A Bayesian Hierarchical Model for Statistical Downscaling of Climate Extremes Using Monthly Output from CMIP5 Models

Over the last two decades, extreme weather and climate events in the context of climate change have drawn more and more attentions due to their potential contributions to disasters and significant impacts on environment and human society. Great efforts have been made on the understanding and prediction of climate extremes under climate change. The fifth phase of the Coupled Model Intercomparison Project (CMIP5) has implemented a variety of climate change experiments by means of coupled global climate models (GCMs). Typically, GCMs are run at coarse spatial resolution and are unable to resolve important sub-grid scale features such as clouds and topography. As a result GCM output cannot be used straightforwardly for local impact studies. Statistical downscaling is a widely used method for obtaining high-resolution climate or climate change information from GCMs, based on statistical relationships between observed small-scale (often station level) variables and larger (GCM) scale variables. Future values of the large-scale variables obtained from GCM simulations of future climate change scenarios are then used to drive the statistical relationships to estimate the smaller-scale details of future climate. For climate extremes, particularly the annual extremes however, there is not yet a straightforward method to downscale. Local climate extreme information has to be extracted from downscaled time series of climate variables at daily or sub-daily time scales, which requires heavy computational burden. In this work, we present a Bayesian hierarchical approach to downscale annual climate extremes using time series of observed annual extremes and monthly GCM output only, without a day-by-day downscaling of the whole series of GCM output at daily time scale.

Local annual maxima are assumed as random variables following the generalized extreme value (GEV) distribution with the cumulative distribution function (CDF) having the form

GEV(y; μ, σ, ξ) = exp{-[1+ξ(y-μ)/σ]^-1/ξ} (1)

where y is the annual maximum variable; μ, σ and ξ are location, scale and shape parameters, respectively; -∞ < μ, ξ <∞, σ > 0, 1+ξ(y-μ)/ξ > 0. By replacing y with -y and μ with -μ, the GEV distribution (1) can also be applied to annual minima.

Obviously, there is no one-to-one correspondence between observed and GCM-simulated annual extremes at the daily time scale. However, the monthly GCM output can be thought of as the large-scale background under which the annual climate extremes might be observed. To downscale climate extremes, the location and the log transformed scale parameters of the GEV distribution are assumed to regress to the GCM counterpart of y at monthly scale, while the shape parameter is assumed constant, forming the following random effect regression model

μ[i] = α[m[i], 1]+α[m[i], 2]*x[i]

log(σ[i]) = β[m[i], 1]+β[m[i], 2]*x[i] (2)

ξ = γ₀

where m[i] (1 ≤ m ≤ 12) indicates the month in which the ith annual extreme is observed; x[i] (i = 1, …, N) is the monthly GCM value for that month. When the model is used to estimate the small-scale climate extremes under the future large-scale climate scenarios, any month of the year in which the annual climate extreme would be observed should be considered, so that m should follow a categorical distribution p_j (j = 1, …, 12) subject to Σp_j = 1, resulting in the following model averaging

μ[i] = Σp_j(α[j, 1]+α[j, 2]*x[i, j])

log(σ[i]) = Σp_j(β[j, 1]+β[j, 2]*x[i, j]) (3)

ξ = γ₀

where x[i, j] is the monthly GCM value for the jth month of the ith year in the future.

The model parameter set consists of (α, β, γ₀, p), of which α and β are multivariate and p is categorical. By further assuming priors for these parameters, Equations (1-3) constitute a Bayesian hierarchical model, which can be inferred using Markov Chain Monte Carlo (MCMC) algorithm. The JAGS software is used to implement the model, and is called in the R software environment for the analysis of MCMC samples.

This approach is applied as a demonstration to downscale the annual maximum and minimum surface air temperatures and maximum daily precipitation from BNU-ESM (Beijing Normal University Earth System Model for CMIP5) simulations for Historical, RCP2.6, RCP4.5 and RCP8.5 scenarios at 489 stations in China. Samples of climate extremes under historical and future scenarios are drawn from their posterior predictive distributions. Return levels with different return periods are estimated accordingly, and their linear trends are analyzed for comparison.