# Geolocation and Tracking by TDOA Measurements Based on Space-Air-Ground Integrated Network
## 4 Discussion  
We consider the SAGI passive localization geometry used in [23]. The observer network contains GEO satellite, LEO satellite, airplane and ground station. The initial positions are (105\({}^{\circ}\), 0\({}^{\circ}\), 36,000 km), (101\({}^{\circ}\), 0\({}^{\circ}\), 500 km), (103\({}^{\circ}\), 3\({}^{\circ}\), 10 km) and (105\({}^{\circ}\), 5\({}^{\circ}\), 0 km), respectively. The initial velocity and course angle of the airplane are 2000 km/h and 60\({}^{\circ}\). The initial locations of these observers under the ECEF coordinate system are shown in Table 1.  
In GDOP (geometric dilution of precision) figures, the X-axis is the longitude and the Y-axis is the latitude. The curve in the GDOP figure means the localization precision of targets, whose location is on the curve. The red hexagram denotes the substar of the GEO, the red five-pointed star denotes the substar of the LEO, the red rhombus denotes the sub-point of the airplane, and the red circle denotes the location of the ground station. Scenario 1: The sensor network contains the GEO satellite and the LEO satellite. Simulation results are shown in Figures 1-3. Figure 1 places emphasis on the localiza \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Title 1** & **X (m)** & **Y (m)** & **Z (m)** & **Vx (m/s)** & **Vy (m/s)** & **Vz (m/s)** \\ \hline GEO satellite & \(-\)10,912,884.14 & 40,727,438.07 & \(-\)44,151.31 & 0 & 0 & 0 \\ LEO satellite & \(-\)13,069,917.55 & 675,827.56 & \(-\)7416.31 & \(-\)4793.46 & \(-\)933.62 & \(-\)5380.76 \\ Airplane & \(-\)1435,061.91 & 6,125,936.03 & 332,097.67 & \(-\)264.62 & \(-\)86.94 & 480.47 \\ Ground station & \(-\)1,644,543.41 & 6,137,519.57 & 552,183.96 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The initial locations of the observers in ECEF. The random error of TDOA measurement is 500 ns in Figure 1, which represents the TDOA covariance matrix \(\mathbf{Q}_{t_{j,k}}=diag\bigg{\{}\Big{(}\frac{500\,\mathrm{ns}}{10^{9}}\Big{)}^ {2},\ldots,\Big{(}\frac{500\,\mathrm{ns}}{10^{9}}\Big{)}^{2}\bigg{\}}\). Figure 1a presents the GDOP without any system error. Figure 1b shows the GDOP with 1 us bias of clock synchronization. Figure 1c gives the GDOP with satellite position bias of (2000, 2000, 2000) m. Figure 1d shows the GDOP with 2 km elevation bias of the target. From the simulation results in Figure 1, we can make the following conclusions: (1) The localization performance degrades seriously in the area around the moving direction of the LEO satellite. (2) The effects of different system errors are also diverse. Comparing the four GDOPs in Figure 1, we can observe that clock synchronization bias is the most influential factor for this scenario. The elevation bias of the target is the second most influential factor. The last one is position bias of the satellites. In Figure 2, the random errors of TDOA measurements are variable as \(\sigma_{t}\) ns which means the TDOA covariance matrix is \(\mathbf{Q}_{t_{j,k}}=diag\bigg{\{}\Big{(}\frac{\sigma_{t}}{10^{9}}\Big{)}^{2},\ldots,\Big{(}\frac{\sigma_{t}}{10^{9}}\Big{)}^{2}\bigg{\}}\). The emitter location is fixed on \((95^{\circ},-5^{\circ},0\) km). Figure 2 compares theoretic performances of geolocation with different system errors. \"Theoretical accuracy without system error\" is generated by (25), \"Theoretical accuracy with clock synchronization error\" is generated by (50), \"Theoretical accuracy with target elevation bias\" is generated by (54) and \"ML localization accuracy without system error\" is generated by (58). The effects of different system errors are clearly shown in Figure 2. We can observe that clock synchronization bias is the most influential factor. After that are target's elevation bias and satellites' position bias. The simulation results are similar to those in Figure 1. The localization accuracy of the proposed ML estimator is also provided in Figure 2 for the case of no system error.