# Urban Flood Mapping Using SAR Intensity and Interferometric Coherence via Bayesian Network Fusion
## 2 Methods
Therefore, the non-informative prior probability is used in this paper, \(p(F=1)=\ p(F=0)=0.5\). The term \(p\big{(}C_{j}\big{)}\) in Equation (4) is a vector of weights of K Gaussian components. \(p\big{(}F\big{|}C_{j}\big{)}\) is a conditional probability table (CPT) that contains the flood probability of each component. It is the core part of the whole process and determines the final result. The CPT can be assigned manually by an expert or be learned from the data. The latter strategy is adopted in this paper to make the whole chain automatic. Under the assumption that the existence of floodwater may cause an abrupt change in either intensity or coherence, the CPT is estimated based on the variation between the average of pre-event series and the co-event acquisition for each component centroid (e.g., \(\mu_{jk}\), \(j\in\{i,\ \gamma\}\), and \(k=1,\cdots,\) K). Intensity could decrease or increase due to the specular reflection or double-bounce, whereas coherence drops because of decorrelation caused by floodwater. Let \(\Delta_{j}\), \(j\in\{i,\ \gamma\}\) be the variation vector that we are concerned; we intend to extract the change information related to flooding at this step:  
\[\Delta_{i}=max\big{(}mean\big{(}\mu_{i}^{pre}\big{)},\ \mu_{i}^{\infty}\big{)}- min\big{(}mean\big{(}\mu_{i}^{pre}\big{)},\ \mu_{i}^{\infty}\big{)},\ \ \ \ \Delta_{\gamma}=mean\big{(}\mu_{\gamma}^{pre}\big{)}-\mu_{\gamma}^{\infty}, \tag{5}\]  
where \(max(\cdot,\ \cdot)\) and \(min(\cdot,\ \cdot)\) extract the element-wise maximum and minimum values of two vectors, respectively, \(mean(\cdot)\) is the average operator along the time axis, \(\mu_{j}^{pre}\) and \(\mu_{j}^{\infty}\) represent component centroids of the pre-event series and the co-event acquisition, respectively. The CPT \(p\big{(}F\big{|}C_{j}\big{)}\) is given by the sigmoid function of \(\Delta_{j}\):  
\[p\big{(}F\big{|}C_{j}\big{)}=\ \frac{1}{1+e^{-\beta_{j}(\Delta_{j}-\alpha_{j})}}, \tag{6}\]  
where \(\beta_{j}\) is the steepness of the curve, and \(\alpha_{j}\) is the \(\Delta_{j}\) value corresponding to \(p\big{(}F\big{|}C_{j}\big{)}=0.5\). With relatively steep curves, the final result is not sensitive to the value of \(\beta_{j}\) and \(\beta_{j}=1\) is set as the default value. \(\alpha_{j}\) is the most important parameter and an optimal value should be assigned. After \(\Delta_{j}\) is computed through Equation (5), its values are sorted in descending order. \(\Delta_{jl}\) (\(l=1,\cdots,\ \text{K}-1\)) is the potential value of \(\alpha_{j}\) and it separates the components to a flood-related changed set \(\Phi_{jC}\) (\(\Delta_{jk}\in\Phi_{jC}\), \(k\leq l\)) and an unchanged set \(\Phi_{jU}\) (\(\Delta_{jk}\in\Phi_{jU}\), \(k>l\)). A cost function is defined to find the optimal index of \(l^{*}\) by measuring intraclass compactness and interclass separability of the two sets [50,51]:  
\[L_{l}=\ \frac{\sum_{V=[C,l]}\sum_{V\Delta_{jk}\in\Phi_{jV}}\big{(}\Delta_{jk}-m_ {jV}\big{)}^{2}}{\sum_{V=[C,l]}\frac{N_{\Phi_{jV}}}{K}\big{(}m_{jV}-m_{j} \big{)}^{2}}, \tag{7}\]  
where \(m_{jV}\) is the mean value of set \(\Phi_{jV}\), \(N_{\Phi_{jV}}\) is the component number of set \(\Phi_{jV}\), and \(m_{j}\) is the mean value of the whole \(\Delta_{j}\). A smaller value of \(L_{l}\) achieves higher intraclass compactness and interclass separability, thus \(l^{*}\) is determined via the minima of Equation (7):  
\[l^{*}=\ arg\underset{l}{\min}\{L_{l}\},\ \ l=1,\cdots,\ \text{K}-1 \tag{8}\]  
The CPT \(p\big{(}F\big{|}C_{j}\big{)}\) needs to be further refined after being calculated via Equations (5)-(8), since we should take into account the flood uncertainty in terms of intensity and coherence depends on the land cover types. Generally speaking, the change information from intensity is reliable for flood detection in non-built-up areas and partially reliable for built-up areas. Missed alarms that happen in some particular built-up areas (see the Introduction section) need the complement of coherence information.