A key question in climate science is how the global mean surface temperature (GMST) responds to changes in greenhouse gases or other forcings. Climate sensitivity is determined by complex feedbacks that operate on very different timescales and may depend on the transient climate state. The standard metric for climate sensitivity is the equilibrium climate sensitivity (ECS) (or Charney sensitivity) given as the change in temperature at equilibrium for a doubling of CO2, neglecting long-term feedbacks associated with vegetation changes, carbon feedbacks and ice sheet dynamics. Estimates of the ECS are either based on complex climate models or observations of past climate (Collins et al., 2013; Knutti et al., 2017). The Intergovernmental Panel on Climate Change (IPCC) presented a likely (> 66 % probability) range for ECS of 1.5 to 4.5 ∘C (Collins et al., 2013).

Regarding the Earth as a climate laboratory and the changes in atmospheric composition and land use over the historical record as a perturbation experiment, observationally based analyses of Earth's energy budget have been used to infer the climate sensitivity (Forster, 2016). Since the current climate is in a nonequilibrium state, observationally based methods can only account for the feedbacks operating during the historical period. These methods using the historical period with observations are referred to as inferred estimates (Armour, 2017; Forster, 2016) and only have the capability to derive an effective climate sensitivity and are generally significantly lower than ECS estimates from atmosphere–ocean general circulation models (AOGCMs) (Armour, 2017; Knutti et al., 2017).

Since the IPCC's fifth assessment report (AR5), there has been an improved understanding of the causes of the differences in estimates of climate sensitivity from climate models and observationally based methods for two main reasons. First, recent analysis of time-varying feedbacks in AOGCM simulations from Coupled Model Intercomparison Project Phase 5 (CMIP5) (Proistosescu and Huybers, 2017; Armour, 2017; Andrews et al., 2015) has indicated that in most AOGCMs the net feedbacks become more positive over time as a new equilibrium is approached. This is most likely due to the evolution of the pattern of sea surface temperature increase in the Pacific and Southern oceans and associated cloud feedbacks. Whether this slow warming has manifested itself in the climate record used for the analysis is the difference between effective and equilibrium climate sensitivity (Armour, 2017; Knutti et al., 2017). Second, ECS formally refers to global near-surface air temperature (“tas” in CMIP5 nomenclature), and in observationally based methods observed surface temperature records that are a blend of air temperature over land and sea surface temperature (SST) over ocean are used in the estimation. Several observed surface temperature records exist with different methods to account for gaps in the observations. Differences in historical surface temperature warming among various analyses is more than 0.1 ∘C (Haustein et al., 2017), arising mainly from approaches taken in regions that are missing or have limited spatial coverage of observations. According to Richardson et al. (2016), there is a general bias in the surface temperature records since water heats more slowly than the air above and due to undersampling in fast-warming regions (e.g., the Arctic). Taking both effects into account, Armour (2017) shows that previous estimates of inferred effective climate sensitivity (ECSinf) of about 2.0 ∘C are consistent with estimates of ECS of 2.9 ∘C from climate models.

Although it is now established that the ECS estimated by the use of complex climate models and ECSinf estimated by using historical observations would differ, there is still considerable spread in ECS estimates from models and between observationally based ECSinf estimates. Using observationally based methods and complex models are complementary approaches to quantifying the net effect of the feedbacks that determine the climate sensitivity. Complex climate models include processes that are highly parameterized, in particular the representation of clouds, precipitation and convection, and associated feedbacks, which are crucial for estimating the ECS (Bony et al., 2015; Tan et al., 2016). There is also a large spread in observationally based estimates (Knutti et al., 2017). A better understanding of the feedbacks in the complex models as well as improvements and understanding differences among the observationally based methods are needed.

Observational estimates of climate sensitivity can be improved using longer data series of higher quality (e.g., correcting for observational biases in temperatures or better forcing estimates) (Urban et al., 2014). Estimates can also be improved by including observational data on other climate variables which were not previously available. Several studies indicate that the temporary slowdown in GMST at the beginning of the millennium coexisted with increased accumulation of heat in the deep ocean (e.g., Meehl et al., 2011, 2013; Balmaseda et al., 2013; Watanabe et al., 2013; Chen and Tung, 2014; Lyman and Johnson, 2013). Johansson et al. (2015) found that if ocean heat content (OHC) change below 700 m over this period were included in their observationally based methods, the mean value of ECSinf would increase.

In this study we use our estimation model that was first documented in Aldrin et al. (2012) and further developed in Skeie et al. (2014). Our method is more complex than the common energy-balance-based estimates (Forster, 2016) in that we embed a simple climate model into a stochastic model with radiative forcing time series as input, treating the Northern and Southern Hemisphere (NH and SH) separately and including a vertical resolution of the ocean (40 layers). The radiative forcing time series are linked to the observations of OHC and temperature change through the simple climate model and the stochastic model, using a Bayesian approach. A unique feature with our method is that we use several observational datasets. The method estimates not only the ECSinf but simultaneously also provides posterior estimates of the radiative forcing, as well as posterior uncertainty estimates in the observations datasets and correlations between them. In this study we further develop our estimation model with additional observational datasets, including heating rates of the deep ocean (below 700 m), new forcing time series from the IPCC AR5 as well as extended time series from 2010 to 2014 to update our estimate of ECSinf. We carry out a number of sensitivity experiments to investigate causes of differences in observationally based ECSinf estimates due to differences in the input data (observations of surface temperature, OHC and radiative forcing (RF)).

Here we present our revised estimate of ECSinf by replacing the RF prior with IPCC data, including OHC data below 700 m and extending the time series to 2014 (case B). We consider this analysis using the IPCC forcing estimates, including deep-ocean OHC and extending the length of the input data series as the most trustworthy and physically based case and thus regard it as our main estimate of the ECSinf, with a mean of 2.0 ∘C (median 1.9 ∘C, 90 % CI 1.2–3.1 ∘C). The mean value is similar while the 90 % CI is narrower compared to the refined Skeie14 estimate (Fig. 1a). The individual influence of the four major updates between cases A and B is shown in Fig. S1 and described at the end of this section. The mean value of TCR in case B is 1.4 ∘C (median 1.3 ∘C, 90 % CI 0.9–2.0 ∘C) (Fig. 1b). As for the ECSinf estimate, the TCR mean value is similar and the 90 % CI is narrower compared to the refined Skeie14 estimate (Fig. 1b). The GMST change is well reproduced (Fig. 2, case B), and less of the recent GMST change is attributed to long-term internal variability compared to the refined Skeie14 estimate (Fig. S5a–b).

The rate of change in anthropogenic forcing is larger between 1940 and 1970 using Forc_AR5 compared to Forc_Skeie14 (Fig. 3). The fit to the GMST in the 1980s–1990s improved (Fig. 2 case B vs. A), where the root mean square error between 1980 and 1999 decreased from 0.12 to 0.077 ∘C. Figure S5 shows posterior estimates of the long-term internal variability, the ENSO term and the model errors. Parts of the increase in GMST over the last decades are explained as long-term internal variability, but the amplitude decreases in case B compared to case A (Fig. S5a–b). In case B, the estimated amplitude of the multi-decadal internal variability (about 0.2 ∘C in each hemisphere, cf. Fig. S5) is in good agreement with the decadal trends in global surface temperatures found in unforced control simulations in the multi-model ensemble from CMIP5 (0.2–0.4 ∘C; Palmer and McNeall, 2014).

The prior anthropogenic mean forcing in 2010 increased from 1.5 to 2.3 W m-2 from case A to case B when Forc_AR5 replaced Forc_Skeie14. For case A, the posterior forcing is shifted to higher values compared to the prior, suggesting that the historical data and our method support higher forcing than the Forc_Skeie14 prior. When the prior is changed to Forc_AR5 in case B, the posterior for the anthropogenic forcing is much closer to the prior (Fig. 3), which indicates that the method and observational data are more in accordance with the new prior than the old one. The same holds for the total forcing (Fig. S4). The 90 % CI for the posterior anthropogenic forcing was 1.3 to 2.8 W m-2 in case A compared to 1.3 to 3.4 W m-2 in case B. The upper limit of the 90 % CI is shifted to larger values. The most uncertain part of the forcing time series is associated with aerosols. The difference between the two forcing priors is mainly due to a much weaker aerosol forcing in Forc_AR5 than in Forc_Skeie14 (compare the two dashed–dotted bars in Fig. 4a). While the posterior aerosol forcing was shifted to smaller negative values in case A, the prior and posterior for aerosol forcing are similar in case B (Fig. 4b). A relatively weak aerosol–cloud interaction as included in Forc_AR5 is consistent with the recent findings in Malavelle et al. (2017) on how sulfate aerosols from volcanic emissions influences clouds.

The ERFs in AR5 are based on an assessment of several studies reflecting improved knowledge of the forcing mechanisms compared to the one-model RF results used in Skeie14. The new ERFs gave a better posterior estimate of GMST (Fig. 2) and reduced change from prior to posterior forcing (Fig. 3). Note that the number of forcing time series that can be combined was 18 in Skeie14, including 3 time series for volcanic and 8 for aerosols, compared to only 1 time series for each of these forcing mechanisms in Forc_AR5 (Table S2). This gives less flexibility in the time development of the forcing in case B compared to case A; however, the GMST change is better reproduced in the 1980s–1990s using Forc_AR5 compared to Forc_Skeie14.

Ultimately, global climate change is governed by the radiative imbalance at the top of the atmosphere (TOA) and modulated by the internal variability. Forcing by greenhouse gases and aerosols as well as albedo changes, feedback processes and the radiative responses to temperature changes determine this imbalance. With a positive net imbalance at TOA, energy accumulates in the Earth system, mainly as increasing OHC (Church et al., 2011). Since OHC is the dominant energy storage in the system, these data series have profound influence on the ECSinf estimates (Tomassini et al., 2007; Skeie et al., 2014; Aldrin et al., 2012; Johansson et al., 2015). In case B, we have extended our use of OHC data, so in addition to the three OHC data series above 700 m, we have included the ORAS4 data above and below 700 m (Table 1) as two separate data sources. Including the deep-ocean OHC data gives a stronger constraint on the overall accumulation of heat in the system, and the posterior estimates of the parameters of θ that determine the vertical transport of heat in the ocean – the effective diffusivity and the upwelling velocity – increase by 44 and 31 %, respectively. Having separate data series for the two ocean layers also provides information that influences the balance between negative (by aerosols) and positive forcings, since these forcings have different evolution over time (cf. Sect. 4.1).

In Fig. 5 the observed and fitted OHC for cases A and B are shown. Including data on OHC change below 700 m increases the total heat uptake. The increase in the fitted OHC above 700 m over the last decade is larger in case B compared to case A. In case B the increase in the fitted OHC above 700 m is larger than the observational data, while below 700 m, the observed OHC increase is higher than the fitted one (Fig. 5). This is to be expected since the parameters of θ do not change over time. Thus, the observed rapid change in OHC below 700 m over the last years with corresponding slower warming above 700 m is attributed to long-term internal variability (a part of the nt term) in the model (Fig. S5c–d). Note that the Ishii and Kimoto series is outside the 90 % CI. The reason is that the assumed observational errors for all series are much larger back in time than in the recent years (see Appendix A). Therefore, the various data series are aligned quite closely to each other in recent years, and since the Ishii and Kimoto series has a much weaker trend than the others, it lies above the 90 % CI in the first part of the data history.

The update of the ECSinf from case A to B was done stepwise in four steps (Fig. S1f, g, i and j). The new ERFs were first implemented. The posterior forcing is much closer to the prior using Forc_AR5 instead of Forc_Skeie14, and the fit to the GMST in the 1980s–1990s also improved with a decrease in the root mean square error between 1980 and 1999 from 0.12 to 0.087 ∘C compared to case A. The stronger forcing resulted in a shift of the ECSinf estimate to lower values (Fig. S1f vs. e), with an ECSinf mean value of 1.5 ∘C (90 % CI 0.9–2.3 ∘C). So far, only OHC data in the upper 700 m were used, leaving the model unconstrained with respect to the heating of the deeper ocean.

We then included the ORAS4 data above and below 700 m as two separate data sources. Similar to Johansson et al. (2015) we found that including the OHC change below 700 m increases the total heat uptake and thus the mean value of ECSinf from 1.5 to 1.7 ∘C (Fig. S1g vs. f). The 90 % CI shifted to larger values ranging from 1.0 to 2.8 ∘C.

The last two steps to update the ECSinf estimate from case A to case B were to use the most recent version of the data prior to 2010 and to extend the data series used from 2010 to 2014 (Table 1). Some of the observational data series have been updated by the data suppliers, so first we use refined data up to 2010 before we extend the data series to 2014 (cf. Appendix B). Using the refined data up to 2010, the estimated mean ECSinf increased from 1.7 to 2.0 ∘C (Fig. S1i) and the 90 % CI was shifted again to larger values ranging from 1.1 to 3.3 ∘C. Further, when the data series were extended from 2010 to 2014, the upper bound of the 90 % CI decreased from 3.3 to 3.1 ∘C while the lower bound remained unchanged and the mean estimate slightly reduced (Fig. S1j).

In total, the changing from case A to case B did not change the mean value of ECSinf (it is 2.0 ∘C in both cases), but the 90 % CI was reduced from 1.0–3.4 to 1.2–3.1 ∘C. The reduction in ECSinf in the first step of the update is more or less counteracted by the subsequent steps.

We now investigate possible causes of differences in observationally based ECSinf estimates due to the use of input data. We analyze the impacts of different usage of the OHC data (cases C and D) and the treatment of uncertainties in the GMST data (case E).

Causes of differences in observationally based estimates of ECSinf due to the use of input data are analyzed, and an updated ECSinf estimate is presented using our Bayesian estimation model. Adding observational data from 2011 to 2014 and OHC data below 700 m and replacing forcing data with IPCC AR5 ERFs, the ECSinf posterior mean was 2.0 ∘C (median 1.9 ∘C, 90 % CI 1.2–3.1 ∘C). The mean value is similar and the range is slightly narrower than the refined Skeie14 estimated (Fig. 1 case B vs. A). The mean ECSinf estimate is larger than in Skeie14. Although the estimate in cases A and B is quite similar, the ECSinf estimate shifted to lower values when Forc_AR5 replaced Forc_Skeie14 (from a mean ECSinf estimate of 2.0 to 1.5 ∘C), and it shifted to larger values when OHC data below 700 m were included (to a mean ECSinf value of 1.7 ∘C). The ECSinf estimate was very sensitive to the forcing data used, and we showed that the ECSinf estimate was also sensitive to the assumed uncertainties in the GMST data (case E: ECSinf mean value increased from 2.0 to 2.2 ∘C) and how the OHC data were treated (cases C and D, with mean ECSinf of 3.2 and 2.8 ∘C, respectively).

Bayesian methods have recently been reviewed by Annan (2015) and Bodman and Jones (2016), and limitation by assuming constant sensitivity over time, the role of the ECSinf prior distribution and equal efficacy for different forcings have been discussed. Implementing an alternative prior for ECSinf as in Skeie14, where 1 / ECSinf is uniformly distributed, shifted the mean ECSinf to lower values from 2.0 ∘C (median 1.9 ∘C, 90 % CI 1.2–3.1 ∘C) to 1.6 ∘C (median 1.6 ∘C, 90 % CI 0.97–2.5 ∘C). The ECSinf estimate is sensitive to the prior; however, one could argue against this alternative prior because it has a high probability of low climate sensitivities that may not be realistic, with 76 % probability for ECSinf being lower than the pure black-body radiation sensitivity of 1.1 ∘C (Aldrin et al., 2012; Skeie et al., 2014). Recently, studies have suggested that assuming equal efficacy for all forcings biases the ECS estimate low (Marvel et al., 2015; Shindell et al., 2015) even when ERFs are used. In our approach, the efficacy is implicitly included in the forcing uncertainty and thus accounted for. However, if we apply an efficacy of 1.5 for ozone, surface albedo, BC on snow and aerosols, which is the efficacy found in the analysis of Shindell (2014), the probability density function of the ECS is shifted to larger values (Fig. S1l), with a 90 % CI ranging from 1.2 to 3.7 ∘C.

The fit to the temperature data in the 1980s and 1990s improved using Forc_AR5 instead of Forc_Skeie14, indicating that the forcing trend over this period is better represented in Forc_AR5 compared to Forc_Skeie14. The trend in the forcing is more uncertain in the first half of the 20th century due to less dominance of CO2, and in our method the same relative uncertainty for the prior forcing is used over the entire time period. A sensitivity simulation omitting observations before 1950, similar to making these observations very uncertain, gave better representation of the GMST in the latter part of the 20th century and an increased mean ECSinf. Future work should include uncertainties in the temporal development of the forcing, and there is a clear need for an international effort to establish forcing time series, using a consistent forcing definition and allowing for uncertainties in emissions to give a better representation of the temporal uncertainties.

Including OHC data below 700 m shifted the ECSinf to higher values. The estimated ECSinf was found to be very sensitive to how the OHC data were used. Including four OHC time series but merging the data above and below 700 m (case C), the ECSinf mean value increased from 2.0 to 3.2 ∘C. The probability of ECSinf above 4.5 ∘C increased to 13 %, values that are practically excluded in our main estimate (case B). Previous studies have used total-column OHC data, and due to the simple representation of the ocean one can argue that this might be more appropriate. However, in case C most of the recent increase in OHC in the model occurred in the uppermost 700 m, allowing a stronger aerosol cooling (Fig. 4a) and hence a larger ECSinf, while the observations indicate that the ocean was warming mainly below 700 m. Using only the total-column OHC might therefore overestimate the aerosol forcing strength and hence the ECSinf. We recognize structural uncertainties in the model, and a multi-model intercomparison of observational methods using identical input data would be of great value to investigate these uncertainties.

Using only the Levitus2000 series for OHC for the total-ocean column (case D), the ECSinf 90 % CI was shifted to lower values with a range of 1.5–4.6 ∘C and the range shrunk compared to case C. The historical measurements of ocean temperatures are sparse (Abraham et al., 2013), with large differences between the datasets. The temporal structure of the reported uncertainties differs, and the full uncertainties are often not assessed. Hence, relying on only one OHC series and its reported uncertainty may underestimate the observational uncertainties and hence overestimate the certainties in the estimated OHC with implications for the ECSinf estimate.

Recent studies indicate that the upper-ocean warming is underestimated due to the gap-filling methods (Durack et al., 2014; Li-Jing et al., 2015), in which case the ECSinf will also be underestimated. When refining historical OHC estimates, not only the best value, but also the uncertainty is crucial for observationally based ECSinf estimation.

Other priorities are to improve the GMST series, including uncertainties – not only for the recent trend (Karl et al., 2015; Cowtan and Way, 2014) but also for earlier time periods. Assuming a very large uncertainty prior to 1950, the GMST fit improved and the ECSinf mean increased while the estimated uncertainty ranges increased.

Our ECSinf posterior mean was 2.0 ∘C with a 90 % CI of 1.2 to 3.1 ∘C. This is consistent with a mean ECS of 2.9 ∘C (Armour, 2017), which compares reasonably well with climate model estimates (Andrews et al., 2012; Forster et al., 2013). A final remark is that it is not obvious that the true ECS is a more relevant metric for the climate sensitivity than the ECSinf in a policy context (i.e., the Paris Agreement). The United Nations Framework Convention on Climate Change (UNFCCC) has not adopted a predefined definition of GMST and the stronger long-term feedbacks found in an analysis of CMIP5 simulations (Proistosescu and Huybers, 2017) operate on a timescale longer than the timescale for reaching 2 ∘C.