# Probability Analysis and Control of River Runoff–sediment Characteristics based on Pair-Copula Functions: The Case of the Weihe River and Jinghe River

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Area and Data

#### 2.1.1. Brief Introduction to the Research Area

^{2}. Its total length reaches 818 km. The upper reaches of the Weihe River start at Weiyuan and end at the exit of the Baoji Canyon; the middle reaches start at the Baoji Canyon and end at Xianyang; and the lower reaches of the river are from Xianyang to the Tongguan river estuary. Huaxian Station is the mouth where the Weihe River joins the Yellow River. Four seasons are distinct in the Weihe River basin, and precipitation is not uniformly distributed in the area. It is dry, with little rain, in the winter, with heavy rains in the summer. In this area, the average annual precipitation is 572 mm; however, more than 60% of the annual rainfall is from July to October. The rainfall is scarce in January and December, much less than 1% of the annual precipitation. At Xianyang Station, the annual average runoff is 36.8 × 10

^{8}m

^{3}(1960–2016), and the annual sediment runoff is 0.88 × 10

^{8}t (1960–2016). As for flood season, water and sediment over the whole year take up 59% and 85%, respectively. Downstream at the Huaxian Station, the annual average runoff is 63.4 × 10

^{8}m

^{3}(1960–2016), and average sediment is 2.73 × 10

^{8}t (1960–2016); runoff and sediment in flood season (from July to October) cover 60% and 89% or so, respectively, for the whole year. Mountainous areas of the WRB (Weihe River basin) mark 84% of the total area of the WRB, which is composed of plateau mountains and plain mountains. Plateau mountains of the WRB are mainly located in basins of the Jinghe and upper Weihe Rivers. The topographic feature of WRB is “west high, east low”, where the highest altitude in the west reaches 3495 m. From the west to the east, the terrain gradually becomes lower and the river valley wider. The climate in the WRB is a warm-temperate, semi-wet, and semi-arid monsoon climate, with characteristics of a mountain basin climate. The average temperature of WRB in a year is 12.8 °C [19].

^{2}, which is 33.7% of the WRB. The Jinghe River flows through the Ningxia Hui Autonomous Region, as well as the Gansu and Shaanxi Provinces, and meets the Weihe River in the Gaoling District, Xi’an City, Shaanxi Province. The topographic feature of the Jinghe River Basin (JRB) is “northwest high, southeast low”. With very sparse vegetation (only making 10% of the total area), soil erosion in the JRB area is severe, making it the main sediment source of the Weihe River. Hydrologic stations of the Jinghe River are illustrated in Figure 1.

#### 2.1.2. Data

^{3}/s; ${C}_{si}$ was the average sediment concentration in kg/m

^{3}at the cross-section when it was at i. $\Delta {t}_{i}$ was in h, showing the time interval between ${C}_{si}$ and ${C}_{s\left(i+1\right)}$; n was the times for measuring the index within a day. (2) In cases where the index sediment concentration was obtained only once a day, we took the average sediment concentration of the cross-section derived from the index sediment concentration as the daily average sediment concentration of the cross-section, then multiplied it by the daily average river flow, and the daily average discharge could be received for the day; (3) when taking the index sediment concentration once every few days, and the index sediment concentration was not measured, the average sediment concentration of the cross-section for each different day was acquired by straight-line interpolation for the cross-section sediment concentration before and after the index sediment concentration had been measured; after that, we multiplied it by the average river flow of each different day, and the daily average sediment discharge for each day was obtained. The formula for calculating annual sediment discharge in this paper was ${W}_{s}=(86400{{\displaystyle \sum}}_{i=1}^{m}\overline{{Q}_{s}})/{10}^{11}$, and in it, ${W}_{s}$ was the annual discharge in 10

^{8}t, there were 86,400 s for a day, and m was the days of a year. The aforementioned data were provided by the Shaanxi Provincial River Reservoir Administration. The data were strictly controlled in quality before release, since the Bureau was authorized for data statistics, publishing for all the rivers and reservoirs in the province.

#### 2.2. Research Methods

#### 2.2.1. Rescaled Adjusted Partial Sums (RAPS) Method

#### 2.2.2. Pair-Copula Function

_{j}has one node connected to n − 1 edges. Figure 2 indicates the structure of a 4-dimensional C-vine [37].

#### 2.2.3. Three-Dimensional C-Vine Pair-Copula Function Probability Calculation

#### 2.2.4. Calculation of the Synchronous and Asynchronous Probability of High–Low Runoff–Sediment

_{1}, x

_{2}, x

_{3}) to be the joint distribution function for the annual runoff of the 3 different stations, F(0.25, 0.25, 0.25) could be calculated by using formula (11); therefore, the value presents the probability when the annual runoff of the stations are all at their lowest. Consequently, the probabilities could also be available when they at their highest and at a normal level for the stations. Adding up the three probabilities, we derive the synchronous probability of annual runoff; then, the asynchronous probability is obtained as 1 minus the synchronous probability. Similarly, the synchronous and asynchronous probability of sediment could be calculated in the same way.

## 3. Results and Discussion

#### 3.1. Division of the Study Intervals

#### 3.2. Establishment of the Marginal Distribution Function

#### Establishment of the 2D Joint Distribution Functions

#### 3.3. Analysis on the Runoff–Sediment Relationships between the Three Stations through Joint Distribution Functions

#### 3.3.1. Analysis on Runoff–Sediment Probability of the Three Stations through Pair-Copula Function

#### 3.3.2. Probability Analysis on High–Low runoff and Sediment Yield of the Three Stations

#### 3.3.3. Probability Control of Runoff and Sediment Yield of the Three Stations

_{runoff}< 0.6. Then, if the runoff frequencies of the Xianyang and Zhangjiashan Stations, which are located upstream of the Huaxian Station, are set as 0.4 < p

_{runoff}< 0.6, then the probability that the runoff frequency of the Huaxian Station is 0.4 < p

_{runoff}< 0.6 (40.82 × 10

^{8}m

^{3}< X

_{runoff}< 48.1 × 10

^{8}m

^{3}) will be 0.778. Similarly, in low runoff years, considering the regulation abilities upstream of the Huaxian Station, the runoff frequency is 0.1 < p

_{runoff}< 0.25, and the calculation results are listed in Table 6. If further soil and water conservation works are strengthened upstream of the Huaxian Station, as well as decontamination of the river channels on a regular basis, the sediment yield of the downstream Weihe River will reduce significantly. Considering the present measures upstream of the Huaxian Station, and applying the marginal distribution function and C-vine copula function built using the 1996–2016 sediment yields, the adjusted frequencies of sediment yields of the three stations are measured and calculated, respectively, with 0.4 < p

_{sediment}< 0.6 in high sediment years, and 0.1 < p

_{sediment}< 0.25 in low sediment years. See Table 6 for the calculation results.

^{8}m

^{3}and 14.50 × 10

^{8}m

^{3}, while that of the Zhangjiashan Station is between 2.30 × 10

^{8}m

^{3}and 5.60 × 10

^{8}m

^{3}, then the probability of the annual runoff yield of the Huaxian Station being between 17.80 × 10

^{8}m

^{3}and 32.50 × 10

^{8}m

^{3}will be 70.55%. Similarly, in high runoff years, if the annual runoff yield of the Xianyang Station is adjusted to somewhere between 18.70 × 10

^{8}m

^{3}and 26.50 × 10

^{8}m

^{3}, while that of the Zhangjiashan Station is between 7.00 × 10

^{8}m

^{3}and 10.10 × 10

^{8}m

^{3}, then the probability of annual runoff yield of the Huaxian Station being between 40.82 × 10

^{8}m

^{3}and 48.10 × 10

^{8}m

^{3}will be 77.8%. If the annual sediment yield of the Xianyang Station is adjusted to somewhere between 0.22 × 10

^{8}m

^{3}and 1.21 × 10

^{8}m

^{3}, then the probability of annual sediment yield of the Huaxian Station being between 0.92 × 10

^{8}m

^{3}and 1.49 × 10

^{8}m

^{3}is 81.56%. In practical applications, different conditional probabilities can also be calculated, as needed, through the established joint distribution functions and conditional probability formulas.

## 4. Conclusions

- (1)
- The pair-copula function is flexible in adoption and simple in parameter solution. It has obvious advantages, in terms of building multidimensional joint distribution functions. Kernel distribution theory and C-vine pair-copula functions were used to obtain the annual runoff and sediment marginal distribution functions and the runoff–sediment joint distribution functions of the three stations of the Weihe River, in different time intervals, accurately.
- (2)
- The synchronous and asynchronous encounter probabilities of high–low runoff between the three stations, and between each pair of stations, were accurately calculated. Meanwhile, through conditional probability formulas, the probabilities that the high, normal, and low status of runoff and sediment yield of downstream hydrologic stations were caused by different statuses of different upstream stations were also calculated.
- (3)
- The designed runoff–sediment yields of upstream stations were estimated, in order to guarantee the runoff–sediment yields of downstream stations being in a certain range, in high and low runoff years, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Figure of annual runoff, sediment, and precipitation yield of Xianyang Station, Zhangjiashan Station, and Huaxian Station. (

**a**) RAPS figure of annual runoff. (

**b**) RAPS figure of annual sediment. (

**c**) Figure of annual precipitation and runoff of Xianyang Station.

**Figure 4.**The marginal distribution function diagram of annual runoff and sediment yield of the Three Stations: (

**a**) Runoff of Xianyang Station. (

**b**) Runoff of Zhangjiashan Station. (

**c**) Runoff of Huaxian Station. (

**d**) Sediment of Xianyang Station. (

**e**) Sediment of Zhangjiashan Station. (

**f**) Sediment of Huaxian Station.

**Figure 6.**Probability of synchronous high–low runoff–sediment of the two stations. Note: HH stands for synchronous high runoff/sediment of the two stations; NN stands for normal runoff/sediment of the two stations; LL stands for synchronous low runoff/sediment of the two stations; and 60–90 stands for the time interval of 1960 to 1990, and similarly for the others. (

**a**) is the probability of synchronous high–low runoff of the two stations, with unit of %; and (

**b**) is the probability of synchronous high–low sediment of the two stations, with unit of %.

Runoff | Year | 1960–1990 | 1990–2016 | 1960–2016 | |||

D | p-Value | D | p-Value | D | p-Value | ||

Xianyang | 0.0972 | 0.2903 | 0.1266 | 0.0610 | 0.0652 | 0.5564 | |

Zhangjiashan | 0.0888 | 0.4695 | 0.1062 | 0.2348 | 0.0507 | 0.9680 | |

Huaxian | 0.0933 | 0.3636 | 0.1018 | 0.3157 | 0.0663 | 0.5824 | |

Sediment | Year | 1960–1996 | 1996–2016 | 1960–2016 | |||

D | p-Value | D | p-Value | D | p-Value | ||

Xianyang | 0.0982 | 0.1818 | 0.1171 | 0.3157 | 0.1156 | 0.0718 | |

Zhangjiashan | 0.0681 | 0.9161 | 0.0995 | 0.6723 | 0.0646 | 0.6923 | |

Huaxian | 0.0986 | 0.1858 | 0.1062 | 0.4715 | 0.0642 | 0.6703 |

**Table 2.**Selection and test of joint distribution functions of the runoff and sediment yields of the three stations.

C-Vine | Year | Variables | Copula | θ | p-Value |
---|---|---|---|---|---|

Runoff | 1960–1990 | 3, 2 | Frank | 11.83 | 0.42 |

3, 1 | Frank | 36.22 | |||

1, 2; 3 | Frank | −0.69 | |||

1990–2016 | 3, 2 | Gumbel | 1.35 | 0.53 | |

3, 1 | Frank | 29.21 | |||

1, 2; 3 | Frank | −0.91 | |||

1960–2016 | 3, 2 | Gumbel | 2.39 | 0.31 | |

3, 1 | Gumbel | 8.14 | |||

1, 2; 3 | Frank | −0.03 | |||

Sediment | 1960–1996 | 3, 1 | Gumbel | 2.50 | 0.07 |

3, 2 | Gumbel | 4.59 | |||

2, 1; 3 | Frank | −7.73 | |||

1996–2016 | 3, 1 | Clayton | 2.82 | 0.40 | |

3, 2 | Clayton | 7.18 | |||

2, 1; 3 | Frank | −4.96 | |||

1960–2016 | 3, 1 | Frank | 10.27 | 0.31 | |

3, 2 | Frank | 16.35 | |||

1, 2; 3 | Frank | −4.44 |

**Table 3.**Probabilities of synchronous and asynchronous high–low runoff–sediment of the three stations.

Year | LLL | NNN | HHH | Synchronous | Asynchronous | |
---|---|---|---|---|---|---|

Runoff | 1960–1990 | 0.1815 | 0.3626 | 0.1815 | 0.7256 | 0.2744 |

1990–2016 | 0.0886 | 0.2477 | 0.1082 | 0.4445 | 0.5555 | |

1960–2016 | 0.1427 | 0.3146 | 0.1720 | 0.6293 | 0.3707 | |

Sediment | 1960–1996 | 0.1154 | 0.2833 | 0.1541 | 0.5528 | 0.4472 |

1996–2016 | 0.1778 | 0.2723 | 0.0879 | 0.5320 | 0.4680 | |

1960–2016 | 0.1485 | 0.3075 | 0.1596 | 0.6156 | 0.3844 |

**Table 4.**Probabilities of high runoff (sediment) of the Huaxian Station caused, respectively, by high, normal, and low runoff (sediment) of the Xianyang and Zhangjiashan Stations.

Year | P(H_{xy}|H_{hx}) | P(H_{zjs}|H_{hx}) | P(N_{xy}|H_{hx}) | P(N_{zjs}|H_{hx}) | P(L_{xy}|H_{hx}) | P(L_{zjs}|H_{hx}) | |
---|---|---|---|---|---|---|---|

Runoff | 1960–1990 | 0.9236 | 0.7744 | 0.0764 | 0.2248 | 0 | 0.0008 |

1990–2016 | 0.9052 | 0.4732 | 0.0948 | 0.4128 | 0 | 0.1140 | |

1960–2016 | 0.9244 | 0.7232 | 0.0756 | 0.2636 | 0 | 0.0132 | |

Sediment | 1960–1996 | 0.7364 | 0.8624 | 0.2528 | 0.1372 | 0.0108 | 0.0004 |

1996–2016 | 0.5648 | 0.7488 | 0.4264 | 0.2512 | 0.0088 | 0 | |

1960–2016 | 0.7452 | 0.8784 | 0.2528 | 0.1216 | 0.0020 | 0 |

_{xy}represents the high annual runoff/sediment yield of Xianyang Station; H

_{hx}represents the high annual runoff/sediment yield of Huaxian Station; N

_{zjs}represents the normal annual runoff/sediment yield of Zhangjiashan Station; and L

_{zjs}represents the low annual runoff/sediment yield of Zhangjiashan Station; similarly for the others.

**Table 5.**Probabilities of low runoff (sediment) of the Huaxian Station caused, respectively, by high, normal, and low runoff (sediment) of Xianyang and Zhangjiashan Stations.

Year | P(H_{xy}|L_{hx}) | P(H_{zjs}|L_{hx}) | P(N_{xy}|L_{hx}) | P(N_{zjs}|L_{hx}) | P(L_{xy}|L_{hx}) | P(L_{zjs}|L_{hx}) | |
---|---|---|---|---|---|---|---|

Runoff | 1960–1990 | 0 | 0.0008 | 0.0764 | 0.2248 | 0.9236 | 0.7744 |

1990–2016 | 0 | 0.1140 | 0.0948 | 0.4916 | 0.9052 | 0.3944 | |

1960–2016 | 0 | 0.0132 | 0.1160 | 0.3596 | 0.8840 | 0.6272 | |

Sediment | 1960–1996 | 0.0108 | 0.0004 | 0.3472 | 0.2020 | 0.642 | 0.7976 |

1996–2016 | 0.0088 | 0 | 0.2064 | 0.0920 | 0.7848 | 0.9080 | |

1960–2016 | 0.0020 | 0 | 0.2416 | 0.1808 | 0.7564 | 0.8192 |

_{xy}, L

_{hx}, H

_{zjs}, N

_{xy}, N

_{zjs}, L

_{xy}, L

_{hx}, and L

_{zjs}are the same as in Table 4.

Typical Conditional Probability | |
---|---|

Runoff | P(40.82 < Runoff_{hx} < 48.10|26.50 > Runoff_{xy} > 18.70, 10.10 > Runoff_{zjs} > 7.00) = 0.7780 |

P(32.50 < Runoff_{hx} < 17.80|14.50 > Runoff_{xy} > 5.80, 5.60 > Runoff_{zjs} > 2.30) = 0.7055 | |

Sediment | P(1.49 > Sedi_{hx} > 0.92|0.22 > Sedi_{xy} > 0.12, 1.21 > Sedi_{zjs} > 0.80) = 0.8156 |

P(0.57 > Sedi_{hx} > 0.41|0.074 > Sedi_{xy} > 0.06, 0.61 > Sedi_{zjs} > 0.22) = 0.9517 |

_{hx}represents the annual runoff yield of the Huaxian Station, Runoff

_{xy}represents the annual runoff yield of the Xianyang Station, Runoff

_{zjs}represents the annual runoff yield of the Zhangjiashan Station; all the units are 10

^{8}m

^{3}. Sedi

_{hx}represents the annual sediment yield of the Huaxian Station, Sedi

_{xy}represents the annual sediment yield of the Xianyang Station, Sedi

_{zjs}represents the annual sediment yield of the Zhangjiashan Station; all the units are 10

^{8}m

^{3}.

© 2019 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**

You, Q.; Jiang, H.; Liu, Y.; Liu, Z.; Guan, Z.
Probability Analysis and Control of River Runoff–sediment Characteristics based on Pair-Copula Functions: The Case of the Weihe River and Jinghe River. *Water* **2019**, *11*, 510.
https://doi.org/10.3390/w11030510

**AMA Style**

You Q, Jiang H, Liu Y, Liu Z, Guan Z.
Probability Analysis and Control of River Runoff–sediment Characteristics based on Pair-Copula Functions: The Case of the Weihe River and Jinghe River. *Water*. 2019; 11(3):510.
https://doi.org/10.3390/w11030510

**Chicago/Turabian Style**

You, Qiying, Hao Jiang, Yan Liu, Zhao Liu, and Zilong Guan.
2019. "Probability Analysis and Control of River Runoff–sediment Characteristics based on Pair-Copula Functions: The Case of the Weihe River and Jinghe River" *Water* 11, no. 3: 510.
https://doi.org/10.3390/w11030510