## 3 Formulation of the Coloration Model
### Irregular Reflection  
#### 3.3.3 Derivation of the Formula for Radiance  
Consider an irregular reflection from an area segment located at the pond bottom point T\({}_{1}\)(x\({}_{1}\), y\({}_{1}\), z\({}_{1}\)) where the unit normal vector is \(\mathbf{n_{1}}\). Now, we discuss the light intensity under the wavelength of \(\lambda_{i}\). Let L\({}_{1}\) be the distance from the pond surface incident point T\({}_{2}\)(x\({}_{2}\), y\({}_{2}\), 0) to the pond bottom reflection point T\({}_{1}\)(x\({}_{1}\), y\({}_{1}\), z\({}_{1}\)), and L\({}_{2}\) be the distance from the pond bottom reflection point to the pond surface emission point T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0). We assume that the absorption coefficient of water is A(\(\lambda_{i}\)), density fluctuation scattering coefficient of water b\({}_{\mathrm{w}}\)(\(\lambda_{i}\)), a white the Mie scattering coefficient of suspended solids b\({}_{\mathrm{Mie}}\), phenomenological attenuation coefficient k\({}_{1}\), the area segment \(\Delta\)S with sides along the x and y axes at the pond bottom, and \(\Delta\)x\(\Delta\)y is the normal projection of \(\Delta\)S onto the horizontal plane. For the pond bottom area segment \(\Delta\)S, as shown in Figure 10, if the angle between the pond bottom unit normal vector \(\mathbf{n_{1}}\) and the vertical unit vector \(\mathbf{k}\) is \(\theta_{2}\), then the normal projection of \(\Delta\)S onto the horizontal plane becomes the increment \(\Delta\)x\(\Delta\)y. Hence, \(\Delta\)S is expressed as \(\Delta\)x\(\Delta\)y divided by cos\(\theta_{2}\), factor of the slope effect of the bottom. That is:  
\[\cos\theta_{2}=\mathbf{n_{1}}\mathbf{\cdot k}=\mathbf{n_{1z}}, \tag{24}\]  
\[\Delta\mathrm{S}=\frac{\Delta\mathrm{x}\Delta\mathrm{y}}{\cos\theta_{2}}= \frac{\Delta\mathrm{x}\Delta\mathrm{y}}{\mathbf{n_{1z}}}. \tag{25}\]  
Now, let P\({}_{\mathrm{Irr}}\)\({}^{\prime}\)(x, y, \(\lambda_{i}\)) (dimension: [wavelength]\({}^{-1}\)) be the observable quantity of irregularly reflected light from the pond bottom. If the light intensity P\({}_{\mathrm{Irr}}\)(\(\theta\), \(\phi\), x\({}_{1}\), y\({}_{1}\), z\({}_{1}\), \(\lambda_{i}\)) (dimension: [wavelength]\({}^{-1}\)) corresponding to the irregular reflection point is expressed as follows, by taking the Fresnel reflection and transmission coefficient at the pond surface emission point as D(\(\varphi\)\({}^{\prime}\),\(\varphi\)) [16]:  
\[\mathrm{P_{Irr}}(\theta,\phi,x_{1},y_{1},z_{1},\lambda_{i})=\frac{\mathrm{L_{ Irr}}\cdot\mathrm{D}(\varphi\text{${}^{\prime}$},\varphi)}{\mathrm{B}\cdot \mathrm{n_{12}^{2}}}\cdot\mathrm{exp}[-\mathrm{L_{2}}\{\mathrm{A}(\lambda_{i})+ \mathrm{b_{w}}(\lambda_{i})+\mathrm{b_{Mie}}+\mathrm{k_{1}}\}], \tag{26}\]  
where the radiance L\({}_{\mathrm{Irr}}\) (dimension: [light flux] [length]\({}^{-2}\) [solid angle]\({}^{-1}\) [wavelength]\({}^{-1}\)) in this equation is calculated next. Let f(\(\lambda_{i}\)) be the spectral solar irradiance and D(\(\theta\), \(\theta\)\({}^{\prime}\)) be the Fresnel reflection and transmission coefficient with the solar radiation incidence into the pond surface. Using the solar irradiance f(\(\lambda_{i}\)) and a color matching function \(\overline{y}\)(\(\lambda_{i}\)), the illuminance in fine weather E\({}_{\mathrm{fine}}\) can be obtained by Equation (27):  
\[\mathrm{E_{fine}}=\mathrm{K_{m}}\sum_{\lambda_{i}}\mathrm{f}(\lambda_{i})\cdot \overline{y}(\lambda_{i})\Delta\lambda. \tag{27}\]  
Assuming that the measured illuminance at the time of taking an image of the pond surface is E\({}_{\mathrm{obs}}\), the effect of brightness of solar radiation can be incorporated with multiplication of the theoretical quantity by the factor E\({}_{\mathrm{obs}}\)/E\({}_{\mathrm{fine}}\).