As the main means of the Global Navigation Satellite System (GNSS) signal enhancement, the flex power impacts the robustness and availability of the positioning, navigation, and timing (PNT) service. However, routine code bias products provided by the agencies with low temporal resolution often fail to provide accurate values. To overcome the vulnerability of space-based PNT system and enrich resilient PNT theory, a resilient code bias concept is proposed to address limitations of the routine code bias. Firstly, the performance of two categories of routine code products, namely postprocessing code bias product and broadcast ephemeris parameters, is analyzed. Then, the concept of the resilient code bias is presented to solve the code bias problems related to defect prevention and system recovery under uncertainty in the complex environment such as flex power. Subsequently, a method for resilient code bias estimation is developed and four code bias states are defined, that are normal, flex, warning, and exceptional states. A resilient index (RI) is developed and estimated in real-time in the resilient code bias product to characterize the code bias features regarding availability, continuity, integrity, robustness, and accuracy. Resilient code bias is compatible with the characteristics of the routine code bias product in the normal state but switches the flex state in the flex power scenario. The characteristics of the resilient code bias product are analyzed for both normal and flex power scenarios, demonstrating its advantages across different states in the case of flex power for the BDS-2, BDS-3, and GPS systems.

Deep space navigation faces significant challenges due to the extreme distances, long communication delays, and the limitations of standalone systems during the cruise phase. Standalone deep space navigation systems cannot meet the demanding precision and reliability requirements of long-duration missions in highly dynamic and extreme environments. To address the issue, we propose the resilient deep space navigation framework that combines X-ray pulsar, Very Long Baseline Interferometry (VLBI), and celestial navigation by establishing the dynamic models considering gravitational perturbations from the Sun and all planets, alongside detailed measurement models for each navigation model. The resilient navigation framework is implemented through the federated designed in both feedback-free and fused-feedback modes to efficiently handle the disparate data rates between astronomical and VLBI subsystems. The performance of various algorithms, including Extended Kalman Filter (EKF), game theory-based H infinity filter, and Robust Student's t Kalman Filter (RSTKF), is evaluated through the simulations of cruise orbits at 1, 20, 50, and 100 astronomical units (AUs) under both ideal Gaussian and time-varying noise conditions. Simulation results demonstrate that while the EKF delivers optimal performance under Gaussian noise, the RSTKF exhibits superior robustness against measurement outliers and non-Gaussian noise, which ensures reliable state estimation. The proposed integrated system effectively mitigates the individual weaknesses of each navigation method, which provides a high-precision, high-reliability, and resilient navigation solution that is crucial for the success of future deep space exploration missions. (c) 2026 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

IntroductionDeep space exploration, as a critical means for humanity to understand and investigate the evolutionary history of the universe, has entered a phase of rapid development. A single deep space navigation mode struggles to meet the high-precision and high-reliability requirements for long-duration mission operations in highly dynamic and extreme environments. Hence, multi-mode resilient navigation will be the inevitable future approach for deep space exploration.MethodsThe study proposes a set of deep space resilient navigation models specifically for gravity assist phase, including X-ray pulsar navigation, Very Long Baseline Interferometry (VLBI) navigation, celestial body navigation, and combined navigation using these methods. Various navigation filtering algorithms applicable to deep space models are introduced, and the performance of the navigation models under different conditions is analyzed through simulation. For six gravity assist phases with the flight sequence proceeding as Earth, Venus, Venus, Earth, Jupiter, Neptune and heliosphere bottom, three navigation filtering methods, which are Extended Kalman Filter (EKF), Game theory-based H∞ filter, and Robust Student’s t Kalman Filter (RSTKF), exhibit consistent performance for X-ray pulsar navigation. Besides, ground-based VLBI observations can be utilized throughout the gravity assist phase and exhibit superior navigation performance compared to X-ray pulsar navigation.ResultsThe performance of the celestial body navigation is significantly worse than that of X-ray pulsar and VLBI navigation, with the accuracy degrading as mission distance increases. The positioning errors of the resilient navigation approach are within 10 km across all tested scenarios. Moreover, the feedback-free resilient navigation demonstrates slightly better performance than the fusion feedback mode, though the latter maintains high-frequency output consistency with astronomical navigation subsystems.DiscussionThese results provide the significant theoretical support and practical reference for advancing deep space navigation.

Accurate global ionospheric modeling is challenged by the limited accuracy of traditional Carrier-to-Code Leveling (CCL) methods and uneven GNSS station distribution. To address it, we propose a novel approach by integrating Slant Total Electron Content (STEC) derived from undifferenced and uncombined Precise Point Positioning (PPP) with the International Reference Ionosphere-2000 (IRI-2020) model. A multi-frequency multi-GNSS PPP model is employed for STEC extraction, while Virtual Observation Stations are established to incorporate IRI model in GNSS-sparse regions. Systematic biases between PPP-derived STEC and IRI observations are addressed through Virtual Bias estimation. The STEC derived from the PPP method demonstrates significantly superior accuracy and stability compared to that from the CCL method. Four modeling approaches were defined based on the STEC extraction method and whether IRI model data was integrated, including CCL, CCL-IRI, PPP, and PPP-IRI. Compared with the IGS final products, the RMS differences on day of year (DOY) 173 (summer solstice) of 2020 were 2.3, 1.4, 2.0, and 1.2 TECU, respectively. The corresponding RMS differences for DOY 173 of 2024 were 7.2, 3.9, 6.8, and 3.6 TECU, respectively. Validation using the dSTEC method indicated that the PPP-IRI approach improved accuracy by 19.8%, 8.3%, and 12.7% compared to the CCL, CCL-IRI, and PPP methods, respectively. The study concludes that the fusion of PPP-derived STEC and IRI data, combined with Virtual Bias estimation, significantly enhances ionospheric modeling outcomes. (c) 2026 Published by Elsevier B.V. on behalf of COSPAR.

As a key method to enhance the anti-jamming capability of Global Navigation Satellite System (GNSS), flex power technology enables ground-based commands to dynamically adjust satellite signals by redistributing signal components, thereby strengthening specific transmissions and improving service robustness in interference environments. Both the Global Positioning System (GPS) and the BeiDou Navigation Satellite System (BDS) support flex power functionality. Activating and deactivating of flex power can significantly impact the various aspects of GNSS performance and introduce new technical challenges. In this paper, we first analyze the flex power operational modes of GPS and BDS, then review existing detection methods to propose a novel detection approach applicable to both GPS and BDS. The proposed method employs carrier-to-noise density ratio (C/N0) and hardware delay as complementary indicators to achieve high detection accuracy with low false alarm rates. Subsequently, we investigate the impacts of flex power on cycle slip detection, code bias, satellite clock offset, phase bias, ionospheric corrections, and Precise Point Positioning (PPP). The results show that flex power affects several GNSS parameters with BDS exhibiting much greater sensitivity compared to GPS. To address these effects and advance resilient Positioning, Navigation, and Timing (PNT) theory, we propose the optimized estimation strategies for resilient code bias, satellite clock offset, and phase bias, along with an enhanced data processing framework for ionospheric modeling and PPP. The effectiveness of the proposed approaches is validated, demonstrating clear improvements in PNT service reliability. This study provides valuable insights and practical methodologies for enhancing the robustness of GNSS PNT services in flex power operations.

Water vapor plays a crucial role in maintaining global energy balance and water cycle, and it is closely linked to various meteorological disasters. Precipitable water vapor (PWV), as an indicator of variations in atmospheric water vapor content, has become a key parameter for meteorological and climate monitoring. However, due to limitations in observation costs and technology, traditional atmospheric monitoring techniques often struggle to accurately capture the distribution and variations in space–time water vapor. With the continuous advancement of Global Navigation Satellite System (GNSS) technology, ground-based GNSS monitoring technology has shown rapid development momentum in the field of meteorology and is considered an emerging monitoring tool with great potential. Hence, based on the GNSS observation data from July 2023, this study retrieves PWV using the Global Pressure and Temperature 3 (GPT3) model and evaluates its application performance in the “7·31” extremely torrential rain event in Beijing in 2023. Research has found the following: (1) Tropospheric parameters, including the PWV, zenith tropospheric delay (ZTD), and zenith wet delay (ZWD), exhibit high consistency and are significantly affected by weather conditions, particularly exhibiting an increasing-then-decreasing trend during rainfall events. (2) Through comparisons with the PWV values through the integration based on fifth-generation European Centre for Medium-Range Weather Forecasts (ERA-5) reanalysis data, it was found that results obtained using the GPT3 model exhibit high accuracy, with GNSS PWV achieving a standard deviation (STD) of 0.795 mm and a root mean square error (RMSE) of 3.886 mm. (3) During the rainfall period, GNSS PWV remains at a high level (>50 mm), and a strong correlation exists between GNSS PWV and peak hourly precipitation. Furthermore, PWV demonstrates the highest relative contribution in predicting extreme precipitation, highlighting its potential value for monitoring and predicting rainfall events.

To meet the high demand for convergence speed in real-time precise point positioning (RTPPP), this paper comprehensively considers both satellite clock offset services and the PPP user perspective, deriving the inter-system bias (ISB) in detail. Using the precise satellite clock offset products from Wuhan University along with self-computed products, we analyze the datum issues related to satellite clock offsets/ISB, taking into account the effects of daily boundary changes, and propose a reliable ISB product workflow. We examine the statistical characteristics of ISB for different receiver types, highlighting their potential to enhance RTPPP performance as prior constraints. Experimental results indicate that after datum unification, the discontinuity of ISB caused by daily boundary issues can be avoided; the ISB values for the same type of receivers are closely aligned, and the statistical characteristics of ISB remain stable. When utilizing ISB products as prior constraints in the Global Positioning System + BeiDou Navigation Satellite System dual-system RTPPP, the root mean square and convergence time in the East-Up-North direction improve by 18%, 10%, 2%, 8%, 3%, and 1% compared to the original PPP, respectively. For the Global Positioning System + Galileo navigation satellite system, the corresponding improvements are 28%, 26%, 17%, 39%, 31%, and 17%.

The economic and strategic value of high-latitude maritime regions is increasingly significant, yet traditional Global Navigation Satellite Systems remain constrained by unfavorable geometric configurations and slow convergence speeds at high latitudes, failing to meet the growing demand for real-time centimeter-level high-precision positioning in these areas. Benefiting from their rapid motion and superior coverage over high-latitude zones, Low Earth Orbit (LEO) satellites offer an effective means to enhance positioning performance in such regions. This paper uses the real BDS data collected by an unmanned surface vessel in the high-latitude waters of the Southern Hemisphere, jointly simulates polar and medium-inclination LEO constellations, and systematically assess the enhancement effects of LEO augmentation on Precise Point Positioning (PPP) and Real-Time Kinematic (RTK) techniques. The results demonstrate that the polar-orbiting constellation markedly improves the observation environment, increasing the number of visible satellites by 70.2% and reducing the Position Dilution of Precision from 2.4 to 1.7, whereas the medium-inclination orbit constellation offered negligible improvement due to insufficient visibility. The rapid geometric change brought by LEO constellations is the core key to achieving fast convergence. Incorporating LEO observations drastically shortened the BDS PPP convergence time from 45.3 min to under 1 min, achieving a reduction of over 97%. Simultaneously, it improved the three-dimensional Root Mean Square accuracy by 54.7%, from 0.086 m to 0.039 m. Convergence within one minute was consistently achieved when at least 5.4 LEO satellites were included in the solution. Moreover, the addition of LEO signals increased the fixed solution rate of short-baseline RTK from 96.5% to 100%, while improving horizontal and vertical accuracy by 31.5% and 12.3%, respectively. This study confirms that LEO constellations, especially those in polar orbits, can substantially enhance BDS precise positioning performance in high-latitude maritime environments, thereby providing critical technical support for related navigation applications.

As a pivotal spatiotemporal infrastructure in the modern information society, satellite navigation provides global users with high-precision, all-weather, and round-the-clock Positioning, Navigation, and Timing (PNT) services [...]

The legacy Global Navigation Satellite System (GNSS) satellite clock offsets obtained by the dual-frequency undifferenced (UD) ionospheric-free (IF) model absorb the code and phase time-variant hardware delays, which leads to the inconsistency of the precise satellite clock estimated by different frequencies. The dissimilarity of the satellite clock offsets generated by different frequencies is called the inter-frequency clock bias (IFCB). Estimates of the IFCB typically employ epoch-differenced (ED) geometry-free ionosphere-free (GFIF) observations from global networks. However, this method has certain theoretical flaws by ignoring the receiver time-variant biases. We proposed a new undifferenced model coupled with satellite clock offsets, and further converted the IFCB into the code and phase time-variant mixed observable-specific signal bias (OSB) to overcome the defects of the traditional model and simplify the bias correction process of multi-frequency precise point positioning (PPP). The new model not only improves the mixed OSB performance, but also avoids the negative impact of the receiver time-variant biases on the satellite mixed OSB estimation. The STD and RMS of the original OSB can be improved by 7.5–60.9% and 9.4–66.1%, and that of ED OSB (it can reflect noise levels) can be improved by 50.0–87.5% and 60.0–88.9%, respectively. Similarly, the corresponding PPP performance for using new mixed OSB is better than that of using the traditional IFCB products. Thus, the proposed pseudorange and phase time-variant mixed OSB concept and the new undifferenced model coupled with satellite clock offsets are reliable, applicable, and effective in multi-frequency PPP.

With the completed establishment of the Beidou global navigation satellite system (BDS-3), the BDS constellation is qualified for the global positioning, navigation and timing (PNT) services. The simultaneous observed B1I/B3I signals of the BDS-2 and BDS-3 satellites facilitate the BDS-2/BDS-3 joint precise point positioning (PPP) solution. For reasons related to the modulation modes, signal property, receiver front end design, precise satellite clock products arising from the different analysis centers and so on, BDS pseudorange biases are absorbed by the pseudorange measurements, including the pseudorange bias variation, signal distortion bias (SDB) and time delay bias (TDB). The large magnitude of the BDS pseudorange biases will degrade the BDS PPP performance. Herein, the simultaneous estimation of the BDS pseudorange biases are applied in BDS PPP and totally six pseudorange biases processing strategies are presented. The magnitudes of the existed BDS SDB and TDB are analyzed. The statistics showed that the BDS PPP performance in term of the convergence time and positioning accuracy can be remarkably improved by considering the BDS pseudorange biases. By comparing the BDS PPP performance in different scenarios, we recommend three processing schemes that are reducing weight of the BDS-2 pseudorange observations, estimating the BDS TDB as constant and neglecting the SDB, and estimating the BDS SDB as the constant per satellite. The analytical results can facilitate the optimal BDS PPP performance.

Pseudorange bias handling is essential in satellite positioning, navigation and timing (PNT) services and ionospheric modeling. Differential code bias (DCB) is commonly utilized and parameterized as the differential form of the pseudorange bias. To overcome the challenges of the inflexible and inconvenient extendable DCB calibration for the multi-frequency observations with multiple signal modulations, the pseudorange observable-specific signal bias (OSB) is used alternatively as the observation-specific individual bias representation. The study estimates the pseudorange OSBs of all the observation channels with two rigorous and modified multi-frequency approaches: multi-frequency modified carrier-to-code leveling (MFMCCL) and multi-frequency modified precise point positioning (MFMPPP) approaches. Both two approaches extract the slant ionospheric observables by considering the time-variant characteristic of the receiver pseudorange bias, after that estimating the pseudorange OSBs with certain reliability. After solving the various linear combinations of the pseudorange OSBs by two approaches, including the slant ionospheric observables and satellite-plus-receiver (SPR) pseudorange biases, all possible multi-frequency pseudorange OSBs are estimated simultaneously. Both two approaches are discussed for the GNSS observations with the code division multiple access (CDMA) and frequency division multiple access (FDMA) models. The two methods are validated with one-month data to estimate 31 types of pseudorange OSBs for the GPS, BDS, GLONASS and Galileo systems. Compared with the Chinese Academy of Sciences (CAS) pseudorange OSB product, the results indicated that both two proposed approaches can estimate the reliable multi-frequency multi-GNSS pseudorange OSBs with high stability and consistency.

A common practice adopted for the pseudorange bias estimation and calibration assumes that Global Navigation Satellite System satellite-dependent pseudorange biases vary gently over time. Whereupon satellite pseudorange biases are routinely estimated and provided as the products with low temporal resolution, e.g., hourly or daily, by the agencies. The story sounds unquestionably perfect under the acquainted assumption. To validate the inadequacy of the above hypothesis we herein present an approach to the estimate the BeiDou Navigation Satellite System (BDS) pseudorange biases with high temporal resolution. Its feasibility, affecting factors, and necessity are discussed. Concretely, the Geometry-Free function models are first constructed to retrieve the linear combination of the pseudorange biases; then the pseudorange Observable-specific Signal Bias (OSB) values with respect to baseline frequencies ( e.g. , BDS C2I/C6I) are estimated along with the ionosphere modeling; subsequently, all multi-frequency pseudorange OSBs are determined by using the ionospheric information with constraint conditions; finally, the possible Differential Code Bias sets are attainable with the estimated pseudorange OSBs. Using the observation data of four months when the estimated BDS pseudorange biases are stable, their reliability is demonstrated with the stability at the level of sub-nanosecond and the BeiDou-3 Navigation Satellite System (BDS-3) values more stable than that of BeiDou-2 Navigation Satellite System (BDS-2). The comparison between the estimated pseudorange biases and the Chinese Academy of Sciences products reveals that the accuracy of the estimated pseudorange biases is 0.2–0.4 ns. Moreover, the large magnitude of the short-term pseudorange bias variation in the tens of nanoseconds for the BDS-2 and BDS-3 are found in years 2021 and 2022, which are affected by two types of the satellite flex power for the BDS-2 and BDS-3, respectively. We stress that it’s necessary to estimate the BDS pseudorange biases with high temporal resolution in the case of the satellite flex power and the products currently provided by the agencies cannot reflect the true quantity under the circumstance.

The Global Navigation Satellite System (GNSS) multi-frequency observations are widely used in positioning applications, while precise orbit determination and precise clock estimation (PCE) techniques typically employ dual-frequency undifferenced (UD) ionospheric-free (IF) observations from global networks. To fully utilize the multi-frequency GNSS observations for generating satellite products at the server-end, we develop some new five-frequency PCE models based on Galileo data, which are five-frequency uncombined FFUC model, FFIF0 model combining the E1/E5a, E1/E5b, E1/E5 and E1/E6 IF observables, FFIF1 model combining E1/E5a and E1/ E5a/E5b/E5/E6 IF observables, respectively. The traditional dual-frequency UD IF and triple-frequency PCE models are also introduced for comparison. The new multi-frequency PCE models can not only make full use of the modern GNSS multi-frequency observations, but also obtain satellite clock offsets and inter-frequency clock bias (IFCB) at the same time, which can better support multi-frequency precise point positioning (PPP) appli-cations. The multi-frequency models can improve the stability of GNSS satellite clock estimation, the precision of satellite clock offsets can be improved by 8-12% for triple-frequency models, and 19-27% for five-frequency PCE models compared with the traditional dual-frequency IF model. The PPP positioning accuracy using only multi -frequency satellite clock offsets can be improved by 7-18% for dual-frequency PPP, 8-15% for triple-frequency PPP, 4-16% for five-frequency PPP. The positioning accuracy can be further improved by 11-28% for triple -frequency PPP and 7-20% for five-frequency PPP. Therefore, the new multi-frequency PCE models are demonstrated to support PPP applications with better performance.

As the significant error sources influencing the Global Navigation Satellite System (GNSS) positioning, navigation, and timing (PNT) services, ionospheric delay and satellite hardware delay should be properly calibrated. Particularly, the pseudorange observable-specific signal bias (OSB) is convenient and can be directly corrected in the raw pseudorange measurement. In this work, we present a novel single-frequency ionospheric-free-half precise point positioning (PPP) (SFPPP2) approach for ionospheric studies, in which the ionospheric vertical total electron content (VTEC) and satellite OSB are isolated from the slant ionospheric observables by means of the ionospheric multilayer mapping function (MF). The computation and parameterization methods of the ionospheric VTEC and satellite pseudorange OSB are present. To validate the effectiveness and reliability of the novel method, we investigate and compare the performance of the single-frequency ionospheric-float PPP (SFPPP1), SFPPP2, and dual-frequency ionospheric-float PPP (DFPPP1) solutions for ionosphere sensing. The analytical results indicate that the novel approach can extract the slant ionospheric observables with the accuracy of submeters. The accuracy of the estimated ionospheric VTEC by the single-frequency PPP approaches is in the submeter level, which exhibits a slightly worse accuracy than that from the dual-frequency PPP solution. The estimated ionospheric VTEC accuracy is improved with the multilayer MF compared with the single-layer MF. The estimated BeiDou Navigation Satellite System (BDS) pseudorange OSB with the proposed SFPPP2 approach is stable, reliable, and of the high accuracy, and the rms’s with respect to Chinese Academy of Sciences (CAS) product for C2I and C6I signals with single-layer and multilayer MFs are 0.40, 0.41, 0.60, and 0.63 ns, respectively. The proposed PPP approach can retrieve the VTEC and satellite pseudorange OSB by mass-market receivers for the GNSS users.

Global Navigation Satellite Systems (GNSSs) can provide high-precision positioning services, which can be applied to fields including navigation and positioning, autonomous driving, unmanned aerial vehicles and so on. However, GNSS signals are easily disrupted in complex environments, which results in a positioning performance with a significantly inferior accuracy and lengthier convergence time, particularly for the single GNSS system. In this paper, multi-GNSS precise point positioning (PPP) with tightly integrating ultra-wide band (UWB) technology is presented to implement fast and precise navigation and positioning. The validity of the algorithm is evaluated by a set of GNSS and UWB data. The statistics indicate that multi-GNSS/UWB integration can significantly improve positioning performance in terms of the positioning accuracy and convergence time. The improvement of the positioning performance for the GNSS/UWB tightly coupled integration mainly concerns the north and east directions, and to a lesser extent, the vertical direction. Furthermore, the convergence performance of GNSS/UWB solution is analyzed by simulating GNSS signal interruption. The reliability and robustness of GNSS/UWB solution during GNSS signal interruption is verified. The results show that multi-GNSS/UWB solution can significantly improve the accuracy and convergence speed of PPP.

Estimates of satellite clock offsets typically employ dual-frequency undifferenced (UD) ionospheric-free (IF) observations from global network. The third-generation BeiDou Navigation Satellite System (BDS-3) can transmit B1I (1561.098 MHz), B3I (1268.52 MHz), B1C (1575.42 MHz), B2a (1176.45 MHz), B2b (1207.14 MHz) and B2ab (1191.795 MHz) signals. To make full use of the advantage of BDS-3 multi-frequency signals and improve BDS-3 service performance, we present some new quad-frequency satellite clock estimation techniques using B1I/B3I/B1C/B2a signals, which are QFIF0 model combining B1I/B3I and B1C/B2a IF observables, QFIF1 model combining the B1I/B3I, B3I/B1C and B1I/B2a IF observables, QFIF2 model combining B1I/B3I and B1I/B3I/B1C/B2a IF observables, and quad-frequency uncombined QFUC model, respectively. These new techniques only improve the performance of satellite clock estimation by fully utilizing the BDS-3 multi-frequency observations on the premise of ensuring the dual-frequency IF datum but also obtain the corresponding inter-frequency clock bias (IFCB) simultaneously. The quad-frequency satellite clock offsets are evaluated in terms of the clock offset precision, the modified Allan deviation (MDEV) and precise point positioning (PPP) performances. The new methods can improve the performances of the estimated clock offsets compared with the traditional dual-frequency IF model. The precision for the estimated clock offsets using quad-frequency satellite clock estimation models can be improved by 13–26

In precise satellite clock estimation, the satellite clock offsets absorb the pseudorange and carrier phase time-variant hardware delays. The dissimilarity of the satellite clock estimated with observations at different frequencies is termed the inter-frequency clock bias (IFCB). The bias inconsistency suggests that the simple ionospheric-free satellite clock cannot directly be applied to the multi-frequency carrier phase observations in multi-frequency precise point positioning (PPP). We propose the carrier phase time-variant observable-specific signal bias (OSB) concept and the corresponding estimation approach to solve this. The definition, rationality, reliability and validity of the carrier phase time-variant OSB are clarified. The new concept advantage is that a set of the carrier phase time-variant OSB values can directly amend on the carrier phase observations, and thereafter, the IFCB effect can be eliminated, which provides the flexibilities for the GNSS carrier phase observation handing. Datasets collected from 144 Multi-GNSS Experiment (MGEX) stations are adopted for the carrier phase time-variant OSB estimation and an analysis of its effect on the GNSS multi-frequency PPP performance. The various multi-frequency PPP models are tested and evaluated considering the carrier phase time-variant OSB correction. The results indicate that the GPS, BDS-2 and BDS-3 carrier phase time-variant OSB time series have the obvious amplitudes and the amplitudes of the Galileo and QZSS carrier phase time-variant OSB are small. The GPS and BDS-2 multi-frequency PPP performance is significantly enhanced when correcting the carrier phase time-variant OSB. The GPS-only kinematic ionospheric-float PPP exhibits the positioning accuracy of 1.0 cm, 2.2 cm and 2.6 cm in the north, east and up components when correcting the carrier phase time-variant OSB, whereas the positioning accuracy of the case without the correction is 1.4 cm, 2.8 cm and 3.7 cm in three directions, respectively. The mean convergence time of two dual-frequency and three triple-frequency BDS-2-only kinematic PPP is reduced by 5.0%, 4.9%, 5.4%, 4.7% and 4.6%, respectively, with the carrier phase time-variant OSB correction. The carrier phase time-variant OSB improvement on BDS-3-only multi-frequency PPP is not obvious owing to the relatively few available and stable carrier phase time-variant OSB values. The reliability, suitability and effectiveness of the GNSS carrier phase time-variant OSB are demonstrated.

BeiDou Navigation Satellite System (BDS) on-orbit satellites contain BDS-2 and BDS-3 satellites. Due to the existence of ISB between BDS-2 and BDS-3 on receiver side, the traditional BDS precise clock estimation (PCE) and precise point positioning (PPP) models for the BDS-2 and BDS-3 combined processing will reduce the accuracy and stability of solutions. To improve the BDS service performance, the ISB between BDS-2 and BDS-3 for both old (B1I/B3I) and new signals (B1C/B2a) and its impact on PCE and PPP are investigated in this contribution. The BDS-2 and BDS-3 integrated PCE and PPP models with and without ISB estimation are presented. The combined processing is comprehensively assessed in terms of the precision of clock offsets and PPP performances. The result demonstrates that the ISB is stable for both old and new signals, and estimating ISB can effectively avoid the confusion of receiver clock datum. The integration of BDS-2 and BDS-3 indeed improves the precision of satellite clock offsets estimations based on the proper PCE models. The average STD for BDS-2 and BDS-3 clock offsets using old signals is improved by 15.8% and 11.1% compared with BDS-2-only and BDS-3-only solutions, respectively. For new signals, the improvement for BDS-3 clock offsets is 14.6% from 0.081 ns to 0.069 ns. The BDS clock offsets estimated by the proposed PCE model with ISB estimation (PCE0) can well support PPP applications. The positioning accuracy for old signals can be improved by 40.4%, 20.0% and 35.4% compared with those of using GFZ rapid products. Similarly, the positioning performance for new signals is slightly better than GFZ PPP. The PCE0 model is the optimal BDS-2 and BDS-3 integrated satellite clock offsets determination model for the server, and the PPP model with ISB estimation (PPP0) is the optimal PPP model for the client. The cooperation between PCE0 and PPP0 can improve the BDS service performance.

BeiDou global navigation satellite system (BDS) began to provide positioning, navigation, and timing (PNT) services to global users officially on 31 July, 2020. BDS constellations consist of regional (BDS-2) and global navigation satellites (BDS-3). Due to the difference of modulations and characteristics for the BDS-2 and BDS-3 default civil service signals (B1I/B3I) and the increase of new signals (B1C/B2a) for BDS-3, a systemically bias exists in the receiver-end when receiving and processing BDS-2 and BDS-3 signals, which leads to the inter-system bias (ISB) between BDS-2 and BDS-3 on the receiver side. To fully utilize BDS, the BDS-2 and BDS-3 combined precise time and frequency transfer are investigated considering the effect of the ISB. Four kinds of ISB stochastic models are presented, which are ignoring ISB (ISBNO), estimating ISB as random constant (ISBCV), random walk process (ISBRW), and white noise process (ISBWN). The results demonstrate that the datum of receiver clock offsets can be unified and the ISB deduced datum confusion can be avoided by estimating the ISB. The ISBCV and ISBRW models are superior to ISBWN. For the BDS-2 and BDS-3 combined precise time and frequency transfer using ISBNO, ISBCV, ISBRW, and ISBWN, the stability of clock differences of old signals can be enhanced by 20.18%, 23.89%, 23.96%, and 11.46% over BDS-2-only, respectively. For new signals, the enhancements are −50.77%, 20.22%, 17.53%, and −3.69%, respectively. Moreover, ISBCV and ISBRW models have the better frequency transfer stability. Consequently, we recommended the optimal ISBCV or suboptimal ISBRW model for BDS-2 and BDS-3 combined precise time and frequency transfer when processing the old as well as the new signals.