Implications of recent multimodel attribution studies for climate sensitivity

Abstract

Equilibrium climate sensitivity (ECS) is inferred from estimates of instrumental-period warming attributable solely to greenhouse gases (AW), as derived in two recent multi-model detection and attribution (D&A) studies that apply optimal fingerprint methods with high spatial resolution to 3D global climate model simulations. This approach minimises the key uncertainty regarding aerosol forcing without relying on low-dimensional models. The “observed” AW distributions from the D&A studies together with an observationally-based estimate of effective planetary heat capacity (EHC) are applied as observational constraints in (AW, EHC) space. By varying two key parameters—ECS and effective ocean diffusivity—in an energy balance model forced solely by greenhouse gases, an invertible map from the bivariate model parameter space to (AW, EHC) space is generated. Inversion of the constrained (AW, EHC) space through a transformation of variables allows unique recovery of the observationally-constrained joint distribution for the two model parameters, from which the marginal distribution of ECS can readily be derived. The method is extended to provide estimated distributions for transient climate response (TCR). The AW distributions from the two D&A studies produce almost identical results. Combining the two sets of results provides best estimates (5–95 % ranges) of 1.66 (0.7–3.2) K for ECS and 1.37 (0.65–2.2) K for TCR, in line with those from several recent studies based on observed warming from all causes but with tighter uncertainty ranges than for some of those studies. Almost identical results are obtained from application of an alternative profile likelihood statistical methodology.

Appendix: Further details of method used to estimate ECS

As stated in Sect. 2, the method closely follows that in Lewis (2014). One observable used, T _A, represents the estimated GMST change over the full analysis period attributable to greenhouse gases, based on the linear trend in GMST from the simulated response to GHG-only forcing. In the case of the observationally-constrained estimate of T _A, derived from GCM simulations, the simulated linear trend has been scaled by the GHG regression coefficient estimated in the attribution analysis. The EBM-simulation trend estimate of T _A is unscaled. The other observable, effective heat capacity C _H, is taken to be the same for GHG-only forcing as it is for historical warming (Frame et al. 2006). Model accuracy is assumed, with there being a ‘true’ setting $(S^{\text{t}} ,K_{\text{v}}^{\text{t}} )$ of model parameters that simulates ‘true’ (error-free) values $(T_{\text{A}}^{\text{t}} ,C_{\text{H}}^{\text{t}} )$. Estimated (posterior) probability density functions (PDFs) for the true values of the observables (those that would have been observed in the absence of error and internal variability), $p_{{T_{\text{A}}^{\text{t}} }} (T_{\text{A}}^{{}} )$ and $p_{{C_{\text{H}}^{\text{t}} }} (C_{\text{H}}^{{}} )$, are derived respectively from the uncertainty ranges given in the relevant attribution analysis and from observational data error distributions, as described in Sect. 2. Since C _H is a measure of heat capacity, it should be independent of the change in GMST and hence of T _A. The joint density for T _A and C _H, $p_{{T_{\text{A}}^{\text{t}} ,C_{\text{H}}^{\text{t}} }} (T_{\text{A}}^{{}} ,C_{\text{H}}^{{}} )$, is accordingly obtained by multiplying their individual densities. As the EBM is a deterministic rather than a statistical model, the dispersion of the estimated PDFs for S must entirely reflect uncertainties in the observationally-constrained estimates of T _A and C _H.

The EBM used has a diffusive ocean below a 75-m-deep mixed layer (Andrews and Allen 2008, Eq. 8). EBM simulations are run using all parameter value combinations lying on a grid that is uniformly spaced in terms of S and K _v and sufficiently large for there to be negligible probability of the true values of S or K _v lying outside it. Four hundred values of S from 0.05 to 20 K, and sixty values of K _v from 0 to 2.95 cm s^−0.5, were used. For notational convenience, here K _v represents the square root of effective ocean vertical diffusivity, which controls C _H in an approximately linear manner (Sokolov et al. 2003). The annual EBM-simulation time series are used to compute $T_{\text{A}}^{m}$ and $C_{\text{H}}^{m}$. $T_{\text{A}}^{m}$ is calculated as the linear-regression-based change in modelled global surface temperature over the period used in the relevant attribution analysis (that is, the product of the regression trend per annum and the length of the period in years). $C_{\text{H}}^{m}$ is calculated as the ratio of changes in modelled ocean heat content and global temperature between the means of 11-year periods centred on 1963 and 2006, thus using data spanning 1958–2011, the period covered by the observational data used.

The observables and parameters are both bivariate and, with the EBM being deterministic, the model $T_{\text{A}}^{m}$ and $C_{\text{H}}^{m}$ values are smooth differentiable functions of the parameters. Moreover, each pair of ($T_{\text{A}}^{m}$, $C_{\text{H}}^{m}$) values corresponds to a unique pair of $(S^{\text{m}} ,K_{\text{v}}^{\text{m}} )$ values and vice versa. Accordingly, there is an invertible both-ways differentiable one-to-one relationship between joint model parameter settings, $(S^{\text{m}} ,K_{\text{v}}^{\text{m}} )$, and joint values of ($T_{\text{A}}^{m}$, $C_{\text{H}}^{m}$). Given the assumption of model accuracy the same relationship exists between the true joint values $(T_{\text{A}}^{\text{t}} ,C_{\text{H}}^{\text{t}} )$ and $(S^{\text{t}} ,K_{\text{v}}^{\text{t}} )$. The estimated joint posterior PDF for $(S^{\text{t}} ,K_{\text{v}}^{\text{t}} )$ is therefore directly and uniquely related to that for $(T_{\text{A}}^{\text{t}} ,C_{\text{H}}^{\text{t}} )$ through the standard formula (Mardia et al. 1979) for converting PDFs upon a transformation of variables:

$$p_{{S^{\text{t}} ,K_{\text{v}}^{\text{t}} }} (S,K_{\text{v}} ) = p_{{T_{\text{A}}^{\text{t}} ,C_{\text{H}}^{\text{t}} }} (f(S,K_{\text{v}} )) \, J_{f}$$

(1)

where f is the functional relationship between $(S^{\text{m}} ,K_{\text{v}}^{\text{m}} )$ and $(T_{\text{A}}^{\text{m}} ,C_{\text{H}}^{\text{m}} )$ and $J_{f}$ is the absolute Jacobian determinant, given by:

$$J_{f} = \left. {{\text{absolute value of }}\left| {\left( {\begin{array}{*{20}c} {\frac{{\partial T_{\text{A}}^{\text{m}} }}{{\partial S^{\text{m}} }}} & {\frac{{\partial T_{\text{A}}^{\text{m}} }}{{\partial K_{\text{v}}^{\text{m}} }}} \\ {\frac{{\partial C_{\text{H}}^{\text{m}} }}{{\partial S^{\text{m}} }}} & {\frac{{\partial C_{\text{H}}^{\text{m}} }}{{\partial K_{\text{v}}^{\text{m}} }}} \\ \end{array} } \right)} \right|} \right|_{{S^{\text{m}} = S,K_{\text{v}}^{\text{m}} = K_{\text{v}} }}$$

(2)

The EBM simulations provide the values of $(T_{\text{A}}^{\text{m}} ,C_{\text{H}}^{\text{m}} )$ at each $(S^{\text{t}} ,K_{\text{v}}^{\text{t}} )$ combination, and hence the form of f and, by numerical differentiation, of $J_{f}$. Once the joint posterior PDF for $(S^{\text{t}} ,K_{\text{v}}^{\text{t}} )$ has been calculated, a marginal PDF for $S^{\text{t}}$ is obtained by integrating out K _v, following standard Bayesian methodology. Uncertainty ranges reflect the percentile points of the corresponding cumulative probability distribution function (CDF), and are termed credible intervals.

The frequentist validity, in terms of repeated sampling properties, of the credible intervals provided by the main objective Bayesian method is checked by also deriving confidence intervals for S using a profile-likelihood approach (Allen et al. 2009). Profile likelihood is an alternative objective parameter inference method giving confidence intervals with at least approximate frequentist validity (Pawitan 2001). Profile likelihoods for AW and EHC are derived, and multiplicatively combined to form a joint (AW, EHC) likelihood. That joint likelihood is then restated in $(S,K_{v} )$ coordinates (likelihoods, unlike PDFs, being unaffected by a change in variables), and the profile likelihood for S computed. Confidence intervals are then calculated using the signed root log-likelihood ratio (SRLR) method.