# Characteristic Analysis and Uncertainty Assessment of the Joint Distribution of Runoff and Sediment: A Case Study of the Huangfuchuan River Basin, China

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Sources

^{2}. The basin is located inland, influenced by the southeast monsoon, and has a continental monsoon climate with an average precipitation of 350–450 mm per year, and the distribution is extremely uneven during the year, with more than 80% of annual precipitation concentrated in June to September. Huangfuchuan belongs to a sandy and coarse sediment river, with an average annual sediment discharge of 61.2 million tons and an average annual sediment modulus of 19.1 million tons/(km

^{2}·a), which is one of the most serious zones of soil erosion in the Loess Plateau area and transports a large amount of coarse sediment to the Yellow River. The Huangfuchuan hydrologic station is close to the basin outlet, controlling a basin area of 3199 km

^{2}. The main tasks of the station include precipitation, water level, flow, sediment testing, etc. It is a national basic hydrological station.

#### 2.2. Mutation Analysis

#### 2.2.1. The Cumulative Anomaly Method

#### 2.2.2. The Double Mass Curve Method

#### 2.3. Trend Analysis

#### 2.3.1. The M-K Trend Test

#### 2.3.2. The TFPW—MK Test

- (1)
- Construct the trend-free series Y
_{t}:

- (2)
- Calculate the first-order autocorrelation coefficient of the series Y
_{t}, and perform a significance test. When ${r}_{1}$ is not significant, a detrended series is used for trend detection. On the contrary, the detrended series is pre-whitened, and the residual series is recorded as ${Z}_{t}$.

- (3)
- The residual sequence is reorganized with the trend term to form a trend-free pre-whitening series ${X}_{t}^{*}$:

- (4)
- Perform an M-K trend test on series ${X}_{t}^{*}$.

#### 2.4. Marginal Distribution Model

_{0}) and the overall sample distribution does not obey a particular distribution (H

_{1}).

#### 2.5. Copula Joint Distribution Model

#### 2.6. Runoff-Sediment Encounter Probability Analysis

#### 2.7. Return Period Analysis

## 3. Results

#### 3.1. Runoff and Sediment Characteristics

^{2}of the fitted relational equation for each period is above 0.97, which is a good fit.

^{2}= 0.8449 between runoff and sediment. The vast majority of the correlation analysis point distances between runoff and sediment transport in the Huangfuchuan River Basin are densely distributed near the linear fitting line. The 95% prediction band (estimated to be the area where 95% of the data points belong) relatively evenly includes the linear fitting line, and the point distances of each era are distributed on both sides of the linear fitting line. This indicates that there is no significant systematic deviation in the relationship between water and sediment, and there is a strong correlation between runoff and sediment transport at the Huangfuchuan River basin from 1954 to 2015.

#### 3.2. Marginal Distributions and Copula Joint Distributions

#### 3.2.1. Selection of Marginal Distributions

#### 3.2.2. Choice of Copula Joint Distribution

^{2}and according to the principle of maximum R

^{2}the copula function can be determined as the optimal function. It can be seen from the Figure 8 that all the three copula functions of the three phases fall near the 45° line, and the correlation coefficients are relatively high (all are greater than 0.9593). The Gumbel copula’s fit is the highest for the stage ${T}_{a}$ and the stage ${T}_{b}$. The Clayton copula’s fit is the highest for the stage ${T}_{c}$. These are consistent with the conclusions of OLS, AIC, and BIC tests. That is, Gumbel copula is selected as the optimal copula function to connect the optimal marginal distribution functions of runoff and sediment in the Huangfuchuan River basin in the stages ${T}_{a}$ and ${T}_{b}$, and the Clayton copula is selected as the optimal copula function to connect the optimal marginal distribution functions of runoff and sediment in the Huangfuchuan River basin in the stage ${T}_{c}$.

#### 3.3. Runoff-Sediment Encounter Probability Analysis

#### 3.4. Return Period Analysis

## 4. Uncertainty Analysis

#### 4.1. Uncertainty Analysis of Margin Distributions

#### 4.2. Uncertainty Analysis of Parameters

#### 4.3. Uncertainty Analysis of Copula Functions

^{3}and the annual sediment discharge is 91.037 million tons in that year. Three copula functions are used to construct the joint distribution function of runoff and sediment, and the runoff and sediment discharge are also subject to the six marginal distributions mentioned above. When the Gumbel copula function is obeyed, the range of the co-occurrence return period is 5.06~13.64 years; when the Clayton copula function is obeyed, the range of the co-occurrence return period is 6.38–30.88 years; when the Frank copula function is obeyed, the range of the co-occurrence return period is 5.66–19.14 years. From the variation range of the co-occurrence return period, it can be seen that the corresponding results of the three Copula functions are quite different. The optimal marginal distribution functions of runoff and sediment are generalized extreme value (GEV) distribution and gamma (GAM) distribution, and the optimal copula function is the Gumbel copula in the stage ${T}_{a}$. The co-occurrence return period of runoff and sediment corresponding to the optimal marginal distribution and the optimal copula function is 10.89 years, while the empirical co-occurrence period of runoff and sediment in 1961 is 10.16 years, which are close to each other, indicating that the selection of the copula function is correct. Therefore, it is significant to select the appropriate copula function for constructing the optimal joint distribution function.

## 5. Discussions

^{3}, and the proportion of dam-controlled area in the basin reached 70%. Therefore, compared with the stage ${T}_{b}$, the synchronous frequency of runoff-sediment (especially the frequency of the same rich and same normal) in stage ${T}_{c}$ significantly decreased, while the asynchronous frequency of runoff–sediment increased during the same period, indicating that soil and water conservation measures in this stage not only reduced runoff and sediment, but also effectively reduced the sediment content of rich runoff, reduced co-occurrence probability, and adjusted the probability of extreme runoff and sediment events.

## 6. Conclusions

- (1)
- Based on the cumulative anomaly method and the double mass curve method, 1979 and 1996 are identified as mutation points of runoff and sediment series in the Huangfuchuan River basin, and the runoff and sediment series in the Huangfuchuan River basin are divided into three stages. The runoff and sediment in all three stages show a strong correlation.
- (2)
- Based on the results of K-S, OLS, and AIC criteria tests, the runoff in the Huangfuchuan River basin obeys the GEV distribution and the sediment obeys the GAM distribution in the stage ${T}_{a}$; the runoff obeys the GEV distribution and the sediment obeys the EXP distribution in the stage ${T}_{b}$; the runoff and sediment obey the GEV distribution in the stage ${T}_{c}$. Based on the results of OLS, AIC, and BIC criteria tests, the Gumbel copula is selected as the optimal joint distribution function of runoff and sediment in the stages ${T}_{a}$ and ${T}_{b}$, and the Clayton copula is selected as the optimal joint distribution function of runoff and sediment in the stage ${T}_{c}$.
- (3)
- The synchronous probabilities in the three stages are 69.84%, 84.82%, and 70.72%, respectively, which are much greater than the asynchronous probabilities, indicating a strong consistency of runoff and sediment in the River basin. The synchronous probabilities in the stages ${T}_{b}$ and ${T}_{c}$ have increased compared with those in stage ${T}_{a}$, indicating that under the combined influence of climate change and human activities, the impact mechanism of runoff–sediment has become more complex and the relationship between runoff and sediment may have changed to a certain extent.
- (4)
- The contour maps of the joint return period and the co-occurrence return period of the two-dimensional joint distribution of runoff and sediment discharge are drawn. On this basis, the typical years of runoff and sediment can be selected, and the same frequency amplification method can be used to obtain the runoff and sediment ratio system with different designed runoff hydrographs and designed sediment discharge hydrographs with the same return period.
- (5)
- The uncertainties and sensitivities of the joint distribution function construction process are analyzed, showing that the optimal marginal distribution functions, the optimal copula functions, and the parameters selected by the great likelihood estimation method can better fit the runoff–sediment relationship in the river basin and reduce the process uncertainties.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, L.; Wu, K.; Jiang, E.; Yin, H.; Wang, Y.; Tian, S.; Dang, S. Evaluating Runoff-Sediment Relationship Variations Using Generalized Additive Models That Incorporate Reservoir Indices for Check Dams. Water Resour. Manag.
**2021**, 35, 3845–3860. [Google Scholar] [CrossRef] - Zhang, X.; Lin, P.; Chen, H.; Yan, R.; Zhang, J.; Yu, Y.; Liu, E.; Yang, Y.; Zhao, W.; Lv, D.; et al. Understanding land use and cover change impacts on run-off and sediment load at flood events on the Loess Plateau, China. Hydrol. Process.
**2018**, 32, 576–589. [Google Scholar] [CrossRef] - Phillips, M.C.; Solo-Gabriele, H.M.; Piggot, A.M.; Klaus, J.S.; Zhang, Y. Relationships between sand and water quality at recreational beaches. Water Res.
**2011**, 45, 6763–6769. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guo, A.; Chang, J.; Wang, Y.; Huang, Q.; Zhou, S. Flood risk analysis for flood control and sediment transportation in sandy regions: A case study in the Loess Plateau, China. J. Hydrol.
**2018**, 560, 39–55. [Google Scholar] [CrossRef] - Tian, P.; Zhai, J.; Zhao, G.; Mu, X. Dynamics of Runoff and Suspended Sediment Transport in a Highly Erodible Catchment on the Chinese Loess Plateau. Land Degrad. Dev.
**2015**, 27, 839–850. [Google Scholar] [CrossRef] - Li, R.; Zhu, A.; Song, X.; Cui, M. Seasonal Dynamics of Runoff-Sediment Relationship and Its Controlling Factors in Black Soil Region of Northeast China. J. Resour. Ecol.
**2010**, 1, 345–352. [Google Scholar] - Thanh Son Ngo, D.B.N.P.S.R. Effect of land use change on runoff and sediment yield in Da River Basin of Hoa Binh province, Northwest Vietnam. J. Mt. Sci.
**2015**, 12, 1051–1064. [Google Scholar] - Guo, A.; Chang, J.; Wang, Y.; Huang, Q. Variations in the Runoff-Sediment Relationship of the Weihe River Basin Based on the Copula Function. Water
**2016**, 8, 223. [Google Scholar] [CrossRef] [Green Version] - Wen, Y.; Yang, A.; Kong, X.; Su, Y. A Bayesian-Model-Averaging Copula Method for Bivariate Hydrologic Correlation Analysis. Front. Environ. Sci.
**2022**, 9, 744462. [Google Scholar] [CrossRef] - Trinh, T.; Ishida, K.; Kavvas, M.L.; Ercan, A.; Carr, K. Assessment of 21st century drought conditions at Shasta Dam based on dynamically projected water supply conditions by a regional climate model coupled with a physically-based hydrology model. Sci. Total Environ.
**2017**, 586, 197–205. [Google Scholar] [CrossRef] - Ho, C.; Trinh, T.; Nguyen, A.; Nguyen, Q.; Ercan, A.; Kavvas, M.L. Reconstruction and evaluation of changes in hydrologic conditions over a transboundary region by a regional climate model coupled with a physically-based hydrology model: Application to Thao river watershed. Sci. Total Environ.
**2019**, 668, 768–779. [Google Scholar] [CrossRef] [PubMed] - Todorovic, P. Stochastic Models of Floods. Water Resour. Res.
**1978**, 14, 345–356. [Google Scholar] [CrossRef] - Remesan, R.; Shamim, M.A.; Han, D.; Mathew, J. Runoff prediction using an integrated hybrid modelling scheme. J. Hydrol.
**2009**, 372, 48–60. [Google Scholar] [CrossRef] - Fan, Y.; Huang, G.; Zhang, Y.; Li, Y. Uncertainty Quantification for Multivariate Eco-Hydrological Risk in the Xiangxi River within the Three Gorges Reservoir Area in China. Engineering
**2018**, 4, 617–626. [Google Scholar] [CrossRef] - Goel, N.K.; Seth, S.M.; Chandra, S. Multivariate Modeling of flood flows. J. Hydraul. Eng.
**1998**, 124, 146–155. [Google Scholar] [CrossRef] - Yue, S.; Rasmussen, P. Bivariate frequency analysis: Discussion of some useful concepts in hydrological application. Hydrol. Process.
**2002**, 16, 2881–2898. [Google Scholar] [CrossRef] - Vinnarasi, R.; Dhanya, C.T. Bringing realism into a dynamic copula-based non-stationary intensity-duration model. Adv. Water Resour.
**2019**, 130, 325–338. [Google Scholar] [CrossRef] - Zhang, L.; Chen, Y.; Wang, Y.; Bai, Y. Probabilistic analysis of the controls on groundwater depth using Copula Functions. Hydrol. Res.
**2020**, 51, 406–422. [Google Scholar] - Abdollahi, S.; Akhoond-Ali, A.M.; Mirabbasi, R.; Adamowski, J.F. Probabilistic Event Based Rainfall-Runoff Modeling Using Copula Functions. Water Resour. Manag.
**2019**, 33, 3799–3814. [Google Scholar] [CrossRef] - Grimaldi, S.; Petroselli, A.; Salvadori, G.; De Michele, C. Catchment compatibility via copulas: A non-parametric study of the dependence structures of hydrological responses. Adv. Water Resour.
**2016**, 90, 116–133. [Google Scholar] [CrossRef] - Xiong, L.; Yu, K.-X.; Gottschalk, L. Estimation of the distribution of annual runoff from climatic variables using copulas. Water Resour. Res.
**2014**, 50, 7134–7152. [Google Scholar] [CrossRef] - Perz, A.; Sobkowiak, L.; Wrzesiński, D. Probabilistic Approach to Precipitation-Runoff Relation in a Mountain Catchment: A Case Study of the Kłodzka Valley in Poland. Water
**2021**, 13, 1229. [Google Scholar] [CrossRef] - Sugimoto, T.; Bárdossy, A.; Pegram, G.G.S.; Cullmann, J. Investigation of hydrological time series using copulas for detecting catchment characteristics and anthropogenic impacts. Hydrol. Earth Syst. Sci.
**2016**, 20, 2705–2720. [Google Scholar] [CrossRef] [Green Version] - Golian, S.; Saghafian, B.; Farokhnia, A.; Rivard, C. Copula-based interpretation of continuous rainfall–runoff simulations of a watershed in northern Iran. Can. J. Earth Sci.
**2012**, 49, 681–691. [Google Scholar] [CrossRef] - Bacchi, B.; Balistrocchi, M. Derivation of flood frequency curves through a bivariate rainfall distribution based on copula functions: Application to an urban catchment in northern Italy’s climate. Hydrol. Res.
**2017**, 48, 749–762. [Google Scholar] - Requena, A.I.; Flores, I.; Mediero, L.; Garrote, L. Extension of observed flood series by combining a distributed hydro-meteorological model and a copula-based model. Stoch. Environ. Res. Risk Assess.
**2015**, 30, 1363–1378. [Google Scholar] [CrossRef] [Green Version] - Chang, J.; Li, Y.; Wang, Y.; Yuan, M. Copula-based drought risk assessment combined with an integrated index in the Wei River Basin, China. J. Hydrol.
**2016**, 540, 824–834. [Google Scholar] [CrossRef] - Li, Y.; Cai, Y.; Li, Z.; Wang, X.; Fu, Q.; Liu, D.; Sun, L.; Xu, R. An approach for runoff and sediment nexus analysis under multi-flow conditions in a hyper-concentrated sediment river, Southwest China. J. Contam. Hydrol.
**2020**, 235, 103702. [Google Scholar] [CrossRef] - Dodangeh, E.; Shahedi, K.; Pham, B.T.; Solaimani, K. Joint frequency analysis and uncertainty estimation of coupled rainfall–runoff series relying on historical and simulated data. Hydrol. Sci. J.
**2019**, 65, 455–469. [Google Scholar] [CrossRef] - Salvadori, G.; Durante, F.; De Michele, C.; Bernardi, M.; Petrella, L. A multivariate copula-based framework for dealing with hazard scenarios and failure probabilities. Water Resour. Res.
**2016**, 52, 3701–3721. [Google Scholar] [CrossRef] [Green Version] - Razmkhah, H.; AkhoundAli, A.-M.; Radmanesh, F. Correlated Parameters Uncertainty Propagation in a Rainfall-Runoff Model, Considering 2-Copula; Case Study: Karoon III River Basin. Environ. Model. Assess.
**2017**, 22, 503–521. [Google Scholar] [CrossRef] - Zimmermann, H.J. Fuzzy Set Theory—And Its Applications; Kluwer Academic Pub: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Serinaldi, F. Can we tell more than we can know? The limits of bivariate drought analyses in the United States. Stoch. Environ. Res. Risk Assess.
**2015**, 30, 1691–1704. [Google Scholar] [CrossRef] - Wang, J.; Shi, B.; Zhao, E.; Yuan, Q.; Chen, X. The long-term spatial and temporal variations of sediment loads and their causes of the Yellow River Basin. Catena
**2022**, 209, 105850. [Google Scholar] [CrossRef] - Liu, X.; Zhang, Y.; Liu, Y.; Zhao, X.; Zhang, J.; Rui, Y. Characteristics of temperature evolution from 1960 to 2015 in the Three Rivers’ Headstream Region, Qinghai, China. Sci. Rep.
**2020**, 10, 20272. [Google Scholar] [CrossRef] - Yi, W.; Feng, Y.; Liang, S.; Kuang, X.; Yan, D.; Wan, L. Increasing annual streamflow and groundwater storage in response to climate warming in the Yangtze River source region. Environ. Res. Lett.
**2021**, 16, 084011. [Google Scholar] [CrossRef] - Rozbeh, S.; Adamowski, J.; Darabi, H.; Golshan, M.; Pirnia, A. Using the Mann–Kendall test and double mass curve method to explore stream flow changes in response to climate and human activities. J. Water Clim. Chang.
**2019**, 10, 725–742. [Google Scholar] - Kendall, M.G. Rank Correlation Methods; Charles Griffin: London, UK, 1975. [Google Scholar]
- Mann, H.B. Nonparametric Tests Against Trend. Econometrica
**1945**, 13, 245–259. [Google Scholar] [CrossRef] - Yue, S.; Pilon, P.; Phinney, B.; Cavadias, G. The influence of autocorrelation on the ability to detect trend in hydrological series. Hydrol. Process.
**2002**, 16, 1807–1829. [Google Scholar] [CrossRef] - Akalke, H. A New Look at the Statistical Model Identification. IEEE Trans. Autom. Control
**1974**, 19, 716–723. [Google Scholar] - Sklar, A. Fonctions de Repartition a n Dimensions et Leurs Marges; Publications de l’Institut Statistique de l’Université de Paris: Paris, France, 1959. [Google Scholar]
- Silva, R.d.S.; Lopes, H.F. Copula, marginal distributions and model selection: A Bayesian note. Stat. Comput.
**2008**, 18, 313–320. [Google Scholar] [CrossRef] - Quinton, A.V.J. Sensitivity analysis of EUROSEM using Monte Carlo simulation I:hydrological, soil and vegetation parameters. Hydrol. Process.
**2000**, 14, 915–926. [Google Scholar] - Borges, C.; Palma, C.; Dadamos, T.; Bettencourt da Silva, R.J.N. Evaluation of seawater composition in a vast area from the Monte Carlo simulation of georeferenced information in a Bayesian framework. Chemosphere
**2021**, 263, 128036. [Google Scholar] [CrossRef] [PubMed] - Tang, X.-S.; Li, D.-Q.; Rong, G.; Phoon, K.-K.; Zhou, C.-B. Impact of copula selection on geotechnical reliability under incomplete probability information. Comput. Geotech.
**2013**, 49, 264–278. [Google Scholar] [CrossRef] - Nguyen, D.D.; Jayakumar, K.V. Assessing the copula selection for bivariate frequency analysis based on the tail dependence test. J. Earth Syst. Sci.
**2018**, 127, 92. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Interannual variation characteristics of precipitation, runoff, and sediment from 1954 to 2015.

Copulas | $\mathit{C}\left(\mathit{u},\mathit{v}\right)$ | Parameters |
---|---|---|

Gumbel | $\mathrm{exp}\left[-{\left({\left(-\mathrm{ln}u\right)}^{\theta}+{\left(-\mathrm{ln}v\right)}^{\theta}\right)}^{1/\theta}\right]$ | θ ≥ 0 |

Clayton | ${\left({u}^{-\theta}+{v}^{-\theta}-1\right)}^{-1/\theta}$ | θ ≠ 0 |

Frank | $-\frac{1}{\theta}\mathrm{ln}\left[1+\left(\frac{\left({e}^{-\theta u}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-\theta}-1}\right)\right]$ | θ ≥ 1 |

Runoff | ||||
---|---|---|---|---|

Rich | Normal | Poor | ||

Sediment | Rich | ${P}_{1}=p\left(R\ge {r}_{pf},S\ge {s}_{pf}\right)$ | ${P}_{2}=p\left({r}_{pk}<R<{r}_{pf},S\ge {s}_{pf}\right)$ | ${P}_{3}=p\left(R\le {r}_{pk},S\ge {s}_{pf}\right)$ |

Normal | ${P}_{4}=p\left(R\ge {r}_{pf},{s}_{pk}<S<{s}_{pf}\right)$ | ${P}_{5}=p\left({r}_{pk}<R<{r}_{pf},{s}_{pk}<S<{s}_{pf}\right)$ | ${P}_{6}=p\left(R\le {r}_{pk},{s}_{pk}<S<{s}_{pf}\right)$ | |

Poor | ${P}_{7}=p\left(R\ge {r}_{pf},S\le {s}_{pk}\right)$ | ${P}_{8}=p\left({r}_{pk}<R<{r}_{pf},S\le {s}_{pk}\right)$ | ${P}_{9}=p\left(R\le {r}_{pk},S\le {s}_{pk}\right)$ |

Statistic | ${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{c}}$ | 1954~2015 | ||||
---|---|---|---|---|---|---|---|---|

Runoff | Sediment | Runoff | Sediment | Runoff | Sediment | Runoff | Sediment | |

${Z}_{s\_MK}$ | −0.9576 | −0.4437 | −1.1363 | −0.8333 | −2.0991 ** | −2.7289 *** | −5.7825 *** | −5.3209 *** |

${Z}_{S\_TFPW}$ | −0.3721 | −0.0758 | −0.5355 | −0.1236 | −1.8939 * | −2.6514 *** | −5.4948 *** | −5.1339 *** |

Stage | Characteristic Variables | Mean | Maximum | Minimum | Standard Deviation | ${\mathit{C}}_{\mathit{v}}$ | $\mathit{S}\mathit{K}$ |
---|---|---|---|---|---|---|---|

${T}_{a}$ | Runoff ($MCM$) | 185.437 | 508 | 41.080 | 115.247 | 0.621 | 1.534 |

Sediment ($MT$) | 58.006 | 171.071 | 5.224 | 41.639 | 0.718 | 1.153 | |

${T}_{b}$ | Runoff ($MCM$) | 134.775 | 435.940 | 25.110 | 96.965 | 0.719 | 1.699 |

Sediment ($MT$) | 43.726 | 147.507 | 5.155 | 39.608 | 0.906 | 1.337 | |

${T}_{c}$ | Runoff ($MCM$) | 37.676 | 103.449 | 0 | 34.475 | 0.915 | 0.886 |

Sediment ($MT$) | 8.667 | 29.076 | 0 | 9.702 | 1.119 | 1.058 |

Correlation Coefficient | ${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{c}}$ |
---|---|---|---|

γ | 0.844 | 0.970 | 0.924 |

τ | 0.673 | 0.895 | 0.801 |

ρ | 0.850 | 0.973 | 0.919 |

Margin | Parameters and Tests | Stage | |||||
---|---|---|---|---|---|---|---|

${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{c}}$ | |||||

Runoff | Sediment | Runoff | Sediment | Runoff | Sediment | ||

Normal | μ | 1.854 | 5800.600 | 1.348 | 4372.600 | 0.377 | 866.706 |

σ | 1.176 | 4249.700 | 0.998 | 4075.600 | 0.354 | 996.833 | |

K-S | 0 | 0 | 0 | 0 | 0 | 0 | |

OLS | 0.123 | 0.077 | 0.089 | 0.097 | 0.096 | 0.121 | |

AIC | −100.778 | −123.977 | −82.909 | −79.885 | −84.932 | −76.369 | |

Gamma | ${\mathsf{\alpha}}_{1}$ | 3.194 | 1.800 | 2.151 | 1.200 | 0.538 | 0.300 |

${\mathsf{\alpha}}_{2}$ | 0.581 | 3257.600 | 0.627 | 3609.500 | 0.700 | 2867.300 | |

K-S | 0 | 0 | 0 | 0 | 0 | 0 | |

OLS | 0.082 | 0.041 | 0.051 | 0.055 | 0.0600 | 0.087 | |

AIC | −120.950 | −155.961 | −103.278 | −100.280 | −103.138 | −88.927 | |

Exponential | μ | 1.8544 | 5800.600 | 1.348 | 4372.600 | 0.377 | 866.706 |

K-S | 1 | 0 | 0 | 0 | 0 | 0 | |

OLS | 0.136 | 0.071 | 0.083 | 0.039 | 0.047 | 0.142 | |

AIC | −95.876 | −128.155 | −85.526 | −112.704 | −112.415 | −70.276 | |

Weibull | ${\mathsf{\alpha}}_{1}$ | 2.099 | 6386.900 | 1.501 | 4549.100 | 0.326 | 514.919 |

${\mathsf{\alpha}}_{2}$ | 1.753 | 1.400 | 1.494 | 1.100 | 0.703 | 0.426 | |

K-S | 0 | 0 | 0 | 0 | 0 | 0 | |

OLS | 0.093 | 0.043 | 0.054 | 0.053 | 0.051 | 0.085 | |

AIC | −114.895 | −153.718 | −101.288 | −101.493 | −108.963 | −89.825 | |

Generalized Extreme Value | $\kappa $ | 0.216 | 0.100 | 0.213 | 0.800 | 0.488 | 1.256 |

α | 0.666 | 2850.600 | 0.591 | 1712.900 | 0.187 | 265.187 | |

ξ | 1.294 | 3724.400 | 0.865 | 1787.800 | 0.165 | 167.711 | |

K-S | 0 | 0 | 0 | 0 | 0 | 0 | |

OLS | 0.054 | 0.043 | 0.048 | 0.062 | 0.043 | 0.067 | |

AIC | −142.254 | −152.881 | −105.071 | −95.941 | −115.704 | −98.983 | |

Lognormal | μ | 0.453 | 8.359 | 0.048 | 7.917 | −2.143 | 4.486 |

σ | 0.580 | 0.886 | 0.757 | 1.067 | 3.123 | 5.306 | |

K-S | 0 | 0 | 0 | 0 | 0 | 1 | |

OLS | 0.062 | 0.052 | 0.049 | 0.053 | 0.139 | 0.161 | |

AIC | −135.023 | −143.847 | −104.694 | −101.602 | −71.133 | −65.324 |

Copula | Parameters and Tests | Stage | ||
---|---|---|---|---|

${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{c}}$ | ||

Gumbel | ${\mathsf{\theta}}_{1}$ | 2.824 | 6.062 | 3.674 |

OLS | 0.026 | 0.039 | 0.061 | |

AIC | −178.881 | −112.607 | −102.371 | |

BIC | −176.444 | −110.826 | −100.482 | |

Clayton | ${\mathsf{\theta}}_{2}$ | 1.984 | 3.575 | 3.638 |

OLS | 0.044 | 0.058 | 0.057 | |

AIC | −151.894 | −98.698 | −105.086 | |

BIC | −149.457 | −96.917 | −103.197 | |

Frank | ${\mathsf{\theta}}_{3}$ | 10.937 | 23.927 | 11.485 |

OLS | 0.027 | 0.040 | 0.061 | |

AIC | −175.828 | −112.037 | −102.050 | |

BIC | −173.390 | −110.256 | −100.161 |

Stage | Synchronous Probability (%) | Asynchronous Probability (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Rich–Rich | Normal–Normal | Poor–Poor | Total | Rich–Normal | Rich–Poor | Normal–Rich | Normal–Poor | Poor–Rich | Poor–Normal | Total | |

${T}_{a}$ | 29.84 | 11.46 | 28.54 | 69.84 | 6.12 | 1.54 | 6.12 | 7.42 | 1.54 | 7.42 | 30.16 |

${T}_{b}$ | 34.04 | 17.48 | 33.30 | 84.82 | 3.39 | 0.07 | 3.39 | 4.13 | 0.07 | 4.13 | 15.18 |

${T}_{c}$ | 28.02 | 11.57 | 31.13 | 70.72 | 8.25 | 1.22 | 8.25 | 5.17 | 1.22 | 5.17 | 29.28 |

Item | ${\mathit{T}}_{\mathit{a}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{c}}$ | |||
---|---|---|---|---|---|---|

Precipitation | Temperature | Precipitation | Temperature | Precipitation | Temperature | |

Runoff | 0.859 ** | −0.137 | 0.773 ** | −0.008 | 0.619 ** | −0.173 |

Sediment | 0.773 ** | −0.235 | 0.745 ** | −0.051 | 0.498 * | −0.065 |

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## Share and Cite

**MDPI and ACS Style**

Huang, X.; Qiu, L.
Characteristic Analysis and Uncertainty Assessment of the Joint Distribution of Runoff and Sediment: A Case Study of the Huangfuchuan River Basin, China. *Water* **2023**, *15*, 2644.
https://doi.org/10.3390/w15142644

**AMA Style**

Huang X, Qiu L.
Characteristic Analysis and Uncertainty Assessment of the Joint Distribution of Runoff and Sediment: A Case Study of the Huangfuchuan River Basin, China. *Water*. 2023; 15(14):2644.
https://doi.org/10.3390/w15142644

**Chicago/Turabian Style**

Huang, Xin, and Lin Qiu.
2023. "Characteristic Analysis and Uncertainty Assessment of the Joint Distribution of Runoff and Sediment: A Case Study of the Huangfuchuan River Basin, China" *Water* 15, no. 14: 2644.
https://doi.org/10.3390/w15142644