Traditionally, an elevation-angle-dependent weighting method is usually used for Global Navigation Satellite System (GNSS) positioning with a geodetic receiver. As smartphones adopt linearly polarized antenna and low-cost GNSS chips, different GNSS observation properties are exhibited. As a result, a carrier-to-noise ratio (C/N0)-dependent weighting method is mostly used for smartphone-based GNSS positioning. However, the C/N0 is subject to the effects of the observation environment, resulting in an unstable observation weight. In this study, we propose a combined elevation angle and C/N0 weighting method for smartphone-based GNSS precise point positioning (PPP) by normalizing the C/N0-derived variances to the scale of the elevation-angle-derived variances. The proposed weighting method is validated in two kinematic PPP tests with different satellite visibility conditions. Compared with the elevation-angle-only and C/N0-only weighting methods, the combined weighting method can effectively enhance the smartphone-based PPP accuracy in a three-dimensional position by 22.7% and 24.2% in an open-sky area, and by 52.0% and 26.0% in a constrained visibility area, respectively.

The carrier phase multipath effect is one of the most significant error sources in the precise positioning of BeiDou Navigation Satellite System (BDS). We analyzed the characteristics of BDS multipath, and found the multipath errors of geostationary earth orbit (GEO) satellite signals are systematic, whereas those of inclined geosynchronous orbit (IGSO) or medium earth orbit (MEO) satellites are both systematic and random. The modified multipath mitigation methods, including sidereal filtering algorithm and multipath hemispherical map (MHM) model, were used to improve BDS dynamic deformation monitoring. The results indicate that the sidereal filtering methods can reduce the root mean square (RMS) of positioning errors in the east, north and vertical coordinate directions by 15%, 37%, 25% and 18%, 51%, 27% in the coordinate and observation domains, respectively. By contrast, the MHM method can reduce the RMS by 22%, 52% and 27% on average. In addition, the BDS multipath errors in static baseline solutions are a few centimeters in multipath-rich environments, which is different from that of Global Positioning System (GPS) multipath. Therefore, we add a parameter representing the GEO multipath error in observation equation to the adjustment model to improve the precision of BDS static baseline solutions. And the results show that the modified model can achieve an average precision improvement of 82%, 54% and 68% in the east, north and up coordinate directions, respectively.

The precise point positioning (PPP) technique generally takes tens of minutes to converge, severely limiting its use. This longer convergence time is mainly due to the slower variation of satellite geometry in space and the stronger correlation of unknown parameters to be estimated. Fortunately, the lower orbit altitude of Low Earth Orbit (LEO) satellites contributes to the fast variation of the satellites’ spatial geometry. In addition, high-precision atmospheric delay information has become readily available, which can help decrease unknown parameters’ correlation. This study proposes a double-augmentation PPP approach with accelerated convergence by tightly integrating the LEO/atmosphere-augmented information. The GNSS observations in both mid-latitude and low-latitude areas, and simulated LEO observations under a Walker/polar mixed constellation, are used to validate the double-augmentation PPP approach. Test results in both areas indicate that the double-augmentation PPP can converge within 0.8 min, improving the convergence time by over 73%, and over 83% compared to the LEO-only augmented PPP and atmosphere-only augmented PPP.

Two methods are widely used for evaluating the precision of satellite clock products, namely the single-satellite method (SSM) and the multi-satellite method (MSM). In the satellite clock product evaluation, an important issue is how to eliminate the timescale difference. The SSM selects a reference satellite to eliminate the timescale difference by between-satellite differencing, but its evaluation results are susceptible to the gross errors in the referenced satellite clock offsets. In the MSM, the timescale difference is first estimated and then removed. Unlike the GPS, the BeiDou Navigation Satellite System (BDS) consists of three types of satellites, namely geosynchronous earth orbit (GEO), inclined geosynchronous orbit (IGSO), and medium earth orbit (MEO) satellites. The three types of satellites have uneven orbital accuracy. In the generation of satellite clock products, the orbital errors are partly assimilated into the clock offsets. If neglecting the orbital accuracy difference of the three types of BeiDou satellites, the MSM will obtain biased estimates of the timescale difference and finally affect the clock product evaluation. In this study, an improved multi-satellite method (IMSM) is proposed for evaluating the real-time BDS clock products by removing the assimilated orbital errors of the three types of BDS satellites when estimating the timescale difference. Three real-time BDS clock products disseminated by three different International GNSS Service (IGS) analysis centers, namely CLK16, CLK20, and CLK93, over a period of two months are used to validate this method. The results indicate that the assimilated orbital errors have a significant impact on the estimation of the timescale difference. Subsequently, the IMSM is compared with the SSM in which the referenced satellite is rigorously chosen, and their RMS difference is only 0.08 ns, which suggests that the evaluation results obtained by the IMSM are accurate. Compared with the traditional MSM, the IMSM improves the RMS by 0.16, 0.11, and 0.07 ns for CLK16, CLK20, and CLK93, respectively. Finally, three real-time BDS clock products are evaluated using the proposed method, and results reveal a significant precision difference among them.

The existence of colored noise in kinematic positioning will greatly degrade the accuracy of position solutions. This paper proposes a Kalman filter-based quad-constellation global navigation satellite system (GNSS) navigation algorithm with colored noise mitigation. In this algorithm, the observation colored noise and state colored noise models are established by utilizing their residuals in the past epochs, and then the colored noise is predicted using the models for mitigation in the current epoch in the integrated Global Positioning System (GPS)/GLObal NAvigation Satellite System (GLONASS)/BeiDou Navigation Satellite System (BDS)/Galileo navigation. Kinematic single point positioning (SPP) experiments under different satellite visibility conditions and road patterns are conducted to evaluate the effect of colored noise on the positioning accuracy for the quad-constellation combined navigation. Experiment results show that the colored noise model can fit the colored noise more effectively in the case of good satellite visibility. As a result, the positioning accuracy improvement is more significant after handling the colored noise. The three-dimensional positioning accuracy can be improved by 25.1%. Different satellite elevation cut-off angles of 10º, 20º and 30º are set to simulate different satellite visibility situations. Results indicate that the colored noise is decreased with the increment of the elevation cut-off angle. Consequently, the improvement of the SPP accuracy after handling the colored noise is gradually reduced from 27.3% to 16.6%. In the cases of straight and curved roads, the quad-constellation GNSS-SPP accuracy can be improved by 22.1% and 25.7% after taking the colored noise into account. The colored noise can be well-modeled and mitigated in both the straight and curved road conditions.

In the precise point positioning/real-time kinematic (PPP-RTK) technique, high-precision ionospheric delay correction information is an important prerequisite for rapid PPP convergence. The commonly used ionospheric modeling approaches in the PPP-RTKs only take the trend term of the ionospheric total electron content (TEC) variations into account. As a result, the residual ionospheric delay still affects the positioning solutions. In this study, we propose a two-step regional ionospheric modeling approach that involves combining a polynomial fitting model (PFM) and a Kriging interpolation (KI) model. In the first step, a polynomial fitting method is used to model the trend term of the ionospheric TEC variations. In the second step, a KI method is used to compensate for the residual term of the ionospheric TEC variations. Datasets collected from continuously operating reference stations (CORSs) in Hunan Province, China, are used to validate the PFM/KI method by comparing with a single PFM method and a combined PFM and inverse distance weighting interpolation (IDWI) method. The experimental results show that the two-step PFM/KI modeled ionospheric delay achieves an average root mean square (RMS) error of 1.8 cm, which is improved by about 48% and 23% when compared with the PFM and PFM/IDWI methods, respectively. Regarding the positioning performance, the PPP-RTK with the PFM/KI method takes an average of 1.8 min or 4.0 min to converge to a positioning accuracy of 1.3 cm or 2.5 cm in the horizontal and vertical directions, respectively. The convergence times are decreased by about 18% and 14% in the horizontal direction and 9% and 5% in the vertical direction over the PFM and the PFM/IDWI methods, respectively.