Bayesian hierarchical modelling of rainfall extremes

Understanding weather and climate extremes is important for assessing, and adapting to, the potential impacts of climate change. The design of hydraulic structures such as dams, drainage and sewers, for instance, relies in part on accurate information regarding patterns of extreme rainfall occurring at various locations and at different durations. Deriving this kind of information is challenging from a statistical viewpoint because a lot of information must be extracted from very little data. In this paper, we describe the use of a spatial Bayesian hierarchical model (BHM) for characterising rainfall extremes over a region of interest, using historical records of precipitation data from a network of rainfall stations. The rainfall extremes are assumed to have a generalised extreme value (GEV) distribution, with the shape, scale and location parameters representing the underlying variables of the BHM’s process layer. These parameters are modelled as a linear regression over spatial covariates (latitude and longitude) with additive spatially-correlated random process. This spatial process leads to more precise estimates of rainfall extremes at gauged locations, and also allows the inference of parameters at ungauged locations. Furthermore, it also mitigates the limitations imposed by short rainfall records in that it allows the model to “borrow strength” from neighbouring sites, thereby reducing the uncertainty at both gauged and ungauged locations. Making use of r-largest order statistics in the data layer further allows the integration of multiple yearly rainfall amounts instead of the annual maximum only. The proposed BHM uses a parametric representation that links the GEV scale parameter obtained for different accumulation durations. This approach leads to two additional process parameters, and allows the use of pluviometer data accumulated over a range of durations, thereby also increasing the amount of data available for inference. A main advantage of the Bayesian approach is that measures of variability arise naturally from the framework. These uncertainty measures represent information of crucial importance for a subsequent use of the estimated quantities. We demonstrate this Bayesian approach using a dataset of pluviometer measurements recorded at 252 meteorological stations located on the Central Coast of New South Wales, Australia. For each station, the rainfall data is accumulated over 12 different durations ranging from 5 minutes to 72 hours, from which the two largest annual maxima are selected. Exploratory analyses of this rainfall dataset are carried out for various purposes, including: (i) basic quality control (removal of erroneous data values), and (ii) to provide insight into the relevance of the model structure and associated assumptions. The proposed model is fitted using Markov chain Monte Carlo (MCMC) simulation, with several types of diagnostics plots used to assess the convergence properties of the resulting chains. We present numerical examples of estimated parameters resulting from the fitted model (regression coefficients, sill and range of the spatial correlation function) together with confidence intervals. Further results from this study are provided by calculating intensity–duration–frequency (IDF) curves for a few sites of interest (both gauged and ungauged) with associated estimates of uncertainty. These results are shown to be in good agreement with station-based maximum likelihood estimates, while achieving smoother curves with tighter uncertainty bands.