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 _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.