# Urban Flood Mapping Using SAR Intensity and Interferometric Coherence via Bayesian Network Fusion
## 2 Methods  
A Bayesian network is a probabilistic graphical model for compactly specifying joint probability distribution over a fixed set of random variables. It is a directed acyclic graph (DAG), where nodes represent variables and links between nodes represent dependencies between them [46]. The DAG specifies conditional independence statements of variables on their ancestors--namely which ancestors are direct \"causes\" for the variable [47]. Assuming \(x_{1},\ \cdots,\ x_{N}\) are the random variables, the joint distribution under a Bayesian network is given by  
\[p(x_{1},\ \cdots,\ x_{N})=\ \prod_{i=1}^{N}p\Big{(}x_{i}\left|\mathrm{pa} \left(x_{i}\right)\right\rangle, \tag{1}\]  
where \(\mathrm{pa}(x_{i})\) represent the parental variables of variable \(x_{i}\), with an arrow pointing from a parent variable to child variable in the DAG. The Bayesian network thus provides information about the underlying process and any conditional distribution(s) can be expressed via inference. The structure of the Bayesian network for flood mapping based on the fusion of intensity and coherence is visualized in Figure 1a. It combines information from both the backscatter intensity and interferometric coherence time series. The shaded nodes in Figure 1 indicate observed variables whereas open nodes mask unknown variables. In detail, the random variable F indicates the flood state (e.g., \(\mathrm{F}\ =\ 1\) for flood state and \(\mathrm{F}\ =\ 0\) for non-flood state, respectively) of each pixel; this is our target variable for which we want to infer its posterior probability conditioned on all the other variables. The variable D corresponds to the combination of intensity and coherence time series: stacked imagery of multitemporal intensity and coherence. C is a hidden variable, which links the influence of variable F on the observed image series D. It is difficult to find a simple causality between flood state and the observed SAR signatures, especially in urban areas associated with the complex backscattering mechanisms due to varying land covers. Therefore, it is necessary to introduce an intermediate variable C with K possible states that represent different temporal behaviors of the area of interest (AOI), corresponding to different land covers from a SAR (e.g., intensity and coherence) point of view [43; 48]. Intensity and coherence characterize different physical properties of a scene: intensity provides information about surface roughness and permittivity, whereas coherence measures the random variation of individual scatterers between two SAR acquisitions, indicating temporal similarity within a cell. Therefore, intensity and coherence could be sensitive to and provide useful information on different flooded land covers. For instance, intensity is sensitive to flooded bare soils and flooded vegetation, both intensity and coherence are sensitive to flooded built-up areas, and coherence is also sensitive to some flooded built-up areas that are insensitive for intensity (as described in the Introduction section). Considering this phenomenon, the hidden variable C is decoupled to two variables of C\({}_{i}\) and C\({}_{\gamma}\) corresponding to the intensity and coherence temporal signature,respectively (Figure 1b), as the flood state of a given class \(\mathrm{C}=k,k=1,\cdots,\mathrm{K}\), should be evaluated from the perspective of intensity (e.g., \(p(F|\mathrm{C}_{i}=k)\)) and coherence (e.g., \(p\big{(}F\big{|}\mathrm{C}_{\gamma}=k\big{)}\)). The variable \(\mathrm{D}\) is partitioned to intensity (e.g., variable \(\mathrm{D}_{i}\)) and coherence series (e.g., variable \(\mathrm{D}_{\gamma}\)) accordingly. The land cover segmentation (e.g., variable \(\mathrm{C}\)) is implemented under the conjunction of intensity and coherence time series (e.g., variable \(\mathrm{D}\)) to retain the internal dependencies between these two data sources and preserve compact clusters of the AOI.