## 3 The Radiative Transfer Equation
### RTE Analytical Solutions with Simplification
#### 3.3.4 Small-Scattering-Angle Approximation  
The SSAA is the most familiar approximation assumption for obtaining an analytical RTE solution. This assumption concludes that \(sin\theta\ \cong\ tan\theta=1\) and \(cos\theta\ \cong\ 0\) at very small scattering angle \(\theta\). The main logic behind such an assumption is that the bulk of the scattering occurs at minimal scattering angles, expressed in Figure 5 (in terms of travelled photon steps) and Figure 6 (in terms of high SPF values); \(g\ \cong\ 1\) for forward scattering. This approximation is particularly useful when the scattering is predominantly forward-peaked, such as in Rayleigh or Mie scattering for small particles. The downside of this simplification is that the temporal information is lost, as scattered photons are considered to travel nearly the same distance (see Figure 5) in the same timeframe as those that have not been scattered. However, such an analytical approach still provides more information about the system than the core IOP, as it gives the radial distribution of photons at a given linkspan, also known as BSF.  
The BSF \(F(r,\ v)\) can be expressed as a Fourier-Bessel transform [99]:  
\[f(r,z)=\frac{1}{2\pi}\int\limits_{0}^{\infty}F_{\mathrm{s}}(v,z)\ F_{w}(v,z)\ J_{0(r\tau)}\ v\ dv \tag{50}\]  
Figure 8: Radiative flux component prediction according to the TSA method.  
where \(r\) is the radial distance, \(J_{0}\) is the zeroth-order Bessel function, and \(F(.)\) is an amplitude transfer function, where the subscripts s and w represent the source and water, respectively. The amplitude transfer function of water is given as follows [99]:  
\[F_{w}(v,z)=e^{[-cz+\int_{0}^{z}B(v\ \frac{r}{z})d\tau]} \tag{51}\]  
where \(B(v)\) is the Fourier-Bessel transform (FBT) of the VSF, \(\beta(\theta)\). The amplitude transfer function of a source is found as the FBT of the radial amplitude of light intensity, typically being Gaussian for laser diodes. The impacts of the underlying simplification are as follows: First, the photons' travel steps are practically equal, leading to the received photons having the same arrival time. Consequently, the received optical pulse will not have a detectable temporal profile. Second, the FOV is limited to less than \(10^{\circ}\). Instead of assuming that all photons arrive at the receiver simultaneously, as with the small-angle approximation, temporal spreading can be estimated statistically and included.  
In the context of radiative transfer, the small-angle approximation simplifies the integration of the SPF, leading to more tractable analytical solutions or numerical calculations. To implement the SSAA, as shown in Figure 9, we need to go through the following steps:  
1. Define the Scattering Phase Function: We must choose a scattering phase function appropriate for our modelling medium. The HG-SPF is commonly used for small-angle approximation due to its simplicity and applicability for forward-peaked scattering.
2. Apply the Small-Angle (i.e., LOS configuration) Approximation: We need to express the SPF as a linear function of the scattering angle \(\theta_{s}\) for small angles, as illustrated in Equation (49). Hence, the HG-SPF is \[\widetilde{\beta}_{HG}(\theta_{s})\ \approx 1+\theta_{s}\cdot g\] (52)
3. Integrate the Phase Function: With the SSAA, the integration of the SPF simplifies, making it easier to compute the radiative transfer quantities such as diffuse flux or reflectance.