# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## Theories and Modeling
### Total Propagated Uncertainty (TPU)  
The TPU in this simulator is calculated using the following direct georeferencing equation:  
\[\mathbf{X}_{P}^{LTF}=\mathbf{R}_{\text{IBF}}^{LTF}\Big{(}\mathbf{R}_{\text{SBF}}^{\text{ IBF}}\cdot\rho\cdot\mathbf{d}_{P}^{\text{SBF}}+\mathbf{X}_{O_{\text{SBF}}}^{\text{IBF}} \Big{)}+\mathbf{X}_{O_{\text{IBF}}}^{LTF}. \tag{20}\]  
This is the equation that describes the position of a point \(P(x,y,z)\) in the LTF coordinate. The direct georeferencing equation that calculates the position in LTF indeed includes all the major sources of errors, such as boresighting, scan angle, GNSS position, and IMU angles. Regarding TPU, it is called \"total\" because multiple error sources are combined and it is called \"propagated\" because the errors are propagated through Equation (20). The TPU as a statistical uncertainty is computed based on the linear approximation and by ignoring the covariances between multiple parameters. Currently, the assumption of negligible covariances is accepted in the lidar community. As the TPU is still in its infancy, no major lidar manufacturers (e.g., Teledyne Optech, Hexagon Leica, Riegl) currently provide direct georeferencing proprietary software that computes TPU. The only known exception is the CZMIL bathymetric lidar system that computes TPU with support from the US Army Corps of Engineers (USACE). The TPU model in this research utilizes 14 parameters. Thus, the number of covariance combination is \(14\times 14\)/2, considering the symmetry of the covariance matrix. While most covariances are likely to be negligible, potential exceptions of non-negligible covariances could be the position and orientation parameters. However, even the Applanix IMU systems do not have covariance specifications due to the measurement difficulties, but it is not impossible. Numerous U.S. Geological Survey (USGS) internal lidar accuracy assessments indicate that the uncertainties from airborne lidar data in terms of smooth surface precision [2] are very close to the computed TPU based on negligible covariances, which justifies the assumption that the negligible covariance is acceptable. To formulate the TPU computation equations, Equation (20) is rewritten as follows:  
\[\left[ \begin{array}{c}x\\ y\\ z\end{array} \right]=\mathbf{R}_{IBF}^{LTF}\cdot(\mathbf{R}_{SBF}^{IBF}\cdot\mathbf{\rho}\cdot\bm {d}_{P}^{SBF}+\left[ \begin{array}{c}x_{0}\\ y_{0}\\ z_{0}\end{array} \right])+\left[ \begin{array}{c}x_{1}\\ y_{1}\\ z_{1}\end{array} \right]. \tag{21}\]  
The TPU in \(x\) position in LTF is given by the following equation, which is derived from Equation (21) (The TPU for y and z components is also computed in the same manner):  
\[ \begin{array}{c}\sigma_{x}^{2}=\left(\frac{\partial x}{\partial x_{0}} \sigma_{x_{0}}\right)^{2}+\left(\frac{\partial x}{\partial y_{0}}\sigma_{y_{0} }\right)^{2}+\left(\frac{\partial x}{\partial z_{0}}\sigma_{z_{0}}\right)^{2} +\left(\frac{\partial x}{\partial z_{1}}\sigma_{x_{1}}\right)^{2}+\left(\frac {\partial x}{\partial y_{1}}\sigma_{y_{1}}\right)^{2}+\left(\frac{\partial x }{\partial z_{1}}\sigma_{z_{1}}\right)^{2}+,\\ \left(\frac{\partial x}{\partial\rho}\sigma_{\rho}\right)^{2}+\left(\frac{ \partial x}{\partial\theta}\sigma_{\theta}\right)^{2}+\left(\frac{\partial x }{\partial R}\sigma_{R}\right)^{2}+\left(\frac{\partial x}{\partial\theta} \sigma_{\rho}\right)^{2}+\left(\frac{\partial x}{\partial\theta}\sigma_{\mu} \right)^{2}+\left(\frac{\partial x}{\partial\theta}\sigma_{\
u}\right)^{2}+ \left(\frac{\partial x}{\partial\theta}\sigma_{\
u}\right)^{2}+\left(\frac{ \partial x}{\partial\theta}\sigma_{\
u}\right)^{2}.\end{array}  \tag{22}\]  
To solve Equation (22), two sets of data are needed. First, the uncertainty, \(\sigma\), of each component, which is given as a priori standard deviation. The second is the partial derivative, which is the Jacobian matrix element. The Jacobian matrix elements for lever-arm displacement and position vector of IBF origin can be derived as follows:  
\[\left[