# Response of Runoff to Meteorological Factors Based on Time-Varying Parameter Vector Autoregressive Model with Stochastic Volatility in Arid and Semi-Arid Area of Weihe River Basin

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{3}/s R) data of the Linjiacun hydrological station, located in China, from 1 May to 30 September 2013 was selected. Precipitation (mm P), temperature (°C T), and evaporation (mm E) data from meteorological stations in the same period, from the daily value dataset of China’s surface climate data, was used in this paper. The daily observation indexes of the station are drawn in Figure 1.

^{3}/s. It increased significantly in summer and reached a peak of 1810 m

^{3}/s on 22 July. The average daily precipitation is 2.7 mm, of which more than 2 mm occurs five times. Five days before the runoff reaches the peak, the daily precipitation is greater than 30 mm twice and peaks on 22 July. The average temperature is 22.4 °C, which fluctuates periodically with the seasons. The average evaporation is 3.7 mm with more peaks.

#### 2.2. Principle of TVP-SV-VAR Model

**y**

_{t}is k × 1 dimension measured variable vector,

**A**,

**F**

_{1}…

**F**

_{s}are k × k-order coefficient matrix, respectively,

**u**

_{t}is k × 1 dimension structural impact. Suppose

**u**

_{t}obeys N (0,

**ΣΣ**) distribution, where

**A**can be expressed as

**y**

_{t}can be written as

**B**

_{i}can be further written as k

^{2}s × 1 dimension vector

**β**. In addition, defining

**A**

_{t}as ${\mathit{a}}_{t}=({a}_{21},{a}_{31},{a}_{32},\dots ,{a}_{k,k-1})$ and ${\mathit{h}}_{t}=({h}_{1t},\dots ,{h}_{kt})$, satisfying ${h}_{jt}=\mathrm{log}{\sigma}_{jt}^{2}$. Assuming that the parameters in Equation (7) satisfy the random walk process, ${\mathit{\beta}}_{t+1}={\mathit{\beta}}_{t}+{u}_{\beta t}$, ${\mathit{a}}_{t+1}={\mathit{a}}_{t}+{u}_{at}$, ${\mathit{h}}_{t+1}={\mathit{h}}_{t}+{u}_{ht}$ satisfy the distribution conditions

#### 2.3. TVP-SV-VAR Model of Runoff Response to Meteorological Factors

**,**and $\omega $. Sequential sampling $\mathit{\beta}|\mathit{a},\mathit{h},{\Sigma}_{\beta}$,

**y**; ${\mathsf{\Sigma}}_{\beta}|\mathit{\beta}$;

**a**$|\mathit{\beta},\mathit{h},{\mathsf{\Sigma}}_{a}$,

**y**; ${\mathsf{\Sigma}}_{a}|\mathit{a}$; $\mathit{h}|\mathit{\beta},\mathit{a},{\mathsf{\Sigma}}_{h}$,

**y**; ${\mathsf{\Sigma}}_{h}|\mathit{h}$. Then, repeat the sampling process.

- (1)
- Sample
**β**

**β**from the conditional posterior distribution, the state space model with respect to

**β**

_{t}as the state variable is written as

- (2)
- Sample
**a**

**a**

_{t}can be expressed as

- (3)
- Sample
**h**

**h**, there is

## 3. Results

#### 3.1. Parameter Estimation and Model Verification

#### 3.2. A Posteriori Estimation of Stochastic Volatility and Simultaneous Impulse Response Analysis

## 4. Discussion

#### 4.1. Pulse Response Analysis with Different Delays

#### 4.2. Pulse Response Analysis with Different Time Points

^{3}/s Node 49), mean value (150 m

^{3}/s Node 106), and maximum value (1810 m

^{3}/s Node 83), respectively. As shown in Figure 8, several conclusions can be reached. Firstly, runoff has a positive impulse response to precipitation, while it has a negative impulse response to temperature and evaporation. The increase in precipitation leads to an increase in runoff, while the increase in temperature and evaporation leads to a decrease in runoff. These are consistent with the actual hydrological characteristics, and the feasibility of the model is well demonstrated. Secondly, the response of different runoffs to precipitation and evaporation is similar, but the amount of runoff plays a decisive role in its response to temperature variability. When the runoff is small, the negative impact of temperature variability on runoff is large. When the runoff is large, the influence of temperature on runoff is not obvious. Finally, the influence of degree of precipitation, evaporation, and temperature on runoff increases first and then decreases with time. The impact of precipitation on runoff reaches the inflection point when t = 2, while the impact of evaporation on runoff reaches the inflection point when t = 3. Hence, the response of runoff to precipitation is more sensitive than that of evaporation. The sensitivity of runoff to temperature depends on runoff. To a certain extent, the impact of temperature on the runoff reaches the peak earlier when the runoff is large. The above model results are consistent with the actual hydrological characteristics of the Weihe River Basin.

#### 4.3. Implications of TVP-SV-VAR Model

#### 4.4. Limitations of TVP-SV-VAR Model

## 5. Conclusions

- (1)
- The posterior estimates of the stochastic volatility of runoff, precipitation, temperature, and evaporation vary significantly with time, and the variance fluctuations of runoff and precipitation have strong synchronicity.
- (2)
- The impact of precipitation and evaporation on the simultaneous pulse of runoff is close to 0. The simultaneous impulse response between temperature and evaporation is the largest.
- (3)
- Runoff has a positive impulse response to precipitation, which decreases with the increase in lag time. It has a negative impulse response to temperature and evaporation, which fluctuates greatly. The response speed is precipitation > evaporation > temperature.
- (4)
- When the runoff has different statistical values, the response curves to precipitation and evaporation are similar, and the response to temperature variability is more complex.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Runoff, precipitation, temperature, and evaporation data of a control basin above Linjiacun from 1 May to 30 September 2013.

**Figure 3.**Posterior estimates of stochastic fluctuations in runoff, precipitation, temperature, and evaporation.

**Figure 5.**Effects of runoff variation on precipitation, temperature, and evaporation under different lag periods.

**Figure 6.**Impulse response of runoff to precipitation, temperature, and evaporation under different lag periods.

**Figure 7.**Impulse response among precipitation, temperature, and evaporation under different lag periods.

**Figure 8.**Pulse response of runoff to precipitation, temperature, and evaporation at different time nodes.

Variable | ADF | 5% Critical Value | Logical Value | Conclusion |
---|---|---|---|---|

Runoff | −6.801 | −1.942 | 1 | stable |

Precipitation | −9.947 | −1.942 | 1 | stable |

Temperature | −3.588 | −1.942 | 1 | stable |

Evaporation | −6.623 | −1.942 | 1 | stable |

Parameter | Mean | Std | 95% Interval | Geweke | Inefficiency |
---|---|---|---|---|---|

${\left({{\displaystyle \sum}}_{\beta}\right)}_{1}$ | 0.0041 | 0.0012 | (0.0025, 0.0070) | 0.384 | 41.65 |

${\left({{\displaystyle \sum}}_{\beta}\right)}_{2}$ | 0.0041 | 0.0012 | (0.0024, 0.0071) | 0.609 | 36.97 |

${\left({{\displaystyle \sum}}_{a}\right)}_{1}$ | 0.0056 | 0.0014 | (0.0036, 0.0090) | 0.000 | 61.45 |

${\left({{\displaystyle \sum}}_{a}\right)}_{2}$ | 0.0056 | 0.0016 | (0.0034, 0.0098) | 0.235 | 44.10 |

${\left({{\displaystyle \sum}}_{h}\right)}_{1}$ | 1.0629 | 0.1051 | (0.8813, 1.2879) | 0.600 | 29.34 |

${\left({{\displaystyle \sum}}_{h}\right)}_{2}$ | 3.9061 | 0.2497 | (3.4574, 4.4399) | 0.000 | 75.99 |

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## Share and Cite

**MDPI and ACS Style**

Zeng, W.; Song, S.; Kang, Y.; Gao, X.; Ma, R.
Response of Runoff to Meteorological Factors Based on Time-Varying Parameter Vector Autoregressive Model with Stochastic Volatility in Arid and Semi-Arid Area of Weihe River Basin. *Sustainability* **2022**, *14*, 6989.
https://doi.org/10.3390/su14126989

**AMA Style**

Zeng W, Song S, Kang Y, Gao X, Ma R.
Response of Runoff to Meteorological Factors Based on Time-Varying Parameter Vector Autoregressive Model with Stochastic Volatility in Arid and Semi-Arid Area of Weihe River Basin. *Sustainability*. 2022; 14(12):6989.
https://doi.org/10.3390/su14126989

**Chicago/Turabian Style**

Zeng, Wenying, Songbai Song, Yan Kang, Xuan Gao, and Rui Ma.
2022. "Response of Runoff to Meteorological Factors Based on Time-Varying Parameter Vector Autoregressive Model with Stochastic Volatility in Arid and Semi-Arid Area of Weihe River Basin" *Sustainability* 14, no. 12: 6989.
https://doi.org/10.3390/su14126989