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Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations
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Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations Haojun Li, Jingxin Xiao, Ling Yang ⇑ College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, PR China Received 28 July 2019; received in revised form 11 October 2019; accepted 12 October 2019
Abstract Considering the time-varying characteristic of the GPS differential code bias between C1 and P1 observations (DCB (C1-P1)), the method for estimating the time-varying DCB (C1-P1) is presented. The time-varying characteristic of the DCB (C1-P1) is analyzed using the estimated values with the twelve-month (1 Jan-31 Dec 2018) GPS data set from 120 International GNSS Service (IGS) stations. The estimated results show that the DCB (C1-P1) has the evident variation with the periods of 12, 8, 6, 4, 3.4286 h. Based on the noticed periods, the estimated single-day DCB (C1-P1) series is modeled with a harmonics-based function. The results show that the initial phase offsets (hi ði ¼ 1 5Þ) have the characterization of linear variation and these variations are very evident, while the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time. The time-varying DCB (C1-P1) is used in single point positioning (SPP) to validate its effect on positioning. The SPP results show that the three-dimensional positioning results can be improved, when the time-varying part of the DCB (C1-P1) is corrected. The results for the 120 IGS stations indicate that the average three-dimensional positioning improvement for three days is about 3 cm. These improvements further advise that the time-varying part of DCB (C1-P1) should be corrected in GPS data processing and be serviced by IGS. The prediction performance shows that the correction rates of the predicted results for all satellites reach more than 40% accuracy and the mean prediction accuracy can reach 70%. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Hardware delay bias; Differential code bias; GPS satellite clock; Single point positioning
1. Introduction The original GPS design has two codes: the coarse/ acquisition (C/A) code, which is freely available to the public, and the precision (P) code reserved for military applications. For the codes and navigation message to travel from the satellite to the receiver, they are modulated to a carrier wave. The C/A code is transmitted on the L1 frequency as a 1.023 MHz signal using a bi-phase shift keying (BPSK) modulation technique. The P(Y)-code is transmitted on both the L1 and L2 frequencies as a 10.23 MHz signal using the same modulation. In electronics, there are ⇑ Corresponding author.
E-mail address: [email protected] (L. Yang).
differences between the used components and signal structure for the different types of observations (Leick et al., 2015). Therefore, there is no doubt that the effects of space environment and hardware on the (C/A) and P codes are different. Due to effects, there is a bias between (C/A) and P codes and it is named as differential code biases (DCB) from C1 and P1 observations on one frequency. The bias of DCB (C1-P1) contains receiver and satellitedependent parts and is estimated with the geometry-free code combination form with C1 and P1 observations. The time-variant characteristic of the satellite-dependent part is analyzed based on the estimated series in Gao et al. (2001). The time-varying characteristic for the most common bias of the DCB (P1–P2) between the P1 and P2 observations (Sardon et al., 1994; Goodwin and
https://doi.org/10.1016/j.asr.2019.10.018 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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H. Li et al. / Advances in Space Research xxx (2019) xxx–xxx
Breed, 2001; Otsuka et al., 2002) was analyzed in Li et al. (2019) and the main periods of 12, 8, 6, 4, 4.8 and 2.67 h are noticed. But, the current International GNSS Service (IGS) (Dow et al., 2009; Li et al., 2010) still provides the DCB products with a daily or monthly constant. The precise point positioning (PPP) is realized based on the single-receiver dual-frequency observations and the IGS products of the satellite clock error and orbit (Zumberge et al., 1997). In the satellite clock estimating, the reference observations of the ionosphere-free phase (L1/L2) and code (P1/P2) combinations are used in the IGS products computing. Li et al. (2017) shows that the satellite-dependent biases in the code and phase observations are absorbed by the estimated, reparameterized satellite clock error. Thus, the code bias not only affects the accuracy of the code observation but also the highquality float ambiguity solution. To improve the accuracy of the PPP positioning, the satellite DCB (C1-P1) bias should be corrected according to its characteristic at the user side, which observes the C1 observation. It is obvious that the current service of the satellite DCB (C1-P1) with the constant characteristic is bad for improvement of the positioning accuracy and it neglects the effect of the timevarying part of the satellite DCB (C1-P1) on the positioning results. Considering these discussions, the time-varying and periodic characteristics of the satellite DCB (C1-P1) is studied and the estimated DCB (C1-P1) series is modeled with the presented mathematical function to enhance the service ability (Li et al., 2018). The application of the estimated and modeled bias in positioning is implemented to validate its effect on the positioning. Finally, the research findings and outlooks are summarized. 2. Estimation of the time-varying DCB (C1-P1) The method for estimating the time-varying DCB (C1P1) series is presented. Based on the time-varying and the periodic characteristics of the DCB (C1-P1), it is modeled and its application in the single point positioning (SPP) is discussed to validate the effect of the time-varying part
on positioning and to improve the service of the DCB (C1-P1) product. 2.1. Estimation method for the satellite-dependent DCB (C1-P1) For a GPS user, the raw C1 and P1 observations are written as: P 1 ¼ q þ bPr bsP þ dr ds I sr T sr þ xP
ð1Þ
C 1 ¼ q þ bCr bsC þ dr ds I sr T sr þ xc
where q is the satellite-to-receiver range, T sr is tropospheric delay, I sr is the slant ionosphere delay of C1 and P1 observations, dr and ds are the receiver and satellite clock errors in meter; bCr and bPr are the receiver hardware delay biases of P1 and C1 observations; bCs and bPs are the satellite hardware delay biases of P1 and C1 observations; xP is the noise of P1 and xc is the noise of C1 observation. The DCB (C1-P1) is generally estimated with the geometryfree code combination form with C1 and P1 observations: C 1 P 1 ¼ bCr bPr bsC bsP þ xC xP s
¼ DCBðC1 P1Þr DCBðC1 P1Þ þ xC xP ð2Þ Eq. (2) indicates that it is impossible to separate the receiver or satellite DCB (P1-C1) from their combination. In general, the satellite-differenced algorithm is used by selecting a reference satellite. However, the visible time span of each satellite during one day is limited. As a result, the proper reference satellites need to be selected for different periods of time, which complicates the computation. According to S-system theory and rank-deficiency free network adjustment theory, a reference constraint can be added to provide a datum, forming a full-rank model (Teunissen, 1985, Odijk et al., 2016). In this data processing, the reference is determined: Pd j ð3Þ j¼1 DCBðC1 P1Þr ¼ 0 where DCBðC1 P1Þjr is the receiver DCBðC1 P1Þ at observation station j; d is the number of the observation stations. When this reference is introduced, the parameterization equation of the DCB (C1-P1) is expressed as: 2
e1d I dd et1
2 3 r 3 0 ðDCBðC1 P1Þ Þ1 6 7 6 ðC1 P1Þ 7 .. . 1;1 7 6 7 6 7 6 7 6 6 ðDCBðC1 P1Þr Þd 7 6 7 ... 01t 6 7 6 7 ¼ s 7 6 ðC1 P1Þ 7 ð DCB ð C1 P1 Þ Þ ed1 I tt 6 6 6 1;t 7 17 6 7 6 7 4 5 4 5 ... .. . s ðC1 P1Þd;t ðDCBðC1 P1Þ Þt ð4Þ
Fig. 1. Distribution of the 240 IGS stations tracking GPS signals. Red dots are stations for estimated DCB (C1-P1). Blue dots are SPP stations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where I dd and I tt are identity matrix, e1d is a vector with each element is 1, t is the satellite number. The estimator of least-square adjustment is implemented to compute the epoch-wise DCB (C1-P1). In processing, the elevationdependent weight is used:
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
H. Li et al. / Advances in Space Research xxx (2019) xxx–xxx
( wm ¼
30 hm 90
1
2sinðhm Þ 10 hm < 30
) ð5Þ
where hm is the satellite elevation at m epoch. The references of the estimated satellite DCB (C1-P1) for each satellite are consistent, even if the different constraint is used. This reference can be absorbed by the receiver clock in positioning so that the positioning result is not affected.
2.2. Modeling and application of the estimated satellitedependent DCB (C1-P1) Currently, the satellite DCB (C1-P1) is serviced by IGS with a daily or monthly constant. It is obvious that the time-varying characteristic of the satellite DCB (C1-P1) is not considered. According to the estimated DCB (C1-P1) series and its periodic variation, the DCB (C1-P1) series is modeled: P DCBðC1 P1Þs ðtÞ ¼ a þ e t þ y i 5i¼1 sinð2p t þ hi Þðt ¼ 0 24hÞ Ti ð6Þ
where a is a constant; e is the coefficient of linear term; i is the order of the harmonics; T i is the period; hi is initial phase offset, and y i is the amplitude. The satellitedependent biases are absorbed by the estimated, reparameterized satellite clock error in the satellite clock estimating (Li et al., 2017). Thus, the satellite DCB (P1-C1) is very important for computing the reparameterized satellite clock of the C1 observation and its corresponding combinations. The satellite clock error of the ionosphere-free code combination formed with C1 and P2 should be: f2
dsC1 =P 2 ¼ ds1;2 þ f 2 f1 2 DCBðC1 P1Þ 1
2
s
ð7Þ
3
where dsC1 =P 2 is the estimated, reparameterized satellite clock error of L1/L2 and C1/P2 and the ds1;2 is the IGS satellite clock product. 3. Data processing The twelve-month (1 Jan-31 Dec 2018) GPS data set from 120 IGS stations is processed to analyze the timevarying and periodic characteristic of the code bias and validate the presented approach. Data is sampled at 30 s and cut-off elevation is set to 10 degrees. Three days, 1–3 Jan 2018 of GPS data from 120 IGS stations is processed to analyze the performances of estimated time-varying DCB (C1-P1) in single point positioning (SPP). The IGS stations are shown in Fig. 1. 3.1. Estimated DCB (C1-P1) According to the presented method, one-year single-day data is processed and the one-year single-day time-varying DCB (C1-P1) series are estimated. The estimated results of the series for 1–3 Jan 2018 are shown in Fig. 2. From the Fig. 2, it is observed that the DCB (C1-P1) series varies with time and has the significant periodic variations. The significant periodic variations are not involved, although the time-varying characteristics of the DCB (C1-P1) series are validated in Gao et al. (2001). One-year single-day DCB (C1-P1) values estimated with the presented method for each satellite are analyzed using a FFT (fast Fourier transformation). The periods for the satellites of G03, G07, G16 and G23 are shown in Fig. 3. Fig. 3 shows that the DCB (C1-P1) values have the significant periods of 12, 8, 6, 4, 3.4286 h. As to the significant periodic variations, it is explained that the satellite hardware is affected by the periodic environment including the sun illumination (Li et al., 2013). Based on the noticed periods and Eq. (6),
Fig. 2. Series (unit: m) of the single-day DCB (C1-P1) for G01 and G10 with G24 as reference satellite, for the time 1–3 Jan 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 3. Periods (unit: h) of the single-day DCB (C1-P1) for G03, G07, G16 and G23, for the time 1 Jan-31 Dec 2018.
One-year single-day DCB (C1-P1) is modeled and the corresponding modeling parameters are analyzed. One-year single-day amplitudes (y i ði ¼ 1 5Þ) and initial phase offsets (hi ði ¼ 1 5Þ) of the model for satellites of G03 and G07 are illustrated in Figs. 4 and 5, respectively. The results in Figs. 4 and 5 show that the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time, which is not similar to the initial phase offsets (hi ði ¼ 1 5Þ). The initial phase offsets demonstrate the characterization of linear variation and these variations are very evident. The irregular variation of the amplitudes can be explained that the GPS code measurement is affected by the space environment including the multipath effect. The estimated results of the amplitudes and initial phase offsets for all satellites show the same characteristics.
3.2. Validation in positioning It is well known that some GNSS receivers just tracked C1 measurement. Thus, the correction of DCB (C1-P1) should be considered and implemented. The constant part of DCB (C1-P1) is considered in the current data processing. It is obvious that this correction neglects the effect of the time-varying part of the satellite DCB (C1-P1) on positioning results. To validate effect of the time-varying part of the satellite DCB (C1-P1) on positioning, the GPS data from 120 IGS stations is processed in SPP mode and the settings for the SPP processing are shown in Table 1. In data processing, two methods of #1 and #2 are used and implemented. In method of #1, the constant part of DCB (C1-P1) from IGS is used to correct the C1 observation,
Fig. 4. Amplitudes (y i ði ¼ 1 5Þ) (unit: m) of the model for satellites of G03 and G07, for the time 1 Jan-31 Dec 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
H. Li et al. / Advances in Space Research xxx (2019) xxx–xxx
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Fig. 5. Initial phase offsets (hi ði ¼ 1 5Þ) of the model for satellites of G03 and G07, for the time 1 Jan-31 Dec 2018. Table 1 Settings for the SPP processing.
Measurements
Corrections
Parameters
Model
Settings
Ionosphere-free code combination Adjustment Weighting DCB(C1-P1) Tides corrections PCV Relativity Station coordinates Troposphere Receiver clock error
Least square filter Elevation-dependent function #1 and #2 Solid tide and Ocean tide correction Absolute IGS 14 correction mode Corrected Estimated Correction: Saastamoinen model Solved for at each epoch as white noise
while the constant part of DCB (C1-P1) from IGS and the fitted time-varying part of DCB (C1-P1) are used in #2. The RMSs of the difference between the SPP results of the two methods and the IGS results are computed for the three directions of North, East and Up. It is difficult to assess the performance superiority of the different methods via the SPP results at each direction. Thus, the results are assessed using the three-dimensional positioning errors: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ TDP ¼ e2North þ e2East þ e2Up where eNorth is the error in the North direction,eEast is the error in the East direction and eUp is the error in the Up direction. The three-dimensional positioning results for the 120 IGS stations are shown in Fig. 6. The threedimensional positioning difference between the #1 and #2 are Fig. 7. The three-dimensional positioning results in Figs. 6 and 7 indicate that improvement can be obtained when the time-varying part of the DCB (C1-P1) is considered and corrected. The average three-dimensional positioning improvement for three days is about 3 cm. These improvements further validate that the time-varying characteristics of the satellite DCB (C1-P1) and the time-
varying part should be corrected in data processing. These also show that the time-varying part of the satellite DCB (C1-P1) should be serviced by IGS. 3.3. Prediction analysis The SPP results show that the correction of the timevarying part of the satellite DCB (C1-P1) can improve the positioning accuracy. In real-time processing, the satellite DCB (C1-P1) products should be obtained by prediction. Therefore, the prediction performance of the satellite DCB (C1-P1) containing the constant and the time-varying parts is discussed. The satellite DCB (C1P1) from single-day (1 Jan 2018) estimates is used to predict the values for the next days (2–6 Jan 2018). The single-day (1 Jan 2018) values are used to compute the modeling coefficients of each satellite and the computed coefficients of Eq. (6) are used to extrapolate the values of the next days (2–6 Jan 2018). The RMSs of the difference between the extrapolated and fitted results are illustrated in Fig. 8 and the means of 5-day single-day correction rates for each satellite are shown in Fig. 9. The correction rate is defined as:
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 6. The three-dimensional positioning results (unit: m) for 120 IGS stations, for the time 1 Jan 2018 (top), 2 Jan 2018 (middle) and 3 Jan 2018 (bottom).
Fig. 7. The three-dimensional positioning results difference (unit: m) between the #1 and #2 for 120 IGS stations, for the time 1–3 Jan 2018.
Fig. 8. The RMSs of the predicted results, for the time 2–6 Jan 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 9. The Correction rate (%) for the predicted DCB (C1-P1), for the time 2–6 Jan 2018. E RMS D cr ¼ RMSRMS 100% E
ð9Þ
where RMSD is the RMS for the differences between the predicted and fitted values and the RMSE is the RMS of the fitted values. Fig. 8 indicates that the accuracy of the predicted DCB(C1-P1) containing the constant and timevarying parts is better than 4.0 cm. Fig. 9 shows that the predicted results for all satellites reach more than 40% accuracy and the average prediction accuracy can reach 70%.
real-time service of the satellite DCB (C1-P1). The accuracy of the predicted results is better than 4.0 cm. The correction rates for all satellites reach more than 40% accuracy and the mean prediction accuracy can reach 70%. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
4. Conclusion and discussion Acknowledgments The DCB (C1-P1) is the bias between the C1 and P1 observations. The time-varying characteristic of the satellite DCB (C1-P1) and the effect of its time-varying part on the positioning are not discussed in the current studies, although it is time-varying. Considering these, the method for estimating the satellite DCB (C1-P1) series is presented and the estimated series is analyzed using one-year singleday estimates. The estimated single-day DCB (C1-P1) results show the time-varying characteristic of the satellite DCB (C1-P1). One-year single-day satellite DCB (C1-P1) results show that the satellite DCB (C1-P1) varies with the evident periods of the 12, 8, 6, 4, 3.4286 h. Based on the noticed periods, one-year single-day satellite DCB (C1-P1) results are modeled with a harmonics-based function. The results of the modeling coefficients show that the initial phase offsets (hi ði ¼ 1 5Þ) have the characterization of linear variation and these variations are very evident and the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time. The time-varying satellite DCB (C1-P1) is used in single point positioning to validate its effect on the positioning. The SPP results show that the three-dimensional positioning results can be improved, when the time-varying part of the satellite DCB (C1-P1) is corrected. The three-dimensional positioning results for the 120 IGS stations show that the average threedimensional positioning improvement for three days is about 3 cm. The improvement further advises that the time-varying part of the satellite DCB (C1-P1) should be considered and corrected in GPS data processing. These also indicate that the time-varying part of the satellite DCB (C1-P1) should be estimated and serviced by IGS. The prediction analysis is implemented to provide the
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Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations Haojun Li, Jingxin Xiao, Ling Yang ⇑ College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, PR China Received 28 July 2019; received in revised form 11 October 2019; accepted 12 October 2019
Abstract Considering the time-varying characteristic of the GPS differential code bias between C1 and P1 observations (DCB (C1-P1)), the method for estimating the time-varying DCB (C1-P1) is presented. The time-varying characteristic of the DCB (C1-P1) is analyzed using the estimated values with the twelve-month (1 Jan-31 Dec 2018) GPS data set from 120 International GNSS Service (IGS) stations. The estimated results show that the DCB (C1-P1) has the evident variation with the periods of 12, 8, 6, 4, 3.4286 h. Based on the noticed periods, the estimated single-day DCB (C1-P1) series is modeled with a harmonics-based function. The results show that the initial phase offsets (hi ði ¼ 1 5Þ) have the characterization of linear variation and these variations are very evident, while the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time. The time-varying DCB (C1-P1) is used in single point positioning (SPP) to validate its effect on positioning. The SPP results show that the three-dimensional positioning results can be improved, when the time-varying part of the DCB (C1-P1) is corrected. The results for the 120 IGS stations indicate that the average three-dimensional positioning improvement for three days is about 3 cm. These improvements further advise that the time-varying part of DCB (C1-P1) should be corrected in GPS data processing and be serviced by IGS. The prediction performance shows that the correction rates of the predicted results for all satellites reach more than 40% accuracy and the mean prediction accuracy can reach 70%. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Hardware delay bias; Differential code bias; GPS satellite clock; Single point positioning
1. Introduction The original GPS design has two codes: the coarse/ acquisition (C/A) code, which is freely available to the public, and the precision (P) code reserved for military applications. For the codes and navigation message to travel from the satellite to the receiver, they are modulated to a carrier wave. The C/A code is transmitted on the L1 frequency as a 1.023 MHz signal using a bi-phase shift keying (BPSK) modulation technique. The P(Y)-code is transmitted on both the L1 and L2 frequencies as a 10.23 MHz signal using the same modulation. In electronics, there are ⇑ Corresponding author.
E-mail address: [email protected] (L. Yang).
differences between the used components and signal structure for the different types of observations (Leick et al., 2015). Therefore, there is no doubt that the effects of space environment and hardware on the (C/A) and P codes are different. Due to effects, there is a bias between (C/A) and P codes and it is named as differential code biases (DCB) from C1 and P1 observations on one frequency. The bias of DCB (C1-P1) contains receiver and satellitedependent parts and is estimated with the geometry-free code combination form with C1 and P1 observations. The time-variant characteristic of the satellite-dependent part is analyzed based on the estimated series in Gao et al. (2001). The time-varying characteristic for the most common bias of the DCB (P1–P2) between the P1 and P2 observations (Sardon et al., 1994; Goodwin and
https://doi.org/10.1016/j.asr.2019.10.018 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Breed, 2001; Otsuka et al., 2002) was analyzed in Li et al. (2019) and the main periods of 12, 8, 6, 4, 4.8 and 2.67 h are noticed. But, the current International GNSS Service (IGS) (Dow et al., 2009; Li et al., 2010) still provides the DCB products with a daily or monthly constant. The precise point positioning (PPP) is realized based on the single-receiver dual-frequency observations and the IGS products of the satellite clock error and orbit (Zumberge et al., 1997). In the satellite clock estimating, the reference observations of the ionosphere-free phase (L1/L2) and code (P1/P2) combinations are used in the IGS products computing. Li et al. (2017) shows that the satellite-dependent biases in the code and phase observations are absorbed by the estimated, reparameterized satellite clock error. Thus, the code bias not only affects the accuracy of the code observation but also the highquality float ambiguity solution. To improve the accuracy of the PPP positioning, the satellite DCB (C1-P1) bias should be corrected according to its characteristic at the user side, which observes the C1 observation. It is obvious that the current service of the satellite DCB (C1-P1) with the constant characteristic is bad for improvement of the positioning accuracy and it neglects the effect of the timevarying part of the satellite DCB (C1-P1) on the positioning results. Considering these discussions, the time-varying and periodic characteristics of the satellite DCB (C1-P1) is studied and the estimated DCB (C1-P1) series is modeled with the presented mathematical function to enhance the service ability (Li et al., 2018). The application of the estimated and modeled bias in positioning is implemented to validate its effect on the positioning. Finally, the research findings and outlooks are summarized. 2. Estimation of the time-varying DCB (C1-P1) The method for estimating the time-varying DCB (C1P1) series is presented. Based on the time-varying and the periodic characteristics of the DCB (C1-P1), it is modeled and its application in the single point positioning (SPP) is discussed to validate the effect of the time-varying part
on positioning and to improve the service of the DCB (C1-P1) product. 2.1. Estimation method for the satellite-dependent DCB (C1-P1) For a GPS user, the raw C1 and P1 observations are written as: P 1 ¼ q þ bPr bsP þ dr ds I sr T sr þ xP
ð1Þ
C 1 ¼ q þ bCr bsC þ dr ds I sr T sr þ xc
where q is the satellite-to-receiver range, T sr is tropospheric delay, I sr is the slant ionosphere delay of C1 and P1 observations, dr and ds are the receiver and satellite clock errors in meter; bCr and bPr are the receiver hardware delay biases of P1 and C1 observations; bCs and bPs are the satellite hardware delay biases of P1 and C1 observations; xP is the noise of P1 and xc is the noise of C1 observation. The DCB (C1-P1) is generally estimated with the geometryfree code combination form with C1 and P1 observations: C 1 P 1 ¼ bCr bPr bsC bsP þ xC xP s
¼ DCBðC1 P1Þr DCBðC1 P1Þ þ xC xP ð2Þ Eq. (2) indicates that it is impossible to separate the receiver or satellite DCB (P1-C1) from their combination. In general, the satellite-differenced algorithm is used by selecting a reference satellite. However, the visible time span of each satellite during one day is limited. As a result, the proper reference satellites need to be selected for different periods of time, which complicates the computation. According to S-system theory and rank-deficiency free network adjustment theory, a reference constraint can be added to provide a datum, forming a full-rank model (Teunissen, 1985, Odijk et al., 2016). In this data processing, the reference is determined: Pd j ð3Þ j¼1 DCBðC1 P1Þr ¼ 0 where DCBðC1 P1Þjr is the receiver DCBðC1 P1Þ at observation station j; d is the number of the observation stations. When this reference is introduced, the parameterization equation of the DCB (C1-P1) is expressed as: 2
e1d I dd et1
2 3 r 3 0 ðDCBðC1 P1Þ Þ1 6 7 6 ðC1 P1Þ 7 .. . 1;1 7 6 7 6 7 6 7 6 6 ðDCBðC1 P1Þr Þd 7 6 7 ... 01t 6 7 6 7 ¼ s 7 6 ðC1 P1Þ 7 ð DCB ð C1 P1 Þ Þ ed1 I tt 6 6 6 1;t 7 17 6 7 6 7 4 5 4 5 ... .. . s ðC1 P1Þd;t ðDCBðC1 P1Þ Þt ð4Þ
Fig. 1. Distribution of the 240 IGS stations tracking GPS signals. Red dots are stations for estimated DCB (C1-P1). Blue dots are SPP stations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where I dd and I tt are identity matrix, e1d is a vector with each element is 1, t is the satellite number. The estimator of least-square adjustment is implemented to compute the epoch-wise DCB (C1-P1). In processing, the elevationdependent weight is used:
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
H. Li et al. / Advances in Space Research xxx (2019) xxx–xxx
( wm ¼
30 hm 90
1
2sinðhm Þ 10 hm < 30
) ð5Þ
where hm is the satellite elevation at m epoch. The references of the estimated satellite DCB (C1-P1) for each satellite are consistent, even if the different constraint is used. This reference can be absorbed by the receiver clock in positioning so that the positioning result is not affected.
2.2. Modeling and application of the estimated satellitedependent DCB (C1-P1) Currently, the satellite DCB (C1-P1) is serviced by IGS with a daily or monthly constant. It is obvious that the time-varying characteristic of the satellite DCB (C1-P1) is not considered. According to the estimated DCB (C1-P1) series and its periodic variation, the DCB (C1-P1) series is modeled: P DCBðC1 P1Þs ðtÞ ¼ a þ e t þ y i 5i¼1 sinð2p t þ hi Þðt ¼ 0 24hÞ Ti ð6Þ
where a is a constant; e is the coefficient of linear term; i is the order of the harmonics; T i is the period; hi is initial phase offset, and y i is the amplitude. The satellitedependent biases are absorbed by the estimated, reparameterized satellite clock error in the satellite clock estimating (Li et al., 2017). Thus, the satellite DCB (P1-C1) is very important for computing the reparameterized satellite clock of the C1 observation and its corresponding combinations. The satellite clock error of the ionosphere-free code combination formed with C1 and P2 should be: f2
dsC1 =P 2 ¼ ds1;2 þ f 2 f1 2 DCBðC1 P1Þ 1
2
s
ð7Þ
3
where dsC1 =P 2 is the estimated, reparameterized satellite clock error of L1/L2 and C1/P2 and the ds1;2 is the IGS satellite clock product. 3. Data processing The twelve-month (1 Jan-31 Dec 2018) GPS data set from 120 IGS stations is processed to analyze the timevarying and periodic characteristic of the code bias and validate the presented approach. Data is sampled at 30 s and cut-off elevation is set to 10 degrees. Three days, 1–3 Jan 2018 of GPS data from 120 IGS stations is processed to analyze the performances of estimated time-varying DCB (C1-P1) in single point positioning (SPP). The IGS stations are shown in Fig. 1. 3.1. Estimated DCB (C1-P1) According to the presented method, one-year single-day data is processed and the one-year single-day time-varying DCB (C1-P1) series are estimated. The estimated results of the series for 1–3 Jan 2018 are shown in Fig. 2. From the Fig. 2, it is observed that the DCB (C1-P1) series varies with time and has the significant periodic variations. The significant periodic variations are not involved, although the time-varying characteristics of the DCB (C1-P1) series are validated in Gao et al. (2001). One-year single-day DCB (C1-P1) values estimated with the presented method for each satellite are analyzed using a FFT (fast Fourier transformation). The periods for the satellites of G03, G07, G16 and G23 are shown in Fig. 3. Fig. 3 shows that the DCB (C1-P1) values have the significant periods of 12, 8, 6, 4, 3.4286 h. As to the significant periodic variations, it is explained that the satellite hardware is affected by the periodic environment including the sun illumination (Li et al., 2013). Based on the noticed periods and Eq. (6),
Fig. 2. Series (unit: m) of the single-day DCB (C1-P1) for G01 and G10 with G24 as reference satellite, for the time 1–3 Jan 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 3. Periods (unit: h) of the single-day DCB (C1-P1) for G03, G07, G16 and G23, for the time 1 Jan-31 Dec 2018.
One-year single-day DCB (C1-P1) is modeled and the corresponding modeling parameters are analyzed. One-year single-day amplitudes (y i ði ¼ 1 5Þ) and initial phase offsets (hi ði ¼ 1 5Þ) of the model for satellites of G03 and G07 are illustrated in Figs. 4 and 5, respectively. The results in Figs. 4 and 5 show that the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time, which is not similar to the initial phase offsets (hi ði ¼ 1 5Þ). The initial phase offsets demonstrate the characterization of linear variation and these variations are very evident. The irregular variation of the amplitudes can be explained that the GPS code measurement is affected by the space environment including the multipath effect. The estimated results of the amplitudes and initial phase offsets for all satellites show the same characteristics.
3.2. Validation in positioning It is well known that some GNSS receivers just tracked C1 measurement. Thus, the correction of DCB (C1-P1) should be considered and implemented. The constant part of DCB (C1-P1) is considered in the current data processing. It is obvious that this correction neglects the effect of the time-varying part of the satellite DCB (C1-P1) on positioning results. To validate effect of the time-varying part of the satellite DCB (C1-P1) on positioning, the GPS data from 120 IGS stations is processed in SPP mode and the settings for the SPP processing are shown in Table 1. In data processing, two methods of #1 and #2 are used and implemented. In method of #1, the constant part of DCB (C1-P1) from IGS is used to correct the C1 observation,
Fig. 4. Amplitudes (y i ði ¼ 1 5Þ) (unit: m) of the model for satellites of G03 and G07, for the time 1 Jan-31 Dec 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 5. Initial phase offsets (hi ði ¼ 1 5Þ) of the model for satellites of G03 and G07, for the time 1 Jan-31 Dec 2018. Table 1 Settings for the SPP processing.
Measurements
Corrections
Parameters
Model
Settings
Ionosphere-free code combination Adjustment Weighting DCB(C1-P1) Tides corrections PCV Relativity Station coordinates Troposphere Receiver clock error
Least square filter Elevation-dependent function #1 and #2 Solid tide and Ocean tide correction Absolute IGS 14 correction mode Corrected Estimated Correction: Saastamoinen model Solved for at each epoch as white noise
while the constant part of DCB (C1-P1) from IGS and the fitted time-varying part of DCB (C1-P1) are used in #2. The RMSs of the difference between the SPP results of the two methods and the IGS results are computed for the three directions of North, East and Up. It is difficult to assess the performance superiority of the different methods via the SPP results at each direction. Thus, the results are assessed using the three-dimensional positioning errors: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ TDP ¼ e2North þ e2East þ e2Up where eNorth is the error in the North direction,eEast is the error in the East direction and eUp is the error in the Up direction. The three-dimensional positioning results for the 120 IGS stations are shown in Fig. 6. The threedimensional positioning difference between the #1 and #2 are Fig. 7. The three-dimensional positioning results in Figs. 6 and 7 indicate that improvement can be obtained when the time-varying part of the DCB (C1-P1) is considered and corrected. The average three-dimensional positioning improvement for three days is about 3 cm. These improvements further validate that the time-varying characteristics of the satellite DCB (C1-P1) and the time-
varying part should be corrected in data processing. These also show that the time-varying part of the satellite DCB (C1-P1) should be serviced by IGS. 3.3. Prediction analysis The SPP results show that the correction of the timevarying part of the satellite DCB (C1-P1) can improve the positioning accuracy. In real-time processing, the satellite DCB (C1-P1) products should be obtained by prediction. Therefore, the prediction performance of the satellite DCB (C1-P1) containing the constant and the time-varying parts is discussed. The satellite DCB (C1P1) from single-day (1 Jan 2018) estimates is used to predict the values for the next days (2–6 Jan 2018). The single-day (1 Jan 2018) values are used to compute the modeling coefficients of each satellite and the computed coefficients of Eq. (6) are used to extrapolate the values of the next days (2–6 Jan 2018). The RMSs of the difference between the extrapolated and fitted results are illustrated in Fig. 8 and the means of 5-day single-day correction rates for each satellite are shown in Fig. 9. The correction rate is defined as:
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
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Fig. 6. The three-dimensional positioning results (unit: m) for 120 IGS stations, for the time 1 Jan 2018 (top), 2 Jan 2018 (middle) and 3 Jan 2018 (bottom).
Fig. 7. The three-dimensional positioning results difference (unit: m) between the #1 and #2 for 120 IGS stations, for the time 1–3 Jan 2018.
Fig. 8. The RMSs of the predicted results, for the time 2–6 Jan 2018.
Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018
H. Li et al. / Advances in Space Research xxx (2019) xxx–xxx
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Fig. 9. The Correction rate (%) for the predicted DCB (C1-P1), for the time 2–6 Jan 2018. E RMS D cr ¼ RMSRMS 100% E
ð9Þ
where RMSD is the RMS for the differences between the predicted and fitted values and the RMSE is the RMS of the fitted values. Fig. 8 indicates that the accuracy of the predicted DCB(C1-P1) containing the constant and timevarying parts is better than 4.0 cm. Fig. 9 shows that the predicted results for all satellites reach more than 40% accuracy and the average prediction accuracy can reach 70%.
real-time service of the satellite DCB (C1-P1). The accuracy of the predicted results is better than 4.0 cm. The correction rates for all satellites reach more than 40% accuracy and the mean prediction accuracy can reach 70%. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
4. Conclusion and discussion Acknowledgments The DCB (C1-P1) is the bias between the C1 and P1 observations. The time-varying characteristic of the satellite DCB (C1-P1) and the effect of its time-varying part on the positioning are not discussed in the current studies, although it is time-varying. Considering these, the method for estimating the satellite DCB (C1-P1) series is presented and the estimated series is analyzed using one-year singleday estimates. The estimated single-day DCB (C1-P1) results show the time-varying characteristic of the satellite DCB (C1-P1). One-year single-day satellite DCB (C1-P1) results show that the satellite DCB (C1-P1) varies with the evident periods of the 12, 8, 6, 4, 3.4286 h. Based on the noticed periods, one-year single-day satellite DCB (C1-P1) results are modeled with a harmonics-based function. The results of the modeling coefficients show that the initial phase offsets (hi ði ¼ 1 5Þ) have the characterization of linear variation and these variations are very evident and the amplitudes (y i ði ¼ 1 5Þ) of each satellite irregularly vary with time. The time-varying satellite DCB (C1-P1) is used in single point positioning to validate its effect on the positioning. The SPP results show that the three-dimensional positioning results can be improved, when the time-varying part of the satellite DCB (C1-P1) is corrected. The three-dimensional positioning results for the 120 IGS stations show that the average threedimensional positioning improvement for three days is about 3 cm. The improvement further advises that the time-varying part of the satellite DCB (C1-P1) should be considered and corrected in GPS data processing. These also indicate that the time-varying part of the satellite DCB (C1-P1) should be estimated and serviced by IGS. The prediction analysis is implemented to provide the
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Please cite this article as: H. Li, J. Xiao and L. Yang, Modeling and application of the time-varying GPS differential code bias between C1 and P1 observations, Advances in Space Research, https://doi.org/10.1016/j.asr.2019.10.018