# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## 3 Results
### TPU Estimation  
Using Equations (21)-(26), the TPU \(\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)\) associated with every lidar point is computed. The variation of the TPU values comes from the partial derivative, which is related to the magnitude of local variables. Consider the first term in TPU Equation (22) as an example. Although the uncertainty \(\left(\sigma_{x_{0}}\right)\) is constant, the partial derivative of a Jacobian element \(\frac{\partial x}{\partial x_{0}}\) can have a wide range of values depending on the variables used in the computation. Thus, the dynamic range of TPU is determined by the dynamic range of variables, such as lidar range, scan angle range, IMU angle range, and so on.  
Figure 9 shows all the uncertainty sources and the lidar system parameters. The lidar system parameters on the right are the common input. Figure 9a represents horizontal TPU and Figure 9b represents vertical TPU using the high precision IMU system with circular scanner. The range of horizontal TPU is 14-15 cm and the vertical TPU range is 4-5 cm, which is much smaller than the horizontal TPU. The overall magnitude of the TPU is determined mainly by the AGL (1 km) and minor contributions come from the rest of the system specification. The minimal range (1 cm) is due to the IMU angle variation. Figure 9c represents the horizontal TPU and Figure 9d represents the vertical TPU using the low precision IMU system with a linear scanner. The range of horizontal TPU is 37-47 cm and the vertical TPU range is 1-15 cm. The high TPU using low precision IMU is obvious compared to the low TPU using high precision IMU. Figure 9a,b use a circular scanner that has a constant off-nadir angle. Figure 9c,d, however, use a linear scanner with the off-nadir angle between \(-20^{\circ}\) and \(20^{\circ}\). Thus, in addition to the raised TPU baseline caused by the low precision IMU, the large off-nadir scan angle range adds a substantial TPU range.  
It is reasonable to assume that most of the uncertainties are constant numbers, at least during the flight time. However, the GNSS positional uncertainties \(\left(\sigma_{x_{1}},\sigma_{y_{1}},\sigma_{z_{1}}\right)\) can be quite dynamic depending on the satellite conditions. Thus, this situation was simulated by creating a temporary short spike in \(\left(\sigma_{x_{1}},\sigma_{y_{1}},\sigma_{z_{1}}\right)\). Figure 10 shows when the temporary poor position dilution of precision (PDOP) occurs in one swath, while the other swath is in good GNSS condition. The left panel in Figure 10, with a linear galvo-mirror scanner, shows the quick rise and fall of the GNSS uncertainty, while the right panel, with a circular scanner, shows a different pattern. The TPU provides a quality check of the lidar  
Figure 9: The simulated TPU and the uncertainty input components from the lidar simulator GUI: (**a**) Horizontal TPU with a circular scanner, (**b**) vertical TPU with circular scanner, (**c**) horizontal TPU with linear scanner, (**d**) vertical TPU with linear scanner. The corresponding parameters on graphical user interface.  
data. When the abnormal TPU spike area is identified, a decision can be made about recollecting data over this area.