# An Approach to Speed up Single-Frequency PPP Convergence with Quad-Constellation GNSS and GIM
## 1 Introduction  
Precise point positioning (PPP) is a single-receiver absolute positioning technique that uses un-differenced pseudorange and carrier phase observations along with precise satellite orbit and clock products [1]. How to reduce the PPP convergence time has become a key issue for further improving PPP performance [2; 3; 4; 5]. The PPP technique conventionally uses dual-frequency observations to remove the first-order ionospheric delay errors. Given that the majority of mass-market global navigation satellite system (GNSS) users are operating single-frequency receivers, the dual-frequency PPP technique has been extended to single-frequency PPP [6; 7; 8]. In the dual-frequency PPP, the ionospheric delay error can be eliminated by forming an ionosphere-free linear combination between dual-frequency observations. However, in the single-frequency PPP, the ionospheric delay becomes a major error source to be mitigated.  
Two different methods have been proposed for ionospheric delay error mitigation in the single-frequency PPP. One is to apply ionospheric models, such as the global ionospheric map (GIM),to mitigate the ionospheric effect [6; 9; 10], but this method can only achieve decimeter-level positioning accuracy due to the accuracy limitation of the ionospheric model [10]. For instance, the GIM model only has an accuracy of 2-8 TECU (total electron content unit), which is equivalent to a ranging error of 0.32-1.28 m for the GPS L1 frequency signal. The second method is to form an ionosphere-free combined observable between the code and carrier phase observations on a single frequency, which is known as GROup And PHase Ionospheric Correction (GRAPHIC) [11; 12; 13; 14]. Based on this GRAPHIC method, an accuracy of several centimeters is achievable, but a long convergence time of over two hours is required [7]. The long convergence time is attributed to two aspects. On one hand, the GRAPHIC combined observables are dominated by the noise of the code measurements. On the other hand, the GRAPHIC combination from single-frequency code and phase observations produces a rank-deficient mathematical problem [15]. In summary, both methods suffer major limitations where the first method converges more quickly, but can only achieve decimeter-level positioning accuracy while the second method can achieve centimeter-level positioning accuracy, but requires a very long convergence time.  
In this study, an approach to combine both methods is proposed, which can speed up the convergence and achieve high positioning accuracy at the same time. Further, instead of applying the GRAPHIC combination, the ionospheric and hardware delays (IHD) are estimated together as a single unknown parameter. The IHD values obtained from the GIM product and the multi-GNSS differential code bias (DCB) product are used as pseudo-observables to estimate the IHD parameter. A time varying weight scheme has also been proposed for the pseudo-observables to gradually decrease their contribution to the position solutions. Since the multi-GNSS signals have been widely demonstrated to be capable of improving the PPP performance [16; 17; 14; 18; 19],the quad-constellation measurements will be utilized with the proposed approach to further accelerate the convergence. Presently, GPS and GLONASS systems are providing global navigation services. The BeiDou system (BDS) provides a regional navigation service and the Galileo system offers an initial global navigation service.  
The rest of the paper is organized as follows: Firstly, a single-frequency PPP approach of combining quad-constellation GNSS and GIM data is presented. Secondly, the GIM products and the multi-GNSS DCB Products from different agencies are compared and evaluated by GNSS measurements. Finally, the converging performance of the proposed single-frequency PPP approach is assessed.