# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## Theories and Modeling
### Lidar Direct Georeferencing  
While the radiative transfer-based waveform approach is the advanced component of the airborne lidar simulator, the direct georeferencing is the backbone of the simulator. A direction vector of an emitted laser pulse defined in sensor body frame (SBF) is defined first, then, by multiplying with the lidar range, \(\rho\), to a point, \(P\), on the surface, the position vector in the SBF is determined.  
\[\mathbf{X}_{P}^{SBF}=\rho\cdot\mathbf{d}_{P}^{SBF}. \tag{12}\]  
Next, the position vector in SBF \(\mathbf{X}_{P}^{SBF}\) is converted to the position in the IMU body frame (IBF), \(\mathbf{X}_{P}^{IBF}\), using the following relationship:  
\[\mathbf{X}_{P}^{IBF}=\mathbf{R}_{SBF}^{IBF}\mathbf{X}_{P}^{SBF}+\mathbf{X}_{O_{SBF}}^{IBF}, \tag{13}\]  
where \(\mathbf{X}_{O_{SBF}}^{IBF}\) is the lever-arm vector to the SBF origin in the IBF and \(\mathbf{R}_{SBF}^{IBF}\) is a rotation matrix that transforms \(\mathbf{X}_{P}^{SBF}\) to \(\mathbf{X}_{P}^{IBF}\) and is defined using three boresighting angles, \(\omega,\varphi,\kappa\), as follows:  
\[\mathbf{R}_{SBF}^{IBF}=\left[\begin{array}{ccc}\cos\varphi\cos\kappa&\sin\omega \sin\varphi\cos\kappa-\cos\omega\sin\kappa&\cos\omega\sin\varphi\cos\kappa+ \sin\omega\sin\kappa\\ \cos\varphi\sin\kappa&\sin\omega\sin\varphi\sin\kappa+\cos\omega\cos\kappa& \cos\omega\sin\varphi\sin\kappa-\sin\omega\cos\kappa\\ -\sin\varphi&\sin\omega\cos\varphi&\cos\omega\cos\varphi\end{array}\right] \tag{14}\]  
Next, the position vector, \(\mathbf{X}_{P}^{IBF}\), is transformed into a new position vector, \(\mathbf{X}_{P}^{LTF}\), in a local tangential frame (LTF), as follows:  
\[\mathbf{X}_{P}^{LTF}=\mathbf{R}_{IBF}^{LTF}\cdot\mathbf{X}_{P}^{IBF}. \tag{15}\]  
An LTF is defined in reference to a global geodetic frame, which follows a coordinate axes convention of a North-East-Down (NED) reference frame. The orientation of the IBF, with respect to the LTF, is described by three IMU angles, roll \((R)\), pitch \((P)\), and heading \((H)\), as follows:  
\[\mathbf{R}_{IBF}^{LTF}=\left[\begin{array}{ccc}\cos P\cos H&\sin R\sin P\cos H- \cos R\sin H&\cos R\sin P\cos H+\sin R\sin H\\ \cos P\sin H&\sin R\sin P\sin H+\cos R\cos H&\cos R\sin P\sin H-\sin R\cos H\\ -\sin P&\sin R\ \cos P&\cos R\cos P\end{array}\right]. \tag{16}\]  
Figure 2: Interaction between a laser delta-pulse and the object. The 3D view on the left and the 2D top-view on the right.  
Along with the position of the GNSS antenna phase center, the lever-arm distances between the IMU origin and the antenna can be used to calculate the position of the O_IBF (which is the same as O_LTF). Using the latitude, \(\phi\), longitude, \(\lambda\), and ellipsoid height, \(h\), from O_LTF, it is possible to form a rotation matrix that transforms an arbitrary position vector in LTF to the position vector in the Earth Centered Earth Fixed (ECEF) coordinate:  
\[\mathbf{R}_{LTF}^{ECEF}=\left[\begin{array}{ccc}-\sin\varphi\cos\lambda&-\sin \lambda&-\cos\varphi\cos\lambda\\ -\sin\varphi\sin\lambda&\cos\lambda&-\cos\varphi\sin\lambda\\ \cos\varphi&0&-\sin\varphi\end{array}\right] \tag{17}\]  
In addition to the rotation matrix, the geodetic position \((\varphi,\ \lambda,\ h)\) of O_LTF itself can be transformed into the position in the ECEF coordinate using the following relationship:  
\[\mathbf{R}_{O_{LTF}}^{ECEF}=\left[\begin{array}{c}(r_{N}+h)\cos\varphi\cos \lambda\\ (r_{N}+h)\cos\varphi\sin\lambda\\ \left[\left(1-e^{2}\right)r_{N}+h\right]\sin\varphi\end{array}\right],\ r_{N}= \frac{a}{\sqrt{1-e^{2}\sin^{2}\varphi}}, \tag{18}\]  
where \(a\) is the semi-major axis, \(e\) is the eccentricity, and \(r_{N}\) is the position in ECEF coordinates. Now the position vector can be solved in ECEF coordinates, as follows:  
\[\mathbf{X}_{P}^{ECEF}=\mathbf{R}_{LTF}^{ECEF}\cdot\mathbf{X}_{P}^{LTF}+\mathbf{X}_{O_{LTF}}^{ECEF}. \tag{19}\]  
Finally, the ECEF coordinates are converted to the map projection, such as the Universal Transverse Mercator (UTM) or a state plane.