The basic underpinning of previous work is energy balance modelling of the climate system, from which it is anticipated that interannual variability may be informative regarding the equilibrium sensitivity. While previous research was based on analysis of the simplest possible zero-dimensional single-layer planetary energy balance, there is evidence that the behaviour of the climate system over the historical period is poorly modelled by such a system e.g.. Therefore, we use here a slightly more complex two-layer model based on  and . This model has been shown to reasonably replicate the transient behaviour of the CMIP5 ensemble of complex climate models . The model is defined by the two equations: 1CmdTmdt=Ft+λTm-ϵγ(Tm-Td)+Cmδt2CddTddt=γ(Tm-Td).

This is a two-layer globally averaged energy balance model which simulates the mixed (Tm) and deep (Td) ocean temperature anomalies in the presence of time-varying forcing Ft. λ=-F2×/S is the radiative feedback parameter; S is the equilibrium sensitivity, and F2× is the forcing due to a doubling of the atmospheric CO2 concentration. Cm and Cd are the heat capacities of the mixed-layer and deep ocean respectively, and γ represents the ocean heat transfer parameter. The parameter ϵ was introduced by  to represent the deep-ocean heat uptake efficacy, and while it is not important for our analysis, we include it for consistency with the broader literature. In a slight modification to , we add a noise term δt to the first equation to represent the internal variability in the system as was originally introduced in a single-layer energy balance climate model by . Here δt is sampled on an annual (i.e. time step) basis from a Gaussian N(0,σ) where we generally use the value σ=0.05, which generates deviations of order 0.05 ∘C on an annual basis, reasonably compatible with both general circulation model (GCM) results and observations of the climate system. Our conclusions are not sensitive to this choice. The mixed-layer temperature Tm is considered synonymous with the globally averaged surface temperature. The equations are solved via the simple Euler method with a 1-year time step.

The values of the various adjustable parameters are listed in Table . We assume depths of 75 and 1000 m for the mixed and deep ocean layers respectively, which are used to calculate the heat capacities Cm and Cd respectively based on ocean coverage of 70 % of the planetary surface area. The default values for adjustable parameters are given in Table , and the values used here lie close to the mean of those obtained by fitting the model to CMIP5 simulations by . Our aim here is not specifically to replicate or mimic this ensemble but to allow for a reasonable range of parameter values. If we set γ=0 and ignore the deep ocean, then we recover the single-layer model of  which was used by both  and  in their theoretical analyses.

Our investigations are performed within the paradigm of Bayesian estimation. In general, the Bayesian approach provides us with a way to estimate a set of unknown parameters Θ from a set of observations O via Bayes' theorem, 3P(Θ|O)=P(O|Θ)P(Θ)/P(O).

Here P(Θ|O) is the posterior probability distribution of Θ conditioned on a set of observations O; P(O|Θ) is the likelihood function that indicates the probability of obtaining observations O for any particular set of parameters Θ, which in this paper will always contain S and may include other parameters. P(Θ) is a prior distribution for the parameters Θ, and P(O) is the probability of the observations, which is required as a normalising constant in the calculation of the posterior probability distribution.

Formally, the value of the observations is fully summarised by the likelihood function P(O|Θ), but we primarily present our results as posterior probability density functions (pdf's) in order to provide an easily interpreted output which can be directly compared to previously published results. We therefore use a uniform prior in S as this is typically the implicit assumption in emergent constraint analyses . This choice results in the posterior being visually equivalent to the likelihood even though their interpretation is somewhat different. In some experiments, we will consider that only the sensitivity is unknown, but in others we will consider a wider range of parametric uncertainties. The priors that we use for all uncertain parameters are shown in Table .

While this study primarily focusses on the behaviour of the simple energy balance model, we also use and present some data from external sources. In order to perform simulations of the historical period, we force our climate model with annual time series for the major forcing factors based on Annex II: Climate System Scenario Tables,. Our two-layer model with a 1-year time step (and Euler method of numerical integration) reacts rather too strongly to short-term spikes in forcing, and thus we scale the volcanic forcing to 70 % of the nominal value in order to give more realistic simulations. We show some outputs of the model together with surface air temperature observations from HadCRUT  as a purely visual indication of the model's performance. We do not use these real temperature observations in any of our analyses, however.

For comparison with our simple model results, we also present some results calculated from historical simulations performed by climate models in the CMIP5  and CMIP6  ensembles. For CMIP5, we use results from 23 models obtained from the Climate Explorer website (https://climexp.knmi.nl/, last access: 20 May 2019). Where multiple simulations were performed with a single model, we show all results (amounting to 89 model runs in total), and these vary substantially due solely to the sample of internal variability in each simulation. Output from CMIP6 models was provided to the authors by Martin Stolpe. Due to the highly variable size of initial condition ensembles in this set of simulations, we limited use to at most five simulations from each model, resulting in a sample of 117 simulations from 31 models.

and  both summarised the variability in the temperature record with a scalar measure that the authors argued (based on simple energy balance modelling) should be informative regarding the sensitivity. summarised the variability via the characteristic decorrelation time constant τ=τ(Δt)=-Δt/ln⁡(ρΔt), where Δt is an adjustable lag time, and ρΔt is the autocorrelation coefficient of the temperature time series at a time lag of Δt. The method of selecting Δt – and therefore the estimation of τ – was not presented in an entirely objective algorithmic form but for the simple one-layer climate model that was considered; the expected value of τ calculated from a long unforced time series is independent of lag. argued that the function Ψ=σT/-ln⁡ρ1 should be linearly related to the equilibrium sensitivity. In this function, ρ1 is the lag-1 autocorrelation of the time series of annual mean surface temperatures, and σT is the magnitude of interannual variability in these temperatures. Ψ and τ are closely related and covary very similarly over a wide range of sensitivity when other model parameters are held fixed (not shown). Henceforth in this section we focus solely on Ψ as it is more precisely defined and has been recently discussed in some detail . However, very similar results are also obtained when equivalent experiments are performed using τ.

We now present some investigations into the relationship between Ψ and S in unforced simulations of the two-layer model introduced in Sect. . We perform a multifactorial experiment in which 1000-member ensembles of simulations are integrated for both 150- and 1000-year durations, over a range of S from 0 to 10 ∘C, and with γ either set to the default value of 0.7 or, alternatively, set to 0, in which case we recover the single-layer version of the model. All other model parameters are held fixed at standard values in these experiments. Since there is no forced trend in these experiments, we do not include any explicit detrending step in these analyses. However, the results are insensitive to detrending.

Figure  shows the results obtained when Ψ is calculated from the time series of annual mean surface temperatures produced by these simulations. For 150-year simulations using the single-layer model, there is a strong linear relationship between the mean value of Ψ obtained and the sensitivity of the model, just as  argued. However, did not consider sampling variability, that is to say, the precision with which this expected value of Ψ might be estimated from a finite time series. As our results show, there is substantial uncertainty in the value of Ψ obtained by individual runs, and there is also strong heteroscedasticity; that is to say, the variance in each ensemble of Ψ values increases markedly with sensitivity. This variation arises from the sequence of noise terms which generate the internal variability in each simulation of the model and is therefore an intrinsic aspect of the theoretical framework relating Ψ to S. For these unforced simulations, it seems quite possible for a model with its sensitivity set to a value of 5 ∘C or even higher to generate a time series which has a modest value for Ψ of say 0.1, even though the expected value of Ψ from such model simulations would be much higher. Similarly to the results shown by , an accurate diagnosis of Ψ could in principle be made with a sufficiently long time series of internal variability, but the sampling uncertainty only decreases with the square root of the duration of the time series (as expected from the central limit theorem), so this is unlikely to be of use in practical applications with real data.

When we consider the two-layer model using the standard parameter value of γ=0.7, the situation is a little different. In this case the relationship between sensitivity and Ψ is flatter and more curved, with the expected value of Ψ changing slowly for S>4 ∘C. The underlying explanation for this is quite simple. Any small perturbation to the surface temperature is damped on the annual timescale by a relaxation factor which varies in proportion to ϵγ-λ, and ϵγ is equal to 0.91 for standard parameter values. Therefore, when S is large, changes in λ=-3.7/S have relatively little impact on the total damping, and thus both the magnitude and autocorrelation of variability are relatively insensitive to further increases in S. also presented a theoretical analysis of this two-layer model in which it was argued that the response of Ψ was close to linear across the GCM parameter range, and our result confirms this for sensitivity values from around 2 to 4 or even 5 ∘C. However, the gradual curvature for larger values results in a saturation of the response of Ψ to increases in S, and this, together with the increasing sampling uncertainty, has consequences that will be shown in subsequent experiments. In fact the relationship between Ψ and the transient climate response (i.e. the warming observed at the time of CO2 doubling under a 1 % per annum increase) is more close to linear than the relationship between Ψ and S. Thus our work does not challenge the underlying analysis that they presented but augments it with additional details.

We now directly consider the question of how useful an observed value Ψo can be as a constraint on the equilibrium climate sensitivity through Bayesian estimation. It is not trivial to directly calculate the exact value of the likelihood P(Ψo|Θ) for a given observed value Ψo, as Ψ is itself a random variable arising from the stochastic model and thus depends on the sequence of random perturbations that were generated during the numerical integration of the model. Therefore, we use here the technique known as approximate Bayesian computation . This is a rejection-based sampling technique in which samples are drawn from the prior distribution, used to generate a simulated temperature time series, and rejected if the value of Ψ calculated from this time series does not lie within a small tolerance of the observed value. The set of accepted samples then approximately samples the desired posterior. We have no observations of long periods of unforced climate variability with the real climate system, so we perform a number of synthetic tests in which different hypothetical values for Ψo are tested.

Our experiments take the form of a perfect model scenario, where the model is assumed to be a perfect representation of the system under consideration, with no structural imperfections. Our uncertainties here are due solely to unknown parameter values and internal variability noise. In these experiments, we assume that Ψ for the true system is calculated from a 150-year temperature time series of the unforced system, without any observational error whatsoever. The results of four experiments – using values of Ψo which range from 0.05 to 0.2 in regular increments – are shown in Fig. . There is not necessarily an immediate correspondence between these synthetic values and the observationally derived value that  calculated, as we are using unforced model simulations here. Nevertheless, the results are qualitatively interesting. With other model parameters set to the default values, the four values of Ψo used here correspond to the expected value generated by 150-year integrations with sensitivities of approximately 1, 2.5, 5, and 10 ∘C respectively. The figure shows that in this experimental scenario, Ψ can only provide a tight constraint in the first case where the sensitivity is very low. In this case, the 5 %–95 % range of the posterior is an impressively narrow 0.6–1.7 ∘C. For the case Ψo=0.1, the equivalent probability interval is 1.8–8.1 ∘C, and for higher values of Ψ the posterior is very flat indeed with just the very lowest values of S excluded. Similar results are obtained when equivalent values of τ are used as observational constraints.

Strictly, when considering the strength of the constraint obtained from the variability, we should focus on the likelihood P(Ψ|S) rather than the posterior pdf P(S|Ψ), since the latter depends also on the prior, which is in principle independent of the observations. The likelihood for different values of S, which tells us the relative probability of any particular sensitivity value generating the observation, can be directly read off from Fig.  as the height of the appropriate density curve at the specific sensitivity value. In the experiment performed with Ψo=0.1, the maximum likelihood value is achieved at a value of S=2.5 ∘C, and the likelihood drops by a factor of 10 at both S=1.3 and S=7.9 ∘C. suggest that a likelihood ratio of 10 or more between two competing hypotheses could be taken to represent “strong” evidence in favour of one over the other, so if we adopt this linguistic calibration we could say that the observation of Ψo=0.1 represents strong evidence in favour of S=2.5 ∘C versus all values outside of the range 1.3–7.9 ∘C (but conversely, does not represent strong evidence to discriminate between any pair of values inside that range). It is somewhat coincidental that this range seems quite similar to the 5 %–95 % range of the posterior pdf as the philosophical interpretation of the ranges is rather different. There is a strong skew in this range, which extends much further towards higher values of S than lower ones, compared to the maximum likelihood estimate. We stress that this skew is a fundamental property of the physical model and is not due to the Bayesian analysis paradigm.

The pdf's plotted in the top panel of Fig.  assume that all model parameters other than S are known with certainty. In reality, we have significant uncertainty as to what values we should assign to several other parameters. We consider just three of these: the ocean heat uptake parameter γ, the efficacy or pattern effect parameter ϵ, and the internal noise parameter σ. fitted the two-layer model to various GCM outputs in order to estimate parameter values including γ and ϵ, and based on these results we use as priors for these parameters the distributions N(0.7, 0.2) and N(1.3,0.3) respectively, which generously encapsulate their results. did not consider internal variability, and thus we do not have such a solid basis for a prior in σ and assume a comparable relative uncertainty of 20 %, i.e. a prior of N(0.05, 0.01). When we repeat the previous experiments but include these additional parametric uncertainties, then for the experiment where we use Ψo=0.05 as a constraint, the posterior for S widens substantially from the previous spread of 0.6–1.7 ∘C, to 0.6–4.9 ∘C as also shown as the dashed blue line in Fig. . The largest factor generating this substantial increase in uncertainty is due to the uncertainty in σ. The equivalent posteriors using the larger observational values for Ψo also broaden somewhat, but this is less visible in the results as they are of course always constrained by the prior range.

Although the results in Fig.  show that an observation of Ψ taken from a short unforced simulation cannot tightly constrain equilibrium sensitivity in this model (except perhaps in the most exceptional of circumstances), it could still be hoped that a more precise constraint could possibly be gleaned by a more advanced analysis that uses some different diagnostic of the time series. In this section, we show how the total information of the time series can be used. By doing this, we create the most optimistic possible scenario for using internal variability to constrain equilibrium sensitivity of this simple climate model.

This approach requires us to calculate the likelihood for the full set of observations, P(O|Θ), where here O=Tmi, i=1…N, is the full time series of annual surface temperature anomalies. Once the model parameters and forcing are prescribed, the time series of surface temperature anomalies is uniquely determined by the series of random noise perturbations δi. Thus, in the absence of observational error, we can invert this calculation to calculate (up to machine precision) the sequence of annual random noise perturbations δi, i=1…N that are required in order to replicate any given observed temperature time series. This is why we selected a model time step as long as 1 year, as it results in the number of observations being as large the number of noise terms making this exact inversion possible. The probability of the model (with a particular set of parameters) generating the observed sequence is exactly the probability of the required noise sequence being sampled. This value is readily calculated, since the joint density of these independent and identically distributed δi is simply the product of their individual densities. This unusual approach, which we do not believe has been previously implemented in this context, is possible here as we are assuming zero observational uncertainty. With this exact likelihood calculation, the Bayesian estimation process is straightforward. In the case where only sensitivity is considered uncertain, it can be performed by direct numerical integration, sampling the sensitivity on a fine regular grid and calculating the likelihood (and therefore posterior) directly at these values.

It is worth emphasising that this calculation represents an absolute best-case scenario for using the time series of temperature anomalies as a constraint. There can be no diagnostic or statistical summary of the observations that provides more information than the full set of observations themselves contain. Thus, we cannot hope to obtain a better constraint by some alternative analysis of the temperature time series.

Figure  shows results obtained from this approach, in the case where only S is considered uncertain. To aid visual comparability with Fig. , the y-axis scale is fixed at the same value despite cutting the peaks of some pdf's. It is not possible to define what a “typical” noise sequence might look like, and therefore we plot 20 replications with different randomly generated instances of internal variability for each sensitivity value tested. It seems that the results may be a little more precise than was obtained using Ψ alone (as shown by the pdf's generally having higher peak densities), though this depends on the specific sample of internal variability that was obtained. It is still only in the case of the lowest sensitivity value of 1 ∘C that we reliably obtain a tight constraint. With the true sensitivity of 2.5 ∘C, the posterior 5 %–95 % range, averaged over the samples, is 1.9–7.0 ∘C, marginally narrower than the 1.8–8.1 ∘C range obtained previously when an equivalent Ψ value of 0.1 was used. When additional parametric uncertainties are considered in this unforced scenario, the constraints again weaken, though not to quite such an extent as in Sect.  when only Ψ is used as a constraint. We do not show these results here.

Thus, there appears to be the potential for internal variability, as represented by the full temperature time series, to provide a slightly better constraint than that obtained by a summary statistic alone, but the improvement is marginal, and even our optimal calculation, which uses the exact likelihood of the full time series, cannot accurately diagnose equilibrium sensitivity except when the true value is very low. These results again show a skew similar to that obtained when Ψ was used as the constraint in Sect.  and . Thus this non-Gaussian likelihood is again an inherent property of the physical model and not an artefact of the analysis. We mention again that these calculations are made under the three optimistic assumptions that (a) the model is perfect and we have exact knowledge of all other model parameters, (b) we know the forcing to be zero over this time period, and (c) there is no observational error.

In order to assess what we can learn about sensitivity from the variability in historical temperature observations, we first consider the utility of Ψ calculated from simulations over this period. proposed that the effect of forcing over this interval could be effectively removed by a process of windowed detrending. Figure  shows results analogous to the two-layer simulations in Fig.  but using forced simulations of the historical period rather than unforced control simulations and, therefore, with Ψ calculated via the windowed detrending method of . One very minor discrepancy with the calculations presented in that paper is that our simulations only extend to 2012 (this being the limit of the forcing time series that we are using), and therefore we omit the last 4 years of the time period that they used. This does not significantly affect any of our results. We do not consider one-layer simulations here as this version of the model is known to provide poor simulations of historical changes .

Grey dots and error bars indicate results obtained when only S is considered uncertain. These results are qualitatively similar to those obtained by the unforced two-layer simulations in Fig. , in that they exhibit a nonlinear and heteroscedastic relationship that levels off for large values of S. The values of Ψ obtained at each value of S are however somewhat larger in the forced experiments, which supports the claims of  and  that the windowed detrending has not been wholly effective in eliminating all influence of forcing. As mentioned previously, analysis of GCM outputs indicate significant uncertainty in other model parameters, and therefore we have performed additional ensembles of simulations which account for uncertainty both in model parameters and aerosol forcing. These uncertain parameters – and their priors – are the same as in Sect. , with the addition here that we also consider uncertainty in the aerosol forcing through the scaling parameter α. The combined effect of these uncertainties has little systematic effect on the mean estimate of Ψ but slightly increases the ensemble spread. Interestingly, in contrast to our earlier experiments, no single factor appears to have a dominant effect here.

Blue and red crosses also shown in Fig. 5 show results obtained from the CMIP5 and CMIP6 ensembles. The CMIP models appear to generate slightly lower values of Ψ than the two-layer model does with the same sensitivity, although the results seem broadly compatible. Reasons for the difference may include biases in parameters of the two-layer model, structural limitations, or differences in the forcings used. Although we are not replicating the emergent constraint approach here, we do note that there is a significant correlation in the CMIP5 ensemble results shown here between their sensitivities and Ψ values. However, the relationship seen here is weaker than that obtained by  for a different (though overlapping) set of models and explains a lower proportion of the variance. This remains true whether we perform the regression with S as predictor, as suggested by our figure, or when using Ψ as predictor as in . For CMIP6, the correlation is insignificant.

Figure  shows results generated when various hypothetical values for Ψo are used to constrain model parameters for historical simulations. As before, we test sensitivity values of 1, 2.5, and 5 ∘, which here correspond to values for Ψo of 0.1, 0.18, and 0.25 respectively. As in the previous experiments, the smallest value of Ψo generates a tight constraint with a 5 %–95 % range of 0.7–1.7 ∘C when only S is considered uncertain. This grows to 1.8–7.2 ∘C for Ψo=0.18, and the larger value of Ψo=0.25 provides very little constraint. When the additional parametric and forcing uncertainties are considered, the tightest range corresponding to Ψo=0.1 grows a little to 0.7–2.0 ∘C, and the Ψo=0.18 case spreads to 1.8–8.8 ∘C.

When we use the observational value of Ψo=0.13 (calculated from HadCRU data) and include multiple parametric uncertainties, the 17 %–83 % posterior range is 1.3–2.7 ∘C, and the 2.5 %–97.5 % range is 0.9–5.1 ∘C. These ranges are somewhat larger than the equivalent results presented by , which were 2.2–3.4 ∘C and 1.6–4.0 ∘C respectively, despite the highly optimistic perfect model scenario considered here.

Finally, we repeat the approach of Sect.  and use the full information of the time series, by the same method of inverting the model to diagnose the internal variability noise that is required to generate the observed temperature time series. Since we are interested solely in variability, we only consider the temperature residuals after detrending. We use a simple linear detrending over the period 1880–2012, which will leave a signature primarily due both to volcanic events and also the contrasting temporal evolution of negative aerosol forcing and positive greenhouse gas forcing, which both generally increase throughout this period but which exhibit different multidecadal patterns.

The calculation is similar to that of Sect. , but there are some minor details which are worth mentioning. Although the detrending is performed over the interval 1880–2012, we initialise the simulations in 1850 to allow for a spin-up. In contrast to Sect.  where detrending was not performed, knowledge of the residuals after detrending does not actually enable an exact reconstruction of the internal variability in the model simulation, as any random trend in this internal variability will have been removed by detrending. In fact a whole family of different model simulations will be aliased onto the same residuals. Therefore, our inversion only truly calculates the noise perturbations after the removal of the component that generates any linear trend. This is not a significant problem for the likelihood calculation as the effect of this aliasing is minor, and its dependence on model parameters is negligible.

The results of multiple replicates are shown in Fig. , in which we consider first the case where only S is uncertain and then our larger set of parametric uncertainties. In the case where only S is unknown, the full time series of detrended residuals provides strong evidence on S, which can as a result be tightly constrained except perhaps when it takes a high value such as 5 ∘C. For S=2.5 ∘C, the posterior 5 %–95 % range is typically under 2 ∘C in width, with the average of our samples being 1.9–3.5 ∘C. When multiple uncertainties are considered, the constraint is however markedly weakened. For a true sensitivity of S=2.5 ∘C, the posterior 5 %–95 % range is typically over 4∘, at 1.6–6.0 ∘C, with a “likely” 17 %–83 % range of 2.2–4.1 ∘C. As in the unforced experiments, these optimal constraints appear somewhat narrower than can be obtained by using the Ψ statistic but are not necessarily tight enough to be compelling in themselves.