To overcome the limitations of traditional evaluation indicators in determining the optimal wavelet decomposition level, this paper proposes an adaptive method for selecting the best decomposition level by combining the Jarque–Bera test and a composite weighting approach. Firstly, in the noise extraction stage, the Jarque–Bera test is employed to ensure that the extracted noise follows Gaussian white noise characteristics, thereby avoiding issues of insufficient denoising or signal distortion. Secondly, in the evaluation stage of the denoised signal, a comprehensive consideration of the geometric and physical meanings of various evaluation metrics, as well as the Pearson correlation coefficients between them, is undertaken. The RMSE and smoothness are selected as evaluation indicators for the denoising performance. Since these two metrics describe signal characteristics from different dimensions, a weighted combination approach is used to generate a single composite evaluation index. Additionally, to overcome the limitations of using a single weighting method, a composite weighting strategy is proposed by combining the entropy weight method and the coefficient of variation method. The composite coefficient between these two weighting methods is calculated using the variance coefficient method, yielding a new composite evaluation metric. A smaller value of this metric indicates better denoising performance, and the corresponding optimal decomposition level is more accurately determined. The simulation results demonstrate that the proposed comprehensive evaluation method can accurately determine the optimal wavelet decomposition level in both known and unknown truth-value cases, exhibiting a high accuracy and good applicability. Furthermore, the experimental results show that using the optimal decomposition level determined by the proposed method for wavelet denoising leads to smoother peak regions, more stable waveforms and significantly improved denoising performance.

In recent years, several Global Navigation Satellite Systems (GNSSs) have begun offering free satellite-based Precise Point Positioning (PPP) services, including Galileo’s High Accuracy Service (HAS), BDS’s PPP-B2b, and Japan’s Quasi–Zenith Satellite System (QZSS) multi-GNSS advanced orbit and clock augmentation (MADOCA). In real-time positioning correction methods, individual services are prone to interruptions and anomalies. This study proposes combining multiple correction services to enhance correction availability and continuity from the user side, thereby fully leveraging observational data from various satellite positioning services. After detecting outliers in clock corrections and standardizing reference frames across services, a Kalman filter is employed to estimate the combined clock corrections. Since GPS is currently the only GNSS supported by these services, this study incorporates GPS correction data from Galileo’s HAS, BDS’s PPP-B2b and Japan’s QZSS MADOCA in a one-week experiment. By combining satellite-based correction products from different services, users can fully utilize GNSS corrections from each. This study demonstrates that combing corrections from multiple services significantly improves their availability and continuity. The average correction availability increased from approximately 75% for a single service to 99% for combined corrections. Static and simulated kinematic experiments at Multi-GNSS Pilot Project stations show that the combination strategy improves positioning accuracy by approximately 52.7%, 10.6% and 59.1% compared to PPP-B2b, MADOCA and HAS, respectively, with convergence time reductions of 48.5%, 37.1% and 67.1%. Marine experiment results with an unmanned underwater vehicle demonstrate that the combination strategy improves horizontal positioning accuracy by 19.7%, 59.1% and 17.8%, and vertical positioning accuracy by 54.1%, 77.8% and 1.2% compared to PPP-B2b, HAS and MADOCA.

This paper proposes an evaluation method for selecting the optimal linear combination coefficients for five-frequency cycle slip detection in BeiDou-3. First, the essence of the optimal linear combination problem for cycle slip detection is analyzed, and two linear combination constraint strategies are introduced: geometry-free ionospheric-free (GFIF) and geometry-free (GF) constraints. For the GFIF combination, a linear combination index is proposed to assess the cycle slip detection performance of different linear combinations, providing a quantitative measure for evaluating the performance of various combination coefficients. By solving the constrained optimization problem, the optimal linear combinations for phase–only geometry-free ionospheric-free (PGFIF) and phase-code geometry-free ionospheric-free (PCGFIF) are determined. For the GF combination, based on phase biases induced by cycle slips, standard deviations of cycle slip detection measurements, ionospheric delay amplification factors, and the success rate of detecting insensitive cycle slips using GFIF, suitable phase-only geometry-free combination (PGF) combinations are selected. The optimal linear combination for PGF is then determined based on the principle of minimizing the condition number. For cycle slip repair, the least squares method is used to solve for the float solution and its covariance matrix, while the LAMBDA algorithm is employed to search for the integer solution of the cycle slip. Finally, the effectiveness of the proposed method is validated through experiments using static, shipborne kinematic, ionospheric active period, and low-sampling-rate data. The experimental results demonstrate that, under five-frequency conditions, the proposed algorithm can accurately detect and repair all simulated cycle slips, confirming the feasibility and effectiveness of the method.

To address the long convergence time of satellite-based Precise Point Positioning (PPP) using BDS-3 PPP-B2b and Galileo high accuracy service (HAS) in the Asia-Pacific region, this study investigates real-time multi-frequency phase OSB estimation and PPP-AR based on the integration of the two services. In the Asia-Pacific region, real-time orbit and clock corrections from PPP-B2b and Galileo HAS were first recovered and unified in reference. A tightly combined real-time phase OSB estimation method was then developed using observations from the complementary constellations, and a corresponding tightly combined real-time PPP-AR model was established. Four schemes were designed: PPP-B2b BDS-3/GPS, HAS Galileo/GPS, PPP-B2b BDS-3/GPS + HAS Galileo, and HAS Galileo/GPS + PPP-B2b BDS-3. Phase OSB products were estimated and ambiguities were fixed using 10 d of multi-frequency global navigation satellite system observations from 21 reference stations in the Asia-Pacific region. PPP/PPP-AR performance was validated using observations from 11 rover stations, and the influence of GPS precise ephemerides with different accuracies on phase OSB estimation was also analyzed. The results show that the integrated PPP-B2b/Galileo HAS schemes provided better satellite geometry than the single-service schemes. Among the tested schemes, B2b C/G + HAS E achieved the best positioning performance, with mean E, N, and U accuracies of 4.7 cm, 2.9 cm, and 7.6 cm, respectively, for the PPP-AR fixed solution, and a mean convergence time of 5.65 min. Compared with the float solution, the improvements were 27.7%, 17.1%, 16.5%, and 42.0%, respectively. In the Asia-Pacific region, HAS GPS showed data interruptions and a smaller number of visible satellites, which resulted in lower phase OSB sequence stability and PPP/PPP-AR positioning accuracy than PPP-B2b GPS. This explains the better performance of B2b C/G + HAS E than HAS E/G + B2b C. The proposed satellite-based PPP integration scheme uses satellite-broadcast corrections at the user end and does not require real-time correction transmission through terrestrial communication networks, providing a technical option for real-time high-precision positioning in areas with limited network access in the Asia-Pacific region.