# A Three-Parameter Hydrological Model for Monthly Runoff Simulation—A Case Study of Upper Hanjiang River Basin

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. SCS Runoff Model and Generalized Proportionality Hypothesis

#### 2.2. Proportionality Hypothesis Application for Monthly Water Balance

#### 2.3. Model Structure and Solution Method

#### 2.3.1. The Solution of Actual Evapotranspiration

#### 2.3.2. The Solution of Soil Water Content and Runoff

^{3}/s) can be calculated by (Equation (7)):

^{3}/s) is finally calculated by the water balance equation after the calculation of ${S}_{t}$ (mm) as follows (Equation (10)):

#### 2.4. Parameter Optimization

#### 2.5. Model Evaluations

- (1)
- Nash-Sutcliffe Efficiency (NSE)

- (2)
- Kling–Gupta Efficiency (KGE)

## 3. Case Study

#### 3.1. Study Area

^{2}, is located in the upper reaches of Hanjiang River Basin in China, accounting for 60% of the total Hangjiang river basin area. For simplicity, the Upper Hanjiang River Basin is referred to as UHRB, as shown in Figure 1. UHRB includes innumerous rivers and complex terrain with mostly mountainous valleys and small areas of plains. Affected by the subtropical monsoon climate, the mean annual temperature varies from 12 °C to 16 °C, and the mean annual precipitation is approximately 873 mm with nearly 80% of the annual precipitation occurring in the summer. The average annual runoff is 4.11 billion m

^{3}. Correspondingly, the runoff distribution is uneven throughout the year, which is similar to that of precipitation, with great inter-annual differences.

#### 3.2. Data Sources

## 4. Result and Discussion

#### 4.1. Model Performance and Parameter Result

#### 4.2. Comparison of Model Performance with the TWBM Model and the ABCD Model

^{3}/s, whereas at lower values the performance of the three models is almost the same. It becomes more apparent in Figure 5, where the points of TMPH are more concentrated, especially at higher runoff values.

#### 4.3. Sensitivity Analysis of TMPH Model Parameters

## 5. Conclusions

- (1)
- The proposed TMPH model shows good performance in the monthly runoff simulation in UHRB, with a simple model structure and few parameters. Specifically, the values of NSE are 0.79 for the calibration period and 0.83 for the validation period, with the value of KGE are 0.86 for the calibration period and 0.78 for the validation period;
- (2)
- In UHRB, the proposed TMPH model have better performance than the TWBM model, especially in the case of higher runoff values. Specifically, the value of NSE increases from 0.72 to 0.79 for the calibration period, and from 0.72 to 0.83 for the validation period. Meanwhile, the value of KGE increases from 0.84 to 0.86 for the calibration period, and from 0.62 to 0.78 for the validation period;
- (3)
- In UHRB, the TMPH model shows a comparable result with the ABCD model in the calibration period, with NSE values of TMPH and ABCD of 0.79 and 0.80, respectively, the KGE values are both 0.86, respectively. While in the validation period, the TMPH model has improved slightly, with the value of NSE increasing from 0.81 to 0.83, while increasing the KGE from 0.75 to 0.78;
- (4)
- The sensitivity analysis of TMPH model parameters shows that the simulation result is most sensitive to parameter n, followed by parameter SC and parameter $\lambda $.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Monthly Models

#### Appendix A.1. TWBM Model

#### Appendix A.2. ABCD Model

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**Figure 4.**Scatter plots of observed and simulated monthly runoff by TMPH, TWBM, and ABCD in calibration period.

**Figure 5.**Comparison of the simulation performance derived from TMPH, TWBM, and ABCD for the selected years.

Parameters | Defintion | Ranges |
---|---|---|

$\lambda $ | Initial water loss ratio (-) | 0–1 |

SC | Water loss capacity of the basin (mm) | 0–2000 |

n | Evapotranspiration parameter (-) | 0–2 |

Model | Calibration Period | Validation Period | ||
---|---|---|---|---|

NSE | KGE | NSE | KGE | |

TMPH | 0.79 | 0.86 | 0.83 | 0.78 |

TWBM | 0.72 | 0.84 | 0.72 | 0.62 |

ABCD | 0.80 | 0.86 | 0.81 | 0.75 |

Parameters | Changes | Index Value | Percentage Change | ||
---|---|---|---|---|---|

NSE | KGE | NSE | KGE | ||

$\lambda $ | +50% | 0.76 | 0.76 | −3.80% | −11.63% |

−50% | 0.77 | 0.88 | −2.53% | 2.33% | |

SC | +50% | 0.74 | 0.69 | −6.33% | −19.77% |

−50% | 0.64 | 0.71 | −18.99% | −17.44% | |

n | +50% | 0.58 | 0.66 | −26.58% | −23.26% |

−50% | −0.13 | 0.13 | −116.46% | −84.88% |

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**MDPI and ACS Style**

Zou, Y.; Yan, B.; Feng, B.; Zhang, J.; Tang, Y.
A Three-Parameter Hydrological Model for Monthly Runoff Simulation—A Case Study of Upper Hanjiang River Basin. *Water* **2023**, *15*, 474.
https://doi.org/10.3390/w15030474

**AMA Style**

Zou Y, Yan B, Feng B, Zhang J, Tang Y.
A Three-Parameter Hydrological Model for Monthly Runoff Simulation—A Case Study of Upper Hanjiang River Basin. *Water*. 2023; 15(3):474.
https://doi.org/10.3390/w15030474

**Chicago/Turabian Style**

Zou, Yixuan, Baowei Yan, Baofei Feng, Jun Zhang, and Yiwei Tang.
2023. "A Three-Parameter Hydrological Model for Monthly Runoff Simulation—A Case Study of Upper Hanjiang River Basin" *Water* 15, no. 3: 474.
https://doi.org/10.3390/w15030474