolution cells included in the distance: it is an integer. But the SAR signal keep in memory the sub pixel length of the path between the emission and the reception. One way this memory is kept is through the phase of the pixel. Indeed, the pixels of SAR images are complex numbers.

The SAR images are often represented in amplitude only because the information conveyed in the amplitude is the most easily interpretable: the amplitude of the pixels represents the energy of the backscattered signal. A metallic structure as the Eiffel Tower will backscatter most of the emitted pulse and thus be very bright in the image. On the contrary, the Seine river does not scatter signal back to the satellite. It is thus dark in the image. On the other hand, the phase of a unique image is completely random.

Amplitude image of the Eiffel Tower and its phase

The magic happens when the phase of at least two images are compared. The phase difference that is computed can bring information of the height of the targets if the images are acquired from slightly different location. This technique is called interferometry. The phase difference can also bring information on the deformation of the target between the acquisition if the images are acquired at different time. This technique is called differential interferometry. When the images are acquired from slightly different location and at different time, both height and deformation are mixed in the phase difference. The proportion of the mixing depends on the acquisition condition.

When it comes to buildings, deformations are often neglected: the height of the buildings is considered as proportional to the phase difference. Since the phase is 2π-periodic, the height can only be estimated modulo the ambiguity height. The ambiguity height is determined by the acquisition parameters. For example, it is inversely proportional to the distance between the two satellites but proportional to the satellites’ altitude or the wavelength of the pulse. To estimate to absolute height, the phase need first to be unwrapped.

The images below are 2 interferograms between TerraSAR-X Spotlight images (thanks to the DLR projet LAN2708). The unwrapping of the phase can be done roughly by counting the fringes: the color legend is 2π-periodic, each color rainbow represents a fringe. We count approximately  6.5 fringes on the image on the left. With an ambiguity height about 46 meters, the measured height is 278, close to the 320m official height of the Eiffel Tower and its antennas.  On the image on the right, 4 fringes can be counted. But ambiguity height is now of 412 meters leading to a 1600m Eiffel Tower.  This surprising result is caused by the non-deformation hypothesis. Indeed, the Eiffel Tower is moving of few centimeters between the acquisitions, mostly due to thermal effects and this deformation impact the phase.

Left : ambiguity height of 46m , Right ambiguity height of 412m

-π π

The Eiffel Tower is an emblematic monument in Paris. I have studied it through a temporal stack of 100 images acquired by TerraSAR-X in spotlight mode (thanks to the DLR projet LAN2708). See how it looks in SAR imagery. The image has a 1m x 1m resolution: one pixel represents roughly a 1m x 1m ground area. This means that targets that are separated by 1m in the scene will be separated in the image.

To make an image, the SAR sensor emits pulses of electromagnetic waves and record the backscattered signal. On the contrary to most optical satellite, the SAR sensor has an oblic view making the side of the Eiffel Tower visible. We can thus see the different parts of the structure of the Eiffel Tower, as well as some buildings, the Seine river and the Trocadero. As in optic, some objects can block the wave, creating shadow behind them. I have cut the shadow of the Eiffel Tower, but the shadow of nearby buildings is visible.

This image is a good illustration of the layover effect. Since SAR sensor is an echo-localisation technique, the treatment that creates the image, the synthesis, is based on the measure of the distance between objects as the Eiffel Tower and the satellite. Thus all the objects that are at the same distance to the satellite will be in the same pixel.  The 3rd floor of the Eiffel Tower and the Seine river appear in the same pixels as well as the 2nd floor and the trees of the Trocadero.

The synthesis is based on the assumption that all the object in the scene are static. When objects are moving during the acquisition, they can be defocalized or delocalized depending on the direction of their mouvement. A peniche boat that was moving on the Seine river is thus delocalized on the quai Branly.

Due to the interaction of the emited wave with rough objects, speckle is present in SAR images. Speckle looks like a salt and pepper noise on homogeneous areas. Its effect can be removed by averaging pixels through the temporal stack. The average can be quadratic (left) or  geometrical (right). The two averages look mainly alike, even if the contrast is sharper on the quadratic mean. Some differences appear on the Seine river. They are due to changes in few images of the temporal stack: mostly peniche boats that are present only in one date. These changes are erased by the geometrical mean but still impact the quadratic mean.