Intensity-duration-frequency (IDF) curves are widely used for peak discharge estimation in designing hydraulic structures. The traditional Gumbel probability method entails selecting annual maximum rainfall depths (intensities) conditioned on a fixed time window width (which in general will not coincide with the rainfall event duration) from a continuous record to perform a frequency analysis in terms of the marginal distribution. The digitized database contains annual maximum intensities for selected discrete durations. This method presents problems when intensities are required for arbitrary durations which are not part of the selected durations. Accurate interpolated and especially extrapolated intensity values are hard to obtain. The present study offers two methods both involving a joint probability approach to overcome the deficiencies inherent in the traditional method of IDF analysis. The first joint probability approach employs Box-Cox and modulus transformations to transform original data to near bivariate normality. The second method does not require such a transformation. Instead, it uses the closed-form bivariate Burr III cumulative distribution to fit the data. Another advantage of the joint probability approach is that it allows one to gauge the rarity of certain extreme events, such as probable maximum precipitation, in terms of the joint occurrence of its extremely high intensity and a sufficiently long duration (e.g. 24 hours). The joint probability approach is applied to three data sets. The resulting conditional probability intensity estimates are quite close to those obtained by traditional Gumbel IDF analysis. In addition, reliable interpolated and extrapolated intensities are available because the approach essentially fits a flexible surface to the discrete data with the capability of providing a complete probabilistic structure. / Master of Science

Societies in much of the Horn of Africa are affected by variability in two distinct rainy seasons: the March-May (MAM) “long” rains and the October-December (OND) “short” rains. The region is the driest area of the tropics, while its societies are heavily dependent on the rainfall cycle. Especially worrying are anomalously dry conditions, which, together with other factors, contribute to food insecurity in the region. The recent 2020-2023 5-season drought, associated with the concurrent “triple-dip” La Niña and resulting in tens of millions of people facing “high levels of food insecurity” (cf: IGAD), renewed fears of long-term and possibly anthropogenically-forced drying trends, especially during the MAM long rains. A long-term decline in the long rains beginning in the early 1980s and lasting until the 2010s had indeed been noted in studies examining historical station-based observations, satellite observations, and farmer recollections in the region, though seasonal average rainfall has since partially recovered. Consequently, global climate models (GCMs) are increasingly used to project changes in rainfall characteristics under global warming scenarios and associated impacts on societies, such as agricultural production, groundwater resources, and urban infrastructure, in addition to providing seasonal forecasts used for near-term decision-making. However, GCMs uniformly predict long-term wetting in both seasons despite observed drying trends in the long rains, an “East African Paradox” that complicates the ability of decisionmakers to plan for future rainfall conditions. Previous generations of GCMs have known biases in key dynamics of the regional hydroclimate. Decisionmakers relying on projections of future rainfall in the GHA therefore need to know whether current GCM projections are trustworthy. In other words, can we be confident in future modeled wetting trends in both the long and short rains? This thesis pursues this question in three parts. Chapter 2 seeks to understand the fundamental dynamics affecting the East African seasonal rainfall climatology, which is unique for its latitude in both its aridity and for the dynamical differences between its two rainy seasons. I explain these characteristics through the climatology of moist static stability, estimated as the difference between surface moist static energy h? and midtropospheric saturation moist static energy h*. In areas and at times when this difference, h? − h*, is higher, rainfall is more frequent and more intense. However, even during the rainy seasons, h? − h* < 0 on average and the atmosphere remains largely stable, in line with the region’s aridity. The seasonal cycle of h? − h*, to which the unique seasonal cycles of surface humidity, surface temperature, and midtropospheric temperature all contribute, helps explain the double-peaked nature of the regional hydroclimate. Despite tropospheric temperature being relatively uniform in the tropics, even small changes in h* can have substantial impacts on instability; for example, during the short rains, the annual minimum in regional h* lowers the threshold for convection and allows for instability despite surface humidity anomalies being relatively weak. This h? − h* framework can help identify the drivers of interannual variability in East African rainfall or diagnose the origin of biases in climate model simulations of the regional climate. Chapter 3 applies these results to conduct a process-based model evaluation of the ability of GCMs from the 6th phase of the Coupled Model Intercomparison Project (CMIP6, the latest GCM generation) to simulate the historical climatology and variability in the East African long and short rains. I find that key biases from the 5th phase of the Coupled Model Intercomparison Project (CMIP5) remain or are worsened, including long rains that are too short and weak and short rains that are too long and strong. Model biases are driven by a complex set of related oceanic and atmospheric factors, including simulations of the Walker Circulation. h? − h* is too high in models, requiring more instability for the same amount of rainfall than in observations. Biased wet short rains in models are connected with Indian Ocean zonal sea surface temperature (SST) gradients that are too warm in the west and convection that is too deep. Models connect equatorial African winds with the strength of the short rains, though in observations a robust connection is primarily found in the long rains. Model mean state biases in the timing of the western Indian Ocean SST seasonal cycle are associated with certain rainfall timing biases, though both biases may be due to a common source. Simulations driven by historical SSTs (so-called ‘AMIP’ runs) often have larger biases than fully coupled runs. However, models generally respond to teleconnections with the Indian Ocean Dipole and the El Niño Southern Oscillation in particular as expected, maintaining the possibility that trends in the long and short rains may also respond correctly to simulated trends in large-scale dynamics. Finally, Chapter 4 applies these results to directly tackle the East African Paradox by analyzing model trends across the entire observational record to identify under what conditions they fail to reproduce observed trends. Since even with perfect models and observational records model output may differ from observations due to internal variability, I analyze the full spread of CMIP6 output, including Large Ensembles and totalling 598 runs from 47 models. I find that while observed trends are always within the model spread if all runs from all Large Ensembles are considered, the Paradox remains in CMIP6 models, since GCMs substantially underproduce strong drying trends compared to observations. Within the observational record, the Paradox is limited to the time period with the most anomalous drying trends (especially in the years 1980-2010); the recent recovery in rainfall falls comfortably within the range of GCM simulations. The Paradox is not visible in AMIP runs forced with observed historical SSTs, suggesting that biases in simulations of SSTs may be part of the explanation, though clear causality remains elusive. The transition towards more biased trends from SST-forced to coupled runs can also be seen in output from hindcasts from seasonal forecast models, where trends calculated from short-lead-time projections (when the ocean state resembles observations) do not feature the Paradox, while lead times starting with 1.5 months do. More broadly, I show that climate model simulations of observed trends alone cannot be used to reject model predictions of increased (or decreased) precipitation under future forcings. Decision-makers relying on future projections of rainfall trends in East Africa will likely need to consider the possibility of further drying in addition to wetting trends from GCMs.