## 3 Bias Impact Analysis
### Bias Impact Analysis for a Spinning Multi-Beam Laser Scanner
Due to the impact of boresight pitch bias in the vertical direction in addition to the impact along flying direction, it would aid in naturally decoupling \(\delta\Delta\omega\) and \(\delta\Delta Y\), provided there is a significant \(y^{\prime}\)-coordinate variability. But, the \(y^{\prime}\)-coordinate variation could be reduced depending on the nature of the targets used or the sensor configuration. In this case, the \(y^{\prime}\)-coordinate variability is limited by the relatively narrow vertical FOV of the LiDAR unit (\(\pm 10^{\circ}\)), thus making it insufficient to eliminate the correlation. In such cases, there is a need to have a significant variation in the value of \((H\pm\Delta h)\) in order to decouple the impacts of \(\delta\Delta Y\) and \(\delta\Delta\omega\) so as to estimate both these biases accurately. This can be achieved by one of the following ways:  
* _Two different flying heights:_ The shift caused along the flight direction by the bias \(\delta\Delta\omega\) will vary depending on the flying height, whereas the shift due to the bias \(\delta\Delta Y\) 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 \(\Delta Y\) and \(\Delta\omega\) will be. A high variation in \(\frac{\Delta h}{H}\) can be achieved either by having vertical planes at different heights or with a significant variation in the heights along given targets. **Impact of Bias in Boresight Roll (\(\Delta\phi\)):**  
A bias in this component (\(\delta\Delta\phi\)) will cause shifts across the flying direction as well as in the vertical direction. The impact of this bias _across the flying direction_\((\pm\,z^{\prime}\,\delta\Delta\phi=\mp(H\pm\Delta h)\,\delta\Delta\phi)\) is flying direction dependent and its magnitude depends on the height (\(z^{\prime}\)) of the point in question relative to the virtual laser unit coordinate system. This bias would impact vertical planes parallel to the flying direction scanned from two flight lines in opposite directions. The impact of this bias in _the vertical direction_\((-x^{\prime}\,\delta\Delta\phi)\) is flying direction dependent (since \(x^{\prime}\)-coordinates will change signs depending on the flying direction, as shown in Figure 6) and its magnitude depends on the \(x^{\prime}\)-coordinate of the point in question. The resultant discrepancy in the \(Z\)-coordinate on combining two tracks in the same and opposite directions are given by Equations (9) and (10), respectively, according to Figure 7. Here, \(D_{AB}\) denotes the lateral distance between the two tracks and \(X\) denotes the distance of the point in question from the line bisecting the lateral distance between the two flight lines. The analysis reveals that this bias would cause a discrepancy for horizontal planes scanned from two flight lines in the same direction depending on the lateral distance between the tracks. On the other hand, for two tracks in opposite directions, the discrepancy would depend on the lateral location of the planar patch of interest relative to the bisecting direction between the tracks.