# Future Joint Probability Characteristics of Extreme Precipitation in the Yellow River Basin

^{1}

^{2}

^{*}

## Abstract

**:**

_{And}> T

_{Kendall}> T

_{Single-variable}> T

_{Or}. Joint return periods (Or) and co-occurring return periods (And) could be considered as the extreme cases under single-variable return periods, serving as an estimation interval for actual return periods. Under the influence of climate change, the bivariate design values for future periods exhibited a variability increase of 6.76–28.8% compared to historical periods, and this increase grew with higher radiative forcing scenarios, ranking as SSP126 < SSP245 < SSP585. The bivariate design values showed a noticeable difference in variability compared to the single-variable design values, ranging from −0.79% to 18.67%. This difference increased with higher quantile values, with R95P-SDII (95) > R90P-SDII (90) > PRCPTOT-SDII.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(Figure 1). The Yellow River Basin holds significant importance as a crucial agricultural and energy production base in China. This region is situated within China’s arid and semi-arid zones, influenced by the East Asian summer monsoon climate and the Northern Hemisphere’s westerly winds. Precipitation exhibits both temporal and spatial variations, with the majority of annual rainfall occurring between June and August. Spatially, precipitation decreases from southeast to northwest, with nearly a tenfold difference between high- and low-precipitation zones. The annual evaporation rate reaches 1100 mm, while the multiyear average annual precipitation is approximately 476 mm, and the annual average temperature is 8 °C. Over the past 70 years, the Yellow River Basin has experienced a notable warming trend, with an average rate of temperature increase of 0.31 °C per decade, which is twice the global warming rate. This region is considered sensitive to global climate change.

#### 2.2. Datasets

- (1)
- Hydro-Meteorological data

- (2)
- Climate Model Data

#### 2.3. Methodology

#### 2.3.1. Model Establishment and Selection

_{X}(x) and F

_{Y}(y) are determined and the marginal distribution functions are continuous functions, there must exist a unique two-dimensional copula function C

_{θ}(x, y), $F(x,y)={C}_{\theta}({F}_{X}(x),{F}_{Y}(y))$ [27]. Similarly, this definition can be extended to the joint distribution functions in n dimensions. The Sklar theorem demonstrates that the generation process of joint distributions primarily depends on the determination of the copula function and marginal distribution functions. Common copula functions are mainly classified into three categories: elliptical, quadratic, and Archimedean types. Among them, Archimedean copula functions, due to having only one parameter and simplicity in modeling, have been widely used in hydrological research for modeling variable dependence, frequency analysis, obtaining reliable design values for specific return periods, and risk mitigation [28,29]. Consequently, this study focused on the application of copula functions in bivariate analysis by utilizing three common Archimedean copula functions (the Gumbel–Hougaard copula, Clayton copula, and Frank copula) (Table 3) to establish the bivariate joint probability distributions for three extreme precipitation indices (PRCPTOT-SDII, R90P-SDII (90), and R95P-SDII (95)).

#### 2.3.2. Calculation of Return Periods

_{c}:

_{c}is given in Table 4.

#### 2.3.3. Estimation of Design Values

_{x}to obtain the design value. The design value is given as P = 1 − 1/T.

_{x}

^{−1}(P)

_{1}(u, v) represents the marginal density functions of the two variables.

_{m}, v

_{m}) that maximizes the f(u, v) obtained, we then calculated the design values using the inverse functions of their marginal distribution functions. Specifically, x = Fx

^{−1}(u

_{m}) and y = Fy

^{−1}(v

_{m}), where Fx

^{−1}and Fy

^{−1}are the inverse functions of the marginal distribution functions.

## 3. Results

#### 3.1. Optimal Selection and Applicability Analysis of Copula Functions

_{(PRCPTOT-SDII)}> Corr

_{(R90P-SDII (90))}> Corr

_{(R95P-SDII (95))}. With an increase in radiative intensity, the correlation between the precipitation amount and precipitation intensity became stronger, with Corr

_{SSP126 (PRCPTOT-SDII)}> Corr

_{SSP245 (PRCPTOT-SDII)}> Corr

_{SSP585 (PRCPTOT-SDII)}. Spatially, there was noticeable spatial continuity in the correlation between the precipitation amount and precipitation intensity, mirroring the spatial distribution of the precipitation amount by generally exhibiting a southeast-to-northwest decreasing trend. In regions with high precipitation amounts, the correlation between precipitation amount and precipitation intensity was stronger.

^{2}) for the functional fits were all greater than 0.99. The fit of the empirical points for the three two-dimensional joint distributions was similar. The Gumbel copula and Frank copula tended to underestimate the observed points, while the Clayton copula tended to overestimate the observed points (Figure 3). Combining the graphical analysis with the Ordinary Least Squares (OLS) minimum criterion (Figure 4), the results indicate that during the historical period, the Clayton copula provided the best description for the PRCPTOT-SDII index, the Frank copula was optimal for the R90P-SDII (90) index, and the Clayton copula was optimal for the R95P-SDII (95) index. Similarly, the optimal copula functions could be obtained for various indices between other stations within the watershed, both historical and future.

#### 3.2. Bivariate Recurrence Period

_{And}> T

_{Kendall}> T

_{Single-variable}> T

_{Or}. The joint recurrence period (Or) and coincident recurrence period (And) can be considered as the maximum and minimum extremes under single-variable recurrence periods, serving as the estimated range for actual recurrence periods. The patterns during recurrence periods can be explained from the perspective of safety thresholds and danger thresholds. Figure 10 describes the safety threshold ranges identified by three recurrence periods (the Kendall recurrence period, coincident recurrence period, and joint recurrence period). The safety threshold range identified by the joint recurrence period (Or) is the rectangular area in the lower left corner, while the Kendall recurrence period defines the lower part of the curve C(u,v) as the safety zone. The coincident recurrence period, building upon the Kendall recurrence period, further adds the And part as a safety threshold region. It can be observed that the danger threshold range defined by the Kendall recurrence period is not excessively large or small compared to the coincident recurrence period and joint recurrence period. This ensures engineering safety while avoiding excessive costs. Thus, this paper used the Kendall recurrence period as the recurrence period design for two-dimensional joint distribution.

#### 3.3. Bivariate Design Values

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Features of optimal copula functions at Jiuzhi Station: (

**A**,

**D**) PRCPTOT-SDII; (

**B**,

**E**) R90P-SDII (90); (

**C**,

**F**) R95P-SDII (95).

**Figure 5.**Spatial distribution of the extreme precipitation index against the optimal copula joint distribution under historical and future scenarios.

**Figure 6.**Combined distribution percentages of the extreme precipitation index for the optimal copula under historical and future scenarios.

**Figure 12.**Bivariate design values of the three indices in the Yellow River Basin for the 20a recurrence period under historical and future scenarios.

Abbreviation | Name | Definition | Unit |
---|---|---|---|

PRCPTOT | Annual precipitation | ≥1 mm precipitation daily cumulative amount | mm |

SDII | Precipitation intensity | The ratio of total precipitation ≥1 mm to number of days | mm/d |

R95P | Heavy precipitation | The sum of 95% quantile values of intense precipitation | mm |

SDII (95) | Heavy precipitation intensity | The ratio of the sum of heavy precipitation to the number of heavy precipitation days | mm/d |

R90P | Heavy rainfall | The part of precipitation exceeding the 90th percentile in precipitation events | mm |

SDII (90) | Heavy precipitation intensity | The sum of rainfall for heavy rain events exceeding the 90th percentile value divided by the number of days with heavy rain | mm/d |

Numbers | Climate Model | Resolution Ratio | Country |
---|---|---|---|

1 | EC-Earth3 | 100 km | Britain |

2 | EC-Earth3-Veg | 100 km | Sweden |

3 | GFDL-ESM4 | 100 km | America |

4 | MPI-ESM1-2-HR | 100 km | Germany |

5 | MRI-ESM2-0 | 100 km | Japan |

6 | IPSL-CM6A-LR | 100 km | France |

Copula Function | Generating Elements | Density Function | Distribution Function |
---|---|---|---|

G-H | $\phi (t)={(-\mathrm{ln}t)}^{\theta}$ | $\begin{array}{l}\\ {c}_{G}(u,c)=\frac{{(-\mathrm{ln}u)}^{\theta -1}{(-\mathrm{ln}v)}^{\theta -1}[\theta -1+{(-\mathrm{ln}u)}^{\theta}+{(-\mathrm{ln}v)}^{\theta}]}{uv{e}^{{[{(-\mathrm{ln}u)}^{\theta}+{(-\mathrm{ln}v)}^{\theta}]}^{1/\theta}}};\theta \in [1,\infty )\end{array}$ | ${C}_{G}(u,c)=\mathrm{exp}\{-{[{(-\mathrm{ln}u)}^{\theta}+{(-\mathrm{ln}v)}^{\theta}]}^{1/\theta}\};\theta \in [1,\infty )$ |

Clayton | $\phi (t)={t}^{-\theta}-1$ | ${c}_{cl}(u,c)=\frac{(1+\theta ){\mu}^{-1-\theta}{v}^{-1-\theta}}{{({u}^{-\theta}+{v}^{-\theta}-1)}^{\frac{1+2\theta}{\theta}}};\theta \in [1,\infty )$ | ${C}_{cl}(u,c)={({u}^{-\theta}+{v}^{-\theta}-1)}^{-1/\theta};\theta \in [1,\infty )$ |

Frank | $\phi (t)=-\mathrm{ln}\frac{{e}^{-\theta t}-1}{{e}^{-\theta}-1}$ | ${c}_{F}(u,v)=\frac{\theta {e}^{-\theta (u+v)}({e}^{-\theta}-1)}{{({e}^{-\theta (u+v)}-{e}^{-\theta u}-{e}^{-\theta}+{e}^{-\theta})}^{2}};\theta \in R$ | ${C}_{F}(u,v)=-\frac{1}{\theta}\mathrm{ln}[1+\frac{({e}^{-\theta u}-1)({e}^{-\theta v}-1)}{({e}^{-\theta}-1)}];\theta \in R$ |

Function | Relationship between τ and θ | K_{c} |
---|---|---|

Gumbel | $\tau =1-\frac{1}{\theta},\theta \in [0,1)$ | $K=t-\frac{t\mathrm{ln}t}{\theta}$ |

Clayton | $\tau =\frac{\theta}{2+\theta},\theta \in (0,\infty )$ | $K=t-\frac{t({t}^{\theta}-1)}{\theta}$ |

Frank | $\tau =1+\frac{4}{\theta}[\frac{1}{\theta}{\displaystyle {\int}_{0}^{\theta}\frac{t}{\mathrm{exp}(t)-1}}dt-1],\theta \in R$ | $K=t-\frac{({e}^{\theta t}-1)}{\theta}\mathrm{ln}\frac{{e}^{-\theta t}-1}{{e}^{-\theta}-1}$ |

**Table 5.**Bivariate design values and their change rate with univariate design values during 1980–2022 (his) and 2023–2100 (SSP126, SSP245, SSP585).

Index | Recurrence Interval | Design Value | Rate of Change Relative to Historical Period (%) | Rate of Change Relative to Univariate Design Value (%) | |||||
---|---|---|---|---|---|---|---|---|---|

His | SSP 126 | SSP 245 | SSP 585 | His | SSP 126 | SSP 245 | SSP 585 | ||

PRCP TOT (mm) | 100a * | 631.33 | 28.88 | 23.83 | 24.83 | −5.27 | −3.30 | −3.99 | −0.79 |

50a | 619.43 | 25.45 | 23.00 | 24.71 | −5.08 | −2.82 | −3.21 | −2.50 | |

20a | 601.56 | 20.50 | 21.81 | 24.12 | −4.75 | −2.14 | −2.42 | −5.17 | |

10a | 585.02 | 23.90 | 20.76 | 23.47 | −4.45 | −1.74 | −1.92 | −1.60 | |

5a | 529.11 | 31.30 | 27.40 | 30.63 | −9.97 | −1.42 | −1.47 | −1.34 | |

2a | 529.11 | 20.25 | 15.35 | 18.48 | −1.45 | −1.26 | −1.15 | −1.09 | |

SDII (mm/day) | 100a | 4.29 | 18.54 | 16.37 | 18.50 | −7.77 | −2.91 | −3.26 | −0.95 |

50a | 4.21 | 16.18 | 15.91 | 18.33 | −7.46 | −2.45 | −2.80 | −2.29 | |

20a | 4.10 | 15.78 | 15.30 | 18.16 | −7.04 | −1.91 | −2.07 | −1.77 | |

10a | 4.00 | 15.48 | 14.75 | 17.89 | −6.71 | −1.51 | −1.48 | −1.40 | |

5a | 3.88 | 15.19 | 14.17 | 17.53 | −6.42 | −1.16 | −1.16 | −1.09 | |

2a | 3.64 | 14.98 | 13.41 | 16.92 | −6.24 | −0.87 | −0.87 | −0.81 | |

R90P (mm) | 100a | 247.30 | 24.53 | 26.56 | 32.32 | −14.09 | −15.35 | −14.08 | −11.85 |

50a | 237.41 | 23.69 | 24.75 | 30.46 | −13.13 | −13.80 | −12.78 | −11.30 | |

20a | 223.13 | 22.37 | 22.14 | 28.55 | −11.61 | −11.50 | −10.81 | −8.53 | |

10a | 210.01 | 21.40 | 20.08 | 25.20 | −10.39 | −9.65 | −9.24 | −7.99 | |

5a | 185.84 | 25.45 | 22.88 | 23.63 | −12.52 | −7.63 | −7.40 | −9.84 | |

2a | 151.60 | 22.43 | 20.99 | 20.99 | −10.75 | −7.42 | −4.68 | −7.49 | |

SDII (90) (mm/day) | 100a | 13.53 | 8.77 | 11.61 | 13.21 | −8.84 | −7.30 | −7.17 | −6.84 |

50a | 13.22 | 9.28 | 11.50 | 13.18 | −7.46 | −6.27 | −6.16 | −5.79 | |

20a | 12.79 | 9.95 | 11.42 | 12.95 | −5.76 | −4.90 | −4.78 | −4.59 | |

10a | 12.44 | 10.48 | 11.38 | 13.05 | −4.56 | −3.87 | −3.76 | −3.49 | |

5a | 12.04 | 10.99 | 11.31 | 12.78 | −3.28 | −2.76 | −2.69 | −2.56 | |

2a | 11.31 | 12.34 | 11.35 | 12.33 | −1.51 | −0.61 | −1.12 | −1.24 | |

R95P (mm) | 100a | 170.64 | 22.67 | 24.52 | 28.55 | −16.61 | −17.69 | −18.67 | −17.65 |

50a | 161.08 | 22.48 | 23.38 | 25.20 | −15.10 | −16.15 | −16.67 | −17.88 | |

20a | 147.53 | 22.52 | 21.62 | 24.50 | −12.73 | −13.40 | −13.73 | −15.22 | |

10a | 135.68 | 17.66 | 19.82 | 23.20 | −10.70 | −14.63 | −11.50 | −13.27 | |

5a | 121.53 | 22.27 | 17.89 | 20.53 | −8.14 | −8.56 | −8.79 | −11.49 | |

2a | 91.22 | 19.53 | 14.24 | 16.77 | −6.25 | −8.93 | −7.75 | −9.77 | |

SDII (95) (mm/day) | 100a | 16.75 | 8.17 | 10.70 | 11.84 | −10.54 | −10.46 | −10.53 | −10.04 |

50a | 16.30 | 8.52 | 10.82 | 12.10 | −8.96 | −9.00 | −8.73 | −8.35 | |

20a | 15.67 | 9.15 | 11.01 | 12.47 | −6.99 | −7.08 | −6.49 | −6.19 | |

10a | 15.15 | 9.73 | 11.27 | 12.77 | −5.57 | −5.61 | −4.84 | −4.66 | |

5a | 14.58 | 10.20 | 11.44 | 13.05 | −4.05 | −4.23 | −3.29 | −3.12 | |

2a | 14.08 | 6.76 | 6.79 | 9.13 | −1.91 | −2.39 | −2.67 | −1.13 |

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## Share and Cite

**MDPI and ACS Style**

Li, F.; Zhang, G.; Zhang, X.
Future Joint Probability Characteristics of Extreme Precipitation in the Yellow River Basin. *Water* **2023**, *15*, 3957.
https://doi.org/10.3390/w15223957

**AMA Style**

Li F, Zhang G, Zhang X.
Future Joint Probability Characteristics of Extreme Precipitation in the Yellow River Basin. *Water*. 2023; 15(22):3957.
https://doi.org/10.3390/w15223957

**Chicago/Turabian Style**

Li, Fujun, Guodong Zhang, and Xueli Zhang.
2023. "Future Joint Probability Characteristics of Extreme Precipitation in the Yellow River Basin" *Water* 15, no. 22: 3957.
https://doi.org/10.3390/w15223957