**Tomographic Urban Imaging using TerraSAR-X High Resolution Spotlight Data Stacks**
*
Zhu, Xiao Xiang ^{1}; Bamler, Richard^{1}; Wang, Yuanyuan^{2}; Shahzad, Muhammad^{2}
^{1}German Aerospace Center, Remote Sensing Technology Institute, GERMANY; ^{2}TU-Munich, Chair of Remote Sensing Technology, GERMANY
*

A conventional spaceborne or airborne Synthetic Aperture Radar (SAR) maps the three-dimensional (3-D) reflectivity distribution of a scene to be imaged into the 2-D azimuth-range (x-r) plane. This can be seen as a projection along the third radar coordinate, elevation (s). x, r, and s form an orthogonal coordinate system specific to the particular SAR imaging geometry. This projection particularly handicaps the interpretation of SAR images of (i) volumetric scatterers and (ii) of urban areas and man-made objects, i.e. objects with constructive elements oriented at steeper angles than the local incidence angle.

SAR tomography (TomoSAR) extends the synthetic aperture principle of SAR into the elevation direction for 3-D imaging. It uses acquisitions from slightly different viewing angles (the elevation aperture) to reconstruct for every azimuth-range (x-r) pixel the reflectivity function along the elevation direction s. It is essentially a spectral analysis problem. Differential SAR tomography (differential TomoSAR), also referred to as 4-D focusing, obtains a 4-D (space-time) map of scatterers by estimating both the elevation and the motion parameters of multiple scatterers inside an azimuth-range resolution cell.

We work with TerraSAR-X high resolution spotlight data. These very high resolution (VHR) X-band space-borne repeat-pass tomographic data stacks of urban areas have some particular properties: A very detailed view of individual buildings is possible; the density of bright points, like persistent scatterers, is extremely high (40,000-100,000/km2). But also non-linear (e.g. thermally induced) movements of different building parts must be expected and will introduce additional phase errors and require robust inversion methods. Due to the tight orbit tube of TerraSAR-X the elevation aperture is small, i.e. the inherent resolution in elevation is about 50 times worse than in azimuth or range. This extreme anisotropy calls for super-resolution algorithms in the elevation direction. Finally, VHR data are expensive and, hence, data stacks should be kept small.

In this presentation, the potential of the new class of VHR space-borne SAR systems for Tomographic reconstruction, i.e. 3-D and 4-D SAR imaging, in urban environment will be demonstrated, in particularly:

- The first 3-D and 4-D reconstructions of an entire urban area (incl. its radar reflectivity) with very high level of detail from space-borne SAR data by TomoSAR will be presented.

- Examples with promising layover separation in urban area will be shown.

- A compressive sensing (CS) based algorithm ''SL1MMER'' is proposed and compared to the conventional non-parametric linear maxima a posterior MAP reconstruction and parametric nonlinear least square (NLS) method. Its super-resolution properties and point localization accuracies are demonstrated using simulations and real TerraSAR-X data.

- The super-resolution power, i.e. the minimum separable distance between two point scatterers, as a function of SNR, number of measurements N, and amplitude ratio of the scatterers for the typical low SNR and low N cases of TomoSAR, is investigated. The fundamental bound of super-resolution of imaging systems in general will be quantified.

- The ''time warp'' method is proposed. It rewrites the D-TomoSAR system model to an M+1-dimensional standard spectral estimation problem, where M indicates the user defined motion model order and, hence, enables the tomographic motion estimation for all possible complex motion models.

- After time warp, the separation of a combined linear and seasonal motion by D-TomoSAR will be exemplified using TerraSAR-X data.

- Due to the side-looking geometry of SAR, a single stack of SAR images only provides information on one side of a building. To serve the function of urban structure monitoring, TomoSAR results of multiple stacks from different view angles are fused together to provide us with a shadow-free point cloud with high degree of coverage over the entire urban area.

- In order to provide a high quality spatio-temporal 4D city model, object reconstruction from these TomoSAR point clouds is emergent. A 3D view of the reconstructed facades over a test building using fused point clouds from multiple viewing angles, i.e. both ascending and descending orbits, is exemplified.