SLIMMER are extended for mixed repeat- and single-pass data 
stacks in Section III. A systematic approach is proposed for the 
fusion of TerraSAR-X and TanDEM-X data in which the 
different data quality provided by the TerraSAR-X and 
TanDEM-X data are taken into account by introducing a 
weighting according to the noise covariance matrix. The 
proposed approach is evaluated with simulated data in Section 
IV. The simulation result shows that the reconstruction 
accuracy of tomographic SAR inversion can be improved 
significantly by using jointly fused TerraSAR-X and TanDEM- 
X data. 
2. DATA QUALITY ANALYSIS 
2.1 TanDEM-X Data 
Due to the simultaneous data acquisition of the TanDEM-X 
image pairs, the TanDEM interferograms possess much higher 
data quality. For instance, Fig.2 shows the example building, 
the Fashion Show Mall in Las Vegas. The right image is the 
mean intensity map from a stack of high resolution TerraSAR- 
X images. The left image is the corresponding optic image in 
Google-earth. From the optic image, it is evident that the roofs 
of the building blocks are flat. Fig.3 compares the TanDEM-X 
interferogram and TerraSAR-X interferogram generated from 
two images with a time lag of 33 days. Severe phase variation 
on the flat surfaces can be only observed in the right image (i.e. 
repeat-pass TerraSAR-X inteferogram) that indicates the phase 
distortion caused by temperature change induced motion, 
atmosphere and temporal decorrelation. 
2.2 Noise Model Analysis 
In VHR X-band data, the common noise sources of repeat-pass 
and single pass data are: 
- The calibration errors in amplitude: The radiometric 
stability of TerraSAR-X/TanDEM-X, i.e. the amplitude 
variations within one stack, is 0.14 dB and is therefore 
negligible compared to the typical SNR of a SAR system. 
- Thermal noise: It can be estimated from background 
pixels and is typically in the order of -20dB. 
Besides, the repeat-pass data have the following additional 
noise sources: 
- Temporal decorrelation: The corresponding noise 
covariance increases exponentially with the temporal 
baseline. E.g. Fig.4 compares the coherence histogram 
estimated by means of non-local means filter (Deledalle 
et al., 2011) over the whole scene (10km x 5km) of the 
aforementioned single-pass TanDEM-X interferogram 
and repeat-pass TerraSAR-X interferogram. It is obvious 
that the single-pass TanDEM-X interferogram is much 
coherent than the repeat-pass TerraSAR-X interferogram. 
- Phase errors caused by atmospheric delay and unmodeled 
motion: The distribution of such an error is unknown. 
Among them the phase error caused by atomospheric 
delay is normally roughly corrected through the 
Atmospheric Phase Screen (APS) estimates from the PSI 
processing. 
Although the above mentioned noise resources cannot be fully 
modeled as circular Gaussian noise, yet as the best asymptotical 
one the Gaussian model is still favoured for conveniences in the 
estimation (ie. only the covariance is needed) As a 
consequence, the different data quality possessed by repeat- and 
single-pass data can be therefore characterized by different 
noise variance. A key issue is to approperiately estimate the 
noise covariance matrix C,,. This will be addressed in a 
separate study. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
3. TOMOGRAPHIC SAR INVERSION FROM MIXED 
REPEAT- AND SINGLE-PASS DATA STACKS 
3.1 System Model 
In presence of noise £, the discrete-TomoSAR system model 
can be written as: 
g=-Ry+te (1) 
where g is the measurement vector with N elements, y is the 
reflectivity function uniformly sampled in elevation at 
S(I2L...,L). R isan NxL (N «L) iregularly sampled 
discrete Fourier transform mapping matrix and the sampling 
position & is a function of the elevation aperture position bn, 
Le. & - -2b,/(Ar), where A is the wavelength and r is the 
range distance. 
In the space-borne case, the multi-pass acquisitions are taken 
over a time of from several weeks to years (depending on the 
revisiting time of the satellite and the number of stacked 
images). Therefore, the long-term motion of the scattering 
object during the acquisition period must be considered by 
adding a motion-induced phase term to the system model, also 
refered to as D-TomoSAR system model. This renders 
tomographic SAR inversion to higher dimensional spectral 
estimation problem. Of course, its system model can also be 
approximated by a discrete version sharing the same expression 
as eq. (1). 
Tomographic SAR inversion aims at resolving the coherent 
targets y . According to the scattering mechanism, the coherent 
targets, i.e. the signal, to be resolved can be categorized as 
discrete scatterers and volumetric scatterers. The reflected 
power of discrete scatterers can be characterized by several ó- 
functions, i.e. the signal can be described by a deterministic 
model with a few parameters. Volumetric scatterers have a 
continuous backscatter profile associated with completely 
random scattering phases, i.e. the signal can only be described 
by stochastic models. Our target application is urban 
infrastructure monitoring, i.e. to resolve discrete scatterers with 
motion. 
There are numerous tomographic SAR inversion methods, 
including the conventional beamforming (BF), singular value 
decomposition (SVD), adaptive beamforming, multiple signal 
classification (MUSIC), nonlinear least squares (NLS) and 
algorithms exploiting the sparsity of the signal such as M- 
RELAX and the newly developed sparse reconstruction based 
SLIMMER algorithm. Since spatial resolution is essential for 
urban applications, to maintain the full range and azimuth 
resolution, we focus on single-looking methods that are based 
on the stacked measurements of single azimuth-range pixels 
and do not explore the correlation between the surrounding 
pixels. In this section, we will introduce the standard maximum 
a posteriori (MAP) estimator, NLS and the SLIMMER 
algorithm with Gaussian white noise and extend those 
estimators to the colored noise case. The description of the 
methods is based on single polarization TomoSAR. 
3.2 Tomographic SAR Reconstruction with White Noise 
3.2.1 MAP Estimator 
For Gaussian stationary white measurement noise, i.e. 
C.= c?I, and a white prior, i.e. C, =1, the MAP estimator 
for y from equation (1) is given by: 
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