## 5 Results
### TDRS Orbit Determination Results
#### 5.1.3 Scenario #3  
Based on the estimated correlation matrix obtained from previous scenarios, it has been observed that there is a robust correlation between the TT&C bias and the initial state vectors. In these previous scenarios, the operational values of the TT&C bias and its covariances were utilized. The covariance value denotes the degree of uncertainty or correlation between different parameters in the model, and an incorrect value can lead to erroneous results. However, it is possible to weaken these correlations by increasing the covariance value. Nonetheless, it is important to note that if the covariance value is set too high, the batch least squared solution may diverge or overestimate the TT&C bias, resulting in inaccurate \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reference Epoch** **(User Spacecraft)**} & \multirow{2}{*}{**TDRS Satellite**} & \multicolumn{4}{c}{**Comparison with Operational Solution**} \\  & & **Max. 3D Position** & **Max. 3D Velocity** & **Cr Difference** & **TT\&C Bias** \\  & & **Difference (m)** & **Difference (m/s)** & **Cr Difference** & **Difference (m)** \\ \hline \multirow{4}{*}{8 December 2021 (IXPE)} & TDRS-6 & 22 & 1.6 & \(-1.8\times 10^{-6}\) & \(-0.4\) \\  & TDRS-7 & 17 & 1.2 & \(-1.1\times 10^{-3}\) & \(0.04\) \\  & TDRS-10 & 13 & 1.0 & \(-5.2\times 10^{-4}\) & \(0.35\) \\  & TDRS-12 & 22 & 1.6 & \(2.7\times 10^{-4}\) & \(1.27\) \\ \hline \multirow{4}{*}{3 April 2022 (LANDSAT-9)} & TDRS-7 & 11 & 0.8 & \(1.3\times 10^{-3}\) & \(0.67\) \\  & TDRS-10 & 30 & 2.2 & \(1.0\times 10^{-3}\) & \(0.10\) \\  & TDRS-11 & 41 & 3.0 & \(8.6\times 10^{-4}\) & \(2.17\) \\  & TDRS-12 & 68 & 5.0 & \(5.0\times 10^{-5}\) & \(-0.06\) \\ \hline \multirow{4}{*}{28 October 2022 (LANDSAT-8)} & TDRS-6 & 52 & 3.8 & \(1.8\times 10^{-3}\) & \(0.02\) \\  & TDRS-7 & 17 & 1.2 & \(5.2\times 10^{-4}\) & \(0.00\) \\ \cline{1-1}  & TDRS-10 & 31 & 2.3 & \(-3.9\times 10^{-3}\) & \(0.02\) \\ \cline{1-1}  & TDRS-11 & 74 & 5.4 & \(-1.2\times 10^{-3}\) & \(-0.03\) \\ \hline \multirow{4}{*}{11 November 2022 (JPSS-2)} & TDRS-6 & 41 & 3.0 & \(-5.5\times 10^{-4}\) & \(-0.34\) \\ \cline{1-1}  & TDRS-7 & 15 & 1.1 & \(1.8\times 10^{-3}\) & \(-0.11\) \\ \cline{1-1}  & TDRS-10 & 11 & 0.8 & \(-1.9\times 10^{-3}\) & \(-0.01\) \\ \cline{1-1}  & TDRS-12 & 20 & 1.2 & \(9.2\times 10^{-4}\) & \(-0.11\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of TT&C-only and operational TDRS solutions with identical initial conditions. The reference epoch displayed in the table represents the reference time of the TDRS solution. orbit determination. Therefore, it is critical to determine an optimal value for the covariance, particularly when there is inadequate knowledge of the TT&C bias and its covariance. To achieve this, a Monte-Carlo analysis was performed with varying covariance of the TT&C bias. The a priori value of TT&C bias was assumed to be zero meters, and the initial state vectors were perturbed (see Table 1 and Section 4.6). Figures 6-9 provide a comprehensive analysis of the uncertainties in the orbit of the TDRS satellites (solid line marked with circles), including the estimated biases in the TT&C measurements (dotted line marked with diamonds). The reference epoch displayed in the figures indicates both the day when the user spacecraft acquired DOWD measurements and the reference epoch of the TDRS solution. These figures demonstrate the impact of using different TT&C bias a priori covariance values on the accuracy of the TDRS orbit. The positional uncertainties presented in these figures were quantified through a comparative analysis between the operational TDRS solutions and the TT&C-based TDRS solutions acquired during the periods of user spacecraft DOWD events. In certain cases, the estimated TT&C bias and the associated TDRS position uncertainty show a complete correlation, as illustrated by the overlapping lines in the figure. The comparison of the definitive ephemerides with the operational solution enabled the calculation of TDRS orbit uncertainties.