Handout (2.0 MB)

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 = γ_{0}

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 = γ_{0}

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, γ_{0}, 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.