# Range-Doppler Based CFAR Ship Detection with Automatic Training Data Selection
## 6 Experimental Results and Discussion
### Clutter Model Fitting and Performance Evaluation
#### 6.1.2 False Alarm Rate Error  
To further quantify the suitability of the clutter models, the actual false alarm rates are also evaluated for different ranges. The false alarm rate is estimated as the ratio of the number of false detections to the number of samples in the data (cf. (8)). It has to be noted that for estimating the false alarm rate all the bright targets have first to be excluded from the scene for avoiding biases. This can be done using the proposed pre-detection method. The obtained results from all the clutter models using linearly and circularly acquired data are shown in Tables 5 and 6, respectively. From Tables 5 and 6, it can be concluded that the K-Rayleigh distribution gives the least false alarm rate error in near and mid ranges, whereas the chi-square gives the highest error. In far range, the 3 MD, chi-square and K-NLLSQ match well with the set false alarm rate. Based on the analyses in the previous three sections, it can be concluded that for near and mid ranges (\(15^{\circ}\)-\(50^{\circ}\) incidence angle) the K-Rayleigh distribution function is the best suitable choice, whereas for the far range (\(>\)\(50^{\circ}\) incidence angle), the 3 MD model or the chi-square is the preferred choice. This recommendation is at least valid for the RC X-band HH polarized F-SAR data used for the investigations and the current sea state conditions during data acquisitions. It has to be further noted that the detection rate of the proposed CFAR detector is not evaluated in this paper. This is because the vessel size is larger than the spatial resolution of the image which gives several pixel-based detections for a single vessel (cf. Figure 20c). Thus, even with the ground truth information the number of \"true\" detections belonging to a vessel cannot be determined reliably. Therefore, it is difficult when not even impossible to accurately measure the probability of detection using data containing real ships. Even if the detected pixels would be (manually) clustered to a single physical object, it is important to set an appropriate detection criterion to consider it as a \"true\" detection. Therefore, this kind of \"object-based\" detection and the derived \"object-based\" detection rate so far cannot be compared with the established \"pixel-based\" probability of detection used in any CFAR detection framework [(37; 51)]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distribution Functions** & **Near Range** & **Mid-Range** & **Far Range** \\ \hline K-NLLSQ & 63.3 & 74.6 & 12.2 \\ K-Vstat & 30.6 & 38.9 & - \\ K-Xstat & 46.9 & 55.9 & - \\ Chi-square & 422.4 & 234.9 & 9.43 \\
3 MD & 257.9 & 154.7 & **7.73** \\ K-Rayleigh & **2.03** & **2.08** & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: False alarm rate errors for circularly acquired data. The values in the table are the ratio between the estimated actual false alarm rate and the set false alarm rate of \(10^{-6}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distribution Functions** & **Near Range** & **Mid-Range** & **Far Range** \\ \hline K-NLLSQ & 80.5 & 112.1 & 3.08 \\ K-Vstat & 35.1 & 57.1 & - \\ K-Xstat & 56.9 & 86.8 & - \\ Chi-square & 277.4 & 242.9 & 2.43 \\
3 MD & 149.2 & 135.9 & **1.56** \\ K-Rayleigh & **1.31** & **1.68** & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: False alarm rate errors for linearly acquired data. The values in the table are the ratio between the estimated actual false alarm rate and the set false alarm rate of \(10^{-6}\). In Figure 30, the ship signal is not completely visible over the entire observation time. There are some gaps. This is because the F-SAR X-band 3 dB azimuth antenna beamwidth in the order of \(8^{\circ}\) is rather small. Since the antenna also cannot be steered electronically or mechanically, during a circular flight with a ship moving in the circle center, the cross-wind may cause a significant yaw angle so that the ship is not always illuminated.