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A Simplified Model for Predicting Drought Magnitudes: a Case of Streamflow Droughts in Canadian Prairies

Abstract

The model for prediction of drought magnitudes is based on the multiplicative relationship: drought magnitude (M) = drought intensity (I) × drought duration (L), where I, L, and M are presumed to obey respectively the truncated normal probability distribution function (pdf), the geometric pdf, and the normal pdf. The multiplicative relationship is applied in the standardized domain of the streamflows, named as SHI (standardized hydrological index) sequences, which are treated equivalent to standard normal variates. The expected drought magnitude E(M T ), i.e. the largest value of M over a sampling period of T-time units (T-year, T-month, and T-week) is predicted for hydrological droughts using streamflow data from Canadian prairies. By suitably amalgamating E(L T ) with mean and variance of I in the extreme number theorem based relationship, the E(M T ) is evaluated. Using Markov chain (MC), the E(L T ) is estimated involving the geometric pdf of L. The Markov chains up to order one (MC-1) were found to be adequate in the proposed model for the annual to weekly time scales. For a given level of drought probability (q) and a sampling period T-time units; the evaluation of E(M T ) requires only 3 parameters viz. lag-1 autocorrelation (ρ 1 ), first order conditional probability (q q , present instant being a drought given past instant was a drought) in SHI sequences and a parameter ø (value 0 to 1), which were estimated from historical data of streamflows. A major strength of the proposed model lies in the use of simple and widely familiar normal and geometric pdfs as its basic building blocks for the estimation of drought magnitudes.

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References

  1. Ankuz DE, Bayazit M, Onoz B (2012) Markov chain models for hydrological drought characteristics. J Hydrometeorol 13(1):298–309

  2. Bayazit M, Bulu A (1988) Complex Markov models to simulate persistent streamflows. J Hydrol 103:199–207

  3. Bayazit M, Onoz B (2005) Probabilities and return periods of multisite droughts. Hydrol Sci J 50(4):605–615

  4. Bonaccorso B, Cancelliere A, Rossi G (2003) An analytical formulation of return period of drought severity. Stoch Environ Res Risk Assess 17(1):157–174

  5. Burn DH, Wychreschul J, Bonin DV (2004) An integrated approach to the estimation of streamflow drought quantiles. Hydrol Sci J 49(6):1011–1024

  6. Cancelliere A, Salas JD (2010) Drought probabilities and return period for annual streamflows series. J Hydrol 391:77–89

  7. Chung C, Salas J (2000) Drought occurrence probabilities and risks of dependent hydrologic processes. J Hydrol Eng 5(3):259–268

  8. Environment Canada (2009) HYDAT CD-ROM Version 96–1.04 and HYDAT CD-ROM User’s Manual. Surface Water and Sediment Data Water Survey of Canada

  9. Gonzalez J, Valdes J (2003) Bivariate drought occurrence analysis using tree ring reconstructions. J Hydrol Eng 8(5):247–258

  10. Guven O (1983) A simplified semi-empirical approach to probabilities of extreme hydrologic droughts. Water Resour Res 19(2):441–453

  11. Haan CT (1977) Statistical hydrology. Iowa state University Press, Ames

  12. Kotz S, Neumann J (1963) On the distribution of precipitation amounts for periods of increasing length. J Geophys Res 68(2):3635–3640

  13. Lloyd EH (1970) Return periods in the presence of persistence. J Hydrol 10:291–298

  14. Mathier L, Perreault L, Bobee B (1992) The use of geometric and gamma-related distributions for frequency analysis of water deficit. Stoch Hydrol Hydraul 6:239–254

  15. McKee TB, Doeson, NJ, Kiest J (1993) The relationship of drought frequency and duration to time scales. Preprints, 8th conference on applied Climatology, Anheim, California: 179–184

  16. McKee TB, Doeson, NJ, Kiest J (1995) Drought monitoring with multiple time scales. Preprints, 9th conference on applied Climatology, Dallas, Texas: 233–236

  17. Millan J, Yevjevich V (1971) Probabilities of observed droughts. Hydrology Paper 50, Colorado State University, Fort Collins

  18. Mishra AK, Singh VP, Desai VR (2009) Drought characterization: a probabilistic approach. Stoch Environ Res Risk Assess 23:41–55

  19. Nalbantis I (2008) Evaluation of a hydrological drought index. Eur Water 23–24:67–77

  20. Nalbantis I, Tsakiris G (2009) Assessment of hydrological drought revisited. Water Resour Manage 23:881–897

  21. Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models: part 1-a discussion of principles. J Hydrol 10(3):282–290

  22. Onoz B, Bayazit M (2002) Trough under threshold modeling of minimum flows in perennial streams. J Hydrol 258:187–197

  23. Paulo AA, Pereira LS (2007) Prediction of SPI drought class transitions using Markov chains. Water Resour Manag 21:1813–1827. doi:10.1007/s11269-006-9129-9

  24. Salas J, Fu C, Cancelliere A, Dutin D, Bode D, Pineda A, Vincent E (2005) Characterizing the severity and risk of droughts of the Poudre River, Colorado. J Water Resour Plan Manag 131(5):383–393

  25. Sen Z (1977) Run sums of annual flow series. J Hydrol 35:311–325

  26. Sen Z (1980) Statistical analysis of hydrological critical droughts. J Hydraul Eng 106(HY1):99–115

  27. Sharma TC (1997) Estimation of drought severity on independent and dependent hydrologic series. Water Resour Manag 11:35–49

  28. Sharma TC, Panu US (2008) Drought analysis of monthly hydrological sequences: a case study of Canadian rivers. Hydrol Sci J 53(3):503–518

  29. Sharma TC, Panu US (2010) Analytical procedures for weekly hydrological droughts: a case of Canadian rivers. Hydrol Sci J 55(1):79–92

  30. Sharma TC, Panu US (2012) Modeling drought durations using Markov chains: a case study of streamflow droughts in Canadian prairies. Hydrol Sci J 57(4):1–18

  31. Sharma TC, Panu US (2013a) A parsimonious model of drought magnitudes: a case of Canadian streamflow droughts. Water Resour Manag. doi:10.1007/s11269-012-0207-x

  32. Sharma TC, Panu US (2013b) A semi-empirical method for predicting hydrological drought magnitudes in Canadian prairies. Hydrol Sci J 58(3):549–569. doi:10.1080/02626667.2013.772688

  33. Shiau JT, Shen WS (2001) Recurrence analysis of hydrological droughts of different severity. J Water Resour Plan Manag 127(1):30–40

  34. Shiau JT (2006) Fitting drought duration and severity with two dimensional copulas. Water Resour Manag 20:795–815

  35. Tallaksen LM, Madsen S, Clausen B (1997) On the definition and modeling of streamflow drought duration and deficit volume. Hydrol Sci J 42(1):15–33

  36. Todorovic P, Woolhiser DA (1975) A stochastic model of n day precipitation. J Appl Meteorol 14:125–137

  37. Yevjevich V (1967) An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Paper 23, Colorado State University, Fort Collins

  38. Zelenhasic E, Salvai A (1987) A method of streamflow drought analysis. Water Resour Res 23(1):156–158

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Acknowledgments

A partial financial support of the Natural Sciences and Engineering Research Council of Canada for this project is gratefully acknowledged.

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Correspondence to U. S. Panu.

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Sharma, T.C., Panu, U.S. A Simplified Model for Predicting Drought Magnitudes: a Case of Streamflow Droughts in Canadian Prairies. Water Resour Manage 28, 1597–1611 (2014). https://doi.org/10.1007/s11269-014-0568-4

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Keywords

  • Drought magnitude
  • Drought intensity
  • Geometric distribution
  • Markov chain
  • Standardized hydrological index
  • Truncated normal distribution