Search Results (43)

The undifferenced and uncombined (UDUC) model preserves raw code and carrier-phase observations for each frequency, avoiding differencing or ionosphere-free combinations. This approach enables the direct estimation of atmospheric parameters. However, the stochastic characteristics of these parameters, particularly ionospheric delay, are often oversimplified or based on empirical assumptions, limiting the accuracy and convergence speed of Precise Point Positioning (PPP). To address this issue, this study introduces a stochastic constraint model based on the power spectral density (PSD) of ionospheric variations. The PSD describes the distribution of ionospheric delay variance across temporal frequencies, thereby providing a physically meaningful constraint for modeling their temporal correlations. Integrating this PSD-derived stochastic model into the UDUC framework improves both ionospheric delay estimation and PPP performance, especially under disturbed ionospheric conditions. This paper presents a BDS PPP/PPP-AR method that estimates the ionospheric power spectral density (IPSD) in real time. Vondrak smoothing is applied to suppress noise in ionospheric observations before IPSD estimation. Experimental results demonstrate that the proposed approach significantly improves convergence time and positioning accuracy. Compared to the empirical IPSD model, the PPP mode using the estimated IPSD reduced horizontal and vertical convergence times by 11.1% and 13.2%, and improved the corresponding accuracies by 15.7% and 12.6%, respectively. These results confirm that real-time IPSD estimation, coupled with Vondrak smoothing, establishes an adaptive and robust ionospheric modeling framework that enhances BDS PPP and PPP-AR performance under varying ionospheric conditions. Full article

To address the problem of the degraded positioning accuracy of the long-baseline undifferenced network RTK (URTK) under extreme space weather conditions, herein, we propose a user-side atmospheric delay estimation strategy based on the undifferenced network RTK concept to enhance positioning performance in geomagnetic storm environments. First, an ambiguity-resolution model that jointly estimates atmospheric error parameters is used to fix the carrier-phase integer ambiguities for long-baseline reference stations. The accurately fixed inter-station ambiguities are then linearly transformed to recover station-specific undifferenced integer ambiguities; undifferenced observation errors at each reference station are computed to generate corresponding undifferenced correction terms. Lastly, recognizing that ionospheric delays vary sharply during geomagnetic storms and can severely compromise the availability of regional undifferenced correction models, we estimate the residual atmospheric parameters on the user side after applying regional corrections. Experimental results show that the server side is not significantly impacted during geomagnetic storms and can continue operating normally. On the user side, augmenting the solution with atmospheric parameter estimation effectively improves positioning availability. Under strong geomagnetic storms, the proposed mode improves user-station positioning accuracy by 63.7%, 60.7%, and 64.4% in the east (E), north (N), and up (U) components, respectively, relative to the conventional user-side solution; under moderate storm conditions, the corresponding improvements are 16.7%, 10.0%, and 11.1%. Full article

The precise point positioning (PPP) method in GNSS is based on the processing of undifferenced phase observations. For long static sessions, this method provides results characterized by accuracies better than one centimeter, and has become a standard practice in the processing of geodetic permanent stations data. However, a drawback of the PPP method is its slow convergence, which results from the necessity of jointly estimating the coordinates and the initial phase ambiguities. This poses a challenge for very short sessions or kinematic applications. The introduction of new satellites in Low Earth Orbits (LEO) that provide phase observations for positioning, such as those currently provided by GNSS constellations, has the potential to radically improve this scenario. In this work, a preliminary case study is discussed. For a given day, two configurations are analyzed: the first considers only the GNSS satellites currently in operation, while the second includes a simulated constellation of LEO satellites. For both configurations, the geometric quality of a PPP solution is evaluated over different session lengths throughout the day. The adopted quality index is the trace of the cofactor matrix of the estimated coordinates, commonly referred to as the position dilution of precision (PDOP). The simulated LEO constellation demonstrates the capability to enhance positioning performance, particularly under conditions of good sky visibility, where the time needed to obtain a reliable solution decreases significantly. Furthermore, even in scenarios with limited satellite visibility, the inclusion of LEO satellites helps to reduce PDOP values and overall convergence time. Full article

Conventional Global Navigation Satellite System (GNSS) time transfer algorithms typically model receiver clock offsets as white noise for estimation, neglecting the physical characteristics of atomic clocks, which consequently limits the performance of GNSS time transfer. To overcome this limitation, this study proposes an undifferenced GPS/Galileo combined Precise Point Positioning (PPP) time transfer model, incorporating both one-state (only clock offset parameter) and two-state (both clock offset and frequency offset parameters) refined clock models with clock instantaneous re-initialization (CIR) strategy at the day boundary epoch. Using observations from International GNSS Service (IGS) Multi-GNSS Experiment (MGEX) stations equipped with external hydrogen masers, precise time transfer performance under refined clock models was evaluated based on undifferenced GPS/Galileo combined PPP float solutions and PPP ambiguity resolutions. Experimental results demonstrate that, compared to traditional models, the refined clock models improve time transfer accuracy and frequency stability by an average of 6.7% and 25.8%, respectively. The improvement is most significant for short term frequency stability, with a maximum enhancement exceeding 85%. As the averaging time increases, the improvement in long term frequency stability gradually diminishes. Notably, the two-state refined clock model slightly outperforms the one-state model in time transfer performance, with the two-state refined clock model improving short-, medium-, and long term frequency stability by 11.5%, 8.0%, and 0.2%, respectively, compared to the one-state refined clock model. These findings strongly advocate adopting the two-state refined clock model to optimize both time transfer precision and short term stability in high-accuracy applications. Full article

Precise point positioning ambiguity resolution (PPP-AR) is a key technique for high-precision global navigation satellite system (GNSS) observations, with phase bias products playing a critical role in its implementation. The multi-GNSS experiment analysis center at Wuhan University (WUM) has adopted the undifferenced ambiguity resolution (UDAR) approach to generate high-precision orbit, clock, and observable-specific bias (OSB) products to support PPP-AR since day 162 of 2023. This study presents the analysis strategy employed and assesses the impact of the transition to ambiguity resolution on the orbit precision, using metrics such as orbit boundary discontinuities (OBD) and satellite laser ranging (SLR) validation. Additionally, the stability of the OSB products and the overall performance of PPP-AR solutions are evaluated. The OBD demonstrates specific improvements of 7.1% and 9.5% for GPS and Galileo, respectively, when UDAR is applied. Notably, BDS-3 medium Earth orbit satellites show a remarkable 15.2% improvement compared to the double-differenced results. However, for the remaining constellations, the improvements are either minimal or result in degradation. Using GPS and GLONASS solutions from the International GNSS Service (IGS) and other solutions from the European Space Agency (ESA) as references, the orbit differences of WUM solutions based on UDAR exhibit a significant reduction. However, the improvements in SLR validation are limited, as the radial orbit precision is primarily influenced by the dynamic model. The narrow-lane ambiguity fixing rate for static PPP-AR, based on data from approximately 430 globally distributed stations, reaches 99.2%, 99.2%, 88.8%, and 98.6% for GPS, Galileo, BDS-2, and BDS-3, respectively. The daily repeatability of station coordinates is approximately 1.4 mm, 1.9 mm, and 3.9 mm in the east, north, and up directions, respectively. Overall, these results demonstrate the effectiveness and potential of WUM’s undifferenced ambiguity resolution approach in enhancing GNSS data processing and facilitating PPP-AR applications. Full article

Real-time integrity monitoring (IM) is essential for autonomous vehicle positioning, requiring high availability and manageable computational load. This research proposes using precise point positioning real-time kinematic (PPP-RTK) as the positioning method, combined with an improved classification adaptive Kalman filter (CAKF) for processing. PPP-RTK enhances IM availability by allowing undifferenced and uncombined observations, enabling individual observation exclusion during fault detection and exclusion (FDE). The CAKF reduces FDE computational load by using a robustness test instead of traditional FDE methods, improving precision and availability in protection level estimation. Epoch-wise weighting adjustments in the robustness test create a more accurate stochastic model, aided by an adaptive unit weight variance (UWV) calculated with a sliding window, achieving a 7–28% UWV reduction. Three test scenarios with up to four simultaneous faults in code and phase observations, ranging from 1 to 200 m and 0.4 to 20 m, respectively, demonstrated successful identification and de-weighting of faults, resulting in maximum positioning errors of 6 mm (horizontal) and 11 mm (vertical). The method reduced FDE computational load by 50–99.999% compared to other approaches. Full article

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. Full article

The BeiDou Navigation Satellite System (BDS) has developed rapidly, and the combination of BDS Phase II (BDS-2) and BDS Phase III (BDS-3) has attracted wide attention. It is found that there are code ISBs between BDS-2 and BDS-3, which may have a certain impact on the BDS-2 and BDS-3 combined positioning. This paper focuses on the performance of BDS-2/BDS-3 combined B1I single-frequency pseudorange positioning and investigates the positioning performance with and without code ISBs correction for different types of receivers, include geodetic GNSS receivers and low-cost receivers. The results show the following: (1) For geodetic GNSS receivers, the code ISBs of each receiver is about −0.3 m to −0.8 m, and the position deviation is reduced by 7% after correcting code ISBs. The code ISBs in the baseline with homogeneous receivers has a little influence on the positioning result, which can be ignored. The code ISBs in the baseline with heterogeneous receivers is about −0.5 m, and the position deviation is reduced by 4% after correcting code ISBs. (2) The code ISBs in the low-cost receivers are significantly larger than those in the geodetic GNSS receivers, and the impact on the positioning performance of the low-cost receivers is significantly greater than that on the geodetic GNSS receivers. After correcting the code ISBs, the position deviation of low-cost receivers can be reduced by around 12% for both undifferenced and differenced modes. (3) For low-cost receivers, correcting the code ISBs can increase the number of epochs successfully solved, which effectively improves the low-cost navigation and positioning performance. (4) The carrier-phase-smoothing method can effectively reduce the distribution dispersion of code ISBs and make the estimation of ISBs more accurate. The STD values of estimated code ISBs in geodetic GNSS receivers are reduced by about 40% after carrier-phase smoothing, while the corresponding values are reduced by about 7% in low-cost receivers due to their poor carrier-phase observation quality. Full article

The multi-global navigation satellite system (GNSS) undifferenced and uncombined precise point positioning (UU-PPP), as a high-precision ionospheric observables extraction technology superior to the traditional carrier-to-code leveling (CCL) method, has received increasing attention. In previous research, only dual-frequency (DF) or multi-frequency (MF) observations are used to extract slant ionospheric delay with the UU-PPP. To reduce the cost of ionospheric modeling, the feasibility of extracting ionospheric observables from the multi-GNSS single-frequency (SF) UU-PPP was investigated in this study. Meanwhile, the between-satellite single-differenced (SD) method was applied to remove the effects of the receiver differential code bias (DCB) with short-term time-varying characteristics in regional ionospheric modeling. In the assessment of the regional real-time (RT) between-satellite SD ionospheric model, the internal accord accuracy of the SD ionospheric delay can be better than 0.5 TECU, and its external accord accuracy within 1.0 TECU is significantly superior to three global RT ionospheric models. With the introduction of the proposed SD ionospheric model into the multi-GNSS kinematic RT SF-PPP, the initialization speed of vertical positioning errors can be improved by 21.3% in comparison with the GRAPHIC (GRoup And PHase Ionospheric Correction) SF-PPP model. After reinitialization, both horizontal and vertical positioning errors of the SD ionospheric constrained (IC) SF-PPP can be maintained within 0.2 m. This proves that the proposed SDIC SF-PPP model can enhance the continuity and stability of kinematic positioning in the case of some GNSS signals missing or blocked. Compared with the GRAPHIC SF-PPP, the horizontal positioning accuracy of the SDIC SF-PPP in kinematic mode can be improved by 37.9%, but its vertical positioning accuracy may be decreased. Overall, the 3D positioning accuracy of the SD ionospheric-constrained RT SF-PPP can be better than 0.3 m. Full article

Compared to the traditional ionospheric-free (IF) precise point positioning (PPP) model, the undifferenced and uncombined (UU) PPP has the advantages of lower observation noise and the ability to obtain ionospheric information. Thanks to the IGS (International GNSS Service), real-time service (RTS) can provide RT vertical total electron content (VTEC) products, and an enhanced RT UU-PPP based on the RT-VTEC constraints can be achieved. The global performance of the BeiDou Navigation Satellite System-2 (BDS-2) and BDS-3 joint RT UU-PPP using different RTS products was investigated. There is not much difference in the RTS orbit accuracy of medium earth orbit (MEO) satellites among all analysis centers (ACs), and the optimal orbit accuracy is better than 5, 9, and 7 cm in the radial, along-track, and cross-track directions, respectively. The orbit accuracy of inclined geosynchronous orbit (IGSO) satellites is worse than that of MEO satellites. Except for CAS of 0.46 ns, the RTS clock accuracy of MEO satellites for other ACs achieves 0.2–0.27 ns, and the corresponding accuracy is about 0.4 ns for IGSO satellites. In static positioning, due to the limited accuracy of RT-VTEC, the convergence time of the enhanced RT UU-PPP is longer than that of RT IF-PPP for most ACs and can be better than 25 and 20 min in the horizontal and vertical components, respectively. After convergence, the 3D positioning accuracy of the static RT UU-PPP is improved by no more than 8.7%, and the optimal horizontal and vertical positioning accuracy reaches 3.5 and 7.0 cm, respectively. As for the kinematic mode with poor convergence performance, with the introduction of RT-VTEC constraints, the convergence time of RT UU-PPP can be slightly shorter and reaches about 55 and 60 min in the horizontal and vertical components, respectively. Both the horizontal and vertical positioning accuracies of the kinematic RT UU-PPP can be improved and achieve around 7.5 and 10 cm, respectively. Full article

Network RTK (NRTK), one of the primary means of high-precision real-time kinematic positioning (RTK), has been widely used. The key to providing highly accurate positioning is the ambiguity of the reference station being correctly fixed, but the atmospheric errors must be handled carefully, which seriously affects the efficiency of ambiguity fixing. This paper aims to improve the efficiency of ambiguity fixing by studying the time-varying characteristics of atmospheric errors. Once reasonable constraints are imposed on atmospheric parameters in the uncombined observation model, it can better fix ambiguity. Atmospheric parameters are estimated by random walk at the reference station, and the power spectral density (PSD) of atmosphere is determined by real-time observations, instead of using empirical values or empirical models that do not consider atmospheric variations. The experimental results showed that the real-time estimated PSD can improve the ambiguity fixing time by 18.4% and the ambiguity fixing success rate for the reference station by 11.7%, compared with using empirical PSD for atmospheric parameters. Unlike general NRTK positioning based on differential error correction values, undifferenced NRTK estimates the integer ambiguity and undifferenced error correction value at a single reference station, ensuring the independence of the error correction value of each reference station, and it can be easily broadcast and received through the network, which is more convenient for realizing high-precision RTK positioning for users. Full article

The incorporation of multi-frequency signals into global navigation satellite systems (GNSS) has presented new possibilities for precise positioning and rapid ambiguity resolution. Inter-frequency clock bias (IFCB) pertains to the time-varying biases among distinct frequencies within carrier phase observations in GNSS signals. The appropriate handling of IFCB is critical in enhancing the accuracy and convergence time of precise point positioning (PPP) solutions. The focus of this study is on the proper modeling of phase IFCB in multi-GNSS multi-frequency PPP. In this paper, the optimal IFCB power spectral density value of 0.6 m/sqrt(s) is first determined. To obtain the optimal stochastic model for IFCB, a thorough comparison and analysis of the product correction and parameter estimation methods is conducted. Additionally, experiments are conducted on the effect of IFCB modeling on the performance of undifferenced and uncombined PPP using data from 130 multi-GNSS experiment stations across the globe over a period of one week in January 2022. The study reveals that the optimal power spectral density for IFCB is within [60, 0.006] m/sqrt(s), modeling IFCB as a random walk is feasible, and the PPP is comparable for the three IFCB processing schemes of product correction, random walk, and white noise. Meanwhile, it is not reasonable to treat IFCB as a random constant or neglect it in the multi-GNSS multi-frequency PPP. In the absence of product correction or for users who require immediate and continuous positioning solutions, modeling IFCBs as random walks can lead to more reliable positioning results and improved convergence performance. Full article

In 2020, the BeiDou-3 global navigation satellite system (BDS-3) was officially completed and put into service. Currently, network real-time kinematic (RTK) technology is considered the main means through which to improve the positioning accuracy of the BeiDou navigation satellite system (BDS). This paper proposes a long-range undifferenced network RTK (URTK) algorithm, based on multi-frequency observation data of the BDS. First, the multi-frequency phase integer ambiguity resolution (AR) model considering atmospheric error parameters is designed, and the multi-frequency phase integer ambiguity of the long-range BDS reference station is determined. Then, the undifferenced integer ambiguity of each reference station is obtained, using linear variation based on the accurately determined phase integer ambiguity between reference stations, and the undifferenced observation error of each reference station is calculated. Considering the weakening spatial correlation of the observation errors between long-range stations, undifferenced classification error corrections of a reference station network are separated, according to different error characteristics. Finally, the inverse distance weighting method is employed to calculate the classification undifferenced error correction of the rover station. The rover station corrects the observation error through applying the undifferenced error correction to achieve high-precision positioning. The measured data of a long-range continuous operation reference station (CORS) network are selected for an experiment. The results show that the proposed algorithm can quickly and accurately realize the resolution of the BDS integer ambiguity of a reference station network and establish an undifferenced area error correction model in order to achieve accurate classification of undifferenced error correction values for a rover station. In China, the BDS-3 is superior to the global positioning system (GPS) in terms of the satellite number, position dilution of precision (PDOP) value, AR success rate, stability, and convergence time. The results show that the AR success rate, stability, and convergence time increase with the operational frequency, and the BDS-3 can achieve centimeter-level positioning of single-system rover stations without relying on the GPS. Full article

Real-time satellite clock offset is a crucial element for real-time precise point positioning (RT-PPP). However, the elapsed time for undifferenced (UD) multi-global navigation satellite system (GNSS) real-time satellite clock offset estimation at each epoch is increased with the growth of stations, which may fall short of real-time application requirements. Therefore, a rapid estimation method for UD multi-GNSS real-time satellite clock offset is proposed to improve the computation efficiency, in which both the dimension of the normal equation (NEQ) and the number of redundant observations are calculated before adjustment; if these two values are larger than the predefined thresholds, the elevation mask is gradually increased until they are less than the predefined thresholds. Then, the clock offset estimation is conducted; this method is called clock offset estimation using partial observations. Totals of 50, 60, 70 and 80 stations are applied to perform experiments. Compared to clock offset estimation using all observations, the elapsed times of clock offset estimation using partial observations can be reduced from 6.80 to 3.10 s, 7.93 to 2.97 s, 12.04 to 3.14 s for 60, 70 and 80 stations, respectively. By using the proposed method, the elapsed time of the clock offset estimation at each epoch is less than 5 s. The estimated clock offset accuracy for GPS, BDS-3, Galileo and GLONASS satellites are better than 0.04, 0.05, 0.03 and 0.16 ns when using the partial observations to estimate clock offset with 50, 60, 70 and 80 stations, respectively. For the multi-GNSS kinematic PPP using the estimated clock offset from 50, 60, 70 and 80 stations with partial observations, the positioning accuracy at 95% confidence level in the east, north and up direction are better than 2.70, 2.20 and 5.60 cm, respectively. Full article

With the full operation of the global BeiDou navigation satellite system (BDS-3), positioning performance can be further enhanced by BDS-3 combined with the regional BeiDou navigation satellite system (BDS-2). However, due to satellite signals being out of lock and the limited visibility of satellites, the traditional multi-frequency BDS-2/3 precise point positioning (PPP) model is unable to maintain great positioning performance under urban environmental conditions. In this study, a mixed multi-frequency undifferenced and uncombined (UDUC) BDS-2/3 PPP model is presented to improve the positioning performance under urban environmental conditions by making full use of B1I, B1C, B2I, B2a, and B3I signals from all visible BDS satellites. In this model, BDS satellites with single-, dual-, triple- and quad-frequency observations all can participate in PPP. The static and kinematic experiments were carried out using the mixed multi-frequency UDUC BDS-2/3 PPP model to fully assess the positioning performance under urban environmental conditions with comparisons to the multi-frequency model. The static experiments indicated that the mixed multi-frequency UDUC BDS-2/3 PPP could continuously achieve decimeter-level positioning accuracy at a cut-off elevation angle of 40°, but part of the BDS-3 PPP would lose resolution due to limited visible satellites. Furthermore, the initial kinematic vehicle experiment showed that mixed multi-frequency UDUC BDS-2/3 PPP had better satellite geometry and more observation redundancy than the traditional multi-frequency model. Compared with the traditional multi-frequency BDS-2/3 model, the positioning accuracy of the mixed multi-frequency model was improved by 51.6, 35.5, and 39.1%, respectively, in east, north, and up directions. The convergence time was shortened by 40%. Full article