# Emulation of 2D Hydrodynamic Flood Simulations at Catchment Scale Using ANN and SVR

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{2}dominated by forests (about 74%). The elevation ranges from 10.00 to 872.34 m.a.s.l., and the average slope of the catchment along the river is about 0.65%. The mean annual precipitation is approximately 1261 mm, with most of the rainfall occurring between October and March (about 60%). On 2 October 2017, an extreme flood event occurred in the downstream parts of the river and inundated Birkeland. This event was the highest ever recorded flooding of this river. The recorded data of the event can be found in the study by [22].

#### 2.2. Hydrodynamic Simulations and Input Ranges

_{p}values and lower K

_{s}values in river section) were simulated and were added to the total data set. In this study we intended to emulate the numerical flood inundation modeling. Therefore, water level was considered as the output of interest in the simulations and the values were observed in three cross sections (CS), specified along the study area (Figure 1b).

#### 2.3. Emulation Methods

#### 2.3.1. Support Vector Regression (SVR)

#### 2.3.2. Artificial Neural Networks (ANN)

#### 2.3.3. Performance Criteria

#### 2.4. Development of the Emulator

_{T}is the total sum of squares, and SS

_{j}is the sum of squared deviations for each parameter j. Both can be calculated using Equations (11) and (12), respectively:

## 3. Results and Discussion

#### 3.1. Dimensionality Reduction

_{p}) and water level followed by river friction (K

_{s}_R) and forest runoff coefficient (C_F). The findings of the correlation tests and ANOVA test are consistent; however, the ANOVA results are more stable at different locations compared to the correlation tests. According to the correlation values displayed in Figure 7, forest friction (K

_{s}_F) shows a weak correlation with water level at cross sections CS1 and CS2 and a negligible correlation at cross section CS3. Unlike correlation values, the calculated contribution value resulted from the ANOVA test shows an almost constant contribution of forest friction (K

_{s}_F with about 4%) at different cross sections. Accordingly, two alternative approaches were adopted to train the emulators: first, the four highest correlated parameters (rainfall (i

_{p}), forest runoff coefficient (C_F), river friction (K

_{s}_R), and forest friction (K

_{s}_F)) were used to develop the emulators. In the second approach forest friction (K

_{s}_F) was removed from the input set and emulators were trained using only three inputs. The results of the first approach (four inputs) are presented in Table 3. Comparing Table 2 and Table 3 (i.e., fourteen inputs and four inputs, respectively), shows that the performance of both emulators considerably improves (about 50% reduction in error values) when the number of the inputs decreases to four inputs. The results of the second approach (three inputs) showed that the predictive ability of the models considerably deteriorates when the number of inputs is decreased to three parameters. Therefore, the results are only presented for the emulators with four inputs (Table 3).

#### 3.2. Error Structure Analysis

#### 3.3. Sample Size

## 4. Conclusions

- The statistical metrics for the developed emulators confirm the applicability of surrogates for predicting the cross-sectional water level. However, evaluating the results from different aspects (performance metrics, error trends, ranges, and distributions) showed that SVR has a better performance compared to ANN.
- Inclusion of too many variables as inputs can deteriorate the performance of the emulators; thus, simplification of the model structure through dimension reduction techniques can be used to obtain the most accurate model. The implemented correlation tests and ANOVA used in this study provided consistent results and showed that they can be a good choice to reduce the dimension of input data, improving the accuracy of the models.
- The error trend and regression plots for the SVR model and ANN model indicate that the performance of the SVR model for different magnitudes of floods are similar and relatively constant, whereas the ANN model tends to overpredict the smaller floods and underpredict the extreme floods. The best and worst performance of the ANN-based emulator is achieved for the medium size and extreme floods, respectively. Therefore, the application of the ANN model may not be safe for prediction of extreme flood events.
- The normality assumption for errors, which is typically undertaken in hazard assessment and decision making, is not always true and can eventuate to incorrect statistical inferences.
- The findings in this study suggest that the training data set size equal to 70% (or more) of data results in reliable and accurate predictions. The results also showed that it is not reliable to train an emulator with less than 50% of the data (corresponding to 600 simulations).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Panagoulia, D.; Dimou, G. Sensitivity of flood events to global climate change. J. Hydrol.
**1997**, 191, 208–222. [Google Scholar] [CrossRef] - Panagoulia, D. From low-flows to floods under global warming. In EGU General Assembly 2009; EGU General Assembly Conference Abstracts: Vienna, Austria, 2009; p. 4511. [Google Scholar]
- Teng, J.; Jakeman, A.J.; Vaze, J.; Croke, B.F.; Dutta, D.; Kim, S.J.E.M. Software Flood inundation modelling: A review of methods, recent advances and uncertainty analysis. Environ. Model. Softw.
**2017**, 90, 201–216. [Google Scholar] [CrossRef] - Chu, H.; Wu, W.; Wang, Q.; Nathan, R.; Wei, J. An ANN-based emulation modelling framework for flood inundation modelling: Application, challenges and future directions. Environ. Model. Softw.
**2020**, 124, 104587. [Google Scholar] [CrossRef] - Jamieson, R.S.; Lhomme, J.; Wright, G.; Gouldby, B. A highly efficient 2D flood model with sub-element topography. Proc. Inst. Civ. Eng. Water Manag.
**2012**, 165, 581–595. [Google Scholar] [CrossRef] - Davidsen, S.; Löwe, R.; Thrysøe, C.; Arnbjerg-Nielsen, K. Simplification of one-dimensional hydraulic networks by automated processes evaluated on 1D/2D deterministic flood models. J. Hydroinform.
**2017**, 19, 686–700. [Google Scholar] [CrossRef][Green Version] - Razavi, S.; Tolson, B.A.; Burn, D.H. Numerical assessment of metamodelling strategies in computationally intensive optimization. Environ. Model. Softw.
**2012**, 34, 67–86. [Google Scholar] [CrossRef] - Solomatine, D.P.; Ostfeld, A. Data-driven modelling: Some past experiences and new approaches. J. Hydroinform.
**2008**, 10, 3–22. [Google Scholar] [CrossRef][Green Version] - Yu, J.; Qin, X.; Larsen, O. Uncertainty analysis of flood inundation modelling using GLUE with surrogate models in stochastic sampling. Hydrol. Process.
**2015**, 29, 1267–1279. [Google Scholar] [CrossRef] - Castro-Gama, M.E.; Popescu, I.; Li, S.; Mynett, A.; van Dam, A. Flood inference simulation using surrogate modelling for the Yellow River multiple reservoir system. Environ. Model. Softw.
**2014**, 55, 250–265. [Google Scholar] [CrossRef] - Luo, J.; Lu, W. Comparison of surrogate models with different methods in groundwater remediation process. J. Earth Syst. Sci.
**2014**, 123, 1579–1589. [Google Scholar] [CrossRef] - Bacchi, V.; Hamdi, Y.; Foch, M.; Pheulpin, L. Development of a new approach for the assessment of Flood Hazard through a kriging surrogate: Application to the bi-dimensional model of the Loire River. In Advances in Extreme Value Analysis and Application to Natural Hazards, EVAN; CHATOU: Paris, France, 2019. [Google Scholar]
- Villa-Vialaneix, N.; Follador, M.; Ratto, M.; Leip, A. A comparison of eight metamodeling techniques for the simulation of N
_{2}O fluxes and N leaching from corn crops. Environ. Model. Softw.**2012**, 34, 51–66. [Google Scholar] [CrossRef][Green Version] - Ghalkhani, H.; Golian, S.; Saghafian, B.; Farokhnia, A.; Shamseldin, A. Application of surrogate artificial intelligent models for real-time flood routing. Water Environ. J.
**2013**, 27, 535–548. [Google Scholar] [CrossRef] - Xie, S.; Wu, W.; Mooser, S.; Wang, Q.; Nathan, R.; Huang, Y. Artificial neural network based hybrid modeling approach for flood inundation modeling. J. Hydrol.
**2021**, 592, 125605. [Google Scholar] [CrossRef] - Zhang, X.; Srinivasan, R.; Van Liew, M. Approximating SWAT model using artificial neural network and support vector machine 1. JAWRA J. Am. Water Resour. Assoc.
**2009**, 45, 460–474. [Google Scholar] [CrossRef] - Xu, T.; Valocchi, A.J.; Ye, M.; Liang, F. Quantifying model structural error: Efficient Bayesian calibration of a regional groundwater flow model using surrogates and a data-driven error model. Water Resour. Res.
**2017**, 53, 4084–4105. [Google Scholar] [CrossRef] - Liu, Y.; Pender, G. A flood inundation modelling using v-support vector machine regression model. Eng. Appl. Artif. Intell.
**2015**, 46, 223–231. [Google Scholar] [CrossRef] - Bermúdez, M.; Ntegeka, V.; Wolfs, V.; Willems, P. Development and comparison of two fast surrogate models for urban pluvial flood simulations. Water Resour. Manag.
**2018**, 32, 2801–2815. [Google Scholar] [CrossRef] - Bass, B.; Bedient, P. Surrogate modeling of joint flood risk across coastal watersheds. J. Hydrol.
**2018**, 558, 159–173. [Google Scholar] [CrossRef] - Al Kajbaf, A.; Bensi, M. Application of surrogate models in estimation of storm surge: A comparative assessment. Appl. Soft Comput.
**2020**, 91, 106184. [Google Scholar] [CrossRef] - Alipour, S.M.; Engeland, K.; Leal, J. A practical methodology to perform global sensitivity analysis for 2D hydrodynamic computationally intensive simulations. Hydrol. Res.
**2021**, in press. [Google Scholar] [CrossRef] - Ferreira, R.M.; Franca, M.J.; Leal, J.G.; Cardoso, A.H. Mathematical modelling of shallow flows: Closure models drawn from grain-scale mechanics of sediment transport and flow hydrodynamics. Can. J. Civ. Eng.
**2009**, 36, 1605–1621. [Google Scholar] [CrossRef] - Conde, D.A.; Canelas, R.B.; Ferreira, R.M. A unified object-oriented framework for CPU+ GPU explicit hyperbolic solvers. Adv. Eng. Softw.
**2020**, 148, 102802. [Google Scholar] [CrossRef] - LeVeque, R.J. Finite Volume METHODS for Hyperbolic Problems; Cambridge University Press: Cambridge, UK, 2002; Volume 31. [Google Scholar] [CrossRef]
- Geuzaine, C.; Remacle, J.F. Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Arcement, G.J.; Schneider, V.R. Guide for Selecting Manning’s Roughness Coefficients for Natural Channels and Flood Plains; US Government Printing Office: Washington, DC, USA, 1989.
- Dorn, H.; Vetter, M.; Höfle, B. GIS-based roughness derivation for flood simulations: A comparison of orthophotos, LiDAR and crowdsourced geodata. Remote Sens.
**2014**, 6, 1739–1759. [Google Scholar] [CrossRef][Green Version] - Goel, M.K. Runoff Coefficient, in Encyclopedia of Snow, Ice and Glaciers; Singh, V.P., Singh, P., Haritashya, U.K., Eds.; Springer: Dordrecht, The Netherlands, 2011; pp. 952–953. [Google Scholar] [CrossRef]
- Subramanya, K. Engineering Hydrology, 4th ed.; Tata McGraw-Hill Education: New Delhi, India, 2013. [Google Scholar]
- Viglione, A.; Hosking, J.R.; Laio, F.; Miller, A.; Gaume, E.; Payrastre, O.; Salinas, J.L.; N’guyen, C.C.; Halbert, K.; Viglione, M.A. Package ‘nsRFA’. Non-supervised Regional Frequency Analysis. CRAN Repository, 2020. Version 0.7-15. Available online: http://cran.r-project.org/web/packages/nsRFA/ (accessed on 20 March 2021).
- Hanssen-Bauer, I.; Drange, H.; Førland, E.; Roald, L.; Børsheim, K.; Hisdal, H.; Lawrence, D.; Nesje, A.; Sandven, S.; Sorteberg, A. Climate in Norway 2100. In Background Information to NOU Climate Adaptation (In Norwegian: Klima i Norge 2100. Bakgrunnsmateriale til NOU Klimatilplassing); Norsk Klimasenter: Oslo, Norway, 2009. [Google Scholar]
- Reis, D.S., Jr.; Stedinger, J.R. Bayesian MCMC flood frequency analysis with historical information. J. Hydrol.
**2005**, 313, 97–116. [Google Scholar] [CrossRef] - Gaume, E.; Gaál, L.; Viglione, A.; Szolgay, J.; Kohnová, S.; Blöschl, G. Bayesian MCMC approach to regional flood frequency analyses involving extraordinary flood events at ungauged sites. J. Hydrol.
**2010**, 394, 101–117. [Google Scholar] [CrossRef] - Lutz, J.; Grinde, L.; Dyrrdal, A.V. Estimating Rainfall Design Values for the City of Oslo, Norway—Comparison of Methods and Quantification of Uncertainty. Water
**2020**, 12, 1735. [Google Scholar] [CrossRef] - Dalbey, K.; Patra, A.; Pitman, E.; Bursik, M.; Sheridan, M. Input uncertainty propagation methods and hazard mapping of geophysical mass flows. J. Geophys. Res. Solid Earth
**2008**, 113, B05203. [Google Scholar] [CrossRef] - Vapnik, V. The Nature of Statistical Learning Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Dodangeh, E.; Panahi, M.; Rezaie, F.; Lee, S.; Bui, D.T.; Lee, C.-W.; Pradhan, B. Novel hybrid intelligence models for flood-susceptibility prediction: Meta optimization of the GMDH and SVR models with the genetic algorithm and harmony search. J. Hydrol.
**2020**, 590, 125423. [Google Scholar] [CrossRef] - Smola, A.J.; Schölkopf, B. A tutorial on support vector regression. Stat. Comput.
**2004**, 14, 199–222. [Google Scholar] [CrossRef][Green Version] - Shu, C.; Burn, D.H. Artificial neural network ensembles and their application in pooled flood frequency analysis. Water Resour. Res.
**2004**, 40, W09301. [Google Scholar] [CrossRef][Green Version] - Fernando, T.; Maier, H.; Dandy, G. Selection of input variables for data driven models: An average shifted histogram partial mutual information estimator approach. J. Hydrol.
**2009**, 367, 165–176. [Google Scholar] [CrossRef] - Lee Rodgers, J.; Nicewander, W.A. Thirteen ways to look at the correlation coefficient. Am. Stat.
**1988**, 42, 59–66. [Google Scholar] [CrossRef] - Karmakar, B.; Dhawane, S.H.; Halder, G. Optimization of biodiesel production from castor oil by Taguchi design. J. Environ. Chem. Eng.
**2018**, 6, 2684–2695. [Google Scholar] [CrossRef] - Yang, W.p.; Tarng, Y. Design optimization of cutting parameters for turning operations based on the Taguchi method. J. Mater. Process. Technol.
**1998**, 84, 122–129. [Google Scholar] [CrossRef] - Sadrzadeh, M.; Mohammadi, T. Sea water desalination using electrodialysis. Desalination
**2008**, 221, 440–447. [Google Scholar] [CrossRef] - Stahle, L.; Wold, S. Analysis of variance (ANOVA). Chemom. Intell. Lab. Syst.
**1989**, 6, 259–272. [Google Scholar] [CrossRef] - Alwosheel, A.; van Cranenburgh, S.; Chorus, C.G. Is your dataset big enough? Sample size requirements when using artificial neural networks for discrete choice analysis. J. Choice Model.
**2018**, 28, 167–182. [Google Scholar] [CrossRef] - Hjort, J.; Marmion, M. Effects of sample size on the accuracy of geomorphological models. Geomorphology
**2008**, 102, 341–350. [Google Scholar] [CrossRef] - Heckmann, T.; Gegg, K.; Gegg, A.; Becht, M. Sample size matters: Investigating the effect of sample size on a logistic regression susceptibility model for debris flows. Nat. Hazards Earth Syst. Sci.
**2014**, 14, 259–278. [Google Scholar] [CrossRef][Green Version] - Kalantar, B.; Pradhan, B.; Naghibi, S.A.; Motevalli, A.; Mansor, S. Assessment of the effects of training data selection on the landslide susceptibility mapping: A comparison between support vector machine (SVM), logistic regression (LR) and artificial neural networks (ANN). Geomat. Nat. Hazards Risk
**2018**, 9, 49–69. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Map of Tovdal river’s catchment area. (

**b**) The case study area with specified cross sections.

**Figure 2.**Water depth ranges in the calibrated model: (

**a**) computational domain, (

**b**) case study region.

**Figure 3.**Spatial distribution of the precipitation stations and the interpolated precipitation over the catchment.

**Figure 6.**Comparison of observed and predicted water level for test set by ANN and SVR at cross section 1 (

**a**), cross section 2 (

**b**), and cross section 3 (

**c**).

**Figure 7.**Absolute correlation values resulting from Kendall, Pearson, and Spearman tests and percentage of contribution (PC) values resulting from ANOVA test (Equation (10)).

**Figure 8.**Error trend and linear and polynomial regression of error values resulted from SVR and ANN predicted water level at the specified cross sections (

**CS1**to

**CS3**).

**Figure 9.**Empirical distribution of error values versus the normal distribution for the SVR and ANN based emulators. (

**a**) SVR error distribution-CS1, (

**b**) ANN error distribution-CS1, (

**c**) SVR error distribution-CS2, (

**d**) ANN error distribution-CS2, (

**e**) SVR error distribution-CS3, (

**f**) ANN error distribution-CS3.

**Figure 10.**The calculated (

**a**) RMSE and (

**b**) MaxAE box plots for test sets resulting from the models trained by different sample sizes for 5000 times.

Parameters | Acronym | Range | |
---|---|---|---|

Precipitation intensity (mm/h) | i_{p} | 5–8.1 | |

Strickler roughness (m^{1/3}/s) | Urban lands | K_{s}_U | 40–70 |

Forests (broad-leaved, coniferous) | K_{s}_F | 15–40 | |

Moors and heathland | K_{s}_Mo | 18–30 | |

Arable lands | K_{s}_A | 18–37 | |

Agriculture and vegetated areas | K_{s}_Ag | 18–37 | |

River and water courses | K_{s}_R | 20–50 | |

Mineral sites | K_{s}_Mi | 30–60 | |

Runoff coefficient (%) | Urban lands | C_U | 80–90 |

Forests (broad-leaved, coniferous) | C_F | 50–70 | |

Moors and heathland | C_Mo | 60–80 | |

Arable lands | C_A | 40–60 | |

Agriculture and vegetated areas | C_Ag | 40–60 | |

Mineral sites | C_Mi | 70–90 |

Location | SVR | ANN | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | R^{2} | MRAE | MaxAE | RMSE | R^{2} | MRAE | MaxAE | |

CS1 | 0.043 | 0.998 | 0.001 | 0.163 | 0.093 | 0.989 | 0.003 | 0.381 |

CS2 | 0.043 | 0.999 | 0.001 | 0.163 | 0.106 | 0.985 | 0.003 | 0.493 |

CS3 | 0.042 | 0.997 | 0.001 | 0.153 | 0.094 | 0.986 | 0.003 | 0.299 |

SVR | ANN | |||||||
---|---|---|---|---|---|---|---|---|

Location | RMSE | R^{2} | MRAE | MaxAE | RMSE | R^{2} | MRAE | MaxAE |

CS1 | 0.024 | 0.999 | 0.001 | 0.084 | 0.099 | 0.987 | 0.003 | 0.291 |

CS2 | 0.024 | 0.999 | 0.001 | 0.084 | 0.121 | 0.983 | 0.004 | 0.426 |

CS3 | 0.022 | 0.999 | 0.001 | 0.078 | 0.085 | 0.988 | 0.003 | 0.363 |

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**MDPI and ACS Style**

Mirza Alipour, S.; Leal, J.
Emulation of 2D Hydrodynamic Flood Simulations at Catchment Scale Using ANN and SVR. *Water* **2021**, *13*, 2858.
https://doi.org/10.3390/w13202858

**AMA Style**

Mirza Alipour S, Leal J.
Emulation of 2D Hydrodynamic Flood Simulations at Catchment Scale Using ANN and SVR. *Water*. 2021; 13(20):2858.
https://doi.org/10.3390/w13202858

**Chicago/Turabian Style**

Mirza Alipour, Saba, and Joao Leal.
2021. "Emulation of 2D Hydrodynamic Flood Simulations at Catchment Scale Using ANN and SVR" *Water* 13, no. 20: 2858.
https://doi.org/10.3390/w13202858