# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## Theories and Modeling
### Lidar Waveform Solution by Radiative Transfer Theory
Assuming a Gaussian beam the radiance and its Fourier Transform are as follows:  
\[\overline{I}(0,\mathbf{k},z\mathbf{k})=\frac{1}{\left(2\pi\right)^{2}}exp\left(-\mathbf{k}^{ 2}\left(1+\frac{z}{nH_{\mathrm{s}}}\right)^{2}r_{0}^{2}\right). \tag{7}\]  
Here, \(r_{0}\) is effective radius defined by multiplication of half of the beam divergence angle, \(\Theta\), and slant altitude, \(H_{\mathrm{s}}\). The effective attenuation coefficient, \(a_{bs}\), due to the small angle forward-scattering \(b_{\mathrm{s}}\) is given as follows using the analytical forward-scattering phase function model:  
\[a_{bs}(z\mathbf{k})=b_{s}-\frac{\alpha b_{s}}{z\mathbf{k}}ln\left(\left(\frac{\alpha}{ z\mathbf{k}}\right)^{-1}+\left(\left(\frac{\alpha}{z\mathbf{k}}\right)^{-2}+1\right)^{0.5 }\right). \tag{8}\]  
Here, \(\alpha\) is a volume scattering function shape parameter by [PERSON] [13]. The effective attenuation coefficient, \(a_{bs}\), characterizes the contribution from small angle forward scattering to the attenuation of spatial harmonics of the spatial frequency, \(k\), in the initial distribution of radiance within a light beam when propagated to the distance, \(z\), in the medium. Finally, using Equations (7) and (8) the irradiance solution of Equation (6) becomes the following:  
\[ \begin{array}{c}E(z,r)=\frac{exp\left(-a_{\mathrm{s}}z^{\prime}\left(r_{-} \right)\right)}{2\pi}\int f(k;z,r)d\mathbf{k},\\ f(k;z,r)=J_{0}(kr)kexp\left(-\left(\frac{k^{2}}{4}\right)P^{2}-a_{bs}(z\mathbf{k}) z\right)\end{array}  \tag{9}\]  
Here the effective radius is defined as \(P=\left(\frac{\Theta}{2}\right)\left(H_{\mathrm{s}}+\frac{z}{n}\right)\) and \(J_{0}\) is the 0-th order first kind Bessel function and \(z^{\prime}\left(r_{-}\right)\) is the geometrical distance of the slant-incident beam projected along the coordinate parallel to the direction of pulse propagation. Equation (9) is a final irradiance solution for a point at \(r\) away from the center on a circular plane at a distance, \(z\), for a Gaussian laser pulse. Once another irradiance solution for the receiver is calculated in the same manner, the lidar waveform is formed by applying two irradiances in Equation (1) and by computing the numerical integration. Complete details of the theories can be found in [16]. These equations are most generic to be used for laser pulse propagation through any medium. This level of sophistication is critical when the laser pulse propagates through substantially dense medium, like water for bathymetric lidar. The high absorption and scattering by the natural water medium must be properly handled using these equations. However, if a lidar simulation is only for the topographic objects, the equations can be dramatically simplified. Equation (9) will be a concentric 2D Gaussian defined by the beam spot radius, which is determined by the slant sensor altitude, \(H_{s}\), and the full beam divergence, \(\Theta\), at the \(e^{-2}\) level. Assuming the instantaneous field of view (IFOV) is significantly larger than the beam divergence, the two irradiance multiplication in the governing Equation (1) is still a Gaussian distribution, mainly determined by the beam divergence as follows:  
\[E_{E}(z,r)E_{R}(z,r)\propto exp\left(-\left(\frac{r}{\left(H_{\mathrm{s}} \Theta/2\right)/\sqrt{2}}\right)^{2}\right). \tag{10}\]  
The delta-pulse first touches down at a tiny line segment A (Figure 2). The length of the interaction line segment increases to the arrow direction and then decreases toward the corner. \[S(t)\propto\int_{\mathbf{r}_{2}(t)}^{\mathbf{r}_{1(t)}}\beta(z(t),\mathbf{r})E_{E}(z(t), \mathbf{r})E_{R}(z(t),\mathbf{r})d\mathbf{r}. \tag{11}\]The interaction function, \(\beta(z(t),\mathbf{r})\) in this case, is the target surface reflectance. Assuming the homogeneous reflectance property of the target surface, the value is constant all over the target surface. The radiative transfer-based irradiance solution, \(E_{E}E_{R}\), is also a 2D circular plane. However, the plane hits the target surface with a non-zero incident angle to make it a slant plane.