## 3 The Radiative Transfer Equation
### RTE Analytical Solutions with Simplification
#### 3.3.2 Single-Scattering Approximation  
The SSA is a simplified model often used in radiative transfer theory to describe the behaviour of light in scattering media. SSA assumes that each photon undergoes at most one scattering event before being absorbed or escaping the medium. While SSA is not a complete solution to the full RTE, it provides a valid approximation in many scenarios, particularly for optically thin or weakly scattering media. To apply the SSA, we need to assume the following simplifications:* The water is homogeneous. Hence, the IOPs have no depth dependency, which requires that the water is infinitely deep;
* The sea surface is level, which implies no wind speed;
* The optical transmitter is the only source of photons;
* There is no inelastic scattering. Then, Equation (43) evolves into the following:  
\[\frac{dL(z,\diameter)}{c\ dz}=L(z,\diameter)+\omega_{0}\int_{-1}^{1}\int_{0}^{2\pi} \stackrel{{\sim}}{{\beta}}(\ \{\dot{\mu},\diameter^{\prime}\}\rightarrow\{\mu,\diameter)\}L(z,\diameter)\ d \diameter^{\prime}\ d^{\prime} \tag{45}\]  
The standard power series solution method is one of the techniques for solving differential equations, in which higher-order terms of the series are weighted by the powers of a parameter whose magnitude is less than 1. The contribution of the higher-order terms is expected to be less than the sum representing the solution. The albedo of single scattering \(w_{0}\) defined in Equation (2) meets the requirement for an expansion parameter. One of the possible acceptable solutions of Equation (45) is given as follows:  
\[\mu\sum\limits_{i=0}^{N}{(i+1)*\ w_{0}^{i}\ \frac{dL_{(i)}}{dz}}=-\sum\limits_{i=1} ^{N}{(i+1)*\ w_{0}^{i}\ L_{L_{(i)}}(z,\ \diameter)+\ \left[\sum\limits_{i=0}^{N}{(i+1)*\ w_{0}^{i}\ L_{L_{(i)}}(z,\ \diameter)}\right]\int\limits_{-1}^{1}\int\limits_{0}^{2\pi} \stackrel{{\sim}}{{\beta}}(\ \{\dot{\mu},\diameter^{\prime}\} \rightarrow\{\mu,\diameter\})\ d\diameter^{\prime}\ d^{\prime} \tag{46}\]  
\[\left(L_{(i)}\right)\ \text{denotes radiance, where $i$ is the number of scattering events. This implies that }\ I=0\ \text{refers to uncatered radiance. This is consistent with the interpretation of }\ w_{0}\ \text{as the probability of ray survival in interaction with matter, i.e., the probability that an optical beam will be scattered and not absorbed. Equation (46) must hold for any \(0\ \leq\ w_{0}\ <1\) value. The solution of Equation (46) can be expressed as follows:  
\[\frac{dL_{(i)}}{c\ dz}=-L_{(i)}+\int\limits_{-1}^{1}\int\limits_{0}^{2\pi} \stackrel{{\sim}}{{\beta}}(\ \{\dot{\mu},\diameter^{\prime}\}\rightarrow\{\mu,\diameter\})L_{(i-1)}\ d\diameter^{ \prime}\ d^{\prime} \tag{47}\]  
We need to solve a series of equations to obtain the scattered radiance. First, we solve when \(i=0\) to obtain the uncatered radiance. Then, we use the 0 th solution for the 1 st to evaluate the path integral, which becomes the source function for singly scattered radiance. After obtaining the first solution for singly scattered radiance, we can use it to evaluate the path function in the second solution, and so on. This process is known as the successive-order-of-scattering (SOS) solution technique [5; 96]. Figure 7 shows the computation of Equation (47) by using the following reduced format:  
\[L(z,\ \theta)=L_{0}e^{-\tau(z)}+w_{s}\.\ L_{0}\int\limits_{0}^{z}\stackrel{{ \sim}}{{\beta}}_{HG}(\theta)\ S(z,\ z^{\prime})\.\ e^{-\tau(z^{\prime})}\ dz^{\prime} \tag{48}\]  
where \(L(z)\) is the intensity of radiation at depth \(z,\ L_{0}\) is the transmitted intensity, \(\tau(z)\) is the optical depth, \(w_{sUsing a constant value for the scattering kernel in the SSA may not accurately represent the scattering behaviour in the water medium, especially if the scattering properties vary spatially or spectrally. Here are some suggestions to improve the implementation of Equation (47) by incorporating a more realistic scattering kernel:  
1.