# Underwater Optical Path Loss after Passage of a Tropical Storm
## 4 Two-Dimensional RTE  
Different from [6], we used a simple algorithm to solve a two-dimensional RTE rather than a complete package, the U.S. Navy's electro-optical detection simulator (EODES) for optical detection, especially for tactical performance prediction of laser line scan sensors of several AQS-24a systems for mine countermeasures in the fleet. Here, we discretize the differential-integral RTE into a set of algebraic equations and use the classic Gauss-Seidel iteration to get numerical solutions. Let the optical beam axis point face in the \(z\)-direction (i.e., the light propagation direction) with \(\rho\) the radial coordinate, \(\mathbf{n}\) the unit vector representing the projection of direction of a light vector onto the transverse plane, and \(\mathbf{n^{\prime}}\) the corresponding scattered direction. Let \(\theta\) be the scattering angle between \(\mathbf{n}\) and \(\mathbf{n^{\prime}}\), i.e., \(\mathbf{n}\mathbf{e}\mathbf{n^{\prime}}=\cos\theta\). Here, \(\theta\) is the scattering angle. An asymmetry parameter \(g\) (ranging from 0 to 1) describes scattering types with \(g=0\) for dominating isotropic scattering and \(g\) near 1 for peaked scattering. The value of \(g\) is near 0.924 for sea water [16], which indicates the feasibility for underwater optical communication. Scattering and absorption are azimuthally symmetric when they are independent on the azimuthal angle. The phase function is [17]  
\[\beta(\mathbf{n},\mathbf{n^{\prime}})=\frac{1-g^{2}}{2\pi(1+g^{2}-2g\cos \theta)} \tag{2}\]  
for the azimuthally symmetric scattering. For the azimuthally symmetric scattering and absorption, the radiance, \(L(\mathbf{r},\mathbf{n})[\text{Watts/(m${}^{2}$sr)}]\), is independent on the azimuthal angle. Under the small-angle approximation and negligible temporal dispersion, the RTE is a differential-integral equation with the position vector \(\mathbf{r}=(\rho,z)\), and the direction vector \(\mathbf{n}\) being two-dimensional [18],  
\[\mathbf{n}\bullet\
abla L(\rho,z,\mathbf{n})=-cL(\rho,z,\mathbf{n})+b\int_{2 \pi}\beta(\mathbf{n},\mathbf{n^{\prime}})L(\rho,z,\mathbf{n^{\prime}})d \mathbf{n^{\prime}}+S(\rho,z,\mathbf{n}) \tag{3}\]  
which has three independent variables, \(\rho\), z, and \(\theta\). Since the observational beam attenuation coefficient is vertical (z) with a resolution of approximately 1 m, the position coordinates (\(\rho\),z) are discretized into (1 m \(\times\) 1 m) grids:  
\[\rho_{i}=i\Delta\rho,z_{j}=j\Delta z,\ \Delta\rho=\Delta z=1\ \text{m}. \tag{4}\]  
The angular variable \(\theta\) is discretized into \(K\) (=72) directions with an equal angle interval (\(5^{\circ}\)),  
\[\theta_{k}=k\Delta\theta,\Delta\theta=5^{\circ}. \tag{5}\]  
The differential-integral Equation (3) is discretized into a set of algebraic equations,  
\[\sin\theta_{k}\frac{L_{i,j,k}-L_{i-1,j,k}}{\Delta\rho}+\cos\theta_{k}\frac{L_{ i,j,k}-L_{i,j-1,k}}{\Delta z}+c_{j}L_{i,j,k}=b_{j}\sum_{k^{\prime}=1}^{K}w_{k,k^{ \prime}}L_{i,j,k^{\prime}}+S_{i,j,k^{\prime}},\ b_{j}=\omega c_{j} \tag{6}\]  
where \(w_{k,k^{\prime}}\) is the discretized phase function (2) and \(c_{j}\) is the observed beam attenuation coefficient. Note that conversion of the differential-integral RTE (3) into a set of algebraic equations (6) is the major difference between [6] and this research. Following [8] with the same discrete phase function \(w_{k,k^{\prime}}\), the Gauss-Seidel iteration is used to solve the discrete two-dimensional (2D) RTE (6),  
\[L_{i,j,k}^{l+1}=\frac{\frac{\sin\theta_{k}}{\Delta\rho}L_{i-1,j,k}^{l+1}+\frac {\cos\theta_{k}}{\Delta z}L_{i,j-1,k}^{l+1}+b_{j}\sum\limits_{k^{\prime}=1}^{ K}w_{k,k^{\prime}}L_{i,j,k^{\prime}}^{l}+S_{i,j,k}}{\sin\theta_{k}/\Delta\rho+\cos \theta_{k}/\Delta z+c_{j}} \tag{7}\]where \(l\) is the iterative step.