# An Approach to Predict Debris Flow Average Velocity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{50}. In China, the modified Dongchuan empirical equation [29] is specially used for calculating the debris flow velocity in the Dongchuan area, where the Jiangjia gully is located, as well as Koch’s empirical velocity equation [30], both of which are based on the Manning equation [31]. Based on the pulled particle analysis (PPA), Huang [32] proposed an equation considering the research of the initial condition of solid accretion of debris flow to calculate the debris flow velocity. The pulled particle analysis approach is based on the theory of the solid–liquid two phase of debris flow, taking the solid particles of the moving debris flow as the analyzed object, establishing a limit equilibrium equation from the static to the motion state. Then, a new debris flow velocity approach is established. In such a framework, debris flow average velocity is obtained based on field monitoring through instruments or calculation equations. However, it can benefit from computer modeling. Debris flow is a complex and open system; the average velocity prediction approach implies that the use of meaningful variables gathered from past events is appropriate to predict the velocity. This study aims at building nonlinear relationships between the influencing variables and debris flow average velocity using data mining techniques. In the last few decades, artificial neural networks, typically radial basis function (RBF) neural network, have become efficient approaches to provide non-linear relations, especially in the multi component quantitative mixture from different types of data sets [33,34,35]. RBF neural network learning rule is simple and easy to be used on computer. It also has a strong robustness, memory and self-learning ability. Gravitational search algorithm (GSA) is one of common ways in optimizing the architecture, the centers and the width of RBF neural network. By combining the GSA with RBF neural network, a new hybrid algorithm (GSA-RBF) was proposed to predict debris flow average velocity. RBF neural network is sensitive to the initialization, and easy to fall into local optimum, but the new algorithm can avoid these disadvantages effectively. This study proposed a new approach, GSA-RBF neural network, to predict six gullies debris flow average velocity in the Wudongde Dam reservoir area. Xu et al. [36] and Wang et al. [37] have done precious works in this area. In particular, by training observation data of the Jiangjia gully, Xu et al. used BP neural network to establish an approach to predict three debris flow velocities in the Wudongde Dam reservoir area. They selected four variables to predict the velocity: flow depth, gradient of channel, debris flow density, and grain size. They compared the results with results computed by the Dongchuan equation and the modified Manning equation, finding that the BP approach is feasible and can predict the average velocity of a debris flow accurately. However, Poggio and Girosi [38] proved that RBF neural network was the best approximation for continuous functions, but BP was not. Yu [39] also proved that BP has poor prediction ability for testing data. The RBF has its disadvantages, i.e., it easily falls into the local optimal solution and leads the results to be not accurate. In this study, we selected the same variables as Xu et al. and tried to use the RBF neural network to train and predict the debris flow velocity. Meanwhile, GSA is used for optimizing the RBF neural network, which is necessary. In addition, Wang et al. evaluated the susceptibility of debris flows in the Wudongde Dam area using principle component analysis and self-organizing map methodologies. Twelve debris flow influencing factors are selected to evaluate the susceptibility of debris flows. The work of Wang et al. is referential, which will help to validate the results of our work.

## 2. Study Area

## 3. Data Acquisition

_{min})/(X

_{max}– X

_{min}) + 0.1

_{max}and X

_{min}are the maximum and minimum index values, respectively.

_{3}) data in this study. The influencing variables data are shown in Table 2. Figure 6 shows how the density is obtained in the field.

## 4. Methodology

#### 4.1. Radial Basis Function Neural Network

^{n}:

_{i}(x) is output of the ith hidden neuron. X is the n-dimension input vector. C

_{i}is the center vector of the ith neuron. α

_{i}is the basis width vector that can usually be determined experimentally.

_{i}and α

_{i}, it conducts the supervised learning. When the C

_{i}and α

_{i}are determined, RBF neural network becomes a linear function from input to output. The steps are as follows.

- Step 1. Initialize the weights randomly
- Step 2. Calculate the output vector Y by the equation:$${y}_{i}={\displaystyle \sum _{i=1}^{p}{W}_{i}{R}_{i}}$$
_{i}is the weight of the ith hidden neuron to the output node. - Step 3. Calculate the error ε
_{i}for each neuron in the output by the equation:$${\mathsf{\epsilon}}_{i}={y}_{i}-{y}_{i}^{\prime}\text{}i=1,2,\dots ,p$$ - Step 4. Based on the least squares method, determine the weights between the hidden neurons and the output nodes:$$W={e}^{(\frac{p}{{c}_{\mathrm{max}}^{2}}{\Vert X-{C}_{i}\Vert}^{2})}\text{}i=1,2,\dots ,p$$
_{max}is the maximum distance between the selected centers. - Step 5. Update the weights until the error meets the requirement:$${W}_{ij}^{\prime}={W}_{ij}+\mathsf{\mu}{\mathsf{\epsilon}}_{i}{R}_{j}\text{}i=1,2,\dots ,m,\text{}j=1,2,\dots ,p$$
_{ij}is the updated weight and μ is learning rate. When the network clustering center C_{i}and weight W_{i}are determined, we can conduct the predictions with the training model.

#### 4.2. The Gravitational Search Algorithm

_{i}

^{d}means the position of ith in dth space. In the process of the ith iteration, the force of individual j act on i is defined as

_{aj}(t) is active gravitational mass of individual j, M

_{pi}(t) is positive gravitational mass of individual i, $\mathsf{\epsilon}$ is a very small constant, and R

_{ij}(t) means the Euclidean distance between the individual i and individual j.

_{j}is a random number within the range of [0, 1], which increases the randomness of the algorithm.

_{ii}(t) means the inertial mass of the individual i. The position of individual i is updated by Equations (12) and (13).

_{i}

^{d}(t) is the speed of i in the d-dimension, x

_{i}

^{d}(t) is the position of i in the d-dimension, and rand

_{i}is a real number within [0, 1], which can enhance the ability of random search algorithm.

_{0}is the initial value of gravitational constant G, with the increase of iteration times, G will gradually decrease to control the accuracy of the search. G is a function of G

_{0}and t:

_{0}and α are constant, max iter is the maximum number of iterations.

_{i}(t) means the fitness value of i at t, and best (t) and worst (t) are defined, respectively, as Equations (18) and (19):

#### 4.3. The Proposed GSA-RBF Method

_{i}. The position of particles swarm is presented by matrix

**X**

_{n}× M. f(x) is the minimized objective function, the optimized objective function of ith individual is:

_{l}

_{m}(t) is expected output value, and y

_{l}

_{m}(t) is the actual output value.

#### 4.4. The Modified Dongchuan Empirical Equation

## 5. Results and Discussion

^{2}value is 0.836; (ii) Figure 9b shows the measured velocity versus the MDEE velocity, whose R

^{2}value is 0.942; and (iii) Figure 9c shows the measured velocity versus the GSA-RBF velocity, whose R

^{2}value is 0.968. RBF performs the worst among the three methods. The MDEE results and the GSA-RBF results are almost the same, which are similar to the measured velocity. However, the GSA-RBF has the strongest correlation with the measured velocity. Thus, the GSA-RBF algorithm produced the highest quality solution in terms of predicting debris flow average velocity on the testing data in the JJG. The GSA-RBF is able to find a near optimal solution while RBF easily trap into local optima. The GSA improves the quality of solutions found by the RBF neural network. In this study, we used 40 groups of data as training data, while Xu et al. used 45 groups of data as training data. In addition, we selected ten groups as testing data. Xu et al. selected five groups as testing data, which were very different from what we selected. The average relative errors using the GSA-RBF method (3.7%) and Xu and coworkers’ method (1.3%) were very close, and both were acceptable.

_{1}, x

_{2}, x

_{3}, and x

_{4}), but these correlations are weak (R

^{2}values range between 0.54 and 0.27). According to the previous work [26], we would expect a negative correlation between debris flow velocity and grain size. The positive correlation in this data set is almost certainly because of that these speed values do not depend only on grain size but on several variables at the same time (x

_{1}, x

_{2}, x

_{3}and x

_{4}). Furthermore, a single value of grain size cannot be representative of the large grain size distribution that occurs in a debris flow. The x

_{1}, x

_{2}, x

_{3}and x

_{4}values provide an oversimplified depiction of real debris flows. Thus, considering more variables affecting debris flow velocity, the GSA-RBF is more appropriate to predict debris flow average velocity.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Cao, C.; Xu, P.; Chen, J.; Zheng, L.; Niu, C. Hazard assessment of debris-flow along the baicha river in heshigten banner, inner mongolia, china. Int. J. Environ. Res. Public Health
**2016**, 14, 30. [Google Scholar] [CrossRef] [PubMed] - Tang, C.; Zhu, J.; Li, W.L.; Liang, J.T. Rainfall-Triggered debris flows following the wenchuan earthquake. B Eng. Geol. Environ.
**2009**, 68, 187–194. [Google Scholar] [CrossRef] - Cui, P.; Zhu, Y.-Y.; Han, Y.-S.; Chen, X.-Q.; Zhuang, J.-Q. The 12 may wenchuan earthquake-induced landslide lakes: Distribution and preliminary risk evaluation. Landslides
**2009**, 6, 209–223. [Google Scholar] [CrossRef] - Cui, P.; Chen, X.Q.; Zhu, Y.Y.; Su, F.H.; Wei, F.Q.; Han, Y.S.; Liu, H.J.; Zhuang, J.Q. The wenchuan earthquake (May 12, 2008), sichuan province, china, and resulting geohazards. Nat. Hazards
**2011**, 56, 19–36. [Google Scholar] [CrossRef] - Hu, K.H.; Wei, F.Q.; Li, Y. Real-Time measurement and preliminary analysis of debris-flow impact force at jiangjia ravine, china. Earth Surf. Proc. Land
**2011**, 36, 1268–1278. [Google Scholar] [CrossRef] - Li, Y.; Liu, J.J.; Hu, K.H.; Su, P.C. Probability distribution of measured debris-flow velocity in jiangjia gully, yunnan province, china. Nat. Hazards
**2012**, 60, 689–701. [Google Scholar] - Cascini, L.; Cuomo, S.; Pastor, M. Inception of debris avalanches: Remarks on geomechanical modelling. Landslides
**2013**, 10, 701–711. [Google Scholar] [CrossRef] - Bugnion, L.; McArdell, B.W.; Bartelt, P.; Wendeler, C. Measurements of hillslope debris flow impact pressure on obstacles. Landslides
**2012**, 9, 179–187. [Google Scholar] [CrossRef] - Scheidl, C.; Chiari, M.; Kaitna, R.; Mullegger, M.; Krawtschuk, A.; Zimmermann, T.; Proske, D. Analysing debris-flow impact models, based on a small scale modelling approach. Surv. Geophys.
**2013**, 34, 121–140. [Google Scholar] [CrossRef] - Johnson, C.G.; Kokelaar, B.P.; Iverson, R.M.; Logan, M.; LaHusen, R.G.; Gray, J.M.N.T. Grain-Size segregation and levee formation in geophysical mass flows. J. Geophys. Res.-Earth
**2012**, 117. [Google Scholar] [CrossRef] - Murakawa, Y.; Hara, M.; Oguchi, H.; Hamate, Y.; Kuwano, H. Surface acoustic wave based sensors employing ionic liquid for hydrogen sulfide gas detection. Microsyst. Technol.
**2013**, 19, 1255–1259. [Google Scholar] [CrossRef] - Berti, M.; Genevois, R.; LaHusen, R.; Simoni, A.; Tecca, P.R. Debris flow monitoring in the acquabona watershed on the dolomites (Italian alps). Phys. Chem. Earth Part B
**2000**, 25, 707–715. [Google Scholar] [CrossRef] - Hurlimann, M.; Rickenmann, D.; Graf, C. Field and monitoring data of debris-flow events in the swiss alps. Can. Geotech. J.
**2003**, 40, 161–175. [Google Scholar] [CrossRef] - Uddin, M.S.; Inaba, H.; Itakura, Y.; Yoshida, Y.; Kasahara, M. Adaptive computer-based spatial-filtering method for more accurate estimation of the surface velocity of debris flow. Appl. Opt.
**1999**, 38, 6714–6721. [Google Scholar] [CrossRef] [PubMed] - Han, Z.; Chen, G.; Li, Y.; Xu, L.; Zheng, L.; Zhang, Y. A new approach for analyzing the velocity distribution of debris flows at typical cross-sections. Nat. Hazards
**2014**, 74, 2053–2070. [Google Scholar] [CrossRef] - Huang, Y.; Zhang, W.J.; Xu, Q.; Xie, P.; Hao, L. Run-Out analysis of flow-like landslides triggered by the ms 8.0 2008 wenchuan earthquake using smoothed particle hydrodynamics. Landslides
**2012**, 9, 275–283. [Google Scholar] [CrossRef] - Cascini, L.; Cuomo, S.; Pastor, M.; Sorbino, G.; Piciullo, L. Sph run-out modelling of channelised landslides of the flow type. Geomorphology
**2014**, 214, 502–513. [Google Scholar] [CrossRef] - Armanini, A.; Capart, H.; Fraccarollo, L.; Larcher, M. Rheological stratification in experimental free-surface flows of granular-liquid mixtures. J. Fluid Mech.
**2005**, 532, 269–319. [Google Scholar] [CrossRef] - Hotta, N. Basal interstitial water pressure in laboratory debris flows over a rigid bed in an open channel. Natl. Hazards Earth Syst. Sci.
**2012**, 12, 2499–2505. [Google Scholar] [CrossRef] [Green Version] - Iverson, R.M.; Logan, M.; LaHusen, R.G.; Berti, M. The perfect debris flow? Aggregated results from 28 large-scale experiments. J. Geophys. Res.-Earth
**2010**, 115. [Google Scholar] [CrossRef] - Stancanelli, L.M.; Lanzoni, S.; Foti, E. Mutual interference of two debris flow deposits delivered in a downstream river reach. J. Mt. Sci.-Engl.
**2014**, 11, 1385–1395. [Google Scholar] [CrossRef] - Stancanelli, L.M.; Lanzoni, S.; Foti, E. Propagation and deposition of stony debris flows at channel confluences. Water Resour. Res.
**2015**, 51, 5100–5116. [Google Scholar] [CrossRef] - Takahashi, T. Debris Flow: Mechanics, Prediction and Countermeasures; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Egashira, S.; MIYAMOTO, K.; Itoh, T. Constitutive equations of debris flow and their applicability. In Proceedings of the First International Conference Water Resources Engineering Division/ASCE, San Francisco, CA, USA, 2–6 September 2001; pp. 340–349.
- Yang, H.J.; Wei, F.Q.; Hu, K.H. Mean velocity estimation of viscous debris flows. J. Earth Sci.-China
**2014**, 25, 771–778. [Google Scholar] [CrossRef] - Prochaska, A.B.; Santi, P.M.; Higgins, J.D. Relationships between size and velocity for particles within debris flows. Can. Geotech. J.
**2008**, 45, 1778–1783. [Google Scholar] [CrossRef] - Cagnoli, B.; Romano, G.P. Granular pressure at the base of dry flows of angular rock fragments as a function of grain size and flow volume: A relationship from laboratory experiments. J. Geophys. Res. Solid Earth
**2012**, 117. [Google Scholar] [CrossRef] - Julien, P.Y.; Paris, A. Mean velocity of mudflows and debris flows. J. Hydraul. Eng.-Asce
**2010**, 136, 676–679. [Google Scholar] [CrossRef] - Chen, G. Prevention and Control of Debris Flow; China Railway Press: Beijing, China, 1983. [Google Scholar]
- Koch, T. Testing various constitutive equations for debris flow modelling. IAHS Publ.-Ser. Proc. Rep.-Intern Assoc. Hydrol. Sci.
**1998**, 248, 249–258. [Google Scholar] - Rickenmann, D. Empirical relationships for debris flows. Nat. Hazards
**1999**, 19, 47–77. [Google Scholar] [CrossRef] - Huang, R. Model Building and Calculation of the Debris Flow Rate in Dam Site of Wudongde Hydropower Station Based on Ppa. Ph.D. Dessertation, Jilin University, Changchun, China, 2011. [Google Scholar]
- Riverol-Canizares, C.; Pilipovik, V. The use of radial basis function networks (rbfn) to predict critical water parameters in desalination plants. Expert Syst. Appl.
**2010**, 37, 7285–7287. [Google Scholar] [CrossRef] - Khodaveisi, J.; Dadfarnia, S.; Shabani, A.M.H.; Moghadam, M.R.; Hormozi-Nezhad, M.R. Artificial neural network assisted kinetic spectrophotometric technique for simultaneous determination of paracetamol and p-aminophenol in pharmaceutical samples using localized surface plasmon resonance band of silver nanoparticles. Spectrochim. Acta A
**2015**, 138, 474–480. [Google Scholar] [CrossRef] [PubMed] - Rasouli, Z.; Hassanzadeh, Z.; Ghavami, R. Application of a new version of ga-rbf neural network for simultaneous spectrophotometric determination of zn(ii), fe(ii), co(ii) and cu(ii) in real samples: An exploratory study of their complexation abilities toward mtb. Talanta
**2016**, 160, 86–98. [Google Scholar] [CrossRef] [PubMed] - Xu, L.; Wang, Q.; Chen, J. Forcast for average velocity of debris flow based on bp neural network. J. Jilin Univ. (Earth Sci. Ed.)
**2013**, 43, 186–191. [Google Scholar] - Wang, Q.; Kong, Y.Y.; Zhang, W.; Chen, J.P.; Xu, P.H.; Li, H.Z.; Xue, Y.G.; Yuan, X.Q.; Zhan, J.W.; Zhu, Y.J. Regional debris flow susceptibility analysis based on principal component analysis and self-organizing map: A case study in southwest china. Arab. J. Geosci.
**2016**, 9, 718. [Google Scholar] [CrossRef] - Poggio, T.; Girosi, F. Networks for approximation and learning. Proc. IEEE
**1990**, 78, 1481–1497. [Google Scholar] [CrossRef] - Yu, G.; Zhang, M.; Wang, G. Application and comparison of prediction models of support vector machines and back-propagation artificial neural network for debris flow average velocity. J. Hydraul. Eng.
**2012**, 43, 105–110. [Google Scholar] - Zhang, W.; Li, H.-Z.; Chen, J.-P.; Zhang, C.; Xu, L.-M.; Sang, W.-F. Comprehensive hazard assessment and protection of debris flows along jinsha river close to the wudongde dam site in china. Nat. Hazards
**2011**, 58, 459–477. [Google Scholar] [CrossRef] - Niu, C.; Wang, Q.; Chen, J.; Zhang, W.; Xu, L.; Wang, K. Hazard assessment of debris flows in the reservoir region of wudongde hydropower station in china. Sustainability
**2015**, 7, 15099–15118. [Google Scholar] [CrossRef] - Dal Sasso, S.F.; Sole, A.; Pascale, S.; Sdao, F.; Pinzon, A.B.; Medina, V. Assessment methodology for the prediction of landslide dam hazard. Nat. Hazards Earth Syst. Sci.
**2014**, 14, 557–567. [Google Scholar] [CrossRef] [Green Version] - Dang, C.; Cui, P.; Cheng, Z.L. The formation and failure of debris flow-dams, background, key factors and model tests: Case studies from china. Environ. Geol.
**2009**, 57, 1901–1910. [Google Scholar] [CrossRef] - Ermini, L.; Casagli, N.; Farina, P. Landslide dams: Analysis of case histories and new perspectives from the application of remote sensing monitoring techniques to hazard and risk assessment. Ital. J. Eng. Geol. Environ.
**2006**, 1, 45–52. [Google Scholar] - Chen, J.; He, Y.; Wei, F. Debris flow erosion and deposition in jiangjia gully, yunnan, china. Environ. Geol.
**2005**, 48, 771–777. [Google Scholar] [CrossRef] - Cui, P.; Chen, X.; Waqng, Y.; Hu, K.; Li, Y. Jiangjia ravine debris flows in south-western china. In Debris-Flow Hazards and Related Phenomena; Springer: New York, NY, USA, 2005; pp. 565–594. [Google Scholar]
- Tiranti, D.; Bonetto, S.; Mandrone, G. Quantitative basin characterisation to refine debris-flow triggering criteria and processes: An example from the italian western alps. Landslides
**2008**, 5, 45–57. [Google Scholar] [CrossRef] - Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. Gsa: A gravitational search algorithm. Inf. Sci.
**2009**, 179, 2232–2248. [Google Scholar] [CrossRef] - Xu, Y.N. Study on Flow Mechanism for Avalanche Soils and Scour and Silting Characteristics of Debris Flows. Ph.D. Dessertation, Institute of Water Conservancy and Hydroelectric Power Research, Beijing, China, 2001. [Google Scholar]
- Liu, H.J.; Tang, C.; Cui, P. Gis-Based criticality zoning of debris flow in dongchuan district. Arid Land Geogr.
**2005**, 28, 445–449. [Google Scholar] - Specification of Geological Investigation for Debris Flow Stabilization dz/t 0220-2006; China Standard Press: Beijing, China, 2006.
- Gregoretti, C.; Fontana, G.D. The triggering of debris flow due to channel-bed failure in some alpine headwater basins of the dolomites: Analyses of critical runoff. Hydrol. Process
**2008**, 22, 2248–2263. [Google Scholar] [CrossRef] - Wolman, M.G. A method of sampling coarse river-bed material. EOS Trans. Am. Geophys. Union
**1954**, 35, 951–956. [Google Scholar] [CrossRef]

**Figure 4.**Field survey: (

**a**) debris flow scouring in the downstream of flowing area in Zhugongdi; (

**b**) debris flow mud marks in Zhugongdi; (

**c**) rock crevices filled with mud in Xiabaitan; and (

**d**) debris flow scouring in the downstream of flowing area in Zhiligou.

**Figure 9.**Fitness of 10 testing data in the JJG using three methods: (

**a**) measured velocity versus the RBF velocity; (

**b**) measured velocity versus the MDEE velocity; and (

**c**) measured velocity versus the GSA-RBF velocity.

y | x_{1} | x_{2} | x_{3} | x_{4} | y | x_{1} | x_{2} | x_{3} | x_{4} |
---|---|---|---|---|---|---|---|---|---|

8.8 | 150 | 6.3 | 2200 | 1.1 | 3.7 | 40 | 6.3 | 2020 | 0.1 |

7.8 | 140 | 6.3 | 1950 | 0.6 | 4.1 | 70 | 5.8 | 1800 | 0.2 |

3.8 | 40 | 6.3 | 1850 | 0.1 | 3.5 | 50 | 5.8 | 1760 | 0.2 |

6.9 | 202 | 5.5 | 2270 | 1.7 | 8.2 | 130 | 6.6 | 2200 | 0.7 |

7.5 | 168 | 5.5 | 2280 | 1.6 | 4.8 | 93 | 5.8 | 1920 | 0.3 |

8.9 | 175 | 6.3 | 2080 | 0.8 | 9.2 | 372 | 6.6 | 2210 | 1.2 |

7.4 | 200 | 6.3 | 2210 | 1.7 | 9.6 | 220 | 6.6 | 2290 | 1.5 |

7.3 | 90 | 6.3 | 2210 | 1 | 5.8 | 107 | 5.5 | 2290 | 1.2 |

6.6 | 70 | 6.3 | 2190 | 1.2 | 3.9 | 55 | 5.8 | 2070 | 0.8 |

9.4 | 210 | 6.6 | 2210 | 1.2 | 5.6 | 70 | 5.5 | 1920 | 0.3 |

4 | 40 | 6.3 | 2040 | 0.3 | 3.9 | 60 | 5.5 | 1830 | 0.1 |

7.4 | 145 | 5.5 | 2250 | 1.1 | 6.9 | 122 | 5.5 | 2210 | 1 |

5.8 | 103 | 5.5 | 2210 | 0.8 | 9.6 | 275 | 6.6 | 2210 | 1.6 |

4.7 | 60 | 5.5 | 1970 | 0.5 | 5 | 65 | 5.5 | 2240 | 1.1 |

7.7 | 161 | 5.5 | 2250 | 1 | 3.7 | 55 | 5.8 | 1800 | 0.1 |

7.7 | 177 | 5.5 | 2240 | 1.1 | 8.1 | 160 | 6.6 | 2220 | 1.2 |

7.9 | 200 | 6.3 | 2250 | 1.4 | 6.6 | 226 | 5.5 | 2130 | 1.1 |

8.4 | 210 | 6.6 | 2200 | 0.8 | 7.4 | 55 | 6.3 | 2250 | 0.9 |

9.3 | 210 | 6.3 | 2290 | 1 | 7.5 | 170 | 6.6 | 2190 | 1.1 |

3.6 | 58 | 5.8 | 1690 | 0.2 | 6.4 | 109 | 5.5 | 2250 | 1.1 |

10 | 95 | 6.3 | 2160 | 0.6 | 9.3 | 210 | 6.3 | 2210 | 1.1 |

7.6 | 125 | 6.3 | 2100 | 0.6 | 6.9 | 250 | 5.5 | 2220 | 0.9 |

7.6 | 11 | 6.3 | 2070 | 0.7 | 6 | 120 | 5.5 | 2200 | 0.8 |

7.6 | 100 | 6.3 | 2190 | 0.9 | 4.9 | 60 | 5.5 | 1990 | 0.6 |

8.5 | 200 | 6.3 | 2300 | 1.5 | 3.6 | 52 | 5.8 | 1700 | 0.1 |

_{1}is flow depth (cm); x

_{2}is the gradient of channel (%); x

_{3}is the density of debris flow (kg·m

^{−3}); and x

_{4}is the average grain size (cm).

Gully | x_{1} (cm) | x_{2} (%) | x_{3} (kg·m^{−3}) | x_{4} (cm) |
---|---|---|---|---|

Xiabaitan | 200 | 40.7 | 2250 | 3.23 |

Shangbaitan | 150 | 35.8 | 2110 | 3.08 |

Zhugongdi | 180 | 41.8 | 2040 | 2.97 |

Zhuzhahe | 180 | 5.0 | 2120 | 2.15 |

Zhiligou | 170 | 10.2 | 2320 | 3.23 |

Mengguogou | 180 | 5.6 | 2100 | 3.06 |

Measured Value (m/s) | RBF | MDEE | GSA-RBF | |||
---|---|---|---|---|---|---|

Value (m/s) | Relative Error (%) | Value (m/s) | Relative Error (%) | Value (m/s) | Relative Error (%) | |

4.8 | 6.1 | 27.1 | 5.0 | 3.6 | 5.0 | 4.2 |

4.9 | 5.3 | 8.2 | 4.7 | 3.6 | 4.8 | 2.0 |

4.7 | 5.3 | 12.8 | 4.7 | 0.5 | 4.7 | 0.0 |

7.7 | 7.9 | 2.6 | 7.2 | 6.3 | 7.1 | 7.8 |

7.7 | 8.1 | 5.2 | 7.4 | 3.3 | 7.2 | 6.5 |

3.9 | 5.0 | 28.2 | 4.2 | 7.0 | 3.6 | 7.7 |

3.9 | 4.9 | 25.6 | 4.2 | 9.0 | 4.1 | 5.1 |

6.4 | 6.2 | 3.1 | 5.8 | 10.0 | 6.4 | 0.0 |

3.7 | 3.8 | 2.7 | 3.7 | 0.2 | 3.8 | 2.7 |

7.6 | 9.9 | 30.3 | 6.8 | 10.3 | 7.7 | 1.3 |

Average error | - | 14.6 | - | 5.4 | - | 3.7 |

Maximum error | - | 30.3 | - | 10.3 | - | 7.8 |

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Cao, C.; Song, S.; Chen, J.; Zheng, L.; Kong, Y.
An Approach to Predict Debris Flow Average Velocity. *Water* **2017**, *9*, 205.
https://doi.org/10.3390/w9030205

**AMA Style**

Cao C, Song S, Chen J, Zheng L, Kong Y.
An Approach to Predict Debris Flow Average Velocity. *Water*. 2017; 9(3):205.
https://doi.org/10.3390/w9030205

**Chicago/Turabian Style**

Cao, Chen, Shengyuan Song, Jianping Chen, Lianjing Zheng, and Yuanyuan Kong.
2017. "An Approach to Predict Debris Flow Average Velocity" *Water* 9, no. 3: 205.
https://doi.org/10.3390/w9030205