## 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
Figure 7: Conceptual diagram of a point on the object, a point on the screen (pixel) on which they are viewed, and an observation position. What is important in Equation (5) is that the distance from a point on the object to the observation point or its orientation has nothing to do with the observable. This observation theory is now applied to the coloration of the pond surface. In the same way as was aforementioned, consider a light emission point T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0) on the pond surface. Let L\({}^{\prime}\)(x\({}_{3}\), y\({}_{3}\), \(\omega^{\prime}\)) be the radiance from here in the direction \(\omega^{\prime}\). Assuming that this point is observable, in Equation (3), \(\mathbf{s}=\mathrm{h}(\mathbf{p},-\omega_{\mathrm{p}}{}^{\prime})\) where \(\mathbf{s}=\mathrm{(x}_{3}\), y\({}_{3}\), 0), thus the observable quantity of light from this point is given by:  
\[ \begin{array}{rl}\mathrm{P}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{3} \big{)}&=\frac{1}{\mathrm{B}\cdot\mathrm{S}_{\mathrm{p}}}\int\limits_{ \mathrm{S}_{\mathrm{p}}}\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{ 3},\omega^{\prime}\big{)}\mathrm{d}\mathbf{p},\\ &=\frac{\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{3},\omega^{ \prime}\big{)}}{\mathrm{B}\cdot\mathrm{S}_{\mathrm{p}}}\int\limits_{ \mathrm{S}_{\mathrm{p}}}\mathrm{d}\mathbf{p},\\ &=\frac{\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{3},\omega^{ \prime}\big{)}}{\mathrm{B}\cdot\mathrm{S}_{\mathrm{p}}}\cdot\mathrm{S}_{ \mathrm{p}},\\ &=\frac{\mathrm{L}^{\prime}\big{(}\mathrm{x}_{3},\mathrm{y}_{3},\omega^{ \prime}\big{)}}{\mathrm{B}}.\end{array}  \tag{6}\]  
Therefore, we have shown that the observable is the radiance multiplied by the scaling factor of 1/B. Since the case of observation of the light flux from the emission point T\({}_{3}\)(x\({}_{3}\), y\({}_{3}\), 0) on the pond surface has been described so far, a method of extrapolating this to points other than the light emission point is described. When the reverse ray tracing method is applied, reflection or scattering points are scanned and contributions from the corresponding area segments at the pond bottom or volume segments inside the pond, respectively, are superimposed as shown by [PERSON] et al. [17]. Here, the area segments are used in the case of irregular reflection at the bottom, and volume segments are used in the case of scattering in it. The area segments of the pond bottom are taken to be parallel to the x and y axes, and the volume segments in the pond are rectangular parallelepipeds being parallel to the x, y, and z axes. These segments are obtained in grid patterns with \(\Delta\)x and \(\Delta\)y on the pond bottom or with \(\Delta\)x, \(\Delta\)y and \(\Delta\)z in the water body inside of the pond. Here, these increments \(\Delta\)x and \(\Delta\)y indicate the small lengths along x and y axes, respectively, when the area or volume segment is projected onto the horizontal plane. The parameter \(\Delta\)z expresses the vertical increment in volume segments and is introduced in the formulas for the light intensities of the scattered lights, as will be described later. Hence, the division of the pond bottom or the water body into grids can be associated with multiplication of the observables by \(\Delta\)x\(\Delta\)y and summing the quantities on each reflection or scattering point. In order to carry out the above summation, it is necessary to newly multiply the intensity formula by the factor with the dimension of [length]\({}^{-2}\) from the viewpoint of the dimension. On the other hand,  
The sum of the above formula is expressed by:  
\[\Delta x\Delta y\sum\limits_{x_{3},y_{3}}P^{\prime}\left(x_{3},y_{3}\right)=\frac{ \Delta x\Delta y}{B}\sum\limits_{x_{3},y_{3}}L^{\prime}\left(x_{3},y_{3}, \omega^{\prime}\right), \tag{9}\]  
where \(x_{3}\) and \(y_{3}\) are correlated to the area segment on the pond bottom or volume segment inside the pond, and these are taken at interval with \(\Delta x\) and \(\Delta y\), respectively.