a. Model bias and model convergence

Recently, Giorgi and Mearns's reliability ensemble average (REA) method (Giorgi and Mearns 2002) quantified the two criteria of bias and convergence for multimodel evaluation, and produced estimates of regional climate change and model reliabilities through a weighted average of the individual AOGCM results. The REA weights contain a measure of model bias with respect to current climate and a measure of model convergence, defined as the deviation of the individual projection of change with respect to the central tendency of the ensemble. Thus, models with small bias and whose projections agree with the consensus receive larger weights. Models with lesser skill in reproducing the observed conditions and appearing as outliers with respect to the majority of the ensemble members receive less weight. In a subsequent note Nychka and Tebaldi (2003) show that the REA method is in fact a conventional statistical estimate for the center of a distribution that departs from a Gaussian shape by having heavier tails. Thus, although the REA estimates were proposed by Giorgi and Mearns as a reasonable quantification of heuristic criteria, there exists a formal statistical model that can justify them as an optimal procedure. The research in this paper is motivated by this statistical insight. Here we start from the same data used by Giorgi and Mearns (i.e., 30-yr regional climate averages representative of current and future conditions) and propose statistical models that recover the REA framework but are flexible in the definition of bias and convergence and are easily generalizable to richer data structures. The main results of our analysis consist of probability distributions of temperature change at regional scales that reflect a relative weighing of the different AOGCMs according to the two criteria.

b. A Bayesian approach to climate change uncertainty

There is interest in the geosciences in moving from single-value predictions to probabilistic forecasts, in so far as presenting a probability distribution of future climate is a more flexible quantification for drawing inferences and facilitating decision making (Reilly et al. 2001; Webster 2003; Dessai and Hulme 2003). Bayesian methods are not the only option when the goal is a probabilistic representation of uncertainty, but are a natural way to do so in the context of climate change projections.

The choice of prior distributions is crucial, when adopting a Bayesian approach, representing the step in the analysis more open to subjective evaluations. Within the climate community, different viewpoints have been expressed regarding the use of expert judgement in quantifying uncertainty. For example, Wigley and Raper (2001) explicitly chose prior distributions to be consistent with their own expert judgement of the uncertainty of key parameters such as climate sensitivity, and Reilly et al. (2001) argued that such assessments should be part of the Intergovernmental Panel on Climate Change (IPCC) process for quantifying uncertainty in climate projections. Opposing this view, Allen et al. (2001) argued that “no method of assigning probabilities to a 100-year climate forecast is sufficiently widely accepted and documented in the refereed literature to pass the extensive IPCC review process.” (They suggested that there might be better prospects of success with a 50-yr forecast, since over this time frame there is better agreement among models with respect to both key physical parameters and the sensitivity to different emissions scenarios.) Following this philosophy, Forest et al. (2002) stated “an objective means of quantifying uncertainty . . . is clearly desirable” and argued that this was achievable by choosing parameters “that produce simulations consistent with 20th-century climate change.” However, their analysis did not use multimodel ensembles. We view the present approach as an extension of the philosophy reflected in the last two quotations, where we adopt uninformative prior distributions but use both model-generated and observational data to calculate meaningful posterior distributions. Another point in favor of uninformative prior distributions is that they typically lead to parameter estimates similar to non-Bayesian approaches such as maximum likelihood. However, Bayesian methods are more flexible when combining different sources of uncertainty, such as those derived from present-day and future climate model runs, and we regard this as a practical justification for adopting a Bayesian approach.

c. Outline

Section 2 contains the description of the basic statistical model and its extensions. Prior distributions for the parameters of interest, and distributional assumptions for the data, conditional on these parameters, are described. We then present analytical approximations to the posterior distribution in order to gain insight into the nature of the statistical model and its results. In section 3 the model is applied to the same suite of AOGCM experiments as in Giorgi and Mearns (2002), and some findings from the posterior distributions are presented. In section 4 we conclude with a discussion of our model assumptions and what we consider promising directions for extending this work. The Markov chain Monte Carlo (MCMC) method used to estimate the posterior distributions is described in the appendix.

a. The model for a single region

In this section we present the analytical form of our basic statistical model. We list first the distributional assumptions for the data (likelihood), then the priors for all the model parameters. We then present conditional approximations to the posterior that will help to interpret our results. Throughout, let X_i and Y_i denote the temperature simulated by AOGCM i, seasonally and regionally averaged and aggregated into two 30-yr means representative of current and future climate, respectively.

1) Likelihoods

We assume Gaussian distributions for X_i, Y_i:

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where the notation N(μ, λ⁻¹) indicates a Gaussian distribution with mean μ and variance 1/λ. Here μ and ν represent the true values of present and future temperature in a specific region and season. A key parameter of interest will be ΔT ≡ ν − μ, representing the expected temperature change. The parameter λ_i, reciprocal of the variance, is referred to as the precision of the distribution of X_i. To allow for the possibility that Y_i has different precision from X_i, we parameterize its distribution by the product θλ_i where θ is an additional parameter, common to all AOGCMs. A more general model might allow for θ to be different in different models, but that is not possible in the present setup given the limited number of data points.

The assumptions underlying (1) and (2) are that the AOGCM responses have a symmetric distribution, whose center is the “true value” of temperature, but with an individual variability, to be regarded as a measure of how well each AOGCM approximates the climate response to the given set of natural and anthropogenic forcings. The assumption of a symmetric distribution around the true value of temperature for the suite of multimodel responses has been implicitly supported by the Coupled Model Intercomparison Project (CMIP) studies (Meehl et al. 2000) where better validation properties have been demonstrated for the mean of a superensemble rather than the individual members. The additional assumption in our model that the single AOGCM's realizations are centered around the true value could be easily modified in the presence of additional data. For example, if single-model ensembles were available, then an AOGCM-specific random effect could be incorporated.

We model the likelihood of the observations of current climate as

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Here μ is the same as in (1), but λ₀ is of a different nature from λ₁, . . . , λ₉. While the latter are measures of model-specific precision, and depend on the numerical approximations, parameterizations, grid resolutions of each AOGCM, λ₀ is a function of the natural variability specific to the season, region and time average applied to the observations. In our model we fix the value of the parameter λ₀, using estimates of regional natural variability from Giorgi and Mearns (2002). The parameter λ₀ could be treated as a random variable as well if our data contained a long record of observations that we could use for its estimation.

3) Posterior distributions

We apply Bayes's theorem to the likelihood and priors specified above. The resulting joint posterior density for the parameters μ, ν, θ, λ₁, . . . , λ₉ is given by, up to a normalizing constant,

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The distribution in Eq. (4) is not a member of any known parametric family and we cannot draw inference from its analytical form. The same is true for the marginal posterior distributions of the individual parameters, that we cannot compute from (4) through closed-form integrals. Therefore, MCMC simulation is used to generate a large number of sample values from (4) for all parameters, and approximate all the summaries of interest from sample statistics. We implement the MCMC simulation through a Gibbs sampler, and details of the algorithm are given in the appendix.

Here we gain some insight as to how the posterior distribution synthesizes the data and the prior assumptions by fixing some groups of parameters and considering the conditional posterior for the others.

For example, the distribution of μ fixing all other parameters is a Gaussian distribution with mean

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and variance

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In a similar fashion, the conditional distribution of ν is Gaussian with mean

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and variance

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The forms (5) and (7) are analogous to the REA results, being weighted means of the nine AOGCMs and the observation, with weights λ₁, . . . , λ₉, λ₀, respectively. As in the case of the REA method, these weights will depend on the data, but a fundamental difference is that in our analysis they are random quantities, and thus we account for the uncertainty in their estimation. Such uncertainty will inflate the width of the posterior distributions of ν, μ, and thus also ΔT.

An approximation to the mean of the posterior distribution of the λ_is, for i = 1, . . . , 9, is

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The form of (9) shows that the individual weight λ_i is large provided both |X_i − μ̃| and |Y_i − ν̃| are small. These two quantities correspond to the bias and convergence criteria, respectively. Here |Y_i − ν̃| measures the distance of the ith model future response from the overall average response, and so has characteristics similar to the convergence measure in Giorgi and Mearns (2002). The important difference from REA for this model is that the distance is based on the future projection (Y_i) rather than the temperature change (Y_i − X_i). As for the bias term, notice that in the limit, if we let λ₀ → ∞ (we model the observation X₀ as an extremely precise estimate of the true temperature μ), μ̃ → X₀, and the bias term becomes in the limit |X_i − X₀|, the same definition of bias as in the REA analysis. The form of (9) also reveals how by chosing a = b = 0.001 we ensure that the contribution of the prior assumption to (9) is negligible.

b. Introducing correlation between present and future climate responses within each AOGCM and robustifying the model

The model presented in section 2a reproduces the basic features of the REA method. In this section we modify our model in two important directions. First, we relax the assumption of independence between X_i and Y_i, for a given climate model. We do so by linking X_i and Y_i through a linear regression equation; this is equivalent to assuming that (X_i, Y_i) are jointly normal, given the model parameters, with an unknown correlation coefficient that is left free to vary, a priori, between −1 and +1.

Thus, adopting a slightly different notation than in section 2a, we express the statistical assumptions for X_i and Y_i as follows. Let

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where η_i ∼ N(0, λ⁻¹_i), making (10) equivalent to (1). Then, given X_i, μ, and ν, let

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where ξ_i ∼ N(0, λ⁻¹_i).

In this model we use the priors listed in section 2a for the parameters μ, ν, λ₁, . . . , λ₉, θ, and we choose a uniform prior on the real line for β_x.

In Eq. (11) β_x introduces a direct (if positive) or inverse (if negative) relation between X_i − μ and Y_i − ν. By using a common parameter we force the sign of the correlation to be the same for all AOGCMs, a constraint that is required because only two data points (current and future temperature response) per model are available for a given region and season. The value of β_x has also implications in terms of the correlation between Y_i − X_i, the signal of temperature change produced by the ith AOGCM, and the quantity X_i − μ, the model bias for current temperature. A value of β_x equal to 1 translates into conditional independence of these two quantities, while values greater than or less than 1 would imply positive or negative correlation between them. (In the REA analysis, Giorgi and Mearns treat these two quantities as independent, after computing empirical estimates of their correlations and finding that they are for the most part low in absolute value and not statistically significant.) But the most important feature of the model with correlation has to do with the form of the posterior distribution for the λ_i parameters. By an approximation similar to what we use in section 2a for the basic model, we derive that the posterior mean has now the approximate form:

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If β_x ≈ 1 the form in (12) uses a convergence term now similar to the REA method's, measuring the distance between ΔT_i ≡ Y_i − X_i and the consensus estimate of temperature change ν̃ − μ̃. We will compare the results from the basic model and this modified version, in order to assess how differently the two definitions of convergence reflect on the final posterior distributions of regional temperature changes.

A second concern is the sensitivity of our model to outlying data points. This is a consequence of hypothesizing normal distributions for X_i and Y_i, given the model parameters. Traditionally, robustness is achieved by choosing distributions with heavier tails than the Gaussian, and Student's t distributions with few degrees of freedom are often a convenient choice because of flexibility (the degrees of freedom parameter allowing to vary the degree of their departure from the Gaussian) and computational tractability. Thus, we model η_i and ξ_i in Eqs. (10) and (11) above as independently distributed according to a Student's t distribution, with ϕ degrees of freedom. We perform separate analyses for ϕ varying in the set {1, 2, 4, 8, 16, 32, 64}, the lower end of the range being associated with heavier tail distributions, the higher being equivalent to a Gaussian model. We are not a priori labeling some of the AOGCMs as outliers, but we examine the difference in the posterior distributions that ensues as a function of the statistical model assumed for the error terms. Varying the degrees of freedom can accommodate extreme projections to a larger or smaller degree and is equivalent to the convergence criterion being less or more stringent, respectively. This achieves at least qualitatively the same effect as the exponents in the REA's weights, by which the relative importance of the bias and convergence terms could be modified.

a. Data

We analyze temperature responses under the A2 SRES emission scenario (Nakicenovic et al. 2000), just a portion of the dataset analyzed by the REA method in Giorgi and Mearns (2002). Output from nine different AOGCMs (see Table 1 and references therein) consists of present and future average surface temperatures, aggregated over the 22 regions shown in Fig. 1, two seasons [December–January–February (DJF) and June–July–August (JJA)] and two 30-yr periods (1961–90 and 2071–2100). Thus, we will separately treat (X_i, Y_i), i = 1, . . . , 9, for each region and season. Observed temperature means for the 22 regions and the 1961–90 period (X₀) were also taken from Giorgi and Mearns (2002), together with estimates of regional natural variability, interpretable as the standard deviation of the three-decadal average that X₀ represents. We use these estimates (listed in Table 2) to determine the values of the parameter λ₀. (The dataset can be downloaded from www.cgd.ucar.edu/~nychka/REA.)

b. Results for a group of representative regions

We present posterior distributions derived from the basic model (section 2a) and the model introducing correlation between X_i and Y_i (section 2b), both models assuming Gaussian distributions for the likelihoods. We choose to focus on a representative group of six regions [Alaska (ALA), east North America (ENA), southern South America (SSA), northern Europe (NEU), East Africa (EAF), northern Australia (NAU)], for both the winter (DJF) and summer (JJA) seasons.

a. Overview of temperature change distributions

Figures 9 and 10 look at differences in the estimates of temperature change and uncertainty among regions, in the form of a series of box plots, sorted by their median values. This representation is useful for assessing different magnitudes of warming, and different degrees of uncertainty (variability) across regions and between seasons. Notice that all distributions are limited to the positive range, making the case for global warming. Warming in winter is on average higher than in summer, for all regions, and the high latitudes of the Northern Hemisphere are the regions with a more pronounced winter climate change. These all are by now undisputed results from many different studies of climate change (Cubasch et al. 2001). The variability of the distributions is widely different among regions, supporting the notion that for some regions the signal of climate change is stronger and less uncertain than for others. Regions such as ENA, SSA, eastern Asia (EAS), the Mediterranean (MED) in winter and West Africa (WAF) and southern Australia (SAU) in summer show extremely tight distributions predicting the number of degrees of warming with relative certainty. On the other hand, regions like northern Asia (NAS) and central Asia (CAS) in winter, EAF and west North America (WNA) in summer, and the Amazons (AMZ), central North America (CNA), and NEU in both seasons show a wide range of uncertainty in the degrees of warming. As we already mentioned, the width and shape of these distributions may be taken as a signal of the degree of agreement among AOGCMs over the temperature projection in the regions. Problematic regions, characterized by multimodal or simply diffuse distributions may suggest areas that merit special attention by the climate modeling community.

b. Advantages of a Bayesian approach

Recent work (Raisanen and Palmer 2001; Giorgi and Mearns 2003) has addressed the need for probability forecasts, but offered only an assessment of the probability of exceeding thresholds, identified as the (weighted) fraction of the ensemble members by which the thresholds in question were exceeded. Moreover, in the case where each ensemble member is equally weighted, issues of model validation are ignored. A differential weighting of the members on the basis of the REA method is more appropriate but depends on a subjective formulation that is difficult to evaluate. In contrast, we think that the Bayesian approach is not only flexible but facilitates an open debate on the assumptions that generate probabilistic forecasts.

The results in this paper demonstrate how the quantitative information from a multimodel experiment can be organized in a coherent statistical framework based on a limited number of explicit assumptions. Our Bayesian analysis yields posterior probability distributions of all the uncertain quantities of interest. These posterior distributions for regional temperature change and for a suite of other parameters provide a wealth of information about AOGCM reliability and temporal (present to future) correlations. We deem this distributional representation more useful than a point estimate with error bars, because important features such as multimodality or long tails become evident. Also, the uncertainty quantified by the posterior distributions is an important product of this analysis and it is not easily recovered using non-Bayesian methods. We note here that the width of the posterior distributions is also a reflection of the limited number of data points that we used in estimating the parameters of the statistical model, particularly for the AOGCM-specific parameters λ₁, . . . , λ₉. Conversely, it is worth underlining that even if we start from extremely diffuse priors, we are able to estimate informative posterior distributions for all parameters.

c. Sensitivity analysis of the statistical assumptions

The posterior distributions of regional temperature change in many region–season combinations may differ both in variance and shape, depending on the statistical model adopted, that is, when introducing the correlation structure between present and future model projections. These sensitivities suggest the need to enrich the experimental setting, thus formulating the statistical model as free as possible from constraints. For example, imposing a common correlation parameter β_x among models is a strong assumption that could be relaxed if more information were available. Similarly, single-model ensembles would make it possible to estimate the internal variability of the AOGCM and allows for the separation of intramodel versus intermodel variability. In addition, climatological information could be incorporated into the prior. The range of our posteriors for the present and future temperature means would remain the same in the presence of proper, but still uninformative priors (i.e., either uniform distributions limited to a physically reasonable range, or Gaussian distributions concentrating most of their mass over such ranges). However, a modeling group may have confidence that results in a particular region are positively biased, or relatively more accurate than in other regions, on the basis of separate experiments. This kind of information could be applied to the formulation of a different prior for that AOGCM's precision parameter. In our opinion, analyzing the robustness of results to model assumptions is as valuable as analysis of the results themselves, highlighting the weak parts of the statistical formulation, and pointing to the need of closer communication between modelers, climatologists, and statisticians in order to circumscribe the range of sensible assumptions. In this regard the Bayesian model presented here may be viewed as a device to foster more effective collaborations.

A natural extension to the current region-specific model is a model that introduces AOGCM-specific correlation between regions and thus borrows strength from all the regional temperature signals in order to estimate model variability and biases. As we discussed in section 1, working with higher-resolution AOGCM output, both spatially and temporally, will require a much more complex effort. Richer information on AOGCM performance as linked to specific regional/seasonal climate reproductions will have to be incorporated in the likelihood of the AOGCM's responses and/or the priors on the precision parameters. In addition, other climate responses may be considered, jointly with temperature. Precipitation is of course the obvious candidate, and a bivariate distribution of temperature and precipitation is a natural extension.

d. About reliability criteria and statistical modeling

The criteria of bias and convergence established by Giorgi and Mearns (2002, 2003) and adopted in this work are commonly discussed in the climate change literature as relevant to evaluating climate change projections. However, one can question whether these two criteria are the most relevant for evaluating the reliability of projections of climate change. The ability of a model to reproduce the current climate is usually regarded as a necessary, but not sufficient condition for considering the model's response to future forcings as reliable (McAvaney et al. 2001). The convergence criterion has been advocated by some authors (Raisanen 1997; Giorgi and Francisco 2000) and can be theoretically derived by assuming that the observed AOGCMs represent a random sample from a superpopulation of AOGCMs. A pitfall in applying model convergence is that models may produce similar responses due to similarities in model structures, not because the models are converging on the “true” response. Furthermore, one may argue that extreme projections could be the result of a model incorporating essential feedback mechanisms that the majority of other models ignore. Yet, agreement among many models has been viewed as strengthening the likelihood of climate change (Cubasch et al. 2001; Giorgi et al. 2001a, b). As we noted in section 2b both the form of the REA weights (with the use of two different exponents for the bias and convergence terms) and the specification of the likelihood in our statistical model allow the final result to depend to a lesser degree on the convergence criterion. Besides changing the likelihood assumptions one could impose a prior distribution for the θ parameter that concentrates most of its mass on values less than 1. In this way, one posits a lower level of confidence in the precision of the future simulation than of the present, implicitly accepting a less stringent criterion of convergence for the future trajectories. The form of (9) shows that a value of θ less than 1 would down weight large deviations in the convergence term (Y_i − ν)², thus achieving a similar effect to the exponent in the REA weight.

From a more general perspective, however, our goal was to extend the REA method, not to reevaluate its criteria. It is expected that other criteria, and more complex combinations of criteria will evolve over time as this statistical work is refined. Much work is being devoted to more sophisticated and extensive methods of model performance evaluation and intermodel comparison (Hegerl et al. 2000; Meehl et al. 2000) and future analyses may modify or extend the two criteria of bias and convergence to account for them.

In conclusion, we view our results as an illustration of the power of bringing statistical modeling to experiments where quantifying uncertainty is an intrinsic concern. Moreover, we hope this work may foster more deliberate analysis that incorporates scientific knowledge of climate modeling at global and regional scales.