## 3 Demonstration Experiment of MDPO
### Double-Differenced Pseudorange Error Assessment  
Prior to the MDPO demonstration, the relative position vector \(\mathbf{b}\) of the rover with respect to the lander location was estimated with all the visible GPS satellites. Then, we used the result to evaluate the expected accuracy of the double-differenced pseudorange observations.  
In the observation model of Equation (1), the pseudorange error \(\omega_{R}^{S}(t_{i})\) is assumed zero-mean Gaussian noise. Thus, the residual vector \(\mathbf{e}\) in Equation (13) has the following stochastic properties:  
\[Cov(\mathbf{e})=E\Big{(}\mathbf{e}\mathbf{e}^{\mathrm{T}}\Big{)}=\sigma^{2}\mathbf{I}, \tag{16}\]  
where \(E(\cdot)\) and \(Cov(\cdot)\) denote the mean value and the covariance, respectively, and \(\sigma\) is the standard deviation of the double-differenced pseudorange error. The covariance of the least-squared solutions of the relative position vector \(\mathbf{\hat{b}}\) is then written as:  
\[Cov(\mathbf{\hat{b}})=\sigma^{2}\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G}\Big{)}^{-1} \tag{17}\]  
For simplicity of notation, let:  
\[\mathbf{H}=\Big{(}\mathbf{G}^{\mathrm{T}}\mathbf{G}\Big{)}^{-1}. \tag{18}\]  
In the local ENU coordinate frame, it follows from Equation (17) that:  
\[\sigma_{\mathrm{E}}^{2}=\sigma^{2}\mathbf{H}_{11};\;\sigma_{\mathrm{N}}^{2}=\sigma ^{2}\mathbf{H}_{22};\;\sigma_{\mathrm{U}}^{2}=\sigma^{2}\mathbf{H}_{33}, \tag{19}\]  
where \(\mathbf{H}_{ii}\) indicates the \(i\)th entry on the diagonal of \(\mathbf{H}\), and \(\sigma_{\mathrm{E}}\), \(\sigma_{\mathrm{N}}\), and \(\sigma_{\mathrm{U}}\) are the standard deviation of the east, north, and up (vertical) components of the position error, respectively. Equation (17) shows that the distribution of the position error depends entirely on the variance of the range error and the satellite geometry. The so-called horizontal dilution of precision (HDOP) is then given by:  
\[\mathrm{HDOP}=\sqrt{\mathbf{H}_{11}+\mathbf{H}_{22}}. \tag{20}\]  
The distribution of the horizontal position error can be described simply as:  
\[\mathrm{drms}=\sqrt{\sigma_{\mathrm{E}}^{2}+\sigma_{\mathrm{N}}^{2}}=\sigma \cdot\mathrm{HDOP} \tag{21}\]  
where drms stands for the distance root-mean-squared and expresses 2-dimensional positioning accuracy.  
Figure 1 shows the horizontal position errors of the rover and the corresponding HDOP values obtained from the double-differenced pseudorange observations of all the visible GPS satellites. The red circle in the horizontal position error plot has a radius of 2 drms (twice distance rms) from the true rover location. Typically, 95% of all possible position errors scatter inside a 2 drms circle.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**IGS Code** & **Site** & **Latitude (deg)** & **Longitude (deg)** & **Elevation (m)** \\ \hline KGNI & Koganei, Japan & 35.7013 & 139.4881 & 123.5 \\ \hline MTKA & Mitaka, Japan & 35.6795 & 139.5614 & 109.0 \\ \hline \end{tabular}
\end{table}
Table 1: Selected GNSS stations and their locations.  
Table 2 summarizes the results shown in Figure 1. As shown in Equation (21), HDOP is only a value of probability for the geometric effect on horizontal positioning accuracy and is roughly interpreted as the ratio of position error to the range error. That means that the value of 2 drm divided by HDOP should provide an expected 2\(\sigma\) (95%) error of the double-differenced pseudorange observations.  
It should be noted that the range offsets due to the ionosphere and the troposphere, both of which do not exist around the Moon, are major error sources of the pseudorange observations on the ground. Fortunately, they are effectively removed by taking double-differenced measurements and show little influence on the final positioning accuracy.