## 3 Formulation of the Coloration Model
### Calculation of Observables Seen from the Observation Point by the Reverse Ray Tracing Method
#### 3.2.2 Calculation of the Observable Quantities from Radiance  
As shown in Figure 6, let L(**x**, \(\omega\)) be the radiance of the light flux, emitted from the light source at the point **x**, where the reflection or scatterings occurs, in the direction of the solid angle \(\omega\). Next, if this light flux is emitted from the pond surface at a certain point T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0) in the direction \(\omega^{\prime}\), the radiance L\({}^{\prime}\)(x\({}_{3}\), y\({}_{3}\), \(\omega^{\prime}\)) is given by Equation (3) [1]:  
\[\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{3},\omega^{\prime}\big{)} =\frac{\mathrm{L}(\textbf{x},\omega)\cdot\mathrm{D}(\varphi^{\prime},\varphi) }{\mathrm{n}_{12}^{2}}, \tag{3}\]  
where n\({}_{12}\) is the relative refractive index of water with respect to atmosphere, and D (\(\varphi^{\prime}\), \(\varphi\)) is the Fresnel reflection and transmission coefficient [16] when the light flux is emitted from the pond surface. Figure 5: Optical path diagrams in the case of (**a**) irregular reflection at the pond bottom and (**b**) scattering inside the pond. Let us consider the observable quantity given by the next Equation (4), which expresses a differential light flux \(\mathrm{d}^{2}\Psi\) with the radiance of \(\mathrm{L}^{\prime}(\mathrm{x}_{3},\,\mathrm{y}_{3},\,\omega^{\prime})\), an angle of \(\theta_{3}\) between the direction \(\mathbf{B_{2}}\) of the light flux and the vertical unit vector of \(\mathbf{k}\), a differential area of \(\mathrm{ds}\) at the light emission point on the surface, and a differential solid angle of \(\mathrm{d}\omega\) as shown in Figure 6:  
\[\mathrm{d}^{2}\Psi=\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\,\mathrm{y}_{3}, \,\omega^{\prime}\big{)}\mathrm{cos}\,\theta_{3}\mathrm{ds}\mathrm{d}\omega. \tag{4}\]  
This measurement problem was explained by [PERSON] [27]. Its conclusion is that the observable is obtained by multiplying the radiance by the scaling factor with the dimension of \([\mathrm{radiance}]^{-1}\), which is denoted by \(1/\mathrm{B}\) in this article. Now, as shown in Figure 7, let \(\mathbf{e}\) be the observation position, and \(\mathbf{s}\) be the position of the point on the object. Let us consider an observation from this point \(\mathbf{e}\) and look at the pixels on the window at position \(\mathbf{p}\). A solid angle \(\Omega_{\mathrm{P}}\) is taken to be a small solid angle including the point \(\mathbf{s}\) when viewed from the observation point \(\mathbf{e}\). Now suppose that \(\omega_{\mathrm{P}}\) is a solid angle that represents the direction in which the observation point \(\mathbf{e}\) is viewed from \(\mathbf{s}\). Let \(\omega_{\mathrm{p}}{}^{\prime}\) be a solid angle that expresses the direction in which the point \(\mathbf{p}\) on the pixel is viewed from \(\mathbf{s}\). \(S_{\mathrm{P}}\) is assumed to be the area of pixels on the window in the range of solid angle \(\Omega_{\mathrm{P}}\). Furthermore, we define \(\mathbf{y}=\mathrm{h}(\mathbf{x},\,-\omega)\) as a visibility function indicating that the point \(\mathbf{y}\) is in the direction of \(-\omega\) from the point \(\mathbf{x}\). Hence, the point \(\mathbf{s}\) on the object is expressed by \(\mathbf{s}=\mathrm{h}(\mathbf{e},\,-\omega_{\mathrm{P}})\) when viewed from the observation point \(\mathbf{e}\), and \(\mathbf{s}=\mathrm{h}(\mathbf{p},\,-\omega_{\mathrm{P}})\) when viewed from the point \(\mathbf{p}\) on the pixel. Likewise, let \(\mathrm{L}(\mathbf{s},\,\omega)\) be the radiance emitted from \(\mathbf{s}\) in the direction \(\omega\). The conclusion reached by [PERSON] [27] is that, from these definitions, the observable \(\mathrm{P}\) is expressed by:  
\[\mathrm{P}\cong\frac{1}{\mathrm{B}\cdot S_{\mathrm{P}}}\int_{S_{\mathrm{P}}} \mathrm{L}\big{(}\mathrm{h}\big{(}\mathbf{p},\,-\omega_{\mathrm{p}}{}^{ \prime}\big{)},\,\omega_{\mathrm{p}}{}^{\prime}\big{)}\mathrm{d}\mathbf{p}. \tag{5}\]  
Figure 6: Optical path of light flux above and below the pond surface.