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Accurate simulations of river stages during typhoon events are critically important for flood control and are necessary for disaster prevention and water resources management in Taiwan. This study applies two artificial neural network (ANN) models, including the back propagation neural network (BPNN) and genetic algorithm neural network (GANN) techniques, to improve predictions from a one-dimensional flood routing hydrodynamic model regarding the water stages during typhoon events in the Danshuei River system in northern Taiwan. The hydrodynamic model is driven by freshwater discharges at the upstream boundary conditions and by the water levels at the downstream boundary condition. The model provides a sound physical basis for simulating water stages along the river. The simulated results of the hydrodynamic model show that the model cannot reproduce the water stages at different stations during typhoon events for the model calibration and verification phases. The BPNN and GANN models can improve the simulated water stages compared with the performance of the hydrodynamic model. The GANN model satisfactorily predicts water stages during the training and verification phases and exhibits the lowest values of mean absolute error, root-mean-square error and peak error compared with the simulated results at different stations using the hydrodynamic model and the BPNN model. Comparison of the simulated results shows that the GANN model can be successfully applied to predict the water stages of the Danshuei River system during typhoon events.

Accurate predictions of water stages during high flow periods are critically important for water resources management and flood control operations. Water stage forecasting in a river with tidal effects is among the most important outstanding problems of flood management. It is never an easy task, because to develop a hydrodynamic model, the behavior of the physical processes must be known. Flow conditions in a river with tidal effects are rarely steady or uniform. Hydrodynamic models provide a physical basis for modeling and have the capability to simulate a wide range of flow conditions. Several researchers have developed flood routing hydrodynamic models based on either the one-dimensional dynamic wave equation or the diffusive wave equation. With these approaches, researchers consider the floodplain section of the one-dimensional river channel. Examples of these formulations are given by [

Because of the existing difficulties and challenges in the prediction of water stages using the flood routing hydrodynamic model, a relatively novel computational approach, artificial neural networks (ANNs), which has found wide acceptance in many disciplines, provides an alternative method for one-step-ahead understanding and management of hydrological processes. ANNs are well-suited for this application, because of their informative processing characteristics, such as nonlinearity, parallelism, noise tolerance and learning and generalization capabilities [

In this study, a one-dimensional flood routing hydrodynamic model was used to simulate the water stages in the Danshuei River system in northern Taiwan during typhoon events. Two artificial neural network models were subsequently adopted to improve the calculations of the flood routing hydrodynamic model. Three quantitative statistical measures,

The Danshuei River is located in northern Taiwan (^{2}, and the mean annual precipitation is 3001 mm. The total channel length is 327.6 km, and the channel slope ranges from 0.015 to 0.0027. The peak discharges of a 200-year flood are 235.00 m^{3}/s, 103.00 m^{3}/s and 2700 m^{3}/s for the Dahan Stream, the Xindian River and the Keelung River, respectively. The annual mean freshwater discharges at the upstream tidal limits of the Dahan River, the Xindian River and the Keelung River are 62.1 m^{3}/s, 72.7 m^{3}/s and 26.1 m^{3}/s, respectively. The mean tide at the Danshuei River mouth is 2.21 m above the mean sea level. The downstream reaches of all three tributaries are affected by tides [

Layout of the Danshuei River system in northern Taiwan. The transect line represents the cross-section.

The flood routing hydrodynamic model is based on the dynamic wave theory of the Saint-Venant equations, which consist of the one-dimensional continuity and momentum equations:
_{1} is the lateral inflow per unit channel length; _{0} is the channel bottom slope; _{f}_{1} is the longitudinal velocity component of the lateral inflow;

The continuity and momentum equations can be solved numerically, given the initial and boundary conditions. There are three conventional numerical approaches, including finite difference, finite element and finite volume methods used to solve one-dimensional continuity and momentum equations. Of the various implicit schemes that have been developed, the “four-point” schemes appear most advantageous, since they can readily be used with unequal distance intervals. Therefore, a four-point implicit finite-difference approximation scheme was used in this study [

In this study, the model transects were established consistent with the measured cross-sectional profiles at intervals of approximately 0.5 km along the river. The theoretical model transects include 310 transects that cover 11 river reaches (shown in

Cross-sectional number of river reaches and Manning friction factor,

River Reach Number | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Number of cross-section | 71 | 8 | 3 | 13 | 9 | 22 |

Manning friction |
0.025 | 0.033~0.039 | 0.033~0.040 | 0.035~0.045 | 0.033~0.039 | 0.030~0.035 |

Number of cross-section | 22 | 10 | 2 | 137 | 13 | |

Manning friction |
0.022~0.027 | 0.022~0.030 | 0.025 | 0.019~0.090 | 0.023~0.028 |

Note: the unit of manning friction n is ^{1∕3} /

The Danshuei River system layout for the flood routing hydrodynamic model simulation and boundary conditions.

In the present study, two ANN models, including BPNN and GANN, were introduced. The algorithms for these two ANN models are described below.

A back propagation neural network (BPNN) was used to amend the simulated water stage results with the one-dimensional flood routing hydrodynamic model to achieve more accurate predictions. The BPNN proposed by Rumelhart _{i}_{j}_{ij}

In a recent study, Zadeh

A linear transfer function in Equation (5) is applied in the output layer.

To scale the inputs and the targets, the normalized equation, Equation (6), is often used, forcing the data to fall within a specified range.

_{N}_{min} and _{max} are the minimum and maximum values of the data, respectively; and _{min} and _{max} are −1 and 1, respectively.

The process of training a neural network involves tuning the values of the weights and biases of the network to optimize network performance. Training of the ANN includes minimizing the cost function, _{k}

The back-propagation method was adopted in the ANN training, utilizing the Levenberg–Marquardt algorithm [

ANN training methods provide a non-linear mapping between inputs and outputs and are extremely useful in recognizing patterns in complex data. Methods, such as back-propagation (BP), have been improved with the genetic algorithm technique (GA). The training starts with GA, which executes a global search on the net weight range, refining an initial random set of weights to yield a better value, most likely closer to the global optimum. The BP algorithm then progresses the training, refining the solution provided by GA to approach the optimal solution.

The development of GA was inspired by the basic concepts of Darwinian evolution. It is a heuristic method for solving computationally difficult problems [

In the present study, the fitness values of all these chromosomes were evaluated using the fitness function. Some of the chromosomes were selected by elitism. The probabilistic-selection criterion was applied for selecting chromosomes for the crossover and mutation operation. Some poorly fitted chromosomes were eliminated from the chromosome solution to maintain the population size constant. The initial population size was 50; after each generation, poor solutions were eliminated to maintain a population size. This genetic operation was performed until it reached the maximum generation value. After reaching maximum generation, the model provided a set of final solution. The chromosome corresponding to the minimum error value is the best solution for the model.

The flow chart in

Flow chart of the ANN method linked with the GA optimizer. BPNN, back propagation neural network.

To evaluate the performance of the one-dimensional flood routing hydrodynamic model and the ANN models, three different criteria were considered to compare the predicted results with the observed data, mean absolute error (MAE), root-mean-square error (RMSE) and peak error (PE), based on the following equations:
_{m}_{o}_{m}_{,peak} is the predicted peak water stage and _{o}_{,peak} is the observed peak water stage.

Seven data sets were used to evaluate the practical accuracy of the models and to ascertain the predictive capability of the models. Five typhoon events, Typhoon Aere (2004), Typhoon Haima (2004), Typhoon Nockten (2005), Typhoon Matsa (2005) and Typhoon Sepat (2007) (673 hourly water stage data), were used for the flood routing hydrodynamic model calibration (and ANN model training), and two typhoon events, Typhoon Fungwong (2008) and Typhoon Morakot (2009) (371 hourly water stage data), were used for flood routing hydrodynamic model verification (and ANN model verification). The calibration and verification events are independent and are not related to each other.

The terminology for model comparisons,

Comparison of observed and simulated water stages for the flood routing hydrodynamic model, BPNN and genetic algorithm neural network (GANN) model calibration at the Ru-Kuo-Yan station for: (

The performance of the one-dimensional flood routing hydrodynamic model, the BPNN model and the GANN model with respect to predicting water stage during the calibration (training) phase at four stations.

Method | Taipei Bridge | Ru-Kou-Yan | Chung-Cheng Bridge | Da-Zhi Bridge | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | |

Calibration with one-dimensional hydrodynamic model | 0.26 | 0.32 | 7.61 | 0.29 | 0.36 | 10.87 | 0.26 | 0.35 | 5.46 | 0.21 | 0.26 | 8.96 |

Training with BPNN model | 0.11 | 0.15 | 4.77 | 0.12 | 0.17 | 5.00 | 0.19 | 0.26 | 4.09 | 0.15 | 0.19 | 5.41 |

Training with GANN model | 0.10 | 0.13 | 3.07 | 0.11 | 0.14 | 4.19 | 0.14 | 0.19 | 2.81 | 0.14 | 0.18 | 3.16 |

Notes: MAE, mean solute error; RMSE, root mean square error; PE, peak error.

Because of the poor accuracy in simulating the water stage using the one-dimensional flood routing hydrodynamic model, ANN models, including BPNN and GANN, were employed to improve the water stage calculations. The BPNN structure for predicting water stage is shown in

BPNN structures for predicting the water stage.

The prediction of the water stage at the Ru-Kou-Yan station by the BPNN model training for the five typhoon events is also shown in

(

The parameters used in the BPNN model.

Parameters | Taipei Bridge | Ru-Kou-Yan | Chung-Cheng Bridge | Da-Zhi Bridge |
---|---|---|---|---|

Input nodes | 7 | 7 | 7 | 7 |

Hidden nodes | 7 | 14 | 11 | 7 |

Output nodes | 1 | 1 | 1 | 1 |

Learning rate | 0.01 | 0.01 | 0.01 | 0.01 |

Momentum | 0.7 | 0.7 | 0.7 | 0.7 |

Iterations | 2500 | 2500 | 2500 | 2500 |

The configuration of the GA.

Parameters | Taipei Bridge | Ru-Kou-Yan | Chung-Cheng Bridge | Da-Zhi Bridge |
---|---|---|---|---|

Population size | 30 | 30 | 30 | 25 |

Maximum generation | 2500 | 2500 | 2500 | 2500 |

Crossover probability | 1.0 | 0.9 | 1.0 | 1.0 |

Mutation probability | 0.01 | 0.01 | 0.01 | 0.01 |

The results in

The scatter plots of simulated and observed water stages using (

The results in

The verification results with the one-dimensional flood routing hydrodynamic model for simulating the water stages at Taipei Bridge for Typhoon Fungwong and Typhoon Morakot are shown in

A comparison of observed and simulated water stages for the flood routing hydrodynamic model, BPNN and GANN model verification at the Taipei Bridge station for (

The performance of the one-dimensional flood routing hydrodynamic model, the BPNN model and the GANN model with respect to predicting the water stage during the verification phase at four stations.

Method | Taipei Bridge | Ru-Kou-Yan | Chung-Cheng Bridge | Da-Zhi Bridge | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | MAE (m) | RMSE (m) | PE (%) | |

Verification with one-dimensional hydrodynamic model | 0.20 | 0.25 | 10.81 | 0.20 | 0.25 | 13.12 | 0.22 | 0.28 | 9.86 | 0.22 | 0.27 | 11.32 |

Verification with BPNN model | 0.13 | 0.16 | 9.89 | 0.18 | 0.21 | 7.17 | 0.19 | 0.25 | 7.99 | 0.21 | 0.26 | 9.02 |

Verification with GANN model | 0.09 | 0.15 | 6.77 | 0.l7 | 0.20 | 4.92 | 0.18 | 0.23 | 6.91 | 0.20 | 0.24 | 7.95 |

Notes: MAE, mean solute error; RMSE, root mean square error; PE, peak error.

The scatter plots of simulated and observed water stages using (

According to the performance of different approaches, one-dimensional flood routing hydrodynamic model and the combination of one-dimensional flood routing hydrodynamic model and two ANN models (

The simulation results also revealed that GANN model was superior to the BPNN model for predicting water stages. This is the reason that BPNN is a type of neural network that can effectively solve non-linear problems, but there are some problems for BP neural network in the training phase, such as getting into a local extreme, and convergence is slow. To overcome these problems and improve the reliability of the network, the effort of the genetic algorithm is combined with BPNN to avoid local minima and to achieve global convergence quickly and correctly [

The simulation of physical processes is of critical importance to flood control and water resource management in a river system. The one-dimensional flood routing hydrodynamic model is a physically-based model that can be used to predict water stages response to high freshwater discharge into the river during typhoon events. For the ANN model, which is a data-driven technique, predictability could be increased by providing a large number of appropriate input-output data sets during the training and verification phases [

The water stages in the Danshuei River system during typhoon events were simulated using a one-dimensional flood routing hydrodynamic model. The observed freshwater discharges at the upstream boundaries and downstream boundary conditions at the Danshuei River mouth were used to drive the model simulation. Five typhoon events, Typhoon Aere (2004), Typhoon Haima (2004), Typhoon Nockten (2005), Typhoon Matsa (2005) and Typhoon Sepat (2007), were used for model calibration (training). Two typhoon events, Typhoon Fungwong (2008) and Typhoon Morakot (2009), were used for model verification. To determine the performance of the hydrodynamic model, the BPNN model, and the GANN model, three criteria (

The results showed that the flood routing hydrodynamic model cannot satisfactorily mimic the water stages during the typhoon events for the model calibration and verification phases. Therefore, two ANN models, including the BPNN model and the GANN model, were adopted to improve the water stage predictions during typhoon events using the flood routing hydrodynamic model. The simulated results indicate that the performance with the BPNN model and the GANN model is better than with the hydrodynamic model alone. Moreover, the GANN model predicts the water stage well and presents low MAE, RMSE and PE values at Taipei Bridge, Ru-Kou-Yan, Chung-Cheng Bridge and Da-Zhi Bridge compared with the simulated results using the one-dimensional hydrodynamic model and the BPNN model. This study shows that the GANN technique can be successfully applied to predict water stages in the Danshuei River system during typhoon events.

In the present study, we focus on the water stage prediction instead of forecast during the typhoon events. In a future study, different lead-time forecasts in the water stage can be developed to assist the local authorities with preventing flooding effects prior to typhoon events. The soft computing techniques, such as the combining fuzzy optimal model with genetic programming [

This research was conducted with the support of the National Science Council, Taiwan, grant No. 101-2625-M-239-001. This financial support is greatly appreciated. The authors would like to express their appreciation to the 10th River Management Bureau, Water Resources Agency, for providing access to the observational data. The authors would also like to thank two anonymous reviewers for their valuable suggestions and comments.

Wen-Cheng Liu created ideal, supervised the progress, discussed the results, and wrote paper. Chuan-En Chung executed the models.

The authors declare no conflict of interest.