# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## Theories and Modeling
### Airborne Lidar Simulator
Then the direction vector in SBF for the Galvo-mirror and circular scanner is given, respectively, as follows:  
\[d_{P}^{\mathrm{SBF}}=[0,\sin\theta,\cos\theta],\] \[d_{P}^{\mathrm{SBF}}=[\sin(FOV/2)\cdot\cos\theta,\sin(FOV/2) \cdot\sin\theta,\cos(FOV/2)].\]  
The final step is to estimate a range to the DEM surface. Unlike the active real lidar that measures the range, the simulator cannot measure the distance. The simulator can only estimate the range to the DEM surface by following the computation procedure described in the direct georeferencing section above, as follows: Starting from the direction vector in SBF \(\rightarrow\)computing a position in SBF by introducing unknown range \(\rho\rightarrow\) transforming the SBF position into IBF position and translating by a lever-arm displacement \(\rightarrow\) transforming the position in IBF into the position in LTF using (\(R\), \(P\), \(H\)) angles. The solution for the position where the laser pulse hits the target can be computed in two methods. One approach is the analytical solution when a specific geometry of the target yields a forward solution. The other approach is a generic optimization using DEM to get the range solution. Once the beam center interaction point is obtained, it is refined into the waveform solution using the radiative transfer-based irradiance distribution described in the above section. Several examples are presented to demonstrate how the waveform-based solution is different compared to the simple georeferencing of the beam center. The example shown in Figure 6 uses a cube and a beam spot size comparable to the cube dimension, which creates a non-negligible interaction effect. The upper panel in Figure 6 illustrates a sequence of interaction lines between the delta-pulse and the cube surface along the arrow direction. As pulse 1 propagates downward and a little toward the right side, the interaction line touches down on the left beam edge first, and then the interaction line makes a transition to the right side to create a full circle interaction area. Thus, the z-coordinate of the lidar point is exactly the altitude of the cube top surface, which is shown in the P1 waveform in Figure 7.  
Figure 4: Preparation of orientation of simulated flight lines. Figure 5: Preset flight patterns of the airborne lidar simulator. Figure 6: A cube target interacts with an incoming laser pulse at different locations. The 3D point cloud on the right side is colored by the intensity, which represents the relative energy given by the waveform integral in Figure 7. The color scheme follows the typical “rainbow” color table of the lidar point cloud visualization so that the yellowish color represents higher values and the bluish color represents lower values. \[\text{PPM\_al}=(2\text{:}F_{s}/v),\;\text{PPM\_ac}=\text{PRF}/(2\text{:}F_{s} SW).\]0 The beam interaction examples for 3 more points and the waveform near the target. The maximum intensity of each waveform corresponds to the z-altitude of the lidar point computed by direct georeferencing. The estimated relative reflected energy of the pulse and the pulse center altitude for each point are as follows: 1.0 (1.0 m) for point 1, 0.08 (1.043 m) for point 2, 0.54 (0.990 m) for point 3, and 0.14 (0.957 m) for point 4. Pulse 2 hits the upper-left edge as it is moving in a westerly downward direction. Thus, it hits on the right edge of the beam circle first and makes a transition toward the left. As it hits the small area of the corner, the interaction area is a small wedge giving only 8% of the relative energy, which means the total reflected energy, by integrating the low intensity, waveform is about 8%, compared to the full circle interaction of point 1. Additionally, the computed altitude is 4.3 cm higher than the cube surface because the short duration of the interaction occurs when the beam center is still above the surface before moving down to the cube surface level. The rest of the interpretation can be understood in a similar manner.