# The Impact of Eclipsing GNSS Satellites on the Precise Point Positioning
## Materials and Methods
### The Multi-GNSS Precise Point Positioning and Its Strategy
where \(P^{sys,s}_{r,if}\) (m) and \(L^{sys,s}_{r,if}\) (m) are the observed-minus-computed pseudorange and phase observables after correcting necessary errors for satellite s and receiver r, respectively; the superscript \"sys\" denotes satellite system according to RINEX version 3.03 [25] (G = GPS, R = GLONASS, E = GALILEO, C = BDS); \(\mathbf{u}^{sys,s}_{r}\) is the linearized vector from satellite s to receiver r and \(\Delta\mathbf{x}_{r}\) (m) is the coordinate increment vector with respect to the priori value; \(Z_{r}\) (m) is the zenith tropospheric wet delay for receiver r and \(m^{sys,s}_{r}\) is its corresponding mapping function; \(dt^{sys}_{r}\) (m) and \(dt^{sys,s}_{r}\) (m) are the frequency-independent clock offsets for receiver r and satellite s, respectively; \(\lambda^{sys,s}_{r,if}\) denotes the wavelength; \(N^{sys,s}_{r,if}\) (cycle) is the float ambiguity due to the assimilation of the hardware delays; \(O^{sys,s}_{r,if}\) (m) is the satellite PCO correction which influences both the pseudorange and phase observables, while the wind-up correct \(W^{sys,s}_{r,if}\) (m) only influences the phase observables; \(b^{sys,s}_{r,f}\) (m) and \(b^{sys,s}_{r,f}\) (m) are the pseudorange hardware delays of the IF combination for receiver \(r\) and satellite s, respectively; \(B^{sys,s}_{r,f}\) (m) and \(B^{sys,s}_{f}\) (m) are the phase hardware delays of the IF combination for receiver \(r\) and satellite \(s\), respectively; and \(\epsilon^{sys,s}_{r,if}\) (m) and \(\epsilon^{sys,s}_{L,r,if}\) (m) are the unmodeled errors in the pseudorange and phase observables, respectively. Without considering the pseudorange Inter-Frequency Biases (IFBs) of GLONASS, the pseudorange hardware delays of receiver can be treated as satellite-independent [26]. During the subsequent derivation, \(b^{sys}_{r,f}\) is used instead of \(b^{sys,s}_{r,f}\). To solve the rank deficiency caused by the correlations between the clock parameters and hardware delays, in general, the new reparameterized clock is defined:  
\[d\overline{t}^{sys,s}=dt^{sys,s}+b^{sys,s}_{if}, \tag{6}\]  
\[d\overline{t}^{sys}_{r}=dt^{sys}_{r}+b^{sys}_{r,if}, \tag{7}\]  
at this time, the pseudorange hardware delays absorbed into the clock parameters are introduced to the phase observations. Therefore, the new float ambiguity parameters \(A^{sys,s}_{r,f}\) can be expressed as the linear combination of the ambiguities and the hardware delays:  
\[A^{sys,s}_{r,f}=\lambda^{sys,s}_{,if}N^{sys,s}_{r,if}+B^{sys,s}_{r,if}-B^{sys,s}_{if}-(b^{sys}_{r,if}-b^{sys,s}_{if}), \tag{8}\]  
Instead of estimating one receiver clock for one satellite system, an alternative method is to introduce Inter-System Biases (ISBs) to compensate for the time differences among GNSS and GPS is selected as the reference in the GNSS PPP processing. \[ISB^{G*}_{r}=d\overline{t}^{*}_{r}-d\overline{t}^{G}_{r}=\left(dt^{*}_{r}-dt ^{G}_{r}\right)+\left(b^{*}_{r,if}-b^{G}_{r,if}\right), \tag{9}\]  
where the superscript \"\({}^{*}\)\" denotes the abbreviation for satellite systems. The estimable ISBs parameters not only contain the true system time differences, but also the differences of hardware delays of the IF combinations. Eventually, the full rank GNSS PPP model using IF combinations are obtained:  
\[P^{sys,s}_{r,if}=u^{sys,s}_{r}\Delta\mathbf{x}_{r}+m^{sys,s}_{r}Z_{r}+d\overline{ t}^{sys}_{r}-d\overline{t}^{sys,s}+O^{G,s}_{r,if}+ISB^{G*}_{r}+\epsilon^{G,s}_{ r,if}, \tag{10}\]  
\[L^{sys,s}_{r,if}=u^{sys,s}_{r}\Delta\mathbf{x}_{r}+m^{sys,s}_{r}Z_{r}+d\overline{ t}^{sys}_{r}-d\overline{t}^{sys,s}+O^{sys,s}_{r,if}+W^{sys,s}_{r,if}+A^{ sys,s}_{r,if}+ISB^{G*}_{r}+\epsilon^{sys,s}_{L,r,if}, \tag{11}\]  
When multi-GNSS data are available, the prior variance of observations are given as follows [27]:  
\[\sigma^{2}=\sigma_{0}^{2}\Big{(}1+1/\sin^{2}e\Big{)}, \tag{12}\]  
where \(e\) is the satellite elevation. The value of \(\sigma_{0}\) for GPS/GALILEO can be set as 0.003 m and 0.3 m for the raw pseudorange and phase observables, respectively.