# Precise Point Positioning Using Triple GNSS Constellations in Various Modes  
[PERSON]  
1 Department of Civil Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada; [EMAIL_ADDRESS]  
1 Correspondence: [EMAIL_ADDRESS]; Tel. : +1-647-771-9924; Fax: +1-416-979-5122  
differential GNSS techniques is that a relatively long time is required for the solution to converge to sub-decimeter level. This is mainly due to the existence of unmodelled residual errors Generally, PPP solution accuracy and convergence time are influenced by the ability to mitigate all potential error sources in the system. PPP essentially relies on the availability and use of precise satellite products, namely orbital and clock corrections. At present, the IGS-MGEX provides precise satellite orbital and clock corrections and the satellite hardware delays for all GNSS constellations [1]. Several comprehensive studies have been published on the accuracy and convergence time of un-differenced GPS PPP solution [2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. However, a drawback of a single GNSS constellation such as GPS is the limited number of visible satellites in urban areas. The addition of other GNSS systems, such as Galileo and BeiDou, offers more visible satellites to users, which in turn enhances the satellite geometry and is expected to improve the overall positioning solution [8]. This in turn makes the PPP solution more feasible, especially in urban areas. However, combining the observations of multi-GNSS constellations comes at the expense of introducing additional biases to the observation mathematical models. These include the GPS to Galileo time offset, GPS to BeiDou time offset and the hardware delays of both Galileo and BeiDou. Recently, [7] showed that combining the un-differenced GPS and Galileo observations in a PPP model improves the solution convergence time by about 25%, in comparison with the GPS-only counterpart. [12; 13; 14; 15; 16] combined the GPS/BeiDou observations in a PPP model, which reduced the convergence time by 20% in comparison with the GPS-only PPP solution. A quad-constellation (GPS, Galileo, BeiDou, and Glonass) PPP model was developed in [17; 18; 19; 20] and its performance was assessed in static and kinematic modes. Their test results showed improvement in both of the PPP solution precision and convergence time. However, those studies were limited to the post-processing mode. Recently, the focus of PPP has shifted to the determination of an accurate solution in real time. The advent of the IGS real-time Pilot Project in 2007 led to the development of real-time clock and orbit streams [4; 21]. At present, the IGS real-time service provides different types of clock corrections, namely IGS01/IGC01, IGS02, and IGS03 [22; 23]. The development of these correction streams allows the user to determine an accurate real-time PPP solution anywhere in the world. The availability of RT-PPP solution elevates PPP as a potentially viable alternative to differential GNSS techniques, in particular the highly productive and widely used real-time kinematic (RTK) solution. However, the long convergence time remains the main disadvantage of PPP [24; 25; 26]. This paper develops a triple-constellation GNSS (GPS, Galileo, and BeiDou) PPP model, which rigorously accounts for all errors and biases, including the additional biases introduced as a result of combining the observations of different GNSS constellations. These additional biases are lumped together into a new unknown parameter, which is referred to as the inter-system bias in our PPP model. In addition, the receiver differential hardware delays are lumped to the receiver clock error. The hydrostatic component of the tropospheric zenith path delay is modelled through the Hopfield model, while the wet component is considered as an additional unknown parameter [7; 27]. All remaining errors and biases are accounted for using existing models as shown in [28].