## 2 DGPS and FKP  
### DGPS and Spatial Decorrelation  
GPS is a widely known navigation system providing accurate, continuous, worldwide, three-dimensional position, velocity and time information. The positioning accuracy of the low-cost L1 single-frequency GPS receivers is generally about 5 m to 10 m because many sources of possible errors reduce the positioning accuracy [5]. In the space segment, GPS satellite orbit and clock errors exist. In addition, while a signal is being transmitted from a GPS satellite to a user, the atmospheric effects such as ionospheric and tropospheric delays may alter the travel time of the GPS signal. In the user segment, multipath error and noise exist. Except for the errors induced in the user segment, other errors, which are characteristic of the spatial correlation, can be fairly eliminated using the correction from the RS of the DGPS. In other words, the RS of the DGPS, which is positioned at a known location that was previously surveyed, can estimate the errors in its GPS signals and broadcast the correction data to nearby users. Figure 1 shows the range errors between the RS and the users considering only the atmospheric effects. Even if there are common errors in the measurements, they cannot be exactly equal because of the spatial decorrelation. As the separation of the user from the RS increases, the difference in the ionospheric and tropospheric delay between the two sites increases. Table 1 lists the different errors depending on the separation based on recent studies [11; 12]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Delay** & **Average** & **Maximum** \\ \hline Ionospheric & 4 m & 40 m \\ Tropospheric & 2 m & 16 m \\ Ephemeris & \textless{}0.25 m & \textless{}0.25 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantity of spatial decorrelations [12]. The errors due to the spatial decorrelation can be expressed using the following equations. First, the pseudorange measurements of the user and RS are given below. \[ \begin{array}{l}\rho_{u}^{i}=\left(\overline{R}^{i}+\delta\overline{R}^{i}- \overline{R}_{u}\right)\cdot\vec{e}_{u}^{i}-b^{i}+I_{u}^{i}+T_{u}^{i}+B_{u}+ \varepsilon_{u}^{i}\\ \rho_{u}^{j}=\left(\overline{R}^{j}+\delta\overline{R}^{j}-\overline{R}_{u} \right)\cdot\vec{e}_{u}^{i}-b^{j}+I_{u}^{j}+T_{u}^{j}+B_{u}+\varepsilon_{u}^{ j}\\ \rho_{r}^{i}=\left(\overline{R}^{i}+\delta\overline{R}^{i}-\overline{R}_{r} \right)\cdot\vec{e}_{r}^{i}-b^{i}+I_{r}^{i}+T_{r}^{j}+B_{r}+\varepsilon_{r}^{ i}\\ \rho_{r}^{j}=\left(\overline{R}^{j}+\delta\overline{R}^{j}-\overline{R}_{r} \right)\cdot\vec{e}_{r}^{j}-b^{i}+I_{r}^{j}+T_{r}^{j}+B_{r}+\varepsilon_{r}^{ j}\end{array}  \tag{1}\]  
The superscripts \(i\) and \(j\) denote the ith and jth GPS satellites, respectively, and the subscripts \(u\) and \(r\) denote the user and RS, respectively. In addition, \(\overline{R},\overline{e},b,B,I,T,\delta R,\) and \(\varepsilon\) are the position, line of sight vector, satellite clock bias, receiver clock bias, ionospheric delay, tropospheric delay, ephemeris error, and noise, respectively. By subtracting between two satellites single differences (\(i^{\prime}\
abla^{j}\rho_{u}\), \(i^{\prime}\
abla^{j}\rho_{r}\)), the double-difference measurement is obtained as follows. \[ \begin{array}{l}{}_{u}\Delta_{r}i^{\prime}\
abla^{j}\rho&=\left[\left( \overline{R}^{i}+\delta\overline{R}^{i}\right)\cdot\vec{e}_{u}^{i}-\left( \overline{R}^{i}+\delta\overline{R}^{j}\right)\cdot\vec{e}_{u}^{i}\right]- \overline{R}_{u}\cdot\left(\vec{e}_{u}^{i}-\vec{e}_{u}^{i}\right)\\ &+\left(I_{u}^{i}-I_{u}^{j}\right)+\left(T_{u}^{i}-T_{u}^{j}\right)+i^{\prime} \
abla^{j}\varepsilon_{u}\\ &-\left[\left(\overline{R}^{i}+\delta\overline{R}^{i}\right)\cdot\vec{e}_{r}^ {i}-\left(\overline{R}^{j}+\delta\overline{R}^{j}\right)\cdot\vec{e}_{r}^{j} \right]+\overline{R}_{r}\cdot\left(\vec{e}_{r}^{i}-\vec{e}_{r}^{j}\right)\\ &-\left(I_{r}^{i}-I_{r}^{j}\right)-\left(T_{r}^{i}-T_{r}^{j}\right)-i^{\prime} \
abla^{j}\varepsilon_{r}\end{array}