## 3 Numerical Simulations
### Orbit Determination Refinement
As previously mentioned, this trend can be explained considering the large size of the covariance matrix obtained at the end of the UKF, which in turn is mainly affected by the fact that only Doppler shift measurements are used. In order to have a better description of the behavior of the sensor, the OD refinement procedure previously described was applied to all the objects included in the list of 355 objects. For each object, the initial estimate obtained with Equations (3) and (7) is propagated to the epoch of the first passage, an updated estimate is obtained, and this estimate is propagated again, till a new passage is considered or the final simulation epoch (02 August 2019 00:00:00 UTC) is reached. Each object can be therefore associated with a final state estimate, and its errors in position and velocity and standard deviations in position and velocity are obtained. The procedure is repeated for all the objects of the list. The obtained errors and standard deviations are then sorted, and a distribution is obtained. Table 3 shows a comparison of the median of the errors and standard deviations distributions obtained both performing OD refinement (\"_OD_\") and without updating the estimate (\"_free_\"). In addition, the median of the indexes \(I_{p}\) and \(I_{v}\) is shown. These two indexes are computed as  
\[I_{p}=\sigma_{p,OD}/\sigma_{p,free} \tag{29}\]  
\[\hat{\mathbf{\mathcal{C}}}_{t_{0}}^{RSW}=\text{diag}\bigg{(}\ \ \left(\sigma_{p}^{R} \right)^{2}\ \ \ \left(\sigma_{p}^{S}\right)^{2}\ \ \ \ \left(\sigma_{p}^{W}\right)^{2}\ \ \ \left(\sigma_{v}^{R}\right)^{2}\ \ \ \ \left(\sigma_{v}^{S}\right)^{2}\ \ \ \left(\sigma_{v}^{W}\right)^{2}\ \ \ \ \bigg{)} \tag{6}\]0  
and they measure the effectiveness of the UKF process in reducing the size of the estimate uncertainty. As can be seen, the results follow the trend of the previous example: the estimates obtained by performing OD refinement show larger errors and uncertainty. That is, the UKF approach apparently seems to worsen the accuracy of the estimate. The weakness shown by Doppler-only OD refinement can be solved by including slant range measurements. The slant range is the sum of the distance of the transiting object from the receiver and the transmitter. We show here an analysis of the performance of BIRALET considering a future upgrade of the sensor, in which slant range measurements will be available. A 1\(\sigma\) accuracy of 10 m is assumed for the simulated slant range measurements. If slant range measurements are included, the size of the covariance matrix obtained at the end of the UKF process decreases, and this has a positive effect on the accuracy of the propagated estimate. Figure 6 shows the trend of the error in position (Figure 6a) and error in velocity (Figure 6b) obtained without OD \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Epoch** & \(\varepsilon_{p,OD}\) & \(\sigma_{p,OD}\) & \(\varepsilon_{v,OD}\) & \(\sigma_{v,OD}\) & \(\varepsilon_{p,free}\) & \(\sigma_{p,free}\) & \(\varepsilon_{v,free}\) & \(\sigma_{v,free}\) \\
**(UTC)** & **(km)** & **(km)** & **(m/s)** & **(m/s)** & **(km)** & **(km)** & **(km)** & **(m/s)** & **(m/s)** \\ \hline
26/07/2019 00:00:00 & - & - & - & - & 1.76e-2 & 2.45e-1 & 4.16e-1 & 1.73 \\
27/07/2019 21:46:25 & - & - & - & - & 1.26e2 & 5.01e2 & 1.27e2 & 5.08e2 \\
27/07/2019 21:48:09 & 2.46e-1 & 2.86 & 3.41e-1 & 4.17 & 1.26e2 & 5.01e3 & 1.27e2 & 5.08e2 \\
28/07/2019 21:48:09 & 2.35e2 & 1.80e3 & 2.39e2 & 1.83e3 & 1.93e2 & 7.63e2 & 1.96e2 & 7.74e2 \\
29/07/2019 21:48:09 & 8.82e2 & 3.76e3 & 9.01e2 & 3.82e3 & 2.62e2 & 1.03e3 & 2.66e2 & 1.04e3 \\ \hline \end{tabular}
\end{table} 
Table 2: Doppler-only performance for object NORAD ID 00730 at different time epochs, in terms of error in position and velocity (\(\varepsilon_{p}\) and \(\varepsilon_{v}\)) and standard deviation in position and velocity (\(\sigma_{p}\) and \(\sigma_{v}\)) obtained performing OD refinement (“_OD_” subscript) or without updating the estimate (“_free_” subscript).