# Multi-Temporal Sentinel-1 Backscatter and Coherence for Rainforest Mapping
## 3 Method
In Section 5 we present the results of the Random Forests classifier in two different cases, characterized by a different set of input features:  
* case _(ORIG)_: \(\hat{\gamma^{0}}\), \(\hat{\tau}\), \(\hat{\rho}_{\text{LT}}\), and \(\theta_{\text{inc}}\),
* case _(SADH)_: \(\hat{\gamma^{0}}\), \(\hat{\tau}\), \(\hat{\rho}_{\text{LT}}\), \(\theta_{\text{inc}}\), \(SADH_{(1,0)}\), and \(SADH_{(0,1)}\). In case _(ORIG)_ we apply the exact algorithm of [7], which represents our baseline. In case _(SADH)_, we additionally use the whole 18 textures extracted from \(\gamma^{0}\) by using the SADH method. In this context, we preliminary measured the Pearson's correlation coefficient of all the possible combinations of input features and verified that no strong correlated features, either complete decorrelated ones, were present. For this reason, we use all the generated features as input to the Random Forests algorithm. The comparison between the results is based on the evaluation of the average accuracy (\(AA\)) and the overall accuracy (\(OA\)) for all the test areas, considering all the valid pixels in the image under test. Given \(P\) classes, the associated confusion matrix \(C\) assumes in general the following form: \begin{table}
\begin{tabular}{c c c} \hline \hline
**ORIG** & \(SADH_{(1,0)}\) & \(SADH_{(0,1)}\) \\ \hline \(\hat{\gamma^{0}}\) & \(AVE_{(1,0)}\) & \(AVE_{(0,1)}\) \\ \(\hat{\tau}\) & \(CLP_{(1,0)}\) & \(CLP_{(0,1)}\) \\ \(\hat{\rho}_{\text{LT}}\) & \(CLS_{(1,0)}\) & \(CLS_{(0,1)}\) \\ \(\theta_{\text{inc}}\) & \(CON_{(1,0)}\) & \(CON_{(0,1)}\) \\  & \(COR_{(1,0)}\) & \(COR_{(0,1)}\) \\  & \(ENE_{(1,0)}\) & \(ENE_{(0,1)}\) \\  & \(ENT_{(1,0)}\) & \(ENT_{(0,1)}\) \\  & \(HOM_{(1,0)}\) & \(HOM_{(0,1)}\) \\  & \(VAR_{(1,0)}\) & \(VAR_{(0,1)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the 22 features considered in Section 5. Column \(ORIG\) shows the parameters used in [7], while columns \(SADH_{(1,0)}\) and \(SADH_{(0,1)}\) explain the textures extracted by using displacement vectors along both the azimuth \(d=(1,0)\) and the slant-range \(d=(0,1)\) directions, respectively, whose mathematical formulation is presented in Equations (4)–(12). \[C= \begin{bmatrix}c_{1,1}&\dots&c_{1,j}&\dots&c_{1,P}\\ \vdots&\ddots&\vdots&&\vdots\\ c_{i,1}&\dots&c_{i,j}&\dots&c_{i,P}\\ \vdots&&\vdots&\ddots&\vdots\\ c_{P,1}&\dots&c_{P,j}&\dots&c_{P,P}\end{bmatrix} , \tag{13}\]  
where the elements along the main diagonal, \(c_{i,j}\) (\(i=j\)), represent the correctly predicted pixels for each class \(j=1,...,P\), that is, they are called class accuracies, and the sum of the elements along each column \(j\) corresponds to the number of pixels belonging to each class \(j=1,...,P\). The average accuracy defines the mean of each accuracy per single class, that is, the sum of class accuracies divided by the number of classes, while the overall accuracy corresponds to the number of correctly predicted pixels divided by the total number of pixels to predict, that is, the sum of all elements in \(C\). In particular, the respective formulas associated with the two metrics are:  
\[AA=\frac{\sum_{j=1}^{P}c_{j,j}}{P} \tag{14}\]  
and  
\[OA=\frac{\sum_{j=1}^{P}c_{j,j}}{\sum_{i=1}^{P}\sum_{j=1}^{P}c_{i,j}}. \tag{15}\]  
While the overall accuracy assesses the global performance of the classifier, the average accuracy further accounts for accuracy unbalancing between the different classes.