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An enhanced algorithm to estimate BDS satellite’s differential code biases

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Abstract

This paper proposes an enhanced algorithm to estimate the differential code biases (DCB) on three frequencies of the BeiDou Navigation Satellite System (BDS) satellites. By forming ionospheric observables derived from uncombined precise point positioning and geometry-free linear combination of phase-smoothed range, satellite DCBs are determined together with ionospheric delay that is modeled at each individual station. Specifically, the DCB and ionospheric delay are estimated in a weighted least-squares estimator by considering the precision of ionospheric observables, and a misclosure constraint for different types of satellite DCBs is introduced. This algorithm was tested by GNSS data collected in November and December 2013 from 29 stations of Multi-GNSS Experiment (MGEX) and BeiDou Experimental Tracking Stations. Results show that the proposed algorithm is able to precisely estimate BDS satellite DCBs, where the mean value of day-to-day scattering is about 0.19 ns and the RMS of the difference with respect to MGEX DCB products is about 0.24 ns. In order to make comparison, an existing algorithm based on IGG: Institute of Geodesy and Geophysics, China (IGGDCB), is also used to process the same dataset. Results show that, the DCB difference between results from the enhanced algorithm and the DCB products from Center for Orbit Determination in Europe (CODE) and MGEX is reduced in average by 46 % for GPS satellites and 14 % for BDS satellites, when compared with DCB difference between the results of IGGDCB algorithm and the DCB products from CODE and MGEX. In addition, we find the day-to-day scattering of BDS IGSO satellites is obviously lower than that of GEO and MEO satellites, and a significant bias exists in daily DCB values of GEO satellites comparing with MGEX DCB product. This proposed algorithm also provides a new approach to estimate the satellite DCBs of multiple GNSS systems.

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Acknowledgments

The supports from the National Natural Science Foundation of China (Grant No. 41231174, 41325015, 41204029, 41375041, 41274039, 41304034), the National “863 Program” of China (Grant No. 2014AA123101), and the State Key Laboratory of Geodesy and Earth’s Dynamics (Grant No. SKLGED2013-4-2-Z, SKLGED2014-3-1-E) are gratefully acknowledged. This work is also supported by the Hong Kong Research Grants Council (RGC) projects (PolyU 5325/12E, F-PP0F and PolyU 5203/13E, B-Q37X). Zhizhao Liu acknowledges support from the Program of Introducing Talents of Discipline to Universities (Wuhan University, GNSS Research Center), China. Thanks are also due to Yunbin Yuan for valuable suggestions and Zishen Li for assistance with the experiments on IGGDCB part.

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Correspondence to Min Li.

Appendices

Appendix 1: The ionospheric observable derived from UPPP

UPPP uses original observables. The number of observables is doubled compared to the ionosphere-free combination method, and the amplification effect of multipath and measurement noises is avoided. Line of sight (LOS) ionospheric observables are treated as parameters to be estimated in this procedure, together with GNSS station coordinate components, receiver clock error, wet zenith tropospheric delay and ambiguities at both frequencies. Since GPS product and BDS product obtained from different Analysis Centers (CODE and WHU) are used in our experiment, the inconsistency of the clock reference will affect GPS-BDS time offset (i.e., 1.5 ns downward trend existed in GPS-BDS time offset for all stations on the DOY 358), thus separate receiver clock errors for GPST and BDT, instead of a receiver clock and a constant GPS-BDS time offset, are estimated.

The GPS and BDS dual-frequency code and carrier-phase observation equations can be expressed as follows:

$$\begin{aligned} P_{r,i}^{j,s}= & {} \rho _r^{j,s} +c\cdot \delta t_r^s -c\cdot \delta t^{j,s}+T_r^{j,s} \nonumber \\&+\,\mu _i^{s} \cdot I_{r,1}^{j,s} +c\cdot b_{r,i}^{j} +c\cdot b_i^{j,s} +\varepsilon _{i,P}^{j,s} \nonumber \\ {\Phi }_{r,i}^{j,s}= & {} \rho _r^{j,s} +c\cdot \delta t_r^s -c\cdot \delta t^{j,s}+T_r^{j,s} -\mu _i^s \cdot I_{r,1}^{j,s}\nonumber \\&+\,\lambda _i^s \cdot {\varvec{M}}_{r,i}^{j,s} +\varepsilon _{i,{\Phi }}^{j,s}, \end{aligned}$$
(12)

where jris denote satellite PRN, receiver, frequency (\(i=1,2)\), and satellite system (\(s=G\) for GPS and \({s}=C\) for BDS), respectively; \(P_{r,i}^{j,s} \) and \({\Phi }_{r,i}^{j,s} \) represent pseudorange code and carrier-phase measurements at frequency \(f_{s,i} \), respectively; \(\rho _r^{j,s} \) is the geometric range from GPS/BDS satellite to receiver antennas; \(\delta t_r^s \) and \(\delta t^{j,s}\) are receiver and satellite clock errors for the satellite system s; \(T_r^{j,s} \) denotes the LOS tropospheric delays; \(\mu _i^s =f_{s,1}^2 /f_{s,i}^2 \) is a constant factor at frequency \(f_{s,i} \); \(I_{r,1}^{j,s} =\frac{\alpha \cdot \mathrm{sTEC}_r^{j,s} }{f_{s,1}^2 }\) is the dispersive ionospheric effect on GPS/BDS signal propagation at frequency \(f_{s,1} \) where \(\mathrm{sTEC}_r^{j,s} \) is the slant TEC and \(\alpha \) is a constant (\(\alpha =4.028\times 10^{17}\, \mathrm{m}\, \mathrm{s}^{-2}\, \mathrm{TECU}^{-1})\); \(\lambda _i^s \) denotes wavelength at frequency \(f_{s,i} \) on GPS or BDS satellite; \(M_{r,i}^{j,s} \) represents carrier-phase ambiguity including satellite and receiver phase instrumental delays and initial phase bias; \(b_{r,i}^j \) and \(b_i^{j,s} \) denote the hardware delay of satellite and receiver, respectively; \(\varepsilon _{i,P}^{j,s} \) and \(\varepsilon _{i,{\Phi }}^{j,s} \) are the noise and multipath effects of observations in pseudorange code and carrier-phase measurements, respectively.

As the satellite clock error is derived from ionosphere-free combination observables, the hardware delays \(b_1^{j,s} \) and \(b_2^{j,s} \) are introduced to the satellite clock product, as shown in Eq. (13). The frequencies of L1 and L2 are used for GPS satellites whilst the frequencies of B1 and B2 are used for BDS satellites. Thus only two frequencies in GPS and BDS can be used in UPPP to be consistent with the clock products.

$$\begin{aligned} \delta \tilde{t}^{j,s}=\delta t^{j,s}-\frac{\mu _2^s }{\mu _2^s -1}\cdot b_1^{j,s} +\frac{1}{\mu _2^s -1}\cdot b_2^{j,s} \end{aligned}$$
(13)

After organizing the equations, the full-rank linear equation can be formed as shown in Eq. (14):

$$\begin{aligned} \Delta P_{r,i}^{j,s}= & {} -{\varvec{\gamma }} _{r,i}^{j,s} \cdot \Delta {\varvec{r}}+c\cdot \delta {\tilde{t}} _r^s +M_w \cdot T_w +\mu _i^s \cdot \tilde{I}_{r,1}^{j,s} +\varepsilon _{i,P}^{j,s} \nonumber \\ \Delta {\Phi }_{r,i}^{j,s}= & {} -{\varvec{\gamma }} _{r,i}^{j,s} \cdot \Delta \varvec{r}+c\cdot \delta \tilde{t}_r^s +M_w \cdot T_w -\mu _i^s \cdot \tilde{I}_{r,1}^{j,s}\nonumber \\&+\,\lambda _i^s \cdot \tilde{M}_{r,i}^{j,s} +\varepsilon _{i,{\Phi }}^{j,s} , \end{aligned}$$
(14)

where \(\Delta P_{r,i}^{j,s} \) and \(\Delta {\Phi }_{r,i}^{j,s} \) denote the observed minus calculated observations (O \(-\) C) for the code and carrier-phase. The errors related to satellite (i.e., phase center offset of GPS and BDS satellites and phase center variation for GPS satellites), propagation path (i.e., phase wind-up and dry tropospheric delay) and the ground stations (i.e., solid earth tide and ocean loading tide) have been corrected using empirical models; \({\varvec{\gamma }} _{r,i}^{j,s} \) is the unit direction vector from receiver to satellite; \(\Delta {\varvec{r}}\) is the correction to the approximate coordinates of the receiver; \(T_w \) and \({M}_w \) denote zenith wet tropospheric delay and its projection function related to the satellite’s zenith distance; the formation of other symbols is shown in Eq. (15).

$$\begin{aligned}&{\delta \tilde{t}_r^s =\delta t_r^s +b_{r,if}^s } \nonumber \\&{\tilde{I}_{r,1}^{j,s} =\frac{\alpha }{f_{s,1}^2 }\cdot \mathrm{sTEC}_r^{j,s}-\frac{c}{\mu _2^s -1}( {B_{12}^{j,s} +B_{r,12}^s })} \nonumber \\&{\tilde{M}_{r,i}^{j,s} =M_{r,i}^{j,s} -\frac{c\cdot b_{r,if}^s }{\lambda _i^s }-\frac{c\cdot b_{if}^{j,s} }{\lambda _i^s }+\frac{c}{\lambda _i^s ( {\mu _2^s -1})}( {B_{12}^{j,s} +B_{r,12}^s })} \nonumber \\&{b_{r,if}^s =\frac{\mu _2^s }{\mu _2^s -1}\cdot b_{r,1}^s -\frac{1}{\mu _2^s -1}\cdot b_{r,2}^s } \nonumber \\&{b_{if}^{j,s} =\frac{\mu _2^s }{\mu _2^s -1}\cdot b_1^{j,s} -\frac{1}{\mu _2^s -1}\cdot b_2^{j,s} }, \end{aligned}$$
(15)

where \(\delta \tilde{t}_r^s \), \(\tilde{I}_{r,1}^{j,s} \) and \(\tilde{M}_{r,i}^{j,s} \) are receiver clock error, LOS slant ionospheric observable, and ambiguity to be estimated, respectively; \(B_{12}^{j,s} =b_1^{j,s} -b_2^{j,s} \) and \(B_{r,12}^s =b_{r,1}^s -b_{r,2}^s \) denote satellite and receiver DCBs. It can be seen that the new ambiguity \(\tilde{M}_{r,i}^{j,s} \) is no longer an integer. For each GPS or BDS satellite, four observation equations can be formed from dual-frequency code and carrier-phase observables and three unknown parameters need to be estimated: slant ionosphere \(\tilde{I}_{r,1}^{j,s} \) on frequency \(f_{s,1} \), and carrier-phase ambiguities \(\tilde{M}_{r,i}^{j,s} \) on two frequencies (\(i=1,2)\). In conclusion, if the numbers of GPS and BDS satellites observed at one epoch are p and q, respectively, the number of observation equations is \(4( {p+q})\). The total number of unknown parameters is \(( {6+3( {p+q})})\). The six parameters common to all satellites are: three coordinate components, two receiver clock errors for GPST and BDT, respectively, and one ZTD parameter of the station. The three parameters for each satellite are slant ionosphere on frequency \(f_{s,1} \) and ambiguities on two frequencies.

A further conversion from length unit (m) to TEC unit (TECU) needs to be handled through multiplying \(\tilde{I}_{r,1}^{j,s} \) in Eq. (15)  by \(\frac{f_{s,1}^2 }{\alpha }\), thus the ionospheric observable can be expressed as a new variable shown in Eq. (16).

$$\begin{aligned} \hat{I}_{r,1,\mathrm{UPPP}}^{j,s} =\mathrm{sTEC}_r^{j,s} -A\cdot ( {B_{12}^{j,s} +B_{r,12}^s }), \end{aligned}$$
(16)

where A is a constant value used to convert time unit (s) to TEC unit (TECU), defined as \(A=\frac{c}{\alpha ( {f_{s,2}^{-2} -f_{s,1}^{-2} })}\); \(\hat{I}_{r,1}^{j,s} \) is used for DCB separation in further processing. The subscript UPPP is used to indicate that the ionospheric observable is derived from UPPP.

The Kalman filter is often used for parameter estimation. Considering moderate variations during short periods (Abdel-salam 2005), the ionospheric observable \(\hat{I}_{r,1,\mathrm{UPPP}}^{j,s} \) can be modeled as random walk during parameter estimation process and the variance of \(\hat{I}_{r,1,\mathrm{UPPP}}^{j,s} \) at each epoch can also be derived in this process.

Appendix 2: The ionospheric observable derived from GFPSR

The geometry-free linear combination of code-delay and carrier-phase at frequencies \(f_{s,i} \) and \(f_{s,k} \) can be formed:

$$\begin{aligned} P_{r,ik}^{j,s}= & {} P_{r,i}^{j,s} -P_{r,k}^{j,s} =-\alpha \cdot ( {f_{s,k}^{-2} -f_{s,i}^{-2} })\cdot \mathrm{sTEC}_r^{j,s} \nonumber \\&+\,c\cdot ( {B_{r,ik}^s +B_{ik}^{j,s} })+\varepsilon _{i,P}^{j,s} -\varepsilon _{k,P}^{j,s} \nonumber \\ {\Phi }_{r,ik}^{j,s}= & {} {\Phi }_{r,i}^{j,s} -{\Phi }_{r,k}^{j,s} =\alpha \cdot ( {f_{s,k}^{-2} -f_{s,i}^{-2} })\cdot \mathrm{sTEC}_r^{j,s} \nonumber \\&+\,( {\lambda _i^s \cdot M_{r,i}^{j,s} -\lambda _k^s \cdot M_{r,k}^{j,s} })+\varepsilon _{i,{\Phi }}^{j,s} -\varepsilon _{k,{\Phi }}^{j,s} \end{aligned}$$
(17)

The definition of variables in Eq. (17) is same as Eq. (12). It should be noted that the correlation between different frequencies for either code or carrier-phase observation is not considered.

It can be seen from Eq. (17) that the geometry-free combination of carrier-phase contains unsolved ambiguities that impede the direct use of carrier-phase observables. To get high-precision code observables, they are smoothed by the carrier-phase ones. In this procedure, the code observables are actually replaced by the carrier-phases, shifted by the average value of code minus phase in a continuous arc (Dach et al. 2007), as shown in Eq. (18).

$$\begin{aligned} \tilde{P}_{r,i}^{j,s}= & {} {\Phi }_{r,i}^{j,s} +\bar{P}_{r,i}^{j,s} -{\bar{\Phi }}_{r,i}^{j,s} +\frac{2f_{s,k}^2 }{f_{s,i}^2 -f_{s,k}^2 }\nonumber \\&\times \left( {\left( {{\Phi }_{r,i}^{j,s} -{\bar{\Phi }}_{r,i}^{j,s} })-( {{\Phi }_{r,k}^{j,s} -{\bar{\Phi }}_{r,k}^{j,s} }\right) }\right) \nonumber \\ \tilde{P}_{r,k}^{j,s}= & {} {\Phi }_{r,k}^{j,s} +\bar{P}_{r,k}^{j,s} -{\bar{\Phi }}_{r,k}^{j,s} +\frac{2f_{s,i}^2 }{f_{s,i}^2 -f_{s,k}^2 }\nonumber \\&\times \left( {\left( {{\Phi }_{r,i}^{j,s} -{\bar{\Phi }}_{r,i}^{j,s} })-( {{\Phi }_{r,k}^{j,s} -{\bar{\Phi }}_{r,k}^{j,s} }\right) }\right) , \end{aligned}$$
(18)

where \(\bar{P}_i^j =\frac{\mathop \sum \nolimits _{n=1}^N P_{r,i,n}^j }{N}\) and \({\bar{\Phi }}_i^j =\frac{\mathop \sum \nolimits _{n=1}^N {\Phi }_{r,i,n}^j }{N}\) are the mean code observations and carrier-phase in an arc; \(P_{r,i,n} \) and \({\Phi }_{r,i,n} \) are the code and phase observables at the nth epoch, respectively.

The smoothed code observables are used to form geometry-free combination. Thus, the actual ionospheric observable can be obtained as shown in Eq. (19):

$$\begin{aligned} \tilde{P}_{r,ik}^{j,s}= & {} \tilde{P}_{r,i}^{j,s} -\tilde{P}_{r,k}^{j,s} =-( {{\Phi }_{r,i}^{j,s} -{\Phi }_{r,k}^{j,s} })\nonumber \\&+\,\frac{\mathop \sum \nolimits _{n=1}^N ( {( {P_{r,i,n}^{j,s} -P_{r,k,n}^{j,s} })+( {{\Phi }_{r,i,n}^{j,s} -{\Phi }_{r,k,n}^{j,s} })})}{N}.\nonumber \\ \end{aligned}$$
(19)

Taking Eq. (17) into Eq. (19), the derived ionospheric observable is shown in Eq. (20):

$$\begin{aligned} \tilde{P}_{r,ik}^{j,s}= & {} -\alpha \cdot ( {f_{s,k}^{-2} -f_{s,i}^{-2} })\cdot \mathrm{sTEC}_r^{j,s}\nonumber \\&+c\cdot ( B_{ik}^{j,s}+B_{r,ik}^s )+\varepsilon _{ik,{\tilde{P}}}^{j,s}. \end{aligned}$$
(20)

It can be seen that the right side of Eq. (20) is the same as that of Eq. (17) except noise \(\varepsilon \), indicating that the ionospheric observables (slant TEC plus combination of satellite and receiver DCBs) derived from the raw code and smoothed code have the same form but different precisions.

Multiplying \(\tilde{P}_{r,ik}^s \) by \(\frac{-1}{\alpha ( {f_{s,k}^{-2} -f_{s,i}^{-2} })}\), the ionospheric observable is converted from length unit (m) to TECU, which can be expressed as shown in Eq. (21):

$$\begin{aligned}&\hat{I}_{r,ik,\mathrm{GFPSR}}^{j,s} =\mathrm{sTEC}_r^{j,s} -A\cdot ( {B_{ik}^{j,s} +B_{r,ik}^s })+\varepsilon _{ik,{\tilde{I}}}^{j,s}\nonumber \\&\varepsilon _{ik,{\tilde{I}}}^{j,s} =\Delta \varepsilon _{\Phi } +\Delta \bar{\varepsilon }_P +\Delta \bar{\varepsilon }_{\Phi }, \end{aligned}$$
(21)

where the subscript GFPSR indicates that the ionospheric observable is derived from the smoothed code; \(\bar{\varepsilon }=\frac{\mathop \sum \nolimits _{n=1}^N \varepsilon _N }{N}\) and N is the number of continuous measurements contained in the arc; \(\Delta \) is the difference operator between frequencies i and k; the rest variables have the same definitions as Eq. (16). It can be clearly seen that the precision of GFPSR is affected by the length of a continuous arc.

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Shi, C., Fan, L., Li, M. et al. An enhanced algorithm to estimate BDS satellite’s differential code biases. J Geod 90, 161–177 (2016). https://doi.org/10.1007/s00190-015-0863-8

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  • DOI: https://doi.org/10.1007/s00190-015-0863-8

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