## 4 Discussion  
The analysis of the impact of the third frequency on PPP and PPP-AR with a flexible model was not shown in the literature. Analysis involving multiple frequencies tends to be focused on comparing only satellites with the same number of frequencies (for example Galileo, or GPS-IIF) [9; 17; 18; 36]. In this manuscript, a comprehensive performance analysis is done by not rejecting satellites that only broadcast on two frequencies, allowing for the processing of many healthy satellites that would be otherwise rejected. This flexible processing allows for a realistic assessment of PPP performance in the current multi-GNSS multi-frequency context, and it proves the benefits of including as many satellites as available, as PPP performance is further improved compared to a solution where only dual-frequency or only triple-frequency satellites are processed. The impact of simultaneous processing and fixing of dual-frequency and triple-frequency satellites is demonstrated. On the one hand, given that not all satellites broadcast on three frequencies, allowing for the mix of dual- and triple-frequency measurements leads to an increase of the number of processed satellites, as opposed to rejecting satellites without a third frequency. On the other hand, the addition of a third frequency is shown to improve the PPP-AR solutions. This improvement is attributed to the fixing becoming more reliable. For instance, the addition of the third frequency allows for the formation of the extra-widelane (EWL) ambiguity, which has metre-level wavelength and can be fixed instantaneously. Being able to fix the EWL helps in the fixing of the widelane ambiguity, which in turn helps the fixing of the narrowlane. Ref. [37] demonstrated the improvements that the third frequency ambiguities bring to the correct fixing of the narrowlane ambiguities based on simulated GPS data. Limited impact is noticed from the addition of a third frequency on the PPP float solutions. Similar observations on the limited effect of additional frequencies on the float solution were noticed in [36] using the BeiDou constellation and the effect of the third frequency was expected to be more significant in the case of poor measurement quality. The limited effect of the third frequency was also mentioned in [19], where the Galileo constellation was used, and in [9] where multiple constellations were processed. Despite this limited effect of the third frequency measurements on the float solutions, it is important  
Figure 11: Float and fixed dual-frequency and dual-/triple-frequency horizontal (**a**) and vertical (**b**) errors for open sky kinematic dataset collected near York University on DOY 151, 2021. Car is static in the first 7 min, and kinematic in the rest of dataset. not to reject any satellite with only two frequencies, as that would lead to the performance seen in the triple-frequency results. The proposed algorithms are also tested on datasets collected with geodetic hardware placed on a car. The vehicle results, shown in the previous section, demonstrate the benefits of processing a mix of dual- and triple-frequency measurements from all four major GNSS constellations. Doing so opens the door to instantaneous precise positioning that only requires globally estimated satellite products. The analysis of the impact of PPP-AR in general, and multi-frequency PPP-AR specifically, on vehicle datasets is limited in the literature. For instance, ref. [38] analyzed the performance of a triple-frequency four-constellation solution where only triple-frequency satellites were processed and where the average number of satellites was 12. The solution achieved rms of 0.29 m, 0.35 m, and 0.77 m in the east, north, and up components, respectively, although only widelane and extra-widelane ambiguities were fixed. Ref. [39] analyzed the impact of an inertial navigation system (INS) on PPP-AR re-convergence using GPS and GLONASS on two frequencies, and was able to achieve convergence (time to first fix (TTFF)) in approximately 30 min. Ref.