## 4 Zero-Baseline Instantaneous RTK Positioning with Smartphones in Lying Down and Upright Positions
For S20, however, Figure 5b shows that there is now an increase in the formal correctly fixed STDs (green lines at bottom) when compared to Figure 5a, in particular just before epoch 14,400. Figure 5: **Zero baseline with smartphones in a (a) lying down and (b) upright position (cf. Figure 2):** Horizontal (N, E) positioning scatter (first row) and corresponding vertical (U) positioning error time-series (second row) for 12 h of data (cf. Table 2) with GP4 (left column) and S20 (right column). In this RTK model GPS + Galileo + QZSS +BDS L1 + L5, E1 + E5a, L1 + L5, B1 observations have been used. A zoom-in window is shown to depict the two orders of magnitude when going from incorrectly fixed (red dots) and ambiguity-float (gray dots), to correctly fixed solutions (green dots). The 95% empirical and formal CIs and ellipses for the correctly fixed solutions are shown as blue and red lines, respectively. The number of visible satellites and the correctly fixed formal up STDs (bottom row) are shown as black and green lines, respectively. The black dashed lines indicate the zoom-in period between epoch 13,800 and 25,000, see further Figure 7, with ILS SRs of 99.9% in both (**a,b**) in the left column for GP4 and 31.0% in (**a**) and 97.4% in (**b**) in the right column for S20, respectively. (**a**) Smartphones used in lying down positions: with GP4 (left): ILS SR 99.9% and S20 (right column): 79.4%. (**b**) Smartphones used in upright positions: with GP4 (left): ILS SR 99.9% and S20 (right column): 97.8%. This increase in the formal STDs can be explained by the drastic drop in the number of satellites (cf. Table 3) and the poorer receiver-satellite geometry strength. We found that the vertical dilution of precisions (VDOPs) reached values close to 7 for this period. If we inspect the horizontal positioning scatter in Figure 5a further, we can see that GP4 and S20 in a lying down position setup shows a poor fit both in size and orientation between the empirical (blue lines) and formal (red lines) 95% confidence ellipses, respectively, and a poor fit between the corresponding Up CIs as well. This poor fit implies the use of an overly optimistic stochastic model (cf. Table 2). In contrast, the GP4 and S20 smartphones in an upright position in Figure 5b show a good agreement between the formal and empirical confidence ellipses/intervals. This implies that the stochastic model settings for GP4 and S20 are now realistic. To further investigate the distribution of the correctly fixed RTK positioning solutions, we depict in Figure 6 the corresponding North error histograms as green bars. On top of the histograms, we show the corresponding empirical and formal theoretical normal distributions, in terms of their probability density functions (PDFs), depicted as blue and red lines, respectively. These PDFs are based on the empirical mean and empirical and formal STDs of the North positioning errors. The GP4 smartphones are shown in the left column and S20 in the right column, with the top row showing the results when the smartphones are lying down and the bottom row showing the corresponding upright-position results. Finally, we remark that the East and Up positioning errors behave in a similar manner and that their corresponding histograms are thus not shown for brevity. Figure 6 shows that the empirical normal PDF (blue lines) fit the GP4 histograms very well even when the smartphones are in a lying down position at the top row and left column, but with a poor fit between the formal (red lines) and empirical PDF. However, when the GP4 smartphones are in an upright position, the North errors become more  
Figure 6: **Zero baseline with smartphones in a (a) lying down and (b) upright position (cf. Figure 2):** North error histograms (green bars) together with their empirical (blue lines) and formal (red lines) PDFs, with GP4 in the left column and S20 in the right column. (**a**) Smartphones used in lying down positions: with GP4 (left) and S20 (right column) using an external antenna.