Although ionosphere-free (IF) combination is usually employed in long-range precise positioning, in order to employ the knowledge of the spatiotemporal ionospheric delays variations and avoid the difficulty in choosing the IF combinations in case of triple-frequency data processing, using uncombined observations with proper ionospheric constraints is more beneficial. Yet, determining the appropriate power spectral density (PSD) of ionospheric delays is one of the most important issues in the uncombined processing, as the empirical methods cannot consider the actual ionosphere activities. The ionospheric delays derived from actual dual-frequency phase observations contain not only the real-time ionospheric delays variations, but also the observation noise which could be much larger than ionospheric delays changes over a very short time interval, so that the statistics of the ionospheric delays cannot be retrieved properly. Fortunately, the ionospheric delays variations and the observation noise behave in different ways, i.e., can be represented by random-walk and white noise process, respectively, so that they can be separated statistically. In this paper, we proposed an approach to determine the PSD of ionospheric delays for each satellite in real-time by denoising the ionospheric delay observations. Based on the relationship between the PSD, observation noise and the ionospheric observations, several aspects impacting the PSD calculation are investigated numerically and the optimal values are suggested. The proposed approach with the suggested optimal parameters is applied to the processing of three long-range baselines of 103 km, 175 km and 200 km with triple-frequency BDS data in both static and kinematic mode. The improvement in the first ambiguity fixing time (FAFT), the positioning accuracy and the estimated ionospheric delays are analysed and compared with that using empirical PSD. The results show that the FAFT can be shortened by at least 8% compared with using a unique empirical PSD for all satellites although it is even fine-tuned according to the actual observations and improved by 34% compared with that using PSD derived from ionospheric delay observations without denoising. Finally, the positioning performance of BDS three-frequency observations shows that the averaged FAFT is 226 s and 270 s, and the positioning accuracies after ambiguity fixing are 1 cm, 1 cm and 3 cm in the East, North and Up directions for static and 3 cm, 3 cm and 6 cm for kinematic mode, respectively.

The performance of low-cost smart terminals is limited by the performance of their low-cost Global Navigation Satellite System (GNSS) hardware and chips, as well as by the impact of complex urban environments, which affect the positioning accuracy and stability of GNSS services. To this end, this paper proposes a robust adaptive Kalman filter for different environments that can be applied after data preprocessing. Based on the Kalman filter algorithm, a robust estimation approach is introduced into real-time kinematic (RTK) positioning to make judgments on the abnormal observation values of low-cost smart terminals, which amplifies the variance and covariance of the outlier observation equation, and reduces the impact of outliers on positioning performance. The Institute of Geodesy and Geophysics III (IGG III) function is used for regulation purposes, where prior information is modified and refreshed using the equivalent weight matrix and adaptive factors, thus reducing the impact of system model errors on system state estimation results. In addition, a robust factor is defined to adjust positioning deviation weighting between the pre- and post-test robust estimates. The experimental results show that after robust RTK positioning in the static experiments, the overall improvement in positioning accuracies of the Xiaomi 8, Huawei P40, Huawei mate40, and low-cost M8 receiver reached 29.6%, 31.3%, 32.1%, and 30.7%, respectively. Similarly, after applying the proposed robust method in the dynamic experiments, the overall positioning accuracies of the Xiaomi 8, Huawei P40, Huawei mate40, and the low-cost M8 receiver improved by 28.3%, 32.9%, 35.4%, and 26.2%, respectively. The experimental results reveal that an excellent positioning effect of a smartphone is positively correlated with robust RTK positioning performance. However, it is worth noting that when the positioning accuracy reaches a high level, such as the positioning results achieved using low-cost receivers, the robustness performance shows a relatively decreasing trend. This finding suggests that under the condition of high positioning accuracy, the sensitivity of specific positioning equipment to interference sources may increase, resulting in a decline in the effect of robust RTK positioning.

Concerning the triple-frequency ambiguity resolution, in principle there are three different realizations. The first one is to fix all the ambiguities of the original frequencies together. However, it is also believed that fixing the combined integer ambiguities with longer wavelength, such as extra-wide-lane (EWL), wide-lane (WL), should be advantageous. Also, it is demonstrated that fixing sequentially EWL, WL and one type of original ambiguities provides better results, as the previously fixed ambiguities increase parameters’ precision for later fixings. In this paper, we undertake a comparative study of the three fixing approaches by means of experimental validation. In order to realize the three fixing approaches from the same information in terms of adjustment, we developed a processing strategy to provide fully consistent normal equations. We first generate the normal equation with the original undifferentiated carrier phase ambiguities, then map it into that with the combined and double-differenced ambiguities required by the individual approach for fixing. Four baselines of 258 m, 22 km, 47 km and 53 km are selected and processed in both static and kinematic mode using the three ambiguity-fixing approaches. Indicators including time of first fixed solution (TFFS), the correct fixing rate, positioning accuracy and RATIO are used to evaluate and investigate results. We also made a preliminary theoretical explanation of the results by looking into the decorrelation procedure of the ambiguity searching algorithm and the intermediate results. As conclusions, integrated searching of original ambiguities or combined ambiguities has almost the same fixing performance, whereas the sequential fixing of EWL, WL and B1 ambiguities overperforms the integrated searching. By the way, the third-frequency data can shorten the TFFS significantly but can hardly improve the positioning.

Multi Global Navigation Satellite System (GNSS) Precise Point Positioning (PPP) has become the mainstream of PPP technology. Due to the differences in the coordinates and time references of each GNSS, multi-GNSS PPP must include additional Inter-System Bias (ISB) parameters to ensure compatibility between different GNSSs. Therefore, research on the characteristics of ISB is also essential. To analyze the short- and long-term time characteristics of multi-GNSS ISBs, as well as their relationship with receiver type and receiver antenna type, the Undifferenced and Uncombined (UDUC) PPP model of Global Positioning System (GPS), BeiDou navigation satellite system (BDS), and Galileo satellite navigation system (Galileo) was rigorously derived, and the physical of ISBs was elaborated in depth. ISB parameters were estimated and analyzed using 31 days of data from the 31 Multi-GNSS Experimental stations (MGEX). The results indicate that: (1) the ISB value is dependent on the station receiver type, receiver antenna type, analysis center product utilized, and GNSS system. (2) The short-term time characteristics of ISB-COM, ISB-WUM, and ISB-GBM are similar for the same station but not for the long term. In addition, ISBs are more stable in the short term. (3) There is little correlation between the ISB time characteristics, the receiver type, and the receiver antenna type, and the day-boundary discontinuity(DBD) on the ISB can be ignored for the concussive days’ process.

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.

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.