Focus Article
Bayesian estimation of climate sensitivity using observationally constrained simple climate models Roger W. Bodman* and Roger N. Jones Edited by Eduardo Zorita, Domain Editor, and Mike Hulme, Editor-in-Chief
One-dimensional simple climate models (SCMs) play an important role within a hierarchy of climate models. They have largely been used to investigate alternative emission scenarios and estimate global-mean temperature change. This role has expanded through the incorporation of techniques that include Monte Carlo methods and Bayesian statistics, adding the ability to generate probabilistic temperature change projections and diagnose key uncertainties, including equilibrium climate sensitivity (ECS). The latter is the most influential parameter within this class of models where it is ordinarily prescribed, rather than being an emergent property. A series of recent papers based on SCMs and Bayesian statistical methods have endeavored to estimate ECS by using instrumental observations and results from other more complex models to constrain the parameter space. Distributions for ECS depend on a variety of parameters, such as ocean diffusivity and aerosol forcing, so that conclusions cannot be drawn without reference to the joint parameter distribution. Results are affected by the treatment of natural variability, observational uncertainty, and the parameter space being explored. In addition, the highly simplified nature of SCMs means that they contain a number of implicit assumptions that do not necessarily reflect adequately the true nature of Earth’s nonlinear quasi-chaotic climate system. Differences in the best estimate and range for ECS may be partly due to variations in the structure of the SCMs reviewed in this study, along with the selection of data and the calibration details, including the choice of priors. Further investigations and model intercomparisons are needed to clarify these issues. © 2016 The Authors. WIREs Climate Change published by Wiley Periodicals, Inc. How to cite this article:
WIREs Clim Change 2016, 7:461–473. doi: 10.1002/wcc.397
INTRODUCTION
C
omputer-based climate models are useful tools for addressing a range of questions about the Earth’s climate system, including the extent to which *Correspondence to:
[email protected] Victoria Institute of Strategic Economic Studies, Victoria University, Melbourne, Australia Conflict of interest: The authors have declared no conflicts of interest for this article.
human activities are changing the climate and how it may change in the future. A wide variety of such models, from complex to simple, play a key role in climate science and thus to climate policy. Here, we focus on one-dimensional simple climate models (SCMs) along the lines described by the Intergovernmental Panel on Climate Change (IPCC)1, in particular, looking at their role in estimating equilibrium climate sensitivity (ECS) using Bayesian statistical methods. These models typically represent the deep ocean separately from the surface and represent land and ocean and/or north and
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south hemisphere surfaces separately; they do not resolve the surface latitudinally and represent the atmosphere as a single layer. SCMs have been applied to a variety of different roles, including emulating more complex models, investigating model structural and parameter uncertainty, and the effects of different forcings and feedbacks.2 SCMs can provide rapid results, investigate the implications and uncertainties associated with different emission scenarios, and play the part of ‘temperature change calculator’ within integrated assessment models. In addition to prognostic applications (e.g., estimating temperature change and sea level rise), some SCMs have a diagnostic capability, particularly in estimating model parameters such as climate sensitivity. The latter is critical to projections with a SCM because it is a prescribed parameter, rather than being an emergent property of the physical processes represented within more complex models. It is also by far the biggest single contributor to uncertainty in global-mean temperature projections for a given emission scenario.3 Climate sensitivity has been the subject of extensive research, with estimates based on a variety of studies using palaeorecords, observations, and climate models.4–8 In the IPCC Fourth Assessment Report (AR4), a wide range of models was used to inform the likely range for ECS of 2–4.5 C, with a best estimate of 3 C.7 This result drew on studies of past climate change over both palaeo time scales and the instrumental record, along with the spread in climate sensitivity derived from ensembles of complex models. Models constrained by past records included studies with intermediate complexity models,9–15 energy balance observations,16,17 a simple statistical model,18 and a simple one dimensional climate model19 (refer AR4 Box 10.27). For the Fifth Assessment Report (AR5), the ECS estimate was revised, reducing the lower limit of the likely range to 1.5 C, with no best estimate given.8 This result was also based on a variety of studies (see AR5 TFE.68 and the chapter sections and other resources indicated therein). Apart from those estimates based on complex models, the ECS results for the instrumental period were largely based on intermediate complexity models with Bayesian methods,20–23 zero-dimensional models,24–27 and SCMs.28,29 Of the latter, only Aldrin et al.28 used Bayesian statistics. Previous studies that estimated climate sensitivity within a Bayesian framework were largely based on EMICs (Earth System Models of Intermediate Complexity).9,10,12,22,30 Statistical emulators have also been adopted to address the computational challenge of iterating through a large parameter space with EMICs.31 462
Several recently published studies combine instrumental observations and a Bayesian MCMC/ MCMH (Monte Carlo Markov Chain/Monte Carlo Metropolis–Hastings) data assimilation framework with SCMs to estimate key model parameters, including climate sensitivity. This review examines five of these studies, discusses the main features, and raises some concerns. Questions about whether or not these models are an appropriate tool for diagnosing ECS are discussed, arriving at the conclusion that the differences between such models and in the analyses and calibration methods used, along with the limitations in such models, mean that the estimates presented to date from such models require careful interpretation.
SHIFT FROM PROGNOSTIC TO DIAGNOSTIC MODE One-dimensional SCMs have recently (since around 2009) been used for estimating ECS and other key model parameters, partly to develop the capability to generate probabilistic temperature change projections as well as probability density functions (pdfs) for ECS. This is typified by the evolution of MAGICC,2,32–35 a prominent upwelling–diffusion/ energy balance model (UD/EBM) that has played a role in all five IPCC assessment reports. The most comprehensive calibration of this model was completed in 2009 when Meinshausen et al. undertook a Bayesian calibration using the then recently developed model version 636 (hereinafter MM2009). Features included the use of a comprehensive set of model parameters together with historical observations of surface temperatures covering each of the model’s four regions (northern hemisphere land and ocean, southern hemisphere land and ocean), ocean heat content (OHC) as a linear trend and a range of radiative forcing estimates. Natural variability from Atmosphere Ocean General Circulation Model (AOGCM) control runs was analyzed and used to produce a covariance matrix allowing for internal variability and measurement errors. The carbon cycle was calibrated against the C4MIP models,37 not observations. Posterior parameter distributions were illustrated, but only a brief summary was provided for ECS, given as a 90% confidence interval (CI) of 2.1–7.1 C (from a uniform prior interval of 0.05–20.0 C).36 Since MM2009, a number of papers based on SCMs and the Earth’s energy balance have been produced,20,21,24,27,38,39 but few were observationally based calibration studies employing Bayesian methods. A further four were identified for this
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Bayesian estimation of climate sensitivity using observationally constrained SCMs
review: Aldrin et al.,28 Bodman et al.,3 Skeie et al.40 and Johansson et al.41 (hereinafter respectively Aldrin2012, Bodman2013, Skeie2014, and Johansson2015). The main features of these studies are presented in the following subsections along with a summary of ECS results. Key issues are addressed in third section followed by additional discussion and conclusions. Bayesian estimation has also been applied in related studies. Ring and Schlesinger42 is a development from Ring et al.38 using a root-mean-square error method to estimate ECS, ocean heat diffusivity (K), and direct sulfate aerosol forcing (FSA) parameters with an SCM.42 Synthetic observations were employed to test a Bayesian estimation procedure to see if ‘true’ values of ECS, ocean diffusivity, and aerosol forcing could be determined. They found that ECS and K did converge given enough time but FSA was not found due to a low signal-to-noise ratio for the interhemispheric temperature difference used to constrain FSA. However, their present day, 90% CI of around 0.5 C for the uncertainty band about the ECS central estimate appears to be quite small in comparison to other similar studies. Urban et al.43 used an SCM to test the rate at which uncertainties might reduce with additional observations of temperature and ocean heat, combining real observations to date and synthetic observations to extend the record. They used the same parameters as above (ECS, K, and FSA), adding parameters addressing uncertainties in initial values for temperature, ocean heat, and their standard deviation and autocorrelation. Although their focus was on the potential to reduce uncertainty in ECS with future observations, the results to 2012 show a posterior distribution of approximately 3.0 C for the mean (2.1–4.7 C 90% CI, estimated from their Figure 3).
Aldrin2012 Aldrin201228 focus on Bayesian estimation of ECS using a somewhat simpler energy balance model (EBM) than MAGICC, with only northern and southern hemisphere ocean boxes, with no land and no carbon cycle. Calibration of seven climate system parameters was accomplished using observations that included northern hemisphere and southern hemisphere annual mean surface temperature anomalies along with OHC to 700 m depth. Radiative forcings were not calculated by the model, but supplied using data from the IPCC Fourth Assessment Report7 and treated as uncertain priors as part of the MCMH algorithm.
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Natural variability was represented with a stochastic autoregressive model based on an El Ninõ– Southern Oscillation (ENSO) index. Uniform priors were used for the climate model parameters and a number of experiments carried out to test how sensitive the results were to the choice of priors and observations. They compared model results and observations but presented no future projections. The calibration step was performed using the MCMH algorithm with very long runs. Two chains were run, one with 50-million iterations and one with 100-million iterations. A burn-in period was allowed for, with the first 6-million iterations discarded, while the remaining iterations were used to estimate the posterior distribution but thinned, retaining each 5,000th iteration.
Bodman2013 Using MAGICC V6,2 Bodman2013 combined climate and carbon cycle observations to calibrate and test the potential to reduce the spread in projected temperature change. This approach differed in a number of respects from MM2009. Historical CO2 concentrations were used to inform the use of carbon cycle parameters rather than rely on those from the C4MIP models (although these did help inform some of the parameter priors). Fewer parameters (11 rather than 82) were used, based on analysis of their contribution to global temperature uncertainty at 2100. Observational time series were arranged in different ways, using a single data source for surface temperatures (HadCRUT344). These were organized as decadal averages for globalmean surface temperature (GMST), a northern minus southern hemisphere index and a land minus ocean temperature index (two of the ‘Braganza’ indices45). Three time series for OHC at 100, 300, and 700 m depths46 together with an ocean vertical temperature change profile47 were also utilized. Natural variability was dealt with by using decadal averages (similar to Sansó and Forest31) rather than a stochastic noise generating process. Priors were truncated normal distributions rather than uniform and based on existing information. A MCMH algorithm was also adopted, but with a single chain of 50,000 iterations, no burn-in period and all of the accepted prior parameter sets (almost 38,000) were used to arrive at a posterior parameter distribution. Conversion of aerosol emissions to radiative forcing was modeled by specifying values for the different aerosol components’ radiative forcing as parameters for a given year, after which radiative forcings
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were scaled in proportion to emissions. All aerosol parameters were left at their default values except for direct sulfate aerosol forcing, which was then used as the variable parameter. Indirect aerosol effects were scaled from the direct effect. Examples of forward projections using the posterior parameter distribution were included, since the potential to reduce uncertainty in future temperature change was a major motivation behind this study.
Skeie et al 2014 Skeie2014 is an extension to Aldrin2012 with a similar model setup but with some modifications including observations updated to 2010. Three simultaneous time series were used for OHC rather than one and additional sensitivity tests performed. This study arrived at an even narrower posterior for ECS than Adrin2012, largely attributed to the inclusion of additional OHC observations. The ECS distribution is claimed to be ‘a lower and better constrained estimate of the climate sensitivity compared to the majority of previous estimates.’ ECS estimates for observational periods ending in 2000 and 2010 were included, noticeably changing the posterior distribution. The estimate to 2000 has a mean of 2.3 C, 90% CI of 0.6–5.6 C, whereas that to 2010 has a posterior mean of 1.8 C (90% CI of 0.9–3.2 C). However, the median ECS values remain much the same at about 1.7 C, while the 90% CI has narrowed, particularly at the upper end. Transient climate response (TCR) based on a 1% per year increase in CO2 was also estimated. Model parameters were the joint posterior parameter distribution arrived at through the calibration process, obtaining a posterior mean TCR estimate of 1.4 C (90% CI of 0.8–2.2 C). Observed and modeled temperature distributions were illustrated for northern and southern hemisphere surface temperature 1850–2010, along with OHC 1945–2010. No future projections runs were modeled, although it would be interesting to see temperature change projections based on the posterior parameter distribution of Skeie2014 as compared to other AOGCM and ESM projections, along with results from other SCM studies, such as Meinshausen et al.2 and Rogelji et al.48
Johansson2015 This study is also based on a UD/EBM arrangement, although differs in having a single land and ocean (i.e., not hemispherically resolved). The paper investigates how ECS evolves under cumulative 464
observations. It uses global land and global ocean annual mean surface temperature anomalies plus a 2000 m OHC time series along with estimates for radiative forcing components. Natural variability is treated using an ENSO index and a stochastic autoregressive process. A Bayesian approach uses the Metropolis algorithm, with 500,000 iterations, a burn-in of 20,000 iterations and retaining every 20th parameter sample from the posterior distribution. Six climate model parameters were calibrated in this way and a sensitivity analysis performed. A key feature is how ECS changes with the period of the data assimilation, which, after 1986, proceeds in 5-year increments to 2011. The peak of the resulting pdf declines to 1996, increases to 2006, and then declines again in 2011, following the evolution of the global warming trend over decadal timescales. However, it does become more constrained over time, though the cause is unclear.41 ECS is not constant and may be only partially independent of the pattern and magnitude of the forcing.49–53 Results will also depend on the available observational data and its time-varying uncertainty. These variable ECS values reflect the limitations in the observational record but could also be reflecting limitations in the methodology (the SCM and MCMH setup). This issue is addressed further in ECS Affected by Observations and Model Structure section. Johansson2015 also state the importance of OHC, a finding that is consistent with Skeie2014. However, other studies with intermediate complexity models have found that climate sensitivity may be only weakly constrained by observations of OHC12 and influenced by the data source, depth covered, and uncertainties.54 This issue is returned to in Sensitivity to Choice of Parameters and Priors section.
Summary of ECS Findings ECS estimates from the five studies outlined above are presented in Table 1. MM2009 and Bodman2013 are both based on MAGICC version 6, with results broadly consistent with estimates from the IPCC fourth4 and fifth8 assessment reports. The other three papers, Aldrin2012, Skeie2014, and Johansson2015, find lower values for the mean and 90% CI. The principle reasons for these differences rest in: • Variations in the historical datasets together with the periods covered; • Variations in the model structures, with different configurations of large-scale regions;
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Bayesian estimation of climate sensitivity using observationally constrained SCMs
TABLE 1 | Equilibrium Climate Sensitivity, C, Estimated from Five-Bayesian Calibration of SCMs (Published Results Have Been Rounded to One Decimal Place)
Source:
MM200936
Aldrin201228
Bodman20133
Skeie201440
Johansson201541
Priors:
Uniform, 0.05 to 20
Uniform, 10−4 to 20
Truncated normal, interval 0 to 18
Uniform, 0 to 20
Uniform, 0.37 to 11.1
2.0
3.2
1.8
2.6
3.1
1
1.7
1.5–5.2
0.9–3.2
Posteriors: Mean Median 90% CI
2.1–7.1
1.2–3.5
1.9–3.3
CI, confidence interval; SCMs, simple climate models. 1 Median estimated from posterior median indicated in Figure 2 (a), Skeie2014.
• The form of the prior distributions adopted; • The treatment of natural variability; • The approach adopted for measurement and observational uncertainties. These and other related issues are addressed further in Key Issues section.
KEY ISSUES This section discusses a range of issues that arise from differences in the calibration approaches used in the studies summarized above.
ECS Affected by Observations and Model Structure Skeie2014 and Johansson2015 show how mean estimates and distributions for ECS vary with observations, with distributions narrowing for longer records. However, the median values vary less in Skeie2014 and the changes in distribution profiles indicate that this is likely true for Aldrin2012 and Johansson2015. The differences in ECS estimates between MM2009 and Bodman2013 as compared to Aldrin2012, Skeie2014, and Johansson2015 may be in part due to data selection and length of coverage. The definition of ECS, as the equilibrium temperature increase from a sustained doubling of atmospheric CO2 concentrations implies a single value. The available observational record provides only a limited constraint on that value. It is also possible that ECS is not a constant, but affected by the changing climate state with different (nonlinear) feedback processes at work. This has implications for how SCMs project future changes. MAGICC V6 introduced a parameter that allows ECS to change with the climate state that, in a Bayesian setup, complicates the interpretation of its results. MM2009
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enabled this parameter as one of the 82 used in their calibration exercise, however, Bodman2013 disabled it. How large-scale regions are configured, for example, two (land and ocean or northern and southern hemispheres) or four (northern land, northern ocean, southern land, southern ocean) regions, along with parameter section will also influence ECS estimates. Resolving the northern and southern hemispheres separately is important for estimating aerosol forcing. The changing ECS estimate over time, as seen in Johansson2015, may partly be due to not using hemispherically resolved data, thus overestimating aerosol forcing strength and consequently overestimating ECS. The selection of observational data can also affect parameter estimates; for example, the choice of surface temperature data has been shown to influence estimates for ECS.23 The MCMH algorithm treats observational data as a set of independent and identically distributed random variables with (typically) normally distributed uncertainty. Observational uncertainty (e.g., measurement errors and natural variability), autocorrelation and parameter uncertainty are addressed in the MCMH formulation. It is less clear how multiple sources of GMST in the same calibration, as in Aldrin2012 and Skeie2014, affect the results. See also the discussion on OHC (Co-linear Observational Data and Role of OHC section).
Calibration Yields a Joint Parameter Distribution As produced by these Bayesian calibration processes, the distributions for ECS are not stand-alone results but are part of joint parameter distributions. ECS estimates are presented as stand-alone results by Aldrin2012, Skeie2014, and Johansson2015, who do not remind the reader of this limitation. Any further discussion of the results in the literature also needs to
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account for this. The complete sets of posterior parameters may provide a reasonable basis for projections within the model that produced them, but they do not provide reliable, independent results for ECS that can be used in other contexts. The ECS distribution will be highly correlated with the aerosol forcing and there will be cross correlations with the other parameters of varying strengths. The different studies exhibit noticeable differences in their posterior distributions for aerosol forcing, even when they yield similar median ECS values. For example, Skeie2014 has a posterior mean that is less negative than Johansson2015 despite starting with an even more negative prior. For producing temperature change projections, the joint posterior parameter distribution therefore should only be used with the SCM that generated it. Studies should present or make available the full joint posterior parameter distributions. It may also be appropriate to provide the joint likelihood, so that users of the study can investigate the effects of substituting their own prior distributions, or of applying profile likelihood techniques to derive frequentist CIs.
Model Evaluation As others have said before,55 comparison of model results to observations is important for developing confidence in a model’s ability to generate projections for climate change. This relates to both the fidelity of the model to observations but applies also, in the case of a parameter distribution, to the measure of uncertainty. Mention has been made of the fit between different calibrations and observations in ECS Affected by Observations and Model Structure section. Evaluation of the model to observations fit is a multidimensional problem with different sets of model diagnostics and observations together with model and observational uncertainties. How can the posterior parameter distribution be evaluated? Furthermore, does a ‘good’ fit imply that the model is suited to projecting future change? The provision of terms in the MCMH algorithm to account for model errors and noise arising from natural variability (if adopted) may provide a statistical description during the calibration period but, potentially lacking adequate processes, the SCM may not perform outside of its calibration range.56 This may also apply to more complex models, including AOGCMs, as, for example, the representation of clouds and convection 466
processes are considerably.57
also
parameterized
and
vary
Colinear Observational Data and Role of OHC There are typically very strong correlations (r > 0.9) between the historical observations used in these Bayesian calibrations, for example, between different surface temperature records, and between surface temperatures and OHC. Using different input datasets can therefore provide only limited extra information to the calibration procedure. These strong correlations are the reason behind Bodman2013 using the ‘Braganza’ indices,45 that is, the hemispheric temperature and land–ocean temperature differences that complement the GMST data. In addition, the ocean vertical temperature change profile was found to contribute more information than simple OHC. Ocean heat uptake is said to be more informative in some of the studies; both Skeie2014 and Johansson2015 state the importance of OHC whereas other studies with intermediate complexity models that have found ocean heat to be a weak constraint on climate sensitivity.12,54 Sokolov et al.54 suggest that this weak constraint results from the strong correlation between surface temperatures and 0–700 m OHC and the large natural variability of the upper ocean (as well as the large uncertainty). As more OHC data becomes available, particularly at 2000 m depth (as used in Johansson2015) this picture may change. To assess the role of OHC observations, we used our Bayesian setup (Bodman2013) and tested some additional model calibration runs, changing the OHC observations. In addition to the reference case, which used Domingues et al.46 (hereinafter Dom08) OHC with three time series for 100, 300, and 700 m OHC, decadally averaged over 5 decades (1951–2000), three alternative cases were tested: (1) in which the 100 and 300 m data were not included in the MCMH data assimilation, (2) with the Dom08 700 m OHC replaced by Levitus et al.58 data (hereinafter Lev12), and (3) where both Dom08 and Lev12 700 m were included together. The resultant impact on the posterior parameter values is summarized in Table 2. Not including the Dom08 100 and 300 m data has little impact on the ECS estimate, although there are small changes to the 90% CI along with some differences in the other tabled climate parameters. The switch from Dom08 to Lev12 700 m OHC time series results in a lower median ECS estimate by
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Bayesian estimation of climate sensitivity using observationally constrained SCMs
TABLE 2 | Posterior Parameter Distributions for OHC Variations: the Bodman2013 Reference Case Which Used 100, 300, and 700 m Dom08 Decadal OHC Observations Then Three Alternate Arrangements: (1) with Dom08 700 m only, (2) Lev12 700 m OHC instead of Dom08 and, (3) both Dom08 and Lev12 700 m OHC.
Parameter
Median
90% Interval
Std Dev
Reference case Climate sensitivity
2.92
1.54–4.76
0.99
Ocean diffusivity
0.72
0.22–1.51
0.39
Land ocean heat exchange
1.65
0.68–2.86
0.66
Land ocean warming ratio
1.57
1.28–1.87
0.18
(1) With Dom08 700 m OHC only Climate sensitivity
2.93
1.44–5.02
1.10
Ocean diffusivity
0.75
0.24–1.51
0.39
Land ocean heat exchange
1.81
0.69–3.05
0.73
Land ocean warming ratio
1.59
1.32
0.18
Climate sensitivity
2.67
1.37–4.72
1.03
Ocean diffusivity
0.72
0.21–1.43
0.38
Land ocean heat exchange
1.81
0.69–3.08
0.72
Land ocean warming ratio
1.59
1.29–1.93
0.20
1.90
(2) With Lev12 700 m OHC instead
(3) With both Dom08 and Lev12 700 m OHC Climate sensitivity
2.63
1.37–4.60
0.99
Ocean diffusivity
0.72
0.24–1.49
0.38
Land ocean heat exchange
1.78
0.73–2.99
0.68
Land ocean warming ratio
1.61
1.31–1.94
0.19
OHC, ocean heat content.
0.25 C. Using both Dom08 and Lev12 700 m data produces parameter estimates close to the Lev12 only case. The Lev12 data has smaller values for estimated uncertainty and, consequently, appears to effectively sideline the influence of the Dom08 data. Having the two in parallel only marginally reduces the 90% CI for ECS; it does not generate the type of result found by, for example, Skeie2014. In their earlier study, Aldrin2012 indicated that ECS is sensitive to the OHC data, but proposed that including three time series ‘would be sensible’ (the approach Aldrin2013 employed included three surface temperature time series); Skeie2014 subsequently enacted this proposal. Skeie2014 performed some sensitivity tests with OHC, using some additional data to 2000 m and using a single 700-m time series. The 2000 m test found little change in ECS, whereas the single 700-m OHC test (with data to 2000 rather than 2010) substantially changed the results, with a mean ECS of 4.5 C, median 3.0 C (approximately, estimated from Figure 2(g), Skeie2014) with a 90% CI of 1.1–14.5 C. Since the three 700-m OHC time
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series datasets are for the same observable they would appear to expand the overall range of observational uncertainty, yet, in effect, give an unwarranted greater weight to the central tendency. The MCMH algorithm, in essence, tests the modelobservations mismatch; it is not clear how three time series for the same quantity can help reduce this mismatch and provide a stronger constraint on ECS. However, there is an assumption here that the Levitus and Domingues model-observation residuals are independent, which may warrant further investigation. The calibrated ocean diffusivity parameters have distinctly different results across the models. The posterior mean is not given in MM2009, although the CMIP3 mean value is 1.1 cm2/second. Bodman2013 arrived at a posterior mean of 0.8 cm2/ second after a prior mean of 0.7 cm2/second, using a truncated normal distribution. Aldrin2012 started with a mean of 0.34 cm2/second (uniform interval 0.11–0.69 cm2/second), resulting in a posterior mean of 0.30. Skeie2014 used a prior mean of 0.43 cm2/
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second (uniform 0.06–0.8 cm2/second) that led to a posterior mean 0.38 cm2/second. These three (Bodman2013, Aldrin2012, and Skeie2014) show little change between their prior and posterior values, possibly indicating the data providing little constraint (difficult to judge without the pdfs). Johansson2015 began with a prior mean of 2.15 cm2/second (uniform 0.3–4 cm2/second) and arrived at a posterior mean of 0.77 cm2/second. Assuming these values for the ocean diffusivity are comparable, given the similarities in model design, then these are significant differences. Ocean diffusivity is connected to ECS: for a given amount of temperature change, a higher ECS will need to be balanced by more heat transfer into the ocean. There appears to be scope for further investigation and testing of the role of OHC data in calibrating model parameters, with a need for data at greater depths. Bodman2013 found an ocean vertical temperature change profile provided information in its calibration work. It would be interesting to see if others found this to be the case as well.
Sensitivity to Choice of Parameters and Priors The SCM studies discussed here seek to calibrate model parameters using Bayesian inversion techniques. MM2009 tackled this with 82 parameters and Bodman2013 reduced this to 11 parameters (four climate, one aerosol forcing, and six carbon cycle), finding that the smaller set was representative of the larger. Aldrin2012 used seven climate parameters as did Skeie2014, and Johansson2015 selected six for calibration. Bodman2013 showed that only a small number of the climate system parameters affected uncertainty in temperature change projections; some made very little difference to the results and could be left at their nominal default settings. The expected benefit from using fewer parameters was to apply the limited observational data to constraining the key parameters and thereby maximize the information gained. The inclusion of the carbon cycle can also influence the representation of equilibrium sensitivity, which is defined independently of the carbon cycle. However, in Earth system models, of which MAGICC is a simplified version, the carbon cycle does play a role when emissions rather than concentrations (or forcings based on concentrations) drive the models. There is some interaction between temperature change and the carbon cycle that affects ECS estimates. In that case, perhaps rather than equilibrium or effective climate sensitivity, it is an Earth 468
system sensitivity that is being calibrated (see also discussion in Previdi et al.6). Calibration results are sensitive to the choice of prior parameter distributions. The choice of priors also appears to be a source of some misunderstandings, particularly in the context of Bayesian theory. Bayes’ theorem implies the use of pre-existing knowledge and expert judgment in the selection of priors. Given that climate sensitivity has been studied extensively, information is available that can help inform these priors. Many studies of this type have instead elected to use uniform distributions, although this may not necessarily be the best choice.59 Even if a uniform distribution is chosen, it is difficult to justify a lower bound of less than 1 C and it is hard to see how a value over 6 C is plausible given past climate change.60 It is also hard to see how these limits could be considered as equally probable as say 2.5 or 3.0 C. Furthermore, high upper bounds in a uniform distribution promote the right-hand skew seen in many of the published pdfs for ECS. Uniform distributions might appear to be noninformative priors, but that is not the case.61 Bodman2013 used a truncated normal prior for ECS with a nominal mean of 3.0 C. However, the posterior distribution is affected by this choice. This can be viewed as being due to weak observational constraints, but for a particular set of data and model, different priors will also lead to different posteriors.62 Alternatives to the subjective Bayesian perspective are robust Bayesian analysis or objective Bayesian methods that employ noninformative priors based on, for example, reference prior theory or the maximum entropy approach.63,64 None of the studies reviewed here have attempted to use noninformative priors although there are examples of this type in the literature using EMICs20,22,23 and a one-dimensional diagnostic model.65,66 This does not avoid the inevitable subjective elements such as choice of data and model parameters62 along with model selection and model structure.
Efficacy/Effectiveness, Feedbacks, and Nonlinearity The different agents that give rise to radiative forcing exhibit different temporal and spatial characteristics. Concerns about this have been expressed through the concepts of efficacy67 and effectiveness.8 The treatment of aerosols and ozone will impact on estimates for ECS derived from SCMs.68 In addition, these forcings give rise to feedbacks that are heterogeneously distributed, with, for example, different feedbacks
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over land and ocean, with variable cloud feedbacks57,69 and also regional feedbacks.52,70 Land in the northern hemisphere will behave differently to that in the south, since the former is where most industrial pollution arises. SCMs that do not account for these factors may be inadequate.71 Differences in the temperature response to aerosols and ozone may assist in explaining some of the differences between simple and complex models. Another issue is the nonlinearity of the climate system. The principal equation that utilizes the climate feedback parameter (generally the inverse of ECS, depending on usage) is a linear relationship, with higher order terms neglected on the basis that the temperature response from feedback interactions is small.72 This can still give rise to an asymmetric probability distribution due to uncertainties in observed climate forcing and the feedback responses.73,74 This linear assumption may underestimate the risk of high warming.75 When the top-ofatmosphere energy imbalance is relatively small, such as in the first half of the 20th century, a linear relationship between radiative forcing and temperature change is an acceptable approximation. As the radiative forcing increases, this linear relationship may no longer hold and second or higher order terms become necessary. The asymmetry seen in most estimates for ECS probability distributions (the long right-hand tails) may be partially explained by this nonlinearity.75 The use of uniform priors also gives rise to the skewed distributions seen in the pdfs for ECS, more so than with normal or noninformative priors.
Aerosols: Forcing Uncertainty, Distribution, and Feedback Treatment of aerosol components and forcing uncertainty is critical to the calibration results. As noted in Calibration Yields a Joint Parameter Distribution section, ECS in SCMs depends heavily on aerosol forcing. The distribution of aerosols over land and ocean will also affect climate feedbacks (Section Efficacy/Effectiveness, Feedbacks and NonLinearity), processes that may not be captured within some SCMs. MAGICC V6 attempts to address these issues by: (a) separating land and ocean climate feedback parameters, (b) having different aerosol forcings across its four regions based on a spatial pattern of optical thickness, and c) providing a comprehensive set of aerosol forcing components including sulfates, nitrates, and dust with both direct and indirect effects. However, in calibrating MAGICC, it is difficult to resolve all of these components. Bodman2013 had the direct sulfate forcing parameter that
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represent the other direct aerosol forcings and then scaled the indirect effect from that. Other arrangements would be possible, but it is not clear what would be gained by adding complexity.
Treatment of Natural Variability Internal, unforced natural variability is a significant feature of the climate system that may affect estimates of climate sensitivity on short-time scales (annual, decadal, and multidecadal). One EMIC-based study found that, assuming a perfect model and perfect observations, considerable uncertainty in ECS would remain as a consequence of internal natural variability.76 A similar study found that natural variability varied the estimate for climate sensitivity by about 0.4–0.8 C.77 The slow-down in the rate of temperature change over the past 10–15 years, the so-called ‘hiatus,’ has been attributed to a range of causes that include natural variability.8,78–81 Among the different contributing factors proposed are processes that affect ocean heat uptake, such as wind driven circulation and heat transfer due to ocean currents, which can transfer heat between upper and lower ocean regions irrespective of changes in radiative forcing. These internal oceanic processes are not represented by the diffusivity equation typically used in SCMs. The treatment of natural variability is therefore a particular challenge for SCMs since they lack any representation of the physical processes that generate this feature of the climate system. The majority of the Bayesian SCM studies reviewed here used annual time series, introducing a stochastic process to generate noise that substitutes for natural variability. MM2009 used internal variability from an AOGCM preindustrial control run to allow for uncertainty arising from natural internal variability. Bodman2013 used decadal means as an alternative method. Whether these approaches are adequate is another outstanding question.
CONCLUSION The ability to diagnose ECS over the 20th century helps inform our understanding of the climate system, and the development of probabilistic and diagnostic methods within SCMs is one way to do this. The focus on Bayesian statistical methods linked to SCMs allows the interplay between observations and model parameters to manage key uncertainties such as ECS. Scientific curiosity will ensure that such explorations are embarked upon, but it is important to understand what conclusions can be taken from the results.
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Of the calibration studies reviewed in this paper, those based on the MAGICC SCM (MM2009 and Bodman2013) produce estimates for ECS around 3 C, whereas the others (Aldrin2012, Skeie2014, and Johansson2015) reported noticeably lower ECS values. The lower estimates are consistent with other results derived using energy balance calculations directly.27,82 Values around 2 C are markedly lower than the estimates associated with AOGCMs and the IPCC. The major differences between MAGICC and the other SCMs are the simpler land–ocean–hemisphere configurations and the absence of a carbon cycle. It is important to understand the reasons behind the differences between the complex and simple models as well as between the simple models. There may be a level of simplicity below which such models should not go. The investigation of the joint probability space and how energy is partitioned within that space taking account of reservoir effects (e.g., OHC), potential nonlinearity and feedbacks appears to be an appropriate starting point. The advantage of SCMs is that they can explore uncertainty spaces relatively quickly, but these spaces need to be physically valid within the model structure. The newness of the theme means that results for ECS are presented in different ways, making intercomparisons difficult. For example, mean values may be presented rather than the more appropriate median because means are higher in the generally right-hand skewed pdfs. Furthermore, the distributions for ECS are not independent in these results but are part of joint posterior pdfs. The linearity of the forcing to temperature relationship is also a subject of debate in the literature. SCMs use climate sensitivity to represent the combined effects of a number of different nonlinear climate feedback processes, including water vapor and lapse rate, clouds, changes in ice sheets, and albedo. These feedbacks vary across interannual and decadal timescales83 and spatially. This may not be factor in SCMs that estimate ECS on multidecadal timescales but the critical question about learning is whether the informational content of new observational data is a linear signal with associated noise, or part of a
nonlinear signal with some noise. This implies a minimum record length, below which SCMs cannot be assumed to represent theoretical uncertainty. For example, the 30-year period for means is a convention constructed for managing uncertainty in stationary climates but may not be adequate for nonstationary climates assuming linearity. All of this leaves an interesting open question— is the posterior parameter distribution from an observationally constrained SCM a reliable basis for assessing future temperature change? Based on posterior parameter distributions from the non-MAGICC SCM studies, for equivalent forcing scenarios, projected future GMST changes would almost certainly be lower than those obtained from MAGICC and almost all CMIP5 AOGCMs. This is partly because effective climate sensitivity increases over time in most CMIP5 models and because their model physics generally produce higher ECS and TCR values than the observationally constrained SCMs studies. It may be that the simple models can provide information on what the climate sensitivity was over the historical period as a result of the feedbacks operating in the climate system then, but those feedbacks may not accurately represent how the climate system will behave in the future. We have asked some questions about the utility of SCMs, their ability to self-diagnose model parameters and to use those parameters for projecting temperature change. There is clearly scope for further analysis and investigation. Open source software and more comprehensive documentation would also help, making it easier to evaluate and compare studies of this type, enabling replication of their findings and for testing sensitivities. SCMs will continue to play important roles in understanding and informing both climate science and climate policy, but it is important to realize that results from such models, even though observationally constrained, are not necessarily correct. As AOGCMs and ESMs become more complex, SCMs help our understanding of key processes at work.84 They will remain an essential component in integrated assessment models and as a flexible tool for evaluating emission scenarios.
FURTHER READING McGuffie K, Henderson-Sellers A. A Climate Modelling Primer. 4th ed. Chichester: John Wiley & Sons; 2014. Kruscke JK. Doing Bayesian Data Analysis: A Tutorial with R. JAGS, and Stan. 2nd ed. London: Academic Press; 2015.
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