## 3 Results
### Analysis of the Point Cloud Horizontal and Vertical Positional Precision for Flight A
To investigate if this heading bias contributed to the horizontal position differences being considerably different for the north and south flow flight lines, it was used to adjust the single-antenna GNSS/INS heading estimates. Then, the horizontal position differences between the matched points of the single-antenna and dual-antenna point clouds were recomputed. Figure 15 illustrates the horizontal position differences between the single and dual-antenna generated point clouds after adjusting for the heading bias. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Region**} & \multicolumn{2}{c}{**Horizontal Position Differences**} & \multirow{2}{*}{**Mean Lidar**} & \multirow{2}{*}{**No. of Points**} \\ \cline{2-2} \cline{5-5}  & **Minimum** & **Maximum** & **Mean** & & **Range** \\ \hline Top Edge Points & 0.006 m & 0.354 m & 0.137 m & 68.2 m & 263,266 \\ Bottom Edge Points & 0.012 m & 0.300 m & 0.105 m & 68.8 m & 258,259 \\ Left Edge Points & 0.000 m & 0.302 m & 0.092 m & 68.1 m & 285,623 \\ Right Edge Points & 0.033 m & 0.339 m & 0.145 m & 68.2 m & 286,220 \\ Middle Points & 0.001 m & 0.164 m & 0.051 m & 64.6 m & 297,850 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Quantitative measures for the horizontal position differences of points within the five regions of interest for flight A.  
Figure 14: Horizontal position differences between the single and dual-antenna generated point clouds for the five regions of points of flight A. An inspection of Figure 15 indicates that the horizontal position differences have been reduced, are more consistent, and are less correlated with the direction each flight line was flown after adjusting for the single-antenna GNSS/INS trajectory heading bias. This suggests that the single-antenna GNSS/INS heading estimates reported by the APX-18 could potentially be improved through data-driven calibrations and this will be the focus of future research. Using the theoretical point cloud horizontal precision model presented in Section 2.3 (see Equation (15)), an estimate for the horizontal precision of a point observed at the edge of the Mid-40\({}^{\prime}\)s circular FoV was computed. Based on the mean single-antenna heading precision of 0.369\({}^{\circ}\), the horizontal precision for an edge FoV point observed at a lidar range of 68 m was 0.15 m, which closely aligns with the maximum horizontal position differences computed after adjusting for the heading bias (Table 8). Equation (15) also indicates that the horizontal position of a point at the middle of the FoV is not affected by the precision of the heading estimate. Therefore, the expected horizontal position difference between the two point cloud regions would be zero. However, Figure 15 and Table 8 indicate a mean horizontal position difference for the points in the middle region of 0.043 m. This is the suspected result of not accounting for the remaining trajectory estimates, calibration parameters, and lidar observations containing errors (i.e., the assumption that these terms are error free was used within the development of Equation (15), which is invalid). Figure 16 illustrates the vertical position differences between the single and dual-antenna generated point clouds based on the points located within the five regions of the Mid-40's FoV. In the figure, note that the heading precisions for the single-antenna GNSS/INS trajectory are shown as the top dashed line and reference the secondary vertical axis. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Horizontal Position Differences**} \\ \cline{2-4}
**Region** & **Minimum** & **Maximum** & **Mean** \\ \hline Top Edge Points & 0.009 m & 0.205 m & 0.087 m \\ Bottom Edge Points & 0.004 m & 0.168 m & 0.080 m \\ Left Edge Points & 0.001 m & 0.179 m & 0.075 m \\ Right Edge Points & 0.001 m & 0.201 m & 0.078 m \\ Middle Points & 0.004 m & 0.104 m & 0.043 m \\ \hline \hline \end{tabular}
\end{table}