*4.3. Implementation*

For the modulation of the auxiliary variables *X*1, *X*2, *X*3, *X*<sup>4</sup> and *X*5, several parameters had to be calculated. The sedimentation rate in the unit stream power equation of Yang Equation (4) was determined by means of Zanke's [78] formula:

$$
\omega = 11 \cdot \nu \cdot \left(\sqrt{1 + 0.01D^3} - 1\right) / d\_{\rm ch} \tag{16}
$$

where ν is the water kinematic viscosity (m2/s); *D\** is the Bonnefille number; and *dch* the characteristic grain diameter (m). The Bonnefille number, *D\**, is given by:

$$D^\* = \left(\rho' \cdot \lg / \nu^2\right)^{1/3} \cdot d\_{\rm ch} \tag{17}$$

$$
\rho' = (\rho\_F - \rho\_W) / \rho\_W \tag{18}
$$

In the above relation, ρ*<sup>F</sup>* is the density of sediment (kg/m3) and ρ*<sup>W</sup>* is the density of water (kg/m3). The kinematic viscosity, ν, of water is given by the equation:

$$\nu = 1.78 \cdot 10^{-6} / (1 + 0.0337 \cdot T + 0.00022 \cdot T^2) \tag{19}$$

where *T* ( ◦C) is the temperature of the water.

The shear velocity, *V*∗, was determined by means of the following formula:

$$V\_\* = \sqrt{\text{ghS}}\tag{20}$$

where *g* is the gravitational acceleration (m/s2); *h* (m) is the flow depth; and *S* is the energy slope (m/m). In Equation (20), the hydraulic radius is replaced approximately by the flow depth. In the case of uniform flow, the energy slope equals the bed slope.

When the auxiliary variables *X*1, *X*2, *X*3, *X*<sup>4</sup> and *X*<sup>5</sup> Equation (21) are introduced into the fuzzified version of the Yang's equation Equation (4), then Equation (22) results in:

$$\begin{cases} X1 = \log(\omega \cdot d\mathbf{5}0/\nu) \\ X2 = \log(V\_\*/\omega) \\ X3 = \log(V \cdot S/\omega - Vcr \cdot S/\omega) \\ X4 = \log(V \cdot S/\omega - Vcr \cdot S/\omega) \cdot \log(\omega \cdot D\mathbf{5}0/\nu) \\ X5 = \log(V \cdot S/\omega - Vcr \cdot S/\omega) \cdot \log(V\_\*/\omega) \end{cases} \tag{21}$$

$$\log \overline{C}\_{\overline{\mathbf{F}}, \mathbf{j}} = \overline{A}\_0 + \overline{A}\_1 \cdot X\_{1, \mathbf{j}} + \overline{A}\_2 \cdot X\_{2, \mathbf{j}} + \overline{A}\_3 \cdot X\_{3, \mathbf{j}} + \overline{A}\_4 \cdot X\_{4, \mathbf{j}} + \overline{A}\_5 \cdot X\_{5, \mathbf{j}} \tag{22}$$

The dependent variable is the logarithm of the concentration of the total load, *CF*, which is produced as fuzzy symmetric triangular number, as well. By introducing the above auxiliary variables *X*<sup>1</sup> to *X*<sup>5</sup> for numeric data, the problem of non-linear fuzzy regression is reduced to a linear fuzzy regression problem. In the fuzzy linear regression model, the coefficients of the independent variables are fuzzy numbers that were determined using the Matlab program.

Furthermore, a simplified version of the Yang's Equation (4), which contains only the exerted unit stream power minus the critical unit stream power, is investigated:

$$
\log \overline{C}\_{F,j} = \overline{A}\_0 + \overline{B} \cdot X\_{\mathfrak{Z},j} \tag{23}
$$

A criterion for the successfulness of this simplification will be the produced uncertainty. More analytically, if the uncertainty is increased significantly, this will indicate an irrational simplification (undertraining behavior).
