## 2 Materials and Methods
### Determination of the Projected Trajectory and Bias for the Actual Trajectory  
The reference line was defined in the middle of the white road lane on the paved track, being measured every 5 m using 30-epoch GNSS RTK observations with CROPOS corrections (Figure 3). The average coordinates from the 30 epochs were calculated for each location to achieve the highest possible positioning accuracy of the projected trajectory while maintaining time and cost efficiency [32]. The projected trajectory was determined according to the properties of the chosen track and agricultural machinery, based on the offset from the reference line (1):  
\[o_{total}=o_{track}+o_{machinery} \tag{1}\]  
Figure 3: Definition of the projected trajectory according to the reference line. The \(o_{track}\) is the sum of corrections specific for the offset of the projected trajectory according to the properties of the study area. In this case, it represented half of the width of the central dividing line for unclassified tracks in Croatia, amounting to 0.060 m. The \(o_{machinery}\) is the sum of corrections specific for the offset caused by the dimensions of the agricultural machinery, its mechanical properties relative to the operator, and the placement of the GNSS antenna. A reference for the steering of the agricultural machinery in this study was determined using the width of the front left tire. Therefore, \(o_{machinery}\) equaled the length of the front semi-axle corrected by the tire width, which amounted to 0.675 m. A projected trajectory was determined according to the total offset \(o_{total}\) from the reference line, which was calculated as the sum of previously determined corrections and amounted to 0.735 m.  
Fifty auxiliary points with equal relative distance were generated 10 m apart from each other on the projected trajectory and were used as a reference for the accuracy assessment of GNSS positioning (Figure 4). Accordingly, the positioning error per observation (\(bias_{i}\)) was determined according to Equation (2):  
\[bias_{i}=\sqrt{\Delta\mathrm{E}_{i}^{2}+\Delta\mathrm{N}_{i}^{2}}=\sqrt{( \mathrm{E}_{pi}-\mathrm{E}_{qi})^{2}+(\mathrm{N}_{pi}-\mathrm{N}_{qi})^{2}}, \tag{2}\]  
where \(i\) represents one epoch of the GNSS observation, E and N represent the eastern and northern coordinates in the HTRS96/TM projection, \(p\) represents the observed location on the actual trajectory, and \(q\) represents the perpendicular location on the projected trajectory closest to the observed location \(p\). The statistical analysis of the observed GNSS field results was performed using R v4.0.3 in RStudio v1.3 (R Foundation, Vienna, Austria). The fundamental properties of two samples of bias determined for each of the repetitions were analyzed using descriptive statistics. Median values were used to evaluate the relationship of the actual GNSS accuracy in the field with the positioning accuracy declared by the provider of the corrections. Minimum and maximum values were used to establish a range of positioning accuracy, while the coefficient of variation (CV) quantified the variability of the positioning accuracy within the repetition. The lower CV values indicated a more uniform accuracy, which benefits the application of a particular GNSS correction in the field due to the increased repeatability [33]. The normal distribution of input samples was evaluated using the Shapiro-Wilk test to determine the suitability of a parametric (\(t\)-test) or non-parametric test (Wilcoxon test). The null hypothesis of the Shapiro-Wilk test that the tested GNSS positioning bias is normally distributed was rejected for every \(p\) value lower than 0.05 [34]. In these cases, the Wilcoxon non-parametric test was determined as optimal instead of the default selection of the parametric \(t\)-test. The difference from the median values of two samples created from the GNSS observations in two repetitions was tested to assess the repeatability of the positioning accuracy for the tested GNSS corrections [35].