## 1. Introduction

[2] Projections of regional climate change forced by rising greenhouse gases concentrations made by many authors and in the Intergovernmental Panel on Climate Change Assessment Reports, including the chapter on Global Climate Projections in the Working Group I Fourth Report (AR4) by *Meehl et al.* [2007], have been largely based on the simulations of comprehensive global atmosphere/ocean climate models (GCMs). The theory behind such projections is still being developed, however (see *Collins* [2007] for an overview). It is usually assumed, although not always stated, that the ensemble of available GCM results, for a specific period of the 21st century and a specific forcing scenario, gives an indication of the range of possible changes of the real world for that case. While it is possible that the models may have a systematic bias, it is often assumed that this is small. The best estimate of the forced change is then essentially the average or multimodel mean, as presented by *Meehl et al.* [2007] for the ensemble of models submitted to the CMIP3 database (see http://www-pcmdi.llnl.gov/ipcc/about_ipcc.php). Variations in this estimate among studies can relate to whether some climate models are given more weight than others based on their assessed reliability [e.g., *Giorgi and Mearns*, 2003], and how that assessment is made and applied.

[3] With regard to simplifying the presentation of changes for several time slices or periods and for various forcing scenarios, it has long been recognized [e.g., *Santer et al.*, 1990] that for a particular model, the spatial pattern of change for important variables like surface air temperature (*T*) and precipitation (*P*) is often similar. Indeed, at many places changes “standardized” by dividing or scaling by the global mean warming at the time are rather constant, aside from the effects of unforced or internal variability. The “pattern scaling” approach can thus be used to provide quite useful estimates of regional or local changes, based on an assessment of a single case, as shown by *Mitchell* [2003] for simulations with rising *CO*_{2}.

[4] The spatial pattern of forced change can vary in some regions due to changing aerosol distributions [e.g., *Harvey*, 2004]. *Watterson* [2003] and *Harris et al.* [2006] consider further differences in patterns, between those from transient climate simulations and those from simulations of warming at near-equilibrium, which are associated with oceanic heat uptake and heat transport changes. Nevertheless, *Meehl et al.* [2007] show that linear pattern scaling is generally useful for temperature and also precipitation, with respect to the CMIP3 multimodel means for the Special Report on Emissions Scenarios forcing scenarios, particularly later in the 21st century. Note that the similarity of the global patterns relative to their spatial variation is a little less for precipitation (see Table S10.2 of the supplementary material available online at http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-chapter10-supp-material.pdf) than for temperature.

[5] A further advantage of pattern scaling is that the problem of projecting the global mean warming can be separated from that of estimating the standardized regional change. *Dessai et al.* [2005], *Luo et al.* [2005], and others used simpler models to estimate global warming over a wider range of possibilities than is available from full GCMs. Assuming that these warming values can be combined arbitrarily with any scaled change, the product of any choice of the two factors provides a possible regional net (or total) change. The Commonwealth Scientific and Industrial Research Organisation (CSIRO) Climate Impact Group has used this approach for a number of years in providing estimates of the range of change over Australia, as reviewed by *Whetton et al.* [2005]. They noted, however, a limitation of this linear method for a nonnegative variable like rainfall, when larger warmings are combined with strong scaled decreases. *Mitchell* [2003] also discussed related nonlinearities.

[6] Given the spread of model results for many cases, it is clear that there is considerable uncertainty in such projections, even for a particular forcing scenario. Some studies have attempted to quantify this uncertainty and to express it in probabilistic terms [e.g., *Giorgi and Mearns*, 2003; *Harris et al.*, 2006], such as in the form of “probability density functions” (PDFs). In the pattern scaling approach, the uncertainty in the net regional change can be related to both uncertainty in the global mean warming (or sensitivity of the climate system) and uncertainty in the local response factor. *Dessai et al.* [2005] and others have estimated PDFs for the net change using the Monte Carlo method of random sampling, from the ranges of both warming and scaled change. The basic rationale for this new article is to present a simple mathematical framework by which the scaling approach can be more precisely applied and the resulting PDFs accurately calculated (conditional, of course, on the assumptions made).

[7] In many of the above studies a key assumption is still that the range of uncertainty in the real-world value is directly given by the range of model results (allowing for weighting, perhaps). Typically, the probability distribution that is derived would apply to the case of randomly choosing one of the models. Hence this assumption is effectively considering the real world to be a single sample from the models, even if these are from a limited “ensemble of opportunity” such as CMIP3. Naturally, such sample-dependent PDFs can only be approximations to the PDF that might represent the full “space of all possible models” [*Collins*, 2007]. (In that context, “PDF” will be used loosely here.) The uncertainty in the real-world warming so obtained would not depend directly on the number of models in the ensemble. Recently, other researchers have used Bayesian methods based on the assumption that it is the model results that are a sample from a PDF and, in the case of *Tebaldi et al.* [2005] and *Furrer et al.* [2007] at least, it is assumed that this is unbiased: The mean of the PDF is the unknown real-world change. Typically in statistical inference under such assumptions, the uncertainty in the estimate of this value tends to diminish as the number of independent samples increases. It is unclear to what extent GCMs can be regarded as independent (especially when important components are shared), but in any case the PDFs from these Bayesian analyses do tend to be relatively narrow.

[8] Whether the range of the PDF for a real-world change should diminish as more GCMs are assessed is a key issue, for both global mean change and regional change, but it is only just being explored in the literature [e.g., *Lopez et al.*, 2006]. In the case of the approach that combines scaled local change with global mean change, it is conceivable that some reduction in range should apply to both factors. This issue will not be pursued here, except through an example of scaled change with a deliberately narrowed PDF. Neither will we specifically include additional unquantifiable uncertainty, as described by *Jones* [2000], such as due to processes that are poorly represented in GCMs or from unknown biases. While the intention is to represent the real-world change here, in practice, the factor PDFs used in the examples are smoothed distributions of GCM results.

[9] The following section presents the mathematical framework that attempts to advance the approach of *Dessai et al.* [2005] for producing PDFs. The linear scaling assumption is presented, and the probabilistic formulation described. Two related methods for calculation of the PDF for net change are given.

[10] In section 3 the approach is illustrated by its application to the case of summer temperature change in southern Australia, as simulated by the CMIP3 models. These include the new CSIRO model Mark 3.5, recently submitted to CMIP3. Plausible weights of the models are used, calculated as described in Appendix A. Several representations of global warming under the A1B scenario are developed, including one using the beta distribution, which allows skewness. Net change PDFs are calculated. The results for this regional case can be compared with those from *Dessai et al.* [2005], as well as with PDFs generated from the “raw” model data.

[11] In section 4 the focus is on winter precipitation change in central Australia, which is considered using both absolute and relative changes from the GCMs. Problems with the linear assumption emerge, with the larger decreases exceeding the base climate rainfall, which is clearly unphysical. An advance made here is the use of a exponential assumption, which avoids this problem. Section 5 summarizes the methods used and makes several recommendations.

[12] It should be noted that the original motivation for the present study was to develop a method that could be readily applied to the provision of PDFs for change in a large range of quantities, for all seasons, several periods and scenarios, and at all locations in Australia. Some aspects of the application were extensions of previous CSIRO practice, as in the recent analysis by *Suppiah et al.* [2007]. The dependence of the results on some choices is discussed here. Further results and their context can be seen in the publication “Climate Change in Australia” [*Commonwealth Scientific and Industrial Research Organisation and Bureau of Meteorology*, 2007] (hereinafter referred to as CSIRO 2007). Please see also the acknowledgments.