# Geolocation and Tracking by TDOA Measurements Based on Space-Air-Ground Integrated Network
## 4 Discussion
It can be seen from Figure 2 that localization accuracy of the proposed ML estimator agrees with the CRLB very well when TDOA noise \(\sigma_{t}\) is less than 0.7 us. Figure 3 compares localization accuracy using the GEO and LEO satellites by varying clock synchronization error. The random error of TDOA measurement is 500 ns, which  
Figure 3: Localization accuracy comparison using the GEO and LEO satellites by varying clock synchronization error. represents the TDOA covariance matrix \(\mathbf{Q}_{t_{j,k}}=diag\bigg{\{}\Big{(}\frac{500\,\mathrm{ns}}{10^{9}}\Big{)}^{2 },\ldots,\Big{(}\frac{500\,\mathrm{ns}}{10^{9}}\Big{)}^{2}\bigg{\}}\). Other errors are assumed nonexistent. The emitter location is fixed on \((95^{\circ},-5^{\circ},0\,\mathrm{km})\). The effect of clock synchronization error is clearly shown in Figure 3. The localization performance of the proposed ML estimator agrees with the theoretical accuracy very well. Scenario 2: The sensor network contains the LEO satellite and the airplane. Simulation results are shown in Figures 4-6. The random errors of observers' positions are (100, 100, 100) m which represents the covariance matrix \(\mathbf{Q}_{s_{j,k}}=diag\Big{\{}\big{(}100\,\mathrm{m}\big{)}^{2},(100\, \mathrm{m}\big{)}^{2},(100\,\mathrm{m})^{2}\Big{\}}\). The target is on the surface of Earth with zero height. The detection number \(K\) is 10. The Monte Carlo number of the ML estimation is 1000. The random error of TDOA measurement is 500 ns in Figure 4, 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 4a presents the GDOP without any system error. Figure 4b shows the GDOP with 1 us bias of clock synchronization. Figure 4c gives the GDOP with satellite position bias of (2000, 2000, 2000) m. Figure 4d shows the GDOP with 2 km elevation bias of the target. From the simulation results in Figure 3, we can observe the following conclusions: (1) The localization accuracy in the area around the sub-satellite point is more precise than the area around the sub-airplane point. (2) There are two ambiguous location areas around the sub-airplane point. (3) The effects of different system errors are also diverse. In Figure 5, the random errors of TDOA measurements are variable as \(\sigma_{1}\,\mathrm{ns}\) which means the TDOA covariance matrix is \(\mathbf{Q}_{t_{j,k}}=diag\bigg{\{}\Big{(}\frac{\sigma_{1}}{10^{9}}\Big{)}^{2 },\ldots,\Big{(}\frac{\sigma_{1}}{10^{9}}\Big{)}^{2}\bigg{\}}\). The target's location is fixed on \((101^{\circ},3^{\circ},0\,\mathrm{km})\). \"Theoretical accuracy without system error\" is generated by (25), \"Theoretical accuracy with clock synchronization error\" is generated by (50), \"Theoretical accuracy with observer position bias\" is generated by (52), \"Theoretical accuracy with target elevation bias\" is generated by (54) and \"ML localization accuracy without system error\" is generated by (58). Figure 4: GDOPs of position estimation using the LEO and airplane. Figure 5: Localization accuracy comparison using the LEO and airplane by varying TDOA measurement noise. Figure 6: Localization accuracy comparison using the LEO and airplane by varying observer position bias. We can observe from Figure 5 that clock synchronization bias is the most influential factor. After that are elevation bias and position bias of the satellites. The simulation results are similar to those in Figure 2. The localization accuracy of the proposed ML estimator is also provided in Figure 5 for the case of no system error. It can be seen from Figure 5 that localization accuracy of the proposed ML estimator agrees with the CRLB very well when TDOA noise \(\sigma_{t}\) is less than 0.5 us. Figure 6 compares localization accuracy using the LEO and airplane by varying observer position bias.