# Nonstationary Annual Maximum Flood Frequency Analysis Using a Conceptual Hydrologic Model with Time-Varying Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}/s to 8335 m

^{3}/s for the past decades. This study can serve as a reference for flood risk management in WRB and possibly for other basins undergoing drastic changes caused by intense human activities.

## 1. Introduction

## 2. Study Area and Data

^{2}, as shown in Figure 1. The elevation within WRB ranges from 340 to 3671 m, with a decrease from northwest to east.

## 3. Methodology

#### 3.1. GR4J Model: Stationary and Nonstationary

#### 3.2. Parameter Estimation and Model Selection

^{3}/s for the model evaluation procedure, thus placing greater emphasis on the simulation performance of high flow [70]. This censoring threshold was chosen because roughly 40% days from 1951 to 2011 witnessed a flow below 100 m

^{3}/s for WRB.

#### 3.3. Deriving Flood Frequency Distribution through Monte Carlo Simulation

## 4. Results

#### 4.1. Simulation Performance of the Stationary and Nonstationary GR4J Models

_{AMF}in Figure 4.

_{AMF}= 0.808) when compared to model zero (NSE

_{AMF}= 0.721) and model one (NSE

_{AMF}= 0.748), together with comparative narrower uncertainty bounds. This also proves the benefit and advantage of incorporating reservoir storage RS and soil-water conservation project SWC that are related to local regulation strategies as covariates.

#### 4.2. Dynamic Variation of Time-Varying Parameter

#### 4.3. Performance of the Stochastic Rainfall Model

#### 4.4. Flood Frequency Derivation Using the Stationary and Nonstationary Model

^{3}/s (see Figure 7). For model one, the corresponding flood magnitudes were continuously on a downward trend. This is expected since the parameter ${\theta}_{1}$ of model one was time-dependent only. An FFA of the Huanxian station using model two (see Figure 8) shows that the flood quantiles for an annual exceedance probability of 0.01 range from 4187 m

^{3}/s (in 2011) to 8335 m

^{3}/s (in 1960). A noticeable decline occurred in the late 1970s when the reservoir storage and soil-water conservation area were undergoing drastic changes, indicating a more reasonable interpretation of the variation of flood magnitudes over time compared to model one. The variation in flood quantiles is in line with the work of Xiong et al. [25] and Hesarkazzazi et al. [74].

## 5. Discussion

## 6. Conclusions

- Improved model performance can be achieved when the parameter ${\theta}_{1}$ (representing the production storage capacity) is treated as time-dependent or as a function of external variables that reflect changes in the hydrological responses within WRB;
- Incorporating the watershed conditions as covariates for the model parameter (model two) can better describe nonstationarity in the flood frequency and magnitude in WRB than the over-simplified time-dependent model (model one);
- The nonstationary model can achieve a more rational description of variations in the frequency and magnitude of floods over time, which reveals the deficiency when applying a stationary model under changing watershed conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of Weihe River Basin (WRB) in China (

**a**) and stations with hydrological and meteorological measurements (

**b**).

**Figure 2.**Annual and the 11-year moving average value of 6 covariates related to human activities. (

**a**–

**f**) represent population, gross domestic product, cultivated land area, irrigated land area, reservoir storage and soil-water conservation land area, respectively.

**Figure 3.**Comparison between the observed and simulated flow-duration curves for the whole period (

**a**) and autumn season (

**b**) from model zero, one and two.

**Figure 4.**Comparison of the observed and simulated annual maximum flow. (

**a**–

**c**) represent model zero, one and two, respectively.

**Figure 6.**Comparison of empirical probability distributions of observed and model-generated annual maximum rainfall series for Tianshui station.

**Figure 7.**Variation of the estimated annual maximum floods for an exceedance probability of 0.01 based on model zero and one.

**Figure 8.**Variation of the estimated annual maximum floods for an exceedance probability of 0.01 based on model zero and two.

Parameter | Description | Unit | Feasible Range |
---|---|---|---|

${\theta}_{1}$ | production storage capacity | mm | 20–1200 |

${\theta}_{2}$ | groundwater exchange coefficient | mm | −5–3 |

${\theta}_{3}$ | one day ahead maximum capacity of the routing store | mm | 20–500 |

${\theta}_{4}$ | time base of unit hydrograph | days | 1–5 |

Case | Covariate | Equation for the Time-Varying Parameter ${\mathit{\theta}}_{1,\mathit{t}}$ | NSE [-] | KGE [-] | RE [%] | |
---|---|---|---|---|---|---|

RS | SWC | |||||

C0 | ${\theta}_{1,t}=542.1$ | 0.711 | 0.801 | 11.5 | ||

C1 | ✓ | ${\theta}_{1,t}=290.3+195.6\frac{R{S}_{t}}{\overline{RS}}$ | 0.753 | 0.845 | 8.9 | |

C2 | ✓ | ${\theta}_{1,t}=296.8+152.4\frac{SW{C}_{t}}{\overline{SWC}}$ | 0.748 | 0.839 | 9.4 | |

C3 | ✓ | ✓ | ${\theta}_{1,t}=304.5+102.1\frac{R{S}_{t}}{\overline{RS}}+69.2\frac{SW{C}_{t}}{\overline{SWC}}$ | 0.765 | 0.853 | 8.5 |

Model | Covariates | Calibration Period (1951–1990) | Validation Period (1991–2011) | ||||
---|---|---|---|---|---|---|---|

NSE [-] | KGE [-] | RE [%] | NSE [-] | KGE [-] | RE [%] | ||

Model zero | none | 0.711 | 0.801 | 11.5 | 0.692 | 0.791 | 13.5 |

Model one | time | 0.743 | 0.832 | 9.8 | 0.714 | 0.812 | 10.1 |

Model two | RS, SWC | 0.765 | 0.853 | 8.5 | 0.716 | 0.818 | 9.8 |

**Table 4.**Event characteristics of the observed rainfall series (OBS) and the synthetic rainfall series (STOCH) for the meteorological station Tianshui.

Event Characteristics | Unit | OBS | STOCH |
---|---|---|---|

Number of rainfall events per year | - | 54 | 48 |

Mean of event volume | mm | 9.54 | 10.08 |

Standard deviation of event volume | mm | 14.76 | 13.58 |

Skewness of event volume | - | 3.24 | 3.52 |

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**MDPI and ACS Style**

Zeng, L.; Bi, H.; Li, Y.; Liu, X.; Li, S.; Chen, J. Nonstationary Annual Maximum Flood Frequency Analysis Using a Conceptual Hydrologic Model with Time-Varying Parameters. *Water* **2022**, *14*, 3959.
https://doi.org/10.3390/w14233959

**AMA Style**

Zeng L, Bi H, Li Y, Liu X, Li S, Chen J. Nonstationary Annual Maximum Flood Frequency Analysis Using a Conceptual Hydrologic Model with Time-Varying Parameters. *Water*. 2022; 14(23):3959.
https://doi.org/10.3390/w14233959

**Chicago/Turabian Style**

Zeng, Ling, Hongwei Bi, Yu Li, Xiulin Liu, Shuai Li, and Jinfeng Chen. 2022. "Nonstationary Annual Maximum Flood Frequency Analysis Using a Conceptual Hydrologic Model with Time-Varying Parameters" *Water* 14, no. 23: 3959.
https://doi.org/10.3390/w14233959