Predicting Drought Magnitudes: A Parsimonious Model for Canadian Hydrological Droughts
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 <Subscript> T </Subscript>) over a period of T-year (or month). The prediction of E(M <Subscript> T </Subscript>) 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 <Subscript> c </Subscript>) 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 <Subscript> T </Subscript>) and the mean duration, L <Subscript> m </Subscript> 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 <Subscript> T </Subscript>) 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 <Subscript> T </Subscript>) 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 <Subscript> T </Subscript>)=E(I) × E(L <Subscript> T </Subscript>) i.e. without considerations of the extreme number theorem and the pdf of M yielded satisfactory results. Copyright Springer Science+Business Media Dordrecht 2013
