Storm Surge Forecast Using an Encoder–Decoder Recurrent Neural Network Model

Abstract

This study presents an encoder–decoder neural network model to forecast storm surges on the US North Atlantic Coast. The proposed multivariate time-series forecast model consists of two long short-term memory (LSTM) models. The first LSTM model encodes the input sequence, including storm position, central pressure, and the radius of the maximum winds to an internal state. The second LSTM model decodes the internal state to forecast the storm surge water level and velocity. The neural network model was developed based on a storm surge dataset generated by the North Atlantic Comprehensive Coastal Study using a physics-based storm surge model. The neural network model was trained to predict storm surges at three forecast lead times ranging from 3 h to 12 h by learning the correlation between the past storm conditions and future storm hazards. The results show that the computationally efficient neural network model can forecast a storm in a fraction of one second. The neural network model not only forecasts peak surges, but also predicts the time-series profile of a storm. Furthermore, the model is highly versatile, and it can forecast storm surges generated by different sizes and strengths of bypassing and landfalling storms. Overall, this work demonstrates the success of data-driven approaches to improve coastal hazard research.

1. Introduction

Hurricanes and tropical storms alone have overwhelmingly been the costliest of all natural hazards in the US. There have been 21 tropical storms, and seven of them became hurricanes in the recent 2021 Atlantic Hurricane Season, for which the total aggregate cost was more than USD 50 billion, and more than 161 people lost their lives. An efficient and accurate coastal storm forecast is vital to reduce storm damage and causalities. It is well-known that the current capability to forecast coastal storms using physics-based coastal models faces a dilemma in achieving both efficiency and accuracy [1]. Accordingly, there is a critical and urgent need to develop a coastal storm forecast system that is fast and accurate so that a sufficient amount of lead time from reliable forecasts is obtained to inform the public of the storm risk.

Two major physics-based coastal models have been used to simulate storm surges and/or wind waves associated with coastal storms in the coastal engineering community. However, neither has achieved efficiency and accuracy at the same time. The Sea Lake and Overland Surges from Hurricanes (SLOSH) model [2] is a coastal numerical model developed by the National Hurricane Center (NHC). SLOSH is a physics-based model that solves the shallow water equations with an explicit scheme on a numerical grid. SLOSH considers storm wind effects using a parametric wind model to treat storm winds as a stress effect [3,4]. Thanks to its simplifications, the SLOSH model can be run quickly. However, it is well-known that the storm surge simulated by SLOSH has an error range from 0 to 20% [5], which can be a significant amount for shallow or low-lying areas, e.g., the US Gulf Coast.

In addition to SLOSH, the Advanced Circulation (ADCIRC) model [6] has been used to simulate coastal storms [7,8]. ADCIRC accounts for the impact of wind-generated waves on storm surges by solving the wind–wave equations in addition to the shallow water equations. ADCIRC can utilize a triangular mesh to depict study areas in high resolution. Although ADCIRC can generate extremely high resolution, detailed, and accurate predictions, it requires significant computation resources [9]. To be useful for decision making during coastal storms, predictions must be generated in a matter of hours or even less. Unfortunately, very few computer centers in the US have the available capacity to perform a sizable ensemble of high-resolution ADCIRC runs in this short period [1].

In addition to physics-based coastal models, neural network models have been developed to study coastal storms. For example, Kim et al. [10] developed a feed-forward and back-propagation neural network model to predict storm surges in the Gulf of Mexico. Their neural network model gave relatively accurate prediction before storm landfall and relatively poor prediction post landfall. Hashemi et al. [11] trained a neural network model using the Levenberg–Marquardt algorithm to predict peak storm surge using a storm surge dataset. Their work was only limited to peak surge height, and a time series of storm surge profiles was not provided. Bezuglov et al. [12] proposed multiple-layer feed-forward neural networks to predict time-series storm surge profiles at selected locations in North Carolina. Their model predicted unrealistic surge profiles due to the lack of state variables in their model, and they pointed out that the recurrent network model would address the limitation [12]. Kim et al. [13] proposed a systematic and objective selection procedure to develop a Levenberg–Marquardt type neural network model for storm surge forecast in Sakai Minato, Japan. They found that neural network performance was influenced by the combination of input parameters and the neural network training units. Lee et al. [14] proposed an artificial neural network model that consists of a one-dimensional convolutional neural network model combined with principal component analysis and k-means clustering to predict peak storm surges. However, this study did not predict the time-series profile of a storm. Ramos-Valle et al. [15] developed a feed-forward and back-propagation model and a recurrent neural network model to study storm surge forecast. They found that a recurrent neural model could improve peak surge prediction. However, their univariate neural network forecast model only predicts the surge water level but no storm-induced velocity. Recently, Ayyad et al. [16] demonstrated that an artificial intelligence model trained with a large-scale storm surge dataset could predict peak storm surge height with similar accuracy as a physics-based model but much more time efficient.

The above review shows that there are a few knowledge gaps in the area of coastal storm forecast. First, physics-based coastal models either trade efficiency for accuracy (e.g., SLOSH) or accuracy for efficiency (e.g., ADCIRC). Improving the science of storm forecast should aim to achieve accuracy and efficiency. Second, artificial intelligence models are promising to address the difficulties faced by physics-based coastal models. However, some neural network models are limited to peak surge prediction, while others are inherently univariate and limited to predicting surge water level without storm-induced velocity, which is a critical factor in determining nearshore processes (e.g., rip currents and sediment transport) under a storm condition. This study aims to improve coastal storm forecast using a neural network model. Specifically, we develop a recurrent network model that forecasts the time-series storm profile, including peak surges. Furthermore, our multivariate recurrent neural network can forecast the storm water level and storm-induced velocity in a single framework.

This paper is organized as follows. We first introduce the storm surge dataset, storm surge parameters for neural network model training, and the model forecast scenarios in Section 2. Second, we explain the background of the artificial neural network model and the model architecture in Section 3. Next, Section 4 presents the training metrics, computational efficiency, and forecast skills for different lead times of the storm surge forecast model, and it further examines the neural network capability to forecast bypassing and landfalling storms on the US North Atlantic Coast. Finally, the conclusion is drawn in Section 5.

2. Storm Surge Dataset

This section first explains the background of the storm surge dataset used to develop our storm surge forecast model. Then we present the selected study locations along the US Atlantic Coast. Furthermore, we introduce the definitions of storm surge parameters. Finally, we present the storm surge forecast scenarios.

2.1. Introduction to North Atlantic Coast Comprehensive Study

This study utilizes a storm surge dataset: the North Atlantic Comprehensive Coastal Study (NACCS), completed by the US Army Corps of Engineers (USACE) [17,18]. NACCS aimed to quantify coastal storm hazards for Virginia to Maine coastal region. In total, 130 master storm tracks were developed by considering historical tropical storms in this region. Of these, 89 were landfall tracks, and the others were bypassing tracks. Furthermore, 1050 synthetic storms were developed based on those master tracks by varying the storm heading direction, central pressure, the radius of the maximum winds, translation speed, and the Holland B parameter, a scaling parameter to define the shape of pressure and wind fields of a storm [19].

Synthetic tropical storm wind and pressure fields were generated for each of the 1050 storms using the Planetary Boundary Layer model [20]. These wind and pressure data were used by ADCIRC [6] to simulate storm surge impact on the US Atlantic Coast. The storm surge dataset used by this study consists of the results induced by 1031 storms. This study randomly divides the 1031 storms into 981 training storms and 50 test storms. The neural network model did not see the test storms during the model training stage. The simulation results were saved at approximately 19,000 locations, which is referred to as save points by NACCS. Save points were chosen to ensure the coverage of probabilities of coastal storm hazards in the study region. Figure 1 shows the storm tracks and the selected three save points where storm surge forecast models are developed in this work. The selected three save points cover the lower limit of the study area in Virginia to the upper limit of the study area in Maine and the third one in the middle location in New York.

Table 1. The index number, water depth, and location information for the selected three save points in the NACCS storm surge dataset.

Save Points	Water Depth (m)	State	Location (Lat, Lon)
SP17771	40.0	Virginia	37.061 N 75.048 W
SP13205	74.0	New York	40.537 N 71.606 W
SP06815	60.7	Maine	43.580 N 70.000 W

lists the identification number, water depth, coastal state, and the coordinates of three save points.

2.2. Synthetic Tropical Storm Parameters

This study utilized synthetic storm data to predict storm surge impact at three save points. Although there are many parameters that can be used to train a storm surge model, as discussed in [15] and experimented with by several studies (e.g., [13,14]), the storm surge forecast model in this study only utilizes four parameters, including the location of the storm center (i.e., latitude and longitude), the central pressure of the storm, and the radius of the maximum winds, as described in

Table 2. The definitions of parameters used for storm surge forecast model development.

Variables	Description
Latitude (deg)	The latitude of the storm center
Longitude (deg)	The longitude of the storm center
Central pressure (hPa)	The pressure in the center of the storm.
Radius of maximum winds (km)	The radius to the maximum wind from the storm center.
Water level (m)	Storm induced water level.
X-velocity (m/s)	Storm-induced depth-averaged velocity in the x direction.
Y-velocity (m/s)	Storm-induced depth-averaged velocity in the y direction.

. The motivation to use them is due to the fact that the four variables are common storm parameters provided in storm forecast advisory, e.g., by NOAA and in hope that our model will be adapted to predict real storm hazards in the future. They will be referred as storm parameters in the following sections. The forecast parameters at each save point include the storm surge water level and the induced velocity in both the x and y directions. In addition, the distance between the save point and the storm center was calculated as a derived variable for developing the storm surge forecast model, as justified by the correlation analysis below.

To better understand the correlation between storm parameters and induced storm surges and velocities, a correlation analysis was carried out by using the Pearson standard correlation coefficient (r), as defined below:

$$r=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2{(y_i-\overline{y})}^2}}$$

(1)

where n is the number of samples; i is the sample index; $x_i$ and $y_i$ are the two interested variables; $\overline{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$; and $\overline{y}=\frac{1}{n}{\sum }_{i=1}^n{y}_i$. The Pearson correlation coefficient is a measure of the linear correlation of variables. A perfect positive relationship between two variables (e.g., x increases as y increases) is indicated by r = 1. A perfect negative relationship between two variables (e.g., x increases as y decreases) is indicated by r = −1. The absence of a linear relationship between two variables is indicated by r = 0. The correlation calculation utilizes the time-series storm surge dataset, and we analyze the correlation of storm parameters at the same time step.

Figure 2 shows the correlation between storm parameters and the induced surges and velocities. The correlation between the distance variable (i.e., the distance between the storm center and the save point) and the surge water level is r = −0.35, indicating that there is a negative correlation since when a stronger surge is usually generated by when the storm approaches the interested location. There is a positive correlation between the distance variable and induced storm velocity since r = 0.01 and 0.13 for velocity in x and y directions, respectively. The latitude and longitude of the storm center, which are used to derive the distance, also have a negative correlation with respect to the surge water level. It is noted that the longitude has a stronger correlation than the latitude of the storm center. The central pressure of storm is negatively correlated to the surge water level (r = −0.24) and positively correlated to the induced velocities (r = 0.04 and r = 0.18 for velocity in the x and y directions, respectively). The radius of the maximum wind has correlations with surge water levels (r = 0.11) and velocities (r = −0.06 and r = −0.08 for velocity in the x and y directions, respectively). The correlation analysis shows that there are different levels of correlation between storm parameters and storm surges, thus justifying the use of them for storm surge forecast model development. On the other hand, the Pearson correlation coefficient only measures a linear correlation. The neural network storm surge forecast model will learn the nonlinear correlation between storm parameters and storm surges.

2.3. Example Storm Surges from a Synthetic Storm

Figure 3 shows the time-series plots of synthetic storm 1011. Its storm track was shown in Figure 1. The readers are reminded that the time information related to synthetic storms was made up and does not reflect real historical events. This storm passed by the three save points from Virginia to Maine studied in this work. The latitude of the storm center steadily increased over time, while its longitude first decreased as the storm approached the coast and then increased as it bypassed it. The synthetic storm had a constant central pressure of 960 hPa, and the radius of the maximum winds was 29 km before 16:00 14 July 2000. Then these values increased progressively as the storm weakened.

Figure 3. The time-series profiles of the storm center (i.e., longitude and latitude), the central pressure, and the radius of maximum winds of storm 1011 in the NACSS dataset.

Figure 3. The time-series profiles of the storm center (i.e., longitude and latitude), the central pressure, and the radius of maximum winds of storm 1011 in the NACSS dataset.

Figure 4 shows the corresponding storm surge results at the three save points caused by synthetic storm 1011 in Figure 3. Figure 1 shows that the save point SP17771 in Virginia is on the left side of the storm track. The ADCIRC simulation shows that the peak surge water level was about 0.55 m. Both velocity components increased first but there was a sharp drop in the velocity in the x direction as the storm approached. Next, the velocity in the y direction dropped from positive to negative as the storm passed, indicating that the water was pushed in the south direction. The save point SP13205 in New York is at the right side of the storm track. Figure 3 shows that the storm had a constant strength till it became weak around 16:00 14 July 2000. Therefore, the strength of the storm in New York was similar to its strength in Virginia. As a result, similar peak surge water level was simulated by ADCIRC. However, both velocity components steadily increased since the storm pushed the water northeast. The storm weakened as it headed toward the North Atlantic coast. As a result, the simulated storm surge water level by ADCIRC at save point SP06815 in Maine was smaller than the other two save points. Similarly, the simulated velocity in Maine was also smaller than that of the other two save points. Note that the storm surge dataset used by this study does not consider the tidal effects in the ADCIRC simulation. As a result, the storm surge water levels in Figure 4 do not include fluctuating tidal patterns.

Overall, both the input storm parameters (e.g., Figure 3) and storm surge results (e.g., Figure 4) are very complicated. We could explain storm surges roughly by relating storm parameters, storm strengths, storm trajectory, and the location of interested areas. Then the research question becomes, can a neural network model figure out these complicated physical processes? The research goal of this project was to develop a neural network storm surge forecast model that can learn complex patterns hidden among storm parameters and the induced surge results.

Figure 4. The corresponding storm surge water level (top row), velocity in the x-direction (middle row), and velocity in the y-direction (bottom row) induced by storm 1011 at the three save points: SP17771, SP13205, and SP06815 from the left to right, respectively. The results were simulated by using the ADCIRC model.

Figure 4. The corresponding storm surge water level (top row), velocity in the x-direction (middle row), and velocity in the y-direction (bottom row) induced by storm 1011 at the three save points: SP17771, SP13205, and SP06815 from the left to right, respectively. The results were simulated by using the ADCIRC model.

2.4. Storm Surge Forecast Lead Times

The forecast lead time is critical for emergency management during a coastal storm. This study proposes predicting the potential storm surge impact before the arrival of a storm. This is different from the lead time obtained by physics-based storm surge models, which achieve lead time by accelerating the computational speed of the coastal model (e.g., [9]) or by solving simplified governing equations (e.g., [2]). In practice, a longer forecast lead time in numerical weather prediction becomes more challenging and less accurate. This study proposes testing three forecast lead times: 3 h, 6 h, and 12 h. In other words, we try to predict storm surges, e.g., 12 h, from now by using past storm conditions till this moment. There was no consensus about the proper length of past data to forecast future events. In Wei [21], we experimented with four ratios of the input hours to forecast hours, including 1:1, 3:1, 5:1, and 7:1. We found that a ratio of 1:1 is the best option in terms of forecast accuracy and efficiency. Therefore, we used the same ratio to develop our storm surge forecast model. For example, a forecast of storm water level and velocities in the future 6 h was made with the past 6 h storm parameters.

3. Storm Surge Forecast Model

This section presents the neural network model for storm surge forecast on the US North Atlantic Coast. We first briefly review the background of the encoder–decoder neural network model and the basics of the LSTM model, which was the fundamental recurrent neural network model used in this study. Next, we describe the proposed neural network architecture in detail. Finally, we explain how to prepare storm surge data to be fed into the model.

3.1. Basics of Encoder–Decoder Neural Network

The basic idea of an encoder–decoder neural network model consists of two steps. The neural network model first reads an input sequence and encodes it into a fixed-length internal state. Then it applies a decoder neural network to translate the internal state to an output sequence (e.g., [22]). Different neural networks can be used as the encoder and decoder models, and a recurrent network model (e.g., LSTM) is the most popular option. The encoder–decoder neural network model has been widely used for language translation (e.g., [23]). Figure 5 shows the proposed encoder–decoder neural network model for storm surge forecasting. We propose using LSTM as the encoder and the decoder neural network, which will be introduced in the next section. The input sequence of our neural network model is the time-series storm parameters as listed in Table 2, and then an LSTM model is used to encode the time-series input into an internal state. Then we apply another LSTM model to predict storm surge results, including surge water level and velocities in both x and y directions. It should be noted that the input length and output length of the neural network model are the same since the input and output ratio is 1:1 in this study. However, it is possible to have varying input and output sequences [21].

3.2. The LSTM Model

As described above, the LSTM model is the fundamental network in our storm surge forecast model. The LSTM network diagram is shown in Figure 6. The state of each LSTM cell consists of a long-term state/memory ${\mathbf{c}}_{\left(t\right)}$ and a short-term state/memory ${\mathbf{h}}_{\left(t\right)}$. The input vector ${\mathbf{x}}_{\left(t\right)}$ and the previous short-term state ${\mathbf{h}}_{(t-1)}$ are fed into four fully connected (FC) neural layers. The main layer output ${\mathbf{g}}_{\left(t\right)}$ using the tanh function. There are three gate controllers. Specifically, the forget gate (${\mathbf{f}}_{\left(t\right)}$) controls the long-term state that should be erased; the input gate (${\mathbf{i}}_{\left(t\right)}$) controls which part of ${\mathbf{g}}_{\left(t\right)}$ should be added to the long-term memory; and the output gate (${\mathbf{i}}_{\left(t\right)}$) controls the long-term memory that should be read and output to ${\mathbf{h}}_{\left(t\right)}$ and ${\mathbf{y}}_{\left(t\right)}$. In addition, the mathematical equations of the LSTM model (e.g., [22,24]) are given below:

$${\mathbf{i}}_{\left(t\right)}=\sigma \left({\mathbf{W}}_{xi}·{\mathbf{x}}_{\left(t\right)}+{\mathbf{W}}_{hi}·{\mathbf{h}}_{(t-1)}+{\mathbf{b}}_i\right)$$

(2)

$${\mathbf{f}}_{\left(t\right)}=\sigma \left({\mathbf{W}}_{xf}·{\mathbf{x}}_{\left(t\right)}+{\mathbf{W}}_{hf}·{\mathbf{h}}_{(t-1)}+{\mathbf{b}}_f\right)$$

(3)

$${\mathbf{o}}_{\left(t\right)}=\sigma \left({\mathbf{W}}_{xo}·{\mathbf{x}}_{\left(t\right)}+{\mathbf{W}}_{ho}·{\mathbf{h}}_{(t-1)}+{\mathbf{b}}_o\right)$$

(4)

$${\mathbf{g}}_{\left(t\right)}=tanh\left({\mathbf{W}}_{xg}·{\mathbf{x}}_{\left(t\right)}+{\mathbf{W}}_{hg}·{\mathbf{h}}_{(t-1)}+{\mathbf{b}}_g\right)$$

(5)

$${\mathbf{c}}_{\left(t\right)}={\mathbf{f}}_{\left(t\right)}\otimes {\mathbf{c}}_{(t-1)}+{\mathbf{i}}_{\left(t\right)}\otimes {\mathbf{g}}_{\left(t\right)}$$

(6)

$${\mathbf{y}}_{\left(t\right)}={\mathbf{h}}_{\left(t\right)}={\mathbf{o}}_{\left(t\right)}\otimes tanh\left({\mathbf{c}}_{\left(t\right)}\right)$$

(7)

among them, the logistic function ($\sigma$) and hyperbolic tangent function (tanh) are defined as

$$\sigma \left(x\right)=\frac{1}{1+e^{-x}}$$

(8)

$$tanh\left(x\right)=2\sigma \left(2x\right)-1$$

(9)

The four matrices ${\mathbf{W}}_{xf}$, ${\mathbf{W}}_{xg}$, ${\mathbf{W}}_{xi}$, ${\mathbf{W}}_{xo}$ are the weights for the four layers that connect to input vector $x_{\left(t\right)}$. Next, ${\mathbf{W}}_{hf}$, ${\mathbf{W}}_{hg}$, ${\mathbf{W}}_{hi}$, ${\mathbf{W}}_{ho}$ are the weight matrices of these four layers for the connection to the previous short-term state $h_{(t-1)}$. Finally, ${\mathbf{b}}_f$, ${\mathbf{b}}_g$, ${\mathbf{b}}_i$, and ${\mathbf{b}}_o$ are the bias terms of the four layers. Thanks to the long-term memory ${\mathbf{c}}_{\left(t\right)}$ and short-term memory ${\mathbf{h}}_{\left(t\right)}$ cells, the LSTM model can recognize important features from the input data, store them in the long-term state, preserve them for as long as needed, and use them for prediction whenever it is needed [22].

3.3. Model Architecture

We developed the storm surge forecast model using the Python package Keras (https://keras.io/, accessed on 15 May 2021), which is an open-source software library that provides a Python interface for artificial neural networks [25]. The computation backend for Keras used in this study is TensorFlow [26] since Keras does not carry out the computation. The software version of Keras 2.4.3 was used in this study. We used Keras Tuner (https://keras.io/keras_tuner/, accessed on 15 May 2021) of [27] to select optimized hyperparameters (e.g., the number of neural network units per layer) to minimize the validation loss.

The proposed model architecture can be found in

Table 3. Summary of the proposed encoder–decoder storm surge forecast model with a lead time of 6 h. The first column is the layer index, the second is the layer type, the third is the output shape of each layer, and the fourth is the number of training parameters.

Layer #	Layer Type	Output Shape	# of Parameters
1	InputLayer	(none, 6, 8)	0
2	LSTM	(none, 50)	12,200
3	Flatten	(none, 50)	0
4	RepeatVector	(none, 6, 50)	0
5	LSTM	(none, 6, 50)	20,200
6	Dense	(none, 6, 50)	2550
7	Dense	(none, 6, 3)	153

. This example shows the forecast lead time of 6 h with an input length of 6 h. The total number of trainable parameters is 35,103. The same model architecture was applied to other forecast lead times (i.e., 3 h and 12 h) by replacing the length “6” with other lengths accordingly. A further description of the neural network model is provided below.

• The input layer instantiates a 3D tensor that has the shape/size of the input training data. The first dimension of (None, 6, 8) is the batch size of the training sample. Since it is determined at the run-time and not pre-defined, a “none” indicates that the size will be determined later. The second dimension is the number of input data lengths (i.e., 6 h) in the training data. The third dimension is the number of input features. Table 2 shows the seven variables, and the eighth variable is the distance between the storm center and the save point. As a result, the number of features is eight in this study.
• The second layer is the encoder layer: LSTM. This study applied 50 neural units. The activation function is the Rectified Linear Unit function (ReLU). The kernel is initialized using the He initialization of [28], and the l2 kernel regularization with a value of 0.01 is also applied in this layer.
• The third layer is a flatten layer, which transforms a multiple-dimensional matrix of features into one-dimensional data.
• The fourth layer is a RepeatVector operation, which repeats the input from the previous layer by a factor that matches the length of the output sequence.
• The fifth layer is the decoder layer: LSTM. This study specified 50 units in the LSTM layer. As a result, the output dimension has the shape of (none, 6, 50) (both “none” and “6” have the same meaning explained in previous layers).
• The sixth hidden layer is a fully connected dense layer, and it has 50 units, as indicated in the output shape. We also applied the TimeDistributed operation to preserve the order of data sequence, which is essential for time-series forecast.
• Finally, the last layer is a fully connected dense layer that matches the target output sequence shape (6, 3). The output length is 6 for this example, and “3” indicates the target storm surge forecast variables: surge water level and velocity in the x and y directions, respectively.

3.4. Data Preparation

It is noted that we have only 981 training storms to develop the storm surge forecast model. We divided each time-series data, including storm parameters and storm surge results, into many sub-samples according to the input and output length by following the data preparation approach proposed in Wei [21]. We will illustrate it by considering a 5-day storm with 120 records with one hour per record given a forecast lead time of 6 h. The first sample has input data with indices of 1 to 6 and target data with indices of 7 to 12. The next sample has input data with indices of 2 to 7 and target data with indices of 8 to 13. As a result, we would generate 114 samples with the limited 120 records.

In addition, we need to address the issue that the input data have different magnitudes (e.g., the magnitude of surge water level is a few meters, but the pressure is around a thousand hPa). To improve the training convergence of our storm surge model, we applied a data normalization step by first removing the mean and then dividing it by its standard deviation. The readers are referred to Wei [21] and Wei and Davison [29] for more details. This data pre-processing step does not change the statistical distribution of input variables. We will re-scale the results back to the real magnitude during the model forecast stage.

4. Results

This section first presents the training metrics of the neural network storm surge forecast model. Then its computational efficiency is discussed. Next, we present the aggregate model forecast skills for three lead times using 50 test storms. Finally, we examine the forecast of time-series storm profiles for both bypassing and landfalling storms.

4.1. Training Metrics

We first present our storm surge forecast model’s training metrics to demonstrate the model development’s convergence. Two metrics monitored the neural network model training: root mean squared error and R squared ($R^2$).

The definition of root mean squared error (RMSE) is given below:

$$RMSE=\sqrt{\left(\frac{1}{n}\right)\sum _{i=1}^n{(y_i-f_i)}^2}$$

(10)

where y is the true value; f is the prediction or forecast value; i is the index; and n is the number of samples.

The coefficient of determination, or $R^2$ is defined as

$$R^2=1-\frac{S{S}_{res}}{S{S}_{tot}}$$

(11)

where the sum of squares of residuals $S{S}_{res}$ is defined as

$$S{S}_{res}=\sum _{i=1}^n{\left(y_i-f_i\right)}^2$$

(12)

and the total sum of squares $S{S}_{tot}$ is defined as

$$S{S}_{tot}=\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2$$

(13)

with $\overline{y}=\frac{1}{n}{\sum }_{i=1}^n{y}_i$.

Figure 7 shows the comparison of neural network model training and validation metrics concerning the number of epochs for save point SP13205 in New York. We also present the training results for three forecast lead times ranging from 3 h to 12 h. The results for the three scenarios show that the RMSE gradually reduces as the number of epochs increases, while the $R^2$ gradually increases as the number of epochs increases. The final convergent RMSE and $R^2$ values differ slightly among the three lead times. It is seen that the shorter the forecast time, the smaller the RMSE, and the higher the $R^2$. This is because a long-term forecast becomes more challenging. This is especially true when a storm can rapidly change its path and strength over a longer period, e.g., 12 h. Nevertheless, the results show that both training and validation curves reach a similar converged value for both RMSE and $R^2$, indicating that the storm surge model does not have under-fitting or over-fitting issues.

4.2. Computational Efficiency

We present the computational cost of our neural network model in this section. In general a neural network model is more computationally efficient than a traditional physics-based model (e.g., [29]). The storm surge forecast model was developed based on a computing system with a CPU model of Intel(R) Xeon(R) CPU @ 2.00 GHz. The CPU clock speed is 2K MHz, and the CPU cache size is 39,424 KB. We used two CPUs for neural network model training and forecast.

Table 4. The average computation time to train the neural network model for 50 epochs and the computation time to run the neural network model for 50 test storms using three forecast lead times (i.e., 3 h, 6 h, and 12 h).

Forecast Lead Time (h)	Model Training Time (s)	Model Test Time (s)
3	1275.38	78.55
6	1897.52	36.05
12	2625.88	18.43

shows the computation time for model training and model forecast. We present the results for three forecast lead times (i.e., 3 h, 6 h, and 12 h) at save point SP13205 in New York. Regarding the computation time to train the model using 981 storms, it only took about 20 min to train the neural network model for 50 epochs for the forecast lead time of 3 h and about 40 min for a longer forecast lead time of 12 h. Figure 7 shows that the model training converged very quickly. Therefore, it is likely that 25 epochs would have been sufficient to train a skillful storm surge model. As a result, we expect the training time to be even shorter. Table 4 also shows that the total computation time to forecast 50 test storm ranges from 18 s to 80 s, or 0.36 s to 1.6 s per storm. Considering a 5-day storm, it would take 10 forecasts to complete using a forecast lead time of 12 h, while it will take 40 forecasts if the lead time is 3 h. Therefore, the computation forecast time is shorter when using a longer forecast lead time. This is indeed a significant achievement to forecast a storm in a fraction of one second using a neural network since it would require hours of computation to run ADCIRC on a high-performance computing cluster for a single storm.

4.3. Model Forecast Skills

The developed storm surge forecast model was tested with 50 storms which were not seen by the model during the training stage. Time-series profiles of storm induced water level, velocities in both x and y directions were predicted. The complete storm surge results for 50 test storms were included as the supplementary data of this article. This section presents the aggregate results using all 50 test storms to demonstrate their forecast skills. The ADCIRC simulation was used as the reference (or ground truth) to determine the forecast metrics discussed in this section. For the same test storm, the predicted time-series profile length would be different when the lead time changes from, for example, 3 h to 12 h. As a result, the calculation of RMSE and $R^2$ used the common data length so that the comparison is meaningful among different lead times.

Table 5. Summary of root mean squared error (RMSE) between neural network forecast and ADCIRC simulation for water level (WL) in meters, velocity in the x direction (VX) in m/s, and velocity in the y direction (VY) in m/s for three save points using three forecast lead times (i.e., 3 h, 6 h, and 12 h) using 50 test storms.

		3 h	6 h	12 h
SP17771	WL	0.024	0.026	0.029
	VX	0.017	0.021	0.029
	VY	0.018	0.022	0.032
SP13205	WL	0.021	0.023	0.025
	VX	0.009	0.013	0.014
	VY	0.009	0.012	0.016
SP06815	WL	0.029	0.033	0.039
	VX	0.006	0.008	0.011
	VY	0.006	0.010	0.011

shows the summary of RMSE between neural network forecast and ADCIRC simulation for water level, velocity in x and y directions for three save points using three forecast lead times. The RMSE of water level forecast at SP17771 in Virginia ranges from 0.024 m using a 3 h forecast lead time to 0.029 m using a 12 h forecast lead time; the RMSE of velocity forecast in the x direction at SP17771 ranges from 0.017 m/s using a 3 h forecast lead time to 0.029 m/s using a 12 h forecast lead time. The RMSE of velocity forecast in the y direction at SP17771 is between 0.018 m/s using a 3 h forecast lead time and 0.032 m/s using a 12 h forecast lead time. The RMSE difference among the three forecast lead times is relatively small at SP17771. Since the RMSE depends on the magnitudes of water level and velocity at each save point, the RMSE is expected to be smaller if the magnitude is smaller. As a result, we would expect a difference in RMSE when considering another save point. For example, the RMSE of velocity at SP06815 is smaller than that at SP17771 because a storm weakens as it moves from Virginia to Maine. The comparison also shows that the neural network model performs better when the forecast lead time is shorter. This is consistent with our previous finding in Wei [21] where we tested different lead times when forecasting wind waves using an artificial intelligence model.

Table 6. Summary of R squared ($R^2$) between neural network forecast and ADCIRC simulation for water level (WL), velocity in the x direction (VX), and velocity in the y direction (VY) for three save points for three forecast lead times (i.e., 3 h, 6 h, and 12 h) using 50 test storms.

		3 h	6 h	12 h
SP17771	WL	0.97	0.96	0.95
	VX	0.95	0.92	0.85
	VY	0.98	0.96	0.93
SP13205	WL	0.97	0.96	0.96
	VX	0.98	0.96	0.95
	VY	0.97	0.95	0.90
SP06815	WL	0.94	0.92	0.89
	VX	0.98	0.97	0.94
	VY	0.97	0.94	0.92

shows the summary of $R^2$ between neural network forecast and ADCIRC simulation for water level, velocity in x and y directions at three locations using three forecast lead times. The $R^2$ metric has removed the sample’s mean in its calculation. As a result, the magnitudes of water level and velocity do not influence $R^2$. The $R^2$ results at three save points are very similar, and we will use SP06815 in Maine as an example to explain the variation. The $R^2$ ranges from 0.89 using a 12 h forecast lead time to 0.94 using a 3 h forecast lead time. The smallest $R^2$ for velocity in the x direction is about 0.94 using a 12 h forecast lead time, which is very good considering that a forecast becomes more challenging as the forecast lead time increases. Similarly, the forecast of velocity in the y direction has the smallest $R^2$ of 0.92 using a 12 h forecast lead time, and the largest $R^2$ is about 0.97 when using a 3 h forecast lead time. Overall, the $R^2$ comparison shows that the storm surge simulated by ADCIRC is well-replicated by the neural network storm surge forecast model.

Furthermore, we evaluate the storm surge model forecast skill using scatter plots in Figure 8. The simulated storm surge results by ADCIRC are presented on the horizontal axis against the neural network model forecast on the vertical axis. Each subplot includes the comparison for all three save points. This analysis used time-series storm surge results for all 50 test storms. The three columns from left to right compare storm-induced surge level, velocity in the x direction, and velocity in the y direction. The three rows from top to bottom compare the forecast lead times: 3 h, 6 h, and 12 h, respectively. The water level comparison (left column) shows that the storm surge forecast by the neural network model clusters very well with the simulation by ADCIRC for all forecast lead times. Only a few peak surges were under-predicted by the neural network. The level of agreement at three save points is very similar. The comparison of velocity in the x direction (middle column) shows that the neural network model forecast and ADCIRC simulation match well for SP13205 in New York and SP06815 in Maine. At the same time, there is a bigger discrepancy when using a 12 h forecast lead time for SP17771 in Virginia. The comparison of velocity in the y direction (right column) has a similar agreement as the velocity in the x direction. It also shows that the velocity has a wider range of [−0.5, 0.5] m/s at SP17771 than [−0.3, 0.3] m/s at SP13205 and SP06815. In summary, the scatter plot comparison shows that the neural network storm surge forecast model is capable of predicting storm surges, including water levels and velocities, as simulated by the physics-based storm surge model ADCIRC.

4.4. Bypassing and Landfalling Storm Forecast

This section presents the neural network forecast capability of bypassing and landfalling storms with a focus on time-series prediction. The neural network forecast time-series results is compared with ADCIRC simulation at three save points.

4.4.1. Storm Selection

This section selects four test storms and presents the detailed storm surge forecast results. Figure 9 shows the selected four storms tracks, and

Table 7. Four representative test storms for time-series storm surge analysis at three save points. There are one bypassing storm and three landfalling storms, with each making landfall near one save point.

Storm ID	Storm Type	Save Points
1010	Bypassing	SP17771, SP13205, SP06815
33	Landfalling	SP17771
418	Landfalling	SP13205
705	Landfalling	SP06815

lists the storm type (either bypassing or landfalling) and the save points impacted by each storm. Storm 1010 was a bypassing storm that passed by three save points from Virginia to Maine. We further select three landfalling storms. Storm 33 made landfall near SP17771 in Virginia, storm 418 made landfall near SP13205 in New York, and storm 705 made landfall near SP06815 in Maine.

4.4.2. Bypassing Storm Results

Figure 10 shows the time-series storm center, the central pressure, and the radius of the maximum winds of the bypassing storm 1010. The storm had a relatively low central pressure of about 950 hPa, and the radius of the maximum winds was about 150 km. The storm maintained a constant strength, and it started to weaken around 12:00 on 14 July 2020. Figure 11 shows the comparison of the forecast storm surge water levels and velocities with ADCIRC simulation. The time-series results for all save points are presented because this storm passes by all of them. Furthermore, we also compare the difference among three forecast lead times (i.e., 3 h, 6 h, and 12 h). The considered duration for this storm is about four days. The water level was constant before the arrival of the storm because the tidal effect was not considered in the storm surge dataset. The storm first impacted SP17771 in Virginia around 12 July 2000. Since then, the water level gradually rose till the storm reached the save point before 14 July 2000. The highest water level induced by this storm was about 0.5 m. The storm caused a continuing increase of velocity toward the east and a minor disturbance of velocity in the north-south direction. Once the storm passed by SP17771, the surge water level dropped very quickly to the pre-storm level. The velocity in the north direction increased as the far-field winds caused the water level to drop below mean sea level and pushed the water to move toward the north. The comparison shows that storm surge forecast using 3 h and 6 h lead times predicts both surge water levels and velocities well. The neural network storm surge forecast model using a longer forecast lead time, e.g., 12 h, does not perform as well as a shorter lead time.

Storm 1010 started to impact save point SP13205 in New York around 13 July 2000, as its water level increased to 0.4 m. Then the water level dropped to 0.15 m around 14 July 2000. It is noted that a drop in velocity in both directions occurred after the drop in water level. The maximum storm surge was about 0.5 m when the storm arrived at the save point. Next, we can see the decrease in water level and the increase in velocity as the storm moved away from the save point. The storm surge forecast model not only predicts the peaks of surge and velocity, but more importantly, it captures the time-varying processes very well, as simulated by ADCIRC.

The save point SP06815 in Maine experienced the impact of storm 1010 shortly after 13 July 2000. The surge pattern is similar to the one at SP13025. The water level dropped before the storm reached the save point. The maximum surge water level was about 0.55 m, which occurred before the storm reached the closest location to the save point. The induced velocities in the x and y directions had a similar pattern, indicating that the water near the save point was pushed toward the southwest. The comparison at this save point shows that the storm surge forecast model correctly predicts the time-varying storm surge water level and velocities.

In summary, the bypassing storm 1010 has caused complex storm surge patterns at three save points from Virginia to Maine. The neural network storm surge forecast model is able to predict both time-varying processes and peak values satisfactorily. Storm surge forecast using a shorter forecast lead time usually performs better. Forecast accuracy reduces when the lead time is longer.

4.4.3. Landfalling Storm Results

Figure 12 shows the time-series storm center position, the central pressure, and the radius of the maximum winds of three landfalling storms, 33, 418, and 705. It shows that the three storms had different strengths and sizes before they became weakening. Specifically, the central pressure was about 965, 965, and 955 hPa for storms 33, 418, and 705, respectively. Storm 33 had the smallest radius of the maximum winds of 51 km, and the radius of the maximum winds for storm 418 was about 94 km. However, the radius of the maximum winds for storm 705 was about 151 km, about three times of storm 33, and about one and half times of storm 418.

Figure 13 shows the comparison of forecast storm surge water levels and velocities with ADCIRC simulation for three landfalling storms. Storm 33 approached SP17771 from the southeast direction, passed by this save point around 18:00 on 14 July 2020, and made landfall in Virginia. The storm surge continued to increase as the storm moved closer to the save point, while the velocities decreased in both directions, indicating that the water was pushed toward the southwest. The maximum surge of 0.7 m was generated when the storm passed by the save point. Then storm surge decreased as the storm passed, and the far-field winds pushed the water toward the northeast. The neural network storm surge forecast model with three lead times correctly predicted the induced surge and velocities. In particular, the forecast using a short lead time (e.g., 3 h) captured surge peaks very well.

Storm 418 made landfall in New York, and its center passed by SP13205 around 19:00 on 14 July 2000. The maximum surge generated by this storm was about 0.7 m. The strong wind field first pushed the water toward the southeast and then northeast. In general, the induced storm surge pattern for water levels and velocities is very similar to the one at SP17771. Despite some differences among forecast lead times, the neural network forecast model was able to predict storm-induced water levels similar to the ADCIRC simulation. Storm 705 made landfall in Maine, and its center passed by SP06815 around 0:00 on 15 July 2020. The induced peak storm surge was about 0.65 m. The velocity comparison shows that the water was pushed southwest before the arrival of the storm. The neural network model was able to predict the surge and velocity well.

In short, landfalling storms caused different surge and velocity patterns from Virginia to Maine. The neural network storm surge forecast model captured the time-varying and peak surge and velocity induced by landfalling storms.

5. Discussion and Conclusions

This study presented an artificial neural network model to predict storm surges on the US North Atlantic Coast from Virginia to Maine. A storm surge dataset from the North Atlantic Comprehensive Coastal Study was used to develop and evaluate the neural network storm surge forecast model. The neural network model consists of an encoder layer and a decoder layer. The encoder layer encodes the input sequence into an internal state, while the decoder layer decodes the internal state to the output sequence. The long short-term memory model was used as the encoder and decoder layers. The model was trained with 981 synthetic storms. The input features of the neural network model include times-series storm position (i.e., latitude and longitude), the central pressure, and the radius of the maximum winds. The neural network model forecasts storm-induced water levels and velocities. Three forecast lead times were tested. The trained model was evaluated using 50 test storms for time-series forecast. In particular, we analyzed the model forecast for both bypassing and landfalling storms. The conclusions and major contributions of this study are summarized as follows:

• Computational efficiency. The neural network convergence was evaluated by two metrics, root mean squared error and R squared. The results showed that the model converged quickly, and does not have over-fitting or under-fitting issues. The neural network model is very computationally efficient, takes less than one second to forecast a storm.
• Time-series forecast. The neural network model not only forecasts peak surges, but also predicts the time-varying process of a storm, including the development and recession of storm surges. What is more, the neural network model can forecast storm-induced velocity, which is important to determine storm damage, such as shoreline and beach erosion.
• Forecast lead times. Forecast lead time was achieved by training the neural network model to understand the correlation between past storm conditions and future storm hazards. Three forecast lead times (i.e., 3 h, 6 h, and 12 h) were tested. The results showed that the neural network model is skillful in both short-term and long-term forecasts.
• Forecast skills. Both bypassing and landfalling storms were used to evaluate the neural network model. The results showed that the model is highly versatile, and it is able to forecast a wide range of storms, including different tracks, sizes, and strengths. The neural network model was evaluated by calculating the root mean squared error and R squared using its forecast and ADCIRC simulation. The comparison showed that the neural network model is able to reproduce storm surges and velocities as simulated using a physics-based model reasonably well.

In conclusion, the proposed neural network storm surge forecast model is a promising technique for forecasting storm surges. Its accuracy is comparable with a physics-based storm surge model, while it is much more computationally efficient than traditional coastal models. Furthermore, this work demonstrated the success of data-driven approaches to transform coastal hazard research.

Nevertheless, there are a few limitations that need be improved in the future. First, the neural network was trained to forecast storm surges in the nearshore zone, where the time-series profile is continuous, and effort is needed to enhance it to predict storm surge overland where the surge profile is, in general discontinuous. Second, the neural network model cannot consider the tidal effect because the storm surge dataset did not consider this effect. Third, the way to divide the dataset into model training and testing may influence the forecast skills since an AI model depends on the training dataset. These issues will be addressed in our future studies.