# Multivariate Drought Risk Analysis for the Weihe River: Comparison between Parametric and Nonparametric Copula Methods

^{1}

^{2}

^{*}

## Abstract

**:**

^{OR}, T

^{AND}, and T

^{Kendall}) were computed through copula modeling, providing valuable insights into the co-occurrence of extreme drought events. For the SPI and SPEI with a 50-year return period, the T

^{OR}values range from 25.5 to 37.9 years, the T

^{AND}values fluctuate between 73.4 and 1233 years, and the T

^{Kendall}values range from 60.61 to 574.71 years, indicating a high correlation between the SPI and SPEI in the study area. The spatial analysis revealed varying patterns across the basin with some regions more prone to experiencing simultaneous drought conditions characterized by both the SPI and SPEI. Furthermore, our results indicated that the SPEI exhibited more severity in drought characterization than the SPI due to its consideration of temperature effects. The disparities in the spatial features of the SPI and SPEI underscore the importance of incorporating multiple meteorological factors for a comprehensive drought risk analysis. This research contributes to a better understanding of the drought patterns and their joint risks in the Wei River basin, offering valuable information for drought preparedness and water resource management.

## 1. Introduction

## 2. Methodology

#### 2.1. Copula Method

_{1}, X

_{2}, …, X

_{d}with their marginals denoted as F

_{1}, …, F

_{d}, their joint cumulative distribution function (CDF) F(.) can be constructed through a copula function as follows [24]:

_{i}(i = 1, 2, …, d) is continuous. The multivariate probability density function (PDF) f(.) would also be formulated as follows [25]:

_{i}= F

_{i}(x

_{i}). ${f}_{i}\left({x}_{i}\right)$ is the marginal PDF for random variable X

_{i}.

#### 2.2. Parametric Copulas

^{−1}(.) = 0 when u

_{2}≥ φ(0); θ is the parameter hidden in the generating function [26]. The most commonly used Archimedean copulas include the Clayton, Gumbel, and Frank copulas.

^{−1}is the inverse standard Gaussian CDF.

#### 2.3. Nonparametric Copulas

_{i}

_{1}, U

_{i}

_{2}), i = 1, …, n, from a bivariate copula C, the corresponding density function c(u

_{1}, u

_{2}) can be estimated through the kernel density estimator as follows [27]:

_{b}(·) = K(·/b)/b. The kernel function K is typically a symmetric, bounded probability density function on R

^{2}, and b

_{n}> 0 is the smoothing or bandwidth parameter. The estimator in Equation (5) will result in a considerable amount of probability mass outside the unit square, leading to ${\widehat{c}}_{n}\left({u}_{1},{u}_{2}\right)$ not being a valid density function on [0, 1]

^{2}due to its integral not equaling one [27]. Furthermore, the estimator will also suffer from severe bias at the boundaries [27]. Some approaches have been developed to tackle the above challenges, including the mirror-reflection method, the beta kernel method, and the transformation method [29,30]. The kernel density estimator allows us to estimate the bivariate copula density function nonparametrically, avoiding the need to assume a specific functional form and providing a flexible approach to capture the underlying dependence structure of the data.

#### 2.4. Primary and Secondary Return Periods

_{C}is Kendall’s distribution associated with the theoretical Copula function C(.). For the Archimedean copulas, K

_{C}can be expressed as follows [26,31,32,33]:

## 3. Case Study

#### 3.1. Overview of Wei River Basin

#### 3.2. Data Collection and Drought Identification

_{i}is the monthly precipitation, n refers to the sample size, and $\overline{x}$ is the average of the precipitation samples. The probability of the random variable x less than x

_{0}can be derived as follows:

_{0}= 2.515517, c

_{1}= 0.802853, c

_{2}= 0.010328, d

_{1}= 1.432788, d

_{2}= 0.189269, and d

_{3}= 0.001308. The Gamma distributions for the monthly precipitation were tested with the Anderson–Darling test with their p-values provided in Table S1. The results suggest that the Gamma distributions can pass the statistical test for most months at each station except that some rejections occurred in December at some stations. However, the Gamma distributions would still be applicable in this study since only the annual minimum SPI values were analyzed, which seldom occurred in December.

_{mean}is the average air temperature (°C); T

_{max}and T

_{min}are the maximum and minimum air temperatures (°C), respectively; and R

_{a}is the daily net radiation on the land surface (MJ m

^{−2}d

^{−1}). The water balance can then be obtained as follows:

_{i}= P

_{i}− PET

_{i}

_{i}, P

_{i}, and PET

_{i}respectively denote the water balance, monthly precipitation, and monthly potential evapotranspiration. In this study, the GEV distribution was employed to normalize the water balance series (i.e., D

_{i}) with the density function expressed as follows [38]:

_{0}= 2.515517, c

_{1}= 0.802853, c

_{2}= 0.010328, d

_{1}= 1.432788, d

_{2}= 0.189269, and d

_{3}= 0.001308. The GEV distributions were tested with the Anderson–Darling test with their p-values provided in Table S2 [40]. The results suggest that the GEV distributions can pass the statistical test for all months at each station, indicating its applicability to derive the SPEI values.

## 4. Results Analysis

#### 4.1. Probability Estimation of Individual Drought Index

#### 4.2. Quantification of Interdependence between SPI and SPEI through Both Parametric and Nonparametric Copulas

#### 4.3. Primary and Joint Return Period of SPI and SPEI

^{OR}, T

^{AND}, and T

^{Kendall}, obtained through copula modeling provide valuable insights into the co-occurrence of extreme drought events. T

^{OR}represents the time of occurrence for droughts when either the SPI or SPEI falls below their respective 50-year RP thresholds. As shown in Equation (6), u

_{1}= u

_{2}= 0.98 since both the SPI and SPEI have an RP of 50 years. T

^{OR}can then be derived from the obtained copula model based on Equation (6). For instance, at station 52986, the Gumbel copula was selected; thus, we can have T

^{OR}= $1/[1-\mathrm{e}\mathrm{x}\mathrm{p}\{-{[{(-\mathrm{l}\mathrm{n}{(u}_{1}))}^{\theta}+{(-\mathrm{l}\mathrm{n}{(u}_{2})}^{\theta}]}^{1/\theta}\}]$ = 33.18 years, where u

_{1}= u

_{2}= 0.98 and θ = 1.669, obtained in Section 4.2. In summary, the T

^{OR}values range from 25.5 to 37.9 years, indicating the timing of individual drought events below the SPI or SPEI with a 50-year RP. On the other hand, the T

^{AND}values fluctuate between 73.4 and 1233 years, representing the time of occurrence for droughts when both the SPI and SPEI are simultaneously below their 50-year RP thresholds.

^{Kendall}values, ranging from 60.61 to 574.71, indicate the likelihood of compound drought occurrences where both the SPI and SPEI experience extreme droughts simultaneously. These joint return periods emphasize the importance of considering the interdependence between the SPI and SPEI for a comprehensive understanding of drought risks in the Wei River basin.

^{AND}return periods less than 400 years. In contrast, the central-southeastern part has relatively fewer chances of experiencing simultaneous SPI- and SPEI-based droughts with the T

^{AND}return period potentially exceeding 800 years. The spatial variations in TOR present a different feature compared to T

^{AND}. It indicates that the western, northeastern, and southeastern regions are less likely to encounter a 50-year drought represented solely by either the SPI or SPEI, while the central-southeastern and northwestern regions are more likely to experience a 50-year SPI or SPEI drought. Regarding T

^{Kendall}, as shown in Figure 5c, its spatial variations are similar to those of T

^{AND}but with relatively shorter return periods. This suggests that the basin may experience compound drought events with shorter recurrence intervals, highlighting the possibility of concurrent extreme drought occurrences based on the interdependence between the SPI and SPEI.

## 5. Conclusions

^{OR}, T

^{AND}, and Kendall’s return period (T

^{Kendall}), provided valuable insights into the co-occurrence of extreme drought events. We observed varying spatial patterns in the basin with certain regions more prone to experiencing concurrent drought conditions characterized by both the SPI and SPEI. Our spatial analysis also revealed that the SPEI exhibited more severity in drought characterization than the SPI, highlighting the significance of considering both precipitation and temperature factors in drought assessments. The disparities in the spatial features of the SPI and SPEI underscore the need for a comprehensive approach that incorporates multiple meteorological variables to enhance drought risk analysis accuracy. Overall, our study contributes to a better understanding of the drought patterns and their joint risks in the Wei River basin. The copula-based approach demonstrated its effectiveness in quantifying the interdependence between the SPI and SPEI, providing valuable information for water resource management and drought resilience planning in the region. The insights gained from this research can serve as a basis for informed decision-making and the development of targeted drought mitigation and adaptation strategies. Policymakers and water resource managers can utilize this knowledge to implement region-specific measures and policies to combat the increasing drought risks in the basin effectively.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wilhite, D.A. (Ed.) Drought as a natural hazard: Concepts and definitions. In Drought: A Global Assessment; Routledge: London, UK, 2000; Chapter 1; Volume I, pp. 3–18. [Google Scholar]
- Mishra, A.K.; Singh, V.P. A review of drought concepts. J. Hydrol.
**2010**, 391, 202–216. [Google Scholar] - McKee, T.B.; Doesken, N.J.; Kleist, J. The relationship of drought frequency and duration to time scales. In Proceedings of the 8th Conference on Applied Climatology, Anaheim, CA, USA, 17–22 January 1993. [Google Scholar]
- Vicente-Serrano, S.M.; Beguería, S.; López-Moreno, J.I. A multiscalar drought index sensitive to global warming: The standardized precipitation evapotranspiration index. J. Clim.
**2010**, 23, 1696–1718. [Google Scholar] [CrossRef] - Beguería, S.; Vicente-Serrano, S.M.; Reig, F.; Latorre, B. Standardized precipitation evapotranspiration index (SPEI) revisited: Parameter fitting, evapotranspiration models, tools, datasets and drought monitoring. Int. J. Climatol.
**2014**, 34, 3001–3023. [Google Scholar] [CrossRef] - Li, X.; Sha, J.; Wang, Z.L. Comparison of drought indices in the analysis of spatial and temporal changes of climatic drought events in a basin. Environ. Sci. Pollut. Res.
**2019**, 26, 10695–10707. [Google Scholar] [CrossRef] [PubMed] - Chong, K.L.; Huang, Y.F.; Koo, C.H.; Ahmed, A.N.; El-Shafie, A. Spatiotemporal variability analysis of standardized precipitation indexed droughts using wavelet transform. J. Hydrol.
**2022**, 605, 127299. [Google Scholar] [CrossRef] - Huang, Y.F.; Ahmed, A.N.; Ng, J.L.; Tan, K.W.; Kumar, P.; El-Shafie, A. Rainfall Variability Index (RVI) analysis of dry spells in Malaysia. Nat. Hazards
**2022**, 112, 1423–1475. [Google Scholar] [CrossRef] - Yong, S.L.S.; Ng, J.L.; Huang, Y.F.; Ang, C.K.; Mirzaei, M.; Ahmed, A.N. Local and global sensitivity analysis and its contributing factors in reference crop evapotranspiration. Water Supply
**2023**, 23, 1672–1683. [Google Scholar] [CrossRef] - Hao, Z.; AghaKouchak, A. Multivariate standardized drought index: A parametric multi-index model. Adv. Water Resour.
**2013**, 57, 12–18. [Google Scholar] [CrossRef] - Kao, S.C.; Govindaraju, R.S. A copula-based joint deficit index for droughts. J. Hydrol.
**2010**, 380, 121–134. [Google Scholar] [CrossRef] - Serinaldi, F.; Kilsby, C.G. Stationarity is undead: Uncertainty dominates the distribution of extremes. Adv. Water Resour.
**2015**, 77, 17–36. [Google Scholar] - Wang, F.; Wang, Z.; Yang, H.; Di, D.; Zhao, Y.; Liang, Q. A new copula-based standardized precipitation evapotranspiration streamflow index for drought monitoring. J. Hydrol.
**2020**, 585, 124793. [Google Scholar] [CrossRef] - De Michele, C.; Salvadori, G.; Vezzoli, R.; Pecora, S. Multivariate assessment of droughts: Frequency analysis and dynamic return period. Water Resour. Res.
**2013**, 49, 6985–6994. [Google Scholar] [CrossRef] - Xiao, M.; Yu, Z.; Zhu, Y. Copula-based frequency analysis of drought with identified characteristics in space and time: A case study in Huai River basin, China. Theor. Appl. Climatol.
**2019**, 137, 2865–2875. [Google Scholar] [CrossRef] - Huang, K.; Fan, Y.R. Parameter Uncertainty and Sensitivity Evaluation of Copula-Based Multivariate Hydroclimatic Risk Assessment. J. Environ. Inform.
**2021**, 38, 131–144. [Google Scholar] [CrossRef] - Fan, Y.R.; Yu, L.; Shi, X.; Duan, Q.Y. Tracing uncertainty contributors in the multi-hazard risk analysis for compound extremes. Earth’s Future
**2021**, 9, e2021EF002280. [Google Scholar] [CrossRef] - Zhou, X.; Huang, G.; Wang, X.; Fan, Y.; Cheng, G. A coupled dynamical-copula downscaling approach for temperature projections over the Canadian Prairies. Clim. Dyn.
**2018**, 51, 2413–2431. [Google Scholar] [CrossRef] - Salvadori, G.; De Michele, C. On the Use of Copulas in Hydrology: Theory and Practice. J. Hydrol. Eng.
**2007**, 12, 369–380. [Google Scholar] [CrossRef] - Song, S.; Singh, V.P. Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data. Stoch. Environ. Res. Risk Assess.
**2010**, 24, 425–444. [Google Scholar] [CrossRef] - Kolev, N.; dos Anjos, U.; de M. Mendes, B.V. Copulas: A Review and Recent Developments. Stoch. Models
**2006**, 22, 617–660. [Google Scholar] [CrossRef] - Latif, S.; Simonovic, S.P. Nonparametric Approach to Copula Estimation in Compounding the Joint Impact of Storm Surge and Rainfall Events in Coastal Flood Analysis. Water Resour. Manag.
**2022**, 36, 5599–5632. [Google Scholar] [CrossRef] - Latif, S.; Simonovic, S.P. Trivariate Joint Distribution Modelling of Compound Events Using the Nonparametric D-Vine Copula Developed Based on a Bernstein and Beta Kernel Copula Density Framework. Hydrology
**2022**, 9, 221. [Google Scholar] [CrossRef] - Zhang, L.; Singh, V.P. Copulas and Their Applications in Water Resources Engineering; Cambridge University Press: New York, NY, USA, 2019. [Google Scholar]
- Fan, Y.R.; Huang, W.W.; Li, Y.P.; Huang, G.H.; Huang, K. A coupled ensemble filtering and probabilistic collocation approach for uncertainty quantification of hydrological models. J. Hydrol.
**2015**, 530, 255–272. [Google Scholar] [CrossRef] - Fan, Y.R. Bivariate hydrologic risk analysis for the Xiangxi River in Three Gorges Reservoir Area, China. Environ. Syst. Res.
**2022**, 11, 18. [Google Scholar] [CrossRef] - Nagler, T. kdecopula: An R Package for the Kernel Estimation of Bivariate Copula Densities. J. Stat. Softw.
**2018**, 84, 1–22. [Google Scholar] [CrossRef] - Aitken, C.G.G.; Lucy, D. Evaluation of Trace Evidence in the Form of Multivariate Data. J. R. Stat. Soc. C
**2004**, 53, 109–122. [Google Scholar] [CrossRef] - Gijbels, I.; Mielniczuk, J. Estimating the Density of a Copula Function. Commun. Stat.-Theory Methods
**1990**, 19, 445–464. [Google Scholar] [CrossRef] - Charpentier, A.; Fermanian, J.D.; Scaillet, O. The Estimation of Copulas: Theory and Practice. In Copulas: From Theory to Application in Finance; Rank, J., Ed.; Risk Books: London, UK, 2007. [Google Scholar]
- Graler, B.; van den Berg, M.J.; Vandenberghe, S.; Petroselli, A.; Grimaldi, S.; De Baets, B.; Verhoest, N.E.C. Multivariate return periods in hydrology: A critical and practical review focusing on synthetic design hydrograph estimation. Hydrol. Earth Syst. Sci.
**2013**, 17, 1281–1296. [Google Scholar] [CrossRef] - Salvadori, G.; De Michele, C.; Kottegoda, N.T.; Rosso, R. Extremes in Nature: An Approach Using Copula; Springer: Dordrencht, The Netherlands, 2007; p. 292. [Google Scholar]
- Sraj, M.; Bezak, N.; Brilly, M. Bivariate flood frequency analysis using the copula function: A case study of the Litija station on the Sava River. Hydrol. Process.
**2015**, 29, 225–238. [Google Scholar] [CrossRef] - Zou, L.; Xia, J.; She, D. Analysis of Impacts of Climate Change and Human Activities on Hydrological Drought: A Case Study in the Wei River Basin, China. Water Resour. Manag.
**2018**, 32, 1421–1438. [Google Scholar] [CrossRef] - 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] - Wen, Y.; Yang, A.; Fan, Y.; Wang, B.; Scott, D. Stepwise cluster ensemble downscaling for drought projection under climate change. Int. J. Climatol.
**2023**, 43, 2318–2338. [Google Scholar] [CrossRef] - Wang, Q.; Zhang, R.; Qi, J.; Zeng, J.; Wu, J.; Shui, W.; Wu, X.; Li, J. An improved daily standardized precipitation index dataset for mainland China from 1961 to 2018. Sci. Data
**2022**, 9, 124. [Google Scholar] [CrossRef] [PubMed] - Wang, Q.; Zeng, J.; Qi, J.; Zhang, X.; Zeng, Y.; Shui, W.; Xu, Z.; Zhang, R.; Wu, X.; Cong, J. A multi-scale daily SPEI dataset for drought characterization at observation stations over mainland China from 1961 to 2018. Earth Syst. Sci. Data
**2021**, 13, 331–341. [Google Scholar] [CrossRef] - Droogers, P.; Allen, R.G. Estimating Reference Evapotranspiration Under Inaccurate Data Conditions. Irrig. Drain. Syst.
**2002**, 16, 33–45. [Google Scholar] [CrossRef] - Şen, Z.; Almazroui, M. Actual Precipitation Index (API) for Drought classification. Earth Syst. Environ.
**2021**, 5, 59–70. [Google Scholar] [CrossRef]

Copula Name | Function [C(u_{1}, u_{2})] | Parameter Range | $\mathbf{Generator}\mathbf{Functions}[\mathit{\phi}(\mathit{t})]$ |
---|---|---|---|

Gaussian | $\mathsf{\Phi}({\mathsf{\Phi}}^{-1}\left({u}_{1}\right),{\mathsf{\Phi}}^{-1}\left({u}_{2}\right)|\mathsf{\Sigma})$ | $\mathsf{\Sigma}\in (-1,1)$ | |

Joe | $1-[{\left(1-{u}_{1}\right)}^{\theta}+{\left(1-{u}_{2}\right)}^{\theta}$ $-{\left(1-{u}_{1}\right)}^{\theta}{\left(1-{u}_{2}\right)}^{\theta}{]}^{1/\theta}$ | $\mathsf{\theta}\in [1,\mathrm{\infty})$ | $-\mathrm{l}\mathrm{n}(1-{\left(1-t\right)}^{\theta})$ |

Gumbel | $\mathrm{e}\mathrm{x}\mathrm{p}\{-{[{(-\mathrm{l}\mathrm{n}{(u}_{1}))}^{\theta}+{(-\mathrm{l}\mathrm{n}{(u}_{2})}^{\theta}]}^{1/\theta}\}$ | $\mathsf{\theta}\in [1,\mathrm{\infty})$ | $-\mathrm{l}\mathrm{n}({t}^{\theta})$ |

Frank | $-\frac{1}{\theta}\mathrm{l}\mathrm{n}\{1+\frac{({e}^{-\theta {u}_{1}}-1)({e}^{-\theta {u}_{2}}-1)}{{e}^{-\theta}-1}\}$ | $\mathsf{\theta}\in [-\mathrm{\infty},\mathrm{\infty})\backslash \{0\}$ | $-\mathrm{l}\mathrm{n}[\frac{{e}^{-\theta t}-1}{{e}^{-\theta}-1}]$ |

ID | Lat (°C) | Lon (°C) | Elevation (m) |
---|---|---|---|

52986 | 35.36667 | 103.8667 | 1886.6 |

52996 | 35.38333 | 105 | 2450.6 |

53738 | 36.83333 | 108.1833 | 1272.6 |

53817 | 35.96667 | 106.75 | 1753.2 |

53821 | 36.58333 | 107.3 | 1255.6 |

53845 | 36.6 | 109.5 | 957.6 |

53903 | 35.93333 | 105.9667 | 1901.3 |

53915 | 35.55 | 106.6667 | 1346.6 |

53923 | 35.73333 | 107.6333 | 1421.9 |

53929 | 35.2 | 107.8 | 1206.3 |

53942 | 35.81667 | 109.5 | 1158.3 |

56093 | 34.43333 | 104.0167 | 2314.6 |

57034 | 34.3 | 108.0667 | 505.4 |

57037 | 34.93333 | 108.9833 | 719 |

57046 | 34.48333 | 110.0833 | 2064.9 |

57134 | 33.53333 | 107.9833 | 1179.2 |

57144 | 33.43333 | 109.15 | 1098.6 |

57143 | 33.86667 | 109.9667 | 742.2 |

Station ID | SPI | SPEI | Kendall between SPI and SPEI | ||
---|---|---|---|---|---|

Mean | Sd | Mean | Sd | ||

52986 | −1.419 | 0.624 | −1.592 | 0.505 | 0.386 |

52996 | −1.490 | 0.505 | −1.584 | 0.455 | 0.472 |

53738 | −1.384 | 0.818 | −1.657 | 0.527 | 0.317 |

53817 | −1.412 | 0.568 | −1.677 | 0.489 | 0.291 |

53821 | −1.355 | 0.660 | −1.603 | 0.563 | 0.535 |

53845 | −1.377 | 0.595 | −1.590 | 0.554 | 0.403 |

53903 | −1.371 | 0.643 | −1.620 | 0.524 | 0.369 |

53915 | −1.402 | 0.524 | −1.611 | 0.507 | 0.432 |

53923 | −1.445 | 0.422 | −1.551 | 0.512 | 0.555 |

53929 | −1.411 | 0.532 | −1.548 | 0.496 | 0.435 |

53942 | −1.538 | 0.505 | −1.622 | 0.583 | 0.584 |

56093 | −1.309 | 0.756 | −1.592 | 0.589 | 0.527 |

57034 | −1.475 | 0.465 | −1.577 | 0.467 | 0.483 |

57037 | −1.568 | 0.449 | −1.574 | 0.495 | 0.370 |

57046 | −1.486 | 0.454 | −1.546 | 0.462 | 0.698 |

57134 | −1.427 | 0.485 | −1.629 | 0.384 | 0.391 |

57144 | −1.446 | 0.539 | −1.546 | 0.537 | 0.418 |

57143 | −1.500 | 0.468 | −1.601 | 0.514 | 0.515 |

Station ID | SPI | SPEI | ||||
---|---|---|---|---|---|---|

Distribution | p-Value (KS) | AIC | Distribution | p-Value (KS) | AIC | |

52986 | GEV | 0.9948 | −278.27 | Gamma | 1.0000 | −336.81 |

52996 | GEV | 0.9942 | −310.82 | Weibull | 0.9944 | −305.30 |

53738 | GEV | 0.9933 | −278.37 | Weibull | 0.9936 | −291.65 |

53817 | GEV | 0.9959 | −288.31 | Gumbel | 1.0000 | −363.26 |

53821 | GEV | 0.9942 | −319.34 | GEV | 1.0000 | −327.56 |

53845 | GEV | 0.9950 | −304.78 | GEV | 1.0000 | −327.82 |

53903 | GEV | 0.9942 | −301.65 | Weibull | 0.9944 | −293.43 |

53915 | Weibull | 1.0000 | −348.92 | GEV | 1.0000 | −342.69 |

53923 | Weibull | 1.0000 | −305.77 | GEV | 0.9952 | −309.03 |

53929 | GEV | 0.9479 | −293.35 | GEV | 0.9952 | −319.24 |

53942 | GEV | 0.9950 | −311.08 | GEV | 1.0000 | −357.52 |

56093 | GEV | 0.9933 | −268.81 | GEV | 0.9377 | −267.38 |

57034 | GEV | 1.0000 | −345.67 | GEV | 0.9959 | −306.45 |

57037 | GEV | 0.9493 | −282.44 | GEV | 0.9523 | −301.61 |

57046 | Weibull | 0.9513 | −299.15 | GEV | 1.0000 | −339.08 |

57134 | GEV | 1.0000 | −331.43 | GEV | 0.9959 | −324.45 |

57144 | GEV | 0.9468 | −270.23 | GEV | 1.0000 | −337.80 |

57143 | GEV | 1.0000 | −333.67 | GEV | 1.0000 | −347.42 |

Station ID | Parametric Copula | Nonparametric Copula | |||||
---|---|---|---|---|---|---|---|

Option | p-Value (KS) | RMSE | AIC | p-Value (KS) | RMSE | AIC | |

52986 | Gumbel | 0.9357 | 0.0316 | −309.03 | 0.9395 | 0.0322 | −290.72 |

52996 | Gumbel | 0.8041 | 0.0281 | −312.21 | 0.9999 | 0.0255 | −292.88 |

53738 | Gaussian | 0.9903 | 0.0333 | −290.58 | 0.9901 | 0.0341 | −271.71 |

53817 | Gaussian | 0.9942 | 0.0348 | −306.83 | 0.9945 | 0.0282 | −310.41 |

53821 | Frank | 0.9925 | 0.0268 | −316.37 | 0.9920 | 0.0263 | −302.81 |

53845 | Gaussian | 0.9373 | 0.0296 | −314.85 | 0.9389 | 0.0284 | −303.40 |

53903 | Gaussian | 0.9999 | 0.0244 | −324.92 | 0.9928 | 0.0261 | −302.56 |

53915 | Frank | 0.9921 | 0.0227 | −338.76 | 0.9414 | 0.0210 | −326.53 |

53923 | Gaussian | 0.9401 | 0.0275 | −321.57 | 0.9401 | 0.0276 | −305.10 |

53929 | Gaussian | 0.9999 | 0.0338 | −302.91 | 0.9389 | 0.0316 | −292.81 |

53942 | Gumbel | 0.9935 | 0.0219 | −341.91 | 0.9939 | 0.0318 | −304.00 |

56093 | Gumbel | 0.9295 | 0.0415 | −271.63 | 0.9910 | 0.0491 | −253.37 |

57034 | Joe | 1.0000 | 0.0227 | −346.27 | 1.0000 | 0.0255 | −321.08 |

57037 | Joe | 0.8196 | 0.0271 | −330.07 | 0.6459 | 0.0356 | −302.80 |

57046 | Joe | 0.9947 | 0.0247 | −338.62 | 0.9424 | 0.0257 | −317.68 |

57134 | Frank | 0.8116 | 0.0274 | −329.08 | 0.9441 | 0.0241 | −326.43 |

57144 | Joe | 0.9936 | 0.0315 | −309.24 | 0.9933 | 0.0442 | −275.99 |

57143 | Gumbel | 1.0000 | 0.0292 | −323.20 | 1.0000 | 0.0259 | −318.21 |

Station ID | SPI | SPEI | T^{OR} (Year) | T^{AND} (Year) | T^{Kendall} (Year) |
---|---|---|---|---|---|

52986 | −2.30 | −2.86 | 33.18 | 101.43 | 78.62 |

52996 | −2.35 | −2.48 | 34.30 | 92.21 | 76.22 |

53738 | −2.64 | −2.68 | 28.54 | 201.35 | 120.48 |

53817 | −2.25 | −3.00 | 26.88 | 357.73 | 170.65 |

53821 | −2.15 | −2.89 | 26.97 | 341.57 | 187.97 |

53845 | −2.35 | −2.77 | 28.50 | 203.35 | 121.07 |

53903 | −2.40 | −2.70 | 27.63 | 262.44 | 173.61 |

53915 | −2.32 | −2.66 | 26.18 | 555.44 | 264.55 |

53923 | −2.14 | −2.60 | 30.46 | 139.40 | 97.28 |

53929 | −2.31 | −2.48 | 29.16 | 175.15 | 126.26 |

53942 | −2.31 | −2.78 | 37.33 | 75.67 | 67.84 |

56093 | −2.29 | −2.58 | 36.99 | 77.14 | 62.58 |

57034 | −2.61 | −2.69 | 25.68 | 948.29 | 458.72 |

57037 | −2.21 | −2.52 | 25.52 | 1233.03 | 574.71 |

57046 | −2.39 | −2.55 | 26.31 | 501.76 | 273.22 |

57134 | −2.35 | −2.56 | 25.99 | 656.36 | 362.32 |

57144 | −2.34 | −2.57 | 37.92 | 73.39 | 60.61 |

57143 | −2.45 | −2.53 | 27.92 | 238.92 | 147.93 |

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## Share and Cite

**MDPI and ACS Style**

Liu, F.; Wang, X.; Chang, Y.; Xu, Y.; Zheng, Y.; Sun, N.; Li, W.
Multivariate Drought Risk Analysis for the Weihe River: Comparison between Parametric and Nonparametric Copula Methods. *Water* **2023**, *15*, 3283.
https://doi.org/10.3390/w15183283

**AMA Style**

Liu F, Wang X, Chang Y, Xu Y, Zheng Y, Sun N, Li W.
Multivariate Drought Risk Analysis for the Weihe River: Comparison between Parametric and Nonparametric Copula Methods. *Water*. 2023; 15(18):3283.
https://doi.org/10.3390/w15183283

**Chicago/Turabian Style**

Liu, Fengping, Xu Wang, Yuhu Chang, Ye Xu, Yinan Zheng, Ning Sun, and Wei Li.
2023. "Multivariate Drought Risk Analysis for the Weihe River: Comparison between Parametric and Nonparametric Copula Methods" *Water* 15, no. 18: 3283.
https://doi.org/10.3390/w15183283