# Hydrological Modelling in Data Sparse Environment: Inverse Modelling of a Historical Flood Event

^{*}

## Abstract

**:**

## 1. Introduction

- check the flood-peak-magnitude plausibility from the hydrological perspective and
- understand and quantify the flood generating processes and
- differentiate between model output uncertainty caused by uncertain model parameters and inputs, especially in a data sparse environment.

## 2. Material and Methods

#### 2.1. Study Area

#### 2.2. Data

#### 2.3. Hydrometeorological Conditions before and during the Flood Event

#### 2.4. Hydrological Model Setup

#### 2.5. Hydrological Model Calibration

#### 2.6. Historical Precipitation Simulation

- match the observed precipitation measured at the observation locations;
- have values which match the distribution of the observed values of the same day;
- have spatial variability which does not contradict the observations and have the variogram of a typical day of the season;
- and using them as input for the hydrological model, the resulting calculated discharge matches the observed discharge well.

## 3. Results

#### 3.1. Calibration and Validation Performance

**bold**in Table 1). $C{M}_{TE}$ converged to zero for DE while $C{M}_{PR}$ converged to 0.35. $FC$, $PWP$ and $\beta $ converged to 16, 16, and 1.6 respectively. These values show considerable departure from values that are common for this area, based on our experience. For example, a value of 16 for $PWP$ means that the amount of model evapotranspiration equals the PET if soil moisture is more than 16 mm. This almost always results in maximum evapotranspiration, a scenario that is highly unlikely.

#### 3.2. Inverted Precipitation Performance

#### 3.3. Inverted Precipitation Plausibility

#### 3.4. Snowmelt Contribution to the Peak

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DE | Differential Evolution |

HBV | Hydrologiska Byråns Vattenbalansavdelning model |

IDW | Inverse distance weighting |

NSE | Nash-Sutcliffe efficiency |

OK | Ordinary Kriging |

PET | Potential evapotranspiration |

ROPE | Robust Parameter Estimation |

W.E. | Water Equivalent |

## References

- Benito, G.; Lang, M.; Barriendos, M.; Llasat, M.C.; Francés, F.; Ouarda, T.; Thorndycraft, V.; Enzel, Y.; Bardossy, A.; Coeur, D.; et al. Use of Systematic, Palaeoflood and Historical Data for the Improvement of Flood Risk Estimation. Review of Scientific Methods. Nat. Hazards
**2004**, 31, 623–643. [Google Scholar] [CrossRef] - Glaser, R.; Riemann, D.; Schönbein, J.; Barriendos, M.; Brázdil, R.; Bertolin, C.; Camuffo, D.; Deutsch, M.; Dobrovolný, P.; van Engelen, A.; et al. The variability of European floods since AD 1500. Clim. Chang.
**2010**, 101, 235–256. [Google Scholar] [CrossRef] - Sudhaus, D.; Seidel, J.; Bürger, K.; Dostal, P.; Imbery, F.; Mayer, H.; Glaser, R.; Konold, W. Discharges of past flood events based on historical river profiles. Hydrol. Earth Syst. Sci.
**2008**, 12, 1201–1209. [Google Scholar] [CrossRef] [Green Version] - Herget, J.; Roggenkamp, T.; Krell, M. Estimation of peak discharges of historical floods. Hydrol. Earth Syst. Sci.
**2014**, 18, 4029–4037. [Google Scholar] [CrossRef] [Green Version] - Bürger, K.; Seidel, J.; Dostal, P.; Imbery, F.; Barriendos, M.; Mayer, H.; Glaser, R. Hydrometeorological reconstruction of the 1824 flood event in the Neckar River basin (southwest Germany). Hydrol. Sci. J.
**2006**, 51, 864–877. [Google Scholar] [CrossRef] - Seidel, J.; Imbery, F.; Dostal, P.; Sudhaus, D.; Bürger, K. Potential of historical meteorological and hydrological data for the reconstruction of historical flood events—The example of the 1882 flood in southwest Germany. Nat. Hazards Earth Syst. Sci.
**2009**, 9, 175–183. [Google Scholar] [CrossRef] - Bomers, A.; van der Meulen, B.; Schielen, R.M.J.; Hulscher, S.J.M.H. Historic Flood Reconstruction with the Use of an Artificial Neural Network. Water Resour. Res.
**2019**, 55, 9673–9688. [Google Scholar] [CrossRef] [Green Version] - Fischer, S.; Schumann, A. Spatio-temporal consideration of the impact of flood event types on flood statistic. Stoch. Environ. Res. Risk Assess.
**2020**, 34, 1331–1351. [Google Scholar] [CrossRef] - Kirchner, J.W. Catchments as simple dynamical systems: Catchment characterization, rainfall-runoff modeling, and doing hydrology backward. Water Resour. Res.
**2009**, 45. [Google Scholar] [CrossRef] [Green Version] - Herrnegger, M.; Nachtnebel, H.P.; Schulz, K. From runoff to rainfall: Inverse rainfall–runoff modelling in a high temporal resolution. Hydrol. Earth Syst. Sci.
**2015**, 19, 4619–4639. [Google Scholar] [CrossRef] [Green Version] - Wright, A.J.; Walker, J.P.; Pauwels, V.R.N. Estimating rainfall time series and model parameter distributions using model data reduction and inversion techniques. Water Resour. Res.
**2017**, 53, 6407–6424. [Google Scholar] [CrossRef] - Boudhraâ, H.; Cudennec, C.; Andrieu, H.; Slimani, M. Net rainfall estimation by the inversion of a geomorphology-based transfer function and discharge deconvolution. Hydrol. Sci. J.
**2018**, 63, 285–301. [Google Scholar] [CrossRef] - Wright, A.J.; Walker, J.P.; Pauwels, V.R.N. Identification of Hydrologic Models, Optimized Parameters, and Rainfall Inputs Consistent with In Situ Streamflow and Rainfall and Remotely Sensed Soil Moisture. J. Hydrometeorol.
**2018**, 19, 1305–1320. [Google Scholar] [CrossRef] - Grundmann, J.; Hörning, S.; Bárdossy, A. Stochastic reconstruction of spatio-temporal rainfall patterns by inverse hydrologic modelling. Hydrol. Earth Syst. Sci.
**2019**, 23, 225–237. [Google Scholar] [CrossRef] [Green Version] - Brázdil, R.; Kundzewicz, Z.W.; Benito, G. Historical hydrology for studying flood risk in Europe. Hydrol. Sci. J.
**2006**, 51, 739–764. [Google Scholar] [CrossRef] - DWD. 2020. Available online: https://opendata.dwd.de/climate_environment/CDC/observations_germany/climate/daily/ (accessed on 18 November 2019).
- LUBW. 2020. Available online: https://udo.lubw.baden-wuerttemberg.de/public/ (accessed on 6 February 2020).
- Wackernagel, H. Ordinary Kriging. In Multivariate Geostatistics: An Introduction with Applications; Springer: New York, NY, USA, 2004; Chapter 11; pp. 74–81. [Google Scholar]
- Hargreaves, G.; Samani, Z. Estimating potential evapotranspiration. J. Irrig. Drain. Div.
**1982**, 108, 225–230. [Google Scholar] - Shepard, D. A Two-Dimensional Interpolation Function for Irregularly-Spaced Data. In Proceedings of the 1968 23rd ACM National Conference, ACM ’68, New York, NY, USA, 27–29 August 1968; Association for Computing Machinery: New York, NY, USA, 1968; pp. 517–524. [Google Scholar] [CrossRef]
- Bergström, S. The HBV Model: Its Structure and Applications; SMHI Reports Hydrology; SMHI: Norrköping, Sweden, 1992. [Google Scholar]
- Das, T.; Bárdossy, A.; Zehe, E.; He, Y. Comparison of conceptual model performance using different representations of spatial variability. J. Hydrol.
**2008**, 356, 106–118. [Google Scholar] [CrossRef] - Francés, F.; Vélez, J.I.; Vélez, J.J. Split-parameter structure for the automatic calibration of distributed hydrological models. J. Hydrol.
**2007**, 332, 226–240. [Google Scholar] [CrossRef] - Anwar, F.; Bárdossy, A. HBV in C++ and Cython. 2020. Available online: https://github.com/faizan90/hydmodeling/tree/master/models/hbvs/ (accessed on 6 August 2020).
- Nash, J.; Sutcliffe, J. River flow forecasting through conceptual models. 1. A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential Evolution—A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Bárdossy, A.; Singh, S.K. Robust estimation of hydrological model parameters. Hydrol. Earth Syst. Sci.
**2008**, 12, 1273–1283. [Google Scholar] [CrossRef] [Green Version] - Cullmann, J.; Krausse, T.; Saile, P. Parameterising hydrological models—Comparing optimisation and robust parameter estimation. J. Hydrol.
**2011**, 404, 323–331. [Google Scholar] [CrossRef] - Scott, D. Multivariate Density Estimation: Theory, Practice, and Visualization; A Wiley-interscience publication; Wiley: Hoboken, NJ, USA, 1992. [Google Scholar]
- Hörning, S.; Sreekanth, J.; Bárdossy, A. Computational efficient inverse groundwater modeling using Random Mixing and Whittaker-Shannon interpolation. Adv. Water Resour.
**2019**, 123, 109–119. [Google Scholar] [CrossRef] - Beven, K.; Binley, A. GLUE: 20 years on. Hydrol. Process.
**2014**, 28, 5897–5918. [Google Scholar] [CrossRef] [Green Version] - Bárdossy, A.; Das, T. Influence of rainfall observation network on model calibration and application. Hydrol. Earth Syst. Sci.
**2008**, 12, 77–89. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Maps showing (

**a**) the location of the study area in Germany, (

**b**) the modelling domain with the sites of the historical precipitation gauges and (

**c**) the catchment area of the gauge at Kirchentellinsfurt.

**Figure 2.**Well performing model parameters (N = 441) for the present day calibration period using ROPE.

**Figure 5.**Observed (red) and calculated (gray) discharges for November and December 1882 using parameters from the present calibration period.

**Figure 6.**Observed (red) and calculated (gray) discharges for November and December 1882 using the interpolated precipitation and 441 different parameter sets.

**Figure 7.**Observed (red) and (gray) calculated discharges for November and December 1882 using parameters from the historical period.

**Figure 10.**Gridded inverted precipitation fields for 24 December 1882. The upper two panels show the most dissimilar simulated pair, the lower the most similar pair.

**Figure 11.**Gridded simulated precipitation fields for 25 December 1882. The upper two panels show the most dissimilar simulated pair, the lower the most similar pair.

**Figure 12.**Observed (red) and calculated (gray) discharges for the November and December 1882 using the 100 simulated realizations.

**Figure 13.**Observed (red) and calculated (gray) discharges for November and December 1882 using the 100 precipitation realizations and 441 different parameter sets.

**Figure 14.**Observed precipitation (

**left**) interpolated from observed data on 11 May 2009 and interpolated (

**right**) using simulated values (VV7) at the observed locations on 24 December 1882.

**Table 1.**HBV model parameter values for present and historical calibrations using DE. Parameter values of suspicious magnitude are in

**bold**.

Parameter | Units | Minimum | Maximum | 1991–2000 | 1882 |
---|---|---|---|---|---|

$TT$ | ${}^{\circ}$C | −1 | 1 | 0.21 | 0.84 |

$C{M}_{TE}$ | mm/day/${}^{\circ}$C | 0 | 4 | 2.94 | 0 |

$C{M}_{PR}$ | mm/day/${}^{\circ}$C/mm | 0 | 2 | 0.14 | 0.31 |

$FC$ | mm | 1 | 700 | 385 | 16 |

$\beta $ | - | 0 | 7 | 2.44 | 1.62 |

$PWP$ | mm | 1 | 700 | 323 | 16 |

$UT$ | mm | 0 | 100 | 7 | 26 |

${K}_{uu}$ | 1/day | 0 | 0.7 | 0.27 | 0.70 |

${K}_{ul}$ | 1/day | 0 | 0.6 | 0.14 | 0.35 |

${K}_{d}$ | 1/day | 0 | 0.7 | 0.30 | 0.65 |

${K}_{ll}$ | 1/day | 0 | 0.3 | 0.08 | 0.06 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bárdossy, A.; Anwar, F.; Seidel, J.
Hydrological Modelling in Data Sparse Environment: Inverse Modelling of a Historical Flood Event. *Water* **2020**, *12*, 3242.
https://doi.org/10.3390/w12113242

**AMA Style**

Bárdossy A, Anwar F, Seidel J.
Hydrological Modelling in Data Sparse Environment: Inverse Modelling of a Historical Flood Event. *Water*. 2020; 12(11):3242.
https://doi.org/10.3390/w12113242

**Chicago/Turabian Style**

Bárdossy, András, Faizan Anwar, and Jochen Seidel.
2020. "Hydrological Modelling in Data Sparse Environment: Inverse Modelling of a Historical Flood Event" *Water* 12, no. 11: 3242.
https://doi.org/10.3390/w12113242