## 2 Algorithm
### Multi-Epoch Double-Differenced Pseudorange Observations (MDPO) Algorithm
\begin{array}{ccc}R(t_{k})&\cdots&R(t_{k+N-1})\end{array} \right]^{T} \tag{18}\]  
\[\Delta X=\left[\Delta x,\Delta y,\Delta z\right] \tag{19}\]  
\[w=\left[ \begin{array}{ccc}-\Delta\
abla\omega(t_{k})&\cdots&-\Delta\
abla \omega(t_{k+N-1})\end{array} \right]^{T} \tag{20}\]  
\[G=\left[ \begin{array}{ccc}\frac{\partial\Delta\
abla r(t_{k})}{\partial x}& \frac{\partial\Delta\
abla r(t_{k})}{\partial y}&\frac{\partial\Delta\
abla r (t_{k})}{\partial z}\\ \vdots&\vdots&\vdots\\ \frac{\partial\Delta\
abla r(t_{k+N-1})}{\partial x}&\frac{\partial\Delta \
abla r(t_{k+N-1})}{\partial y}&\frac{\partial\Delta\
abla r(t_{k+N-1})}{ \partial z}\end{array} \right] \tag{21}\]where \(G\) is known as an observation matrix. By solving the least-square problem that minimizes the residual error \(|R-G\Delta X|\), an estimated value of \(\Delta X\), defined as \(\Delta\hat{\Delta X}\), is obtained:  
\[\Delta\hat{X}=\left(G^{T}G\right)^{-1}G^{T}R \tag{22}\]  
Then, a new estimated value \(X_{R}^{1}(t_{k})=\left(x_{R}{}^{1}(t_{k})\), \(y_{R}{}^{1}(t_{k})\), \(z_{R}{}^{1}(t_{k})\right)\) is given by Equation (23), which provides a better fit to the observation. \[X_{R}^{1}(t_{k})=X_{R}^{0}(t_{k})+\Delta\hat{X} \tag{23}\]  
This estimation process continues (\(X_{R}^{1}\), \(X_{R}^{2}\)) until the number of iterations reaches the designed value, i.e., \(n\), and then final estimated value \(X_{R}^{n}(t_{k})\) is acquired. To estimate the rover's three- dimensional position \(X_{R}=\left(x_{R},y_{R},z_{R}\right)\), the number of multi-epoch observations must be larger than 3 (N \(>\) 3). Next, we characterize the quality of the estimates. We can write an expression for the error as  
\[\Delta\hat{X}-\Delta X=\left(G^{T}G\right)^{-1}G^{T}(G\Delta X+w)-\Delta X= \left(G^{T}G\right)^{-1}G^{T}w \tag{24}\]  
Suppose that \(w\) follows a white Gaussian distribution that has a mean value of zero and covariance matrix C: the covariance of \(\Delta\hat{X}-\Delta X\), defined as \(P\), is given by  
\[P=\left(G^{T}G\right)^{-1}\!\!G^{T}CG\!\left(G^{T}G\right)^{-1} \tag{25}\]  
The expression becomes much simpler if the components of \(w\), i.e., \(-\Delta\
abla\omega(t_{i})\), are uncorrelated and have an identical variance, i.e., \(C=\sigma_{\Delta\
abla\omega}^{2}I\):  
\[P=\sigma_{\Delta\
abla\omega}^{2}\left(G^{T}G\right)^{-1} \tag{26}\]  
where \(\sigma_{\Delta\
abla\omega}^{2}\) is the variance of double-differenced receiver observation errors. In GNSS terminology, \(\left(G^{T}G\right)^{-1}\) is known as the dilution of precision (DOP) matrix, which is used to specify error propagation as a mathematical effect of navigation satellite geometry on positional measurement precision. We define the DOP matrix and its elements \(\sigma_{DOP}\) as  
\[DOP=\left[ \begin{array}{ccc}\sigma_{DOP\ 11}&\cdots&\sigma_{DOP\ 1N}\\ \vdots&\ddots&\vdots\\ \sigma_{DOP\ 1N}&\cdots&\sigma_{DOP\ NN}\end{array} \right]=\left(G^{T}G \right)^{-1} \tag{27}\]  
where \(\sigma_{DOP}\) is elements of \(DOP\). By substituting Equation (27) into Equation (26), theoretically, the achievable rover position error, i.e., \(\Delta\hat{X}-\Delta X\), at a time of \(t_{k}\) is given by  
\[\mathit{UPE}(t_{k})=\left|\Delta\hat{X}(t_{k})-\Delta X(t_{k})\right|=\ \sqrt{\sum_{j=1}^{N}\!\left(\sigma_{DOP\ jj} \right)^{2}}\times\sigma_{\Delta\
abla\omega} \tag{28}\]  
where \(\mathit{UPE}\) represents the distance between the rover's true position and an estimated rover position. We define GDOP as  
\[GDOP=\ \sqrt{\sum_{j=1}^{N}\!\left(\sigma_{DOP\ jj}\right)^{2}} \tag{29}\]  
Then Equation (28) is written as  
\[\mathit{UPE}(t_{k})=GDOP\times\sigma_{\Delta\
abla\omega} \tag{30}\]As mentioned in the previous section, we assume receiver observation errors follow a normal distribution with a zero mean (i.e., Gaussian white noise). As such, UPE also follows a 1-D Gaussian distribution, and 95 percent of it lies inside the interval from \(-2s\) to \(+2s\), where \(s\) is the standard deviation.