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... Triple-frequency combination is a generalization of the dual-frequency combination L4, which is free of geometric parameters and widely used in cycle slips detection and ionospheric monitoring (Simsky 2006). Moreover, the first-order ionospheric delays can also be removed by a simple linear combination of triple-frequency carrier phases (Montenbruck et al. 2010;Hauschild 2017): ...

... Assuming each carrier phase has the same measurement noise (e.g., 0 = 1 = 2 = 3 = 1 cm ), the normalizing condition 2 1 + 2 2 + 2 3 = 1 ensures that the noise of the triple-frequency combination will match that of the individual carrier phases (i.e., M = 1 cm ; Montenbruck et al. 2010;Hauschild 2017). For comparison of different linear combinations, we evaluate the snow depth retrieval performance of all possible triple-frequency phase combinations of Galileo by simulation tests. ...

... In most cases, there is no dominant peak in the frequency domain. As investigated by Montenbruck et al. (2010) and Hauschild (2017), for a specific triple-frequency combination, the linear combination is dominated by signals which have proximate frequencies and similar weights, while the carrier phase multipath error of the other signal is strongly attenuated. For different combinations, the carrier phase multipath errors and corresponding PSD are determined by the wavelengths of each of the three carriers. ...

Snow is an important water resource and plays a critical role in the hydrologic cycle. Accurate measurements of snow depth are needed by scientists to set up a more refined meteorology–hydrology model. Recently, the Global Navigation Satellite System Reflectometry (GNSS-R) has been developed and applied for snow depth monitoring, with low cost and high resolution. We propose an improved snow depth retrieval method using a combination of GNSS triple-frequency carrier phase. The topographic feature of the reflecting surface is considered for estimating the snow depth by using the density-based spatial clustering of applications with noise algorithm and normalization method. Observables from the GNSS station in Alaska, USA, are used to monitor snow depth and compared with the ground-truth measurements. Compared with the traditional triple-frequency snow depth retrieval method, the new approach has better performance for Galileo and BDS. The RMSE of the snow depth estimate reduces by nearly 40%, and the correlation coefficient increases from 0.93 to 0.97 for Galileo and from 0.91 to 0.95 for BDS, respectively. The research findings show no notable deviations on snow depth average estimation between Galileo and BDS observations compared to the GPS ones. Moreover, the solution with the proposed method results in improving spatial resolution due to the increasing number of satellites and better azimuth coverage.

... However, with the widespread use of triplefrequency observations, the impact of periodic variation in satellite phase hardware delay on the triple-frequency data is becoming significant [10,11]. Montenbruck et al. [12] found that the carrier phase observations of L1, L2, and L5 of the GPS have an inconsistency of 20 cm, which was labeled as the inter-frequency clock bias (IFCB). Table 1. ...

... Remark PRN GPS Block IIF (12) G01, G03, G06, G08, G09, G10, G24, G25, G26, G27, G30, G32 Block III (5) G04, G11, G14, G18, G23 ...

Multi-frequency observations are now available from GNSSs, thereby bringing new opportunities for precise point positioning (PPP). However, they also introduce new challenges, such as inter-frequency clock bias (IFCB) between the new frequencies and the original dual-frequency observations due to triple-frequency observations, which severely impact the PPP. In this paper, we studied the estimation and correction methods of uncombined inter-frequency clock bias of GPS, BDS-3, and Galileo, analyzed the time-varying characteristics and short-term stability of IFCB, and analyzed the influence of IFCB on the positioning of the GPS, BDS-3, and Galileo, based on a triple-frequency un-differential non-combined PPP model. The obtained results show that the amplitude of Block IIF satellites of the GPS can reach up to 10–20 cm, and the IFCB in BDS-3, Galileo, and GPS Block III satellites can be neglected. After correction by IFCB, the 3D positioning accuracy of the GPS triple-frequency PPP was 1.73 cm and 4.75 cm in the static and kinematic modes, respectively, while the convergence time was 21.64 min and 39.61 min. Compared with the triple-frequency GPS PPP without any correction with IFCB, the static and kinematic 3D positioning accuracy in this work was improved by 27.39% and 17.34%, and the corresponding convergence time was improved by 10.55% and 15.22%, respectively. Furthermore, the delayed IFCB was also used for positioning processing, and it was found that a positioning performance comparable to that of the same day can be obtained. The standard deviation of IFCB for a single satellite was found to be no more than 1 cm, when the IFCB value of a neighboring day was subtracted from the IFCB value of same day, which proves the short-term stability of IFCB.

... Despite the benefits of multi-frequency signals, observations with more than two frequencies will introduce an apparent inconsistency among different carrier phase observations (Montenbruck et al. 2010). This inconsistency was B Yidong Lou ydlou@whu.edu.cn 1 difference IFCB. ...

... Since differential phase wind-up effects in the tri-carrier combination amount to 2 mm, it can therefore be ignored for IFCB analysis (Montenbruck et al. 2012). The ionosphere-free and geometry-free combination of triple-frequency phase observations at epoch t can be defined as (Montenbruck et al. 2010): ...

Triple-frequency observations will introduce an inter-frequency clock bias (IFCB) between the new frequency and the original dual-frequency observations. It has been verified that satellite IFCB can reach dozens of centimeters and several centimeters for GPS Block IIF satellite and BDS satellite, respectively. The existence of satellite IFCB will significantly affect undifferenced triple-frequency data processing. Based on 4-year data collected from 80 globally distributed stations, the long-term characteristics of IFCB coefficients obtained by using harmonic analysis have been studied.
The results demonstrate that the coefficients of IFCB periodic model cannot be well fitted only by using sun elevation angle. Also, coefficients have obvious periodic characteristics and their periods differ among different satellites. Thus, a new linear-plus-periodic model is proposed to fit the long-term coefficients. Then, IFCB empirical correction models for 12 GPS Block IIF satellites and BDS GEO and IGSO satellites are built. In order to validate the correction model, IFCB standard deviation (STD), triple-frequency precise point positioning (PPP) and undifferenced extra-wide-lane (EWL) ambiguity resolution are employed. The results based on more than 4-year observations show that, with correction model applied, the average IFCB STD decreases by about 65.5% and 45.5% for GPS and BDS satellites, respectively. Compared to triple-frequency PPP without IFCB correction, triple-frequency PPP results with IFCB correction show that Up, North and East components accuracy are improved by 12.3%, 16.0% and 13.2%, respectively. Besides, IFCB correction will greatly improve the consistence of EWL fractional cycle bias among different stations and improve the success rate of EWL ambiguity resolution.

... Despite the fact that the satellite flawlessly passed all in-orbit tests and perfectly meets all spacecraft and navigation payload specifications, two surprising aspects were noted by the scientific community during the commissioning phase. First, an apparent inconsistency of the L 1 , L 2 , and L 5 carrier phase measurements at the 10-cm level was identified based on a geometry-and ionosphere-free linear combination of triplefrequency observations shortly after the permanent L5 activation on June 28, 2010 (Montenbruck et al. 2010a). This inconsistency can best be understood by a thermally dependent inter-frequency bias. ...

... Obviously, both the L1/L2 clock offset and the inter-frequency bias are subject to thermal variations with peak positive values related to a cooling of equipment and peak negative values reached after equipment warm-up. This is consistent with observations reported after first activation of the L5 payload, where a gradual decrease of the triplecarrier combination was encountered during the early days of operation (Montenbruck et al. 2010a). ...

The Block IIF satellites feature a new generation of high-quality rubidium clocks for time and frequency keeping and are the first GPS satellites transmitting operational navigation signals on three distinct frequencies. We investigate apparent clock offset variations for the Block IIF-1 (SVN62) spacecraft that have been identified in L1/L2 clock solutions as well as the L1/L5-minus-L1/L2 clock difference. With peak-to-peak amplitudes of 10–40 cm, these variations are of relevance for future precision point positioning applications and ionospheric analyses. A proper characterization and understanding is required to fully benefit from the quality of the new signals and clocks. The analysis covers a period of 8 months following the routine payload activation and is based on GPS orbit and clock products generated by the CODE analysis center of the International GNSS Service (IGS) as well as triple-frequency observations collected with the CONGO network. Based on a harmonic analysis, empirical models are presented that describe the sub-daily variation of the clock offset and the inter-frequency clock difference. These contribute to a better clock predictability at timescales of several hours and enable a consistent use of L1/L2 clock products in L1/L5-based positioning.

... While triple-frequency signals have many benefits, new bias has also been introduced. Montenbruck et al. [12] first demonstrated the existence of a bias between L1/L2 and L1/L5 in ionosphere-free (IF) combination, including a periodic line bias between the satellite and the signal, based on geometry-free and ionosphere-free (GFIF) phase combination. The inconsistency of L1/L2/L5 was defined as inter-frequency clock bias (IFCB) [13]. ...

The currently available triple-frequency signals give rise to new prospects for precise point positioning (PPP). However, they also bring new bias, such as time-varying parts of the phase bias in the hardware of receivers and satellites due to the fact that dual-frequency precise clock products cannot be directly applied to triple-frequency observation. These parameters generate phase-based inter-frequency clock bias (PIFCB), which impacts the PPP. However, the PIFCBs of satellites are not present in all GNSSs. In this paper, various IF1213 PPP models are constructed for these parts, namely, the triple-frequency PIFCB (TF-C) model with PIFCB estimation, the TF inter-frequency bias (IFB) (TF-F) model ignoring the PIFCB, and the TF-PIFCB-IFB (TF-CF) model with one system PIFCB estimation. Additionally, this study compares these IF1213 PPP models with the dual-frequency ionosphere-free (DF) model. We conducted single system static PPP, dual-system static and kinematic PPP experiments based on BDS/GPS observation data. The GPS static PPP experiment demonstrates the reliability of the TF-C model, as well as the non-negligibility of the GPS PIFCB. The BDS static PPP experiment demonstrates the reliability of the TF-F and TF-CF models, and that the influence of the BDS-2 PIFCB can be neglected in BDS. The BDS/GPS PPP experimental results show that the third frequency does not significantly improve the positioning accuracy but shortens the convergence time. The positioning accuracy of TF-C and TF-CF for static PPP is better than 1.0 cm, while that for kinematic PPP is better than 2.0 cm and 4.0 cm in the horizontal and vertical components, respectively. Compared with the DF model, the convergence time of the TF-C and TF-CF models for static PPP is improved by approximately 23.5%/18.1%, 13.6%/9.7%, and 19.8%/12.1%, while that for kinematic PPP is improved by approximately 46.2%/49.6%, 33.5%/32.4%, and 35.1%/36.1% in the E, N and U directions, respectively. For dual-system PPP based on BDS/GPS observations, the TF-C model is recommended.

... That is, an additional bias on the third-, quad-, or more frequency observations need to be processed, which is called the inter-frequency clock bias (IFCB) [13,14]. Montenbruck et al. found that the changing characteristics of the GPS IFCB time series were related to the relative position of the satellite-Earth-Sun by analyzing the law of period changes of the GPS IFCB, and the peak amplitudes reached over a dozen centimeters [13,15]. As for BeiDou-2, small bias variations were also recognized, but they were generally confined to peak amplitudes of approximately 2 cm. ...

The time-varying biases within carrier phase observations are integrated into satellite clock offset parameters for precise clock estimation. Consequently, when the precise satellite clock bias is applied to the third frequency observation for precise point positioning (PPP), a new type of inter-frequency clock bias (IFCB) with satellite dependence should be noticed. If the IFCB is estimated together with the receiver coordinates, tropospheric wet delay, ambiguity and other parameters, it will increase the computational burden and lead to more time consumption. In order to solve this problem, the IFCB of GPS Block IIF satellites were estimated using 162 global uniformly distributed Multi-GNSS Experiment (MGEX) stations. By analyzing the time-varying characteristic of each satellite IFCB and combining the lag characteristics of the final ephemeris products, a modeling method of short-term IFCB prediction based on the epoch-by-epoch sliding Pearson autocorrelation function is proposed. The feasibility of this method was verified through the Student’s t-distribution, comparison with the measured IFCB, the posteriori residual of the third frequency carrier phase and the kinematic/static PPP solutions. The results showed that since the IFCB period was not a complete 24 h, the difference in the IFCBs time series on different days was increasingly significant with the passage of lag time, and the correlation constantly decreased. The peak-to-peak amplitudes of the IFCB difference reached 1.13, 3.44, 6.86 and 11.25 cm when the lag time was 1, 9, 19 and 29 days, respectively. In addition, based on the lag characteristic of final precise ephemerides released by the International GNSS Service (IGS) analysis centers, the prediction accuracy of the IFCB was evaluated with a time lag of 7 days. The root mean square of the posteriori residuals at the third-frequency observation decreased by approximately 51.3% compared to that without considering for IFCB correction. The triple-frequency uncombined PPP in the horizontal and vertical directions improved by approximately 33.2% and 17.2% for the static PPP solutions and 50.2% and 39.7% for the kinematic PPP solutions, respectively. In general, the accuracy and convergence time of the triple-frequency uncombined PPP were equivalently improved when the predicted IFCB and the measured IFCB were used.

... Although multi-frequency PPP has many advantages, some problems arise from the integration of multiple signals. Montenbruck et al. (2010) assessed that there was a time-varying and frequency-dependent bias within carrier phase measurements. Moreover, Pan et al. (2017b) showed that the bias was caused by a frequency-dependent phase hardware delay at the satellite. ...

Due to inter-frequency clock bias (IFCB), the dual-frequency precise clock products cannot be directly applied to BDS-2 triple-frequency precise point positioning (PPP). In this study, the datasets collected at 195 ground tracking stations for a whole year are employed to investigate the modeling and prediction of BDS-2 IFCB. The modeling periods of IFCB are consistent with orbital repeat periods, namely a week for medium earth orbit (MEO) satellites and a day for geostationary orbit (GEO) and inclined geosynchronous orbit (IGSO) satellites. The harmonic analysis results show that the IFCB of MEO satellites has seven significant periods of (168, 84, 56, 42, 33.6, 28, 12.9) h, while there are six obvious periods of (24, 12, 8, 6, 4.8, 4) h for GEO and IGSO satellites. Two function models composed of a linear term and a harmonic term with seven orders for MEO satellites and six orders for GEO and IGSO satellites are established to describe the IFCB variations, and the model coefficients are fitted by least squares. The IFCB modeling accuracies are 4 mm for MEO satellites and 2 mm for GEO and IGSO satellites. The established model can predict the IFCBs with single-day prediction accuracies of 5.0 and 3.9 mm for GEO and IGSO satellites and single-week prediction accuracies of 5.0 mm for MEO satellites, respectively. After applying these predicted IFCBs to BDS-2 triple-frequency PPP, the positioning accuracies are improved by 14–29%.

... The code IFB at the frequency which is used for satellite clock estimation can be absorbed into the ionospheric delay, while the others are absorbed by setting an individual receiver clock parameter for each one. And the phase IFB can be absorbed into ambiguity (Montenbruck, et al. 2010;Pan et al., 2019). Another ...

PPP-RTK which takes full advantages of both Real-Time Kinematic (RTK) and Precise Point Positioning (PPP), is able to provide centimeter-level positioning accuracy with rapid integer Ambiguity Resolution (AR). In recent years, with the development of BeiDou Navigation Satellite System (BDS) and Galileo navigation satellite system (Galileo) as well as the modernization of Global Positioning System (GPS) and GLObal NAvigation Satellite System (GLONASS), more than 140 Global Navigation Satellite System (GNSS) satellites are available. Particularly, the new-generation GNSS satellites are capable of transmitting signals on three or more frequencies. Multi-GNSS and multi-frequency observations become available and can be used to enhance the performance of PPP-RTK. In this contribution, we develop a multi-GNSS and multi-frequency PPP-RTK model, which uses all the available GNSS observations, and comprehensively evaluate its performance in urban environments from the perspectives of positioning accuracy, convergence and fixing percentage. In this method, the precise atmospheric corrections are derived from the multi-frequency and multi-GNSS observations of a regional network, and then disseminated to users to achieve PPP rapid AR. Furthermore, a cascade ambiguity fixing strategy using Extra‐Wide‐Lane (EWL), Wide-Lane (WL) and L1 ambiguities is employed to improve the performance of ambiguity fixing in the urban environments. Vehicle experiments in different scenarios such as suburbs, overpasses, and tunnels are conducted to validate the proposed method. In suburbs, an accuracy of within 2 cm in the horizontal direction and 4 cm in the vertical direction, with the fixing percentage of 93.7% can be achieved. Compared to the GPS-only solution, the positioning accuracy is improved by 87.6%. In urban environments where signals are interrupted frequently, a fast ambiguity re-fixing can be achieved within 5 s. Moreover, multi-frequency GNSS signals can further improve the positioning performance of PPP-RTK, particularly in the case of small amount of observations. These results demonstrate that the multi-frequency and multi-GNSS PPP-RTK is a promising tool for supporting precise vehicle navigation.

... In this study, we will develop the UC model for real-time BDS-3 clock estimation with multi-frequency signals and assess the consistency of BDS-3 multi-frequency signals with the GFIF (Geometry-Free Ionosphere-Free) combination [18]. Since receivers with the B2b signal track capability are quite limited for the current IGS network, the following models and experiments will be restricted to the other four signals, i.e., B1I/B3I/B1C/B2a. ...

The global system of BDS (BeiDou Navigation Satellite System), i.e., BDS-3, is characterized with a multi-frequency signal broadcasting capability, which was demonstrated as beneficial for GNSS (Global Navigation Satellite System) data processing. However, research on real-time BDS-3 clock estimation with multi-frequency signals is quite limited, especially for the new B1C and B2a signals. In this study, we developed models for BDS-3 multi-frequency real-time data processing, including the uncombined model for clock estimation and the GFIF (Geometry-Free Ionosphere-Free) combined model for IFCB (Inter-Frequency Clock Bias) determination. Based on the models, simulated real-time numerical experiments with about 80 global IGS (International GNSS Service) network stations are conducted for validation and analysis. The results indicate that: (1) the uncombined model with multi-frequency signals can achieve comparable accuracy with the traditional dual-frequency IF model in terms of clock estimation, and the double-differenced clock STDs (Standard Deviations) are generally less than 0.05 ns with post-processed clocks as a reference; (2) unlike the B1C and B1I/B3I signals, the satellite IFCBs generated from multi-frequency clock estimation show apparent temporal variations for B2a and B1I/B3I signals, further investigation with GFIF models confirm the variations mainly result from the errors of receiver antenna corrections. Therefore, we addressed the feasibility of the uncombined model and the importance of accurate antenna information in the multi-frequency data processing.

... Since the L4 and triple-frequency observables are considered to be the combination of multipath errors for each carrier, it can be recognized that the corresponding spectral peak frequencies are dominated by multiple signals of different frequencies (Montenbruck et al. 2010;Hauschild 2017). Such complex signals comprising several periodic oscillation components contribute to the difficulty associated with dynamic correction and quality control. ...

Many types of geophysical and meteorological studies rely on the accurate measurement of sea level. Ground-based global navigation satellite system reflectometry (GNSS-R) has demonstrated strong potential for absolute sea level retrieval within a global reference frame at a low cost with a high resolution. In this paper, a combination of single-frequency code and a carrier phase (code-minus-carrier phase), referred to as CMC observable, is used to estimate sea level changes for the first time. The CMC method is free of geometric delays, while the ionospheric terms can be mitigated by adopting the moving average filtering method. The relationship between the antenna height and the spectral peak frequency of the CMC time series is described as a linear model. In addition, we combine CMC and SNR measurements from four constellations (GPS/GLONASS/Galileo/BDS) by adopting the robust regression strategy with iteratively reweighted least squares. GNSS datasets collected from three sites with different tidal amplitudes were used to monitor sea level changes and compare them with tide gauge records. The research findings show that the root-mean-squared errors (RMSEs) of the CMC method across three study sites are between 6.95 and 11.39 cm, and the corresponding correlation coefficients exceed 0.99, which is comparable to that of the current SNR method. The sea level retrieval accuracy can be further improved by combining the CMC and SNR observations with RMSE between 3.78 and 9.06 cm. We also compare the performance of GNSS-R retrieval with different dynamic models. Because vertical acceleration has been considered, the second-order dynamic model reports a better performance than the first-order dynamic model.

... In (2) and (3), g 1i = f 2 1 / f 2 i , g 23 = g 13 /g 12 ; d t r and d t s are the transformed receiver and satellite clock errors, respectively; I j 1 is the transformed slant ionospheric delay at the first frequency and I j i = g 1i I j 1 ; N j i is the transformed float ambiguity; DCB r,12 denotes the receiver differential code bias (DCB), defined as the difference between B r,1 and B r,2 ; and DCB In addition, an extra receiver and satellite-dependent code bias, termed the interfrequency bias (IFB), is added for L5 code observations [17]. Moreover, satellite-dependent inter-frequency clock biases (IFCBs), defined as the differences between precise satellite clock errors estimated based on L1/L2 and L1/L5 IF combinations [18], are added for L5 phase observations [14]. Then, the new code noise ε P j r,i can be expressed as ...

We investigate the estimation of the fractional cycle biases (FCBs) for GPS triple-frequency uncombined precise point positioning (PPP) with ambiguity resolution (AR) based on the IGS ultra-rapid predicted (IGU) orbits. The impact of the IGU orbit errors on the performance of GPS triple-frequency PPP AR is also assessed. The extra-wide-lane (EWL), wide-lane (WL) and narrow-lane (NL) FCBs are generated with the single difference (SD) between satellites model using the global reference stations based on the IGU orbits. For comparison purposes, the EWL, WL and NL FCBs based on the IGS final precise (IGF) orbits are estimated. Each of the EWL, WL and NL FCBs based on IGF and IGU orbits are converted to the uncombined FCBs to implement the static and kinematic triple-frequency PPP AR. Due to the short wavelengths of NL ambiguities, the IGU orbit errors significantly impact the precision and stability of NL FCBs. An average STD of 0.033 cycles is achieved for the NL FCBs based on IGF orbits, while the value of the NL FCBs based on IGU orbits is 0.133 cycles. In contrast, the EWL and WL FCBs generated based on IGU orbits have comparable precision and stability to those generated based on IGF orbits. The use of IGU orbits results in an increased time-to-first-fix (TTFF) and lower fixing rates compared to the use of IGF orbits. Average TTFFs of 23.3 min (static) and 31.1 min (kinematic) and fixing rates of 98.1% (static) and 97.4% (kinematic) are achieved for the triple-frequency PPP AR based on IGF orbits. The average TTFFs increase to 27.0 min (static) and 37.9 min (kinematic) with fixing rates of 97.0% (static) and 96.3% (kinematic) based on the IGU orbits. The convergence times and positioning accuracy of PPP and PPP AR based on IGU orbits are slightly worse than those based on IGF orbits. Additionally, limited by the number of satellites transmitting three frequency signals, the introduction of the third frequency, L5, has a marginal impact on the performance of PPP and PPP AR. The GPS triple-frequency PPP AR performance is expected to improve with the deployment of new-generation satellites capable of transmitting the L5 signal.

... All GLONASS-K satellites and the last seven satellites of the GLONASS-M series can simultaneously transmit signals on the G1, G2, and G3 frequencies. However, with the wide usage of triple-frequency observations, Montenbruck et al. (2010) found that the carrier phase observations of L1, L2 and L5 have obvious inconsistency of about 10 cm, which was defined as interfrequency clock bias (IFCB) in their later study (Montenbruck et al. 2012). ...

Usually, the difference between the satellite clocks computed with L1/L2 and clocks computed with L1/L5 is defined as inter-frequency clock bias (IFCB). It is critical to correct its L5 time-variant portion in the GNSS triple-frequency precise positioning. Using two years of observations from more than 100 stations worldwide, we use the epoch-differenced method to estimate IFCB for all available 12 GPS BLOCK-IIF satellites, and analyze its short-term and long-term variations. The experimental results indicate that the IFCB variations are clearly consistent for two satellites located in the same orbital plane, which perhaps means that the variations of IFCB are dependent on the orbital plane. We found that the IFCB of each Block-IIF satellite shows repetition characteristics over two years. The annual repetition cycle of 352 days of IFCB is consistent with the GPS year 351.4 days may originate from the rotation of satellites around the earth. GPS triple-frequency uncombined PPP is carried out using 9 globally distributed MGEX stations from June 1 to 30, 2018. The experimental results indicate that compared to the PPP solutions without IFCB corrections, GPS triple-frequency PPP can achieve an accuracy of 2.2, 3.8 and 11.4 mm in the north, east, and up components after correcting IFCB, which is an accuracy increase in 31.3%, 17.4%, and 13.0%, respectively. The average RMS of the phase posteriori residuals for each frequency is also reduced significantly, especially 79.1% for L5 frequency.

... However, the improvement may not very significant since only 12 GPS block IIF and 3 GPS III satellites broadcast the multi-frequency signals currently. And it should be noted that the GPS triple-frequency measurement suffers inter-frequency clock bias (Montenbruck et al. 2010;Pan et al. 2019), which must be taken into account in the multi-frequency PPP processing. The performance of multi-frequency and multi-GNSS PPP WAR still needs further investigation. ...

As of December 2019, the core constellation of BeiDou Navigation Satellite System with global coverage (BDS-3) has been fully deployed with 24 MEO, 3 IGSO and 1 GEO satellites. In addition to the legacy B1I and B3I signals, BDS-3 satellites are capable of transmitting several new navigation signals, including B1C, B2a, B2b and B2a + b. The multi-frequency signals of BDS-3 bring new opportunities for fast ambiguity resolution (AR) of BDS precise point positioning (PPP). In this contribution, the BDS five-frequency PPP AR method was developed and the benefits of BDS-3 multi-frequency signals for precise positioning were investigated. The multi-frequency uncalibrated phase delay (UPD) products were estimated firstly using observations from 222 Multi-GNSS Experiment stations. The extra-wide-lane, wide-lane and narrow-lane UPDs of BDS-3 exhibit high stability with standard deviations less than 0.1 cycle. With the derived UPD products, five-frequency PPP AR was conducted at 12 static stations and a moving vehicle. The time to first fix of BDS PPP AR was shortened to 25.5 min with multi-frequency observations, showing improvement of 25.4% when compared to dual-frequency PPP AR. Moreover, instantaneous PPP wide-lane ambiguity resolution (WAR) can be achieved with success rate of over 90%. A fast decimeter-level accuracy can be achieved for BDS five-frequency PPP WAR with positioning accuracy improved by 68%-82% compared to PPP float solutions. The vehicle-borne experiment further demonstrates that BDS PPP WAR is potential to maintain decimeter-level positioning accuracy even in a complex environment.

... Satellite phase biases (SPB) on each carrier-phase frequency signal include the time-invariant and time-variant parts [16,17]. The time-invariant parts of satellite phase biases, which are commonly defined as uncalibrated phase delays (UPD) or just SPB, are respectively absorbed into each frequency float ambiguity parameters when conducting float UDUC-PPP [18]. ...

The new generations of global navigation satellite system (GNSS) space vehicles can transmit three or more frequency signals. Multi-frequency observations bring a significant improvement to precise point positioning ambiguity resolution (PPP AR). However, the multi-frequency satellite code and phase biases need to be properly handled before conducting PPP AR. The traditional satellite bias correction methods, for example, the commonly used differential code biases (DCB), are limited to the dual-frequency ionosphere-free (IF) case and become more and more difficult to extend to multi-GNSS and multi-frequency cases. In this contribution, we propose the observable-specific signal bias (OSB) correction method for un-differenced and uncombined (UDUC) PPP AR. The OSB correction method, which includes observable-specific satellite code and phase bias correction, can directly apply kinds of OSBs to GNSS original observation data, thus, it is more appropriate for multi-GNSS and multi-frequency cases. In order to verify the performance of multi-frequency UDUC-PPP AR based on the OSB correction method, triple-frequency GPS observation data provided by 142 Multi-GNSS Experiment (MGEX) stations were used to estimate observable-specific satellite phase biases at the PPP service end and some of them were also used to conduct AR at the PPP user end. The experiment results showed: the averaged time-to-first-fix (TTFF) of triple-frequency GPS kinematic UDUC-PPP AR with observable-specific satellite code bias (SCB) corrections could reach about 22 min with about 29% improvement, compared with that without observable-specific SCB corrections; TTFF of triple-frequency static UDUC-PPP AR with observable-specific phase-specific time-variant inter-frequency clock bias (IFCB) corrections took about 15.6 min with about 64.3% improvement, compared with that without observable-specific IFCB corrections.

... This kind of inconsistency leads to the difference of satellite clocks estimated with different observations, which is defined as IFCB. While the IFCBs of Galileo and BDS satellites show a much smaller amplitude, they are non-negligible for the GPS Block IIF satellites, which experience a peak-to-peak amplitudes of 10-40 cm (Montenbruck et al., 2010;Montenbruck et al., 2012;Li et al., 2016;Zhao et al., 2017;Pan et al., 2017;Guo and Geng, 2018;Pan et al., 2018;. In order to solve the inconsistency in GPS triple-frequency observations, three methods currently exist in general (Pan et al., 2019), namely the clock method (Guo and Geng, 2018), the IFCB method (Montenbruck et al., 2012), and the modified IFCB method (Li et al., 2016). ...

Real-time precise Global Navigation Satellite System (GNSS) satellite orbit and clock products are the prerequisite of real-time GNSS-based applications. With the modernization of Global Positioning System (GPS) constellation and construction of other GNSS constellations, the number of GNSS satellites transmitting multi-frequency signals is increasing rapidly. While the benefit of multi-frequency to Precise Point Positioning (PPP) and Precise Clock Estimation (PCE) has been evaluated by many researchers, the impact of multi-frequency on orbit and clock estimation is still limited, especially in real-time mode. Different from the dual-frequency model, the Inter-Frequency Clock Bias (IFCB) must be treated carefully in the triple-frequency model. In this contribution, the triple-frequency model for real-time GPS satellite orbit and clock estimation is derived and the influence of L5 frequency observations on real-time GPS satellite orbit and clock are analyzed. With observation data from globally distributed stations spanning January 1 to 14 of 2018, real-time GPS orbit and clock are estimated with both dual-frequency and triple-frequency observations. Numerical experiments indicate that, compared to the International GNSS Service (IGS) final orbit, the L5 frequency observations improve the consistency of the normal component with about 0.2 cm, however lead to a decrease of consistency in the radial component with about 0.6 cm; the influence on the tangential direction varies with different number of stations. The potential reason for this phenomenon may partly be attributed to the absence of accurate antenna information for L5 frequency. However, due to the tight constraint of dynamic force model on orbit, the overall differences for each component is less than 1.0 cm, which means the contribution of the third frequency observation on the orbit is limited.

... Both the code-and phase-specific IFB differ among satellites. According to Montenbruck et al. (2010), the GPS phase-specific IFB varies with time. As such, in precise point positioning (PPP) processing, the IFB cannot be absorbed by the receiver clock offset or phase ambiguity parameters. ...

The GLONASS SVNs 702K (R09), 755 (R21) and 701K (R26) satellites currently provide G1, G2 and G3 signals. The difference between satellite clocks calculated by G1/G2 and G1/G3 ionospheric-free combinations, termed inter-frequency bias (IFB), is identified. The presence of IFB limits the application of G3 signal in precise positioning. The IFB is investigated using the datasets from 70 stations with a global distribution spanning 30 consecutive days. The epoch-wise phase-specific IFB (PIFB) estimates show periodic variations with a period of eight days and an average peak-to-peak amplitude of 0.107, 0.327 and 1.663 m for the R09, R21 and R26 satellites, respectively. The daily stable code-specific IFB (CIFB) estimates also show 8-day periodic signal. The day-to-day scattering of daily stable CIFB is 0.060–0.085 m. The estimation accuracy and prediction accuracy of PIFB are 0.025 and 0.019 m, respectively, while the corresponding statistics for the daily stable CIFB are 0.452 and 0.056 m, respectively. A modified estimation approach is developed to derive the time-varying epoch-wise CIFB. The epoch-wise CIFB and PIFB shows sub-daily periodic variations with the most notable periods of 5.625 and 11.250 h, respectively. The correction rate is 32% in terms of the prediction of the time-varying part of the epoch-wise CIFB. In addition, the signal quality is assessed from such aspects as carrier-to-noise density ratio, measurement noise and multipath errors.

... This threat causes either a step or a rate of change between the code and carrier broadcast from the satellite. This threat has never been observed on the GPS L1 signal, but has been observed on SBAS geostationary satellite signals and on the GPS L5 signal (Gordon et al. 2010;Montenbruck et al. 2010). This threat is relevant in SBAS only if carrier smoothing of the code is used. ...

As Intelligent Transport Systems (ITS) become more automated and more demanding, ITS positioning integrity is becoming a key performance parameter. ITS relies on GNSS technology for absolute positioning. In order to develop efficient models and methods that can provide high levels of integrity, it is necessary to study the vulnerabilities of the GNSS-based positioning systems intended for use in ITS applications, in particular those which require positioning accuracy at the sub-metre level. These vulnerabilities are attributed to several sources and include biases and errors in the GNSS measurements, and in the corrections applied to the measurements for augmented performance, as well as those induced by the operating environment. The vulnerabilities also comprise possible anomalies that may affect each component of the system, including disturbances or disruption in the communications between the service provider and users, data latency, to name a few. In this paper a preliminary overview of possible vulnerabilities is presented for two widely-used GNSS positioning techniques envisioned for ITS applications: the Satellite-Based Augmentation System (SBAS) and low-cost RTK. Some examples are given, including the source of these errors, e.g. satellite or receiver hardware, environment, external communications, the error magnitude, temporal and spatial behaviour, their deterministic and stochastic characteristics, and their impact on estimated positions. Furthermore, some of the corresponding mathematical models that can be used to describe these vulnerabilities in the integrity monitoring algorithms are presented.

... Although the joint use of multi-frequency signals can bring great benefits, there still exist difficulties in multifrequency integrated PPP. Montenbruck et al. (2010) first identified the presence of time-, signal-and satellitedependent line biases in carrier phase observations based on a geometry-free and ionospheric-free (GFIF) carrier phase combination, namely the difference between L1/L2 and L1/L5 ionospheric-free (IF) combined carrier phase observations. In precise clock estimation (PCE), the time-varying phase biases will be grouped with satellite clock offset parameters. ...

The time-varying biases within carrier phase observations will be integrated with satellite clock offset parameters in the precise clock estimation. The inconsistency among signal-dependent phase biases within a satellite results in the inadequacy of the current L1/L2 ionospheric-free (IF) satellite clock products for the GPS precise point positioning (PPP) involving L5 signal. The inter-frequency clock bias (IFCB) estimation approaches for triple-frequency PPP based on either uncombined (UC) observations or IF combined observations within a single arbitrary combination are proposed in this study. The key feature of the IFCB estimation approaches is that we only need to obtain a set of phase-specific IFCB (PIFCB) estimates between the L1/L5 and L1/L2 IF satellite clocks, and then, we can directly convert the obtained L1/L5 IF PIFCBs into L5 UC PIFCBs and L1/L2/L5 IF PIFCBs by multiplying individual constants. The mathematical conversion formula is rigorously derived. The UC and IF triple-frequency PPP models are developed. Datasets from 171 stations with a globally even distribution on seven consecutive days were adopted for analysis. After 24-h observation, the UC and IF triple-frequency PPP without PIFCB corrections can achieve an accuracy of 8, 6 and 13 mm, and 8, 5 and 13 mm in east, north and up coordinate components, respectively, while the corresponding positioning accuracy of the cases with PIFCB consideration can be improved by 38, 33 and 31%, and 50, 40 and 23% to 5, 4 and 9 mm, and 4, 3 and 10 mm in the three components, respectively. The corresponding improvement in convergence time is 17, 1 and 22% in the three components in UC model, respectively. Moreover, the phase observation residuals on L5 frequency in UC triple-frequency PPP and of L1/L2/L5 IF combination in IF triple-frequency PPP are reduced by about 4 mm after applying PIFCB corrections. The performance improvement in UC triple-frequency PPP over UC dual-frequency PPP is 7, 4 and 2% in terms of convergence time in the three components, respectively. The daily solutions of UC triple-frequency PPP have a comparable positioning accuracy to the UC dual-frequency PPP.

... • Code-carrier incoherency, see [17]; ...

... A key issue in triple-frequency PPP is to align the satellite clocks, namely IFCB consideration. The IFCB of GPS Block IIF satellites has been investigated by many researchers in terms of sources, dependence, periods, prediction, modeling, variations and amplitude limits [13,14,20,21]. The GPS multipath, which can be extracted using the ionospheric-free geometric linear combination, is a function of satellite elevation angle and antenna height, etc. [25][26][27]. ...

The joint use of multi-frequency signals brings new prospects for precise positioning and has become a trend in Global Navigation Satellite System (GNSS) development. However, a new type of inter-frequency clock bias (IFCB), namely the difference between satellite clocks computed with different ionospheric-free carrier phase combinations, was noticed. Consequently, the B1/B3 precise point positioning (PPP) cannot directly use the current B1/B2 clock products. Datasets from 35 globally distributed stations are employed to investigate the IFCB. For new generation BeiDou Navigation Satellite System (BDS) satellites, namely BDS-3 satellites, the IFCB between B1/B2a and B1/B3 satellite clocks, between B1/B2b and B1/B3 satellite clocks, between B1C/B2a and B1C/B3 satellite clocks, and between B1C/B2b and B1C/B3 satellite clocks is analyzed, and no significant IFCB variations can be observed. The IFCB between B1/B2 and B1/B3 satellite clocks for BDS-2 satellites varies with time, and the IFCB variations are generally confined to peak amplitudes of about 5 cm. The IFCB of BDS-2 satellites exhibits periodic signal, and the accuracy of prediction for IFCB, namely the root mean square (RMS) statistic of the difference between predicted and estimated IFCB values, is 1.2 cm. A triple-frequency PPP model with consideration of IFCB is developed. Compared with B1/B2-based PPP, the positioning accuracy of triple-frequency PPP with BDS-2 satellites can be improved by 12%, 25% and 10% in east, north and vertical directions, respectively.

... Despite the advantages of triple-frequency signals, an apparent inconsistency of the L 1 , L 2 and L 5 carrier phase measurements at the 10 cm level was identified according to a geometry-and ionospheric-free linear combination of triple-frequency observations (Montenbruck et al. 2010). This inconsistency was understood to be caused by a thermally dependent inter-frequency satellite hardware phase bias and defined as the difference between the current clock products computed with L1/L2 and the satellite clocks computed with L1/L5, termed inter-frequency clock bias (IFCB) (Montenbruck et al. 2012). ...

The latest generation of GPS satellites, termed Block IIF, provides a new L5 signal. Multi-frequency signals open new prospects for precise positioning and fast ambiguity resolution and have become the trend in Global Navigation Satellite System (GNSS) development. However, a new type of inter-frequency clock bias (IFCB), i.e., the difference between the current clock products computed with L1/L2 and the satellite clocks computed with L1/L5, was noticed. Consequently, the L1/L2 clock products cannot be used for L1/L5 precise point positioning (PPP). In order to solve this issue, the IFCB should be estimated with a high accuracy. Datasets collected at 129 globally distributed Multi-GNSS Experiment (MGEX) stations from 2015 are employed to investigate the IFCB. The results indicate that the IFCB is satellite dependent and varies with the relative sun–spacecraft–earth geometry. Other factors, however, may also contribute to the IFCB variations according to the harmonic analysis of the single-day IFCB time series. In addition, the results show that the IFCB exhibits periodic signal with a notable period of 43,080 s and the peak-to-peak amplitude is 0.023–0.269 m. After considering a time lag of 240 s, the average cross-correlation coefficient between the IFCB series of two consecutive days is 0.943, and the prediction accuracy of IFCB is 0.006 m. A triple-frequency PPP model that takes the IFCB into account is proposed. When using 3-h datasets, the positioning accuracy of triple-frequency PPP can be improved by 19, 13 and 21 % compared with the L1/L2-based PPP in the east, north and up directions, respectively.

... We didn't observe similar variation in any of the other signals. We found explanation in [5]. The carrier phase signal delay at satellite varies with temperature changes as the satellite is exposed to the Sun. ...

Simultaneous time transfer in three frequency channels can be performed after implementing new GPS signals. It leads to improving the accuracy on the one hand by increasing the number of measurements entering the statistical processing, and on the other hand by improving the characteristics of the time transfer based on ionospheric free solution.
This paper describes the design of optimal methods for processing the results of measurements in several frequency channels based on the Best Linear Unbiased Estimation (BLUE). The optimal estimate was derived for a number of elementary models (no ionospheric delay, first-order ionospheric delay, first- and second-order ionospheric delay), and the contribution of using measurements in three frequency channels was evaluated compared to procedures used up to now. We show that the optimal estimates are in some cases equal to already used linear combination of measurements (e.g. L3P) and thus these classical procedures can be considered optimal. We confirm the fact that the ionospheric free solution is always redeemed with significant increase of impact of random errors. The impact is increased by factor of three concerning the measurements in two frequency channels compared to measurement in a single frequency channel. We show that improvement of 15 % can be achieved if measurements in three frequency channels are performed.
Results of experimental measurement are also presented. The measurement aimed to look into the real behavior of random errors when using five GPS signals broadcast in three frequency channels, namely the amount of their correlation, which plays an important role in processing the results of several simultaneous measurements. We show that the correlation between measurement errors caused by multipath propagation is low despite the small span between the L2 and L5 frequency channels, and that one can benefit by including the measurements in the L5 frequency channel in the processing.

... For satellite phase delays (UPDs) the situation is inconclusive. Some satellites, such as the two Galileo experimental satellites (GIOVE-A/B), do not show significant variations of the delays between the different carrier frequencies (see [13]), whereas the modernised GPS Block IIF satellites are affected by periodic variations up to several decimetres between L1/L2 and the L5 frequency (see [5]). Different approaches, such as [2], based on FCB or [3], [1], [4] based on integer recovery clocks (IRCs) have been proposed and were implemented to estimate and provide satellite UPD corrections. ...

In the recent years, GPS only dual-frequency GNSS receivers are increasingly developing into multi-constellation, multi-frequency receivers. This development poses a new set of technical challenges. A key factor for precise GNSS applications is the stability of receiver induced inter-frequency/signal biases.
The consequences of receiver induced bias instabilities on GNSS analyses are of particular importance for analyses applying more than two signals types per link. Certainly, most up-to-date applications maintain the common ionosphere free (dual-frequency) processing. This implementation allows the absorption of these biases by ionosphere and clock estimates. However, to releasing the full potential of the multi-constellation, multi-frequency GNSS environment a joint processing of all available signals is inevitable.
Based on experimental results the presented paper shows different sources for bias instabilities and changes as among other internal receiver temperature antenna etc. It shows that some receiver types perform well, whereas others may lead to serious problems for multi-frequency processing. The results and their impacts are discussed and compared for dual-frequency GPS only and multi-constellation, multi-frequency processing.

The traditional inter-frequency clock bias (IFCB) prediction method based on the harmonic model has large jumps at the junction of prediction arcs. An improved method is proposed for GPS IFCB short-term prediction, which fits linear and sinusoidal terms of the harmonic model adopting epoch-differenced (ED) IFCB and corrects constant term using the precision IFCB with the nearest epoch to the initial prediction epoch. The jumps in adjacent prediction arcs are reduced effectively and the IFCB short-term prediction accuracy is improved. The results illustrate that the average prediction accuracy of the proposed method is improved by 62.5%, 47.1%, 35.5% and 30% for 3, 6, 12 and 24 h, respectively. Meanwhile, the positioning performance of GPS triple-frequency PPP is improved and the accuracy is improved by 5.08%, 2.3% and 5.18% over the traditional prediction method in the three directions of E, N and U, respectively.

The antenna phase center offset and ranging error caused by its variation can reach several cm or even dm level, which is a key error correction term that must be considered for GNSS high-precision applications. China’s independently self-built Beidou navigation satellite system (BDS) has opened global services, but there is no absolute antenna phase center correction model of Beidou-3 in the field of GNSS that is consistent with the definition of IGS. In this paper, based on a high-precision robot, the absolute phase center field calibration method is studied, and the geodetic-grade e antenna of the monitoring station of the international GNSS Monitoring & Assessment Service (iGMAS) is calibrated for the first time based on the absolute antenna correction platform of Wuhan University. The preliminary results show that the antenna phase model has a frequency binning pattern, and the phase model characteristics with similar frequencies are relatively close. The internal conformity accuracy between the calibration results of multiple time periods is better than 1 mm, and the short baseline positioning verification consistency accuracy can reach 1.5–3 mm.

As the development of the global navigation satellite system (GNSS), the navigation satellite transmitting multi-frequency signals becomes a prevailing trend. Currently, the usual GNSS precise orbit determination (POD) strategy uses dual-frequency (DF) ionosphere-free (IF) linear combination to build up the observation equation. In the area of precise positioning, it has been validated that the third frequency can decrease the convergence time of ambiguity resolution and has a subtle improvement on the positioning accuracy. However, there is no report about the benefits of the third frequency on GNSS POD. Although the precision of dual-frequency POD can meet requirements for generating different types of products, an obvious advantage of multi-frequency observations is that more robust results can be derived. Besides, with the third frequency observations, the improvements of orbits and clocks need to be validated. Thus, a triple-frequency (TF) uncombined (UC) POD method of GNSS satellites is developed. The hardware delay of carrier phase is divided by time-invariant and -variant components. Then the UC observation model is given by re-parameterizing the unknown parameters. The datum of satellite clocks is aligned to IGS products. The step-by-step ambiguity fixing method, i.e. the extra-wide-lane, wide-lane and narrow-lane ambiguities being fixed in sequence, is deduced by using double-differenced ambiguities in a network. We select 74 ground stations and 12 GPS BLOCK IIF satellites to process POD with a length of 20 days. Results of TF-UC POD show that about 10% improvements on orbits and clocks are received compared to L1/L2 DF-IF POD. The time-variant characteristic of phase biases from satellites and stations at the third frequency is analyzed, which shows that the bias at the station has a slight influence on the derived products. These results imply that the precision of orbits and clocks can be improved with observations from the third frequency added.

The Japanese Quasi-Zenith Satellite System (QZSS) constellation has added three new Block-II satellites. Two of these satellites have been launched into inclined geosynchronous orbits and one into a geostationary orbit. All three spacecraft broadcast ranging signals on GPS L1, L2 and L5 frequencies from their main L-band antenna together with the centimeter-level augmentation service (CLAS) signal L6 (formerly LEX) on the Galileo E6 frequency band. Like on the Block-I satellite, a sub-meter level augmentation service (SLAS) signal is transmitted from a separate antenna on the GPS L1 frequency. A new feature is the addition of the position technology verification service (PTV) Signal on the L5 frequency from yet another antenna. After determination of the antenna baseline vector, differential processing of measurements from different observations allows for an estimation of the satellite’s yaw attitude. The L1 SLAS and the L1 C/A-code signals have been used to estimate the yaw attitude with an accuracy of less than 1°. Differential carrier-phase center variation maps have been derived for this signal combination. Yaw estimation results are presented for periods of special interest, for example 360° yaw rotations, orbit correction maneuvers and the satellite’s eclipse period, where a special pseudo yaw steering attitude mode is applied. The second part of the paper introduces a new concept using triple-frequency signals from two different antennas for attitude determination. This method is demonstrated with QZSS measurements, but is also applicable to other satellite navigation system, like the enhanced GLONASS-M satellites with L3 signal capabilities.

Multi-constellation GNSS (multi-GNSS) and multi-frequency signals open new prospects for fast ambiguity resolution (AR) of precise point positioning (PPP). Currently, all the BDS and Galileo satellites are capable of transmitting signals on three or more frequencies. In this contribution, we investigate the triple-frequency PPP ambiguity resolution with B1, B2 and B3 observations from BDS satellites and E1, E5a and E5b observations from Galileo satellites and evaluate the contribution of BDS + Galileo combination to triple-frequency PPP AR. The uncalibrated phase delay (UPD) products are estimated based on triple-frequency observations, and the temporal characteristic as well as the residual distributions are analyzed. Our results show that the extra-wide-lane (EWL) and wide-lane (WL) UPDs for BDS and Galileo satellites are both stable during the 30 days and the daily narrow-lane (NL) UPD series are also steady with no obvious fluctuation. The Galileo UPDs exhibit better performance than BDS UPDs due to the high-quality observations. It is also interesting to find that the EWL UPD corrections for all Galileo satellites are very close to the zero. With the precise UPD products, the triple-frequency PPP AR with BDS and Galileo observations was implemented in both static and kinematic modes. Compared to the ambiguity-float solution, the performance can be significantly improved by triple-frequency PPP AR with the positioning accuracy improved by 30–70% in both static and kinematic modes. Moreover, the triple-frequency PPP fixed solutions also present better performance than the dual-frequency PPP fixed solutions in terms of time to the first fix and positioning accuracy, especially for the Galileo-only and BDS + Galileo solutions. And the fusion of multi-GNSS (BDS and Galileo) can further improve the position estimations compared to the single system with more satellites and better spatial geometry.

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.

This chapter introduces the concept of observation combinations, commonly used, for example, to compute positioning solutions with measurements from multiple frequencies or to study measurement noise, multipath, or ionospheric effects. Based on a generic parametrization for pseudorange and carrier-phase observations, a general expression for linear combinations is introduced. The impact of the coefficients on the properties and the noise of the combined observable is explained. The chapter covers combinations using measurements from a single satellite observed by one receiver. The discussion will then be extended to differential observations from two satellites, receivers and epochs.

Based on the measured data of IGS, the characteristics of QZSS signal, namely, carrier to noise ratio, noise and multipath characteristics, inter-frequency bias, were studied in this paper. At the same time, the results were compared with those of other satellite navigation systems, such as GPS and Galileo, to analyze the reason for the differences. Methods and results provided in this paper have important reference values for the study and construction of satellite navigation systems.

As the understanding of our Earth system grows, the importance of comprehending the structure and processes in the remote stratosphere is intensified and the interest in stratospheric observations mushrooms. Despite its great potential, Radio Occultation (RO) data have been underused in exploiting the stratosphere. A major reason for the underutilization is the imperfections in pre-existing RO data processing methods. We propose an advanced stratospheric RO data processing, where the variational method provides a general framework in which multiple-frequency RO measurements of different quality are effectively combined with the aid of a priori. The variational combination (VAR) is designed to extract the most information from RO measurements, where a priori plays a role of enhancing the observation and attenuating measurement noise. The signal-to-noise ratio (SNR) is found to be a universal quality indicator, which concisely describes the uncertainty of RO measurements in diverse conditions. The measured SNR is used to parameterize a dynamic observation error, which is essential for the VAR to use the observation optimally. Tests with real data show that VAR significantly improves the accuracy of the RO retrieval even in the upper stratosphere, where the RO data were once considered to possess little observational value. When compared with independent radiosonde observations, for instance, the VAR-produced data are more accurate than the analysis from the European Center for Medium-Range Weather Forecasts (ECMWF) for which the radiosonde data have been assimilated. The VAR-produced data are also precise enough to reveal the systematic error of the radiosonde data.

We present two efficient approaches, namely the epoch-differenced (ED) and satellite- and epoch-differenced (SDED) approaches, for the estimation of IFCBs of the two Block IIF satellites. For the analysis, data from 18 stations from the IGS network spanning 96 d is processed. Results show that the IFCBs of PRN25 and PRN01 exhibit periodical signal of one orbit revolution with a magnitude up to 18 cm. The periodical variation of the IFCBs is modeled by a sinusoidal function of the included angle between the sun, earth and the satellite. The presented model enables a consistent use of L1/L2 clock products in L1/L5-based positioning. The algorithm is incorporated into the MGPSS software at SHAO (Shanghai Astronomical Observatory, Chinese Academy of Sciences) and is used to monitor the IFCB variation in near real-time.

Over the coming years GPS and GLONASS will be modernised, whilst at the same time new systems like QZSS, Galileo, and Compass are launched. The modernisations of the existing and the deployment of new Global Naviagation Satellite Systems (GNSS) will make a whole range of new signals available to the users. The anticipated improvements will strongly depend on our understanding and handling of the biases that will inevitably exist between the different systems and signals. Furthermore the extremely high stability of the future satellite clocks means, that any form of differencing observations to cancel out the satellite clock offsets, effectively leads to a very significant loss of information. The fundamentally new aspect of our approach for GNSS analysis in a multi-GNSS and multi-signal environment is that it avoids the formation of differences as well as of linear combinations. Thus all available observations from all GNSS systems as observed by all the receivers in a network are incorporated in the parameter estimation. The fact that all observations are analysed without any pre-selection of observation types, needed for linear combinations or observation differences, leads to an enormous simplification of the processing.

The passive hydrogen maser (PHM) of GIOVE-B and the
latest generation of Rubidium clocks for GPS Block IIF
satellites have both demonstrated a superior and highly
competitive stability in ground tests. In practice, however,
the apparent clock performance for GNSS users is limited
by measurement errors and imperfections of the signal
chain that affect the clock variance at different time
scales. Within this paper, we provide a direct comparison
of the apparent clock performance for GIOVE-B and the
first Block IIF satellite (SVN62). The analyses are based
on observations of the IGS and CONGO network.
Periodic errors in the apparent clocks of both satellites are
analyzed and an effort is made to separate the impact of
orbit determination errors from physical clock or line bias
variations. For SVN62 an empirical clock correction
model is discussed, which offers a notable reduction of
the Allan variance at orbital time scales. Furthermore,
triple-frequency observations are used to demonstrate the
presence of thermally induced line bias variations and to
quantify the resulting inter-frequency clock biases.

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