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 minima based on such a process. To accomplish this, we derive a stochastic model and use it to simulate 10 ×10⁶ 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, as is the case for many moving-window drought indices. The resulting distribution of annual minima follows 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.

Hydrologic models are instrumental in understanding the behavior of the Nile River Basin (NRB), yet their effectiveness is often limited by the basin’s complex hydrology and sparse observational records. This study applies a basin-scale hydrological modeling approach to simulate near-natural, pre-reservoir flow conditions in the NRB, while incorporating lake and wetland submodels. The basin was discretized into 34 sub-watersheds with an outlet at Aswan. The conceptual GR4J rainfall–runoff model was implemented within the Raven modeling framework, chosen for its parsimony and suitability for data-limited conditions. Multi-objective calibration used discharge data from the Global Runoff Data Centre (GRDC), supplemented by digitized historical records to improve spatial and temporal coverage. A stepwise calibration strategy was applied at 18 sites, focusing on pre-reservoir periods to capture natural flow dynamics. The calibrated model reproduces observed discharges with high skill. At the Aswan outlet, Nash–Sutcliffe Efficiency (NSE) values were 0.87 (calibration) and 0.80 (validation), with percent bias (PBIAS) values of 6.1% and 5.0%, respectively. Model performance was strongest in the Blue Nile, White Nile headwaters, and the Nile main stem. The model also successfully simulated the hydrological step-change observed in Lake Victoria during the 1960s, underscoring its robustness in simulating regional hydroclimate disruptions. This calibrated model enables reconstruction of historical Nile discharge and simulation of past hydrologic disturbances, including those driven by major volcanic eruptions over the past millennia.

The transboundary Nile River Basin (NRB) occupies a tenth of the African continent and supports the daily livelihood of approximately 300 million people in 11 riparian countries. The NRB is hydrologically complex: two major watersheds, the Blue and White Nile, contribute about 85% of the total annual discharge; more than 50 % of the White Nile’s water is lost over the Sudd wetland; and the Blue and White Nile watersheds produce dramatically different seasonal hydrologic responses due to differences in hydroclimate and lake/wetland storage. The Inter-Tropical Convergence Zone (ITCZ) drives anomalies in temperature and precipitation; however, this atmospheric driver likely produces distinct hydrologic responses that depend on the spatial center over the Blue or the White Nile headwaters. Quantifying this effect requires a well-calibrated hydrologic model of the entire watershed under near-natural conditions, including hydrologic routing through major lakes and the Sudd wetland. This study aims to calibrate such a hydrologic model and recreate the hydrologic response of the major watersheds in the NRB using recovered records that extend to 1901, thereby greatly increasing the period used for model calibration and approximating near-natural responses prior to construction of several modern reservoirs. We employed GR4J, a parsimonious rainfall-runoff hydrologic model, because of its flexibility and minimal data requirements to match the NRB’s limited data availability, particularly during the early 1900s. Climate drivers, including precipitation and daily minimum/maximum temperatures were based on the Global Soil Wetness Project Phase 3 (GSWP3). Discharge data for model calibration were acquired from the Global Runoff Data Centre (GRDC) and through digitization of long discharge records from Hurst (1958). The NRB was discretized into 36 hydrological response units (HRUs), and calibrated using stepwise, multi-objective approach at 16 gauge locations between 1901 and 1964. In addition to avoiding the effects of several modern reservoirs, this early calibration period also avoided the most severe effects of climate change, as supported by the lack of discernible trends using the Mann-Kendall test. Our results show a successful calibration of the GR4J hydrological model to reasonably reproduce discharge at multiple locations across the NRB, with Nash-Sutcliffe Efficiencies of 0.83 and 0.64 at the outlets of the Blue and the White Nile, respectively. Additionally, the calibrated model accurately captured an abrupt change of Lake Victoria levels during the 1960s, affirming its reliability in simulating regional climate disruptions and lake dynamics. The model can therefore be used to study the sensitivity of major watersheds in the NRB and serves as a benchmark for understanding anthropogenic-induced departures from the natural hydrological behavior of the Nile River.Keywords: Nile River Basin, GR4J, Calibration, Hydrology

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.

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.

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.