## 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
When the segments are projected onto the pond surface, they give a spread of \(\Delta x\Delta y\) at T\({}_{3}\)(x\({}_{3}\), \(y_{3}\), 0). Now, let us consider a simple example, as shown in Figure 8. What we examine next is that there should be no multiplication factor other than scaling factor of 1/B in the formulas for the observables. It is assumed that the incident light enters a rectangular parallelepiped of a virtual pond from the zenith. Let \(S_{\text{H}}{}^{\prime}\) be the bottom area of the rectangular parallelepiped. An irregular reflection point at the rectangle pond bottom or a scattering point at a rectangular parallelepiped in the pond with a thickness of one layer is assumed, and the pond bottom or scattering layer is gridded by \(\Delta x\) and \(\Delta y\). Suppose that the light observed from the irregular reflection point or scattering point (\(x_{\text{m}}\), \(y_{\text{m}}\), \(z_{\text{m}}\)) is generated. The formula which holds for the grid is as follows:  
\[S_{\text{H}}^{\prime}=\sum\limits_{x_{\text{m}},y_{\text{m}}}\Delta x\Delta y. \tag{10}\]  
For the sake of simplicity, it is assumed that the irregularly reflected light or scattered light is observed from a sufficiently far point above the pond surface, and that these irregular reflection or scattering points provide a radiance L\({}_{\text{H}}\) of the uniform intensity from the area of \(S_{\text{H}}{}^{\prime}\). If the light intensity \(P^{\prime}(x_{3},y_{3})\) ejected from the light emission point is distributed by a uniform 2D probability density of \(p_{\text{H}}\) over the pond surface with the area of \(S_{\text{H}}\), then the light intensity distribution with the dimension of [length]\({}^{-2}\) is expressed as follows:  
\[P_{\text{H}}=\frac{1}{S_{\text{H}}}. \tag{11}\]  
Applying the idea of Equation (7),  
\[ \begin{array}{ll}P^{\prime\prime}\left(x,y\right)&=\sum\limits_{x_{\text{m} },y_{\text{m}}}\Delta x\Delta y\frac{L_{\text{H}}\cdot P_{\text{H}}}{B},\\ &=\frac{L_{\text{H}}}{B\cdot S_{\text{H}}}\sum\limits_{x_{\text{m}},y_{\text{ m}}}\Delta x\Delta y,\\ &=\frac{L_{\text{H}}\cdot S_{\text{H}}}{B\cdot S_{\text{H}}}.\end{array}  \tag{12}\]  
In the rectangular parallelepiped we discuss now, the incident light enters from the zenith and the resulted lights are observed from directly above, so \(S_{\text{H}}=S_{\text{H}}{}^{\prime}\). Therefore, the following equation holds:  
\[P^{\prime\prime}\left(x,y\right)=\frac{L_{\text{H}}}{B}. \tag{13}\]  
Figure 8: Simple model for light emission. This means that the observable which is obtained by extrapolation from the emission points on the pond surface can be calculated from the radiance with simple multiplication by the scaling factor of 1/B. In the above simple case, we have taken a uniform probability density of \(\rm p_{H}=1/S_{H}\), and have obtained the formula for the extrapolated observable. This consideration shows that we can express the extrapolated light intensity observables from the radiances belonging to the grids using the probability distribution function with the dimension of \(\rm[length]^{-2}\). Thus, the density function for the 2D normal distribution will be one of the candidates owing to its dimension and normalization possibility. Consequently, it is considered that the observable at any point on the pond surface can be calculated by extrapolation by Equation (7).