## 2 Methods
### SAR Interferometric Coherence  
The first method is based on a by-product of SAR interferometry, the coherence, which is here used for another application: fast disaster mapping of damaged buildings and infrastructure. Coherence is a measure of correlation and describes the preservation of the phase in an interferogram generated by two SAR images [75]. Coherence can be used to test the quality of an interferogram, e.g., for DEM generation or displacement mapping (see Section 1.3), as low coherence (de-correlation) corresponds to noisy interferograms, which often makes phase unwrapping complicated. The coherence \(\gamma\) ranging from 0 to 1 is calculated according to Equation (5) [76, 77, 78]:  
\[\gamma=\frac{E(c_{1}c_{2}^{*})}{\sqrt{E(c_{1}c_{1}^{*})E(c_{2}c_{2}^{*})}} \tag{5}\]  
Figure 4: Schematic flow chart of the two methods used for damage assessment: At least three SAR acquisitions are required: Two images acquired before the event (time t\({}_{1}\) and t\({}_{2}\)) and at least one SAR acquisition shortly after the disaster at time t\({}_{3}\) (as described in Figure 3). First, all images are co-registered to one common master (here image t\({}_{2}\) is selected as master image). Then, both slave images are resampled to the master’s reference grid. For both, the SAR intensity correlation, as well as for the coherence (calculated from the interferograms), two image pairs are generated: one pre-event (t\({}_{1}\) and t\({}_{2}\)) and one co-event (t\({}_{2}\) and t\({}_{3}\)) image pair. The damage is assessed by detecting the change between the corresponding image pairs (see Section 2.3). where \(c_{1}\) and \(c_{2}\) are the corresponding complex pixel values of the two acquisition dates t\({}_{1}\) and t\({}_{2}\), \(c^{\ast}\) is the complex conjugate of \(c\), and \(E\) indicates the expected value. Both, the coherence, as well as the intensity correlation defined in Equation (9), are calculated over a certain window of pixels. There are several types of de-correlation (loss of coherence [79]):  
\[|\gamma|=\gamma_{thermal}\;\gamma_{misreg}\;\gamma_{spat}\;\gamma_{dopp}\; \gamma_{atm}\;\gamma_{temp} \tag{6}\]  
* Thermal de-correlation \(\gamma_{thermal}\) is caused by uncorrelated noise inside the radar sensor itself [80]. * Miss-registration de-correlation \(\gamma_{misreg}\) is due to inaccurate registration of the two SAR images [29]. * Spatial de-correlation \(\gamma_{spat}\) occurs for too large baselines [74]. For \(B\bot>B_{critical}\) the difference of the incidence angle \(\theta\) (see Figure 2) at both acquisitions becomes too large. This leads to a de-correlation of the SAR echoes from both SAR sensors, _i.e._, the coherence strongly reduces (Equation (7) [81]). Furthermore, \(\gamma_{spat}\) also depends on the topography of the area of interest (AoI). Especially slopes facing the SAR sensor show strong de-correlation with increasing slope angle (foreshortening and layover effect). At flat terrain, common band filtering (usage of overlapping parts of the spectrums) corrects for the effect of \(\gamma_{spat}\). However, this correction reduces the range resolution [74]. \[B_{critical}\;=\;\frac{\lambda R_{s}}{2r(\cos\theta)^{2}}\] (7) with \(r\) representing the SAR sensor's spatial range resolution. For instance, \(B_{critical}\) of the European Remote Sensing (ERS) satellite is _ca._ 1100 m [82]. Therefore, shorter baselines are preferable for SAR interferometric applications [8], e.g., [83, 84] suggest \(B\bot<300\) m when using ERS data. Modern SAR sensors such as TerraSAR-X have a larger bandwidth and the effect of \(\gamma_{spat}\) is therefore nowadays less important compared to the early SAR missions [85]. * Doppler centroid de-correlation \(\gamma_{dopp}\), which is similar to spatial de-correlation, is caused by too large differences of the squint angle between both radar acquisitions. According to [PERSON] and [PERSON] [29], this effect can be avoided by a proper antenna steering. Moreover, range adaptive azimuth common band filtering mitigates the effect of \(\gamma_{dopp}\).