## 1 Introduction  
With respect to signal propagation, the atmosphere is subdivided into two main layers of the troposphere (also referred to as neutral atmosphere) and the ionosphere. When satellite signals pass through the atmosphere and reach the receiver antenna, the satellite electromagnetic signal will be delayed. The delay can be divided into ionospheric delay and tropospheric delay. Ionospheric delay and tropospheric delay can result in distance errors for international global navigation satellite system (GNSS) satellite users who require high precision measurements [1]. In the case of severe ionospheric disturbances, the pseudorange measurement error increases to about 100 meters [2]. The tropospheric delay contains a hydrostatic component and a wet component. In the zenith direction, the hydrostatic component reaches around 2.3 m [3,4] and the wet component varies from centimeters (or less) in the arid regions to as large as 0.35 m in the humid regions [3,5]. Therefore, for precise point positioning (PPP), it is necessary to correct these two delays. Using the global ionospheric maps (GIMs) and tropospheric zenith path delays announced by the international global navigation satellite system (GNSS) service (IGS) can improve the GNSS positioning performance. In 1998, the IGS organization established an ionospheric working group to continuously monitor ionospheric changes and provide daily GIMs products with a standard deviation of 2-8 TECU (total electron content units) [6]. For dual-frequency precise point positioning (PPP), the ionospheric-free (IF) combination can be used to eliminate the influence of the first order term of the ionospheric delay [7], but the noise is amplified, and the ionospheric delay is solved as a parameter by undifferentiated and uncombined analysis, then avoiding the noise amplification [8]. Of course, it is also possible to estimate the ionospheric delay by using a prior ionospheric model [9,10]. Reference [11] through experimental analysis shows that by using constraint through GIMs products the convergence time of dual-frequency PPP is reduced by 30%. Reference [12,13] applied ionospheric-constrained to inertial navigation system (INS) assisted PPP to overcome the adverse effects of PPP in kinematic mode. Compared with dual-frequency PPP, single-frequency (SF) PPP has received more attention due to its low cost, but ionospheric delay is still the main error. Similar to the dual-frequency (DF) IF, the single-frequency (SF) observation also has an IF combination. The SF IF is a combination between the code and carrier phase observations, which is called group and phase ionospheric correction (GRAPHIC) [14, 15, 16]. Based on GRAPHIC, the positioning accuracy of SF PPP can be improved, but the convergence time is extended. Several scholars use GIMs constraints in single-frequency PPP to reduce convergence time [10,17, 18, 19]. However, they did not mention the influence of tropospheric constraints on SF PPP convergence time. Some scholars have studied tropospheric constraints. Reference [20] developed a new global positioning system (GPS)/Beidou navigation satellite system (BDS) real time kinematic (RTK) algorithm that is constrained by tropospheric delay parameters from a numerical weather prediction (NWP) model for medium/long-range baselines, which is better than 3 cm in the horizontal component and better than 5 cm in the vertical component. Reference [21] developed a numerical weather model (NWM)-constrained PPP processing system to improve the multi-GNSS PPP, and with the NWM-constrained, the convergence time was shortened by 20.0%, 32.0%, and 25.0% for the north, east, and vertical components, respectively. After the troposphere constraint was used for the fixed PPP/INS adaptive covariance model to converge for one hour, the positional accuracy of the north, east, and height components were increased by 19.51%, 61.11%, and 23.53%, respectively, compared with the conventional PPP solution [22].