## 3 The Radiative Transfer Equation
### RTE Numerical Solutions
#### 3.4.2. Invariant-Imbedding Method  
The invariant-imbedding method is a numerical solution based on analytical estimation. It creates a model by converting the RTE, a two-point boundary value problem, into a regular differential equation with an initial condition [90]. For this reason, it can only solve one-dimensional problems with a boundary condition array of two elements. The overall solution will be a two-component solution corresponding to the transmitted (\(I_{tr}\)) and the reflected (\(I_{re}\)) intensities. Hence, the scattered beam intensity will be (\(I_{tr}-I_{re}\)). This implies that the solution highly depends on the seawater type (i.e., the attenuation coefficient \(c(\lambda)\) and scattering albedo (\(w_{s}\)) and the SPF model. The invariant-imbedding method transforms the RTE into a set of coupled differential equations, which can be solved using numerical integration. The team of [74] indicated that this numerical approach is complex mathematically and challenging to program. However, we devised an implementation approach using the Runge-Kutta Python library to solve the differential equations in our implementation. Setting the most relevant boundary conditions to obtain a reliable outcome was challenging. Hence, the availability of experimental data is essential rather than critical to setting the most realistic boundary conditions to reduce the implementation run time. As in any numerical computation, reducing the mean square error could require injecting a bias to achieve a more realistic solution profile, equivalent to a curve-smoothing operation. Figure 11 shows the implementation of an invariant-imbedding method in \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & & **Advantages** & & **Disadvantages** & **Ref.** \\ \hline \multirow{8}{*}{DOM} & \multirow{8}{*}{\(\bullet\)} & \multirow{8}{*}{High efficiency in simulation} & \(\bullet\) & Programming is difficult & \multirow{8}{*}{[74]} \\  & & & \(\bullet\) & Cannot perform well & & \\  & & \(\bullet\) & High efficiency in simulation & & with strongly peaked & \\  & & \(\bullet\) & Quick for calculating irradiance & & SPF & \\  & & & and homogeneous water & & \(\bullet\) & Extremely mathematical & \\  & & & \(\bullet\) & Solves only & & \\  & & & & & \multicolumn{1}{c}{one-dimension problems} & \\ \hline \multirow{8}{*}{Invariant Imbedding} & \multirow{8}{*}{\(\bullet\)} & \multirow{8}{*}{Covers all multiple-scattering orders} & \multirow{8}{*}{\(\bullet\)} & Extremely mathematical & \multirow{8}{*}{[74]} \\  & & & No statistical errors in computed radiance & & \(\bullet\) & Programming is difficult & \\  & & \(\bullet\) & High efficiency in simulation & & \(\bullet\) & Solves only & \\  & & \(\bullet\) & High efficiency in simulation & & one-dimension problems & \\  & & \(\bullet\) & Extremely fast & & \\ \hline \multirow{8}{*}{Stochastic Model} & \multirow{8}{*}{\(\bullet\)} & \multirow{8}{*}{High efficiency in simulation} & \multirow{8}{*}{\(\bullet\)} & Programming is difficult & \multirow{8}{*}{[24,25,109]} \\  & & & \(\bullet\) & Lack of experimental & & \\ \hline \multirow{8}{*}{MCS} & \multirow{8}{*}{\(\bullet\)} & \multirow{8}{*}{It is based on conceptually} & \multirow{8}{*}{\(\bullet\)} & \multirow{8}{*}{Statistical errors of} \\  & & & essential physics and simulates & & \\  & & how nature absorbs and scatters & & & \\  & & idealistic light beams & & & computed radiances can \\  & & Completely generic; can handle & & be minimised by tracing & \\  & & time-dependent and & & many photons, which & \\  & & three-dimensional problems of & & needs much running & \\  & & any geometry & & computer time & \\  & & \(\bullet\) & Mathematical tricks can be & & \\  & & applied to accelerate & & & \\  & & computations & & & \\  & & \(\bullet\) & Simple to program & & \\  & & \(\bullet\) & A precise solution with a great & & \\  & & degree of flexibility & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: A summary of commonly used numerical approaches for solving the RTE. terms of scattered optical radiance.