## 5 Vessel Classification Algorithm
\(\Delta j\) is the image cell extend of the slant-range (for the \(x\) coordinate) and azimuth (for the \(y\) coordinate) dimension (see Equation 8). It gives a measure of the uncertainty on estimating the proper coordinates of PePS and, so, it is used to ponder the difference error. For the \(z\) coordinate  
\[ \begin{array}{ccc}e_{z,n}^{p}=1&if&\frac{|<z_{n}>-z_{n}|}{\Delta z}\geq 1\\ e_{z,n}^{p}=\frac{|<z_{n}>-z_{n}|}{\Delta z}&if&\frac{|<z_{n}>-z_{n}|}{\Delta z }<1\end{array}  \tag{12}\]  
where \(z_{n}\) plays the same role than \(j_{n}\) in Equation 11 and \(\Delta z\) corresponds to the height bias experimented by the system for a particular phase error (\(\sigma_{\phi}\))  
\[\Delta z=\frac{\lambda\sin\phi r_{o}}{4\pi B_{\perp}}\sigma_{\phi} \tag{13}\]  
Finally, for the polarimetric error \[e_{p,n}^{p}=1\quad if\quad<p_{n}>\
eq p_{n} \tag{14}\] \[e_{p,n}^{p}=0\quad if\quad<p_{n}>=p_{n}\]  
where \(p_{n}\) is the weight of the normalized Pauli significance as defined in Equations 4 and 5. In Equation 9, the factors (\(0\geq W_{j}\geq 1\) for \(j\in\{1\dots 4\}\)) are weights that give different \"significance\" to each error in the identification process. Empirical analysis have shown that the following values provide the best overall results (\(W_{x}=0.15\ W_{y}=0.15\ W_{z}=0.35\ W_{p}=0.35\)). They reduce the influence of azimuth and range errors due to the image distortions induced by sea surface [1, 32]. In summary, the main steps of VCA are [2]:  
1. Coregistrate single-pass interferometric pairs and generate Pauli interferograms. 2. Isolate all the local maxima (\(m\)) present in all the three Pauli interferograms for a fixed dynamic range (\(DR_{i}\)) and find the values of the four components defining the feature set. In this step, a ship bearing estimation methodology based on the Radon transform has been used for making the required coordinate system transformation (see Section 4. Equation 8). 3. For each pattern with \(M\) PePS, test the similarity of the \(m\cdot\frac{M!}{(M-m)! }\) permutations possible. Here, the formulation from Equation 9 to 14 applies with the _suitability scatter_ step. This implies that, for a permutation \(\kappa\), \(r\) could be lower than \(M\) if exists one or more local maxima that present, in at least two of the four components of the feature set vector, errors equal to 1. 4. That pattern with that permutation providing the highest similarity value is selected to classify the observed ship for the dynamic range \(DR_{i}\). 5. Repeat steps 2-4 for different dynamic ranges in order to reach the maximum similarity and/or the best discrimination among the different models. The model which feature set has been selected the highest number of times to classify the observed ship becomes the final decision of the algorithm. The final similarity value is the value providing the best discrimination among models. Note that this rule is empirical and it has been motivated by the information that the algorithm has to deal with. It has been adopted because it allows to evaluate the performances of the proposed method in an easy and quick way. Obviously, better decision rules may be developed in the future.