## 3 Formulation of the Coloration Model
### Density Fluctuation Scattering  
Let \(\mathrm{L_{3}}\) be the distance from the pond surface incident point \(\mathrm{T_{2}(x_{2},y_{2},0)}\) to the scattering point \(\mathrm{T_{5}(x_{n},y_{n},z_{n})}\) inside the pond, and \(\mathrm{L_{4}}\) be the distance from the scattering point to the pond surface emission point \(\mathrm{T_{3}(x_{3},y_{3},0)}\). The volume segment of the scattering point is parallel to the horizontal plane and the vertical direction, and the volume of this rectangular parallelepiped having sides of \(\Delta x\), \(\Delta y\) and \(\Delta z\) along the \(x\), \(y\), and \(z\) axis directions, respectively, is \(\Delta x\Delta y\Delta z\). In the same argument as was in pond bottom irregular reflection, the light intensity with the dimension of [wavelength]\({}^{-1}\) is expressed for each scattering point as follows:  
\[\mathrm{P_{ES}(\theta,\phi,x_{n},y_{n},z_{n},\lambda_{i})=\frac{\mathrm{L_{ES }}\cdot D(\varphi^{\prime},\varphi)}{\mathrm{B}\cdot\mathrm{n_{12}^{2}}}\cdot \exp[-\mathrm{L_{4}}\{\mathrm{A}(\lambda_{i})+\mathrm{b_{w}}(\lambda_{i})+ \mathrm{b_{Mie}}+\mathrm{k_{1}}\}]. \tag{32}\]  
Figure 11: Relationship between unit incident light vector \(\mathbf{A}\) and unit normal vector \(\mathbf{n_{1}}\). In the case of water density fluctuation scattering, if the phase function of scattering is \(\widetilde{\beta_{\rm w}}(\Theta_{\rm S})\) and \(\Theta_{\rm S}\) is the scattering angle, then the volume scattering function \(\beta_{\rm w}(\lambda_{\rm i},\,\Theta_{\rm S})\) is expressed by:  
\[\beta_{\rm w}(\lambda_{\rm i},\Theta_{\rm S})={\rm b}_{\rm w}(\lambda_{\rm i}) \widetilde{\beta_{\rm w}}(\Theta_{\rm S}), \tag{33}\]  
where the phase function is given by [PERSON] et al. [29] as follows:  
\[\widetilde{\beta_{\rm w}}(\Theta_{\rm S})=\frac{3}{8\pi}\frac{1+\delta_{1}+(1 -\delta_{1})\cos^{2}\Theta_{\rm S}}{2+\delta_{1}}\,\left({\rm sr}^{-1}\right), \tag{34}\]  
where \(\delta_{1}\) is a depolarization factor. As shown in Figure 12, the scattering angle \(\Theta_{\rm S}\) is given by the following equation including the unit direction vector \(\mathbf{A^{\prime\prime}}\) of the light flux before scattering and the unit direction vector \(\mathbf{B_{1}}\) of the light flux after scattering:  
\[\cos\Theta_{\rm S}=\mathbf{A^{\prime\prime}}\cdot\mathbf{B_{1}}. \tag{35}\]  
Incident light on the volume segment at the scattering point is received in the horizontal plane of the segment. Its unit normal vector is equal to the vertical unit vector. The effect caused by incident angle of light can be considered by analogy to the case where the light intensity of irregular reflection light is related to the incident angle as shown in Figure 11. We replace the unit normal vector \(\mathbf{n_{1}}\) shown in Figure 11 with the vertical unit vector \(\mathbf{k}\), and we take \(\mathbf{A}=(\rm A_{x},\,A_{y},\,A_{z})\) as:  
\[\cos\theta_{\rm 4}=\mathbf{k}\cdot\mathbf{A}=\rm A_{z}, \tag{36}\]  
where  
\[\rm A_{z}=\sqrt{1-\frac{1}{n_{12}^{2}}\cos^{2}\theta}. \tag{37}\]  
From the above considerations, the radiance \(\rm L_{ES}\) emitted from the scattering point is written as follows:  
\[\rm L_{ES}={\rm b}_{\rm w}(\lambda_{\rm i})\widetilde{\beta_{\rm w}}(\Theta_{ \rm S})\Delta z\cdot f(\lambda_{\rm i})D\big{(}\theta,\theta^{\prime}\big{)} A_{z}\cdot\exp[-L_{3}\{A(\lambda_{\rm i})+{\rm b}_{\rm w}(\lambda_{\rm i})+{ \rm b}_{\rm Mie}+{\rm k}_{1}\}]. \tag{38}\]  
Therefore, the light intensity \(\rm P^{\prime}_{ES}(x,\,y,\,\lambda_{\rm i})\) of water density fluctuation scattered light is written by Equation (39). In the same way as the pond bottom irregular reflection, the products of the intensity given by Equation (39) are taken with the color matching functions, integrated along the wavelength, and thus the tristimulus values are obtained, and these are linearly transformed to give the linear RGB values.