## 3 Simulation
### Simulation Overview
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Value** & **Unit** & **Remarks** \\ \hline Simulation Period & 15,000 & min & Approximately two weeks in \\  & & & Earth time \\ Range measurement resolution of & 0.4 & m & Minimum observable range by the \\ the user pseudorange receivers & 0.4 & m & rover and lander receivers \\ Latitude of Initial & \(-90\) & deg & \\ Rover/Lander Position & 90 & deg & \\  & & & Total observation period of one \\ Interval of & 0.5 & min & MDPO estimation is equivalent to \\  & & & 1 min when the number of \\  & & & multi-epoch observations is 2. \\ Rover traveling distance between & 3.75 & m & The rover travels at 7.5 m/min for \\ MDPO observations & 3.75 & m & 0.5 min between MDPO estimations \\ Rover traveling direction & Random & deg & Heading direction is selected from \\  & & & three values (\(+\frac{\pi}{3},-\frac{\pi}{3},~{}0\)) randomly. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 3: Simulation overview. Tables 4-6 show the systematic errors used in the simulation. Table 4 summarizes the satellite orbit determination error value used in the simulation, i.e., \(\Delta Along,\Delta Radial\) and \(\Delta Cross\) defined in Equation (38). The value was chosen by adding a sufficient margin to the reference data from the LRO project [21]. Table 5 summarizes the time tag error used in the simulation, i.e., \(d\pi_{R}\) defined in Equation (39). The value is based on the assumption that the rover clock is calibrated periodically by a navigation message every orbital period. Through the navigation message, the rover clock can be synchronized to the satellite clock with the uncertainty of the signal traveling time from the satellite to the rover and lander receivers, which was modeled as white noise in the simulation. The time tag error also contains an additive-type noise, which was modeled as a random walk noise that is reset to zero periodically. Table 6 summarizes the DEM model error value used in the simulation, i.e., \(\Delta z_{R\ DEM}\) defined in Equation (41). Currently, the accuracy of the best existing DEM data in a vertical direction is about 3 m within a \(\pm\)60-deg latitude and about 10 m near polar regions [17; 18]. The same parameters were used in the following simulations unless otherwise mentioned. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Items** & **Type** & **Value** & **Unit** & **Remarks** \\ \hline \hline \multirow{4}{*}{Satellite Orbit Determination Error in the Along direction} & \multicolumn{4}{c}{\(\Delta Along(t_{i})\)= \(\omega_{OD-Along}\)\((t_{i})+c_{OD-Along}\)} \\ \cline{2-5}  & White Gaussian noise \(\omega_{OD-Along}\) & 100.0 & m & \(\omega_{OD\ t}\ =\ Valure\times\) a random scalar drawn from the standard normal distribution. \\  & Bias noise \(c_{OD-Along}\) & 200.0 & m & Bias \(c_{OD}\) is a random number that is greater than or equal to \(-Value\) and less than \(Value\) \\ \hline \hline \multirow{4}{*}{Satellite Orbit Determination Error in the Radial direction Bias noise\(c_{OD-Radial}\)\((t_{i})\)= \(\omega_{OD-Radial}\)\((t_{i})+c_{OD-Radial}\)} \\ \cline{2-5}  & White Gaussian noise\(\omega_{OD-Radial}\) & 10.0 & m & Same as above \\  & Bias noise \(c_{OD-Radial}\) & 20.0 & m & \\ \hline \hline \multirow{4}{*}{Satellite Orbit Determination Error in the Cross direction Bias noise\(c_{OD-Cross}\)\((t_{i})\)= \(\omega_{OD-Cross}\)\((t_{i})+c_{OD-Cross}\)} \\ \cline{2-5}  & White Gaussian noise\(\omega_{OD-Cross}\) & 100.0 & m & Same as above \\ \cline{1-1} \cline{2-5}  & Bias noise \(c_{OD-Cross}\) & 200.0 & m & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Overview of satellite orbit determination error used in the simulation.