# Multistep Flood Inundation Forecasts with Resilient Backpropagation Neural Networks: Kulmbach Case Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Data and Structure of Artificial Neural Networks

#### 2.2. Hyperparameter Tuning in ANN

^{−6}in this case).

#### 2.3. Prediction of the First Interval of Flood Events

#### 2.4. Real-Time Forecasting for Sequential Multistep Forecast Intervals

#### 2.5. Model Evaluation

- T is the predicted value, water depth from the ANN model in our case.
- S is the observed value, water depth from the hydraulic model (HEC-RAS) in our case.
- To assess the general conduct of the model over the training and validation dataset, the average RMSE is also calculated for the average accuracy among all the events in the testing dataset.

## 3. Study Area and Database

#### 3.1. Study Area

^{2}. An extreme flood event hit the city on 28 May 2006. A flood mitigation plan was prepared by local stakeholders to mitigate future events. In the ANN model, the above seven streams are taken as the input boundary conditions. The goal of the ANN modeling is to replace the hydraulic processes within the marked study area to enable fast real-time forecasts (see Figure 4).

#### 3.2. HEC-RAS and Synthetic Event Database

## 4. Results

#### 4.1. Assessment of the Prediction of Water Depths of the First Intervals (time 0) of Flood Events

#### 4.1.1. Synthetic Flood Events

#### 4.1.2. Historical Flood Events

^{3}/s was selected as the forecast threshold to initiate the forecasts since this value is crossed before the beginning of the flooding in all three historical events. The forecast threshold is chosen slightly bigger than the average discharge of 9.2 m

^{3}/s of White Main [35] to avoid the low discharges from triggering flood warnings.

#### Historical flood events 2006

#### Historical flood events 2013

^{3}/s; therefore, the start of the prediction at 9 h is marked with a red line when one discharge hits the forecast threshold. Figure 8 compares the prediction of the inundation map of the first intervals of 3 h, 6 h, 9 h and 12 h with the inundation map from the hydraulic model of the historical flood event 2013. Table 3 shows the performance of the prediction for historical event 2013, evaluated by average RMSE for each individual ANN. The forecast performance is slightly better than that of the event 2006.

#### Historical flood events 2005

#### 4.2. Assessment of Real-time Forecasting of Water Depths for Multistep Flood Forecast Intervals, 1–5 h

#### Historical flood events 2006

#### Historical flood events 2013

#### Historical flood event 2005

#### 4.3. Forecast of the Inundation Extent

## 5. Discussion

#### 5.1. Assessment of the Prediction of Water Depths of the First Intervals of Flood Events

#### 5.1.1. Synthetic Flood Events

#### 5.1.2. Historical Flood Events

#### Historical flood event 2006

#### Historical flood event 2013

^{3}/s for a second time. The red line in Figure 7 marks the new start of the forecast (red line). For this event, it is nine hours later after the first forecast signaled by the ANN. Table 3 shows that the ANN model achieved high accuracy for the flood event in 2013. For the 3 h prediction, 96% of the grids have RMSE less than 0.2 m. 6 h prediction has 82% grids with RMSE less than 0.2 m. From 3 h and 6 h prediction of event 2013, the ANN performs better than the event 2006. Overall, the event of 2013 is also well predicted, with over 78% of grids having RMSE less than 0.3 m. Similar to event 2006, the predicted flood inundation maps of 3 h and 6 h intervals are similar to the hydraulic inundation simulations (see Figure 10).

#### Historical flood event 2005

#### 5.2. Assessment of Real-time Forecasting of Water Depths for Multistep Flood Forecast Intervals, 1–5 h

#### Historical flood event 2006

#### Historical flood event 2013

^{3}/s for the second time, namely 9 hours after time 0 (the first time the forecast window was activated). From the second starting point, all other forecasts are done for every forecast for X h + 1 h to X h + 5 h. Table 6 shows the forecast accuracy of the historical flood event of 2013. From this table, the ANN model performs as similar to the flood event in 2006. The forecast of the 2013 flood event has good results in 3 h, 6 h and most of 9 h (over 70% of the grid with RMSE < 0.3 m).

#### Historical flood event 2005

#### 5.3. Forecast of the Inundation Extent

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The forward-feed neural network setup in the forecast study. The input layer is fed with discharge inflows of certain time interval windows. The output layer generates the flood inundation for that interval. Resilient backpropagation is applied for training this network.

**Figure 2.**Training of artificial neural networks (ANN) forecast model. Four ANN models for 3 h, 6 h, 9 h, 12 h first interval predictions are set up in this work, trained with the discharges from each synthetic flood event. After this, the models are to predict the corresponding first intervals for other events.

**Figure 3.**Shift of ANN forecast models for multistep forecast intervals. The yellow color shows the forecast of the first interval (forecast interval same as the training interval, i.e., at time 0). The green color shows applying the original 3 h forecast network for 1 h later forecast from 1–4 h. The orange color shows applying the original 3 h forecast network for 2 h later forecast from 2–5 h. The black box shows the general case of applying the original X h forecast network for S h later forecast, from S h to X h + S.

**Figure 4.**Map of the study area. It shows the location of Kulmbach in Germany. The blue curves represent the river network. The shaded region is the study area with its topography represented. On the marked boundary, the red points represent the seven inflows on the boundary (three rivers and four smaller streams).

**Figure 5.**Hydrographs of the flood event in 2006. Seven discharge curves of three rivers and four streams are shown in different colors. Time 0 marks the start of the prediction. The dash lines upon the discharge curves mark the different discharge sections for prediction inputs.

**Figure 6.**Inundation maps from the prediction of water depths of the first intervals in flood event 2006. (

**a**) ANN inundation map 3 h; (

**b**) hydrodynamic inundation map 3 h; (

**c**) ANN inundation map 6 h; (

**d**) hydrodynamic inundation map 6 h; (

**e**) ANN inundation map 9 h; (

**f**) hydrodynamic inundation map 9 h; (

**g**) ANN inundation map 12 h; (

**h**) hydrodynamic inundation map 12 h.

**Figure 7.**Hydrographs of the flood event in 2013. Seven discharge curves of three rivers and four streams are shown in different colors. The red line time marks the new start of the prediction at 9 h, where one discharge first exceeds the forecast threshold of 10 m

^{3}/s. The dash lines upon the discharge curves mark the different discharge sections for prediction inputs.

**Figure 8.**Inundation maps from the prediction of water depths of the first intervals in flood event 2013. (

**a**) ANN inundation map 3 h; (

**b**) hydrodynamic inundation map 3 h; (

**c**) ANN inundation map 6 h; (

**d**) hydrodynamic inundation map 6 h; (

**e**) ANN inundation map 9 h; (

**f**) hydrodynamic inundation map 9 h; (

**g**) ANN inundation map 12 h; (

**h**) hydrodynamic inundation map 12 h.

**Figure 9.**Hydrographs of the flood event in 2005. Seven discharge curves of three rivers and four streams are shown in different colors. Time 0 marks the start of the prediction. The dash lines upon the discharge curves mark the different discharge sections for prediction inputs.

**Figure 10.**Inundation maps from the prediction of water depths of the first intervals in flood event 2005. (

**a**) ANN inundation map 3 h; (

**b**) hydrodynamic inundation map 3 h; (

**c**) ANN inundation map 6 h; (

**d**) hydrodynamic inundation map 6 h; (

**e**) ANN inundation map 9 h; (

**f**) hydrodynamic inundation map 9 h; (

**g**) ANN inundation map 12 h; (

**h**) hydrodynamic inundation map 12 h.

**Figure 11.**Performance of the forecast of inundation extent growths by three indices. (

**a**) probability of detection (POD) in flood event 2006; (

**b**) false alarm ratio (FAR) in flood event 2006; (

**c**) critical success index (CSI) in the flood event 2006; (

**d**) POD in flood event 2013; (

**e**) FAR in the flood event 2013; (

**f**) CSI in flood event 2013; (

**g**) POD in flood event 2005; (

**h**) FAR in flood event 2005; (

**i**) CSI in flood event 2005.

**Table 1.**Number of wet grids and grid percentages of different large error thresholds for testing synthetic flood events (60 events, #121~#180).

Prediction Time (h) | Wet ANN Grid | ANN Grid with Average RMSE > 0.2 m | ANN Grid% with Average RMSE ≤ 0.2 m | ANN Grid with Average RMSE > 0.3 m | ANN Grid% with Average RMSE ≤ 0.3 m | ANN Grid with Average RMSE > 0.4 m | ANN Grid% with Average RMSE ≤ 0.4 m |
---|---|---|---|---|---|---|---|

3 | 300 | 47 | 84.33% | 18 | 94.00% | 10 | 96.67% |

6 | 417 | 174 | 58.27% | 78 | 81.29% | 27 | 93.53% |

9 | 474 | 106 | 77.64% | 37 | 92.19% | 15 | 96.84% |

12 | 483 | 50 | 89.65% | 12 | 97.52% | 7 | 98.55% |

**Table 2.**Numbers of wet grids and accurate grid percentage for event 2006. A wet grid is with the water level over 0.1 m; any water depth below this cutoff value is eliminated. Table shows grid numbers with a larger root-mean-square error (RMSE) and their percentages to the total wet grids.

Prediction Time (h) | Wet ANN Grid | ANN Grid with RMSE > 0.2 m | ANN Grid% with RMSE ≤ 0.2 m | ANN Grid with RMSE > 0.3 m | ANN Grid% with RMSE ≤ 0.3 m | ANN Grid with RMSE > 0.4 m | ANN Grid% with RMSE ≤ 0.4 m |
---|---|---|---|---|---|---|---|

3 | 280 | 46 | 83.57% | 20 | 92.86% | 6 | 97.86% |

6 | 405 | 84 | 79.26% | 42 | 89.63% | 25 | 93.83% |

9 | 474 | 134 | 71.73% | 64 | 86.50% | 36 | 92.41% |

12 | 483 | 157 | 67.49% | 85 | 82.40% | 47 | 90.27% |

**Table 3.**Numbers of wet grids and accurate grid percentages for the flood event in 2013. A wet grid is with the water level over 0.1 m; any water depth below this cutoff value is eliminated. The table shows grid numbers with larger RMSE and their percentages to the total wet grids.

Prediction Time (h) | Wet ANN Grid | ANN Grid with RMSE > 0.2 m | ANN Grid% with RMSE ≤ 0.2 m | ANN Grid with RMSE > 0.3 m | ANN Grid% with RMSE ≤ 0.3 m | ANN Grid with RMSE > 0.4 m | ANN Grid% with RMSE ≤ 0.4 m |
---|---|---|---|---|---|---|---|

3 | 285 | 9 | 96.84% | 2 | 99.30% | 2 | 99.30% |

6 | 405 | 72 | 82.22% | 27 | 93.33% | 8 | 98.02% |

9 | 474 | 134 | 71.73% | 65 | 86.29% | 25 | 94.73% |

12 | 483 | 175 | 63.77% | 104 | 78.47% | 56 | 88.41% |

**Table 4.**Numbers of wet grids and accurate grid percentages for the flood event in 2005. A wet grid is with the water level over 0.1 m; any water depth below this cutoff value is eliminated. Table shows grid numbers with larger RMSE and their percentages to the total wet grids.

Prediction Time (h) | Wet ANN Grid | ANN Grid with RMSE > 0.2 m | ANN Grid% with RMSE ≤ 0.2 m | ANN Grid with RMSE > 0.3 m | ANN Grid% with RMSE ≤ 0.3 m | ANN Grid with RMSE > 0.4 m | ANN Grid% with RMSE ≤ 0.4 m |
---|---|---|---|---|---|---|---|

3 | 280 | 65 | 76.79% | 36 | 87.14% | 19 | 93.21% |

6 | 405 | 165 | 59.26% | 115 | 71.60% | 74 | 81.73% |

9 | 474 | 216 | 54.43% | 148 | 68.78% | 93 | 80.38% |

12 | 483 | 244 | 49.48% | 168 | 65.22% | 107 | 77.85% |

**Table 5.**Forecast accuracy percentages for the flood event in 2006. This table shows the grid percentage to the total wet grids with average RMSE within 0.3 m. The forecast begins by the starting point, several hours later than the event beginning for the real-time forecast.

Starting Point (h) | Prediction Interval (h) | |||
---|---|---|---|---|

3 | 6 | 9 | 12 | |

+1 | 98.93% | 92.10% | 83.12% | 79.09% |

+2 | 98.94% | 90.86% | 77.85% | 79.92% |

+3 | 96.94% | 89.38% | 76.58% | 78.26% |

+4 | 95.11% | 86.95% | 70.89% | 75.36% |

+5 | 86.60% | 69.29% | 66.88% | 68.12% |

**Table 6.**Forecast accuracy percentages rate for the flood event in 2013. This table shows the grid percentage to the total wet grids with average RMSE within 0.3 m. The forecast begins by the starting point, several hours later than the event beginning for the real-time forecast.

Starting Point (h) | Prediction Interval (h) | |||
---|---|---|---|---|

3 | 6 | 9 | 12 | |

+1 | 99.30% | 92.59% | 84.18% | 72.67% |

+2 | 98.95% | 89.88% | 78.90% | 70.39% |

+3 | 96.30% | 89.17% | 75.74% | 67.91% |

+4 | 94.17% | 82.51% | 68.78% | 67.29% |

+5 | 91.28% | 77.34% | 66.46% | 66.67% |

**Table 7.**Forecast accuracy percentages for the flood event in 2005. This table shows the grid percentages to the total wet grids with average RMSE within 0.3 m. The forecast begins by the starting point, several hours later than the event beginning for the real-time forecast.

Starting Point (h) | Prediction Interval (h) | |||
---|---|---|---|---|

3 | 6 | 9 | 12 | |

+1 | 83.74% | 69.95% | 66.89% | 63.15% |

+2 | 81.67% | 68.23% | 64.77% | 61.70% |

+3 | 76.31% | 67.00% | 63.08% | 60.25% |

+4 | 74.85% | 66.50% | 60.97% | 60.25% |

+5 | 70.97% | 61.08% | 58.65% | 60.25% |

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**MDPI and ACS Style**

Lin, Q.; Leandro, J.; Gerber, S.; Disse, M. Multistep Flood Inundation Forecasts with Resilient Backpropagation Neural Networks: Kulmbach Case Study. *Water* **2020**, *12*, 3568.
https://doi.org/10.3390/w12123568

**AMA Style**

Lin Q, Leandro J, Gerber S, Disse M. Multistep Flood Inundation Forecasts with Resilient Backpropagation Neural Networks: Kulmbach Case Study. *Water*. 2020; 12(12):3568.
https://doi.org/10.3390/w12123568

**Chicago/Turabian Style**

Lin, Qing, Jorge Leandro, Stefan Gerber, and Markus Disse. 2020. "Multistep Flood Inundation Forecasts with Resilient Backpropagation Neural Networks: Kulmbach Case Study" *Water* 12, no. 12: 3568.
https://doi.org/10.3390/w12123568