*3.1. Residuals Analysis and PCV Estimation to Obtain Corrected ANTEX Entries*

First, we analyzed the residuals as a crucial step to correct for existing phase patterns caused by the receiving antenna. The residuals are derived from the static RTKLIB solution. The RTKLIB output contains the residuals for each satellite, epoch and frequency on code and phase. Table 4 shows the mean residuals and their respective RMSE.

The LEIAR25.R3 LEIT antenna demonstrated the lowest phase RMSE on both frequencies. It is also evident that the L2 phase residuals are generally higher than the L1 residuals for all antennas. The Trimble AV28 and U-blox ANN-MB-00 show generally higher phase RMSE values on both frequencies compared to the other antennas. The phase RMSE is considerably smaller when using a circular metallic ground plane. The antennas in Table 4 are ordered by approximated acquisition costs. A quality difference with generally lower phase residuals for the upper price category antennas compared to the less expensive ones is evident. As known from GNSS basics, the code residuals have higher deviations than the phase measurements. The mean code error is not zero because they are less precise and the mean single-difference phase residuals are utilized as reference in computing the double-differences. Though interesting, the code residuals are less important for the analysis because precise applications employ phase measurements. L1 and L2 mean phase residuals averaged over the elevation angle are shown in Figure 5.



**Figure 5.** Elevation based antenna pattern on the phase residuals. L1 residuals are depicted left and L2 on the right. The continuous lines (0.5◦∅) show the mean residual over 0.5 degree elevation bins, the dotted lines (0.5◦*σ*) in the background its respective standard deviation and the dashed line (5◦∅) the smoothed 5 degree bins corresponding to the ANTEX format.

Figure 5 shows the averaged elevation-based pattern for L1 and L2 observations with all tested antennas using a bin size of 0.5 degrees (continuous line), its standard deviations (thin dotted line in the background) and the residuals smoothed over 5 degree bins to correspond to the ANTEX format (thick dashed line). Generally apparent are higher variations at lower elevation angles for all antennas and a higher standard deviation of L2 compared to L1. A clear signal is evident for the Trimble AV28 and U-blox ANN-MB-00 antennas using the rectangular metal bracket as base. This effect can be seen on L1 with satellites between 30 and 50 degrees and on L2 at about 25 and 40 degrees. Since these results evoked further investigation, the experiment was repeated with a circular ground plane. The identified pattern could be reduced when using a circular ground plane. However, on L2 the observed variations above 60 degrees are noticeable greater compared to the other antennas. Evident is that both antennas, although being produced by a different manufacturer and having an inherently different design, demonstrate a comparable elevation-based pattern. The reference antenna LEIAR25.R3 LEIT demonstrated no clear elevation-based phase bias at any elevation angle since both, base and rover antenna are of the same model and the resulting residual error in differential analysis mostly cancels out. The antennas GA530 and TRM55971.00 also demonstrate a slightly visible mean phase pattern.

Typically, azimuth dependent residuals are stacked over several weeks or months of data or over several days by rotating the antenna by some degree. This was not performed in this study. Instead, with the limited observation days available we stacked the azimuth and elevation dependent residuals over the selected observation days. Since the antenna phase residuals of AV28 and ANN-MB-00 with different mountings evokes additional attention, Figure 6 compares the elevation and azimuth dependent L1 and L2 phase residuals for the reference antenna LEIAR25.R3 LEIT and the antennas Trimble AV28, U-blox ANN-MB-00 with rectangular bracket and circular plane.

Figure 6 demonstrates the performance of a reference antenna (LEIAR25.R3 LEIT) compared to the two low-cost antennas Trimble AV28 and U-blox ANN-MB-00 with a rectangular bracket and a circular plane as base. Note that the zenith angle is used instead of elevation angle to correspond with the ANTEX format and more intuitive interpretation of the polar plots. Generally noteworthy are the fewer satellite tracks on L2 which is caused by not all GPS satellites yet transmitting the L2C signal. As previously illustrated in the elevation based residuals, the LEIAR25.R3 antenna (Figure 6a) depicts generally low residuals over the full horizon with only a few higher values close to the ground. The patterns from the antennas AV28 and ANN-MB-00 are characterized by negligible differences between them. Noticeable differences are slightly lower residuals at low elevation in South-West (225◦) direction on L1 for the AV28 (Figure 6b left) compared to the ANN-MB-00 (Figure 6d left) antenna using a rectangular bracket. Evident are also the strong negative residuals on L1 at low elevation in North (0◦) direction and the stronger positive residuals on L2 in the same direction. The described L1 pattern at about 45 degree elevation (AV28 rectangular and ANN-MB-00 rectangular in Figure 5) is also visible with the strongest signal in East and West direction. The errors using a circular plane are generally smoother for both antennas. Regions with increased residuals are, however, still evident. For further processing, the elevation and azimuth dependent residuals were averaged in 5 degree bins to comply with the ANTEX standard. To overcome possible outliers in the data and since only limited observation days were used, we also applied a polynomial fit over the azimuth dependent signal. Based on the base antenna's (LEIAR25.R3 LEIT) ANTEX PCV entries, the elevation and azimuth-elevation dependent phase residuals were used to generate absolute PCVs to append or replace existing ANTEX entries for the tested antennas.

(**a**) Satellite phase residuals on L1 (top left) and L2 (top right) for the reference antenna LEIAR25.R3 LEIT.

(**b**) AV28 L1 with a rectangular bracket (left) and a circular plane (right)

(**d**) ANN-MB-00 L1 with a rectangular bracket (left) and a circular plane (right)

(**c**) AV28 L2 with a rectangular bracket (left) and a circular plane (right)

(**e**) ANN-MB-00 L2 with a rectangular bracket (left) and a circular plane (right)

**Figure 6.** Satellite phase residuals on L1 (left) and L2 (right) for the selected antennas (**a**) LEIAR25.R3 LEIT (top frame), (**b**) AV28 L1, (**c**) AV28 L2, (**d**) ANN-MB-00 L1 and (**e**) ANN-MB-00 L2. The figures (**b**–**e**) depict on the left the phase residuals using a rectangular metallic bracket as base and on the right with a circular metallic plane of 10 cm diameter.
