# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## 3 Results
### Accuracy Assessment of 3D Point Cloud  
The lidar simulator is an important research tool for 3D accuracy assessment. For example, consider a parking lot with painted lines on the pavement, as shown in Figure 11a. A common practice in performing a check point survey is to place a GNSS rover on the intersection point of two painted lines. For the 3D accuracy assessment, the intersection position (\(x\), \(y\), \(z\)) in the airborne lidar point cloud must be identified. Once identified, the difference from the survey point yields an error vector (\(\Delta x\), \(\Delta y\), \(\Delta z\)). Using multiple error vectors, a statistical estimate of the 3D accuracy \(\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)\) can be made. However, the position identification from the airborne point cloud is difficult, mainly because of the low point density. Thus, the usual practice is to ignore horizontal accuracy and find the location in the airborne point cloud using the surveyed horizontal position and interpolate the z-value for that position. The difference against the surveyed z-value is used to calculate the root mean square error in z (RMSEz). Identifying the intersection point from two crossing lines is challenging. Figure 11b,c illustrates the random cases of lidar point cloud intensity over the modeled painted lines with the circular scanner  
Figure 11: Simulated intensity point cloud on the painted lines on the parking lot: (**a**) Parking lot paint line for 3D accuracy assessment, (**b**) simulated intensity point cloud of the 2 m line, (**c**) simulated intensity point cloud of the 6 m line. Figure 10: Horizontal TPU with temporary poor position dilution of precision (PDOP) with a linear (**left**) and circular (**right**) scanner. at 9 PPSM, which is the USGS quality level 1 category. The color intensity ranges from yellow (full spot hit), green, cyan, and blue (minimal hit at the line edge). Intensity is used because the elevation of the painted lines and the background pavement is not distinguishable. The background dotted lines are the painted lines with 20 cm width. Only the simulator, however, can show the true painted lines. Figure 11b simulates a 2 m long line and Figure 11c simulates a 6 m long line. Depending on the location in a swath, the point density, and various other conditions, some cross lines will be under sampled, e.g., the upper right in Figure 11b. The 6 m line example in Figure 11c demonstrates that the number of points is enough to describe the cross line properly. An airborne lidar simulator is a useful tool in addressing the required conditions (line length, width, point density, and intensity contrast) of the crossing lines for a 3D accuracy assessment. About 200 cross painted lines were distributed in the two-swath overlap area. As demonstrated in Figure 11, to pinpoint the intersection point requires an unrealistically large point density. Thus, directly identifying a conjugate point is not a correct approach. Instead, 3D line regression modeling is needed so that the intersection point can be determined from two regression lines. The difference (\(\Delta x\), \(\Delta y\), \(\Delta z\)) is computed against the true intersection point. Using a large number of error vectors, a statistical estimate of 3D accuracy \(\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)\) can be made. The uncertainty plot (Figure 12b) illustrates the uncertainty of \(x\) and \(y\) error \(\left(\sigma_{x},\sigma_{y}\right)\) as a function of painted line length. As the length of the painted line increases, the uncertainty becomes smaller. Figure 12a,c represents the (\(\Delta x\), \(\Delta y\)) distribution for the 2 m and 6 m lines, respectively, and the uncertainty \(\sigma\) values are 3 cm and 1.5 cm, respectively. The research results (like Figure 12b) obtained from the simulation will provide a guideline for a minimum line length to meet the uncertainty requirement in the 3D accuracy assessment.