## 1. Introduction

Global warming has been shown to be almost independent of emissions pathway and approximately proportional to cumulative carbon dioxide emissions in simulations of the response to a range of emissions scenarios with previous generation carbon climate models (Allen et al. 2009; Matthews et al. 2009; Zickfeld et al. 2009). This suggests that a model's climate response to carbon dioxide emissions can be well characterized by the ratio of the temperature increase to the cumulative carbon emissions. This ratio is a combined measure of both a model's carbon cycle response to emissions and its physical climate response to an increase in atmospheric CO_{2}, making it a useful benchmark for model intercomparison. Because it directly relates warming to emissions, it also has the potential to be of use in climate change mitigation policy; for example, this ratio can be used to calculate the cumulative emissions consistent with a 2°C warming target (Allen et al. 2009; Matthews et al. 2009). We note, however, that the transient climate response to cumulative carbon emissions (TCRE) is not directly applicable to emissions of non-CO_{2} greenhouse gases and other forcing agents. Matthews et al. (2009) propose that the TCRE of a model is defined as the ratio of warming to cumulative CO_{2} emissions in a simulation with a prescribed 1% yr^{−1} increase in CO_{2} at the time when CO_{2} reaches double its preindustrial concentration, paralleling the definition of the transient climate response (TCR), which is defined as the warming at this time. Defined in this way, Matthews et al. (2009) find a TCRE range of 1.0–2.1 K EgC^{−1} in 11 carbon–climate models that took part in the Coupled Carbon Cycle Climate Model Intercomparison Project (C^{4}MIP) project. Note that TCRE is the same as the carbon–climate response defined by Matthews et al. (2009), but we use the name TCRE to more clearly distinguish this metric from a similar measure based on peak warming (Allen et al. 2009).

As well as a metric for the comparison of carbon–climate models, TCRE may also be assessed from observations. Because it is approximately independent of cumulative emissions, Matthews et al. (2009) estimate TCRE by dividing an observationally constrained estimate of CO_{2}-attributable warming over the twentieth century by an estimate of cumulative CO_{2} emissions over the same period and obtain a 5%–95% confidence range of 1.0–2.1 K EgC^{−1}. Allen et al. (2009) report a ratio of warming to cumulative emissions in 2000 of 1.4–2.5 K EgC^{−1}. They note that the ratio of peak warming to cumulative emissions could be higher than this if climate sensitivity is relatively high.

Many of the models included in phase 5 of the Coupled Model Intercomparison Project (CMIP5) are earth system models (ESMs), which include coupled carbon cycles, allowing TCRE to be directly assessed for these models (Arora et al. 2013). This study therefore consists of two parts: the first part describes the calculation of the TCRE of 15 CMIP5 models and examines the extent to which warming is proportional to cumulative emissions in each model. The second part derives a new observationally constrained estimate of TCRE, first by using CMIP5-simulated temperature and temperature observations to estimate CO_{2}-attributable warming to 2010 and then by dividing this by an estimate of cumulative CO_{2} emissions up to this time. Our analysis takes advantage of the most recent surface temperature observations, new estimates of historical CO_{2} emissions, and the newly available CMIP5 output.

## 2. Models and data

We use output from prescribed CO_{2} simulations from 24 CMIP5 models in this study. Of these, 15 are ESMs providing the output necessary to directly diagnose CO_{2} emissions and hence calculate TCRE (see Table 1 for further details). Monthly-mean atmosphere–ocean and atmosphere–land biosphere carbon fluxes were globally integrated with appropriate area weighting and summed over time to derive cumulative atmosphere–ocean and atmosphere–land fluxes. The atmospheric CO_{2} concentration in the 1% yr^{−1} CO_{2} increase (1PCT) simulations starts at its preindustrial value of close to 285 ppm (the actual specified preindustrial CO_{2} concentration differs by at most 1.3 ppm from this value in the simulations used) and increases by 1% yr^{−1}. Summing the increase in atmospheric CO_{2} in petagrams of carbon and the cumulative globally integrated atmosphere–land and atmosphere–ocean CO_{2} fluxes gives the derived cumulative emissions: anthropogenic emissions that when specified in a free CO_{2} simulation would have given the same evolution of atmospheric CO_{2}. We focus mainly on the 1PCT simulations here because TCRE is a metric of the response to CO_{2} only and it is conventionally defined using such simulations (Matthews et al. 2009). We also examine the response in simulations in which the CO_{2} concentration is instantaneously quadrupled and then held constant (4×CO_{2}) for 10 CMIP5 models for which the necessary output is available (Table 1). Note that we diagnose CO_{2} emissions purely from the carbon flux data available on the CMIP5 archive and the specified CO_{2} concentration and disregard any contribution from any other prescribed changes in CO_{2}, such as a restoring term specified in the GFDL-ESM models, which is small relative to other terms (Dunne et al. 2013).

CMIP5 output used in this study. The second column indicates those models for which TCRE was directly calculated: these are all ESMs including coupled carbon cycles. Models for which the necessary output was available to diagnose cumulative emissions in the abrupt 4×CO_{2} simulation are indicated with an “A.” The third column indicates those models for which a detection and attribution analysis was performed: these models had ALL, GHG, and NAT simulations to 2010, and their respective ensemble sizes and the length of control simulation are indicated. The fourth column indicates those models for which the ratio of TCR to 1861–2005 GHG-induced warming was calculated: these models all had GHG simulations to 2005 and the ensemble size is indicated.

For the analysis of observationally constrained TCRE, we use a detection and attribution analysis first to obtain a multimodel-mean scaling factor on the greenhouse gas response, based on a fit to temperature observations. For such an analysis, we require near-surface air temperature from the historical simulations including all major anthropogenic and natural forcings (ALL), simulations with only greenhouse gas changes (GHG), simulations with only natural forcing changes (NAT), preindustrial control simulations (CTL), and because we estimate TCR from these same models, 1PCT simulations. To be able to use observations up to 2010 in our analysis, we used only those models with simulations extended to at least 2010 for the detection and attribution analysis: such output was available from nine models (Table 1). These nine models have larger GHG ensemble sizes on average than those whose single forcing simulations end in 2005 (Table 1), providing further motivation for our decision to restrict attention to these models in the detection and attribution analysis. Historical simulations were extended from 2005 to 2012 by merging with historical extension simulations where available (CNRM-CM5, GISS-E2-H, GISS-E2-R, and NorESM1-M) or otherwise with representative concentration pathway (RCP 4.5) simulations (CanESM2, CSIRO Mk3.6.0, HadGEM2-ES, and IPSL-CM5A-LR) or, in the case of BCC_CSM1.1, using the full historical simulation, which extended to 2012. To examine the ratio of TCR to the historical GHG-induced warming trend (Table 1), 16 models having 1PCT simulations and GHG simulations to at least 2005 were used.

## 3. Transient response to cumulative emissions in the CMIP5 models

The global-mean temperature anomaly relative to the corresponding control simulation is shown against diagnosed cumulative carbon emissions for the 1PCT simulations in Fig. 1. As in the C^{4}MIP simulations (Matthews et al. 2009), the two quantities are approximately proportional. While most simulations were 140 yr long, extending to the time of a quadrupling of preindustrial CO_{2}, 1PCT simulations from the two GFDL models had atmospheric CO_{2} held constant after year 70 (i.e., at CO_{2} doubling) and therefore only the portion of the simulations up to year 70 is shown to maintain comparability. Dashed lines show the relationship for each model based on its TCRE (i.e., a straight line passing through the origin and the 20-yr-mean temperature anomaly and cumulative emissions centered on year 70), corresponding to doubled preindustrial CO_{2} concentration and approximately 1 EgC cumulative emissions. In absolute terms, the deviations from these straight lines are largest at the end of the simulation, where diagnosed emissions are around 3 EgC, well above the 2 EgC emissions below which Matthews et al. (2009) reported proportionality. In BNU-ESM, CanESM2, and HadGEM2-ES the warming at the end of the simulation is more than 1 K less than that predicted based on each model's TCRE. In fractional terms, these residuals are about 20% of the warming (Fig. 1c). Fractional deviations are also large for cumulative emissions <1 EgC, where the signal-to-noise ratio is lower and there appears to be a tendency for positive residuals at <1 EgC, which is consistent with an overall tendency for the ratio of warming to cumulative emissions to decrease with cumulative emissions in these models and consistent with the findings for C^{4}MIP (Matthews et al. 2009). Linearity is hard to assess for the two GFDL models because atmospheric CO_{2} concentration was not increased beyond double its preindustrial value. The remaining 10 models all show a close to linear relationship between warming and cumulative emissions in the 1PCT simulations. None of the models considered here exhibit an increasing ratio of warming to cumulative emissions as emissions increase beyond CO_{2} doubling. The proportionality of warming to cumulative emissions depends in part on a cancellation of the saturation of carbon sinks with increasing cumulative emissions (leading to a larger airborne fraction of cumulative emissions for higher emissions) and the logarithmic dependence of radiative forcing on atmospheric CO_{2} concentration [leading to a smaller increase in radiative forcing per unit increase in atmospheric CO_{2} at higher CO_{2} concentrations; Matthews et al. (2009)]. Negative deviations from a constant TCRE in some models imply that the radiative saturation effect is dominating.

Under the more extreme conditions of an instantaneous quadrupling of preindustrial atmospheric CO_{2}, deviations of the temperature response from the relationship predicted by each model's TCRE are larger (Fig. 1b). In this case, deviations are of both signs with BNU-ESM and CanESM2 warming substantially less than expected based on their TCRE and GFDL-ESM2G warming up to 50% more than expected based on its TCRE. Although the 4×CO_{2} simulations of BNU-ESM and CanESM2 warm less than predicted based on their TCRE, their temperature responses are similar to those in their 1PCT simulations for the same cumulative emissions. The consistency of the behavior of GFDL-ESM2G in the two simulations cannot be assessed because its 1PCT simulation did not extend beyond double preindustrial CO_{2}. MIROC-ESM shows an inconsistency between the two simulations, with the 4×CO_{2} simulation warming considerably more than the 1PCT simulation for the same cumulative emissions (disregarding the first 20 yr of the 4×CO_{2} simulation). This appears to be related to a much more rapid saturation of the terrestrial carbon sink in this model relative to that of the other models, with the terrestrial biosphere becoming a net carbon source much sooner than in the other models at around year 10 of the simulation. Matthews et al. (2009) found that the University of Victoria (UVic) Earth System Climate Model (an intermediate complexity model) exhibits a consistent proportionality between warming and cumulative emissions in a 4×CO_{2} simulation and a 1PCT simulation: these results demonstrate that such close proportionality in the 4×CO_{2} simulations is not seen in every model.

_{2}between models. Following Matthews et al. (2009), we can write TCRE aswhere

_{2}doubling (i.e., TCR),

_{2}doubling in exagrams of carbon, and

_{2}burden anomaly at CO

_{2}doubling in exagrams of carbon. Thus, TCRE is a product of the temperature response per unit change in the CO

_{2}burden and the cumulative airborne fraction at CO

_{2}doubling. Because

_{2}concentration between models). Figure 2a therefore represents each model's TCRE as a rectangle, with height equal to the model's TCR (i.e., Δ

*T*) and width equal to its airborne fraction of cumulative emissions at CO

_{2}doubling (

*C*/

*E*). The area of the rectangle is thus proportional to its TCRE. All quantities are calculated based on 20-yr means centered on the year of doubled preindustrial CO

_{2}concentration in the 1PCT simulations. The range of TCRE for the models assessed here is 0.8–2.4 K EgC

^{−1}, which is somewhat broader than the C

^{4}MIP range of 1.0–2.1 K EgC

^{−1}(Matthews et al. 2009). If we assume that model TCRE values are all drawn from the same distribution, then the probability

*P*that the TCRE of another model drawn from this distribution will exceed the maximum value among

*N*models is

*r*= 0.17). Thus, we can calculate an ensemble of synthetic values of TCRE by taking the product of the temperature response per unit change in atmospheric CO

_{2}burden for the

*i*th model and the airborne fraction from the

*j*th model:Doing this for all

*i*and

*j*, we obtain a distribution with a 5%–95% range of 1.0–2.4 K EgC

^{−1}. If we make the additional assumption that the real world TCRE is drawn from the same distribution as the models' TCRE, which we will revisit later, then we could interpret this as the confidence range on the real world TCRE.

Variations in TCRE between models are clearly caused both by variations in TCR and variations in airborne fraction, although variations in TCR are the most important (Fig. 2). TCR ranges from 1.0 to 2.6 K in these models with a mean of 1.9 K, which is close to the phase 3 of the Coupled Model Intercomparison Project (CMIP3) model mean of 1.8 K (Meehl et al. 2007). The cumulative airborne fraction is determined by the uptake of carbon by the land biosphere and ocean (Fig. 2b). Variations in land uptake are the dominant factor driving variations in the cumulative airborne fraction in these models (Arora et al. 2013), with the four models with the highest airborne fraction having the CO_{2} fertilization effect limited either by an explicit (NorESM1-ME and CESM1-BGC) or implicit (CanESM2 and MIROC-ESM) parameterization of nitrogen availability (Fig. 2b). The strength of the CO_{2} fertilization effect in CanESM2 is “downregulated” based on the response of plants grown in ambient and elevated CO_{2} following Arora et al. (2009). The MIROC-ESM uses an empirical approach to model photosynthesis, which implicitly includes the response to nutrient limitation (Ito and Oikawa 2002). MPI-ESM-LR has the lowest airborne fraction because of a strong land uptake (Fig. 2b; see also Arora et al. 2013) and INM-CM4.0 has the second lowest airborne fraction and the highest fraction of emissions taken up by the ocean at CO_{2} doubling, which is associated with a much higher atmosphere–ocean CO_{2} flux than in other models and may be associated with strong vertical mixing (E. Volodin 2012, personal communication).

## 4. Constraining the transient response to cumulative emissions with observations

While the TCRE of climate models is a useful summary metric that describes their response to CO_{2} emissions, the most relevant quantity for policymakers and other stakeholders is the TCRE of the real climate system. Because TCRE depends both on the carbon cycle and on the physical climate system, constraining it with observations requires both carbon cycle and temperature observations. TCRE may be constrained from observations in various ways (Allen et al. 2009; Matthews et al. 2009), but perhaps the most direct approach is to first use surface temperature observations to derive an estimate of CO_{2}-attributable warming to the present and then divide this by an estimate of cumulative CO_{2} emissions (Matthews et al. 2009). Estimating TCRE in this way relies on the assumption that TCRE is constant as a function of cumulative emissions between present day cumulative emissions (~0.5 EgC) and cumulative emissions at CO_{2} doubling (~1 EgC). Figure 1 shows that this is a reasonable assumption, though it suggests that TCRE calculated in this way may be biased high by ~10%. Note that we estimate CO_{2}-attributable warming using a fit to 150 yr of temperature observations rather than directly from a single annual mean, making our estimate subject to less noise than those shown in Fig. 1 for a 0.5 EgC cumulative emission. This method also relies on the assumption that the actual increase in atmospheric CO_{2} is the same as that which would have occurred had CO_{2} emissions been the only climate forcing present. Because the historical increases in radiative forcings associated with aerosols and non-CO_{2} greenhouse gases have approximately cancelled each other out (Forster et al. 2007) and large trends in volcanic and solar forcings have not occurred over this period, CO_{2} concentration changes caused by the temperature response to non-CO_{2} forcings are probably small. Although we include an estimate of land use change emissions and its uncertainty range in our analysis, calculating land use change emissions remains a challenge (e.g., Arora and Boer 2010). Anthropogenic land use change not only leads to direct emissions but also reduces the capacity of the terrestrial biosphere to sequester carbon because forests are replaced by croplands, which are frequently harvested, an effect not accounted for here (the 1PCT experiments we use are performed with vegetation from circa 1850). Although our approach relies on several assumptions, it has the virtue of relative simplicity, and it does not depend, for example, on fitting a particular carbon cycle model to observations.

_{2}-attributable warming. We do not attempt to constrain CO

_{2}-attributable warming directly in a detection and attribution analysis because the temperature response to CO

_{2}is unlikely to be sufficiently distinct from the temperature response to other anthropogenic greenhouse gases. An alternative approach is therefore to scale simulated 2010 CO

_{2}-induced warming up or down based on a regression coefficient for the overall greenhouse gas response. Unfortunately, simulations with only historical CO

_{2}variations are available for just one model (CanESM2), although CO

_{2}-only 1PCT simulations are available for all models considered here, allowing an observationally constrained multimodel estimate of TCR to be derived. This may then be scaled by the ratio of 2010 CO

_{2}-induced warming to TCR in CanESM2, a ratio that is likely to be relatively model independent, because both simulations are forced only with monotonically increasing atmospheric CO

_{2}. Using this method we obtain an observationally constrained estimate of TCRE:where

_{2}changes only,

_{2}to 2010. Relative to a corresponding simpler expression using simulations of CanESM2 only

We start by estimating

Although a standard residual test indicated regression residuals were consistent with simulated internal variability in all cases (Allen and Tett 1999; Allen and Stott 2003), Ribes and Terray (2013) argued that this test may be too permissive. We therefore also tested the consistency of the residuals using an alternate approach. First, the standard regression was carried out with one ALL ensemble member withheld, and the residual sum of squares was calculated. Second, the observations were substituted with the withheld ALL ensemble member, and the procedure was repeated. The latter step was repeated using each ALL ensemble member in turn. The residual sum of squares from the regression of the observations was then compared with the sample of residual sums of squares for each ALL ensemble member using a Student's *t* test. Such an approach tests whether the fit to observations is significantly worse than the fit to a model's own ALL simulations. Such an exercise indicated that residual observed variability was too large at the 5% level for regressions using CanESM2, CNRM-CM5, GISS-E2-H, and GISS-E2-R but consistent for the other models.

The results of a simple multimodel analysis (Multi; Gillett et al. 2002) are compared with a multimodel analysis in which the effects of model uncertainty are estimated based on intermodel differences in response [error in variables (EIV); Huntingford et al. 2006; Fig. 4a]. The residual test described above indicated consistent residuals for the multimodel analysis. Unlike in most previous implementations of EIV (Hegerl et al. 2007; Huntingford et al. 2006), in this case model uncertainty considerably inflates the uncertainties in the regression coefficients, perhaps because of large intermodel differences in the OTH response. EIV results were found to be sensitive to the normalization applied to the intermodel covariance matrix (Huntingford et al. 2006): in this case, we weighted this matrix by an estimate of the ratio of the variance attributable to internal variability in the simulated response patterns to that in the observations [

Observational uncertainty is a further source of uncertainty not usually accounted for in detection and attribution analyses (e.g., Hegerl et al. 2007). However, HadCRUT4 consists of an ensemble of 100 realizations that sample over observational uncertainty. We therefore repeated the EIV analysis using each of these realizations (Fig. 5a). The contribution of observational uncertainty was added in quadrature to the calculated regression coefficient uncertainties to give an overall confidence range (bars labeled ObsU in Fig. 4a). While the contribution of observational uncertainty is not negligible, the GHG response was detected using all 100 realizations of HadCRUT4, and observational uncertainty was found to make a smaller contribution to the regression coefficient uncertainty than internal variability.

As might be expected, estimates of GHG-attributable warming (obtained by scaling the simulated GHG-induced warming trend by the GHG regression coefficient) are relatively close in most of the attribution analyses described (Fig. 4b), with the exception of CSIRO Mk3.6.0, for which it is higher, and GISS-E2-H, for which this quantity is not well constrained. An observationally constrained estimate of TCR may also be derived for each model by multiplying its TCR by its GHG regression coefficient (Allen et al. 2000; Hegerl et al. 2007; black bars in Fig. 4b). The assumption is usually made that a model's TCR is proportional to its GHG-induced warming trend over the historical period (Allen et al. 2000; Hegerl et al. 2007). The fact that observationally constrained estimates of GHG-attributable warming derived using HadGEM2-ES, IPSL-CM5A-LR, and NorESM1-M are very similar while the corresponding estimates of TCR are different (Fig. 4b) suggests that this assumption is violated for these models. Because we wish to derive a multimodel estimate of TCR, it is important to investigate this issue further.

Figure 6 shows the TCR of 16 CMIP5 models plotted against the 1861–2005 warming trend in the GHG-only simulations of these models (we use the period to 2005 in order to include additional models for which simulations finished in 2005). Deviations from a linear relationship are larger than can be explained purely by uncertainties in the GHG-induced trend and TCR associated with internal variability (as shown by the uncertainty bars). Deviations from a linear relationship are not explained by the inclusion of tropospheric ozone changes in some models' GHG-only simulations (indicated by a diagonal cross in Fig. 6), because the models exhibiting a relatively high ratio of GHG-induced warming to TCR are not all those that include ozone forcing. The most likely explanation is that the temperature response to the non-CO_{2} GHGs is not proportional to the temperature response to CO_{2} in different models, either because of different non-CO_{2} GHG radiative forcings in different models or different efficacies of the non-CO_{2} GHGs in different models (the efficacy of a greenhouse gas is the ratio of the temperature response per unit change in radiative forcing due to that gas to the temperature change per unit radiative forcing due to CO_{2}). This is consistent with Forster et al. (2007) who show considerable uncertainty in the efficacy of the non-CO_{2} greenhouse gases. This is thus an important source of uncertainty, which we must consider when we use the multimodel GHG regression coefficient to scale the multimodel TCR. Matthews et al. (2009) attempted to account for this source of uncertainty by using a published uncertainty in the efficacies of non-CO_{2} GHGs (Forster et al. 2007). We account for this uncertainty in our analysis by using the spread of the ratio of TCR to GHG-induced warming in the 16 models included in Fig. 6. Specifically, the fractional uncertainty in

The final elements required for our estimate of TCRE are first the CO_{2}-induced warming in 2010 in CanESM2, which was derived from the 2008–12 temperature anomalies in an ensemble of five simulations of CanESM2 with historical variations in CO_{2} only, relative to its preindustrial control simulation. Second, we require an estimate of anthropogenic CO_{2} emissions. Between 1750 and 2010 burning fossil fuels and cement production has resulted in the emission of 357 ± 22 PgC (Boden et al. 2012), while net land use change emissions have resulted in 160 ± 40 PgC (Friedlingstein et al. 2010; Houghton 2008), where a 2*σ* uncertainty range of ±50% has been assumed. This results in a total estimate of 517 ± 46 PgC (uncertainties here are one standard deviation).

Figure 7a shows an observationally constrained multimodel estimate of TCR derived by multiplying the EIV GHG regression coefficient shown in Fig. 4a by the mean TCR in BCC_CSM1.1, CanESM2, CNRM-CM5, CSIRO Mk3.6.0, GISS-E2-H, GISS-E2-R, HadGEM2-ES, IPSL-CM5A-LR, and NorESM1-M (1.9 K). Contributions to the uncertainty in TCR from the uncertainty in the GHG regression coefficient (Fig. 4a), a component caused by observational uncertainty, and the uncertainty in the relationship between TCR and GHG-induced warming, which is assessed directly from the spread of this ratio across models (Fig. 6), are added in quadrature. The relative contributions to the uncertainty are shown in Fig. 7b. The calculated 5%–95% confidence range for TCR of 0.9–2.3 K lies mainly within the 1.0–3.5-K range reported by Hegerl et al. (2007) based on the assessment of previous such analyses and is somewhat narrower than the range of TCR in the models used here (Fig. 7a). It is broader than that determined by Gillett et al. (2012) using a single model, likely first due to our taking account of model uncertainty in the derivation of the regression coefficients and second due to our accounting for uncertainty in the radiative forcings and efficacies.

Last, we apply Eq. (2) to calculate an estimate of TCRE, scaling the multimodel estimate of TCR by the ratio of 2010 warming in a CO_{2}-only simulation of CanESM2 to that model's TCR and dividing by cumulative emissions to 2010. Both the historical CO_{2}-only and 1PCT simulations from CanESM2 contain only CO_{2} forcing so this ratio is not subject to the uncertainty associated with the relative effects of the different GHGs, and we assume its uncertainty is solely due to internal variability, which is assessed from a control simulation. It is difficult to test this assumption without historical CO_{2}-only simulations from more models. We may place an upper bound on this contribution to the uncertainty by examining variations in the ratio of warming at year 70 in the abrupt 4×CO_{2} simulations compared with each model's TCR. The standard deviation in this ratio is ~0.11 of its mean value and assuming a similar spread in the ratio of CO_{2}-only historical warming to TCR, our uncertainty estimate of TCRE is inflated by ~8%. This undoubtedly strongly overestimates the true uncertainty in this ratio because the forcing variations in the 1PCT simulation are much closer to those in a historical CO_{2}-only simulation than they are to the 4×CO_{2} simulations; therefore, multimodel variations in this ratio are likely very much smaller, and we disregard them for the present. In this way, we obtain an estimate of CO_{2}-attributable warming in 2010 relative to preindustrial values of 0.4–1.0 K. Dividing by the estimate of historical cumulative emissions just described gives an estimate of TCRE of 0.7–2.0 K EgC^{−1} (best estimate of 1.3 K EgC^{−1}; Fig. 7) (If we use the standard multimodel regression coefficients rather than the EIV regression coefficients, we obtain a slightly narrower range of 0.8–1.9 K EgC^{−1}.) The uncertainty range is derived by adding in quadrature uncertainties corresponding to the GHG regression coefficient (Fig. 4a), the ratio of TCR to GHG-induced warming (Fig. 6), temperature observations, the cumulative CO_{2} emissions, CanESM2 2008–12 warming, and CanESM2 TCR. Figure 7b shows that the dominant contributions to the overall uncertainty are the uncertainty in the GHG regression coefficient and the uncertainty in the ratio of TCR to GHG-induced warming. This observationally constrained 5%–95% confidence range for TCRE is somewhat narrower than the range spanned by the models themselves, as shown in Fig. 7a, but slightly broader than the observationally constrained range of 1.0–2.1 K EgC^{−1} derived by Matthews et al. (2009). Matthews et al. (2009) derived their estimate of TCRE based on an estimate of CO_{2}-attributable warming and cumulative emissions between the first and last decades of the twentieth century. Their estimate of CO_{2}-attributable warming was in turn based on an estimate of GHG-attributable warming scaled by a ratio of CO_{2} to all GHG forcing and also scaled to account for the efficacies of the non-CO_{2} GHGs. Our approach uses warming and emissions from preindustrial to present and more fully accounts for uncertainty in forcing and response to non-CO_{2} GHGs based on a broader sample of models. Assuming negligible zero emissions commitment, Allen et al. (2009) calculate a value of TCRE of 1.4–2.5 K EgC^{−1}, a somewhat narrower and higher uncertainty range than the range we calculate here. Note that our wider uncertainty range may be explained by our consideration of the uncertainty in the relationship between the temperature response to CO_{2} and the temperature response to all GHGs, a term not considered by Allen et al. (2009).

## 5. Conclusions

The TCRE of 15 CMIP5 models is found to range from 0.8 to 2.4 K EgC^{−1} (interpretable as a 5%–95% range), a somewhat broader range than that of the C^{4}MIP models, whose TCREs range from 1.0 to 2.1 K EgC^{−1} (Matthews et al. 2009), though this difference is not statistically significant. Two CMIP5 models with high TCRE (CanESM2 and MIROC-ESM) have a parameterized limitation on the carbon fertilization effect of the land biosphere, approximating the effect of nitrogen limitation, along with relatively high values of TCR. Both physical climate properties and carbon cycle properties contribute to variations in TCRE across these CMIP5 models, though physical climate properties are the biggest contributor to TCRE variations.

In the second part of the study, the CMIP5 models were used together with updated temperature observations and emissions data to derive an observationally constrained estimate of TCRE. A multimodel attribution analysis applied using a subset of nine CMIP5 models for which the necessary simulations were available indicated a detectable influence of greenhouse gas changes and natural forcings over the 1861–2010 period. Model uncertainty was found to substantially inflate the uncertainty in the regression coefficients. In previous studies deriving observationally constrained estimates of TCR, simulated model-mean TCR has typically been assumed to scale linearly with the response to all GHGs. We find significant deviations from such a linear relationship in a subset of 16 CMIP5 models, likely because the radiative forcings of the various GHGs and the efficacies of the non-CO_{2} GHGs differ between models (Forster et al. 2007). Accounting for this source of uncertainty, together with the uncertainty in the GHG regression coefficient and observational uncertainty, we obtain an estimate of TCR of 0.9–2.3 K. Using simulations from CanESM2 of the response to historical CO_{2} emissions alone, we scale this estimate to determine an observationally constrained estimate of CO_{2}-attributable warming to 2010 and divide this by an estimate of CO_{2} emissions caused by fossil fuel burning and land use change to derive TCRE. Our derived 5%–95% confidence range of 0.7–2.0 K EgC^{−1} is broadly consistent with previous estimates of this metric and has an upper bound below the TCRE of four of the models for which we calculate it directly. This suggests that some of the models (BNU-ESM, CanESM2, MIROC-ESM, and HadGEM2-ES) may have too high a TCRE relative to observations. Overall, we give greater precedence to the observationally constrained range than to the model spread and conclude that TCRE is very likely (*P* > 90%) between 0.7 and 2.0 K EgC^{−1}. As the observational record lengthens and the relative importance of forcings other than CO_{2} declines, it will be possible to more closely constrain TCRE.

We thank BCC and IPSL for supplying us with BCC_CSM1.1 and IPSL-CM5A-LR land–atmosphere carbon flux data directly since the necessary data were either not available on the CMIP5 archive or were corrupted.

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