# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## 1 Introduction  
Airborne lidar point clouds and the derived elevation products are essential geospatial datasets, along with airborne or spaceborne image data. Accordingly, the efforts to ensure high-quality lidar data became increasingly important [1; 2]. Much research contributed to the quality assurance (QA) of lidar data [3; 4; 5]. The direct georeferencing geomatics equation is the backbone of lidar accuracy research. An airborne lidar simulator that computes point cloud solutions by the forward modeling of airborne lidar parameters using the geomatics equation is a useful tool in lidar research. A lidar simulation software can simulate the effect of lidar parameters on the point cloud as well as various advanced lidar research. An airborne lidar simulator consists of a laser module, a scanner module, a global navigation satellite system, an inertial measurement unit (GNSS/IMU) module, the flight parameters, and the environmental parameters. The environmental parameters include the optical property of the propagating medium and the 3D terrain, including 3D objects. The laser module is characterized by pulse repetition frequency (PRF), pulse width, the beam divergence angle at the \(e^{-2}\) level, and the pulse distribution function. The scanner determines the overall scan pattern, such as the linear oscillating pattern from a Galvo mirror scanner and the circular pattern from a rotating prism. The scan frequency is also an important factor to determine the along-track and across-track lidar point distribution. The field of view (FOV) of a scanner along with the sensor above ground level (AGL) altitude determines the swath width. The DEM component describes the terrain over which the lidar system scans. Several airborne lidar simulator software programs were created using mainly direct georeferencing [6; 7; 8; 9]. The simple computation assumes that the beam divergence is effectively zero. Thus, it assumes that the light interaction occurs only at the point where the beam center hits the terrain surface. In case the beam edge touches the target, but the beam center evades the target as shown in Figure 1, the simple georeferencing will not find a target hit. Instead, it will find the point at the background terrain. If the beam spot is substantially large, only the waveform approach based on the radiative transfer of a laser pulse can describe the problem properly. An advanced simulator addresses the beam divergence using a simplified Gaussian beam spot model [10]. The approach presented here simulates the lidar scanner using the waveform from the solution of the radiative transfer equation of a laser pulse propagation, rather than a semi-analytical beam spot model. The governing lidar waveform equation is given by the following [11; 12; 13; 14; 15; 16]:  
\[S(t)=Q\Sigma\,\Omega\frac{c}{2n}\int_{z(t)}dr\beta(z,\mathbf{r})E_{E}(z,\mathbf{r})E_{ R}(z,\mathbf{r}). \tag{1}\]  
The general lidar equation models the returned optical power, \(S(t)\). Here, \(Q\) is the laser pulse energy, \(\sum\) is the receiver pupil area, \(\Omega\) is the solid angle subtended from a target to the receiver, \(c\) is the speed of light, and \(n\) is the refractive index. The governing equation models the optical return as an integral of the two irradiance distributions, emitter irradiance, \(E_{E}\), and the receiver irradiance, \(E_{R}\), in the presence of the interaction function \(\beta\). The integration using Equation (1) is performed over a normal cross-section plane at distance \(z(t)\) after the pulse propagation time \(t\). The vector \(\mathbf{r}\) represents the two-dimensional (2D) position on the cross-section plane. As the emitted laser pulse propagates, the cross-section area expands due to the beam divergence and the light is attenuated by the medium, creating weaker emitter irradiance. The concept of receiver irradiance may not be intuitive because the receiver does not send out any light. However, the receiver sensitivity distribution can be considered as if it is a radiance distribution.