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The hiatus has been attributed to a range of causes, including a reduction in solar forcing⁴, a La Niña-like cooling of the tropical Pacific Ocean with an associated increase in Pacific Ocean heat uptake^{4, 5, 6, 7}, increased ocean heat uptake in the Atlantic Ocean and Southern Ocean⁸, volcanic aerosols^{4, 9} and anthropogenic aerosols¹⁰. Further, studies estimating ECS based on simple climate models and observations extending over the hiatus period have suggested an ECS at the lower end of the likely range given in IPCC AR5 (refs 1, 2, 11, 12). However, disentangling the roles of potential causes of the hiatus from climate system properties such as the ECS is complicated by data and model limitations.

In this study we provide a new estimate of ECS, analyse the effect observations over the hiatus have had on it, and estimate the relative contribution of various factors to the temperature trend during the hiatus. Our analysis differs from previous methodologically related statistical estimates of ECS in that we distinguish observations of global mean near-land surface temperature (GMLST) from those of global mean sea surface temperature (GMSST) and use ocean heat content (OHC) observations continuous over time to a greater depth (2,000m instead of 700 m), a potentially important addition as heat accumulation during the hiatus is thought to be particularly strong at depths below 700m (refs 5, 8, 13). We also consider surface temperature variability induced by the El Niño/Southern Oscillation (ENSO; ref. 14).

By using an energy balance model and a Bayesian approach to statistics we assess how estimates of ECS change with the accumulation of historical observations, similar in some respects to studies focusing on learning about ECS over time^{11, 15, 16, 17}. These studies have primarily focused on the pace of learning about ECS over time in rather general terms whereas we focus on how the shape of the probability density function (PDF) of ECS changes over time as observations accumulate to analyse implications of the hiatus for estimates of ECS. This is done by progressively extending the time horizon when estimating the PDF: the model integrations all start in 1765, but end in 1986, 1991, 1996, 2001, 2006, or 2011.

The model simulations based on parameters sampled from the joint posterior PDF from the full observational history up to 2011 replicate well the observed surface temperature history, including the warming hiatus since the early 2000s (Fig. 1). The mean modelled global average surface air temperature (SAT; ref. 18) exhibits a correlation coefficient of 0.95 with observations over the period 1880–2011. As expected, the relative fit of the model to the observations improves after 1950, when observational uncertainties begin to decline significantly (see also Supplementary Figs 3 and 4).

Mean and 90% uncertainty intervals are used to characterize model output. All time series are normalized to the period 1951–1980. a–d, The SAT anomaly (a) is decomposed into four different contributing factors; NINO3.4 (b), solar irradiance (c), volcanic aerosols (d) and anthropogenic warming (e). The legend in a applies to all panels.

To model forced temperature response we use a land–ocean resolved upwelling diffusion energy balance model (UDEBM). The model, and the use of such models in statistical modelling, is described at greater length in Supplementary Section 1. The UDEBM is forced by estimates of effective radiative forcing (RF) over the period 1765–2011, based on ref. 24 for anthropogenic sources (http://www.pik-potsdam.de/~mmalte/rcps/), ref. 25 for Solar RF, and refs 26, 27 for volcanic aerosol RF (http://hurricane.ncdc.noaa.gov/pls/paleox/f?p=519:1:0::::P1_STUDY_ID:14168 and http://data.giss.nasa.gov/modelforce/strataer, respectively).

Posterior PDFs are calculated based on Bayes theorem

where p(θ) is the prior probability density for parameters θ, L(y|θ) is the likelihood function, and p(θ|y) is the posterior probability density for the parameters θ conditional on observations y. The likelihood function is the probability density of the observations y conditional on the parameters θ; for our application, it is a measure of how well the model with specific parameter combinations replicates the observations, accounting for internal variability and observational error. The statistical methodology is explained in more detail in Supplementary Section 1.

The model is constrained by observations on GMSST and GMLST (ref. 18) (http://www.nature.com/nclimate/journal/v5/n5/full/ftp://ftp.ncdc.noaa.gov/pub/data/mlost/operational/products) and OHC above 2,000 m (ref. 28) (http://www.nodc.noaa.gov/OC5/3M_HEAT_CONTENT).

The UDEBM does not generate internal variability. To compensate for this the variability induced by ENSO is captured by making the modelled GMSST and GMLST dependent on the NINO3.4 index lagged by two months and six months, respectively. The lag is estimated by maximizing the correlation between the residual of temperature observations minus the modelled temperature output (calculated without considering the global temperature impact of ENSO) and the lagged NINO3.4 index (Supplementary Section 1.8). The NINO3.4 index is based on the NOAA Extended Reconstructed Sea Surface Temperature (SST) V3b (http://climexp.knmi.nl/selectindex.cgi). Remaining natural variability in GMLST, GMSST and OHC is assumed to follow a first-order autoregressive (AR1) process, with the autocorrelation and variance estimated from the residuals between the modelled and observed variables (Supplementary Section 2.1–2.3).

The prior probability density distributions are either uniform, normal, truncated normal or lognormal. We consider the following EBM parameters to be probabilistic: climate sensitivity parameter, effective mixed layer depth, effective vertical diffusivity, upwelling rate, relative warming of sinking polar water to global mean sea surface warming and equilibrium land–sea surface warming ratio. The priors for all of these parameters are assumed to be uniformly distributed, with the exception of the equilibrium land–sea surface warming ratio, for which we assume a normal distribution based on output from climate system models². The prior scaling factors for relating the NINO3.4 index to GMSST and GMLST variability and the parameters of the AR(1) processes are uniform and set wide enough to not constrain the posterior estimate. The paths of the different effective radiative forcing contributions (for CO₂, CH₄, N₂O, tropospheric O₃, anthropogenic aerosols, solar, and volcanic aerosols) are fixed, but the magnitudes are scaled with probabilistically treated scaling factors. The prior distributions for scaling parameters for historical effective radiative forcing time series are assumed to be normal, truncated normal or lognormal, with standard deviations that approximately correspond to the effective radiative forcing uncertainty reported in IPCC AR5 (ref. 25). However, the prior scaling factor for volcanic aerosol forcing is based on ref. 29. The forcing estimates and their prior uncertainties are estimated from climate system models, these are primarily not constrained by observations on the global energy balance; hence there should be no major problems of circular reasoning. See Supplementary Section 1.7 for further information about priors.

The posterior PDF is estimated through the use of a Markov Chain Monte Carlo (MCMC) approach using the Metropolis algorithm³⁰. We take 500,000 samples when estimating the posterior PDF for each model set-up, with set-ups differing with regard to the end-year of the observational record used. The proposal distribution for the Metropolis algorithm is a random walk. A new sample for all the uncertain model parameters is taken simultaneously in each proposal. The first 20,000 samples of the posterior distribution are used as ‘burn-in’; that is, they are discarded, and only the subsequent MCMC samples are used to estimate the posterior distribution. Every twentieth sample of the chain is retained, and the PDFs presented in Fig. 2 are estimated from the samples by using a normally distributed kernel function.

The decomposition presented in Fig. 1, and the subsequent estimation of the linear temperature trend due to the different factors presented in Table 1, are constructed by running the UDEBM with only one specific forcer/mechanism at a time. The estimated temperature contribution for each forcer/mechanism is based on 1,000 samples of the posterior PDF estimated when using observations up to 2011. Running the model with only one specific forcer/mechanism at a time is valid because the forcing response in the UDEBM is linear. Hence, the sum of the temperature contributions of each forcer/mechanism is equal to the temperature response of the sum of the forcers/mechanisms.

Corrected online 14 April 2015

In the version of this Letter originally published, in the third paragraph, the section of text including ‘Our analysis differs... at depths below 700m (refs 5,8,13).’ was unclear and should have been:
'Our analysis differs from previous methodologically related statistical estimates of ECS in that we distinguish observations of global mean near-land surface temperature (GMLST) from those of global mean sea surface temperature (GMSST) and use ocean heat content (OHC) observations continuous over time to a greater depth (2,000m instead of 700 m), a potentially important addition as heat accumulation during the hiatus is thought to be particularly strong at depths below 700m (refs 5,8,13).'
Further, in the Methods section, radiative forcing should have been described as effective radiative forcing. These errors have been corrected in all versions of the Letter.

Corrected online 10 June 2015

In the version of this Letter originally published, ref. 9 was incorrectly cited twice; the sentence including the second occurrence should have read: "In contrast to other studies⁴, we find that the recovery from Mt Pinatubo is stronger than the cooling caused by the small volcanoes that has occurred since the early 2000s." This error has been corrected in the online versions of the Letter.

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