# Range-Doppler Based CFAR Ship Detection with Automatic Training Data Selection
## 6 Experimental Results and Discussion
### Clutter Model Fitting and Performance Evaluation
#### 6.1.1 Threshold Error  
In order to know which models give the best results, the actual threshold errors are estimated. The threshold error is computed by calculating the absolute difference between the thresholds estimated from the data CCDF and the model CCDF at a certain CCDF value in the tail region of the histogram. The threshold error \(error_{\text{threshold}}\) is written as  
\[error_{\text{threshold}}=\left|\eta_{\text{data}}-\eta_{\text{model}} \right|_{\text{CCDF}=10^{-4}}. \tag{32}\]  
where \(\eta_{\text{data}}\) and \(\eta_{\text{model}}\) are the thresholds obtained from the data and the clutter model at a specific CCDF value, respectively. The threshold error is computed in the tail region because of two reasons: bright ship target signals lie mostly in that region and the tail region is the region where most of the mismatch between reality and models occurs. The threshold errors (in log scale) at CCDF = 10\({}^{-4}\) and 10\({}^{-5}\) computed using different distribution functions for near, mid and far ranges as well as for linear and circular data are shown in Tables 3 and 4, respectively.  
From Tables 3 and 4 it can be seen that the threshold error is minimum in case of K-Rayleigh distribution and maximum in case of chi-square distribution in near and mid-range. In circular data, for far range the chi-square distribution gives the minimum error.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Clutter Models**} & **Near Range** & **Mid-Range** & **Far Range** \\ \cline{2-4}  & \multicolumn{3}{c}{**CCDF**} \\  & **10\({}^{-4}\)** & **10\({}^{-4}\)** & **10\({}^{-4}\)** \\ \hline K-NLLSQ & 6.89 & 6.02 & \(-\)5.86 \\ K-Vstat & 5.19 & 4.60 & - \\ K-Xstat & 6.19 & 5.49 & - \\ Chi-square & 9.70 & 7.73 & \(\mathbf{-11.65}\) \\
3 MD & 8.94 & 6.94 & \(-\)10.86 \\ K-Rayleigh & \(\mathbf{-6.68}\) & \(\mathbf{2.27}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimated threshold error for different clutter models for circularly acquired F-SAR data.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Clutter Models**} & **Near Range** & **Mid-Range** & **Far Range** \\ \cline{2-4}  & \multicolumn{3}{c}{**CCDF**} \\  & **10\({}^{-4}\)** & **10\({}^{-4}\)** & **10\({}^{-4}\)** & **10\({}^{-5}\)** \\ \hline K-NLLSQ & 3.97 & 8.01 & \(-\)10.34 & \(-\)2.16 \\ K-Vstat & 2.41 & 6.89 & - \\ K-Xstat & 3.23 & 7.61 & - \\ Chi-square & 6.98 & 8.87 & \(-\)10.34 & \(-\)4.98 \\
3 MD & 5.62 & 8.27 & \(-\)10.34 & \(\mathbf{-5.17}\) \\ K-Rayleigh & \(\mathbf{-5.79}\) & \(\mathbf{-0.26}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Estimated threshold errors for different clutter models for linearly acquired F-SAR data.  
In far range of linearly acquired data (cf. Table 3, right), the computed threshold errors are the same for CCDF = \(10^{-4}\) for the K-NLLSQ, chi-square and 3 MD model. Therefore, the threshold errors were additionally computed for CCDF = \(10^{-5}\) and it is found that the 3 MD gives minimum error in far range of linear data.