## 1 Introduction
A third approach is to estimate the ionospheric delay of each GNSS satellite as a process parameter during the SF-PPP process (referring to the uncombined approach hereafter). However, in such an approach, a priori constraints on the ionospheric delays become very critical, since the ionosphere parameters are heavily correlated with other parameters. Thus, external ionospheric models are necessary [28; 29; 30]. The positioning accuracy for a Ublox-like cost-effective single-frequency receiver was 0.4-0.5 m for GPS, 0.7-0.8 m for GLONASS, and 0.3-0.4 m for the combined-system approaches when utilizing the uncombined approach [30]. By combining multi-GNSS observations, the results can be even improved; the SF-PPP with quad-constellation GNSS reduced the convergence time by 40%, 47%, 34% in the east, north, and up directions compared to the GPS-only SF-PPP [29]. SF-PPP performances have been evaluated for smart devices in many studies [1; 31; 32; 33]. Using the GPS observations collected with a Nexus 9 tablet, static SF-PPP was performed and reached 0.37 m and 0.51 m at horizontal and vertical components, respectively [31]. The PPP-wizard application developed in [32] reached sub-meter accuracy in static mode and meter-level in dynamic mode with smartphone GNSS data. To reduce the code measurement noise of the Nexus 9 tablet, an optimized carrier-phase smoothing method was proposed and the resulting positioning accuracy of the static experiment was about 0.6 and 0.8 m in the horizontal and vertical components, respectively [33]. The time difference filtering approach was also adopted to suppress the systematic errors in Nexus 9 tablet data, and the results showed that the RMS error of the static positioning solution was less than 0.6 and 1.4 m in the horizontal and vertical directions, respectively [1]. Similar SF-PPP tests were conducted using the Huawei Mate 30 module, which achieved 1 m precision in the horizontal component [16]. The above studies focused on the algorithms of the joint use of the pseudorange, the carrier phase, and Doppler observation output from smart devices, but lacked the application of different ionospheric treatments in SF-PPP. The focus of this study was to evaluate the performance of GRAPHIC and the uncombined SF-PPP approaches with datasets from Multi-GNSS Experiment (MGEX) stations as well as Huawei P40 smartphones. The MGEX stations, by providing geodetic-grade GNSS observations, can be treated as baselines for our SF-PPP assessment. The P40 can track dual or triple-frequency signals for different satellite systems, explicitly L1/L5 for GPS, B1I/B1C/B2a for BDS, E1/E5a for GALILEO, and L1/L5 for QZSS. However, the data continuity on the second frequency band (L5/B2a/E5a) was severely interrupted due to channel limitations, and only the measurements on the first frequency band were used for the investigation. We collected the P40 datasets both using its self-contained built-in antenna and an external geodetic antenna, thus making it possible to identify data quality problems caused by either the P40 chips or antennas. The paper is organized as follows. In Section 2, two kinds of SF-PPP models are elaborated by formulas and instructions. Then, the effectiveness of the positioning performance of the two models is examined by geodetic receivers in Section 3, both in terms of accuracy and convergence. In Section 4, first, we briefly describe a quality assessment that was conducted with the dataset collected by Huawei P40, and then the SF-PPP processing is examined. Finally, the conclusions are summarized in Section 5.