Skip to main content
SpringerLink
Log in

An entropy and copula-based framework for streamflow prediction and spatio-temporal identification of drought

  • ORIGINAL PAPER
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Reliable and easy-to-implement predictions of hydrometeorological variables are important for policymaking and public security. In this study, we developed a probabilistic framework for the description of hydrometeorological high-dimensional dependence and prediction by first-time coupling the principle of maximum entropy (POME) with C-vine copulas (PC). Two case studies with different emphases were investigated to evaluate the application of the PC framework. In the first case, we tested the PC framework based on a one-month-ahead streamflow forecast at the outlet station of the Jinsha River Basin. Results indicated that: (1) the marginal probability distributions or margins derived from optimal-moment-based POME best represented the current state of knowledge compared with those from traditional parametric distributions, and (2) the PC framework produced more skillful forecasts than did the traditional parametric C-vine (TC) and three data-driven models. The second case verified the performance of the PC framework in nationwide summer drought identification. Results of visual comparison of two typical historical summer drought events indicated that the PC framework captured the spatio-temporal characteristics of droughts. The PC framework combines the respective advantages of POME and C-vine copulas, ensuring its potential in higher-dimensional hydrometeorological modeling and flexibility in extending to other fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data availability

All data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  • Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insur Math Econ 44(2):182–198

    Article  Google Scholar 

  • Aghakouchak A (2014) Entropy–copula in hydrology and climatology. J Hydrometeorol 15(6):2176–2189

    Article  Google Scholar 

  • Al-Sudani ZA, Salih SQ, Yaseen ZM (2019) Development of multivariate adaptive regression spline integrated with differential evolution model for streamflow simulation. J Hydrol 573:1–12

    Article  Google Scholar 

  • Barriopedro D, Gouveia CM, Trigo RM, Wang L (2012) The 2009/10 drought in China: possible causes and impacts on vegetation. J Hydrometeorol 13(4):1251–1267

    Article  Google Scholar 

  • Bedford T, Cooke RM (2002) Vines–a new graphical model for dependent random variables. Ann Stat 30(4):1031–1068

    Article  Google Scholar 

  • Brechmann EC, Hendrich K, Czado C (2013) Conditional copula simulation for systemic risk stress testing. Insur Math Econ 53(3):722–732

    Article  Google Scholar 

  • Burrough PA, McDonnell RA (1998) Principles of geographical information systems. Oxford University Press, Oxford

    Google Scholar 

  • Chen L, Singh VP (2018) Entropy-based derivation of generalized distributions for hydrometeorological frequency analysis. J Hydrol 557:699–712

    Article  Google Scholar 

  • Feng Y, Shi P, Qu S, Mou S, Chen C, Dong F (2020) Nonstationary flood coincidence risk analysis using time-varying copula functions. Sci Rep 10(1):1–12

    Google Scholar 

  • Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 12(4):347–368

    Article  Google Scholar 

  • Gringorten II (1963) A plotting rule for extreme probability paper. J Geophys Res 68(3):813–814

    Article  Google Scholar 

  • Gu L, Yin J, Zhang H, Wang HM, Yang G, Wu X (2021) On future flood magnitudes and estimation uncertainty across 151 catchments in mainland China. Int J Climatol 41:E779–E800

    Article  Google Scholar 

  • Guo A, Chang J, Wang Y, Huang Q, Guo Z (2017) Maximum entropy-copula method for hydrological risk analysis under uncertainty: a case study on the loess plateau. China Entropy 19(11):609

    Article  Google Scholar 

  • Heřmanovský M, Havlíček V, Hanel M, Pech P (2017) Regionalization of runoff models derived by genetic programming. J Hydrol 547:544–556

    Article  Google Scholar 

  • Huang S, Huang Q, Chang J, Zhu Y, Leng G, Xing L (2015) Drought structure based on a nonparametric multivariate standardized drought index across the Yellow River basin, China. J Hydrol 530:127–136

    Article  Google Scholar 

  • Hutchinson M (2013) ANUSPLIN version 4.3. Centre for Resource and Environmental Studies, Australian National University, Canberra. Available online at http://fennerschool.anu.edu.au/publications/software/anusplin.php

  • Jaynes E (1957) Information theory and statistical mechanics. Phys Rev 106(4):620

    Article  Google Scholar 

  • Ji Z, Wan Y (2021) A novel method for socioeconomic data spatialization. Spatial Stat 43:100501

    Article  Google Scholar 

  • Joe H (1997) Multivariate models and multivariate dependence concepts. CRC Press, Boca Raton

    Book  Google Scholar 

  • Joe H, Kurowicka D (eds) (2011) Dependence modeling: vine copula handbook. World Scientific, Singapore

    Google Scholar 

  • Joe H (1996) Families of m-variate distributions with given margins and m (m−1)/2 bivariate dependence parameters. Lecture Notes-Monograph Series, pp 120–141

  • Ju X, Wang Y, Wang D, Singh VP, Xu P, Wu J, Ma J, Liu J, Zhang J (2021) A time-varying drought identification and frequency analyzation method: a case study of Jinsha River Basin. J Hydrol 603:126864

    Article  Google Scholar 

  • Kapur JN, Kesavan HK (1992) Entropy optimization principles and their applications. In: Entropy and energy dissipation in water resources. Springer, Dordrecht, pp 3–20

  • Kuhn M, Wing J, Weston S, Williams A, Keefer C, Engelhardt A, Team, R. C (2020) Package ‘caret.’ RJ 223:7

    Google Scholar 

  • Kurowicka D, Cooke RM (2006) Uncertainty analysis with high dimensional dependence modelling. Wiley, New York

    Book  Google Scholar 

  • Leng G, Tang Q, Rayburg S (2015) Climate change impacts on meteorological, agricultural and hydrological droughts in China. Global Planet Change 126:23–34

    Article  Google Scholar 

  • Li F, Zheng Q (2016) Probabilistic modelling of flood events using the entropy copula. Adv Water Resour 97:233–240

    Article  Google Scholar 

  • Li Z, Shao Q, Tian Q, Zhang L (2020) Copula-based drought severity-area-frequency curve and its uncertainty, a case study of Heihe River basin. China Hydrol Res 51(5):867–881

    Article  Google Scholar 

  • Li H, Wang D, Singh VP, Wang Y, Wu J, Wu J (2021) Developing an entropy and copula-based approach for precipitation monitoring network expansion. J Hydrol 598:126366

    Article  Google Scholar 

  • Liu Z, Zhou P, Zhang Y (2015) A probabilistic wavelet–support vector regression model for streamflow forecasting with rainfall and climate information input. J Hydrometeorol 16(5):2209–2229

    Article  Google Scholar 

  • Liu D, Wang D, Singh VP, Wang Y, Wu J, Wang L, Zou X, Chen Y, Chen X (2017) Optimal moment determination in POME-copula based hydrometeorological dependence modelling. Adv Water Resour 105:39–50

    Article  Google Scholar 

  • Liu Z, Cheng L, Hao Z, Li J, Thorstensen A, Gao H (2018) A framework for exploring joint effects of conditional factors on compound floods. Water Resour Res 54(4):2681–2696

    Article  Google Scholar 

  • Lopez-Paz D, Hernández-Lobato JM, Zoubin G (2013) Gaussian process vine copulas for multivariate dependence. In: International conference on machine learning. PMLR, pp 10–18

  • Lu HJ, Mo XG, Hu S (2012) Spatiotemporal variation characteristics of meteorological droughts in North China Plain during 1960–2009. J Natl Disasters 21(06):72–82 ((in Chinese with English summary))

    Google Scholar 

  • Maity R, Ramadas M, Govindaraju RS (2013) Identification of hydrologic drought triggers from hydroclimatic predictor variables. Water Resour Res 49(7):4476–4492

    Article  Google Scholar 

  • Massey Jr, Frank J (1951) The Kolmogorov-Smirnov test for goodness of fit. J Am Stat Assoc 46(253):68–78

    Article  Google Scholar 

  • McKee TB, Doesken NJ, Kleist J (1993) The relationship of drought frequency and duration to time scales. In: 8th conference on applied climatology. American Meteorological Society, Boston

  • Milly PCD (2008) Climate change: stationarity is dead. Whither Water Manag Sci 319:573–574

    CAS  Google Scholar 

  • Moradkhani H, Hsu KL, Gupta HV, Sorooshian S (2004) Improved streamflow forecasting using self-organizing radial basis function artificial neural networks. J Hydrol 295(1–4):246–262

    Article  Google Scholar 

  • Pan S, Joe H (2022) Predicting times to event based on vine copula models. Comput Stat Data Anal 175:107546

    Article  Google Scholar 

  • Qiu J (2010) China drought highlights future climate threats: Yunnan’s worst drought for many years has been exacerbated by destruction of forest cover and a history of poor water management. Nature 465(7295):142–144

    Article  CAS  Google Scholar 

  • Sarhadi A, Ausín MC, Wiper MP, Touma D, Diffenbaugh NS (2018) Multidimensional risk in a nonstationary climate: Joint probability of increasingly severe warm and dry conditions. Sci Adv. https://doi.org/10.1126/sciadv.aau3487

    Article  Google Scholar 

  • Shannon CE (2001) A mathematical theory of communication. ACM SIGMOBILE Mobile Comput Commun Rev 5(1):3–55

    Article  Google Scholar 

  • Shukla S, Wood AW (2008) Use of a standardized runoff index for characterizing hydrologic drought. Geophys Res Lett. https://doi.org/10.1029/2007GL032487

    Article  Google Scholar 

  • Singh VP (2013) Entropy theory and its application in environmental and water engineering. Wiley-Blackwell, New York

    Book  Google Scholar 

  • Singh VP (2014) Entropy theory in hydrologic science and engineering. McGraw-Hill Education, New York

    Book  Google Scholar 

  • Singh VP, Guo H (1995) Parameter estimation for 2-parameter log-logistic distribution (LLD2) by maximum entropy. Civ Eng Syst 12(4):343–357

    Article  Google Scholar 

  • Singh VP, Rajagopal AK, Singh K (1986) Derivation of some frequency distributions using the principle of maximum entropy (POME). Adv Water Resour 9(2):91–106

    Article  Google Scholar 

  • Singh VP, Guo H, Yu FX (1993) Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome. Stoch Hydrol Hydraul 7(3):163–177

    Article  Google Scholar 

  • Sklar M (1959) Fonctions de repartition an dimensions et leurs marges. Publ Inst Statist Univ Paris 8:229–231

    Google Scholar 

  • Tao Y, Wang Y, Wang D, Ni L, Wu J (2021) A C-vine copula framework to predict daily water temperature in the Yangtze River. J Hydrol 598:126430

    Article  Google Scholar 

  • Tosunoglu F, Singh VP (2018) Multivariate modeling of annual instantaneous maximum flows using copulas. J Hydrol Eng 23(3):04018003

    Article  Google Scholar 

  • U.S. Army Corps of Engineers, (2000). Hydrologic modeling system HEC-HMS

  • Vernieuwe H, Vandenberghe S, De Baets B, Verhoest NE (2015) A continuous rainfall model based on vine copulas. Hydrol Earth Syst Sci 19(6):2685–2699

    Article  Google Scholar 

  • Wang H, Yang ZX, Wang L, Liu YP, Song Y, Cao J (2014) The application of TVDI in drought monitoring over Yunnan province during 2009 to 2010. J Yunnan Univ (natl Sci) 36(1):59–65 ((In Chinese with English summary))

    CAS  Google Scholar 

  • Wei T, Song S (2018) Copula-based composite likelihood approach for frequency analysis of short annual precipitation records. Hydrol Res 49(5):1498–1512

    Article  Google Scholar 

  • Wilks DS (2011) Statistical methods in the atmospheric sciences, vol 100. Academic Press, Cambridge

    Google Scholar 

  • Williams AP, Cook BI, Smerdon JE (2022) Rapid intensification of the emerging southwestern North American megadrought in 2020–2021. Nat Clim Chang 12(3):232–234

    Article  Google Scholar 

  • Wu C, Hu BX, Huang G, Wang P, Xu K (2018) Responses of runoff to historical and future climate variability over China. Hydrol Earth Syst Sci 22(3):1971–1991

    Article  Google Scholar 

  • Wu H, Su X, Singh VP, Feng K, Niu J (2021) Agricultural drought prediction based on conditional distributions of vine copulas. Water Resour Res. https://doi.org/10.1029/2021WR029562

    Article  Google Scholar 

  • Xu K, Yang D, Xu X, Lei H (2015) Copula based drought frequency analysis considering the spatio-temporal variability in Southwest China. J Hydrol 527:630–640

    Article  Google Scholar 

  • Xu Y, Wang L, Ross KW, Liu C, Berry K (2018) Standardized soil moisture index for drought monitoring based on soil moisture active passive observations and 36 years of north American land data assimilation system data: a case study in the southeast United States. Remote Sens 10(2):301

    Article  Google Scholar 

  • Xu C, McDowell NG, Fisher RA, Wei L, Sevanto S, Christoffersen BO, Weng E, Middleton RS (2019) Increasing impacts of extreme droughts on vegetation productivity under climate change. Nat Clim Change 9(12):948–953

    Article  CAS  Google Scholar 

  • Xu P, Wang D, Wang Y, Qiu J, Singh VP, Ju X, Zhang A, Wu J, Zhang C (2021) Time-varying copula and average annual reliability-based nonstationary hazard assessment of extreme rainfall events. J Hydrol 603:126792

    Article  Google Scholar 

  • Yu X, He X, Zheng H, Guo R, Ren Z, Zhang D, Lin J (2014) Spatial and temporal analysis of drought risk during the crop-growing season over northeast China. Nat Hazards 71(1):275–289

    Article  Google Scholar 

  • Yuan W, Cai W, Chen Y, Liu S, Dong W, Zhang H, Yu G, Chen Z, He H, Guo W, Liu D, Liu S, Xiang W, Xie Z, Zhao Z, Zhou G (2016) Severe summer heatwave and drought strongly reduced carbon uptake in Southern China. Sci Rep 6(1):1–12

    Google Scholar 

  • Zhang L, Singh VP (2012) Bivariate rainfall and runoff analysis using entropy and copula theories. Entropy 14(9):1784–1812

    Article  Google Scholar 

  • Zhang L, Singh VP (2019) Copuals and their applications in water resources engineering. Cambridge University Press, London, England

    Book  Google Scholar 

  • Zhang WC, Zheng JM, Ren JZ (2013) Climate characteristics of extreme drought events in Yunnan. J Catastrophol 28(01):59–64 ((in Chinese with English summary))

    CAS  Google Scholar 

  • Zhang XJ, Tang Q, Pan M, Tang Y (2014) A long-term land surface hydrologic fluxes and states dataset for China. J Hydrometeorol 15(5):2067–2084

    Article  Google Scholar 

  • Zhang Q, Yao Y, Li Y, Huang J, Ma Z, Wang Z, Wang S, Wang Y, Zhang Y (2020) Causes and changes of drought in China: research progress and prospects. J Meteorol Res 34(3):460–481

    Article  Google Scholar 

  • Zscheischler J, Michalak AM, Schwalm C, Mahecha MD, Huntzinger DN, Reichstein M et al (2014) Impact of large-scale climate extremes on biospheric carbon fluxes: an intercomparison based on MsTMIP data. Global Biogeochem Cycles 28(6):585–600

    Article  CAS  Google Scholar 

  • Zuo G, Luo J, Wang N, Lian Y, He X (2020) Two-stage variational mode decomposition and support vector regression for streamflow forecasting. Hydrol Earth Syst Sci 24(11):5491–5518

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This study was supported by the second Tibetan Plateau Scientific Expedition and Research Program (STEP), Grant No. 2019QZKK0203, and the open fund of Key Laboratory of Water Science and Engineering, Ministry of Water Resources (2021100108).

Author information

Authors and Affiliations

  1. Key Laboratory of Surficial Geochemistry, Ministry of Education, Department of Hydrosciences, School of Earth Sciences and Engineering, State Key Laboratory of Pollution Control and Resource Reuse, Frontiers Science Center for Critical Earth Material Cycling, Nanjing University, Nanjing, People’s Republic of China

    Xiaopei Ju, Dong Wang, Pengcheng Xu, Along Zhang & Jichun Wu

  2. School of Water Resources and Hydropower Engineering, North China Electric Power University, Beijing, People’s Republic of China

    Yuankun Wang

  3. Department of Biological and Agricultural Engineering, Texas A&M University, College Station, TX, 77843, USA

    Vijay P. Singh

  4. National Water Center, UAE University, Al Ain, UAE

    Vijay P. Singh

  5. College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou, People’s Republic of China

    Pengcheng Xu

  6. Nanjing Hydraulic Research Institute, Nanjing, People’s Republic of China

    Tao Ma, Jiufu Liu & Jianyun Zhang

Authors
  1. Xiaopei Ju
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuankun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vijay P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pengcheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Along Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Jichun Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Tao Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Jiufu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Jianyun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Xiaopei Ju: Methodology, Formal analysis, Writing - Original Draft, Writing - Review & Editing Dong Wang: Conceptualization, Writing - Review & Editing, Supervision Yuankun Wang: Conceptualization, Supervision, Project administration Vijay P. Singh: Conceptualization, Writing - Review Pengcheng Xu: Conceptualization, Supervision Along Zhang: Writing - Review Jichun Wu: Writing - Review & Editing, Project administration Tao Ma: Writing - Review & Editing, Project administration Jiufu Liu: Writing - Review & Editing, Project administration Jianyun Zhang: Writing - Review & Editing, Project administration

Corresponding authors

Correspondence to Dong Wang or Pengcheng Xu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary file1 (PDF 614 kb)

Appendices

Appendix 1

See Table 5.

Table 5 Information of five Copulas used in this study
Full size table

Appendix 2

See Table 6.

Table 6 Information of four candidate parametric distribution functions used in this study
Full size table

Appendix 3

3.1 Standardized drought indexes

The empirical Gringorten plotting position formula (Gringorten 1963) was used to construct the empirical CDF of the selected cumulative climate variables and then the standard drought indicators (SDI) were derived via normal transformation (Eq. 17)

$$SD{I(i)}={N}^{-1}(\frac{i-0.44}{n+0.12})$$
(17)

where \(i\) is the rank of observed climate variable values in descending order at certain scales (e.g., SPI-6; 6-month-cumulative precipitation), \(n\) is the length of series, and \({N}^{-1}\) denotes the normal quantile transformation.

Below is the reference mentioned above:

Gringorten (1963).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ju, X., Wang, D., Wang, Y. et al. An entropy and copula-based framework for streamflow prediction and spatio-temporal identification of drought. Stoch Environ Res Risk Assess 37, 2187–2204 (2023). https://doi.org/10.1007/s00477-023-02388-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-023-02388-2

Keywords

Search

Navigation