Probabilistic projections of transient climate change

Abstract

This paper describes a Bayesian methodology for prediction of multivariate probability distribution functions (PDFs) for transient regional climate change. The approach is based upon PDFs for the equilibrium response to doubled carbon dioxide, derived from a comprehensive sampling of uncertainties in modelling of surface and atmospheric processes, and constrained by multiannual mean observations of recent climate. These PDFs are sampled and scaled by global mean temperature predicted by a Simple Climate Model (SCM), in order to emulate corresponding transient responses. The sampled projections are then reweighted, based upon the likelihood that they correctly replicate observed historical changes in surface temperature, and combined to provide PDFs for 20 year averages of regional temperature and precipitation changes to the end of the twenty-first century, for the A1B emissions scenario. The PDFs also account for modelling uncertainties associated with aerosol forcing, ocean heat uptake and the terrestrial carbon cycle, sampled using SCM configurations calibrated to the response of perturbed physics ensembles generated using the Hadley Centre climate model HadCM3, and other international climate model simulations. Weighting the projections using observational metrics of recent mean climate is found to be as effective at constraining the future transient response as metrics based on historical trends. The spread in global temperature response due to modelling uncertainty in the carbon cycle feedbacks is determined to be about 65–80 % of the spread arising from uncertainties in modelling atmospheric, oceanic and aerosol processes of the climate system. Early twenty-first century aerosol forcing is found to be extremely unlikely to be less than −1.7 W m⁻². Our technique provides a rigorous and formal method of combining several lines of evidence used in the previous IPCC expert assessment of the Transient Climate Response. The 10th, 50th and 90th percentiles of our observationally constrained PDF for the Transient Climate Response are 1.6, 2.0 and 2.4 °C respectively, compared with the 10–90 % range of 1.0–3.0 °C assessed by the IPCC.

Appendix: Specification of covariance

This section describes how the covariance matrix Ψ _m,t appearing in Eq. 12 is estimated. Similarly to Eq. 10 in Sexton et al. (2012), Ψ _m,t can be expressed as

$$ \psi_{m,t} = \Upsigma_{m,t}^{em} (X_{j}^{a} ) + \Upsigma_{m,t}^{\varepsilon } + \Upsigma_{m,t}^{e} $$

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where the three terms represent contributions arising from (1) uncertainty in the emulated estimate derived from our climate model output, (2) uncertainty in the associated discrepancy, and (3) uncertainty in the observed values used to calculate the likelihood. The covariance matrix representing observational uncertainty can be partitioned into the covariance matrix $\Upsigma_{mm}^{e}$ representing uncertainty associated with the recent mean climate observables, and the covariance matrix $\Upsigma_{tt}^{e}$ representing observational uncertainty for our 4 large scale temperature indices:

$$ \Upsigma_{m,t}^{e} = \left( {\begin{array}{*{20}c} {\Upsigma_{mm}^{e} } \hfill & 0 \hfill \\ {0^{T} } \hfill & {\Upsigma_{tt}^{e} } \hfill \\ \end{array} } \right) $$

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The specification of $\Upsigma_{mm}^{e}$ is described in Sexton et al. (2012), while $\Upsigma_{tt}^{e}$ is obtained from the HadCRUT3 data set, which comes with its own estimates of a number of different sources of uncertainty. All of these sources of error, except the uncertainty that arises from lack of global coverage which does not contain any information on autocorrelation, are used to construct N_o = 1,000 spatio-temporal realisations of the observed data set. Sampled values o _ij (representing i = 1, …, 1,000 realisations of the j = 1, …, 4 observables o _t ) are used to estimate $\Upsigma_{tt}^{e} = D^{T} D/(N_{o} - 1)$, where $D_{ij} = (o_{ij} - \bar{o}_{j} )$.

Since our GCM experiments do not support construction of an emulator of transient climate change for any location in the parameter space x ^a (see Sect. 3.2), our methodology does not directly produce estimates for the components of the covariance matrix $\Upsigma_{m,t}^{em} (x^{a} )$ corresponding to emulated values μ _t of the time dependent historical observables o _t . We assume therefore that the covariance structure of emulation errors in the future equilibrium responses, scaled as previously described, will be a good estimate of the required error covariances for historical changes. In practice, $\Upsigma_{m,t}^{em} (x^{a} )$ is estimated by rescaling the correlation matrix of emulated 2 × CO₂ equilibrium responses r _t (x ^a) corresponding to o _t by the time scaling error variances σ _t associated with historical changes in the same variables. For example, if N _m  = 6 eigenvectors are used to represent the mean climate observables o _m , and N _t  = 4 values for r _t (x ^a) are emulated, this specifies a 10 × 10 covariance matrix Cov(μ _i (x ^a), μ _j (x ^a)) for the multivariate equilibrium emulation (Appendix B.2, Sexton et al. 2012). The covariance matrix is then decomposed into the correlation matrix $C_{ij}^{eqm} (x^{a} )$ and standard deviations σ ^eqm _i (x ^a), and rescaled so that the (i,j)th element of $\Upsigma_{m,t}^{em}$ is given by

$$ \Upsigma_{ij}^{em} (x^{a} ) = C_{ij}^{eqm} (x^{a} )\sigma_{i} \sigma_{j} ,\quad \sigma_{i} = \left\{ {\begin{array}{*{20}c} {\sigma_{i}^{eqm} (x^{a} )} \hfill & {i \le N_{m} } \hfill \\ {\sigma_{{t_{k} }} } \hfill & {i > N_{m} ,\;k = i - N_{m} } \hfill \\ \end{array} } \right. $$

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replacing values for σ^eqm by the time scaling standard deviations σ_t for those elements of Σ^em associated with historical climate change.

The discrepancy covariance matrix $\Upsigma_{m,t}^{\varepsilon }$ is estimated from the reconstruction $\mu_{t} (X_{(i)}^{{a^{*} }} )$ of values of o _t simulated by independent CMIP3 models, obtained by finding best analogue parameter sets $X_{(i)}^{{a^{*} }}$ as described in Sect. 3.3. Elements of $\Upsigma_{m,t}^{\varepsilon }$ are obtained by calculating covariances between errors in $\mu_{t} (X_{(i)}^{{a^{*} }} )$, and corresponding errors in emulated reconstructions of the values of recent mean climate (the o _m variables) simulated by the same set of CMIP3 models.