## 3 Bias Impact Analysis
### Bias Impact Analysis for a Spinning Multi-Beam Laser Scanner
\[\text{Same direction}: \delta Z_{m_{A}}-\delta Z_{mB}=\left(-x^{\prime}_{A}+x^{\prime}_{B }\right)\,\delta\Delta\phi=\,-D_{AB}\,\delta\Delta\phi \tag{9}\] \[\text{Opposite directions}: \delta Z_{m_{A}}-\delta Z_{mB}=\left(-x^{\prime}_{A}+x^{\prime}_{B }\right)\,\delta\Delta\phi=\,-2X\,\delta\Delta\phi \tag{10}\]  
Since a bias in lever arm \(\Delta X\) also causes shifts across the flying direction, there is a need to decouple the impacts of \(\delta\Delta X\) and \(\delta\Delta\phi\) so as to estimate both these biases accurately. Due to the impact of boresight roll bias in the vertical direction in addition to the impact across flying direction, it would aid in naturally decoupling \(\delta\Delta\phi\) and \(\delta\Delta X\), provided there are planar patches scanned from flight lines in the same direction with sufficient lateral separation and also, some planar patches located at a significantly high lateral distance from the flight lines. However, in the case of the unavailability of such planar patches located at a high lateral distance, the decoupling of the two parameters can be achieved by ensuring a significant variation in the value of \((H\pm\Delta h)\). This can be achieved by one of the following ways:  
Figure 7: Relationship between \(x^{\prime}\)-coordinates for: (**a**) two tracks in same direction; and (**b**) two tracks in opposite directions. * _Two different flying heights_: The shift caused along the flying direction by the bias \(\delta\Delta\phi\) will vary depending on the flying height, whereas the shift due to the bias \(\delta\Delta X\) will be constant for any flying height. Thus, the two biases can be derived accurately using flight lines at different flying heights. * _Variation in target height w.r.t. flying height_: In case of flight lines at the same flying height, a variation in the height of points along a target primitive would result in varying shifts. The amount of variation required depends on the flying height, i.e., \(H\pm\Delta h=H\Big{(}1\pm\frac{\Delta h}{H}\Big{)}\). So, the higher the value of \(\frac{\Delta h}{H}\) for a given flying height, the better the estimation accuracy of \(\Delta X\) and \(\Delta\phi\) will be. **Impact of Bias in Boresight Heading (\(\Delta\kappa\)):**  
A bias in this component \((\delta\Delta\kappa)\) will cause shifts across and along the flying direction. The impact of this bias _across the flying direction_\((\mp\,y^{\prime}\,\delta\Delta\kappa)\) is dependent on the \(y^{\prime}\)-coordinate variability. So, this would cause a discrepancy for vertical planes parallel to the flying direction for a single track. Moreover, the discrepancy on combining two tracks in same or opposite directions would depend on the \(\pm y^{\prime}\) variability within the points comprising the planes. The impact of this bias _along the flying direction_\((\pm\,x^{\prime}\,\delta\Delta\kappa)\) is flying direction independent since the sign change of \(x\prime\)-coordinate on flight direction change is nullified by the presence of dual sign in the term. Also, the magnitude of this impact is \(x\prime\)-coordinate dependent. This bias would induce a discrepancy in case of vertical planes perpendicular to the flying direction scanned from two flight lines in the same/opposite directions depending on the lateral distance between the tracks, as given by Equations (11) and (12). For the UAV system used in this study, the impact along flying direction will be more pronounced than the impact across flying direction since the LiDAR unit is scanning with the laser beams rotating \(360^{\circ}\) around the \(y^{\prime}\)-axis, thus resulting in a high \(x^{\prime}\)-coordinate variability. However, the \(y^{\prime}\)-coordinate variability is limited by the total vertical FOV of \(\pm 10^{\circ}\) of the Velodyne VLP-16 Puck Hi-Res unit.