# Urban Flood Mapping Using SAR Intensity and Interferometric Coherence via Bayesian Network Fusion
## 2 Methods
Another reason is that the conditional probabilities of \(p(F|\mathrm{C}_{i})\) and \(p\big{(}F\big{|}\mathrm{C}_{\gamma}\big{)}\) are evaluated interactively (described in detail later in this section). Therefore, consistent clusters of intensity and coherence are required. The joint probability of \(p\big{(}\mathrm{D}_{i},\mathrm{D}_{\gamma},\ F\big{)}\) in Figure 1b is given by:  
\[p\big{(}\mathrm{D}_{i},\mathrm{D}_{\gamma},\ F\big{)}=\ \sum\
olimits_{\mathrm{C}_{i}}p( \mathrm{D}_{i}|\mathrm{C}_{i})p(\mathrm{C}_{i}|F)\sum\
olimits_{\mathrm{C}_{ \gamma}}p\big{(}\mathrm{D}_{\gamma}\big{|}\mathrm{C}_{\gamma}\big{)}p\big{(} \mathrm{C}_{\gamma}\big{|}F\big{)}p(F), \tag{2}\]  
and the posterior probability of \(\mathrm{F}=1\) can be expressed as:  
\[p\Big{(}F=1|\mathrm{D}_{i},\mathrm{D}_{\gamma}\Big{)}=\ \frac{\sum \
olimits_{\mathrm{C}_{i}}p(\mathrm{D}_{i}|\mathrm{C}_{i})p(\mathrm{C}_{i}|F= 1)\sum\
olimits_{\mathrm{C}_{\gamma}}p\big{(}\mathrm{D}_{\gamma}\big{|} \mathrm{C}_{\gamma}\big{)}p\big{(}\mathrm{C}_{\gamma}\big{|}F=1\big{)}p(F=1)} {\sum\
olimits_{\mathrm{F}}\sum\
olimits_{\mathrm{C}_{i}}p(\mathrm{D}_{i}| \mathrm{C}_{i})p(\mathrm{C}_{i}|F)\sum\
olimits_{\mathrm{C}_{\gamma}}p\big{(} \mathrm{D}_{\gamma}\big{|}C_{\gamma}\big{)}p\big{(}\mathrm{C}_{\gamma}\big{|}F \big{)}p(F)}, \tag{3}\]  
where \(p(\mathrm{C}_{i}|F)\) and \(p(\mathrm{C}_{\gamma}|F)\) can be calculated via the Bayes rule:  
\[p\big{(}\mathrm{C}_{i}\big{|}F\big{)}=\ \frac{p\big{(}F|\mathrm{C}_{i}\big{)}p \big{(}\mathrm{C}_{j}\big{)}}{\sum\
olimits_{\mathrm{C}_{i}}p\big{(}F|\mathrm{ C}_{i}\big{)}p\big{(}\mathrm{C}_{j}\big{)}}, \tag{4}\]  
with \(j\in\{i,\ \gamma\}\). Each term in Equation (3) can be calculated analytically. The distribution of \(\mathrm{D}=\left( \begin{array}{c}\mathrm{D}_{i}\\ \mathrm{D}_{\gamma}\end{array} \right)\) is estimated by a finite Gaussian Mixture Model (GMM) involving the hidden variable \(\mathrm{C}\). Each assignment of \(\mathrm{C}\) is a Gaussian component, thus \(p(\mathrm{D}|\mathrm{C}=k)=\mathcal{N}(D\big{|}\mu_{k},\ \Sigma_{k})\), \(k=1,\cdots,\mathrm{K}\), and the parameters of each Gaussian component, \((\mu_{k},\ \Sigma_{k})\), \(\mu_{k}=\left( \begin{array}{c}\mu_{ik}\\ \mu_{\gamma k}\end{array} \right)\), \(\Sigma_{k}=\left( \begin{array}{cc}\Sigma_{ik}&\Sigma_{i\gamma k}\\ \Sigma_{\gamma jk}&\Sigma_{\gamma jk\gamma k}\end{array} \right)\), are estimated by the expectation-maximization (EM) algorithm [49]. Therefore, \(p(\mathrm{D}_{i}|\mathrm{C}_{i}=k)\) and \(p\big{(}\mathrm{D}_{\gamma}|\mathrm{C}_{\gamma}=k\big{)}\) are also Gaussian densities and can be calculated by \(p(\mathrm{D}_{i}|\mathrm{C}_{i}=k)=\mathcal{N}(\mathrm{D}_{i}|\mu_{ik},\ \Sigma_{iik})\) and \(p\big{(}\mathrm{D}_{\gamma}|\mathrm{C}_{\gamma}=k\big{)}=\mathcal{N}\big{(} \mathrm{D}_{\gamma}\big{|}\mu_{\gamma k},\ \Sigma_{\gamma jk}\big{)}\), respectively [46]. The number of Gaussian mixtures, \(\mathrm{K}\), depends on the homogeneity of the AOI. A smaller value of \(\mathrm{K}\) is needed for a more homogeneous area. In practice, a relatively large \(\mathrm{K}\) is preferable as under-clustering results in mixed clusters with variable spectral signatures and causes a compromised result, whereas over-clustering does not impact the final result [43]. We select \(\mathrm{K}\) via the Bayesian information criterion (BIC). The flood prior probability \(p(F=1)\) can be approximated  
Figure 1: The Bayesian network structure for flood mapping. The variable \(\mathrm{F}\) indicates flood states, \(\mathrm{C}\) is a hidden variable, and \(\mathrm{D}\) is a variable of observed Synthetic Aperture Radar (SAR) data. The subscript \(i\) and \(\gamma\) indicate intensity and coherence, respectively: (**a**) The original Bayesian network; (**b**) Bayesian network with decoupled variables \(\mathrm{C}\) and \(\mathrm{D}\). by auxiliary data [43]. However, auxiliary data are not always available in emergency response and a SAR data self-consistent approach is more preferable.