# Airborne Waveform Lidar Simulator Using the Radiative Transfer of a Laser Pulse
## Theories and Modeling  
### Lidar Waveform Solution by Radiative Transfer Theory  
Compared to the simple radiometric solution, the radiative transfer theory of a laser pulse and the numerical solution is quite challenging. Thus, only the essential components of the theory are briefly described here. Refer to References [11; 12; 13; 14; 15; 16] for the full theories and the numerical solutions [16]. To solve the governing equation of a lidar system, the irradiance solutions are needed in Equation (1). The irradiance as a function of 2D position vector, \(\mathbf{r}\), on the beam cross section at a given distance, \(z\), is an integration of the radiance distribution over the hemisphere. \[E(\mathbf{r})=\int d\omega(\mathbf{n})\,I(\mathbf{r},\mathbf{n}). \tag{2}\]  
Here, the radiance is integrated over infinitesimal solid angle, \(d\omega(\mathbf{n})\), centered on the direction vector \(\mathbf{n}\). Thus, we need a radiance distribution function \(I(\mathbf{r},\mathbf{n})\), which is obtained as a solution of the laser pulse propagation based on the radiative transfer theory. \[\mathbf{n}\cdot\
abla I(\mathbf{\rho},\mathbf{n})=-cl(\mathbf{\rho},\mathbf{n})+\int_{4\pi}I(\mathbf{ \rho},\mathbf{n^{\prime}})\beta(\mathbf{n^{\prime}}\rightarrow\mathbf{n})d\omega(\mathbf{n^{ \prime}}). \tag{3}\]  
This is a 3D radiative transfer equation (RTE) that describes the net radiance balance at a given position and direction. The value \(\mathbf{\rho}\) is a 3D position vector, \(c\) is an attenuation coefficient, and \(\beta\) is a volume scattering function of the medium. Since there is no analytical solution for Equation (3) in 3D, it is reduced to a radiance distribution on a 2D plane perpendicular to the beam direction using small angle approximation because the volume scattering function of the natural air or water medium is strongly forward-oriented. \[\frac{\partial}{\partial z}I(z,\mathbf{r},\mathbf{n}_{t})+\mathbf{n}_{t}\cdot\
abla_{t}I( z,\mathbf{r},\mathbf{n}_{t})=-cl(z,\mathbf{r},\mathbf{n}_{t})+b_{\rm s}\int I(z,\mathbf{r},\mathbf{n^{ \prime}}_{t})x_{\rm s}\Big{(}\big{|}\mathbf{n}_{t}-\mathbf{n^{\prime}}_{t}\big{|} \Big{)}d\mathbf{n^{\prime}}_{t}. \tag{4}\]  
Here, \(\mathbf{n}_{t}\) is the 2D direction vector defined on the 2D beam cross section and its norm is small according to the small angle approximation, \(b_{\rm s}\) is a small angle forward-scattering coefficient, and the small angle forward-scattering phase function, \(x_{\rm s}\), is defined as a function of difference between incoming and scattered 2D direction vectors. The radiance solution is derived via an analytical Fourier transform. \[\widetilde{I}(z,\mathbf{k},\mathbf{p})=\widetilde{I}(z_{0},\mathbf{k},\mathbf{p}+\mathbf{k}(z-z_{ 0}))exp\Bigg{(}-\int_{z_{0}}^{z}\{c-2\pi b_{\rm s}\cdot\widetilde{x}_{s}(\mathbf{ p}+\mathbf{k}(z-c))\}d\varepsilon\Bigg{)}. \tag{5}\]  
Here, \(\mathbf{k},\mathbf{p}\) is the conjugate frequency variable of the \(\mathbf{r},\mathbf{n}_{t}\) and \(\widetilde{x}_{s}\) is the analytical Fourier transform of \(x_{\rm s}\). Converting Equation (5) into an inverse Fourier transform and plugging it into the Equation (2) yields the following:  
\[E(z,\mathbf{r})=\int\widetilde{I}(0,\mathbf{k},z\mathbf{k})exp(-i\mathbf{k}\cdot\mathbf{r})\,exp[- z^{\prime}(\mathbf{r})a_{\rm s}-a_{\rm bs}(z\mathbf{k})z]d\mathbf{k}. \tag{6}\]  
Here, \(a_{\rm s}\) is the effective absorption coefficient and \(a_{\rm bs}\) is the effective attenuation coefficient due to forward scattering. Two things are needed to solve Equation (6). First, the source laser pulse radiance distribution in the Fourier transform, and, second, the effective attenuation coefficient.