## 2 Algorithm
### Multi-Epoch Double-Differenced Pseudorange Observations (MDPO) Algorithm
Otherwise, the rover position cannot be identified deterministically by the MDPO algorithm and the rover position accuracy changes depending on the quality of other navigation information used during multi-epoch observations. Hereafter, \((x_{R}(t_{k}),\ y_{R}(t_{k}),z_{R}(t_{k}))\) represents a fixed rover position during \(t_{k}-t_{k+N-1}\). The standard approach to solving nonlinear systems, such as Equation (7), is known as the Newton-Raphson method, which is a general iterative method that uses linear regression to find theroot of a function. The idea is to start with rough estimates of the rover position and refine them in stages so that the estimates fit the observations better. First, the range is calculated on an initial estimated value of the rover position \(\mathbf{X}_{R}^{0}(t_{k})=\left(x_{R}{}^{0}(t_{k}),\ y_{R}{}^{0}(t_{k}),z_{R}{}^{0}( t_{k})\right)\):  
\[r_{R}^{s\ 0}(t_{i})=\sqrt{\left(x^{S}\left(\xi_{i}^{s}\right)-x_{R}{}^{0}(t_{k}) \right)^{2}+\left(y^{S}\left(\xi_{i}^{s}\right)-\ y_{R}{}^{0}(t_{k})\right)^{2 }+\left(z^{S}\left(\xi_{i}^{s}\right)-z_{R}{}^{0}(t_{k})\right)^{2}} \tag{8}\]  
\[i=k,\ldots,k+N-1\]  
Practically, the previous position of the rover can be used as an initial guess, i.e., \(\mathbf{X}_{R}^{0}(t_{k})=\mathbf{X}_{R}(t_{k-1})\). Then, the double-differenced range on the initial estimated value of the rover position is calculated as  
\[\Delta\
abla r^{0}(t_{i})=r_{1}^{1\ 0}(t_{i})-r_{1}^{2\ 0}(t_{i})-\left(r_{2}^{1 \ 0}(t_{i})-r_{2}^{2\ 0}(t_{i})\right) \tag{9}\]  
We also define a new parameter \(R\):  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]0  
where \(R\) is the difference between the measured double-differenced pseudorange value and the calculated double-differenced range. By substituting Equation (10) into Equation (7), the difference between the true double-differenced range and the calculated double-differenced range, i.e., \(\Delta\
abla r(t_{i})-\Delta\
abla r^{0}(t_{i})\), can be described:  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]1  
On the other hand, the residual error between the true double-differenced range and the calculated double-differenced range can be written using a Taylor series approximation as:  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]2  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]3  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]4  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]5  
By substituting Equation (11) into (12), the following equation is obtained:  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]6  
Equation (16) for multiple epochs \(t_{k}-t_{k+N-1}\) can be written at once using the following matrix:  
\[r_{R}^{S}(t_{i})=\sqrt{\Big{(}x^{S}\Big{(}t_{i}^{s}\Big{)}-x_{R}(t_{i})\Big{)} ^{2}+\Big{(}y^{S}\Big{(}t_{i}^{s}\Big{)}-y_{R}(t_{i})\Big{)}^{2}+\Big{(}z^{S} \Big{(}t_{i}^{s}\Big{)}-z_{R}(t_{i})\Big{)}^{2}} \tag{2}\]7  
\[R=\left[