GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations

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This study proposes a unified uncombined model to estimate GPS satellite inter-frequency clock bias (IFCB) in both triple-frequency code and carrier-phase observations. In the proposed model, the formulae of both phase-based and code-based IFCBs are rigorously derived. Specifically, satellite phase-based IFCB refers to its time-variant part and it is modeled as a periodic function related to the sun–spacecraft–earth angle. A zero-mean condition of all available GPS satellites that support triple-frequency data is introduced to render satellite code-based IFCB estimable. Three months of data from 40 globally distributed stations of the International GNSS Service Multi-GNSS Experiment are used to test our method. The results show that the four-order periodic function is suitable for eliminating the 12-h, 6-h, 4-h, and 3-h periods that exist in the a posteriori phase residuals when no periodic function is used. For comparison, the geometry-free and ionosphere-free (GFIF) phase combination and differential code bias (DCB) products released by DLR (German Aerospace Center) and IGG (Institute of Geodesy and Geophysics, China) are also used to calculate the satellite phase-based and code-based IFCBs, respectively. The results show that (1) the average root mean square (RMS) of the phase-based IFCB difference between the proposed method and the GFIF phase combination is 4.3 mm; (2) the average RMS in the eclipse period increased by 50% compared with the average RMS in the eclipse-free period; (3) the mean monthly STD for code-based IFCB from the proposed method is 0.09 ns; and (4) the average RMS values of code-based IFCB differences between the proposed method and the DCB products released by DLR and IGG are 0.32 and 0.38 ns. This proposed model also provides a general approach for multi-frequency GNSS applications such as precise orbit and clock determination.

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This work was sponsored by the National Natural Science Foundation of China (Grant Nos. 41804024, 41931075, 41804026, 41574027). The authors are grateful to the editors and reviewers for their valuable comments on improving our manuscript. We would also like to thank International GNSS Service (IGS) for providing GPS data and products, and German Aerospace Center (DLR) and Institute of Geodesy and Geophysics (IGG) for providing multi-frequency DCB products. The GPS observation data and IGS final orbit and clock products are obtained from the CDDIS ( The GPS P1-C1 DCB products are achieved from the CODE ( The GPS multi-frequency DCB products are available at

Author information

C. Shi, L. Fan, and J. Zhang designed the research; L. Fan, G. Jing, and M. Li performed the research; M. Li, C. Wang, and F. Zheng analyzed the data; and L. Fan wrote the paper.

Correspondence to Min Li.

Appendix: The satellite IFCB calculated by GFIF combination

Appendix: The satellite IFCB calculated by GFIF combination

GFIF combination is equivalent to the difference of satellite clock errors determined by ionosphere-free combination of code and carrier-phase observations at different frequencies. It eliminates any geometry-related errors and atmosphere delays except for hardware biases in code or carrier-phase observations (Montenbruck et al. 2012; Li et al. 2012a), as shown in Eq. (A1).

$$ \begin{aligned} & {\text{GFIF}}(P_{r,1}^{s} ,P_{r,2}^{s} ,P_{r,i}^{s} ) = {\text{IF}}(P_{r,1}^{s} ,P_{r,2}^{s} ) - {\text{IF}}(P_{r,1}^{s} ,P_{r,i}^{s} ) \\ & = \underbrace {{\frac{1}{{\mu_{2} - 1}}{\text{DCB}}_{r,2} - \frac{1}{{\mu_{i} - 1}}{\text{DCB}}_{r,i} }}_{{{\text{IFCB}}_{P,r,i} }} + \underbrace {{\frac{1}{{\mu_{2} - 1}}{\text{DCB}}_{2}^{s} - \frac{1}{{\mu_{i} - 1}}{\text{DCB}}_{i}^{s} }}_{{{\text{IFCB}}_{P,i}^{s} }} + \varepsilon_{{{\text{IFCB}}_{P,r,i}^{s} }} \\ & {\text{GFIF}}(\varPhi_{r,1}^{s} ,\varPhi_{r,2}^{s} ,\varPhi_{r,i}^{s} ) = {\text{IF}}(\varPhi_{r,1}^{s} ,\varPhi_{r,2}^{s} ) - {\text{IF}}(\varPhi_{r,1}^{s} ,\varPhi_{r,i}^{s} ) \\ & = \underbrace {{\frac{1}{{\mu_{2} - 1}}\delta {\text{DPB}}_{2}^{s} - \frac{1}{{\mu_{i} - 1}}\delta {\text{DPB}}_{i}^{s} }}_{{\delta {\text{IFCB}}_{\varPhi ,i}^{s} }} + \underbrace {{\frac{1}{{\mu_{2} - 1}}\delta {\text{DPB}}_{r,2} - \frac{1}{{\mu_{i} - 1}}\delta DPB_{r,i} }}_{{\delta {\text{IFCB}}_{\varPhi ,r,i} }} \\ & \quad + \,{\text{GFIF}}(\bar{N}_{r,1}^{s} ,\bar{N}_{r,2}^{s} ,\bar{N}_{r,i}^{s} ) + \varepsilon_{{{\text{IFCB}}_{\varPhi ,r,i}^{s} }} \\ \end{aligned} , $$

where \( i \in Z \) and \( i \in [3,\infty ) \). \( {\text{IFCB}}_{P,i}^{s} \) and \( {\text{IFCB}}_{P,r,i} \) are satellite and receiver code-based IFCBs. \( \delta {\text{IFCB}}_{\varPhi ,i}^{s} \) and \( \delta {\text{IFCB}}_{\varPhi ,r,i} \) are the time-variant part of satellite and receiver phase-based IFCBs.

Since we only focus on a satellite’s IFCB in this paper, we assume that the time-variant part of the receiver IFCB is small enough to be ignored (Li et al. 2012a), that is, \( \delta {\text{IFCB}}_{\varPhi ,r,i} = 0 \). Thus, GFIF phase combination is formed from the ambiguities and the time-variant part of satellite phase-based IFCB.

Based on Eq. (A1), the noise amplification factor of the GFIF combination relative to the raw observation can be expressed as shown in Eq. (A2):

$$ \kappa = \sqrt {\left( {\frac{{\mu_{2} }}{{\mu_{2} - 1}}} \right)^{2} + \left( {\frac{1}{{\mu_{2} - 1}}} \right)^{2} + \left( {\frac{{\mu_{3} }}{{\mu_{3} - 1}}} \right)^{2} + \left( {\frac{1}{{\mu_{3} - 1}}} \right)^{2} } , $$

Taking GPS satellite as an example, the noise of code-based and phase-based IFCBs is approximately four times larger than the noise of raw code and carrier-phase observations. This shows that the GFIF combination will significantly amplify the observation noise.

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Fan, L., Shi, C., Li, M. et al. GPS satellite inter-frequency clock bias estimation using triple-frequency raw observations. J Geod 93, 2465–2479 (2019).

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  • GPS
  • GNSS
  • Inter-frequency clock bias
  • Differential code bias
  • Raw observations
  • Geometry-free
  • Ionosphere-free combination