# Forecasting of Extreme Storm Tide Events Using NARX Neural Network-Based Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Extreme Events

^{2}. Only the 8% is represented by land, including the city of Venice and many smaller islands while about 11% is permanently covered by open water and 80% consists of mudflats, tidal shallows, and salt marshes. The lagoon is connected to the Adriatic Sea by means of three inlets: Lido, Malamocco, and Chioggia (Figure 1). In ordinary conditions, the observed sea level in the Lagoon is little different from that induced by the astronomical tide since meteorological effects are small. In the case of particularly adverse weather conditions, with significant drop pressure and strong sirocco wind, the meteorological effects become significant: if they are in phase with a high astronomical tide, they can lead to the phenomenon of high water, which is conventionally defined as a tide level greater than 80 cm with reference to the Punta della Salute tide gauge. At this tide level, the flooding of the lower part of the city begins. The storm surge may be enhanced by a severe low pressure on the upper Adriatic and contemporary high pressure on the lower Adriatic. The situation worsens further if a strong wind from North-East (Bora) blows in the northern Adriatic together with sirocco wind in middle Adriatic, leading to the convergence of wind-induced marine currents. In addition, due to the shape of the Adriatic Sea, a storm surge leads to the formation of seiches whose amplitude is progressively decreasing [25]. The concomitant action of the above factors can lead to extreme values of storm tide levels, as observed more and more frequently in recent decades.

**Figure 1.**Location of the Punta della Salute tide gauge ( ) and Piattaforma CNR weather station ( ), with a thematic map of the Venice Lagoon [27].

#### 2.2. NARX Model Architectures

_{u}and n

_{y}are the input and output network layers, and f is the non-linear function, approximating by the Feedforward Neural Network (FNN).

_{1}was used, while a linear activation function f

_{2}, with only one neuron n, was used for the output layer. A preliminary analysis was carried out to assess the optimal number of hidden nodes, which was found to be equal to 3 (Figure 2, indicated as h

_{1}, h

_{2}, and h

_{3}). For the output layer, the weight w and bias b of the NARX model were optimized based on the Bayesian Regularization training algorithm. The NARX process was stopped if one of the following parameters was reached: the maximum number of epochs, settled equal to 1000; the Levenberg–Marquardt adjustment parameter, settled equal to 1 × 10

^{−10}; the error gradient was below a minimal value, settled equal to 1 × 10

^{−7}.

^{®}2020a environment [31]: Model A and Model B. The two models were different in the input variables. Both the models included the lagged values of the water level h

_{tide}(t − t

_{a}) among the inputs.

_{astr}(t − t

_{a}), wind speed v

_{wind}(t − t

_{a}), wind direction α

_{wind}(t − t

_{a}), and barometric pressure P

_{atm}(t − t

_{a}) as exogenous input parameters, whereas Model B considered only h

_{astr}(t − t

_{a}) as an exogenous input variable. The astronomical tide h

_{astr}(t − t

_{a}) values were computed through the following:

_{0}is the average sea level, A

_{n}is the amplitude, σ

_{n}the angular frequency, k

_{n}the phase delay of component n, and N is the number of harmonics used to evaluate the astronomical tide height. These values can be found on the Venice Municipality website [32].

_{a}values were considered, in order to assess the performance of the models as t

_{a}increases: 12, 24, 48, and 72 h.

#### 2.3. Evaluation Metrics

^{2}, which assess how well the model replicates observed outcomes and predicts future outcomes, the Mean Absolute Error (MAE), which provides the average error magnitude for the predicted values, and the Relative Absolute Error (RAE), equal to the ratio between absolute error and absolute value of the difference between average and each measured values. These metrics are defined as:

_{tide}equal or greater than 110 cm.

## 3. Results and Discussion

#### 3.1. Training and Testing

_{a}, equal to 12 h (Model A—R

^{2}= 0.950, MAE = 1.96 cm and RAE = 23.03%; Model B—R

^{2}= 0.941, MAE = 2.14 cm, and RAE = 25.13%). The prediction performance reduces as the lag time increases, with the NARX models providing the less accurate forecasts for t

_{a}= 72 h (Model A—R

^{2}= 0.899, MAE = 2.91 cm, and RAE = 34.25%; Model B—R

^{2}= 0.860, MAE = 3.29 cm, and RAE = 38.75%). In Figure 3 the comparison between measured and predicted extreme storm tide levels for both Model A and Model B, and for different lag times is reported, highlighting a slight tendency to underestimate these extreme events. Overall, regardless of the lag time, the NARX network-based models are able to forecast extreme storm tide events, including the exceptional storm tides. Moreover, the model performance was not significantly affected by the additional input parameters: Model B is only slightly less accurate than model A. This obviously does not mean that the weather parameters have little influence on extreme events: the fundamental influence of the weather parameters is taken into account by means of the lagged values of the storm tide level [24].

#### 3.2. Time Series Analysis

_{a}, equal to 12 h, with an overestimation of the storm tide equal to 0.29 cm. Model B also provides very good outcomes: the less accurate forecast was obtained for t

_{a}= 24 h, with an underestimation of 3.08 cm. For the third peak, equal to 129 cm, a very effective prediction was achieved with Model A also for t

_{a}= 72 h, with a slight underestimation of just 0.02 cm. For the same lag time, Model B provides a very accurate forecast too, with an underestimation of 0.61 cm. For the exceptional tide peak = 185 cm, the best forecast was achieved with Model A and t

_{a}= 48 h, with an underestimation of 0.2 cm. However, Model B also provided accurate predictions for all lag times, and the worst forecast was obtained for t

_{a}= 24 h, with an overestimation of 5.99 cm. Following this peak, there is a further exceptional storm tide, corresponding to a 145 cm sea level, the last of the considered period. For the latter, the best forecasts were again achieved with Model A, with an underestimation of 0.27 cm for t

_{a}= 24 h. However, accurate predictions were also obtained with Model B, with an underestimation of 0.28 cm in the case of t

_{a}= 48 h.

#### 3.3. Sensitivity Analysis

^{2}= 0.860, MAE = 3.29 cm, and RAE = 38.75%). As expected, a performance reduction was observed as the length of the time series reduced (Table 2), with the worst predictions obtained with the shorter training time series (R

^{2}= 0.820, MAE = 3.96 cm, and RAE = 40.57%).

_{tide}higher than 140 cm. As confirmed by the metrics, reported in Table 3, both models A and B were able to provide accurate predictions of this event, for all lag times.

_{tide}higher than 110 cm) before and after the exceptional storm tides. Additionally, in this case, predictions were very accurate, with a slight underestimation of the exceptional storm tides for Model B and t

_{a}= 72 h.

#### 3.4. Comparison with Other Models

_{m}indicates the mean error and 2σ the confidence interval, with σ representing the standard deviation of the errors, i.e., the differences between predicted and measured levels.

_{tide}in the range of 80 to 100 cm, SHYFEM model highlights mean errors which, for t

_{a}= 72 h, reach values closer to 0 with respect to the NARX models (Model A—ε

_{m}= −0.9 cm, SHYFEM 6—ε

_{m}= −0.3 cm). However, both NARX-based models exhibit significantly narrower confidence interval 2σ in comparison with SHYFEM and statistical models. In particular, for h

_{tide}between 80 and 100 cm, the lower value of 2σ was computed for Model A and t

_{a}= 24 h, equal to 5.5 cm (Model B and t

_{a}= 24 h—2σ = 6.1 cm). Considering the same h

_{tide}interval and lag time, deterministic and statistical models shows a wider confidence interval, with 2σ equal to 12.9, 16.8, 16.1, and 16.6 cm, respectively, for SHYFEM 5, SHYFEM 6, STAT-1, and STAT-2. Moreover, NARX models do not show a marked increase in 2σ as the lag time increases, with 2σ = 6.1 cm for Model A and t

_{a}= 72 h (Model B and t

_{a}= 72 h—2σ = 6.2 cm). The same cannot be said for the deterministic models that highlight a relevant increase in 2σ as the lag time increases, with 2σ equal to 18.5 cm (SHYFEM 5) and 20.9 cm (SHYFEM 6) for t

_{a}= 72 h.

_{tide}> 120 cm, STAT-1 and STAT-2 show 2σ values, respectively, equal to 27 and 26.2 cm. Even the deterministic models exhibit a widening of the confidence interval. For t

_{a}= 24 h, 2σ equal to 26.2 and 26.1 cm were, respectively, computed for SHYFEM 5 and SHYFEM 6. As the lag time increases, instead, SHYFEM 5 and SHYFEM 6 show an opposite trend, with a narrowing of the confidence interval for the first model (2σ = 16.8 cm) and a widening of the same for the second model (2σ = 47.0 cm). NARX-based models instead are characterized by confidence intervals similar to those computed for h

_{tide}between 80 and 100 cm. As for Model A, 2σ values were equal to 5.1 and 5.2 cm, respectively, for lag times of 24 and 72 h. Model B shows a greater widening of the confidence interval as the lag time increases, passing from 5.3 cm for t

_{a}= 24 h to 7 cm for t

_{a}= 72 h.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Ensemble Model

_{a}= 12 h the performances achieved by each network were always high, with R

^{2}values always higher than 0.93 for both Model A and Model B. A reduction in the performance was observed as the lag time increases. However, R

^{2}values were still higher than 0.76 for all network and for both Model A and Model B highlighting how, despite a long horizon forecasting, NARX modeling was still able to provide accurate predictions.

_{a}= 12 h (Model A—R

^{2}= 0.951, MAE = 1.81 cm, and RAE = 21.73%; Model B—R

^{2}= 0.951, MAE = 1.85 cm, and RAE = 22.13%). In agreement with the training and testing stage reported in Section 3.1, as the lag time increased, a reduction in the prediction performance was observed (for t

_{a}= 72 h, Model A—R

^{2}= 0.882, MAE = 3.06 cm, and RAE = 36.66%; Model B—R

^{2}= 0.863, MAE = 3.25 cm, and RAE = 38.99%). In Figure A2, the comparison between measured and predicted storm tides for the ensemble models (both Model A and Model B) and for different lag times is reported. All individual networks and the ensemble model showed absolutely comparable performance, ensuring the temporal transferability of NARX-based models.

Model | Net | t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE (cm) | RAE (%) | R^{2} | MAE (cm) | RAE (%) | R^{2} | MAE (cm) | RAE (%) | R^{2} | MAE (cm) | RAE (%) | ||

Model A | 1 | 0.942 | 2.00 | 23.92 | 0.892 | 2.82 | 33.76 | 0.840 | 3.44 | 41.17 | 0.839 | 3.17 | 38.00 |

2 | 0.943 | 2.06 | 24.67 | 0.916 | 2.63 | 31.55 | 0.888 | 2.88 | 34.55 | 0.871 | 3.03 | 36.34 | |

3 | 0.955 | 2.16 | 25.89 | 0.911 | 2.72 | 32.57 | 0.877 | 2.96 | 35.46 | 0.841 | 3.15 | 37.72 | |

4 | 0.940 | 2.09 | 25.09 | 0.916 | 2.52 | 30.26 | 0.896 | 2.64 | 31.63 | 0.843 | 3.42 | 41.02 | |

5 | 0.941 | 2.00 | 23.99 | 0.910 | 2.74 | 32.84 | 0.896 | 2.68 | 32.11 | 0.880 | 2.88 | 34.55 | |

6 | 0.942 | 1.96 | 23.44 | 0.905 | 2.96 | 35.52 | 0.902 | 2.88 | 34.56 | 0.832 | 3.24 | 38.87 | |

7 | 0.954 | 2.11 | 25.30 | 0.903 | 2.98 | 35.68 | 0.877 | 2.96 | 35.53 | 0.782 | 3.85 | 46.13 | |

8 | 0.941 | 1.99 | 23.86 | 0.902 | 2.99 | 35.80 | 0.880 | 2.93 | 35.09 | 0.888 | 2.89 | 34.64 | |

9 | 0.941 | 2.09 | 25.03 | 0.913 | 2.69 | 32.18 | 0.887 | 2.86 | 34.26 | 0.886 | 3.03 | 36.31 | |

10 | 0.942 | 2.06 | 24.74 | 0.906 | 2.95 | 35.37 | 0.898 | 2.92 | 34.95 | 0.884 | 2.36 | 28.27 | |

Ensemble | 0.951 | 1.81 | 21.73 | 0.902 | 2.93 | 35.17 | 0.886 | 2.83 | 33.90 | 0.882 | 3.06 | 36.66 | |

Model B | 1 | 0.938 | 2.02 | 24.20 | 0.889 | 2.87 | 34.45 | 0.839 | 3.44 | 41.23 | 0.768 | 4.05 | 48.57 |

2 | 0.940 | 2.14 | 25.66 | 0.883 | 3.08 | 36.96 | 0.839 | 3.44 | 41.23 | 0.838 | 3.19 | 38.19 | |

3 | 0.938 | 2.01 | 24.13 | 0.889 | 2.85 | 34.17 | 0.849 | 3.38 | 40.50 | 0.762 | 4.19 | 50.16 | |

4 | 0.941 | 2.12 | 25.36 | 0.911 | 2.72 | 32.55 | 0.883 | 2.91 | 34.90 | 0.780 | 4.03 | 48.33 | |

5 | 0.939 | 1.99 | 23.90 | 0.899 | 3.03 | 36.25 | 0.876 | 2.98 | 35.69 | 0.820 | 3.62 | 43.35 | |

6 | 0.942 | 2.08 | 24.96 | 0.912 | 2.71 | 32.44 | 0.892 | 2.74 | 32.84 | 0.783 | 3.98 | 47.67 | |

7 | 0.941 | 2.10 | 25.23 | 0.902 | 2.99 | 35.83 | 0.838 | 3.18 | 38.09 | 0.774 | 4.05 | 48.56 | |

8 | 0.939 | 1.99 | 23.87 | 0.900 | 2.99 | 35.88 | 0.891 | 2.82 | 33.79 | 0.840 | 3.16 | 37.88 | |

9 | 0.940 | 2.14 | 25.67 | 0.902 | 2.97 | 35.61 | 0.897 | 2.63 | 31.48 | 0.841 | 3.11 | 37.64 | |

10 | 0.941 | 2.00 | 23.91 | 0.913 | 2.68 | 32.14 | 0.887 | 2.88 | 34.53 | 0.844 | 3.12 | 37.35 | |

Ensemble | 0.951 | 1.85 | 22.13 | 0.894 | 3.02 | 36.17 | 0.882 | 2.79 | 33.42 | 0.863 | 3.25 | 38.99 |

**Figure A2.**Comparison between measured and predicted storm tide for different lag times—Ensemble model.

## Appendix B. NARX Model with Only the Lagged Tide Level as Input Variable

_{tide}(t − t

_{a}) as input. As an example, Figure A3 reports the comparison between measured and predicted sea level for the same event investigated in Section 3.2, for which Models A and B showed good accuracy. Model C clearly failed to provide accurate predictions, leading to significant underestimation of the highest peaks. Table A2 reports some metrics that attest to the poor performance of Model C, with the best results for t

_{a}= 12 h (R

^{2}= 0.397, MAE = 17.77 cm, and RAE = 79.22%) and the worst ones for t

_{a}= 72 h (R

^{2}= 0.310, MAE = 18.72 cm, and RAE = 83.54%).

**Figure A3.**Extreme storm tide event forecasting for the period 11–14 November 2019, comparison between measured time series and predicted values: Model C.

Metric | Model C | |||
---|---|---|---|---|

t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | |

R^{2} | 0.397 | 0.377 | 0.358 | 0.310 |

MAE (cm) | 17.77 | 18.16 | 18.00 | 18.72 |

RAE (%) | 79.22 | 81.08 | 80.39 | 83.54 |

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**Figure 4.**Extreme storm tide events forecasting in the period 11–14 November 2019, comparison between measured time series and predicted values: Model A—t

_{a}= 12 h (

**a**); Model B—t

_{a}= 12 h (

**b**); Model A—t

_{a}= 24 h (

**c**); Model B—t

_{a}= 24 h (

**d**); Model A—t

_{a}= 48 h (

**e**); Model B—t

_{a}= 48 h (

**f**); Model A—t

_{a}= 72 h (

**g**); Model B—t

_{a}= 72 h (

**h**).

**Figure 5.**Comparison between measured and predicted storm tide—Sensitivity analysis to the training time series length.

**Figure 6.**Event 1, extreme storm tide event forecasting for the period 10–14 November 2012, comparison between measured time series and predicted values: Model B, t

_{a}= 12 h (on the

**left**), t

_{a}= 72 h (on the

**right**).

**Figure 7.**Event 2, extreme storm tide event forecasting for the period 27 October–3 November 2018, comparison between measured time series and predicted values: Model B, t

_{a}= 12 h (on the

**left**), t

_{a}= 72 h (on the

**right**).

**Figure 8.**Event 3, extreme storm tide event forecasting for the period 20–29 December 2019, comparison between measured time series and predicted values: Model B, t

_{a}= 12 h (on the

**left**), t

_{a}= 72 h (on the

**right**).

Metric | Model A | Model B | ||||||
---|---|---|---|---|---|---|---|---|

t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | |

R^{2} | 0.950 | 0.923 | 0.911 | 0.899 | 0.941 | 0.905 | 0.888 | 0.860 |

MAE (cm) | 1.96 | 2.46 | 2.91 | 2.91 | 2.14 | 2.55 | 3.05 | 3.29 |

RAE (%) | 23.03 | 28.95 | 34.21 | 34.25 | 25.13 | 29.97 | 35.87 | 38.75 |

Time Series Length for the Training | t_{a} (h) | R^{2} | MAE (cm) | RAE (%) |
---|---|---|---|---|

January 2009–June 2012 (Full length) | 12 | 0.941 | 2.14 | 25.13 |

24 | 0.905 | 2.55 | 29.97 | |

48 | 0.888 | 3.05 | 35.87 | |

72 | 0.860 | 3.29 | 38.75 | |

January 2010–June 2012 | 12 | 0.937 | 2.19 | 25.25 |

24 | 0.905 | 3.14 | 33.66 | |

48 | 0.869 | 3.18 | 36.16 | |

72 | 0.844 | 3.43 | 39.12 | |

January 2011–June 2012 | 12 | 0.924 | 2.61 | 26.14 |

24 | 0.898 | 3.25 | 35.72 | |

48 | 0.863 | 3.23 | 38.03 | |

72 | 0.832 | 3.56 | 40.06 | |

January 2012–June 2012 | 12 | 0.902 | 2.67 | 30.04 |

24 | 0.891 | 3.54 | 37.12 | |

48 | 0.844 | 3.71 | 38.36 | |

72 | 0.820 | 3.96 | 40.57 |

Event | Metric | Model A | Model B | ||||||
---|---|---|---|---|---|---|---|---|---|

t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | t_{a} = 12 h | t_{a} = 24 h | t_{a} = 48 h | t_{a} = 72 h | ||

Event 1 (length 48 h) | R^{2} | 0.995 | 0.993 | 0.990 | 0.989 | 0.995 | 0.989 | 0.989 | 0.975 |

MAE (cm) | 2.01 | 2.66 | 3.35 | 3.56 | 2.04 | 3.53 | 3.44 | 4.68 | |

RAE (%) | 6.59 | 8.71 | 10.99 | 11.68 | 6.69 | 11.58 | 11.28 | 15.33 | |

Event 2 (length 168 h) | R^{2} | 0.992 | 0.989 | 0.987 | 0.984 | 0.987 | 0.982 | 0.975 | 0.974 |

MAE (cm) | 2.43 | 2.70 | 3.24 | 3.53 | 2.76 | 3.65 | 4.17 | 4.09 | |

RAE (%) | 9.30 | 10.33 | 12.37 | 13.49 | 10.57 | 13.96 | 15.95 | 15.62 | |

Event 3 (length 216 h) | R^{2} | 0.995 | 0.993 | 0.992 | 0.992 | 0.992 | 0.988 | 0.988 | 0.986 |

MAE (cm) | 2.46 | 2.85 | 2.75 | 2.95 | 2.90 | 3.26 | 3.46 | 3.30 | |

RAE (%) | 7.88 | 9.13 | 8.80 | 9.44 | 9.28 | 10.42 | 11.07 | 10.54 |

**Table 4.**Accuracy index grouping the measured tides by height classes: comparison between NARX, SHYFEM, and statistical models.

h_{tide} (cm) | t_{a} (h) | I = ε_{m} + 2σ (cm) | STAT-1 | STAT-2 | |||
---|---|---|---|---|---|---|---|

Model A | Model B | SHYFEM 5 | SHYFEM 6 | ||||

80–100 | 24 | −1.2 ± 5.5 | −1.4 ± 6.1 | 2.0 ± 12.9 | 1.0 ± 16.8 | −3.7 ± 16.1 | −4.0 ± 16.6 |

48 | −1.1 ± 6.3 | −1.1 ± 6.4 | 1.6 ± 18.3 | −0.6 ± 18.7 | |||

72 | −0.9 ± 6.1 | −0.9 ± 6.2 | 1.9 ± 18.5 | −0.3 ± 20.9 | |||

101–120 | 24 | −1.6 ± 5.5 | −1.6 ± 5.4 | 0.5 ± 12.5 | −3.6 ± 19.6 | −5.7 ± 20.7 | −5.3 ± 23.3 |

48 | −1.8 ± 5.9 | −1.9 ± 6.1 | −3.1 ± 17.5 | −5.5 ± 24.8 | |||

72 | −1.5 ± 5.8 | −1.6 ± 6.6 | −2.8 ± 20.6 | −6.8 ± 26.4 | |||

>120 | 24 | −1.5 ± 5.1 | −1.1 ± 5.3 | −9.4 ± 26.2 | −0.0 ± 26.1 | −19.4 ± 27.0 | −16.7 ± 26.2 |

48 | −2.2 ± 5.1 | −2.5 ± 6.9 | −8.6 ± 21.2 | 0.2 ± 14.8 | |||

72 | −2.0 ± 5.2 | −2.7 ± 7.0 | −11.1 ± 16.8 | −8.8 ± 47.0 |

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**MDPI and ACS Style**

Di Nunno, F.; Granata, F.; Gargano, R.; de Marinis, G.
Forecasting of Extreme Storm Tide Events Using NARX Neural Network-Based Models. *Atmosphere* **2021**, *12*, 512.
https://doi.org/10.3390/atmos12040512

**AMA Style**

Di Nunno F, Granata F, Gargano R, de Marinis G.
Forecasting of Extreme Storm Tide Events Using NARX Neural Network-Based Models. *Atmosphere*. 2021; 12(4):512.
https://doi.org/10.3390/atmos12040512

**Chicago/Turabian Style**

Di Nunno, Fabio, Francesco Granata, Rudy Gargano, and Giovanni de Marinis.
2021. "Forecasting of Extreme Storm Tide Events Using NARX Neural Network-Based Models" *Atmosphere* 12, no. 4: 512.
https://doi.org/10.3390/atmos12040512