A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang
Abstract
:1. Introduction
2. Unit Stream Power Theory of Yang for Sediment Transport in Natural Rivers
3. “Fuzzy Twin”—The Physical Meaning
4. Materials and Methods
4.1. Experimental Data for the Derivation of Yang’s Formula
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)
4.2. Fuzzy Regression
- The model is as follows:
- Determination of the degree h at which the data [(x1j, x2j, …, xnj), yj] is aimed to be included in the estimated number Yj: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 is defined as follows: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: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: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:
- The problem results in the following linear programming problem: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
5. Results
5.1. Determination of Yang’s Formula Independent Variables
5.2. Multiple Regression Analyses
5.2.1. Multiple Regression Analysis for the Reconstruction of the Unit Stream Power Formula
5.2.2. Fuzzy Regression Analysis
5.2.3. Validation
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
References
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Particle Size, d50 (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×103 | Total Sediment Concentration, Ct (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 (m2/s) | Fall Velocity, ω (m/s), Zanke (1977) | Shear Velocity, V*(m/s) | Dimensionless Critical Velocity, Vcr/ω | X1 | X2 | X3 | X4 | X5 | Total Measured Sediment Concentration, Ct (ppm) | logCt | Total Calculated Sediment Concentration, CF (ppm) | logCF | 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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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
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 StyleKaffas, 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
APA StyleKaffas, K., Saridakis, M., Spiliotis, M., Hrissanthou, V., & Righetti, M. (2020). A Fuzzy Transformation of the Classic Stream Sediment Transport Formula of Yang. Water, 12(1), 257. https://doi.org/10.3390/w12010257