Predicting Drought Magnitudes: A Parsimonious Model for Canadian Hydrological Droughts

• Published in: Water resources management Vol. 27; no. 3; pp. 649 - 664
• Main Authors: Sharma, T. C., Panu, U. S.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2013; Springer; Springer Nature B.V
• Subjects: Approximation; Atmospheric Sciences; Civil Engineering; Drought; Droughts; Earth and Environmental Science; Earth Sciences; Earth, ocean, space; Engineering and environment geology. Geothermics; Environment; Exact sciences and technology; Expected values; Extreme values; Geotechnical Engineering & Applied Earth Sciences; Hydrogeology; Hydrologic modeling; Hydrology; Hydrology. Hydrogeology; Hydrology/Water Resources; Markov analysis; Markov chains; Mathematical models; Natural hazards: prediction, damages, etc; Normal distribution; Probability density functions; Random variables; Stream flow; Studies; Theorems; Time series; Water resources; Water shortages; Canada; Markov chain; Truncated normal distribution; Deficit-volume; Standardized hydrological index; Drought intensity; Drought magnitude; Markov chain analysis; models; probability; North America; autocorrelation; Q; drought; prediction; streamflow
• ISSN: 0920-4741; 1573-1650
• DOI: 10.1007/s11269-012-0207-x

A multiplicative relationship, drought magnitude ( M ) = drought intensity ( I ) × drought duration or length ( L ) is used as a basis for predicting the largest expected value of hydrological drought magnitude, E ( M T ) over a period of T -year (or month). The prediction of E ( M T ) is carried out in terms of the SHI (standardized hydrological index, tantamount to standard normal variate) sequences of the annual and monthly streamflow time series. The probability distribution function (pdf) of I (drought intensity) was assumed to follow a truncated normal. The drought length ( L c ) was taken as some characteristic duration of the drought period, which is expressible as a linear combination of the expected longest (extreme) duration, E ( L T ) and the mean duration, L m of droughts and is estimated involving a parameter ø (range 0 to 1). The drought magnitude (deficit-sum, M ) has been assumed to follow a gamma pdf, in view of the observed behavior of M . The model M = I × L has been invoked via two approximations, viz. Type-1 involves only mean of I and Type-2 involves both mean and variance of I through the theorem of extremes of random numbers of random variables. The E ( L T ) were obtained using the Markov chain (MC) model of an appropriate order, which turned out to be zero order Markov chain (MC-0) at the annual time scale. At the monthly time scale, the E ( L T ) was best represented by MC-0 for SHI sequences with low value of lag-1 autocorrelation ( ρ  < 0.3) and first order Markov chain (MC-1) for SHI sequences with ρ  > 0.3. At low cutoff levels ( q  ≤ 0.2), the trivial relationship E ( M T ) = E ( I ) × E ( L T ) i.e. without considerations of the extreme number theorem and the pdf of M yielded satisfactory results.