Storm-surge prediction at the Tanshui estuary : development model for maximum storm surges

Introduction Conclusions References


Introduction
Storm surges are sudden rises of water levels that can occur during typhoons, leading to the risk of flooding in low-lying coastal areas.Generally, storm surges are predicted using numerical methods.For example, Kawahara et al. (1980), Westerink et al. (1992), Blainetal (1994), andHsu et al. (1999) applied the finite element method, Yen and Chou (1979), Walton and Christensen (1980), Harper and Sobey (1983), and Hwang and Yao (1987) applied the finite difference method with nonlinear shallow water equations to simulate storm surges.
In recent years, neural network technology has become more and more mature.It has been widely applied to non-linear natural phenomena.For example, Grubert (1995) used feed-forward back-propagation neural networks to predict the flow rate at a river mouth.Deo and Naidu (1999) used neural networks to build a model for real-time wave prediction.The results show that these neural networks perform better than that of the AR model for wave prediction.Tsai and Lee (1999)  fitting was not required for that model.Tsai et al. (2002) used the neural network technique for forecasting and supplementing the time series of wave data using neighbor station wave records.Marzenna (2003) used neural networks to predict storm surges and compared the results obtained using different neural network topologies.Lee et al. (2004) used short-term observational data to predict long-term sea levels with a back-propagation neural network and obtained accurate results.Tsai et al. (2005) used back-propagation neural networks for the training of a water level time series model with data from previous typhoons used as training data to predict later typhoons.The prediction results were very good.However, the model was only used to predict overall water levels during storms.Storm surge values obtained by deducting astronomical tides from the predicted overall water levels might not be the same as actual values.In practice, astronomical tides plus storm surges equal the overall water levels during storms.Currently, astronomical tides can be well predicted using the harmonic analysis method or other numerical methods.Moreover, the maximum storm surge plus the highest spring tide may provide the potential for coastal inundation.Thus, the focus of this study is on the maximum storm surges.

Storm surge empirical formula
Storm surges are also called irregular weather water levels.Studies on storm surges at specific spots are more meaningful with higher practical values.Storm surges are also related to weather conditions (e.g., wind speed, wind direction, and minimum atmospheric pressure).Regarding maximum storm surge estimations, Cornner (1957) believed that a larger center of low pressure would lead to higher wind speeds at a station.The pressure can be used to estimate storm surges.Unoki and Isozaki (1966) also believed that maximum storm surges can be expressed in terms of pressure.They Introduction

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Full analyzed all the storm surge data in Japan to obtain an empirical formulation for predicting storm surges at estuaries by the open sea.Horikawa (1978) referenced the analysis results of previous actual data from Japan and suggested that, in addition to pressure, other parameters such as wind speed should also be taken into consideration.Their empirical formula is as follows: where ζ max is the maximum storm surge, ∆P is the maximum pressure difference during the storm surge, V max is the maximum wind speed during the typhoon, θ is the angle between the direction of the wind with the maximum wind speed and the tide-gauge station's normal line, and A and B are the empirical constants at the maximum storm surge.

Multi layer perception network
A multi layer perception (MLP) network is a supervised learning method.This type of topology contains one input layer, one or more hidden layers, and one output layer.Data is sent to the output layer from the input layer using the feedforward method.The formulas are listed below: where y j is the output variable, W i j is the weight between the j th neural layer and the i th neural layer, X i is the input variable as a biomimetic neuron input signal, f (net j ) is the transformation function as a biomimetic non-linear function of the neurons, B j is Introduction

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Full the threshold (bias) for the j th neuron, and net j is the consolidation function for the j th neuron.
A transformation function is usually an S curve called a sigmoid function which increases stability and can be written as: The MLP network learning method, like the back-propagation neural network, is based on iteration.However, it allows for more than one hidden layer, therefore it can handle rather complex functions.Furthermore, with optimized network algorithms, the time and number of iterations required can be reduced.Usually the number of layers and neurons are obtained using a trial and error method.The best topology can thus be found.With enough hidden layers and neurons, any continuous function can be approximated.

Radial basis function network
The topology of the radial basis function (RBF) network is similar to that of the MLP network.Basically, the most advantageous feature of the RBF network is its fast learning speed, making it suitable for application in real-time systems.Its output can be written as: where x is the input vector (x 1 , . .., x p ) T , W j is the weight from the j th layer to the output layer, B j is the threshold (bias) for the j th neuron, ψ j is the basis function of the A common basis function used for the RBF network is a Gaussian function, which can be written as: where σ j is the smoothing parameter (called the width in this study) of the j th neuron which controls the radial basis function, U j is the center of the neurons in the j th radial basis function hidden layer, and x − U j is the Euclidean distance between U j and the input vector.

Maximum storm surge prediction model
According to Murty (1984), over the past century, an average of 3.5 typhoons has struck Taiwan per year.Storm surges are very likely to occur at estuaries in the northern areas of Taiwan, such as Tanshui which is bordered by the Taiwan Strait.Data from the station at the Tanshui Estuary were collected in order to explore these storm surges.

Data sources
This study used storm surge and weather data acquired during typhoons from the station at the Tanshui Estuary (the tide-gauge station outside the southern dock of the

Pre-processing of data and evaluation indexes
Before training a neural network, pre-processing of the input data is very important.Input data must be normalized as discussed below: where D min and D max represent the range of linear mapping, x max and x min are the maximum and minimum values in the series, and x old and x new are the series before and after transformation.Generally, network performances can be efficiently shown by looking at the errors and correlations obtained using two separate statistical indexes, the root mean square errors (RMSE) and correlation coefficients (C.C.).Their definitions follow: where n is the sample size, y k is the observed value of the kth sample point, and ŷk is the kth estimation.

Topology presentation of neural networks
In this study, the topologies of neural networks are presented in the form of "I x H y O z ", where I x represents the number of neurons (factors which influenced storm surges, such as pressure difference, wind speed, etc.) in the input layer, H y represents the number of hidden layers, and O z represents the number of output variables.The number of output neurons was 1 (z =1) because there was only one variable to be predicted, the storm surge.

Empirical formulation
To obtain the empirical formula for finding the maximum storm surge, the generalized least squares method was applied with data for the 22 selected typhoons being used for the regression analysis.The following empirical constants for the Tanshui Estuary were obtained: A = 0.00952 and B = 0.0031.The formula can be written as: Equation ( 10) is applied to the data for the 22 typhoons and the results are summarized in Fig. 1.According to the graph, the estimations obtained with the storm surge empirical formula are mostly lower than the observed values.The reason could be that there was not enough data for empirical calculation of the storm surge, leading to poor precision.The correlation coefficient is 0.565, as shown in Fig. 2.

Empirical verification
Three models were used for discussion based on the pressure difference (∆P ), the wind field factor (U = V

MODEL A
In line with scholars such as Cornner et al. (1957) andIsozaki (1966) who believe that pressure differences can be used to estimate maximum storm surges, this study first utilizes the pressure difference as the only input variable to build a neural network model to predict the maximum storm surge.As can be seen in Table 2, in Model A, the best topologies for the MLP and RBF neural network models are I 1 H 7 O 1 and I 1 H 8 O 1 , respectively.According to the storm surge predictions shown in Figs. 3 and 4, the MLP model could predict some of the maximum storm surges based on a single input variable.
According to Table 2, although there was not sufficient precision, when using the pressure difference as the only input variable, the performance of the neural network model was better than that of the empirical formula.

MODEL B
In Model A, the only input variable was the maximum pressure difference.Then after that, another input variable, the corresponding wind field factor, was added to predict maximum storm surges.According to Table 2, the best network topologies for the MLP and RBF neural network models were I 2 H 7 O 1 and I 2 H 11 O 1 .According to the correlation coefficients in Figs. 5 and 6, the predictions for maximum storm surges in Model B, in which the maximum wind factor was considered, were better than those in Model A.
The correlation coefficient for the MLP model was 0.983, and that of the RBF model was 0.935.Figures 3, 4, 7, and 8 show the maximum storm surge predictions of Models A and B. According to these figures, the predictions obtained using both the MLP model and the RBF model in Model B were more precise than those in Model A. It is also found that the MLP predictions were more precise for typhoons causing larger storm surges.They were similar to the RBF predictions although not as precise as the MLP predictions.It

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Full is obvious that the influence of wind factor should not be ignored in the prediction of the maximum storm surge, especially for typhoons causing larger storm surges.

MODEL C -influences of upstream flow
The storm surges in estuaries can be categorized into those in estuaries near seas and in estuaries near rivers.The storm surges discussed in this study are the former.
The maximum pressure difference and wind factor were used as inputs in Model B.
In addition to the maximum pressure difference and the corresponding maximum wind speed, the upstream flow was input as well in Model C. According to Table 2, the best topologies for the MLP and RBF neural network models were I 3 H 6 O 1 and I 3 H 10 O 1 , respectively.
According to the correlation coefficients in Table 2, the MLP and RBF models performed better for Model C (with upstream flow being input) than for Model B (without the input of upstream flow).However, the correlation for the MLP model in Model C were only 0.003 higher than that in Model B, while the correlation for the RBF model in Model C was lower than those in Model B.
Although according to the predictions of storm surges as shown in Figs.7 and 9, the RMSE for the MLP model in Model B was slightly higher than that in Model C, while it can be seen from Figs. 8 and 10 that the RMSE for the RBF model was slightly lower for Model B than for Model C. Thus, it is likely that although the upstream flow had some influence on storm surge, the influence is far smaller than that of wind field and where the wind field and maximum pressure difference were input.Neural network models can be used to predict highly non-linear influences, thereby improving on the disadvantages of the empirical formula.In Model C, although the upstream flow was a factor of influence on the storm surges, manual observations are required to obtain flow information during typhoons.However, it is very common for there to be missing values or large observation errors.Based on the above review and discussion, the best input factors for Model B would further be applied to build a neural network prediction of storm surges with time series.Introduction

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Full  Full used back-propagation neural networks for real-time tide prediction.The predictions were very accurate and parameter Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

j
th layer, and F (x) is the output function of the network.The transformation function for the neurons in the output layer is a linear function.Discussion Paper | Discussion Paper | Discussion Paper | second fishing port by the Tanshui Estuary) from 1996-2001 and in 2005.Data for 22 typhoons were selected based on their paths and whether they had caused serious storm surges at the Tanshui Estuary.The data used are summarized in Table 1.Data from the Xiulang Station, a hydrological station 25 km downstream from the Tanshui Estuary operated the Water Resources Agency on the Tamsui River were used.Discussion Paper | Discussion Paper | Discussion Paper |

C
y k is the observed value of the kth sample point, ŷk is the kth estimation, ȳk is the average of the estimations, and ȳk is the average of the observed values.Discussion Paper | Discussion Paper | Discussion Paper |

2
max cos θ) at the Tanshui Estuary and the flow rate (Q) from the upstream Xiulang Station (make segmentation).Model A: ζ max = f (∆P ) Model B: ζ max = f (∆P , U) Model C: ζ max = f (∆P , U, Q) Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Figure 1.Predictions obtained by the storm surge empirical formula

Table 2 .
Comparison of models for the prediction of storm surges.