Estimation and analysis of differential code biases for BDS3/BDS2 using iGMAS and MGEX observations

Abstract

In this contribution, the BDS3 differential code biases (DCBs) are estimated by using the iGMAS and MEGX networks and the performance of both satellite and receiver DCBs for BDS3 is evaluated with the observational data during the period of DOY 1–180, 2017. The characteristics of BDS3 and BDS2 DCB are compared, and the code inter-system biases (ISB) between BDS3 and BDS2 are also analyzed in detail. The comparison of our estimated BDS C2I-C6I and C2I-C7I DCBs and the DLR and IGG products shows a good agreement. For BDS2, the mean differences are within $\pm \,0.2\,{\text{ns}}$ and STDs are within 0.15 ns. However, the BDS3 presents a larger difference, with the mean difference of about 0.35 ns, because fewer stations are included in the DLR/IGG processing. The comparison of BDS3 and BDS2 DCB shows that the receiver DCB differences between BDS3 and BDS2 are close to zero for the same network, i.e., iGMAS or MGEX. In other words, there is no significant systematic bias between BDS3 and BDS2 receiver DCB. However, when the iGMAS and MGEX networks are processed together, we found that the receiver DCB differences between BDS3 and BDS2 are not close to zero and present an obvious systematic bias between different networks. The further analysis of code ISB between BDS3 and BDS2 also shows a similar phenomenon. Therefore, the receiver DCB of BDS3 and BDS2 should be separately estimated or calibrated when iGMAS and MGEX networks are processed together. We also analyze the receiver DCB and code ISB between Galileo FOC and IOV satellites and found that there is no such systematic bias between Galileo FOC and IOV satellites. A 180-day analysis of estimated BDS3 and BDS2 DCB shows that the satellite DCBs of BDS3 are fairly stable, with a mean STD of about 0.18 ns. For BDS2, the IGSO DCBs are the most stable with a mean STD of about 0.09 ns, and the GEO DCBs exhibit the worst stability with a mean STD of about 0.18 ns. The mean STDs of receiver DCBs for BDS3 and BDS2 are 0.38 and 0.41 ns, respectively, and the STD of receiver DCBs of BDS3 is smaller than that of BDS2 at most stations.

Appendix: ISB processing

The observation equations for carrier phase and pseudorange can be expressed as follows:

$$ P_{r,j}^{s} = \rho_{r}^{s} - dt^{s} + dt_{r} + c \cdot \left( {d_{r,j} - d_{j}^{s} } \right) + I_{r,j}^{s} + T_{r}^{s} + e_{r,j}^{s} $$

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$$ L_{r,j}^{s} = \rho_{r}^{s} - dt^{s} + dt_{r} + \lambda_{j} \left( {b_{r,j} - b_{j}^{s} + N_{r,j}^{s} } \right) - I_{r,j}^{s} + T_{r}^{s} + \varepsilon_{r,j}^{s} $$

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where s and $r$ represent the satellite and receiver, respectively, j refers to the carrier frequency; ρ ^s _r is the geometric distance between the satellite and receiver; dt^s and dt_r denote the clock biases of satellite and receiver; d_r,j and d ^s _j are the code biases of receiver and satellite, while b_r,j and b ^s _j are the receiver- and satellite-dependent uncalibrated phase delay; λ_j is the wavelength; N ^s _{r, j} is the integer ambiguity; I ^s _{r, j} and T ^s _r represent the ionospheric and tropospheric delay, respectively; e ^s _{r, j} and ɛ ^s _{r, j} refer to the measurement noise and multipath error of the pseudorange and carrier phase observations. To eliminate the first-order ionospheric delays, ionosphere-free (IF) combination observations are usually used in POD

$$ P_{{r,{\text{IF}}}}^{s} = \rho_{r}^{s} - dt^{s} + dt_{r} + c \cdot \left( {d_{{r,{\text{IF}}}} - d_{\text{IF}}^{s} } \right) + T_{r}^{s} + e_{{r,{\text{IF}}}}^{s} $$

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$$ L_{{r,{\text{IF}}}}^{s} = \rho_{r}^{s} - dt^{s} + dt_{r} + \lambda_{\text{IF}} \left( {b_{{r,{\text{IF}}}} - b_{\text{IF}}^{s} + N_{{r,{\text{IF}}}}^{s} } \right) + T_{r}^{s} + \varepsilon_{{r,{\text{IF}}}}^{s} $$

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Since the satellite state vector at t can be calculated by the initial state at t₀ and the state transition matrix, when BDS and GPS are processed together, and the linearized equations for the IF combination observations above can be formed as follows,

$$ \left\{ \begin{array}{*{20}l} P_{{r,{\text{IF}}}}^{C} = \varPsi_{r}^{C} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{C} \cdot O_{0}^{C} - R_{r} } \right) \\ \quad -\, dt^{C} + dt_{r} + c \cdot \left( {d_{{r,C,{\text{IF}}}} - d_{\text{IF}}^{C} } \right) + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + e_{{r,{\text{IF}}}}^{C} \\ P_{{r,{\text{IF}}}}^{G} = \varPsi_{r}^{G} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{G} \cdot O_{0}^{G} - R_{r} } \right) \\ \quad-\, dt^{G} + dt_{r} + c \cdot \left( {d_{{r,G,{\text{IF}}}} - d_{\text{IF}}^{G} } \right) + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + e_{{r,{\text{IF}}}}^{G} \\ \end{array} \right. $$

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$$ \left\{ \begin{array}{*{20}l} L_{{r,{\text{IF}}}}^{C} = \varPsi_{r}^{C} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{C} \cdot O_{0}^{C} - R_{r} } \right) - dt^{C} + dt_{r} + \lambda_{{{\text{IF}},C}} \\ \quad \cdot \left( {b_{{r,C,{\text{IF}}}} - b_{\text{IF}}^{C} + N_{{r,{\text{IF}}}}^{C} } \right) + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + \varepsilon_{{r,{\text{IF}}}}^{C} \\ L_{{r,{\text{IF}}}}^{G} = \varPsi_{r}^{G} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{G} \cdot O_{0}^{G} - R_{r} } \right) - dt^{G} + dt_{r} + \lambda_{{{\text{IF}},G}} \\\quad\cdot \left( {d_{{r,G,{\text{IF}}}} - d_{\text{IF}}^{G} + N_{{r,{\text{IF}}}}^{G} } \right) + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + \varepsilon_{{r,{\text{IF}}}}^{G} \\ \end{array} \right. $$

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$$ \left\{ {\begin{array}{*{20}c} {O_{0}^{C} = \left( {x_{0}^{C} y_{0}^{C} z_{0 }^{C} \dot{x}_{0}^{C} \dot{y}_{0}^{C} \dot{z}_{0 }^{C} {\text{SRP}}_{1}^{C} {\text{SRP}}_{2}^{C} \ldots {\text{SRP}}_{n}^{C} } \right)^{T} } \\ {O_{0}^{G} = \left( {x_{0}^{G} y_{0}^{G} z_{0 }^{G} \dot{x}_{0}^{G} \dot{y}_{0}^{G} \dot{z}_{0 }^{G} {\text{SRP}}_{1}^{G} {\text{SRP}}_{2}^{G} \ldots {\text{SRP}}_{n}^{G} } \right)^{T} } \\ \end{array} } \right. $$

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where superscript C and G represent the BDS and GPS satellites, respectively; $\varPsi$ is the unit vector of the direction from receiver to satellite and $\varPhi \left( {t,t_{0} } \right)$ is the state transition matrix from initial epoch t₀ to epoch t; R_r refers to the increment vector of the receiver position; $m_{{r,{\text{trop}}}}$ represents the wet mapping function; and ${\text{ZTD}}_{r}$ refers to the residual part of the zenith wet tropospheric delay. O₀ is the initial satellite orbit state vector; x₀, y₀, z₀ are the initial positions of satellite; while $\dot{x}_{0}$, $\dot{y}_{0}$$\dot{z}_{0}$ are the velocity; ${\text{SRP}}_{1}^{G} ,{\text{SRP}}_{2}^{G} , \ldots ,{\text{SRP}}_{n}^{G}$ denote to the solar radiation pressure parameters. In one receiver, the code delays of BDS ($d_{{r,C,{\text{IF}}}}$) and GPS ($d_{{r,G,{\text{IF}}}}$) are different, and this difference is a kind of inter-system biases (ISBs). In the process, we usually estimate the GPS receiver clock and the code ISB parameters for BDS relative to GPS. Thus, Eq. (13) can be rewritten as:

$$ \left\{ \begin{array}{*{20}l} P_{{r,{\text{IF}}}}^{C} = \varPsi_{r}^{C} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{C} \cdot O_{0}^{C} - R_{r} } \right) - dt^{C} + dt_{r,G} + c \\\quad\cdot\, \left( {ISB_{{r,G\_C,{\text{IF}}}} - d_{\text{IF}}^{C} } \right) + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + e_{{r,{\text{IF}}}}^{C} \hfill \\ P_{{r,{\text{IF}}}}^{G} = \varPsi_{r}^{G} \cdot \left( {\varPhi \left( {t,t_{0} } \right)^{G} \cdot O_{0}^{G} - R_{r} } \right) - dt^{G} + dt_{r,G} + c \\\quad\cdot\, d_{\text{IF}}^{G} + m_{{r,{\text{trop}}}} \cdot {\text{ZTD}}_{r} + e_{{r,{\text{IF}}}}^{G} \hfill \\ \end{array} \right. $$

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where $dt_{r,G} = dt_{r} + c \cdot d_{{r,G,{\text{IF}}}}$ is the receiver clock of GPS; ${\text{ISB}}_{{r,G\_C,{\text{IF}}}} = b_{{r,C,{\text{IF}}}} - b_{{r,G,{\text{IF}}}}$ is code ISB parameters for BDS relative to GPS. In this study, we performed the GPS + BDS3 and GPS + BDS2 POD processing. The receiver clocks of BDS2 and BDS3 can be obtained as follows:

$$ \left\{ {\begin{array}{*{20}c} {dt_{r,C2} = dt_{r,G} + {\text{ISB}}_{r,G\_C2} } \\ { dt_{r,C3} = dt_{r,G}^{'} + {\text{ISB}}_{r,G\_C3} } \\ \end{array} } \right. $$

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where ${\text{d}}t_{r,C2}$ and dt_r,C3 refer to the receiver clock biases for BDS2 and BDS3, respectively; dt_r,G and dt ^’ _{r, G} are the receiver clock biases for GPS obtained from GPS + BDS2 and GPS + BDS3 POD, respectively; ${\text{ISB}}_{r,G\_C2}$ and ISB_{r,G_C3} are the code ISBs of BDS2 and BDS3 relative to GPS. It should be noted that different receiver clock may be chosen as reference for GPS + BDS2 and GPS + BDS3 POD; the estimated receiver clocks of GPS may be different even for the same station. To remove this bias, a receiver clock is set as reference, and the other receiver clocks are subtracted by the reference clock,

$$ \left\{ \begin{array}{*{20}l} \nabla t_{r,C2} = t_{r,C2} - t_{{{\text{ref}},C2}} = \left( {t_{r,G} + {\text{ISB}}_{r,G\_C2} } \right) \\\quad-\, \left( {t_{{{\text{ref}},G}} + {\text{ISB}}_{{{\text{ref}},G\_C2}} } \right) \\ \nabla t_{r,C3} = t_{r,C3} - t_{{{\text{ref}},C3}} = \left( {t_{r,G}^{'} + {\text{ISB}}_{r,G\_C3} } \right) \\\quad-\, \left( {t_{{{\text{ref}},G}}^{'} + {\text{ISB}}_{{{\text{ref}},G\_C3}} } \right) \\ \end{array} \right. $$

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where Δt_r,C2 and Δt_r,C3 refer to the receiver clocks after removing reference clock for BDS2 and BDS3, respectively. The reference clock only affects the estimated absolute clock of receiver, but it has no effect on the relative clock between the receivers. So the relative receiver clock $t_{r,G} - t_{{{\text{ref}},G}}$ is equal to $t_{r,G}^{'} - t_{{{\text{ref}},G}}^{'}$, although different reference clocks may be applied by them. The difference between BDS2 and BDS3 receiver clocks after removing reference clock can be obtained as:

$$ \nabla t_{r,C2} - \nabla t_{r,C3} = \left( {{\text{ISB}}_{r,G\_C2} - {\text{ISB}}_{r,G\_C3} } \right) - \left( {{\text{ISB}}_{{{\text{ref}},G\_C2}} - {\text{ISB}}_{{{\text{ref}},G\_C3}} } \right) = {\text{ISB}}_{r,C3\_C2} - {\text{ISB}}_{{{\text{ref}},C3\_C2}} $$

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