## 3 The Radiative Transfer Equation
### RTE Numerical Solutions
#### 3.4.1 Discrete Ordinate Model  
The DOM calculates a solution to the RTE by considering the medium as a stack of homogeneous layers, which implies the same attenuation coefficient \(c(\lambda)\) and refractive index \(n(\lambda)\). Propagation layers are divided into a finite number of discrete solid angles \(\varnothing_{s}\), and the RTE is solved individually for each of these solid angles [9; 100; 90]. The following layer will use these solutions as boundary conditions for the next set of solutions. This approach is similar to the SSAA analytical approach discussed above. The main disadvantage of this method is that it cannot handle highly peaked VSFs (like those found in oceanic particulate matter) because of the size of solid angles compared to the region of interest. In summary, in this approach, the angular domain is discretised into a finite number of directions, and the RTE is solved along these discrete ordinates. Hence, the downside of this approach is that it can only be used for homogeneous water bodies [101; 102]. Also, it cannot handle highly peaked VSFs, such as those present in marine particulate matter, due to the magnitude of solid angles in relation to the area of interest [90]. In [103], the researchers created a fast, direct, and efficient numerical RTE solution. This solution leverages the matrix-free Gauss-Seidel iterative approach to compute the received signal power of UOWC systems. It can also handle significantly forward-packed VSF, which DOM cannot do. Based on the simulation outcomes, this approach yields the same precision as the MC method while using significantly less simulation time. The reader is referred to [104; 105; 106; 107] for further information on DOM. Below is a brief illustration of the mathematical expression we used in the DOM for RTE implementation [6; 8; 9] as a one-dimension (1D) RTE as follows:  
\[\frac{dI}{ds}=-a(\lambda)I+\frac{a(\lambda)}{4\pi}\int\limits_{0}^{2\pi}I( \theta)d\Omega \tag{53}\]  
where \(I\) is the intensity of optical radiation, \(s\) is the spatial coordinate, \(a(\lambda)\) is the absorption coefficient, \(\theta\) is the propagation angle, and \(d\Omega\) represents the solid angle. In the DOM approach, we discretise the angular domain into \(N\) discrete angles \(\theta_{n}\), and then we approximate the integral over all angles by a discrete sum:  
\[\frac{dI}{ds}=-a(\lambda)I+\frac{a(\lambda)}{4\pi}\sum\limits_{n=1}^{N}w_{n} \ I(\theta_{n}) \tag{54}\]  
where \(w_{n}\) are the weights associated with each discrete angle \(\theta_{n}\). To solve this equation numerically, we used iterative methods such as the method of successive orders or the method of relaxation. In our Python code implementation, \(I\) would use this discretised form of the RTE and appropriate numerical methods to solve for the intensity \(I\) at each spatial point. Figure 10 shows the output of our implementation of DOM for coastal and harbor water scenarios for downward directions (\(s>0\)) and upward directions (\(s<0\)). It is easy to notice in Figure 10, over the range of the downward directions, that the DOM intensity solution profiles exhibit a gamma-gamma distribution with Rytov standard deviation \(1\leq\sigma\leq 3\)[108], i.e., a strong-to-saturation scattering regime for the coastal waters. This implies that the DOM approach tends to overestimate the scattering. Harbor waters exhibit a near-saturation regime, which leads to an intensity (mW\(\cdot\) mm\({}^{-2}\)) profile that resembles the expected exponential distribution with Rytov standard deviation \(3\leq\sigma\leq 10\)[108]. Nevertheless, at \(s=-0.24\), the DOM solution shows a discontinuity for the coastal and harbor water scenarios. Also, in Figure 10, the DOM solution shows the expected high attenuation of the photons in harbor waters compared to coastal waters. Hence, the outputs of the DOM method should be compared with other benchmark methods like BLL for high-absorption scenarios. In general, DOM can be used to deduce valid solutions.