# Urban Flood Mapping Using SAR Intensity and Interferometric Coherence via Bayesian Network Fusion
## 2 Methods
The change information from coherence is reliable for coherent targets (e.g., built-up areas) and not reliable for non-coherent areas such as vegetated areas - especially when data with large temporal baseline (e.g., from days up to months) are used. Moreover, the different temporal baselines between the pre-event acquisitions and the co-event acquisition can also cause either false alarms or missed alarms in non-coherent areas. Therefore, we first classify the AOI to coherent (built-up) and non-coherent (non-built-up) areas through the relationship between \(mean\Big{(}\mu_{\gamma}^{pre}\Big{)}\) and a threshold \(t\), and subsequently refine \(p(F|C_{i})\) and \(p\Big{(}F|C_{\gamma}\Big{)}\) as:  
\[ \begin{split} p\Big{(}F\Big{|}C_{ik}\in\Big{\{}\Big{[}mean\Big{(} \mu_{\gamma\kappa}^{pre}\Big{)}>t\Big{]}\delta\kappa\Big{[}(\Delta_{ik}<\alpha _{i})\delta\kappa(\Delta_{\gamma\kappa}>\alpha_{\gamma})\Big{]}\Big{\}}\Big{)} \Big{)}=0.5,\\ p\Big{(}F\Big{|}C_{\gamma\kappa}\in\Big{\{}\Big{[}mean\Big{(} \mu_{\gamma\kappa}^{pre}\Big{)}\leq t\Big{]}\delta\kappa\Big{[}\big{[}(\Delta_ {ik}>\alpha_{i})\delta\kappa(\Delta_{\gamma\kappa}<\alpha_{\gamma})\big{]} \Big{]}\Big{[}(\Delta_{\gamma\kappa}>\alpha_{\gamma})\delta\kappa(\Delta_{ik}< \alpha_{i})\big{]}\Big{]}\Big{\}}\Big{)}=0.5,\end{split}  \tag{9}\]  
where & denotes the logical AND, and \(\|\) denotes the logical OR, respectively. Equation (9) refines the flood evidence in terms of intensity and coherence according to the category of each component. For a given component \(k\) which belongs to the coherent (built-up) areas, when the flood evidence is favored by coherence (e.g., \(\Delta_{\gamma k}>\alpha_{\gamma}\)) whereas not by intensity (e.g., \(\Delta_{ik}<\alpha_{i}\)), intensity may fail to capture the flood information. Therefore, \(p(F|C_{ik})=0.5\) is assigned to this component to account for the uncertainty of intensity. For components that belong to non-coherent (non-built-up) areas, on the one hand, the non-flooded one can be misclassified as a flooded component by coherence due to the large temporal variation of the land cover and the (possible) difference of temporal baseline in the dataset. On the other hand, when pre-event coherence of a flooded component is already low (e.g., vegetated areas), neglectable variation can be observed between pre- and co-event coherence, thus it is difficult to detect floodwater by coherence. Therefore, \(p\Big{(}F|C_{\gamma k}\Big{)}=0.5\) is assigned to the non-coherent components which show inconsistent flood evidence from intensity and coherence. After the final \(p\Big{(}F|C_{j}\Big{)}\) is determined, the flood posterior probability of each pixel can be evaluated through Equations (3) and (4). The fully-connected CRF is adopted to refine the flood probability by integrating the long-range spatial information and the final binary flood extent is obtained via the maximum a posteriori (MAP) operation. In this process, the flood posterior probability of each pixel estimated via the Bayesian network is the unary potential term of the fully-connected CRF, and the difference images calculated by Equation (5) are the feature vectors of the appearance kernel [18]. Finally, the binary flood extent can be further classified to the following flood categories: non-obstructed-flood that is characterized by a decreased \(\sigma^{\circ}\) in co-event acquisition; obstructed-flooded non-coherent areas (e.g., flooded vegetation and vegetated built-up areas) that show an increased \(\sigma^{\circ}\) in the co-event acquisition and low pre-event \(\gamma\); obstructed-flooded coherent areas (e.g., flooded predominantly built-up areas) that present an increased/unchanged \(\sigma^{\circ}\) in co-event acquisition and high pre-event \(\gamma\).