Theoretical Annual Exceedances from Moving Average Drought Indices

Abstract. Numerous drought indices originate from the Standardized Precipitation Index (SPI) and use a moving average structure to quantify drought severity by measuring normalized anomalies in hydroclimate variables. This study examines the theoretical probability of annual exceedances from such a process. To accomplish this, we derive a stochastic model and use it to simulate 10 million years of daily or monthly SPI values in order to determine the distribution of annual exceedance probabilities. We believe this is the first explicit quantification of annual extreme exceedances from a moving average process where the moving average window is proportionally large (5–200 %) relative to the year. The resulting distribution of annual minima follow a Generalized Normal distribution, rather than the Generalized Extreme Value (GEV) distribution, as would be expected from extreme value theory. From a more applied perspective, this study provides the expected annual return periods for the SPI or related drought indices with common accumulation periods (moving window length), ranging from 1 to 24 months. We show that the annual return period differs depending on both the accumulation period and the temporal resolution (daily or monthly). The likelihood of exceeding an SPI threshold in a given year decreases as the accumulation period increases. This study provides clarification and a caution for the use of annual return period terminology (e.g. the 100 year drought) with the SPI and a further caution for comparing annual exceedances across indices with different accumulation periods or resolutions. The study also distinguishes between theoretical values, as calculated here, and real-world exceedance probabilities, where there may be climatological autocorrelation beyond that created by the moving average.