# A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Unit Stream Power Theory of Yang for Sediment Transport in Natural Rivers

_{t}is the total sediment concentration (ppm), with wash load excluded; ${V}_{*}$ is the shear velocity (m/s); ν is the water kinematic viscosity (m

^{2}/s); ω is the fall velocity (m/s); and d

_{50}is the median particle diameter (m).

_{F}is the calculated total sediment concentration (ppm); and V

_{cr}S is the critical unit stream power, derived as the product of mean critical flow velocity and energy slope.

## 3. “Fuzzy Twin”—The Physical Meaning

_{50}, and not by its actual diameter, as well as the disregard of turbulence. Going further, the uncertainty raises by the subjectivity in the estimation of the incipient motion criterion [54] and the turbulence impact on sediment transport. To better identify the source of uncertainty in Yang’s formula, the assumption of one-dimensional, uniform and steady flow (especially, in the case of natural rivers), as well as the regression analysis between sediment transport rate and stream discharge, which partly neglects the physical mechanisms of the sediment transport phenomenon, must be considered, as well. The uncertainties would significantly be reduced in the case of an analytical physically based model. The complex nature of sediment transport and the associated uncertainties have been very well documented in literature [54,55,56,57,58,59]. In terms of uncertainty, Kleinhans (2005) [57] compares the notoriety of the sediment transport problem with that of the roughness problem and he stresses the necessity of calibration. In such cases, fuzzy regression, contrarily to conventional solutions such as classic regression, offers an efficient and applicable solution, by producing a fuzzy band within which the measured values are most likely included. Indeed, Azamathulla et al. [60] state that classic regression does not efficiently cope with the uncertainties that dominate both input and output data and instead they use a Fuzzy Inference System (FIS) as a prediction model. Hence, “fuzzy” is justified by the fact that the computed sediment concentration is not a crisp value, as it would be if the classic formula of Yang Equation (4) had been used, but a range of values which is expected to contain the observed data.

## 4. Materials and Methods

#### 4.1. Experimental Data for the Derivation of Yang’s Formula

_{F}, as the dependent variable and the log(ω·d

_{50}/ν), log(${V}_{*}$/ω), log(VS/ω−V

_{cr}S/ω), log(ω·d

_{50}/ν)·log(VS/ω−V

_{cr}S/ω), log(${V}_{*}$/ω)·log(VS/ω−V

_{cr}S/ω), as the independent variables, and applying a multiple regression analysis for 463 sets of data in laboratory flumes.

#### 4.1.1. Nomicos’ Data (1956)

#### 4.1.2. Vanoni and Brooks’ Data (1957)

#### 4.1.3. Kennedy’s Data (1961)

#### 4.1.4. Stein’s Data (1965)

#### 4.1.5. Guy, Simons and Richardson’s Data (1966)

#### 4.1.6. Williams’ Data (1967)

#### 4.1.7. Schneider’s Data (1971)

_{50}, is the median sieve diameter of the sediment, while Guy et al. [65] published their data in terms of fall diameter. According to Yang [22], the difference between these two measurements of particle size is insignificant when either one is smaller than 0.4 mm. The fall diameter was converted, by Yang, into sieve diameter by means of Figure 7 of Report 12 of the Inter-Agency Committee on Water Resources (1957) [69]. The numbers shown in parentheses, in Table 1, refer to the fall diameters for the coarse sand.

#### 4.2. Fuzzy Regression

_{A}(x) the membership function of the fuzzy number A; and R is the set of real numbers.

- The model is as follows:$${\tilde{\mathsf{{\rm Y}}}}_{j}={\tilde{A}}_{0}+{\tilde{A}}_{1}{x}_{1j}+{\tilde{A}}_{2}{x}_{2j}+\dots +{\tilde{A}}_{n}{x}_{nj}$$
_{i}, c_{i}) are symmetric fuzzy triangular numbers selected as coefficients; and x is the independent variable (Figure 1). In addition, n is the number of independent variables; m is the number of data; a is the central value (where μ = 1); and c is the semi-width. - Determination of the degree h at which the data [(x
_{1j}, x_{2j}, …, x_{nj}), y_{j}] is aimed to be included in the estimated number Y_{j}:$${\mu}_{Yj}({y}_{j})\ge h,\text{\hspace{1em}}j=1,\dots ,m$$The constraints express the concept of inclusion in case that the output data are crisp numbers. In the examined case of the widely used model of Tanaka [47], a more soft definition of the fuzzy subsethood is used compared to the Zadeh [42] definition. Hence, the inclusion of a fuzzy set A into the fuzzy set B with the associated degree $0\le h\le 1$ is defined as follows:$${\left[A\right]}_{h}\subseteq {\left[B\right]}_{h}$$In our case, since the data are crisp (for each individual data), the set A is only a crisp value (a point of data which must be included in the produced fuzzy band) and the fuzzy set B is a fuzzy triangular number. Hence, Equation (11) is equivalent to:$$\sum _{i=0}^{n}{a}_{i}{x}_{ij}-(1-h){\displaystyle \sum _{i=0}^{n}{c}_{i}\left|{x}_{ij}\right|}}\le {y}_{j}\le {\displaystyle \sum _{i=0}^{n}{a}_{i}{x}_{ij}+(1-h){\displaystyle \sum _{i=0}^{n}{c}_{i}\left|{x}_{ij}\right|}},\text{\hspace{1em}}j=1,\dots ,m$$It must be clarified that the above equations hold for a specified h-cut and not for every α-cut. Normally, the 0-strongcut is used since greater levels of h lead to a greater uncertainty. - Determination of the minimization function (objective function) J. In the conventional fuzzy linear regression model, the objective function, J, is the sum of the produced fuzzy semi-widths for the data:$$J=\left\{m{c}_{0}+{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{i=1}^{n}{c}_{i}\left|{x}_{ij}\right|}}\right\}$$
_{0}is the semi-width of the constant term; and c_{i}semi-width of the other fuzzy coefficients.Since fuzzy symmetric triangular numbers are selected as fuzzy coefficients, it can be proved that the objective function is the sum of the semi-widths of the produced fuzzy band regarding the available data:$$J=\left\{m{c}_{0}+{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{i=1}^{n}{c}_{i}\left|{x}_{ij}\right|}}\right\}=\frac{1}{2}{\displaystyle \sum _{j=1}^{m}\left({Y}_{j}{}^{+}-{Y}_{j}{}^{-}\right)}$$ - The problem results in the following linear programming problem:$$\begin{array}{l}\text{\hspace{1em}\hspace{1em}}min\left\{m{c}_{0}+{\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{i=1}^{n}\stackrel{}{{c}_{i}}\left|{x}_{ij}\right|}}\right\}\\ {\displaystyle \sum _{i=0}^{n}\stackrel{}{{a}_{i}}{x}_{ij}-}\left(1-h\right){\displaystyle \sum _{i=0}^{n}\stackrel{}{{c}_{i}}\left|{x}_{ij}\right|={y}_{h}^{L}\le {y}_{j}}\\ {\displaystyle \sum _{i=0}^{n}\stackrel{}{{a}_{i}{x}_{ij}}}+\left(1-h\right){\displaystyle \sum _{i=0}^{n}{\stackrel{}{c}}_{i}}|{x}_{ij}|={y}_{h}^{R}\ge {y}_{j}\end{array}\text{\hspace{1em}\hspace{1em}\hspace{1em}}\}$$
_{i}≥ 0, for i = 0, 1, …, n.In addition, many times, when data are classic numbers, we can easily approximate non-linear cases with the fuzzy linear regression model with the help of auxiliary variables. In this case, the total uncertainty (cumulative width) indicates incomplete complexity, whereas non-physical behavior is an indicator of overtraining [77], due to adoption of excessive complexity in non-linear models.

#### 4.3. Implementation

_{1}, X

_{2}, X

_{3}, X

_{4}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:

^{2}/s); D* is the Bonnefille number; and d

_{ch}the characteristic grain diameter (m). The Bonnefille number, D*, is given by:

^{3}) and ${\rho}_{W}$ is the density of water (kg/m

^{3}).

^{2}); 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.

_{1}, X

_{2}, X

_{3}, X

_{4}and X

_{5}Equation (21) are introduced into the fuzzified version of the Yang’s equation Equation (4), then Equation (22) results in:

_{F}, which is produced as fuzzy symmetric triangular number, as well. By introducing the above auxiliary variables X

_{1}to X

_{5}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.

## 5. Results

#### 5.1. Determination of Yang’s Formula Independent Variables

_{cr}/ω.

_{cr}/ω, was calculated, for each set of data, by means of Equations (5) and (6). The ranges of values for all calculated parameters and variables, for each dataset, are provided in Table 2.

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}are the dimensionless variables log(ω·d

_{50}/ν), log(${V}_{*}$/ω), log(VS/ω−V

_{cr}S/ω), log(ω·d

_{50}/ν)·log(VS/ω−V

_{cr}S/ω), log(${V}_{*}$/ω)·log(VS/ω−V

_{cr}S/ω), respectively, C

_{t}is the total measured sediment concentration obtained from the experimental data, and C

_{F}is the total calculated sediment concentration, as obtained from the unit stream power formula, using Yang’s coefficients and by replacing the independent variables with the calculated values of X

_{1}, X

_{2}, X

_{3}, X

_{4}, X

_{5}.

_{t}and logC

_{F}, Nash-Sutcliffe Efficiencies (NSEs) of 0.79 and 0.72 were achieved with Zanke’s and Rubey’s formulas, respectively. Hence, in Table 2, only the results obtained by the use of Zanke’s formula, for fall velocity, are presented.

_{t}and logC

_{F}, and the NSE value of 0.79, it can be said that the approximation between the measured and calculated results, as well as the quality of data is deemed satisfactory.

#### 5.2. Multiple Regression Analyses

#### 5.2.1. Multiple Regression Analysis for the Reconstruction of the Unit Stream Power Formula

^{2}, is equal to:

#### 5.2.2. Fuzzy Regression Analysis

_{3}—is separately investigated (Figure 3). In case that the crisp linear regression is used with the X

_{3}as the only independent variable, the results are similar, and the squared correlation coefficient, r

^{2}, is equal to 0.801. Hence, the linear dependence can be suggested. In this case, Equation (26) becomes:

_{3}as the only independent variable) increases the uncertainty. However, the usefulness is that the emphasis is put on the subtraction of the critical unit stream power from the exerted unit stream power, as a main independent variable. Going in the opposite direction, if only the variable X

_{3}is removed, then the uncertainty of the produced fuzzy band is greater than the above value (J = 396.75).

_{3}as input variable, are depicted. As it can be observed from the figure, the data of the fuzzy fourth-degree polynomial regression and the data of the fuzzy linear regression almost overlap for the most part. However, the fourth-degree polynomial regression presents an “irrational behavior” in the area of low X

_{3}values, from a physical meaning point of view. To better explain this, as the difference “exerted unit stream power minus critical unit stream power” (here, represented by X

_{3}) grows larger, a higher sediment transport, and therefore a higher sediment concentration is expected. Simply put, logC

_{F}and X

_{3}are similar amounts and the increase of one by the decrease of the other is not justified. The negligible reduction of the uncertainty, as well as the “irrational behavior” of the fuzzy fourth-degree polynomial regression, indicates the improperness of the polynomial models for these data. From the above it is concluded that the auxiliary variable X

_{3}is the most significant parameter parameter. The use of high polynomial extension to Equation (28) did not improve the results. Equation (26) results in significantly less uncertainty and should be preferred.

#### 5.2.3. Validation

_{50}) were obtained from granulometric curves, which were constructed upon sieve analysis data, and are in a range between 0.38 mm and 0.88 mm. Along with the sediment data, basic hydraulic parameters, such as flow velocity, flow depth, energy slope and water temperature were available in the same survey. These data were used for the computation of the independent variables in Equations (24) and (26). The independent variables in any of the Equations (4), (24) or (26) represent the geometric and flow characteristics of the stream that they are applied for. A total of 55 sets of data were used for the validation of Equations (24) and (26).

_{1}, is activated if, and only if, the observed data are not included within the produced fuzzy band. This measure was initially proposed by Ishibuchi et al. [87] as a cost function to be minimized in the learning process, regarding a neural network with interval weights.

_{1}= E

_{2}= 0. By applying the validation measures to the data for natural streams, the following values are achieved: E

_{1}= 0.1868, E

_{2}= 2. This means that only two points do not belong to the fuzzy band (E

_{2}= 2), but these points are not far from the produced fuzzy band, as suggested by the low value of the E

_{1}measure.

## 6. Conclusions

_{3}). Nevertheless, a simplification based on the X

_{3}(i.e., only the variable X

_{3}is taken into account) leads to a fuzzy linear curve that can be used to interpret the phenomenon. The produced fuzzy band compared with the central values indicates the good performance of the proposed fuzzy curve. In terms of elaboration of the original data utilized by Yang for the establishment of the unit stream power theory, this research goes the closest possible to what could be called “fuzzy twin” of Yang’s stream sediment transport formula.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

^{2}, and the Nash-Sutcliffe Efficiency, NSE, should be clarified. In case of a multiple regression, R

^{2}and NSE are simply identical. However, the coefficient of determination has a value bounded between 0 and 1 [90], that is ${R}^{2}\in \left[0,\text{\hspace{0.17em}}1\right]$ [82].

^{2}, is equal to $R{\text{\hspace{0.17em}}}^{2}\text{\hspace{0.17em}}=NSE$. The correlation coefficient, r, indicates the strength and the direction of a linear relationship with respect to the data, whilst it cannot imply causation.

^{2}, and obviously non-negative, i.e., $NSE=R{\text{\hspace{0.17em}}}^{2}\text{\hspace{0.17em}}=0.857$. However, when the crisp multiple regression equation is used upon the other validation measurements, from natural streams, then it takes negative values, i.e., $NSE=-1.207$.

_{3}), again for the training data, then the NSE is identical to the R

^{2}and with the squared value of the correlation coefficient, r, $NSE=R{\text{\hspace{0.17em}}}^{2}\text{\hspace{0.17em}}={r}^{2}=0.801$.

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**Figure 2.**Projection of the achieved fuzzy relation regarding the log of the total sediment concentration with respect to (

**a**) X

_{3}, (

**b**) X

_{2}, (

**c**) X

_{4}.

**Figure 3.**Fuzzy linear regression to achieve the log of the total sediment concentration with respect to X

_{3}.

Particle Size, d_{50} (mm) | Channel Length, L (ft) | Channel Width, W (ft) | Water Depth, h (ft) | Temperature, T (°C) | Average Velocity, V (ft/s) | Water Surface Slope, S×10^{3} | Total Sediment Concentration, C_{t} (ppm) | Number of Data, N |
---|---|---|---|---|---|---|---|---|

Nomicos’ data (1956) [61] | ||||||||

0.152 | 40 | 0.875 | 0.241 | 25.0–26.0 | 0.80–2.63 | 2.0–3.9 | 300–5767 | 12 |

Vanoni and Brooks’ data (1957) [62] | ||||||||

0.137 | 60 | 2.79 | 0.147–0.346 | 18.9–27.4 | 0.77–2.53 | 0.7–2.8 | 37–3000 | 14 |

Kennedy’s data (1961) [63] | ||||||||

0.233 | 40 | 0.875 | 0.147–0.346 | 24.5–30.1 | 1.57–3.42 | 2.6–16.0 | 730–34,700 | 14 |

0.549 | 40 | 0.875 | 0.074–0.346 | 24.3–27.0 | 1.65–4.65 | 5.5–27.2 | 1680–35,900 | 14 |

0.233 | 60 | 2.79 | 0.145–0.356 | 23.0–27.3 | 1.35–3.45 | 1.7–22.9 | 490–58,500 | 13 |

Stein’s data (1965) [64] | ||||||||

0.4 | 100 | 4.0 | 0.59–1.20 | 20.0–28.9 | 1.38–5.51 | 0.61–10.79 | 93–24,249 | 42 |

Guy et al. data (1966) [65] | ||||||||

0.19 | 150 | 8.0 | 0.49–1.09 | 12.3–19.7 | 1.04–4.74 | 0.43–9.50 | 29–47,300 | 29 |

0.27 | 150 | 8.0 | 0.45–1.13 | 10.2–18.5 | 1.24–4.93 | 0.46–10.22 | 12–35,800 | 18 |

0.28 | 150 | 8.0 | 0.30–1.07 | 10.2–17.6 | 1.04–4.93 | 0.45–10.07 | 12–42,400 | 33 |

0.48(0.45) | 150 | 8.0 | 0.19–1.00 | 9.0–20.0 | 0.75–6.18 | 0.39–10.10 | 10–15,100 | 34 |

1.2(0.93) | 150 | 8.0 | 0.38–1.11 | 16.7–21.7 | 1.46–6.07 | 0.37–12.80 | 15–10,200 | 32 |

0.32 | 60 | 2.0 | 0.54–0.74 | 7.0–34.3 | 1.24–5.73 | 0.86–16.20 | 55–49,300 | 29 |

0.33 (uniform) | 60 | 2.0 | 0.49–0.52 | 19.8–20.3 | 1.17–6.93 | 0.88–11.40 | 47–18,400 | 12 |

0.33 (graded) | 60 | 2.0 | 0.48–0.53 | 19.6–24.1 | 1.06–6.34 | 0.47–9.80 | 12–22,500 | 14 |

0.50(0.47) | 150 | 8.0 | 0.30–1.33 | 10.7–24.5 | 4.69–17.45 | 0.42–9.60 | 23–17,700 | 50 |

0.59(0.54) | 60 | 2.0 | 0.59–0.89 | 16.9–25.1 | 1.37–6.27 | 0.38–19.28 | 17–50,000 | 35 |

Williams’ data (1967) [66] | ||||||||

1.35 | 52 | 1.0 | 0.094–0.517 | 11.9–30.8 | 1.27–3.49 | 1.10–22.18 | 10–9223 | 37 |

Schneider’s data (1971) [67] | ||||||||

0.25 | – | 8.0 | 1.012–2.822 | 20.4–22.4 | 1.67–6.45 | 0.10–4.97 | 18–17,152 | 31 |

Kinematic Viscosity × 10^{−6}, ν (m^{2}/s) | Fall Velocity, ω (m/s), Zanke (1977) | Shear Velocity, V_{*}(m/s) | Dimensionless Critical Velocity, V_{cr}/ω | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | Total Measured Sediment Concentration, C _{t} (ppm) | logC_{t} | Total Calculated Sediment Concentration, C_{F} (ppm) | logC_{F} | Number of Data, N |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Nomicos’ data (1956) [61] | |||||||||||||

0.88 | 0.02 | 0.04–0.05 | 3.46–4.0 | 0.53–0.54 | 0.27–0.43 | −1.79–(−0.84) | −0.97–(−0.44) | −0.52–(−0.36) | 300–5767 | 2.48–3.76 | 303–7387 | 2.48–3.87 | 12 |

Vanoni and Brooks’ data (1957) [62] | |||||||||||||

0.85–1.04 | 0.015–0.017 | 0.03–0.05 | 3.87–4.77 | 0.29–0.45 | 0.29–0.46 | −1.98–(−0.98) | −0.74–(−0.28) | −0.71–(−0.42) | 37–3000 | 1.57–3.48 | 132–4419 | 2.12–3.65 | 14 |

Kennedy’s data (1961) [63] | |||||||||||||

0.8–0.91 | 0.04–0.04 | 0.04–0.09 | 2.56–3.26 | − | 0.01–0.36 | −1.53–(−0.43) | −1.56–(−0.44) | −0.22–(−0.01) | 730–34,700 | 2.86–4.54 | 1070–27,619 | 3.03–4.44 | 14 |

0.86–0.91 | 0.08–0.09 | 0.05–0.1 | 2.09–2.42 | 1.72–1.75 | −0.24–0.06 | −1.73–(−0.42) | −2.97–(−0.73) | −0.02–0.42 | 1680–35,900 | 3.23–4.56 | 1075–29,057 | 3.03–4.46 | 14 |

0.85–0.94 | 0.036–0.038 | 0.04–0.12 | 2.4–3.34 | 0.96–1.02 | 0.02–0.52 | −1.7–(−0.25) | −1.7–(−0.24) | −0.22–(−0.02) | 490–58,500 | 2.69–4.77 | 619–40,833 | 2.79–4.61 | 13 |

Stein’s data (1965) [64] | |||||||||||||

0.83–1.01 | 0.065–0.069 | 0.04–0.14 | 2.12–2.75 | 1.41–1.52 | −0.19–0.32 | −2.66–(−0.61) | −3.85–(−0.88) | −0.25–0.51 | 93–24,249 | 1.97–4.39 | 55–15,860 | 1.74–4.2 | 42 |

Guy et al. data (1966) [65] | |||||||||||||

1.02–1.23 | 0.01–0.02 | 0.04–0.14 | 2.62–4.94 | 0.09–0.55 | 0.24–0.81 | −2.0–(−0.26) | −1.1–(−0.10) | −0.9–(−0.21) | 29–47,300 | 1.46–4.68 | 122–38,227 | 2.09–4.58 | 29 |

1.05–1.3 | 0.02–0.04 | 0.04–0.14 | 2.67–3.82 | 0.51–0.85 | 0.1–0.79 | −2.38–(−0.22) | −1.79–(−0.12) | −0.66–(−0.17) | 12–35,800 | 1.08–4.55 | 50–42,848 | 1.7–4.63 | 18 |

1.07–1.3 | 0.1–0.04 | 0.04–0.13 | 2.65–4.22 | 0.13–0.99 | −0.06–0.83 | −2.51–(−0.24) | −2.49–(−0.14) | −0.88–0.14 | 12–42,400 | 1.08–4.63 | 43–41,340 | 1.64–4.62 | 33 |

1.01–1.35 | 0.01–0.08 | 0.03–0.11 | 2.26–4.66 | 0.19–1.56 | −0.34–0.92 | −3.72–(−0.37) | −4.92–(−0.22) | −1.04–1.25 | 10–15,100 | 1.0–4.18 | 1–28,845 | 0.15–4.46 | 34 |

0.97–1.1 | 0.05–0.12 | 0.03–0.13 | 2.05–2.54 | 1.5–2.16 | −0.59–0.37 | −3.13–(−0.63) | −6.7–(−1.19) | −0.45–1.84 | 15–10,200 | 1.18–4.01 | 28–15,613 | 1.45–4.19 | 32 |

0.74–1.43 | 0.02–0.06 | 0.04–0.18 | 2.29–3.86 | 0.57–1.46 | −0.13–0.73 | −2.33–(−0.17) | −3.33–(−0.15) | −0.59–0.3 | 55–49,300 | 1.74–4.69 | 105–45,297 | 2.02–4.66 | 29 |

1.0–1.02 | 0.02–0.05 | 0.04–0.13 | 2.34–4.24 | 0.71–1.27 | 0.05–0.5 | −2.05–(−0.27) | −2.11–(−0.3) | −0.63–(−0.09) | 47–18,400 | 1.67–4.27 | 176–37,507 | 2.25–4.57 | 12 |

0.92–1.02 | 0.01–0.05 | 0.03–0.12 | 2.82–5.95 | 0.09–1.19 | −0.2–0.91 | −2.46–0.07 | −2.94–0.03 | −0.82–0.49 | 12–22,500 | 1.08–4.35 | 64–93,556 | 1.81–4.97 | 14 |

0.91–1.28 | 0.03–0.08 | 0.03–0.11 | 2.15–3.54 | 0.82–1.74 | −0.36–0.39 | −1.99–(−0.21) | −2.77–(−0.3) | −0.39–0.647 | 23–17,700 | 1.362–4.25 | 314–48,987 | 2.5–4.69 | 50 |

0.9–1.09 | 0.06–0.09 | 0.03–0.2 | 2.05–3.17 | 1.35–1.79 | −0.45–0.44 | −3.02–(−0.37) | −4.66–(−0.55) | −0.23–1.37 | 17–50,000 | 1.23–4.7 | 17–26,180 | 1.24–4.42 | 35 |

Williams’ data (1967) [66] | |||||||||||||

0.79–1.24 | 0.15–0.16 | 0.03–0.1 | 2.05–2.25 | 2.22–2.43 | −0.7–(−0.2) | −3.33–(−1.07) | −7.75–(−2.53) | 0.23–2.31 | 10–9223 | 1.0–3.97 | 35–8038 | 1.55–3.91 | 37 |

Schneider’s data (1971) [67] | |||||||||||||

– | – | – | – | – | – | – | – | – | 18–17,152 | – | – | – | 31 |

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**MDPI and ACS Style**

Kaffas, K.; Saridakis, M.; Spiliotis, M.; Hrissanthou, V.; Righetti, M. A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang. *Water* **2020**, *12*, 257.
https://doi.org/10.3390/w12010257

**AMA Style**

Kaffas K, Saridakis M, Spiliotis M, Hrissanthou V, Righetti M. A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang. *Water*. 2020; 12(1):257.
https://doi.org/10.3390/w12010257

**Chicago/Turabian Style**

Kaffas, Konstantinos, Matthaios Saridakis, Mike Spiliotis, Vlassios Hrissanthou, and Maurizio Righetti. 2020. "A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang" *Water* 12, no. 1: 257.
https://doi.org/10.3390/w12010257