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2 Supplementary Information for Estimating the transient climate response from observed warming Andrew Schurer, Gabi Hegerl, Aurélien Ribes, Debbie Polson, Colin Morice, Simon Tett Additional details of encoding uncertainty into the observational ensemble timeseries Measurement and sampling uncertainties relating to land regions are encoded into ensemble time series by adding samples drawn from a Gaussian distribution with a zero mean and a standard deviation defined by the uncorrelated measurement and sampling uncertainty term provided in the HadCRUT4 time series files. Measurement and sampling uncertainties for ocean regions in HadCRUT4 exhibit correlation in time arising from uncorrected biases persisting in measurements from individual measurement platforms (Kennedy 2011a&b). Morice et al (2012) models the effect of these in monthly average time series as a first order autoregressive process. In this study we have encoded this additional uncertainty term into the regional time series by adding samples drawn from the autoregressive model described in Morice et al. (2012) onto the observational ensemble series. [See Morice et al. (2012) section 5.2 for further details] References: Kennedy J.J., Rayner, N.A., Smith, R.O., Saunby, M. and Parker, D.E. (2011a). Reassessing biases and other uncertainties in sea-surface temperature observations since 1850 part 1: measurement and sampling errors. J. Geophys. Res., 116, D14103, Kennedy J.J., Rayner, N.A., Smith, R.O., Saunby, M. and Parker, D.E. (2011b). Reassessing biases and other uncertainties in sea-surface temperature observations since 1850 part 2: biases and homogenisation. J. Geophys. Res., 116 Morice, C. P., J. J. Kennedy, N. A. Rayner, and P. D. Jones, 2012: Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: The HadCRUT4 data set. J. Geophys. Res. Atmos., 117

3 Figure S1 - Global mean decadal internal variability of the individual models. A standard deviation is calculated for each of the 150 year sub-samples of picontrol used in this analysis. The distribution of values for samples for each individual model used in the main analysis (coloured line) is compared to distribution for all the picontrol samples used to form the covariance matrix (salmon solid histogram). Mean standard deviation for individual models, coloured vertical line; mean standard deviation for all samples, black vertical line.

4 Figure S2 Imperfect model results with inflated variance. Calculated TCR values plotted against actual TCR values for 77 different models used as observations, for the multi-model mean where the analysis uses picontrol samples with inflated variance. The analysis was repeated with uncertainty covariance matrices calculated from these variance inflated control samples, with the factors which resulted in a failure rate outside the 90% confidence interval closest to 10% used in the final analysis. The values chosen to inflate the variance with, were 2.4 for the global mean, 2.6 for the global mean with hemispheric contrast and for the addition of the seasonal contrast, and 2.8 for the addition of the land-sea contrast.

5 Figure S3 TCR estimate from median observations using just surface air temperatures compared to blended temperature Panels are the same as in figure 10. Solid lines are for blended fingerprints dashed lines for SAT means only. Note, results do not include observational uncertainty, so solid lines are not identical to those in figure 10.

6 Figure S4 Individual model performance. Violin plots show distribution of distances between each of the pseudo-observations real TCR and the estimated TCR values (colours see table 1, outer shape has thickness indicating likelihood, circle shows the median value). The upper text value in the panels is the mean absolute error between the best guess estimate and the truth for each of the models and the lower number is the percentage of times the real answers lies outside the 5-95% range, these are the same performance metrics as those in figure 5. Note the improved performance of the multi-model mean (Shown in black).

7 Control Simulations used to calculate covariance matrix ACCESS1-0 ACCESS1-3 bcc-csm1-1-m bcc-csm1-1 CanESM2 CCSM4 CESM1-BGC CESM1-CAM5 CESM1-FASTCHEM CESM1-WACCM CMCC-CESM CMCC-CM CMCC-CMS CNRM-CM5 CSIRO-Mk3-6-0 GFDL-CM3 GFDL-ESM2G GFDL-ESM2M GISS-E2-H-CC GISS-E2-R-CC GISS-E2-R HadGEM2-CC HadGEM2-ES IPSL-CM5A-LR IPSL-CM5A-MR IPSL-CM5B-LR MIROC5 MIROC-ESM-CHEM MIROC-ESM MPI-ESM-LR MPI-ESM-MR MPI-ESM-P MRI-CGCM3 NorESM1-ME NorESM1-M Table S1 List of CMIP5 models which provide picontrol simulations used in calculating the covariance matrix for the main analysis.