Calculation of probability density functions for temperature and precipitation change under global warming

[1] There remains uncertainty in the projected climate change over the 21st century, in part because of the range of responses to rising greenhouse gas concentrations in current global climate models (GCMs). The representation of potential changes in the form of a probability density function (PDF) is increasingly sought for applications. This article presents a method of estimating PDFs for projections based on the "pattern scaling'' technique, which separates the uncertainty in the global mean warming from that in the standardized regional change. A mathematical framework for the problem is developed, which includes a joint probability distribution for the product of these two factors. Several simple approaches are considered for representing the factors by PDFs using GCM results, allowing model weighting. The four-parameter beta distribution is found to provide a smooth PDF that can match the mean and range of GCM results, allowing skewness when appropriate. A beta representation of the range in global warming consistent with the Intergovernmental Panel on Climate Change Fourth Assessment Report is presented. The method is applied to changes in Australian temperature and precipitation, under the A1B scenario of concentrations, using results from 23 GCMs in the CMIP3 database. Statistical results, including percentiles and threshold exceedences, are compared for the case of southern Australian temperature change in summer. For the precipitation example, central Australian winter rainfall, the usual linear scaling assumption produces a net change PDF that extends to unphysically large decreases. This is avoided by assuming an exponential relationship between percentage decreases in rainfall and warming.