# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 7 Results
### Simulated Scenario Results  
In order to test the observability of the different biases modelled in previous sections, the estimation method will be tested with a simulated scenario. In this scenario there are two radars in position (0, 0) and (50, 0) (in NM). The scenario has 1000 point uniformly distributed in a square area with a side of 400 NM centered at (0, 0). The maximum altitude of these points is 15,000 m.  
In this case the simulated measurements are synchronized and the noise of the measurements are white Gaussian noise with zero mean with standard deviation of 75 m in range and 0.05\({}^{\circ}\) in azimuth. This test is made using Cartesian coordinates with the flat Earth model. The measures obtained by the radars are biased using the previously developed models with all the parameters. The evaluation of the results is made using the RMS error in position of all the corrected measurement in Equation (73). As the ideal position of the points is known, the error will be calculated between the corrected measurement and the ideal position. In this simulated scenario, the RMS error of the measured plots is 434.72 m and the RMS error of the corrected plots is 126.13 m. As in the simulated data the ideal position of the aircraft is known, we can measure the errors produced only by the noise. In this case, the RMS error produced just by the noise is 120.08 m. As the RMS error of the corrected plots is almost the RMS error produced by the noise of the measurements, it is proved the observability of the model when the plots are well distributed. With a bigger number of samples, the RMS error of the estimation is nearer to the RMS error produced by the noise. In the previous simulated scenario the bias parameter used in the generation of the simulated position are shown in Table 1. The next paragraphs will show the estimation of some parameters using different models but always generating the simulated plots with the complete model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **RMS Error** \\ \hline \(\Delta\rho_{0}\) & 100 m \\ \(\alpha_{1}\) & \(10^{-3}\) m/m \\ \(\alpha_{2}\) & \(10^{-9}\) m/m\({}^{2}\) \\ \(\alpha_{3}\) & 1.15 \\ \(\theta_{0}\) & \(0.04^{\circ}\) \\ \(s_{ant}\) & \(0.5^{\circ}\) \\ \(\beta_{axis}\) & \(0.4^{\circ}\) \\ \(\alpha_{axis}\) & \(45^{\circ}\) \\ \(\beta_{enc}\) & \(0.5^{\circ}\) \\ \(\alpha_{enc}\) & \(90^{\circ}\) \\ \(\Delta\text{R}/\text{R}\) & \(5\times 10^{-4}\) \\ \(\alpha_{exc}\) & \(45^{\circ}\) \\ \(\Delta H_{p}\) & \(-500\) m \\ \(\Delta T\) & \(15^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the biases used in the simulation for the radar 1. Figure 12: Estimated bias values for Radar 1 using only the basic model (\(\Delta\rho_{0}\), \(\alpha_{1}\) and \(\theta_{0}\)). (**a**) Range offset; (**b**) Azimuth offset; (**c**) Range gain; (**d**) Temperature bias. In the evaluation of the simulated data, the complete model with the values indicated in Table 1 has been used in the generation of the plots, but different bias models with more or less parameters have been used in the estimation. The next figures shows the mean estimated value of the biases with a blue line and the mean plus/minus the standard deviation of the estimation with a dotted red line. The basic model used classically in the bias estimation literature uses only \(\Delta\rho_{0}\), \(\alpha_{1}\) and \(\theta_{0}\). The obtained results are shown in Figure 12. The temperature bias isn't estimated with this model and in the figure corresponding to this bias the value and the deviation is zero.