## 2 Methods
### Deep Belief Network (DBN)  
#### 2.3.1 Restricted Boltzmann Machine  
RBM is a neural perceptron consisting of visible and hidden layers, and the neurons between the visible layer (\(v_{i}\), \(i\) = 1,..., \(N_{p}\)) and the hidden layer (\(h_{j}\), \(j\) = 1,..., \(N_{h}\)) are bidirectional and fully connected. The basic structure of RBM is shown in Figure 1:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Feature** & **Definition** & **For Mineral** & **For Biogenic** & **For Clean Sea** \\  & & **Oil** & **Slicks** & **Surface** \\ \hline VV intensity & \(S_{VV}^{2}\) & Lower 1  & low & High \\ \hline Entropy (\(H\)) & \(P_{i}=\frac{\lambda_{i}}{\sum\limits_{j=1}^{N}\lambda_{j}}\) & High & Low & Lower \\ \hline Alpha (\(\alpha\)) & \(\alpha=P_{1}\alpha_{1}+P_{2}\alpha_{2}+P_{3}\alpha_{3}\) & High & Low & Lower \\ \hline Degree of Polarization (DoP) & \(P=\frac{\sqrt{S_{0}^{2}+S_{0}^{2}+S_{1}^{2}}}{S_{0}^{2}}\) & Low & High & High \\ \hline Ellipticity (\(\chi\)) & \(\sin(2\chi)=-\frac{s_{0}}{m_{00}}\) & Positive & Negative & Negative \\ \hline Pedestal Height (\(PH\)) & \(NPH=\frac{\min(\lambda_{1}\lambda_{2}\lambda_{3})}{\max(\lambda_{1}\lambda_{2} \lambda_{3})}\) & High & Low & Lower \\ \hline Standard & & & & \\ Deviation of CPD & & & & \\ \hline Conformity & & & & \\ Coefficient (Conf. Co.) & & & & \\ \hline Correlation & & & & \\ Coefficient (Corr. Co.) & & & & \\ \hline Coherence & & & & \\ Coefficient (Conf. Co.) & & & & \\ \hline \hline \end{tabular}  
* Note: “lower” and “higher” mean that the property of the feature on a certain type of surface is close to the other surface that has the property of “low” or “high”, but slightly lower or higher. “Std. copolarized phase difference (CPD)” stands for the standard deviation of CPD.  
\end{table}
Table 1: Features investigated in this study.  
In RBM, \(\mathbf{W}\) represents the weight between any two connected neurons, in which each neuron has a bias coefficient \(b\) (of neurons) and \(c\) (of hidden neurons).  
The energy of the RBM can be represented by:  
\[E(v,h)=-\sum_{i=1}^{Nv}b_{i}v_{i}-\sum_{j=1}^{Nh}c_{i}j_{i}-\sum_{i,j=1}^{Nv,Nh }W_{i,j}v_{i}h_{j} \tag{4}\]  
And the probability of the activation of the hidden layer neuron \(h_{j}\) is:  
\[P\big{(}h_{j}|v\big{)}=\sigma(b_{j}+\sum_{i}W_{i,j}x_{i}) \tag{5}\]  
Similarly, the neurons in the visible layer can also be activated by the bidirectional connected hidden neurons:  
\[P(v_{i}|h)=\sigma(c_{i}+\sum_{j}W_{i,j}h_{j}) \tag{6}\]  
where \(\sigma\) is the activation function, e.g., sigmoid function:  
\[\sigma(x)=\frac{1}{1+e^{-x}} \tag{7}\]  
Since for RBM the neurons of the same layer is not connected, they are independent:  
\[P(h|v)=\prod_{j=1}^{Nh}P\big{(}h_{j}|v\big{)} \tag{8}\]  
\[P(v|h)=\prod_{i=1}^{Nv}P(v_{i}|h) \tag{9}\]  
Based on the input data vector \(x\), the possibility of the activation of each hidden layer neuron can be calculated. Similarly, based on the activation state of hidden layer neurons, the activation state of visible layers can be calculated. Through a contrastive divergence algorithm [28], the parameters of the RBM: (\(b\), \(c\), \(\mathbf{W}\)) can be set based on the input data vector \(x\) iteratively by a Gibbs sampling technique. An RBM can be seen as a feature detector, which is often used for dimensional reduction of the data. The training process of RBM is to find a probability distribution that can best produce training samples.