## 2 DGPS and FKP
### Network RTK and FKP
Therefore, the users can eliminate the effects of the spatial decorrelation using the FKP correction. The FKP is less accurate than the VRS and MAC because it simplifies the range errors into the first-order gradients. Nevertheless, it can be easily modified for pseudorange measurements in the low-cost L1 single-frequency GPS receiver. The parameters of the FKP correction show that the horizontal gradients are linear approximations of the geometric and ionospheric errors in the neighborhood of the RS. The geometric gradient contains the non-dispersive (ephemeris and troposphere residuals) errors whereas the ionospheric gradient contains the dispersive errors. The geographical coordinates \(\varphi_{r},\lambda_{r}\) are the ellipsoidal coordinates of the corresponding RS and the coordinates of the user are \(\varphi\), \(\lambda\). The corrections can be then written as follows. \[ \begin{array}{l}\delta\rho_{0}=6.37\cdot\left(N_{o}(\varphi-\varphi_{r})+E_ {o}(\lambda-\lambda_{r})\cos(\varphi_{r})\right)\\ \delta\rho_{I}=6.37\cdot H\cdot\left(N_{I}(\varphi-\varphi_{r})+E_{I}(\lambda -\lambda_{r})\cos(\varphi_{r})\right)\\ H=1+16(0.53-el/\pi)^{3}\end{array}  \tag{5}\]  
Figure 3: Broadcast FKP correction information. Figure 2: Linear FKP planes for four RSs [19]. The subscripts \(o\) and \(I\) denote the components of the geometric and ionospheric signals, respectively. In addition, \(N,E,el\) are the gradients in the north-south and east-west directions and elevation angle of the satellite at the user position, respectively [(20; 21)]. These parameters are transmitted to the user through the network transport of the RTCM via the Internet protocol (NTRIP) in the form of an RTCM version 3.1 standard message. Based on the RTCM standard document, message type 1034 contains the FKP gradients and message type 1005 contains the station coordinates [(21)]. Regions where the users make use of the FKP service are such as many European countries including Germany, parts of America, Korea, Japan, and Australia [(22)]. The ephemeris error due to the spatial decorrelation is much smaller than the tropospheric delay. Hence, the correction used for the geometric errors can be used for the tropospheric delays. In addition, the sign of the correction for the ionospheric error should be reversed for the pseudorange measurements because the correction in Equation (5) is obtained for the carrier-phase measurements. Accordingly, Equation (5) is modified as follows. \[\delta\rho_{o}=\delta T_{FKP}\simeq\delta T,\;\delta\rho_{I}=-\delta I_{FKP} \simeq\delta I \tag{6}\]  
The subscript \(FKP\) denotes the corrections made using the FKP messages. The residual errors due to the spatial decorrelation in Equation (4) could be then eliminated using the modified FKP corrections as follows. \[ \begin{array}{ll}{}_{u}\Delta_{r}{}^{i}\
abla^{j}\rho-&\left(\overline{R}^{ i}\cdot_{u}\Delta_{r}\bar{e}^{i}-\overline{R}^{j}\cdot_{u}\Delta_{r}\bar{e}^{j}- \overline{R}_{u}\cdot\bar{e}^{i}_{u}+\overline{R}_{u}\cdot\bar{e}^{j}_{u}+ \overline{R}_{u}\cdot\bar{e}^{i}_{u}-\overline{R}_{u}\cdot\bar{e}^{j}_{u} \right)\\ &=\left({}^{i}\
abla^{j}\delta T-_{r}\Delta_{u}{}^{i}\
abla^{j}T^{i}_{\rm model }\right)+\left({}^{i}\
abla^{j}\delta I\right)\\ &\simeq\left({}^{i}\
abla^{j}\delta T_{FKP}-_{r}\Delta_{u}{}^{i}\
abla^{j}T^{ i}_{\rm model}\right)+\left(-{}^{i}\
abla^{j}\delta I_{FKP}\right)\end{array}  \tag{7}\]