# An Index-Flood Statistical Model for Hydrological Drought Assessment

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## Abstract

**:**

## 1. Introduction

## 2. Study Area—Czech Republic

^{2}. The catchments are based on hydrological division of the Czech Republic as provided by the Czech Hydrometeorological Institute, which is also considered in the application of water management policies.

## 3. Data and Methods

#### 3.1. Data

#### Drought Definition

- event severity (deficit volume), D [mm or m
^{3}]; - event length, L [months];
- event intensity, $I=D/L$ [mm/month or m
^{3}/month]; - relative severity (i.e., deficit volume to monthly runoff ratio), $rD$ [-];
- relative event intensity, $rI=rD/L$ [t
^{−1}].

#### 3.2. Statistical Model

#### 3.3. Model Assessment

#### 3.3.1. Ratio Diagrams and Gumbel Plot

#### 3.3.2. Discordance

#### 3.3.3. Anderson–Darling Test

^{2}) test was chosen over goodness-of-fit framework within [22] based on the findings presented by [66] which specifically compares the Anderson–Darling test with methods used in [22] in order to make recommendations for test selection based on the assumed skewness of the data.

^{2}) test is a modification of the Cramér–von Mises test [67,68,69]. It differs from the Cramér–von Mises test in such a way that it gives more weight to the tails of the distribution [70]. A

^{2}is the most powerful empirical distribution function test [71]. The Anderson–Darling test statistic belongs to the quadratic class of the empirical distribution function statistic in which it is based on the squared difference ${({F}_{n}\left(x\right)-F\left(x\right))}^{2}$.

^{2}, are based on bootstrap resampling as suggested by [73] and used by [74].

^{2}calculated from the deficit volumes at catchment $s(s=1,\dots ,S)$ and let ${t}_{b}^{*}\left(s\right)$ be the value of A

^{2}from bootstrap sample b$(b=1,\dots ,B)$ for this catchment. For a chosen significance level ${\alpha}_{LOC}$, the local critical values ${c}^{{\alpha}_{LOC}}\left(s\right)$ are obtained for each catchment as the kth smallest value ${t}_{\left(k\right)}^{*}\left(s\right)$ of the ${t}_{b}^{*}\left(s\right)$, where $k=(1-{\alpha}_{LOC})(B+1)$.

- Fit the statistical model to the original sample.
- Calculate standard normal residuals with the parameter estimates from step using quantile mapping.
- Calculate the average correlation $\widehat{\rho}$ of the standard normal residuals.
- Generate a sample of S equicorrelated standard normal variables with correlation $\widehat{\rho}$.
- Transform the sample from step 4 back to the original scale using the parameter estimates from step 1.
- Fit the statistical model again.
- Calculate the A
^{2}statistics. - Repeat steps 4–7 until the desired number of bootstrap samples is obtained.

## 4. Results and Discussion

#### 4.1. Spatial Pooling

^{2}) tests. The diagrams were constructed by plotting the estimated sample L-moment ratios versus the theoretical L-moment ratio curves for the candidate distributions (Figure 4). From the considered distributions, the estimated L-moment ratios for deficit volumes correspond best to those of the Generalized Pareto Distribution (GPD). In addition, the Anderson–Darling test at the significance level ${\alpha}_{LOC}$ = 0.05 rejected the GPD only at six out of 133 catchments, which is very close to the nominal level of the test.

#### 4.2. Choice of the At-Site Distribution

#### 4.3. Drought Definition

#### 4.4. Reduction of Uncertainty

#### 4.5. Identification of Homogeneous Regions

## 5. Summary and Concluding Remarks

- Regional frequency analysis reduces uncertainty of estimated drought characteristics and parameters of its distribution.
- Use of Generalized Pareto Distribution is appropriate to describe the deficit volumes on majority of catchments, which is not the case for Generalized Extreme Value distribution. However, it is not clear to what extent this result depends on characteristics of the area under study and other parameters of the analysis like the threshold defining drought.
- The most subjective part of the regional frequency analysis is the definition of homogeneous regions—methods such as region of influence or Self Organizing maps could be considered to minimize the subjective decisions within the regional frequency analysis.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**Left panel**): Digital Elevation model of the Czech Republic; (

**Right panel**): Resulting clusters, discordance measure and results of at-site A

^{2}test.

**Figure 2.**Comparison of drought characteristics for observed and simulated runoff. The empirical quantiles of the individual characteristics are indicated on the horizontal axis, the vertical axis shows values of drought characteristics, the polygons correspond to interquartile range.

**Figure 3.**Left panels: Gumbel plot—Continuous black lines represent fitted regional quantile functions, grey points with lines are scaled deficit volumes with probabilities calculated using plotting position, dashed lines highlight discordant catchments; Right panels: Discordance measure showing ratio between coefficient of L-variation and L-skewness, discordant ratios lie outside the notional ellipsis (critical value) that would be drawn around the concordant values.

**Figure 4.**L-moment ratio diagram with highlighted clusters and results of at-site Anderson–Darling test. The dashed lines show the theoretical L-moment ratio for Generalized Pareto Distribution (GPD) and Generalized Extreme Value distribution (GEV) and the points the L-moment ratios for each catchment.

**Figure 5.**Estimated return periods of deficit volumes. Dashed line shows regional quantile function, darker area the 25th and 75th, light area 5th and 95th percent quantile calculated from the bootstrap samples.

**Table 1.**Average values of severity (D), intensity (I), length (L), relative severity (rD) and relative intensity (rI) of deficit events derived from simulated data.

Period | D | I | L | rD | rI |
---|---|---|---|---|---|

1901–1930 | 4.46 | 1.70 | 2.34 | 0.24 | 0.09 |

1931–1960 | 6.01 | 1.97 | 2.76 | 0.36 | 0.11 |

1961–1990 | 6.68 | 2.19 | 2.95 | 0.44 | 0.12 |

1991–2015 | 4.74 | 1.79 | 2.38 | 0.29 | 0.10 |

**Table 2.**Validation of simulated deficit volumes. D: severity, I: intensity, L: length, rD: relative severity and rI: relative intensity.

D | I | L | rD | rI | |
---|---|---|---|---|---|

Observed runoff | 5.25 | 1.94 | 2.29 | 0.24 | 0.09 |

Simulated runoff | 6.15 | 2.35 | 2.36 | 0.28 | 0.10 |

**Table 3.**Mean values of annual precipitation sum (P[mm]) for each cluster, average deficit volumes DV [mm] for each cluster and probabilities p

_{0}of year without drought event per cluster.

P [mm] | DV [mm] | p_{0} | |
---|---|---|---|

Cluster 1 | 993.87 | 21.65 | 0.30 |

Cluster 2 | 699.80 | 10.73 | 0.36 |

Cluster 3 | 574.50 | 6.43 | 0.49 |

ξ | α | κ | A^{2} CriticalValue | |
---|---|---|---|---|

Cluster 1 | −0.01 | 0.86 | −0.15 | 2.42 |

Cluster 2 | −0.02 | 0.83 | −0.19 | 2.64 |

Cluster 3 | −0.04 | 0.71 | −0.32 | 2.79 |

α | κ | 2_{yr} | 50_{yr} | |
---|---|---|---|---|

Cluster 1 | 99.86 | 69.97 | 67.99 | 66.44 |

Cluster 2 | 99.84 | 75.03 | 74.95 | 72.82 |

Cluster 3 | 99.40 | 55.94 | 56.28 | 52.04 |

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**MDPI and ACS Style**

Strnad, F.; Moravec, V.; Markonis, Y.; Máca, P.; Masner, J.; Stočes, M.; Hanel, M.
An Index-Flood Statistical Model for Hydrological Drought Assessment. *Water* **2020**, *12*, 1213.
https://doi.org/10.3390/w12041213

**AMA Style**

Strnad F, Moravec V, Markonis Y, Máca P, Masner J, Stočes M, Hanel M.
An Index-Flood Statistical Model for Hydrological Drought Assessment. *Water*. 2020; 12(4):1213.
https://doi.org/10.3390/w12041213

**Chicago/Turabian Style**

Strnad, Filip, Vojtěch Moravec, Yannis Markonis, Petr Máca, Jan Masner, Michal Stočes, and Martin Hanel.
2020. "An Index-Flood Statistical Model for Hydrological Drought Assessment" *Water* 12, no. 4: 1213.
https://doi.org/10.3390/w12041213