## 1 Introduction  
The Precise Point Positioning (PPP) technique noticeably evolved since its early conception. The technique was intended as a precise positioning alternative that would reduce the computational burden in network static relative baseline processing, and would only require a stand-alone receiver [1, 2], as opposed to, e.g., Real-Time Kinematics (RTK), which requires the presence of and communication with nearby base stations [3, 4]. Because of the latter technique requiring local infrastructure to provide precise positioning, PPP is becoming the de facto global precise positioning technique, as it only requires globally estimated satellite corrections [5]. However, the technique is known to suffer from relatively long convergence times [6, 7] due to its large number of states to be estimated, especially the satellite-dependent ionospheric and ambiguity states. Ambiguity resolution proved to be a valuable tool in reducing convergence time [8, 9, 10], as making use of the integer nature of the ambiguities allows for faster fixing to their correct values instead of waiting for them to converge. To be able to fix ambiguities to their integer values, the classic PPP model has to be modified to remove any biases from the ambiguity estimates, done through between-satellite single-differencing. The PPP-AR model used in this research is the Decoupled Clock Model (DCM). The model was first introduced in [11] and it is based on the assumption that the pseudorange and carrier-phase measurements are not synchronized to the same level of precision. This difference in precision is assumed to be due to the hardware delays that affect differently measurements on the same frequency. Therefore, rather than having the same receiver clock for both sets of pseudorange and carrier-phase measurements, the DCM estimates a separate receiver clock for the pseudorange and carrier-phase measurements. Doing so decouples both measurements and allows for integer estimation of the ambiguities. These ambiguities are single-differenced relative to a reference satellite for which the ambiguities are set to arbitrary integer values. The original DCM was based on an ionosphere-free combination of GPS pseudorange and carrier-phase measurements, as well as the Melbourne-Wubenna (MW) combination, allowing for fixing of widelane (WL) and narrowlane (NL) ambiguities. The DCM was later expanded into an extended version--the Extended DCM--for which the Melbourne-Wubenna combination is split into two separate observables: a narrowlane pseudorange and a widelane carrier-phase [12]. Doing so allows for the estimation of the ionospheric delays, which in turn makes the constraining of said delays possible. This enables the so-called PPP-RTK for which regional ionospheric maps can be used to improve PPP performance and approach RTK levels of accuracy and convergence time [7; 13], as well as the use of global ionospheric maps (GIM), which are less accurate [13; 14]. Recent years saw the addition of a third GPS frequency on L5 (1176.45 MHz) on top of the typical L1 (1575.42 MHz) and L2 (1227.6 MHz), as well as the addition of other constellations with multiple frequencies. The current GNSS landscape consists of four GNSSs: the latest generations of GPS satellites broadcast on L1, L2, and L5, while older generations do not broadcast on L5. Galileo satellites broadcast on four frequencies E1, E5a, E6, and E5b, with the option to track AltBOC (Alternative Binary Offset Carrier), a modulation combining E5a and E5b with reduced measurement noise. BeiDou was also designed to broadcast on multiple frequencies through two generations. The BeiDou-2 generation broadcasts on three frequencies (B1-2 at 1561.098 MHz, B2B at 1207.14 MHz, and B3 at 1268.52 MHz), while the newes BeiDou-3 generation broadcasts on five frequencies (B1 and B2a that coincide with GPS L1 and L5, respectively, and B1-2, B2 and B3, which coincide with BeiDou-2 signals).