## 2 Materials and Methods
### Data and Processing  
The algorithms described above are tested on two sets of data. One worldwide set of stations from the International GNSS Service (IGS) is chosen to assess the global performance of the algorithms, as well as to draw conclusions on areas of improvement. Another set of kinematic data were collected by placing a geodetic receiver on a car and driving in a suburban environment, where the purpose is to highlight the benefits from the additional frequencies to achieve instantaneous precise positioning in the kinematic case. Figure 2 shows the network of stations used in the processing. 32 IGS stations were randomly chosen, the only constraints being that they should track all four major GNSS constellations on up to three frequencies. The stations were processed in 1-h-long independent datasets from day 70-77 of 2021, with a sample interval of 30 s, leading to a total of 6137 individual datasets after filtering out some datasets with missing epochs of observations. This sample size is considered to be significant enough to allow generalization of the conclusions and results to assess the performance of the proposed algorithms and to analyze the impact of the fixing of the third frequency's ambiguities. Figure 1: Reference satellite choice strategy when processing a combination of dual- and triple-frequency satellites. The kinematic testing is done with a geodetic NovAtel PwrPak7 receiver, which was placed, along with its antenna, on a rack on top of the roof of a car. The car was driven to collect two separate datasets, with a sampling rate of 1 Hz: one dataset was collected by driving close to the York University campus in Toronto, Canada on DOY 337, 2020, and a second parking lot dataset where the car was driven in an open-sky parking lot on DOY 151, 2021. The ground tracks of the two datasets are shown in Figure 3. A second NovAtel PwrPak7 receiver was used as a base station at a surveyed location and an RTK (Real Time Kinematics) solution was generated as a reference solution using the commercial RTK software NovAtel Inertial Explorer. The baseline for both datasets was kept under 5 km. The constellations being processed are GPS, GLONASS, Galileo, and BeiDou-2, with ambiguities being resolved for GPS, Galileo, and BeiDou-2. BeiDou-3 was not included in the processing due to satellite bias corrections not being widely available yet, making ambiguity resolution impossible. In terms of frequencies, L1, L2, and L5 were processed for GPS, G1, and G2 for GLONASS, E1, E5b, and E5a for Galileo, and B1-2, B2b, and B3 for BeiDou-2. Although the IGS stations are static, they are processed as if they were kinematic without imposing constraints on the receiver's dynamics. The stations' position coordinates are estimated using process noise of 100 km/h. The same process noise is also used for the two kinematic datasets. For receiver biases, although they tend to only vary by a few  
Figure 3: Ground tracks of datasets used in kinematic processing. Figure 2: IGS stations used in processing. nanoseconds over a day, their behaviour depends on the temperature, location, and quality of the receiver, as they could reach relatively high variations [23; 24; 25]. To be cautious, the DCM does not make assumptions on the behaviour of biases and treats them as clock-like terms, leading to all receiver-related states being estimated as clock-like terms and treated as white noise processes. Receiver and satellite antenna corrections are corrected for using the IGS14 ANTEX file [26]. Because the GPS L5 satellite antenna corrections are not included in the ANTEX file, the corrections on L2 are applied on L5 measurements. A complete description of the processing strategies adopted for the various estimated parameters is provided in Table 2. The processing was performed with the in-house PPP engine developed at York University's GNSS lab [27; 28]. To allow for PPP processing in general, and ambiguity resolution specifically, satellite products are required.