## 3 Simulation
### Simulation Results  
To secure statistical accuracy, a Monte Carlo simulation was conducted 100 times, and averaged data are presented for each specific scenario. Rover trajectory and model errors were renewed and created with every simulation. The simulation results with different receiver observation errors \(\Delta\
abla\omega\) are shown in Table 7. Figure 4 shows an example of the estimated rover trajectory overlaying the true rover trajectory when \(\Delta\
abla\omega\) (\(\mathbf{2\times\sigma_{\Delta\
abla\omega}}\)) is 0.4 m. Figure 5 shows the distribution of the position error between the true rover positions and the estimated rover positions of Figure 4. According to Figure 5, under the condition of the satellite orbital parameters shown in Table 3, the error distribution does not have a large anisotropy but may become more anisotropic for other cases, depending on the satellite orbital parameters. Figure 6 shows the GDOP history of Figure 4. As seen in Figure 6, the GDOP is calculated intermittently when both satellites are visible from both the rover and lander. Furthermore, the value of GDOP changes because of the geolocation of two satellites over the course of the simulation period, as well as within one consecutive observable period. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Item** & **Type** & **Value** & **Unit** & **Remarks** \\ \hline \multirow{4}{*}{DEM Error} & \multicolumn{4}{c}{\(\Delta z_{R}=\omega_{DEM}\,+\,c_{DEM}\)} \\ \cline{2-5}  & White Gaussian noise \(\omega_{DEM}\) & 10.0 & m & \(\omega_{DEM\,t}=Value\times\) & a random scalar \\  & Bias noise \(\omega_{DEM}\) & 5.0 & m & Bias \(c_{DEM}\) is a random number that is greater than or equal to \(-Value\) and less than \(Value\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Overview of DEM error used in the simulation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Receiver Observation Errors**\(\Delta\
abla\omega\)**(\(\mathbf{2\times\sigma_{\Delta\
abla\omega}}\)) [m]** & **Total GDOP** & **Total UPE (2 drms) [m]** \\ \hline
0.4 & 44.3 & 45.6 \\
0.8 & 44.3 & 55.4 \\
1.6 & 44.3 & 89.6 \\
3.2 & 44.3 & 172.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Simulation results: MDPO algorithm performance evaluation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Type** & **Value** & **Unit** & **Remarks** \\ \hline \multirow{4}{*}{Time Tag Error} & \multicolumn{4}{c}{\(d\tau_{R}(t_{i})=\omega_{time\,\,tag}+x_{time\,\,tag}\)} \\ \cline{2-5}  & White Gaussian noise \(\omega_{time\,\,tag}\) & 100.0 & ms & \(\omega_{time\,\,tag}=Value\times\) & a random scalar \\  & \multirow{4}{*}{\(x_{time\,\,tag}\)} & \multirow{4}{*}{0.1} & \multirow{4}{*}{ms/min} & \(\omega_{time\,\,tag}=Value\times\) & a random scalar \\  & & & & & \(\mathrm{drawn}\) from the standard \\  & & & & & normal distribution. \\ \cline{1-1}  & \multirow{2}{*}{\(x_{time\,\,tag}\)} & \multirow{2}{*}{0.1} & \multirow{2}{*}{ms/min} & \(\omega_{time\,\,tag}=Value\times\) & a random walk is a time series model \\  & & & & & \(x_{time\,\,tag}=Value\times\) & such that \\  & & & & & \(x_{time\,\,tag}=x_{time\,\,tag}=x_{time\,\,tag}=x_{t-1}+\omega_{t}\) where \(\omega_{t}\) is a discrete white noise series. Random \\ \cline{1-1}  & & & & & walk noise is reset to zero periodically \\ \cline{1-1}  & & & & & assuming orbit determination takes place \\ \cline{1-1}  & & & & & every orbital period. \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Overview of time tag error used in the simulation. Figure 4: Example of simulation result: (**Red dots**) true rover trajectory; (**Blue dots**) rover positions estimated by MDPO (receiver observation errors \(\Delta\
abla\omega\) (\(2\times\sigma_{\Delta\
abla\omega}\) ) = 0.4 m). Figure 5: Example of position error distributions (receiver observation errors \(\Delta\
abla\omega\) (\(2\times\sigma_{\Delta\
abla\omega}\) ) = 0.4 m).