GPS satellite clock determination in case of inter-frequency clock biases for triple-frequency precise point positioning

Abstract

Significant time-varying inter-frequency clock biases (IFCBs) within GPS observations prevent the application of the legacy L1/L2 ionosphere-free clock products on L5 signals. Conventional approaches overcoming this problem are to estimate L1/L5 ionosphere-free clocks in addition to their L1/L2 counterparts or to compute IFCBs between the L1/L2 and L1/L5 clocks which are later modeled through a harmonic analysis. In contrast, we start from the undifferenced uncombined GNSS model and propose an alternative approach where a second satellite clock parameter dedicated to the L5 signals is estimated along with the legacy L1/L2 clock. In this manner, we do not need to rely on the correlated L1/L2 and L1/L5 ionosphere-free observables which complicates triple-frequency GPS stochastic models, or account for the unfavorable time-varying hardware biases in undifferenced GPS functional models since they can be absorbed by the L5 clocks. An extra advantage over the ionosphere-free model is that external ionosphere constraints can potentially be introduced to improve PPP. With 27 days of triple-frequency GPS data from globally distributed stations, we find that the RMS of the positioning differences between our GPS model and all conventional models is below 1 mm for all east, north and up components, demonstrating the effectiveness of our model in addressing triple-frequency observations and time-varying IFCBs. Moreover, we can combine the L1/L2 and L5 clocks derived from our model to calculate precisely the L1/L5 clocks which in practice only depart from their legacy counterparts by less than 0.006 ns in RMS. Our triple-frequency GPS model proves convenient and efficient in combating time-varying IFCBs and can be generalized to more than three frequency signals for satellite clock determination.

Appendix: Hardware biases assimilated into observation residuals

These equations complement Eqs. (2) and (4) by stating how hardware biases affect the pseudorange and carrier-phase residuals in our model for undifferenced uncombined triple-frequency GPS data processing. Note that parameters e (indices are ignored) do not denote observation residuals, but the time-varying hardware biases that are assimilated into observation residuals. We can find that all pseudorange residuals on L1, L2 and L5 contain both time-varying pseudorange and carrier-phase hardware biases. There are no hardware biases assimilated into carrier-phase residuals except those for L5.

$$\begin{aligned} \left\{ {\begin{array}{ll} e_{1i,P}^k &{}=\delta B_{1i,P} -(\alpha _1 +\alpha _2 )\delta B_{1i,L} +2\alpha _2 \delta B_{2i,L} -\delta b_{1,P}^k \\ &{}\quad +\,(\alpha _1 +\alpha _2 )\delta b_{1,L}^k -2\alpha _2 \delta b_{2,L}^k \\ e_{2i,P}^k &{}=\delta B_{2i,P} -2\alpha _1 \delta B_{1i,L} +(\alpha _1 +\alpha _2 )\delta B_{2i,L} -\delta b_{2,P}^k \\ &{}\quad +\,2\alpha _1 \delta b_{1,L}^k -(\alpha _1 +\alpha _2 )\delta b_{2,L}^k \\ e_{5i,P}^k &{}=\delta B_{5i,P} -(\alpha _1 +m^{2}\alpha _2 )\delta B_{1i,L} +(\alpha _2 +m^{2}\alpha _2 )\delta B_{2i,L} \\ &{}\quad -\,\delta b_{5,P}^k +2m^{2}\alpha _2 \delta b_{1,L}^k -2m^{2}\alpha _2 \delta b_{2,L}^k \\ e_{5i,L}^k &{}=(-\alpha _1 +m^{2}\alpha _2 )\delta B_{1i,L} +(\alpha _2 -m^{2}\alpha _2 )\delta B_{2i,L} +\delta B_{5i,L} \\ \end{array}} \right. \nonumber \\ \end{aligned}$$

(A1)