Last year, a paper of mine (Lewis 2014) showing that the approach used in Frame et al (2005), which argued for using a uniform prior for estimating equilibrium (strictly, effective) climate sensitivity (ECS), in fact led to a unique, objective Bayesian estimate for ECS upon undertaking a simple transformation (change) of variables. The estimate was lower, and far better constrained at the upper end, than the one resulting from use of a uniform prior in ECS, as recommended in Frame et al (2005) when estimating ECS. The only uniform priors involved were those for estimating posterior probability density functions (PDFs) for observational variables with Gaussian (normally distributed) data uncertainties, where they are totally noninformative and their use is uncontroversial. I wrote an article about Lewis (2014) at the time, and a version of the paper is available here.

I’ve now had a new paper that uses an essentially identical method to Lewis (2014), but with updated, higher quality data, published by Climate Dynamics, here. A copy of the accepted version is available on my web page, here.

Like many climate sensitivity studies, the method involves comparing observationally-based and model-simulated temperature data at many differing settings of model parameters, a simple global energy balance model (EBM) with a diffusive ocean being used. But, unusually, surface temperature observational data were not used directly. Like Lewis (2014), my new paper uses observationally-constrained estimates of global mean warming attributable purely to greenhouse gases, separated using detection and attribution methods from temperature changes with other causes, treating them as “observable” data. Effective heat capacity, the ratio of ocean etc. heat uptake to the change in global mean surface temperature (GMST), is used as a second observable. It is estimated using the AR5 planetary heat uptake estimates spanning 1958–2011 and HadCRUT4v2 GMST data.

Detection and attribution studies involve coupled 3D global climate model (GCM) simulation runs with different categories of forcing. They use multiple-regression techniques to estimate what scaling factors to apply to the GCM-simulated spatiotemporal temperature response patterns for the various categories of forcing in order to best match their sum with observational data. The scaling factors (being the regression coefficients) adjust for the GCM(s) under- or over-estimating the responses to the various categories of forcing and/or the forcing strengths. So the estimates of GHG-attributable warming they produce, used as input data in my study, are fully constrained by gridded observational temperature records. This approach is potentially better able to isolate aerosol forcing, the biggest cause of uncertainty when estimating ECS from warming over the instrumental period, than methods using low dimensional models. I used estimates from the same multimodel detection and attribution studies that underlay the main anthropogenic attribution statements in the IPCC fifth assessment Working Group 1 report (AR5), Gillett et al (2013) (open access) and Jones et al (2013), based on their longest analysis periods (respectively 1861-2010 and 1901-2010).

The abstract from my paper reads as follows:

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.

The posterior probability density functions (PDFs) for the two ECS estimates are shown in Figure 1. The exact match of best estimates and uncertainty bounds using the alternative frequentist profile likelihood method confirms that the objective Bayesian method used provides frequentist probability-matching.

Figure 1. The box plots indicate boundaries, to the nearest grid value, for the percentiles 5–95 (vertical bar at ends), 17-83 (box-ends), and 50 (vertical bar in box), and allow for off-graph probability lying between S = 5 K and S = 20 K. Solid line box plots reflect the percentile points of the CDF corresponding to the plotted PDF. Dashed line box plots give confidence intervals derived using the SRLR profile likelihood method (the vertical bar in the box showing the likelihood profile peak).

The revised best (median) estimate for ECS in Lewis (2014) using the objective Bayesian approach, after correcting data handling errors, was 2.2°C. It seems likely that estimate was biased high by the use of temperature data spanning just the 20th century, which started with two anomalously cool decades.

The new study’s best estimate for ECS is almost identical to that of 1.64°C obtained in Lewis and Curry (2014). That study used a simple single-equation energy budget model to compare, between periods spanning 1859–2011, the rise in GMST with forcing and heat uptake estimates given in AR5. As it relied on the expert assessment of aerosol forcing given in AR5, which spans a very wide range, the ECS estimate upper uncertainty bound was higher, at 4.05°C, than in my new study.

The ECS estimate in my new study is also very similar to that in Lewis (2013). That study compared the evolution of surface temperatures in four latitude zones with simulations spanning 1860– 2001 by the MIT 2D global climate model (GCM). Many simulations were performed with differing parameter settings and hence varying model values of equilibrium/effective climate sensitivity (ECS), ocean effective vertical diffusivity (K v ) and aerosol forcing – which can be tightly constrained when zonal rather than GMST data is used. The parameter combination that best fitted the observational data gave a median estimate for ECS of 1.64°C. With non-aerosol forcing etc. uncertainties adequately allowed for, the 5–95% uncertainty range was 1.0–3.0°C.

Figure 2 shows posterior PDFs for the two TCR estimates from my new study. The best estimates are within 0.05°C of each other. Their average is 1.37°C, with a 5–95% range of 0.65–2.2°C. This is within a few percent of the best estimates for TCR in Lewis and Curry (2014), and of those given in Otto et al (2013), of which I was a co-author alongside fourteen AR5 lead authors.

FIG. 2: Estimated marginal PDFs for transient climate response derived, upon integrating out K v , using the transformation of variables method. The box plots indicate boundaries, to the nearest grid value, for the percentiles 5–95 (vertical bar at ends), 17-83 (box-ends), and 50 (vertical bar in box). They reflect the percentile points of the CDF corresponding to the plotted PDF.



