Storm Surge Level Prediction Based on Improved NARX Neural Network

doi:10.21203/rs.3.rs-1945590/v1

Download PDF

Article

Storm Surge Level Prediction Based on Improved NARX Neural Network

https://doi.org/10.21203/rs.3.rs-1945590/v1

This work is licensed under a CC BY 4.0 License

Version 1

posted

You are reading this older preprint version

Read the latest preprint version →

The northern Gulf of Mexico coast is affected by the North Atlantic hurricane season, which brings serious economic losses to the southern U.S. every year; therefore, it is necessary to make an accurate advance prediction of storm surge level. In this paper, a prediction model has been constructed based on Nonlinear Auto-Regressive Exogenous (NARX) Neural Network. Five types of data are selected as the input factors of the model. A neuron pruning strategy based on sensitivity analysis is introduced. Moreover, a modular prediction method is introduced based on the tide harmonic analysis so as to make the prediction results more accurate. At last, a complete storm surge level prediction model, Pruned Modular (PM)-NARX, is constructed. In this paper, the model is trained by using historical data and used for storm surge level prediction along the northern Gulf of Mexico coast in 2020. The simulation test results show that the correlation coefficient is stable above 0.99 at 12 h in advance within one minute. The prediction speed, accuracy, and stability are higher than those of conventional models. The above can prove that the PM-NARX can effectively provide early warning before the storm surge to avoid property damage and human casualties.

NARX Neural Network

Pruning Strategy

Modular Prediction

Tide Harmonic Analysis

Storm surge is an abnormal sea level change phenomenon caused by strong weather systems such as cold fronts, tropical cyclones, temperate cyclones, and sudden changes in air pressure. Offshore areas are more susceptible to extreme storm surges due to the combined effects of barometric, wind, and tidal fields, affecting all aspects of the social and economic environment worldwide. Due to the unique geographical position of the Gulf of Mexico, tropical cyclones generated in the North Atlantic and the Caribbean often enter the Gulf and are gradually strengthened when entering into coastal areas, causing severe damage there. To deal with the damage caused by storm surge disasters, a lot of work has been carried out in storm surge disaster monitoring and surveillance, and prediction forecasting in countries around the world, such as the SLOSH and ADCIRC models in the United States, the SEA model in the UK, the Mike21 model in Denmark, and the DSCM model in the Netherlands [1–4]. On this basis, a numerical prediction model based on artificial features plus linear model fitting has also evolved [5–7]. These models were helpful for predicting storm surges, but they have obvious drawbacks. They do not allow long-term prediction of storm surge activity, the computational process is very complicated and expensive, and the calculation costs is quite high, etc. By reducing the delay and streamlining the process of storm surge prediction, and by quickly and accurately predicting some critical data for each storm surge event, the authorities can be directed to prepare for the disaster and further reduce the damage caused by the disaster.

In recent years, with the development of artificial intelligence, more and more researchers have used data-driven models that introduce the idea of artificial intelligence in predicting storm surges [8–11], and the prediction is accomplished by mapping input and output variables to obtain information from the collected data. Various storm parameters such as sea level height, wind, barometric pressure, and other tropical cyclone characteristics are usually used in this type of prediction method to predict storm surge levels in the future period without directly studying the natural rules that govern the storm surge mechanism. The most commonly used method includes artificial neural networks (ANN) [12–14]. Other methods than that are Long Short Term Memory Network (LSTM) [15–16], Back Propagation Neural Network (BPNN) [17], etc., which still have disadvantages such as low prediction accuracy and lack of generalizability.

When performing storm surge prediction, the observed data can be considered as time series. The structural features of NARX neural network make it have better learning efficiency for time series and higher prediction accuracy. There are several previous examples of successful applications of NARX neural networks [18–21]. In this paper, based on the traditional NARX model, a simple storm surge prediction model, Pruned Modular- Nonlinear Auto-Regressive Exogenous Neural Network (PM-NARX) is constructed. It requires a small amount of computational data, but has fast computing speed, good robustness, high prediction accuracy, and sound generalization; therefore, it is helpful for predicting future storm surge extremes based on the meteorological parameters obtained at the observation stations.

Simulation Materials. In 2020, the North Atlantic produced 31 tropical cyclones, including 13 hurricanes. There are 9 tropical cyclones that made landfall on the northern Gulf Coast, including 7 hurricanes, accounting for 53.8% of the total number of hurricanes. In addition, 4 tropical cyclones are affecting the southeast coast of the United States, and the rest are tropical cyclones affecting southern Mexico (near the equator) or far from land. The North Atlantic hurricane season is mainly concentrated in the Gulf of Mexico, and its impact is evenly distributed among the southern states of the U.S. To verify the prediction effect of PM-NARX on the impact of different tropical cyclones, a number of stations in southeastern Texas, southern Louisiana, and southwestern Florida were selected to predict the storm surge at specific stations. The basic information of the selected stations and the actual observed storm surge data of each station are shown in Table1:

Table 1

Basic information about the selected stations
Station Area	Station Number	Station ID	Longitude	Latitude
Texas	S1	8771341	94°43.5′W	29°21.4′N
	S2	8775792	97°14.2′W	27°38.0′N
	S3	8779770	97°12.9′W	26°3.7′N
Louisiana	S4	8761927	90°6.8′W	30°1.6′N
	S5	8764227	91°20.3′W	29°27.0′N
	S6	8768094	93°20.6′W	29°46.1′N
Florida	S7	8725520	81°52.3′W	26°38.9′N
Florida	S8	8726520	82°37.6′W	27°45.6′N

Accordingly, the storm surge time series extracted from each observation station are listed in Figs. 1, 2 and 3 in the form of line graphs. Because the complete observation data is too long, only one month (720h) of data to be predicted in these figures.

The measured data are sampled every 1h, and the data are listed to obtain the time series of surge level for that period. In this paper, the prediction results are set to a total of 720 data for 30 days (Fig. 9–11), and a longer time span is used to analyze whether the model can accurately predict when the storm surge extremes occur and reduce the predicted values back to the normal range in time. For stations S1-3 in Texas, this paper researches the impact of Tropical Storm BETA on the region, thus selecting the measured storm surge data from September 1, 2020 GMT 0000 to September 30, 2020 GMT 2300 as the prediction output for the simulation prediction test. For stations S4-6 in Louisiana, this paper researches the impact of Hurricane DELTA on the region, thus selecting the observed surge data from October 1, 2020 GMT 0000 to October 30, 2020 GMT 2300; for stations S7-8 in Florida, this paper researches the impact of Hurricane Eta on the region, thus selecting the observed surge data from November 2020 GMT 0000 to October 30, 2020 GMT 2300. And for stations S7-8 in Florida, this paper researches the impact of Hurricane ETA on the region, thus selecting the observed surge data from November 2020 GMT 0000 to October 30, 2020 GMT 2300.

Concerning the selection of training data, it is a highly complex process because the causes and influencing factors of storm surges are very complicated, mainly the joint action of waves and atmosphere, where the atmospheric action is dominant and can be divided into the action of air pressure and wind stress, etc. To this point, five non-linear meteorological factors (wind speed, wind direction, gust speed, air temperature, and pressure) corresponding to the time of storm surge were selected for this paper, and the corresponding historical data are selected for each station to train the PM-NARX and the two comparison models.

In order to compare the prediction data more intuitively, three indicators, Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Correlation coefficient (CC), are introduced as the main basis for judging the prediction effect, and the calculation formula is shown in the Equations(1–3):

$MAE=\frac{1}{n}\sum _{k=1}^{n}\left\|{y}_{k}-{\widehat{y}}_{k}\right\|$	(1)
$RMSE=\sqrt{\frac{\sum _{k=1}^{n}{\left({y}_{k}-{\widehat{y}}_{k}\right)}^{2}}{n}}$	(2)
$CC=\frac{\sum _{k=1}^{n}\left({y}_{k}-{\stackrel{-}{y}}_{k}\right)\left({\widehat{y}}_{k}-{\stackrel{-}{\widehat{y}}}_{k}\right)}{\sqrt{\sum _{k=1}^{n}{\left({y}_{k}-{\stackrel{-}{y}}_{k}\right)}^{2}\sum _{k=1}^{n}{\left({\widehat{y}}_{k}-{\stackrel{-}{\widehat{y}}}_{k}\right)}^{2}}}$	(3)

Where n is the number of samples, or it is interpreted as a time index in time series analysis; ${y}_{k}$ and ${\stackrel{-}{y}}_{k}$ represent the observed values and the average of the observed values, respectively; and ${\widehat{y}}_{k}$ and ${\stackrel{-}{\widehat{y}}}_{k}$ represent the predicted values and the average of the predicted values, respectively.

Simulation Method The NARX neural network is a model proposed by Leontaritis et al. (1985) for describing nonlinear discrete systems [22]. It is one of the most widely used neural networks in non-linear dynamic systems and is suitable for time series prediction. It has been validated and applied to solve non-linear series prediction problems in a variety of fields. Its memory effect on historical data enhances its ability to handle dynamic data and has significantly improved the prediction performance of complex sequences [23–25]. The basic structure of the NARX model is shown in Fig. 4.

As shown in Fig. 4, the standard NARX neural network is divided into a three-layer structure, an input layer containing n neural nodes, a hidden layer containing l neural nodes, and an output layer containing m neural nodes. x(t) represents the external input to the neural network, y(t) represents the output of the neural network at time t, and y(t-1) represents the output of the neural network at time t-1. ${\omega }_{ij}$ represents the association weight of the i-th node in the input layer to the j-th node in the hidden layer, ${\omega }_{jk}$ represents the weight of the j-th node in the hidden layer to the k-th node in the output layer, ${a}_{j}$represents the bias from the input layer to the hidden layer, ${b}_{k}$represents the bias from the hidden layer to the output layer. The model learning rate is $\eta$, the excitation function is $g\left(x\right)$, which takes the tanh function, and the output mapping is between (-1, 1), monotonically continuous, with a limited output range, optimally stable, and faster convergence because it is 0-mean. The output layer uses Identity as an activation function on the final output node for the regression task. tanh and Identity are expressed as follows [26].

$$g\left(x\right)=\frac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$$

$$Identity\left(x\right)=x$$

In addition, the neural network uses a certain indicator as a clue to find the optimal weight parameter, which is called the loss function and indicates the degree of difference between the current predicted output of the neural network and the actual measured data. In this paper, MSE is chosen as the loss function, when the gradient is easy to calculate and has a more stable solution, and the calculation formula is expressed by Eq. (3).

$$loss=\frac{1}{n}\sum _{i=1}^{n}{\left({e}_{i}\right)}^{2}$$

In Eq. (6), n represents the number of data individuals, e_i represents the prediction error of the i-th data. Regarding the selection of training functions in this model, it can be determined that trainlm and trainbr functions are trained optimally after several experimental validations. The former utilizes the Levenberg-Marquardt optimization algorithm to find the extreme value by gradient, which has the advantages of both the gradient method and the Newton method. The latter utilizes Bayesian Regularization, which has better generalization ability and reduces the difficulty of determining the optimal network structure. The difference in training accuracy between the two functions is not significant, but trainlm is finally determined as the training function for this model because of its shorter training time.

The single-step prediction model of the NARX neural network can be expressed as:

$$y\left(t\right)=f\left(y\right(t-1),y(t-2),\dots ,y(t-{n}_{y}),x(t-1),x(t-2),\dots ,x(t-{n}_{x}\left)\right)$$

From Eq. (7), it can be seen that the output of the NARX neural network is influenced by the input and output data of the previous moment, and similarly, the multi-step prediction model with a step ahead can be expressed as:

$$y\left(t\right)=f\left(y\right(t-a),y(t-a-1),\dots ,y(t-{n}_{y}-a+1),x(t-a),x(t-a-1),\dots ,x(t-{n}_{x}-a+1\left)\right)$$

The storm surge level prediction is a multi-step time series prediction like that shown in Eq. (8). However, the standard NARX neural network uses a closed-loop model with parallel structure in the multi-step prediction, as shown in Fig. 5(a). Moreover, the output of the neural network is fed directly to the input, as shown in Fig. 4. Since the prediction is used instead of the actual measured values, prediction errors are accumulated, thus the performance of the model may deteriorate rapidly as the prediction time horizon increases. For this problem, the open-loop model neural network with series-parallel structure shown in Fig. 5(b) is constructed. In this mode, since the expected output of the NARX neural network training is known (The storm surge observation data), the desired output is fed back to the input.

Figure 5 Diagram of two structures of NARX neural network, where TDL means the time delay ladder, y(t) is the known desired output, and Y(t) is the predicted tide height

In this paper, the direct multi-step prediction method is used, and due to the recursive nature of the NARX neural network itself, the final model combines the characteristics of both direct and recursive multi-step prediction methods. In addition, in order to further improve the prediction speed and accuracy, this paper improves the traditional NARX model in terms of the overall structure of the model and the prediction data set, respectively.

Sensitivity pruning strategy There are no clear regulations for the setting of the number of nodes in the hidden layer of the neural networks. For this problem, this paper analyzes the sensitivity of the model prediction output by the contribution of the output weights of the hidden layer neurons, so as to adjust the overall structure of the neural network, without depending on the input data set, and the sensitivity is calculated by the hidden layer nodes of the neural network itself so as to complete the pruning operation, which has general applicability. The detailed operation is to remove neurons whose sensitivity is below a certain threshold, which can effectively remove redundant hidden layer neurons without sacrificing network performance, and improve computational efficiency by reducing neural network dimensionality thereby [27–29].

In this paper, a single hidden layer, single output neural network is constructed so that the nodes’ number of the input layer is i = 1, 2, ... n, the nodes’ number of the hidden layer is j = 1, 2, ... l, and the nodes’ number of output layer is k = m = 1.

At moment t, the weight update formula from the hidden layer to the output layer is:

$${\omega }_{jk}^{{\prime }}\left(t\right)={\omega }_{jk}\left(t\right)+\eta {H}_{j}{\left(t\right)e}_{k}\left(t\right)$$

In Eq. (6), ${H}_{j}\left(t\right)$ is the output of the hidden layer, ${H}_{j}\left(t\right)=g\left({I}_{i}\left(t\right)\right)$, where ${I}_{i}\left(t\right)$ is the input of the hidden layer:

$${I}_{i}\left(t\right)=\sum _{i=1}^{{n}_{x}}{\omega }_{ij}{\left(t\right)x}_{{j}_{x}}\left(t\right)+\sum _{i=1}^{{n}_{y}}{\omega }_{ij}{\left(t\right)y}_{{j}_{y}}\left(t\right)+{a}_{j}, \left\{\begin{array}{c}{j}_{x}+{j}_{y}=j\\ {n}_{x}+{n}_{y}=n\end{array}\right.$$

In addition, ${e}_{k}\left(t\right)$ is the prediction error, ${e}_{k}\left(t\right)=\left({Y}_{k}\left(t\right)-{O}_{k}\left(t\right)\right)$, where ${Y}_{k}\left(t\right)$ is the desired output and ${O}_{k}\left(t\right)$ is the output of the output layer:

$${O}_{k}\left(t\right)=\sum _{j=1}^{l}{H}_{j}{\left(t\right)\omega }_{jk}\left(t\right)+{b}_{k}$$

Since the number of nodes in the output layer of the model constructed in this paper is 1, the weight ${\omega }_{jk}$ from the hidden layer to the output layer can be expressed as ${\omega }_{h}$, h = 1, 2, ... l. Taking ${\omega }_{h}$ as the input quantity for sensitivity analysis, its sensitivity ${S}_{h}$ to the predicted output can be expressed by Eq. (9):

$${S}_{h}=\frac{{Var}_{{\omega }_{h}}\left[E\left(\left.Y\right|{I}_{i}={\omega }_{h}\right)\right]}{{Var}\left(Y\right)}$$

In Eq. (9), Var represents the variance, Y represents the desired output, and E represents the value of Y when ${\omega }_{h}$ is equal to ${I}_{i}$. The final expression for the sensitivity of the output weights of the hidden layer of the neural network can be obtained by performing the Fourier transform on the input and output quantities (Eq. 13), while the resulting sensitivity values are normalized in order to reduce the computational error (Eq. 14):

$${S}_{h}=\frac{\sum _{t=1}^{+\infty }\left({A}_{t{\omega }_{h}}^{2}+{B}_{t{\omega }_{h}}^{2}\right)}{\sum _{t=1}^{+\infty }\left({A}_{t}^{2}+{B}_{t}^{2}\right)}$$

$${S}_{h}^{{\prime }}=\frac{{s}_{h}}{\sum _{h=1}^{l}{s}_{h}}$$

where ${A}_{t}$ and ${B}_{t}$ are represented by Equations (15) and (16), respectively:

$${A}_{t}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sum _{t=0}^{\infty }\left({A}_{t}{cos}\left({\omega }_{j}s\right)+{\beta }_{t}{sin}\left({\omega }_{j}s\right)\right){cos}\left({\omega }_{j}s\right)ds, -\pi \le s\le \pi$$

$${B}_{t}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\sum _{t=0}^{\infty }\left({A}_{t}{cos}\left({\omega }_{j}s\right)+{\beta }_{t}{sin}\left({\omega }_{j}s\right)\right){sin}\left({\omega }_{j}s\right)ds, -\pi \le s\le \pi$$

When the obtained ${s}_{h}^{{\prime }}$ is less than the set threshold, it indicates that the current hidden layer neural node is redundant, and the algorithm will delete the neural node and all the values connected to it. The judgment of redundant neural nodes and the deletion operation are performed during the training process. When the neural network structure is determined, the parameters are then modified using the most rapid descent algorithm.

In this paper, a set of test data were selected to verify the feasibility of the sensitivity pruning strategy. The initial hidden layer neural nodes are set to 30, ${S}_{h}^{{\prime }}$ is set to 0.2, and the results are run under the condition of 10,000 training times as shown below, the deleted hidden layer nodes end up being 6, and the mean square error of the final training result converges to about 0.08m, the test process and result is shown in Figs. 6 and 7:

As seen in the figures above, after the pruning process, the neural network removes some redundant neurons, making the whole network more compact. In addition, to verify the improvement of the traditional NARX neural network by introducing the sensitivity pruning strategy, five tests were performed on the NARX model before and after pruning respectively, and the correlation of operation time and prediction results are shown in Table 2.

Table 2

Correlation of prediction results and time cost between the traditional NARX and the pruned NARX
Model	CC	Time/s
Traditional NARX	0.9878	147
	0.9827	143
	0.9876	106
	0.9880	128
	0.9863	115
Mean value	0.9865	128
Pruned-NARX	0.9818	50
	0.9827	56
	0.9820	47
	0.9847	66
	0.9828	56
Mean value	0.9828	55

As shown in Table 2, the average correlation of the prediction results of the NARX model processed by the sensitivity pruning strategy decreased by 0.38% compared with the original model, which was basically negligible. In comparison. the average prediction time was reduced by 57% compared with the original model, and the computing speed was greatly improved.

Modular prediction The modular prediction method is introduced to improve the prediction dataset. In the field of storm surge prediction, traditional data-driven models generally predict surge levels directly. They cannot be targeted to make accurate predictions using the effects of meteorological factors on surge levels. This paper divides the storm surge data as prediction output into an astronomical component influenced by gravity with mathematical regularity and a non-astronomical component manipulated by extreme meteorological factors. Tidal harmonic analysis is introduced to calculate the former and, the proposed model is used to predict the latter.

It is the most traditional technique in tidal forecasting. Firstly, we separate out the tidal divisions (such as lunar tidal system, solar tidal system, etc.) from the measured tidal records, and find out the amplitude and phase angle of each tidal division. Then, after being revised by astronomical factors, the tidal division's summation constant is obtained, which can be used to calculate the tidal changes within a certain period and analyze the tidal properties of the area. Each tide component can be expressed by Eq. 17:

$$h=R{cos}\left({q}_{t}+{V}_{0}+u-K\right)$$

In Eq. (17), h means the height of the component, R represents the amplitude of the component, q_t means the angular rate of the component (which can be determined according to the component), ${V}_{0}+u$ denotes the phase angle of the imaginary celestial body at zero universal time at the beginning of the observation period, and K means the phase angle at which the high tide lags behind the moon culmination due to seabed friction and inertia.

The tide component expression can be further written as Eq. 18:

$$h=fH{cos}\left({q}_{t}+{V}_{0}+u-K\right)$$

In Eq. (18), $f$, ${q}_{t}$ and ${V}_{0}+u$ are all known. If the values of H and K of each component are obtained, the tide component can be obtained. H and K are called the tide harmonic constants of tide components. If the harmonic constants are known, the tide height of each component can be acquired, and the future tide can be calculated by adding all the components together.

The harmonic analysis data introduced in the model were taken from the NOAA website, and 37 sub-tides were selected for prediction (M2, S2, N2, K1, M4, O1, M6, MK3, S4, MN4, NU2, S6, MU2, 2N2, OO1, LAM2, S1, M1, J1, MM, SSA, SA, MSF, MF, RHO, Q1, T2, R2, 2Q1, P1, 2SM2, M3, L2, 2MK3, K2, M8, MS4). The improved prediction model introduces a modular approach on the basis of pruning by deleting the regular astronomical tide data in the input dataset, replacing the prediction output with the non-astronomical surge data (surge deviation), and finally combining the resulting prediction output with the astronomical tide calculated by the harmonic analysis to derive the final prediction results. It has also been demonstrated in previous studies that the modularized NARX model has better prediction results compared to the conventional model, with improved accuracy and robustness of the model, with a combined degree of improvement of about 23.3% [30].

The improved model structure As shown in Fig. 8, a complete PM-NARX model consists of two parts, the neural network part and the modular prediction part. Firstly, in the neural network part, the input data is evaluated once at the end of each training process on the sensitivity of the output weights of the hidden layer, so as to filter out redundant neural nodes for pruning operations. The pruned neural network is then tested to check whether the test results satisfy the set conditions, and if not, the internal parameters are adjusted and retrained in the first step, and if the test results satisfy the set conditions, the number of iterations is counted once (K + 1), and the number of training iterations is further determined whether the set upper limit is satisfied. When the number of passing test results satisfies the set condition (K ≥ 6) or the number of training iterations is up to the upper limit, then enter the modular prediction part, the model is brought into the meteorological factor data for predicting non-astronomical surges. Finally, the prediction output from pruned-NARX is combined with the astronomical tide data calculated by the tidal harmonic analysis can produce the final storm surge prediction results.

The parameters of each model need to be set before conducting the prediction simulation, respectively. Firstly, the PM-NARX model needs to set the number of neural nodes inside the neural network (input layer, hidden layer, and output layer), the sensitivity threshold ${s}_{h}^{{\prime }}$ of the neural nodes in the hidden layer, the maximum number of training times of the model, and the TDL value. Secondly, the ARIMA model requires setting the values of the p, d, and q parameters. Finally the ε-SVR model requires setting the penalty parameter c, the kernel parameter g, and the tolerance deviation ε. The parameters of PM-NARX in the three models can be applied to all station dataset: since the input dataset contains 5 meteorological factors and the output data is the surge level at a certain moment, the input layer and output layer neural nodes are set to 5 and 1, respectively, and the implied layer neural nodes are initially set to 30, ${s}_{h}^{{\prime }}$ is set to 0.2, and the maximum training times are set to 5000. When determining the TDL, since the non-astronomical surge separated from the actual observed sea level data can be regarded as white noise with basically no autocorrelation, but considering that the predicted output will eventually be superposed with the astronomical tide, the TDL is determined to be 24 according to the tidal period. The other two models need to adjust their internal parameters according to the data of each station separately, as shown in Table 3.

Table 3

The internal parameter settings of the ε-SVR and ARIMA models
Method\Station		S1	S2	S3	S4	S5	S6	S7	S8
ARIMA	p	5	5	3	1	1	1	1	1
	d	5	5	5	5	5	5	2	2
	q	1	1	1	1	1	1	1	1
ε-SVR	c	2.00	11.31	1.41	0.18	0.71	0.71	0.50	0.71
	g	8.00	0.71	5.66	4.00	8.00	16	5.66	11.31
	ε	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05

From the perspective of practical application, the key time for hurricane prevention is 1 to 6 hours before hurricane landfall, and all preparations should be completed within 12 hours before hurricane landfall, so the advance time step a of the simulation is initially set to 12 in this paper. In this way, the preparation for storm surge prediction simulation is basically completed, and the prediction output can be obtained by bringing each station data set into each model, as shown in the figure (9–11).

For S1-3, the former half of the storm surge data shows a cyclical upward trend, and the latter half shows a storm surge extreme and decreases cyclically. Since the Beta rating is TS and the wind is not strong, there is apparent periodicity of the observed data, and the prediction for it is relatively easy. From the prediction results, the prediction outputs of the three models are basically consistent with the actual observed surge data, but the error plots can be used for comparison so as to show that the prediction error of PM-NARX is the smallest and the smoothest, and the prediction error of ARIMA is the largest, with an error of more than 0.2m even in S1, while the prediction effect of ε-SVR lies between the other two.

For S4-6, the storm surge data are relatively stable overall. Still, there is a brief period of extreme surge at each station under the direct influence of C4 Hurricane Delta, which is different from the overall data pattern and is challenging to predict by general data-driven models. All three models were able to accurately identify the moment when such anomalous data appeared in the surge level data, but the gap between the models could be seen in the accuracy of the predicted values, and the prediction results of all three models fluctuated drastically at the anomalous data moment. The ARIMA model has a prediction value that differs from the observed surge level by more than 0.5m, and the error is more in the high prediction value that exceeds the observed value, with a difference of 0.92m at station S6. The error of the ε-SVR model is more in the low prediction value that fails to reach the observed value, with a difference of -0.96m at station S4. In contrast, the PM-NARX model also has prediction fluctuation, but the extent is lower. The prediction results of S4-5 show the obvious advantage of PM-NARX, and only one error of -1m occurred after the anomalous data at S6, which is the same as ε-SVR. However, its prediction of surge level extreme is still the most accurate among the three models.

For S7-8, although hurricane Eta is also C4, its wind has weakened when it moves near the stations, and its intensity has dropped to TS. So, the storm surge data at S7-8 are similar to S1-3, and the prediction effects of the three models are relatively close, and the error fluctuations are all basically controlled within 0.1m. However, compared with the error plot, it can be seen that the prediction error of ARIMA has the most significant fluctuation at station S7. The prediction error of ε-SVR has the most significant fluctuation at station S8, and PM-NARX has an error of more than 0.2 m. This error has also appeared in the previous simulations, appearing a few hours after the storm surge extreme, but it has little impact on the reliability of the model prediction results. From an overall perspective, the PM-NARX model has the slightest and most stable prediction error, and its performance is still dominant.

For the prediction results of the three models 12 h ahead (prediction lead time a = 12), the prediction error evaluation indexes of the three models at the selected stations are compiled and calculated in this paper. In addition to the MAE, RMSE, and CC mentioned above, the extreme of positive and negative errors and the time cost for prediction are additionally included, and are presented in Table 4:

Table 4

Summary of the prediction errors evaluation indicators and related information
No.	Model	Error extreme (positive)/m	Error extreme (negative)/m	MAE/m	RMSE/m	CC	Time/s
S1	ARIMA	0.21	-0.17	0.0342	0.0443	0.9878	84
	ε-SVR	0.10	-0.09	0.0222	0.0286	0.9949	1147
	PM-NARX	0.08	-0.14	0.0167	0.0238	0.9967	56
S2	ARIMA	0.04	-0.03	0.0068	0.0089	0.9989	90
	ε-SVR	0.04	-0.04	0.0070	0.0094	0.9998	775
	PM-NARX	0.02	-0.02	0.0034	0.0043	0.9991	46
S3	ARIMA	0.14	-0.12	0.0239	0.0311	0.9891	93
	ε-SVR	0.12	-0.08	0.0160	0.0211	0.9951	1715
	PM-NARX	0.06	-0.07	0.0141	0.0191	0.9962	63
S4	ARIMA	0.56	-0.39	0.0150	0.0358	0.9736	83
	ε-SVR	0.05	-0.96	0.0145	0.0424	0.9590	1205
	PM-NARX	0.09	-0.37	0.0115	0.0214	0.9900	54
S5	ARIMA	0.63	-0.62	0.0360	0.0620	0.9687	89
	ε-SVR	0.46	-0.44	0.0347	0.0594	0.9714	517
	PM-NARX	0.12	-0.23	0.0130	0.0199	0.9975	57
S6	ARIMA	0.92	-0.88	0.0480	0.0796	0.9436	104
	ε-SVR	0.17	-0.40	0.0273	0.0396	0.9862	263
	PM-NARX	0.46	-1.00	0.0162	0.0478	0.9799	57
S7	ARIMA	0.08	-0.08	0.0182	0.0236	0.9943	94
	ε-SVR	0.11	-0.09	0.0172	0.0230	0.9946	1318
	PM-NARX	0.08	-0.10	0.0135	0.0196	0.9960	59
S8	ARIMA	0.16	-0.16	0.0263	0.0338	0.9929	93
	ε-SVR	0.18	-0.15	0.0326	0.0421	0.9892	565
	PM-NARX	0.11	-0.24	0.0162	0.0244	0.9962	53

According to the comparison of prediction results and the prediction evaluation indicators in Table 4, it can be seen that all three models can produce relatively precise prediction results, but in comparison, the PM-NARX model has the most stable and accurate prediction results at all stations, with a mean CC value of 0.9939, which is higher than the same indicator of the ε-SVR and ARIMA models of 0.9863 and 0.9811, respectively. The information that is not reflected in the prediction results and error comparison figures can be seen by comparing the three evaluation indicators. The increase in MAE may be due to a rise in the number of error extrema points in the prediction error or the appearance of larger error extrema. However, the increase of RMSE may be because the overall stability of the prediction error is reduced, and the error extremes alone have little effect on this indicator, so the state of the prediction error can be judged from the magnitude of these two indicators. For example, PM-NARX has a negative error extreme of -1m at station S6, leading to an increase in MAE values, and because of the simultaneous presence of large positive error extremes, indicating overall fluctuations in the prediction results, leading to a simultaneous increase in RMSE values. By analyzing the error evaluation indicators, it can be seen that PM-NARX has the lowest mean values of MAE and RMSE, and is able to output accurate, reference-value prediction results consistently at all selected stations.

This paper proposes a storm surge level prediction model (PM-NARX) based on an improved NARX neural network. By Using meteorological parameters as input, the neural network structure is automatically determined by integrating a strategy of neuron sensitivity pruning, introducing a modular prediction method for harmonic analysis data, and dividing the observed surge level data into the astronomical tide and non-astronomical surge components to achieve a targeted prediction of storm surge level. The PM-NARX was trained and simulated on a dataset collected from eight stations along the northern coast of the Gulf of Mexico. A comparison of the MAE, RMSE CC values and Times of the prediction results shows that PM-NARX performs high accuracy, quickly and stability. Finally, since PM-NARX is a data-driven model and is not limited to storm surge prediction, it can even be applied to other time-series multi-step prediction problems beyond storm surge prediction as well.

Data Availability Statement

The datasets analysed during the current study are available in the NOAA repository, (https://tidesandcurrents.noaa.gov/).

Funding

This work was supported in part by the National Key R&D Program of China (Grant No. 2021YFC2801005), the Transportation Education Scientific Research Project of China Institute of Communications Education (JTYB20-09), and the Fundamental Research Funds for the Central Universities (Grant No. 3132022142).

Ethical Statement

The manuscript has not been published before and is not being considered for publication elsewhere. All authors have contributed to the creation of this manuscript for important intellectual content and read and approved the final manuscript. We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Storm Surge Level Prediction Based on Improved NARX Neural Network”.

Jelesnianski, C. P. A numerical calculation of storm tides induced by a tropical storm impinging on a continental shelf. Mon Weather Rev93, 343-358 (1965).
Jelesnianski, C. P. SPLASH, Part 1, Landfall Storms. United States: NOAA Natiounal Weather Service. 52-53 (1972).
Jelesnianski, C. P. SPLASH, Part 2, General Track and Variant Storm Conditions. United States: NOAA National Weather Service. 54-55 (1974).
Luettich, R., Westerink, J. ADCIRC: A (parallel) ADvanced CIRCulation model for oceanic, coastal and estuarine waters. Users Manual. 1-115 (2015).
Oliveira, M. M. F. D., Ebecken, N. F. F. COASTAL PROCESSES, A model to predict the coastal sea level variations and surge (ed. Brebbia, C.A., BENASSAI, G. and Rodriguez, G.R.) 126, 75-84; 10.2495/CP090071 (2009).
Salmun, H., Molod, A., Wisniewska, K., Buonaiuto, F. S. Statistical Prediction of the Storm Surge Associated with Cool-Weather Storms at the Battery, New York. J Appl Meteorol Climatol50, 273-282 (2011).
Yanshuang, X., et al. Fast Storm Surge Ensemble Prediction Using Searching Optimization of a Numerical Scenario Database. Weather Forecast36, 1629-1648 (2021).
Lee, T. Predictions of Typhoon Storm Surge in Taiwan Using Artificial Neural Networks. Adv. Eng. Softw.40, 1200-1206 (2009).
Oliveiria, M. M. F. D., Ebecken, N. F. F., Oliveria, J. L. F. D., Santos, I. D. A. Neural network model to predict a storm surge. J Appl Meteorol Climatol 48, 143-155 (2009).
Sung-Hyup, Y., Jang-Won, S. Storm surge prediction using an artificial neural network model and cluster analysis. Nat Hazards51, 97-114 (2009).
Hien, N. T. et al. Genetic Programming for Storm Surge Forecasting. Ocean Eng.215, 107812 (2020).
Wang, Q., Chen, J. & Hu, K. Storm Surge Prediction for Louisiana Coast Using Artificial Neural Networks. In: Hirose A, Ozawa S, Doya K, Ikeda K, Lee M, Liu D, eds. Cham: Springer International Publishing, 2016:396-405.
Lee, H., Kim, D., Jeong, Y. & Hong, S. Development of a Real-Time Storm-Surge Response System for Decision-Making Support on the Korean Coast. J. Coastal Res. 911-915 (2018).
Sahoo, B. & Bhaskaran, P. K. Prediction of Storm Surge and Coastal Inundation Using Artificial Neural Network – a Case Study for 1999 Odisha Super Cyclone. Weather and Climate Extremes. 23, 100196 (2019).
Chen, K. et al. Storm Surge Prediction Based on Long Short-Term Memory Neural Network in the East China Sea. Applied Sciences, 2022.
Chao, W., Young, C., Hsu, T., Liu, W. & Liu, C. Long-Lead-Time Prediction of Storm Surge Using Artificial Neural Networks and Effective Typhoon Parameters: Revisit and Deeper Insight. Water-Sui.12, 2394 (2020).
Zhang, X. & Jiang, S. Study On the Application of BP Neural Network Optimized Based On Various Optimization Algorithms in Storm Surge Prediction. P. I. Mech. Eng. M-J. Eng..
Han, J., Kim, S., Jang, M. & Ri, K. Using Genetic Algorithm and NARX Neural Network to Forecast Daily Bitcoin Price. Comput. Econ.56, 337-353 (2020).
Zubier, K. M. & Eyouni, L. S. Investigating the Role of Atmospheric Variables on Sea Level Variations in the Eastern Central Red Sea Using an Artificial Neural Network Approach. Oceanologia. 62, 267-290 (2020).
Di Nunno, F., Granata, F., Gargano, R. & de Marinis, G. Forecasting of Extreme Storm Tide Events Using NARX Neural Network-Based Models. Atmosphere-Basel. 12, (2021).
Di Nunno, F., de Marinis, G., Gargano, R. & Granata, F. Tide Prediction in the Venice Lagoon Using Nonlinear Autoregressive Exogenous (NARX) Neural Network. Water-Sui.13, 1173 (2021).
Breiman, L. Random Forests, Machine Learning 45. J. Clin. Microbiol.2, 5-32 (2001).
Math Works. MATLA R2012b Neural Network Tool box User Guide www.mathworks.com (2012).
Menezes, J. & Barreto, G. A. Long-Term Time Series Prediction with the NARX Network: An Empirical Evaluation. Neurocomputing. 71, 3335-3343 (2008).
Andalib, A. & Atry, F. Multi-Step Ahead Forecasts for Electricity Prices Using NARX: A New Approach, a Critical Analysis of One-Step Ahead Forecasts. Energ. Convers. Manage.50, 739-747 (2009).
Cybenko, G. V. Approximation by Superpositions of a Sigmoidal Function. Math Cont Sig Syst (MCSS) 2:303-314. Math. Control Signal.2, 303-314 (1989).
Han, H. & Qiao, J. A Structure Optimisation Algorithm for Feedforward Neural Network Construction. Neurocomputing. 99, 347-357 (2013).
Lauret, P., Fock, E. & Mara, T. A. A Node Pruning Algorithm Fulased On a Fourier Amplitude Sensitivity Test Method. IEEE TRANSACTIONS ON NEURAL NETWORKS. 17, 273-293 (2006).
Pastor-Bárcenas, O. et al. Unbiased Sensitivity Analysis and Pruning Techniques in Neural Networks for Surface Ozone Modelling. Ecol. Model.182, 149-158 (2005).
Wu, W. H., Li, L. B., Yin, J. C., Lyu, W. Y. & Zhang, W. J. A Modular Tide Level Prediction Method Based on a NARX Neural Network. IEEE ACCESS. 9, 147416-147429 (2021).

No competing interests reported.

Download PDF

Version 1

posted

You are reading this older preprint version

Read the latest preprint version →

Storm Surge Level Prediction Based on Improved NARX Neural Network

Status:

Version 1

Abstract

Figures

1 Introduction

2 Materials And Methods

3 Simulation And Discussion

4 Conclusion

Declarations

References

Additional Declarations

Status:

Version 1

\(MAE=\frac{1}{n}\sum _{k=1}^{n}\left\|{y}_{k}-{\widehat{y}}_{k}\right\|\)	(1)
\(RMSE=\sqrt{\frac{\sum _{k=1}^{n}{\left({y}_{k}-{\widehat{y}}_{k}\right)}^{2}}{n}}\)	(2)
\(CC=\frac{\sum _{k=1}^{n}\left({y}_{k}-{\stackrel{-}{y}}_{k}\right)\left({\widehat{y}}_{k}-{\stackrel{-}{\widehat{y}}}_{k}\right)}{\sqrt{\sum _{k=1}^{n}{\left({y}_{k}-{\stackrel{-}{y}}_{k}\right)}^{2}\sum _{k=1}^{n}{\left({\widehat{y}}_{k}-{\stackrel{-}{\widehat{y}}}_{k}\right)}^{2}}}\)	(3)