This study developed a low-Earth orbit (LEO) enhanced GPS, Galileo, and Beidou-3 navigation satellite system positioning using an undifferenced and uncombined (UDUC) precise point positioning-ambiguity resolution (PPP-AR) model, evaluating its performance over 7575 km and 145 km reference networks. Atmospheric delays were interpolated from reference stations to users via the modified linear combined model (MLCM) to constrain the UDUC PPP. The results demonstrate that, in both static and kinematic positioning, LEO satellites significantly enhance positioning and AR performance, notably improving convergence speed. While regional atmospheric delay augmentation also yields significant gains, its efficacy decreases with larger network scales, leading the 145 km network to underperform the 75 km network. The study concludes that LEO enhancement combined with atmospheric delay constraints is a powerful tool for high-precision positioning, with network scale being also a factor for optimal performance This study presents the first comprehensive evaluation of the synergistic effects between LEO fixed solutions enhancement and regional atmospheric delay constraints in UDUC static and kinematic PPP-AR, providing novel insights for network scale optimization.

In response to the persistent limitations of multi-system precise point positioning (PPP) in terms of convergence speed, ambiguity fixing rate, and especially positioning performance in kinematic scenarios, this study proposes a low-Earth orbit (LEO) fixed solution-enhanced positioning model, aiming to significantly improve both the positioning performance and ambiguity resolution (AR) performance of PPP. Based on the undifferenced and uncombined (UDUC) PPP framework, GPS, Galileo, and BDS-3 are enhanced using LEO satellites under both static and kinematic positioning modes, and performs AR for the satellites of the participating systems. Compared with the float solution, AR improves both positioning accuracy and convergence speed. After ambiguity resolution, the convergence time for static and kinematic positioning in the three-dimensional (3-D) direction is shortened by 24% and 8%, respectively, and the positioning accuracy is improved by 40% and 30%, respectively. Compared with single-system solutions, the inclusion of more systems improves positioning accuracy, convergence speed, and AR performance. In particular, the enhancement of LEO satellites significantly accelerates the convergence. The enhancement with LEO satellites shortens the 3-D convergence time for static and kinematic positioning by 88% and 91%, improves the positioning accuracy by 35% and 50%, increases the ambiguity fixing rate by 17% and 52%, and shortens the time to first fix (TTFF) by 72% and 79%, respectively. This study confirms that LEO satellite enhancement can effectively advance PPP technology toward rapid ambiguity resolution and real-time high-precision positioning.

In response to the rapid growth in demand for high-precision location-based services driven by the widespread adoption of smart devices, and considering the limited positioning accuracy and stability of global navigation satellite systems (GNSS) in complex environments, this paper proposes a GNSS/fifth-generation (5G) integrated high-precision positioning method designed for smartphone platforms. Specifically, a joint smoothing model is constructed by integrating multi-frequency carrier phase, Doppler shift, and initial GNSS/5G positioning estimates to enhance the continuity and reliability of pseudorange observations in complex environments; inter-satellite and inter-base-station single-differencing techniques are applied to GNSS and 5G observations, respectively, and combined with a non-combined solution model, in order to eliminate receiver clock bias and improve robustness against frequency instability. Results demonstrate that the proposed approach achieves measurable gains in positioning accuracy compared with GNSS-only positioning. In static open-sky experiments, the horizontal RMS error of the Huawei P40 decreased from 0.648 m to 0.385 m, while that of the Xiaomi 11 decreased from 0.692 m to 0.466 m. Under obstructed conditions, 5G integration mitigated GNSS degradation, reducing the vertical RMS error of the Huawei P40 from 2.374 m to 1.762 m and that of the Xiaomi 11 from 2.535 m to 1.900 m. With carrier-phase observations, the inclusion of 5G measurements was still effective in accelerating convergence and enhancing positioning accuracy and stability across both open-sky and obstructed environments. These results validate the feasibility and practical value of GNSS/5G integration for enhancing smartphone positioning accuracy and continuity across diverse environments.

The performance of Global Navigation Satellite Systems (GNSS) in high-latitude marine environments is critical for maritime navigation and scientific research. This study uses dynamic observations. It systematically evaluates GNSS signal quality and precise positioning capabilities in the Southern Hemisphere's high-latitude seas. Results show that the Number of Visible Satellites remains above 11 at a 10 degrees cut-off elevation angle, ensuring sufficient satellite availability. The average Position Dilution of Precision (PDOP) for individual systems is around 2.0, while the combined PDOP across all systems falls below 1.0, indicating optimal satellite geometry. Signal strength is robust, with an average Carrier-to-Noise Ratio exceeding 40 dBHz, and the BeiDou Navigation Satellite System (BDS) demonstrates superior multipath resistance (23 cm) compared to other systems. In Real-Time Kinematic (RTK) positioning, GNSS surpass sub-meter three-dimensional accuracy, with horizontal accuracy better than 2 cm and vertical accuracy better than 8 cm. BDS outperforms Global Positioning System (GPS) in RTK positioning, and the combined GPS/BDS system improves horizontal accuracy by 33.33 %, vertical accuracy by 16.44 %, and fix rate by 1.1 %. For Precise Point Positioning (PPP), GPS achieves 0.124 m three-dimensional accuracy, while multi-system combinations enhance accuracy to centimeter level (horizontal <4 cm, vertical <9 cm) and reduce the convergence time by over 70 %. These findings provide valuable reference for high-precision GNSS applications and marine development in the high-latitude regions of the Southern Hemisphere. (c) 2025 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Integrated navigation software with global navigation satellite system (GNSS) and inertial navigation system (INS) has been widely applied in vehicular navigation. However, the positioning accuracy of different software solutions varies due to differences in algorithms, environmental conditions and devices. To evaluate the positioning performance of different integrated navigation software in various urban environments, this study conducts real-world vehicle experiments and provides a detailed analysis of the positioning accuracy of five software KF-GINS, OB-GINS, IGNAV, GINav and PSINS. The algorithms and reasonable usage strategies of all above five software are given as well. For manned vehicle scenarios, the positioning accuracy of each software can reach the centimeter-level on open road and the decimeter-level on tree occlusion road, while the position errors diverge in tunnel. The attitude accuracy remains within 3 degrees. Positioning accuracy is improved under multi-GNSS mode compared to GPS-only mode. IGNAV achieves the highest accuracy in most scenarios by expanding the dimensions of the extended Kalman filter state and observation equations. KF-GINS assigns higher weights to GNSS positioning results, leading to better accuracy than IGNAV on open road. GINav exhibits relatively large initial state errors which affect the accuracy of subsequent positioning. PSINS achieves a favorable balance between speed and reliability, providing reasonably accurate solutions in a short computation time. OB-GINS utilizes factor graph optimization to make full use of redundant observations, resulting in stronger robustness and more stable accuracy. Except under extremely harsh conditions, all software can provide relatively reliable positioning results. For low-speed unmanned ground vehicle scenarios, the positioning accuracy of various software can also be maintained at the meter level, but they exhibit poor adaptability in attitude determination, failing to output reliable orientation estimates. For computational efficiency of the various software, PSINS achieves the fastest computation speed. (c) 2025 COSPAR. Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The time-variable gravity field solutions from the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On (GRACE-FO) mission are generally contaminated by the correlation errors, specifically parameter correlation (strong parameter coupling) and observation noise correlation (colored instead of white noise). In this context, we propose a decorrelation approach to pursue an improved time-variable gravity solution following a step-wise processing. The step-wise decorrelation approach comprises three steps, standard, parameter decorrelation, and observation noise decorrelation processes. First, the standard process serves to establish a reliable signal reference for the following. Then, the parameter decorrelation is implemented through the separate estimation of orbit and gravity field parameters. Finally, to achieve the goal of observation noise decorrelation, the post-fit residuals obtained from the result of parameter decorrelation are used to estimate a colored noise model, which is considered to determine the final gravity field model, specifically termed the step-wise decorrelation solution. The basic idea is that under the regularization constraints of separate estimation for dynamic parameters, the reduced dynamic parameter space allows certain low-frequency perturbative errors to emerge in post-fit residuals, enabling comprehensive characterization of observation noise correlation. Using this step-wise decorrelation approach, we process monthly GRACE-FO gravity field time series and evaluate the performance of them from the aspects of signal and noise. Spectral and spatial domain analyses of noise levels confirm significant noise suppression in the final solution. For instance, it achieves 66

In this study, a low-Earth orbit (LEO) constellation of 160 LEO satellites was simulated, and simulated LEO observations from American regional stations were used to investigate regional augmented precise point positioning (PPP). For reference networks with scales of 75 km and 165 km, the low-order surface model (LSM), the distance-based linear interpolation method (DIM), and the modified linear combination model (MLCM) were respectively used to interpolate atmospheric delays. For both scales of the reference networks, the root mean square errors (RMSEs) of the inter-satellite single-difference (SD) ionospheric slant delay residuals and the tropospheric zenith wet delays (ZWDs) residuals interpolated by the three methods were respectively less than 2.00 cm and 0.25 cm, and the accuracy of the atmospheric delays interpolated by the three methods was comparable. Within the 75‑km and 165‑km reference networks, using the interpolated atmospheric delays to constrain undifferenced and uncombined (UDUC) PPP improved the positioning performance of both the float and fixed solutions, and also augmented the ambiguity resolution (AR) performance of the fixed solutions. The enhancement effects of the three methods were comparable, and the enhancement effect decreased slightly as the scale of the reference network increased. Among the three methods, LSM performed slightly better than DIM and MLCM in the up direction, since the LSM used in this study is an interpolation model that accounts for the elevation factor.

High‑rate Global Navigation Satellite System (GNSS) observations provide direct measurements of surface coseismic displacements and are a key tool for seismic monitoring. However, the signal‑to‑noise ratio (SNR) of high‑rate GNSS coseismic displacement data is often low in the case of moderate‑magnitude or distant earthquakes, and traditional denoising methods struggle to balance noise suppression with signal preservation. To overcome this problem, this study proposes a new feature-preserving deep-denoising method based on complementary ensemble empirical mode decomposition (CEEMD) and UNet network (CEEMD‑UNet). The proposed method first adaptively decomposes the non‑stationary coseismic displacement via CEEMD, effectively overcoming the limitations of conventional denoising techniques. The resulting intrinsic mode functions (IMFs) are then accurately denoised using a UNet network, achieving an effective balance between noise removal and preservation of coseismic signal features. Tests on synthetic data show that after denoising, the average cross‑correlation coefficients in the E, N, and U directions all exceed 0.85, and the average SNR is improved by factors of 14.72, 12.84, and 38.77, respectively. Validation using real GNSS data from the 2018 Mw 7.0 Anchorage, Alaska earthquake indicates that the average root mean square error across stations is reduced by 41.39%, 48.79%, and 67.90% in the three components, while the SNR is increased by factors of 5.05, 2.89, and 0.96. Compared with CEEMD‑WD and UNet‑only denoising methods, the proposed approach demonstrates significant advantages in waveform consistency, amplitude preservation, and cross‑scenario adaptability, offering a reliable technical solution for processing low‑SNR high‑rate GNSS seismic data.

The reliability of single-Beidou Navigation Satellite System (BDS) modules serves as a fundamental cornerstone for the application of BDS. In this study, based on a dual-antenna carrier-phase altimetry model, a systematic investigation and comparison of the altimetric performance between single-BDS modules (BDS(S)) and multi-constellation Global Navigation Satellite System (GNSS) modules was carried out. To achieve this, field experiments were conducted at two river stations with different flow velocity characteristics along the BA and FENG rivers in Xi’an, Shaanxi, China. These experiments enabled a comprehensive evaluation of the performance of the single-BDS module from multiple aspects, including the number of visible satellites, satellite spatial distribution, Ratio statistics, and water level retrieval accuracy. The results reveal that the BDS(S) scheme presents certain advantages in tracking BDS satellite signals. Moreover, the BDS(S) scheme exhibits dilution of precision (DOP) values and satellite availability comparable to those of the BDS + GPS(M) scheme under the experimental conditions. Regarding ambiguity resolution performance at the BA station, statistical analysis indicates that the BDS(S) scheme achieved an ambiguity-fixing rate of 98.6

The Gravity Recovery and Climate Experiment Follow-on (GRACE-FO) satellites carry a novel technology demonstration instrument, the laser ranging interferometer (LRI), parallel to the microwave interferometer (MWI) for inter satellite ranging. As critical geometric observations for capturing the Earth gravity field variation, the quality of monthly gravity field solutions and the characterization of residual sub-monthly signals are significantly influenced by the performance of the LRI Level-1B (LRI1B) data products. Currently, multiple versions of LRI1B products are independently processed and released by the Albert Einstein Institute (AEI), the Huazhong University of Science and Technology (HUST), the Jet Propulsion Laboratory (JPL), and the Sun Yat-sen University (SYSU). Here, we examine the performance of multiple LRI1B datasets on GRACE-FO time-variable gravity field recovery through data cross comparison, monthly LRI-based gravity solutions, and post-fit residuals from January 2019 to June 2023. All LRI1B data products maintain high data availability of ∼85%. Direct cross comparisons reveal that the JPL v04 product presents systematically higher noise within the high frequency band exceeding 0.1 Hz. As for monthly gravity field solutions, spectral and geospatial analyses demonstrate that all four gravity solutions are consistent with GRACE-FO Science Data System (SDS) products, achieving comparable noise levels with an average open ocean root mean square of 3.06 cm and signal recovery correlations of ∼0.99 for annual mass variations. However, significant divergences are identified in the post-fit range rate and range acceleration residuals. With the low frequency component excluded, the JPL solution exhibits a noise floor of 4 × 10−10 m/s2 for range acceleration residuals, whereas the AEI, HUST, and SYSU solutions maintain noise levels below 2 × 10−10 m/s2. While the high frequency noise does not degrade monthly solutions, it presents a potential limitation for the extraction of subtle sub-monthly geophysical signals. These findings provide a scientific basis for data product selection and laser data processing strategies for the future mission.

Global Navigation Satellite System (GNSS) now provides global users with high-precision positioning, navigation, and timing services. Precise point positioning (PPP) technique has been developed and utilized for international atomic time link computations. However, PPP relies on high-precision satellite products to correct satellite-related errors in measurement observations output by a GNSS receiver. To support applications, precise satellite products were provided either via network transmission or through direct satellite broadcast in real-time. In recent years, network-based products, which are encrypted, internationally standardized, and transmitted in binary format, and satellite-based products, which are fragmented, customized, encoded, and transmitted in binary format, have made it more difficult for users to reliably and continuously recover them. At present, real-time open-source PPP software does not provide these two supports, which limits the application of real-time PPP, particularly for stable and continuous real-time PPP time transfer research and implementation. In this study, two supporting software packages have been developed and released as open source, designed to be embedded into mainstream open-source C/C++ PPP software. RTStreamHub is a support package for receiving and processing encrypted network-based products, designed to ensure reliable, real-time reception of securely transmitted satellite binary products. HASPPP is a dedicated software for Galileo satellite-based PPP, providing integrated support for recovering complex satellite-based products and implementing PPP, while also offering interfaces for seamless integration into mainstream C/C++ software for PPP time transfer applications. The real-time PPP time transfer precision of both network- and satellite-based products was analyzed using the PPP model embedded within the software HASPPP. Taking fiber-optic link results as a reference, the PPP time transfer precision is 125.9 ps based on network-based satellite products and 133.8 ps based on Galileo satellite-based products. The performance of these software packages has been validated, providing support modules for mainstream C/C++ PPP software.

The accuracy and reliability of vehicle navigation are facing severe challenges in urban environments. Due to the time-increasing INS (Inertial Navigation System) errors, traditional GNSS (Global Navigation Satellite System) and INS integrated navigation method cannot maintain high-precision positioning performance under poor GNSS observation conditions for a long time. In order to improve the accuracy of position and attitude estimation, an enhanced GNSS/INS positioning method with position and attitude constraints was proposed based on the traditional loosely coupled GNSS/INS integration. Position constraint information from light detection and ranging (LiDAR) and attitude constraint information from time-difference-carrier-phase and non-holonomic constraints were adopted to correct the integrated navigation errors. The simulation results showed that the positioning accuracy of the proposed method can reach centimeter level. Compared with the traditional GNSS/INS integration, the positioning accuracy was improved by 76

The BeiDou global satellite navigation system (BDS) is an important tool for precise remote time and frequency transfer. The current day-boundary discontinuity in the data process of the BDS-3 carrier phase (CP) time transfer limits the exploitation of the full potential of the BeiDou system, particularly over extended periods (e.g. >1 day). This study focused on the daily discontinuities in BDS-3 ground time and frequency transfer induced by the current satellite orbit and clock products from the individual Multi-GNSS experiment (MGEX) analysis center. An interpolation approach that bridges the daily jump of satellite products is proposed to improve the continuity of BDS-3 time and frequency transfer when using the CP technique. Only third-order polynomial is required to smooth and bridge the impact of the daily jump for medium earth orbit (MEO) and inclined geosynchronous satellite orbit (IGSO) satellite products. Eight MGEX network stations equipped with various types of receivers and antennas with dual-frequency BDS-3 signals were used to establish four time transfer links (i.e. BRUX – PTBB, AMC4–HOB2, LCK4–USUD, and ONS1–LCK3) and evaluate their effectiveness. These results reveal that the current BDS-3 satellite product discontinuity from individual analysis center easily caused the drift of phase residuals near the day-boundary epoch, with mean amplitudes of 0.081 m and 0.434 m for MEO and IGSO satellite products, respectively. The proposed approach can contribute to effectively eliminating this drift in time and frequency transfer. Furthermore, it achieved a noticeable reduction in the percentage of negative day-boundary discontinuities in BDS-3 ground time and frequency transfer results. The average values of the jump were 0.143 ns, 0.367 ns, 0.138 ns, and 0.131 ns for the BRUX – PTBB, AMC4–HOB2, LCK4–USUD, and ONS1–LCK3 time links, respectively, representing improvements of 11.4%, 40.5%, 4.5%, and 12.0%, respectively, compared with the traditional approach.

Global Navigation Satellite System (GNSS)/Inertial Navigation System (INS) integrated navigation is one of the key methods for achieving precise positioning in complex urban environments. However, in some scenarios such as urban canyons, overpasses, and foliage occlusion, GNSS signals are frequently attenuated or interrupted, leading to degraded positioning accuracy when relying solely on INSs. To address this limitation, this study developed an improved GNSS/INS-integrated navigation algorithm based on a hybrid framework that combines a Robust Adaptive Kalman Filter (RAKF) with a Radial Basis Function (RBF) neural network. The RAKF allows a multi-criterion optimization strategy to be created to adaptively adjust the measurement noise covariance matrix according to GNSS data quality indicators such as PDOP, the number of satellites, and signal quality factors. This enhances the filter’s robustness and outlier detection capability under degraded GNSS conditions. Meanwhile, the RBF network is trained to predict pseudo-position increments, which substitute missing GNSS measurements during signal outages to maintain continuous navigation. Real-world vehicular experiments were conducted to evaluate the proposed RBF-aided RAKF (RBF-RAKF) against three other methods: the Extended Kalman Filter (EKF), standard RAKF, and RBF-aided Kalman Filter (RBF-KF). The experimental results demonstrate that during GNSS outages the proposed method achieved root mean square (RMS) positioning errors of 0.94, 1.02, and 0.21 m in the north, east, and down directions, respectively, representing improvements of over 90% compared with conventional filters. Moreover, the algorithm maintained meter-level horizontal accuracy and sub-meter vertical precision under severe GNSS signal degradation. These results confirm that the proposed RBF-RAKF algorithm provides stable and high-precision navigation performance in challenging urban environments.

The atmospheric delay corrections including tropospheric delay corrections and ionospheric delay corrections are critical information in positioning solutions of PPP-RTK (Precise Point Positioning-Real-Time Kinematic) rapid positioning service, in order to achieve rapid convergence of fixed ambiguity and high-precision positioning solutions in a short time. With rapid development of GNSS (Global Navigation Satellite Systems), the availability of multi-frequency and multi-GNSS observations has led to a significantly increase of number of atmospheric delay corrections, which will surely put more pressure on data transmission and reception to PPP-RTK service and potentially improve the cost of operational economy of users. Therefore, we propose a simplified method to process the atmospheric delay corrections, which sacrifices a certain degree of mathematical rigorousness in positioning equations, but significantly reduces the number of corrections while ensuring that the degradation in positioning performance remains in a controllable way. The traditionally used slant tropospheric delay corrections are simplified to zenith tropospheric delay corrections, and the usually used pseudo-range slant ionospheric delay corrections are no longer transmitted and are simply calculated as the opposite of phase slant ionospheric delay corrections at the user station. The experiments results show that the number of atmospheric delay corrections reduced about 60 %, and the differences between the atmospheric delay corrections processed by the proposed simplified method and the typically traditional method are acceptable to a certain extent for users located in networks with small/medium scale and under relatively stable atmospheric conditions. Although the employment of atmospheric corrections processed by the proposed method introduces the longer TTFF (Time To First Fix) and the lower success fixing rate of ambiguity, the positioning solutions with the proposed simplified method is still basically adequately to meet the positioning requirement of user stations. The proposed simplified model is beneficial to some users who prefer to put more emphasis on equipment portability and data transmission economy.

In critical safety domains like civil aviation, Receiver Autonomous Integrity Monitoring (RAIM) supports integrity services from oceanic routes to non-precision approaches (NPA). The expansion of GNSS (Global Navigation Satellite System) civil frequencies improves these services. To evaluate RAIM availability enhancements from Multi-Frequency Multi-System (MFMS) integration, observation data and broadcast ephemeris from 18 global Multi-GNSS Experiment (MGEX) stations were analyzed for GPS/BDS/Galileo (Global Positioning System/BeiDou Navigation Satellite System/Galileo Navigation Satellite System) combinations. The results indicate that MFMS integration significantly improves positioning and integrity performance compared to single-system operation.

Global Navigation Satellite System (GNSS) buoys are widely used to retrieve wave parameters such as significant wave heights (SWHs) and dominant wave periods. In addition to the statistical methods employed to estimate wave parameters, spectral-analysis-based approaches are also frequently utilized to analyze them. This study presents statistical and spectral methods for retrieving wave parameters at GNSS buoy positioning resolution in the Huanghai Sea area. To verify the method’s effectiveness, the zero-crossing method and three spectral analysis techniques (periodogram, autocorrelation function, and autoregressive model methods) were used to estimate wave height and period for comparison. The vertical positioning resolution was decomposed into low-frequency ocean-tide level information and high-frequency wave height and period information with the Complete Ensemble Empirical Mode Decomposition (CEEMD) method and moving average filtering. The horizontal positioning results and velocity parameters were used to determine the wave direction using directional spectrum analysis. The results show that the three spectral methods yield consistent effective wave heights, with a maximum difference of 0.02 s in the wave period. Compared with the zero-crossing method results, the wave height and period obtained through spectral analysis differ by 0.05 m and 0.79 s, respectively, while the average wave height and period differ by 0.09 m and 0.08 s, respectively. The GNSS-derived wave heights also closely match tidal gauge observations, confirming the method’s validity. Directional spectrum analysis indicates that wave energy is concentrated in the 0.2–0.25 Hz frequency band and within a directional range of 0° ± 30°, with a dominant northward propagation trend. These findings demonstrate that the proposed approach can provide high accuracy and physical consistency for GNSS-based wave monitoring under complex sea conditions.

Real-time water level monitoring is of critical significance in flood disaster mitigation and water resource management. This paper proposes a real-time Global Navigation Satellite System Interferometric Reflectometry (GNSS-IR) water level retrieval method based on the hybrid integration of sliding window and Long Short-Term Memory (LSTM). By dynamically updating input sequences through the sliding window mechanism, an LSTM model captures both temporal and nonlinear characteristics of water level variations, enabling high-precision real-time prediction. Experimental results demonstrate that during non-typhoon seasons, the predicted sea level achieves a correlation coefficient of 99.78 % and a root mean square error (RMSE) of 10.81 cm compared to tide gauge measurements. The system still formulates stable predictions for near-real-time sea level monitoring even with 1.31 % data gaps caused by missing values, which satisfies the requirements. During storm surge, the correlation coefficient between predicted and measured data reaches 96.18 %, with a RMSE of 16.55 cm. Notably, the method maintains robust real-time predictive capability even under extreme conditions where wind speeds exceed 30 m/s and retrieval values significantly decrease. These results demonstrate that the proposed method achieves high accuracy under both normal and extreme hydrological conditions, providing an efficient, cost-effective technical pathway for nearshore real-time water level monitoring and disaster early warning.

Precise point positioning–real-time kinematic (PPP-RTK) enables users to achieve rapid centimeter-level absolute positioning accuracy within a few epochs. The interpolation of ionospheric delay corrections at the user end, extracted from reference stations, constitutes a key aspect of the process, which depends not solely on the precision of the interpolation model. This study investigates the recommended number of selected reference stations and proposes a method to mitigate the potential loss of observations due to missing ionospheric corrections. According to the experimental results, the number of reference stations should be determined based on the reference network size. Under normal conditions (terrain is relatively flat and the atmospheric conditions are inactive) where reference stations are approximately evenly distributed in all directions, and using low-order surface interpolation model, for networks with 50 km spacing, four or five reference stations are recommended, while for 100 km networks, six or seven stations are enough to calculate precise corrections. Adding more stations beyond these thresholds provides limited improvement in interpolation accuracy and increases the communication load. In addition, an interpolation basis recovery algorithm is proposed to preserve otherwise excluded satellite observations through intelligent handling of correction data gaps at individual reference stations. Experimental validation demonstrates that the recovered ionospheric delay corrections obtained through the algorithm deviate from the ground-truth interpolated values of no more than ±1 cm, an accuracy level deemed adequate for PPP-RTK applications. Furthermore, approximately 3% of the observations, which would otherwise have been discarded due to the missing corrections from a specific reference station, are retained by the algorithm.

Precise point positioning with ambiguity resolution (PPP-AR) has been developed to restore the integer nature of estimated ambiguities, significantly enhancing time and frequency transfer performance. Since the end of 2019, the International Bureau of Weights and Measures has adopted PPP-AR for calculating integer PPP (IPPP) results in the time links of international atomic time. Dual-frequency (DF) ambiguity resolution method has been applied to IPPP time and frequency transfer. To incorporate multi-frequency observations into IPPP time and frequency transfer, this study adopts cascaded ambiguity resolution and cascaded cycle slip detection to investigate the performance of real-time multi-frequency IPPP in time and frequency transfer. The quality and availability of multi-frequency phase bias products were first assessed. Except for the GPS L5 frequency, the standard deviation (STD) of the phase bias products remained below 0.2 ns, with 78.79% of the narrow-lane fractional part residuals falling within 0.15 cycles, and 95.13% within 0.30 cycles. The IPPP time and frequency transfer performance of DF, triple-frequency (TF), and quad-frequency (QF) was systematically analyzed. Across both zero baselines and various baseline experiments, IPPP demonstrated reduced time transfer error fluctuations compared to float solutions. Furthermore, as the averaging time increased to several hours, IPPP demonstrated improved frequency stability. Following DF ambiguity resolution, the STD value of the zero-baseline common clock difference decreased by 5.5 ps, while the TF improved from 67.53 ps to 64.13 ps. Meanwhile, the frequency stability of the QF ambiguity-fixed solution reached 9.910 x 10-17 at 2.84 d averaging. Furthermore, for both short and long baselines, IPPP led to a significant reduction in time transfer errors. As the averaging time increased, IPPP demonstrated further enhancement in frequency stability.