# 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

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**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.
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**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