# Neural Structures to Predict River Stages in Heavily Urbanized Catchments

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Studies Description

#### 2.2. The Lura River Basin

^{2}at the junction with the Olona River.

^{3}was implemented in the “intermediate Lura” to reduce the peak discharge from 70 m

^{3}s

^{−1}to less than 20 m

^{3}s

^{−1}before entering the urban areas. Despite implementing these structural measures, the lower catchment is still hit by frequent floods due to intense and spatially concentrated storms. As the hydrological concentration time is about 30 min, the current monitoring network detects and transmits the registered data with a 10-min resolution.

#### 2.3. The Laveggio River Basin

^{2}, as shown in Figure 3.

^{3}s

^{−1}corresponding roughly to a 30-year return period event. The erosion of the paved riverbed was observed in several areas, with damage to the embankments, but without significant flooding of the surrounding areas. The best-documented events were on 3 May 2002 and 11 November 2014, respectively, with a peak discharge of approximately 58 m

^{3}s

^{−1}and 60 m

^{3}s

^{−1}. The documentation of these two events shows that the channel sections were almost full in both cases. Then, according to the information from the fire brigade, there was only one point where the water overflowed, namely on the left side into the commercial area above Ponte Via Giuseppe Motta. Sandbags have been placed along the channel as a preventive measure during other flooding events since 1992.

#### 2.4. Data Description

_{a}thus work as multi-output—the forecast is the future sequence of values along the prediction horizon h:

#### 2.5. Multivariable Models: Rain Gauges and Total Rain Scenarios

_{t}are fed by an estimate r(t) of the total precipitation over the catchment, computed from all the gauge measurements together, i.e.,

_{i}(t), contributing individually as model input, as in Equation (3).

#### 2.6. Neural Networks

#### 2.6.1. Convolutional Neural Network

- The convolutional layer. It plays a crucial role in CNNs. Convolution is a linear operation extracting features from inputs through a small moving filter called the kernel. An element-wise product between each kernel element and the input is calculated, obtaining a feature map. The two key hyperparameters that define the convolution operation are the size and number of kernels. Multiple kernels are considered, so this procedure is repeated multiple times, returning different feature maps. The outputs of this linear operation are then passed through a nonlinear activation function, such as a rectified linear unit (ReLU);
- The output feature maps. The previous outputs are transformed into a 1D vector and connected to one or more dense or fully connected (FC) layers in which every input is connected to every output. This last layer maps the extracted features into the final output.

#### 2.6.2. Long Short-Term Memory Neural Network

- The forget gate decides what information shall be filtered from the cell state by looking at the previous hidden state h(t − 1) and the current input x(t);
- The input gate is responsible for updating the cell state c(t). In the input gate, sigmoid and tanh functions combine the previous hidden state h(t − 1) and the current input x(t). The cell state is affected by both forget and input gates (c(t));
- The output gate defines the following hidden state h(t) based on the previous hidden state h(t − 1), current input x(t), and the newly modified cell state c(t). The hidden state h(t) and the new cell state c(t) move forward in the neural chain.

#### 2.7. Model Architecture

_{i}and model predicted value ${\widehat{y}}_{i}$, with N equal to the cardinality of the training dataset. As is well known, MSE emphasizes the contribution provided by more significant errors that correspond to the occurrence of extreme events, in line with the primary objective of the present modelling exercise. Each training takes place twice, randomly varying the weights initialization to avoid being stuck in a particular solution. Models are implemented in Python mainly using the Keras library, with Tensorflow as the back-end.

#### 2.8. Performance Assessment Metrics

_{i}and the model estimation ${\widehat{y}}_{i}$:

- I
_{flood}(t) is positive (P) if floods are present in a sequence of values (either real or predicted). Flood sequences are defined as hourly sequences [y(t), y(t + 1),…, y(t + h)] that contain at least one record of water level equal to or greater than a certain threshold (Table 4). Conversely, a negative value (N) means there is no flood in the sequence.$${I}_{flood}\left(t\right)=\left\{\begin{array}{c}Pify\ge floodthreshold\\ N\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise\end{array}\right.\phantom{\rule{0ex}{0ex}}\mathrm{with}y\in \left[y\right(t),y(t+1),\dots ,y(t+h\left)\right].$$ - I
_{rapid}(t) indicates whether rapid flow increments occur in a (real or predicted) water stage sequence. Rapid increments are defined as hourly sequences [y(t + h − 1), …, y(t + 1), y(t)] containing at least one value greater than the first of the sequence by the rapid increment threshold (Table 4).$${I}_{rapid}\left(t\right)=\left\{\begin{array}{c}Pif\left[y\left(i\right)-y\left(t\right)\right]\ge rapidincrementthreshold,\hspace{1em}i\in \left[t+1,t+h\right]\\ N\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise\hfill \end{array}\right.$$

_{flood}and I

_{rapid}indicators for both Lura and Laveggio domains. These values are set by referring to the statistical characteristics of water stage time series, as well as historical flooding damages. In general, the water stages of the Laveggio River are lower than those of the Lura River. The thresholds reflect this.

## 3. Results

_{rapid}. The confusion matrix relative to rapid increments is the same for all neural models with TP equal to zero. Consequently, all the performance indicators related to rapid increments are also null.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Approximate location of the Lura (red) and Laveggio (blue) river basins in their sub-alpine territorial context. Map data © OpenStreetMap.

**Figure 2.**Location of the Lura River basin and the relative measurement stations. Map data © OpenStreetMap.

**Figure 3.**Location of the Laveggio River basin and the relative measurement stations. Map data © OpenStreetMap.

**Figure 6.**I

_{flood}and I

_{rapid}for autoregressive FF, LSTM, and CNN models. Test data, Lura River. P means a flood or a rapid event is present, N means that it is not.

**Figure 7.**I

_{flood}and I

_{rapid}for multivariable LSTM and CNN models in the rain gauges scenario. Test data, Lura River. P means a flood or a rapid event is present, N means that it is not.

**Figure 8.**I

_{flood}and I

_{rapid}for multivariable LSTM and CNN models in the total rain scenario. Test data, Lura River. P means a flood or a rapid event is present, N means that it is not.

**Figure 9.**NSE over the forecasting horizon h of LSTM (

**a**) and CNN (

**b**) predictors for the floods of the Lura River: dashed lines represent autoregressive models and solid lines represent multivariable models with markers for the rain gauges and total rain scenarios.

**Figure 10.**I

_{flood}and I

_{rapid}for the multivariable LSTM model with 10-min input update frequency. Lura test data, rain gauges scenario. P means a flood or a rapid event is present, N means that it is not.

**Figure 11.**Performance of multivariable LSTM. Rain gauges scenario: test flood event of 21 July 2018, Lura River. Actual (

**green**) and forecasted water levels with 60- (

**red**) and 10-min (

**orange**) input update.

**Figure 12.**I

_{flood}and I

_{rapid}for the LSTM model with 60-min input update frequency. Laveggio test data, rain gauges scenario. P means a flood or a rapid event is present, N means that it is not.

**Figure 13.**I

_{flood}and I

_{rapid}for the LSTM model with 10-min input update frequency. Laveggio test data, rain gauges scenario. P means a flood or a rapid event is present, N means that it is not.

**Figure 14.**Performance of multivariable LSTM in the rain gauges scenario: test flood event of 4 June 2018, Laveggio River. Actual (

**green**) and forecasted water levels with 60- (

**red**) and 10-min (

**orange**) input update.

**Table 1.**Example of the input–output patterns (time step equal to 10 min) for models with an input update frequency of 60 (a) and 10 min (b). In the autoregressive approach, the input vector v only contains previous water stages y; in the multivariable approach, the rainfall records are also included.

(a) | Input | Output | ||||||||||

t
= 6 | v_{01} | v_{02} | v_{03} | v_{04} | v_{05} | v_{06} | $\widehat{y}$_{07} | $\widehat{y}$_{08} | $\widehat{y}$_{09} | $\widehat{y}$_{10} | $\widehat{y}$_{11} | $\widehat{y}$_{12} |

t
= 12 | v_{07} | v_{08} | v_{09} | v_{10} | v_{11} | v_{12} | $\widehat{y}$_{13} | $\widehat{y}$_{14} | $\widehat{y}$_{15} | $\widehat{y}$_{16} | $\widehat{y}$_{17} | $\widehat{y}$_{18} |

… | … | |||||||||||

(b) | Input | Output | ||||||||||

t
= 6 | v_{01} | v_{02} | v_{03} | v_{04} | v_{05} | v_{06} | $\widehat{y}$_{07} | $\widehat{y}$_{08} | $\widehat{y}$_{09} | $\widehat{y}$_{10} | $\widehat{y}$_{11} | $\widehat{y}$_{12} |

t
= 7 | v_{02} | v_{03} | v_{04} | v_{05} | v_{06} | v_{07} | $\widehat{y}$_{08} | $\widehat{y}$_{09} | $\widehat{y}$_{10} | $\widehat{y}$_{11} | $\widehat{y}$_{12} | $\widehat{y}$_{13} |

… | … |

LSTM Hyperparameters | Autoregressive Model | Multivariable Model |
---|---|---|

Batch-size | 64 | 32 |

Learning rate | 0.001 | 0.001 |

Epochs | 15 | 30 |

Activation function | relu | relu |

N. hidden neurons | 10 | 20 |

CNN Hyperparameters | Autoregressive Model | Multivariable Model |
---|---|---|

Batch-size | 32 | 16 |

Learning rate | 0.001 | 0.001 |

Epochs | 20 | 25 |

Activation function | relu | relu |

Convolutional layers | 1 | 2 × n.features |

Kernel size | 3 | 3 |

Filters | 64 | 32 |

Fully connected layers | 1 | 2 |

**Table 4.**Thresholds of floods and rapid events in Lura and Laveggio rivers for I

_{flood}and I

_{rapid}indicators.

River | Flood Threshold (m) | Rapid Increment Threshold (m) |
---|---|---|

Lura | 1.2 | 0.20 |

Laveggio | 0.6 | 0.08 |

**Table 5.**The performance of the Lura River’s autoregressive models: average value of the evaluation metrics over the entire test dataset and flood events only.

River | Model | Data | Model | RMSE (cm) | MSE (cm^{2}) | MAE (cm) | NSE |
---|---|---|---|---|---|---|---|

Lura | Autoregressive | All test data | Persistent | 11.45 | 131.17 | 3.01 | 0.78 |

FF | 7.31 | 53.01 | 2.04 | 0.90 | |||

LSTM | 7.35 | 54.11 | 2.05 | 0.91 | |||

CNN | 7.31 | 53.43 | 2.70 | 0.90 | |||

Extreme events | Persistent | 48.63 | 2393.00 | 34.23 | −0.01 | ||

FF | 41.16 | 1694.01 | 26.01 | 0.24 | |||

LSTM | 40.51 | 1641.08 | 25.91 | 0.27 | |||

CNN | 36.24 | 1314.02 | 23.83 | 0.41 |

**Table 6.**Performance of multivariable models for the Lura River: average value of evaluation metrics over the forecasting horizon (h) in the entire test dataset and flood events (rain gauges and total rain scenarios).

River | Model | Data | Scenario | RMSE (cm) | MSE (cm ^{2}) | MAE (cm) | NSE |
---|---|---|---|---|---|---|---|

Lura | Multivariable | All test data | LSTM—Rain Gauges | 6.20 | 38.53 | 2.01 | 0.93 |

LSTM—Total Rain | 6.02 | 40.23 | 2.13 | 0.93 | |||

CNN—Rain Gauges | 6.52 | 47.08 | 2.98 | 0.91 | |||

CNN—Total Rain | 6.24 | 44.20 | 2.08 | 0.92 | |||

Extreme events | LSTM—Rain Gauges | 29.39 | 977.64 | 20.09 | 0.56 | ||

LSTM—Total Rain | 30.35 | 1042.33 | 20.90 | 0.53 | |||

CNN—Rain Gauges | 33.45 | 1262.23 | 23.36 | 0.44 | |||

CNN—Total Rain | 30.67 | 1110.57 | 21.13 | 0.50 |

**Table 7.**Comparison of multivariable LSTM performance with 60- and 10-min input update frequency for the entire test dataset and flood events only. Rain gauges scenario, Lura River.

Input Update Frequency | |||||||||
---|---|---|---|---|---|---|---|---|---|

60 min | 10 min | ||||||||

River | Data | RMSE (cm) | MSE (cm^{2}) | MAE (cm) | NSE | RMSE (cm) | MSE (cm^{2}) | MAE (cm) | NSE |

Lura | All test data | 6.20 | 38.53 | 2.01 | 0.93 | 3.02 | 9.32 | 1.20 | 0.98 |

Extreme events | 29.39 | 977.64 | 20.09 | 0.56 | 9.52 | 92.16 | 4.57 | 0.96 |

**Table 8.**Comparison of multivariable LSTM performances with 60- and 10-min input update frequency for the entire test dataset and flood events (Laveggio, rain gauges scenario).

Input Update Frequency | |||||||||
---|---|---|---|---|---|---|---|---|---|

60 min | 10 min | ||||||||

River | Data | RMSE (cm) | MSE (cm^{2}) | MAE (cm) | NSE | RMSE (cm) | MSE (cm^{2}) | MAE (cm) | NSE |

Laveggio | All test data | 1.80 | 3.52 | 0.61 | 0.90 | 0.92 | 0.86 | 0.31 | 0.97 |

Extreme events | 13.46 | 200.71 | 8.80 | −0.12 | 7.70 | 0.60 | 4.17 | 0.63 |

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**MDPI and ACS Style**

Chiacchiera, A.; Sai, F.; Salvetti, A.; Guariso, G.
Neural Structures to Predict River Stages in Heavily Urbanized Catchments. *Water* **2022**, *14*, 2330.
https://doi.org/10.3390/w14152330

**AMA Style**

Chiacchiera A, Sai F, Salvetti A, Guariso G.
Neural Structures to Predict River Stages in Heavily Urbanized Catchments. *Water*. 2022; 14(15):2330.
https://doi.org/10.3390/w14152330

**Chicago/Turabian Style**

Chiacchiera, Annunziata, Fabio Sai, Andrea Salvetti, and Giorgio Guariso.
2022. "Neural Structures to Predict River Stages in Heavily Urbanized Catchments" *Water* 14, no. 15: 2330.
https://doi.org/10.3390/w14152330