Data

Instrumental observations are derived from the gridded 5° × 5° CRUTEM3v data set compiled by the University of East Anglia’s Climate Research Unit (ref. 23; data available at http://www.cru.uea.ac.uk/cru/data/temperature/). We use all time series having at least ten years of complete monthly April–September data, that are poleward of 45° N, and that contain a non-zero fraction of land according to a 0.25° × 0.25° land–sea mask (ref. 31; data available at http://ldas.gsfc.nasa.gov/gldas/GLDASvegetation.php). To avoid introducing spurious structure in the time series of spatial standard deviations, which could result from the short 1961–90 reference period used in standardizing the CRUTEM3v data set, we apply a Bayesian ANOVA technique (ref. 28; code available at http://www.ncdc.noaa.gov/paleo/softlib/softlib.html) to estimate and remove means with respect to the entire 1850–2011 interval spanned by the instrumental data set.

The maximum latewood-density data set32,33 is on the same spatial grid as the instrumental data set, and we use only the 96 grid boxes with centres poleward of 45° N. Data files and descriptions are available at http://www.cru.uea.ac.uk/~timo/datapages/mxdtrw.htm. As with the instrumental data, we apply the Bayesian ANOVA technique28 to estimate and remove means with respect to the entire interval spanned by the data set, in this case 1400–1994.

All varve thickness records publicly available from the NOAA Paleolimnology Data Archive (http://www.ncdc.noaa.gov/paleo/paleolim/paleolim_data.html) as of January 2012 are incorporated, provided they meet the following criteria: extend back at least 200 years, are at annual resolution, are reported in length units, and the original publication or other references indicate or argue for a positive association with summer temperature. As is common34, varve thicknesses are logarithmically transformed before analysis, giving distributions that are more nearly normally distributed and in agreement with the assumptions characterizing our analysis (see subsequent section). Records that are complete between 1400 and 1969 are standardized to unit variance and zero mean using the sample statistics computed over that interval. To obtain a more homogeneous normalization for records incomplete between 1400 and 1969, we scale the mean and standard deviation of each incomplete record to equal those statistics of the corresponding data points in the scaled, complete records. Finally, the Bayesian ANOVA technique28 is used to remove means with respect to the entire 1400–2005 time span of the varve data set.

The ice-core data set consists of 14 of the 15 annually resolved δ18O series used in a recent sea ice reconstruction35. We exclude the Mount Logan series, as the original reference36 indicates it is a proxy for precipitation source region, not temperature. To increase the spatial coverage, we additionally use two δ18O series from Svalbard (refs 37, 38; data available at ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/polar/svalbard/svalbard2005d18o.txt) and one each from Baffin Island (refs 39, 40; data file fisher_1998_baffin.ppd available at http://www.ncdc.noaa.gov/paleo/pubs/pcn/pcn-proxy.html) and Devon Island (ref. 41; data available at ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/polar/devon/d7273del_1yr.txt), all with annual observations. The data set spans the 1400–1998 interval; the standardization procedure is the same as for the log-transformed varve series except that it uses the 1400–1974 interval for initial standardization.

Section 1 of Supplementary Information and Supplementary Tables 1 and 2 provide additional details regarding the data sources.

Inference

Bayesian hierarchical models provide a flexible framework for combining models and observations20. They are generally characterized by a process level that represents the structure of the system and a data level that represents the relationships between each data type and the process. Both the process and data levels contain parameters that are uncertain and whose distributions are inferred as part of the analysis. Similarly, each observation is considered uncertain; in the current analysis this includes both the proxy and the instrumental observations. The process targeted for inference is, therefore, never directly observed but must be inferred using uncertain observations and a model with parameters whose distributions must likewise be estimated. Although Bayesian hierarchical models have been proposed and used to infer past climate12,13,14,42,43, the analysis presented here is, to our knowledge, the first application of such a model to infer surface temperatures as a function of space and time from a multiproxy data set.

In this analysis, we use the Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time (BARCAST)13,14. In our application, the process level describes the evolution of the average April–September temperature anomaly field, T, as,

where μ is the mean of the temperature anomaly field, α is the coefficient of a first-order autoregressive process, the subscript t indexes the year, and 1 is a column vector of ones. The temperature field is represented on a 5° × 5° grid, and we consider only continental locations that are north of 45° N. The innovation vector for each year, , is assumed to be an independent draw from a mean-zero multivariate normal distribution with an exponentially decaying spatial correlation: ∼ N(0, Σ), where Σ ij = σ2exp(–φ |s i −s j |), and |s i −s j | is the distance between the ith and jth elements of the field vector, T t .

The data level describes the relationships between the true temperature anomalies and the instrumental and proxy observations of these anomalies. Instrumental observations, W 0,t , are modelled as noisy versions of the true anomalies at the corresponding locations,

The noise terms are assumed to be independent draws from a multivariate normal distribution, e 0,t ∼ N(0,τ 0 2I), where I is the identity matrix, and H 0,t is a selection matrix of zeros and ones that picks out elements of T t for which there are instrumental observations.

The types of proxy observations included are tree-ring density chronologies, ice-core δ18O series, and log-transformed lake-varve thickness series. Each type, k, is assumed to have a linear relationship with the local value of the true temperature, where β k ,1 and β k ,0 are respectively the slope and intercept terms and H k ,t is, as above, a selection matrix. The noise terms are once more assumed to be independent normal draws, e k ,t ∼ N(0,τ k 2I). The regression parameters vary between proxy types, but are constant for all proxies of a given type.

Prior distributions are placed on each of the parameters included in the model: α, μ, φ, σ2, τ2 k = 0…3 , β k = 1…3,1 , β k = 1…3,0 , as well as for T in the year before observations become available. Priors are selected to be proper, weakly informative, and—where possible—conjugate20. Bayes’ rule is used to calculate the posterior distribution of the parameters and field given the observations and priors. A Gibbs sampler with a single Metropolis step (for φ) is then used to draw from the posterior distribution20. Further details of the inference are available in an earlier publication13, code is available at http://www.ncdc.noaa.gov/paleo/softlib/softlib.html, and convergence of the Gibbs sampler is discussed in Section 3.5 of the Supplementary Information.

The result of the analysis is an ensemble of draws of both the parameters and the temperature anomalies, each of which is equally likely given the data, priors and modelling assumptions. Furthermore, each ensemble member will have variability similar to the actual temperature anomalies14, insomuch as the model and data are correctly represented. The median across the ensemble (Fig 1a, b; Supplementary Fig. 5), however, has attenuated variability14, especially in data-poor parts of the reconstruction. This attenuation provides a more accurate estimate of the past temperature, though not the variability of that temperature, and is generally used as our best estimate.

For the purposes of comparing the climatic information content of the different proxies (for example, Fig. 2c), it is possible to run BARCAST in a reduced mode, using a subset of the data to update the temperature field while sampling all model parameters from the posterior distributions resulting from the analysis with the full data set. That is, at each iteration of the Gibbs sampler, a vector of parameters is drawn from the posterior distribution derived using all data, and then the draw of the temperature field is updated using these parameters and a given subset of the data. It is likewise possible to simulate the natural variability of the temperature field (for example, Fig. 3) by not applying any of the data constraints. In this case, the process level model has parameters constrained by the data but the specific evolution of the temperature field is unconstrained. Comparison of the variability between the constrained and unconstrained simulations indicates the extent to which the data controls the solution (see Supplementary Information Sections 2.5 and 6).

Assumptions and implications

The stationary, isotropic and exponentially decaying spatial covariance model specified for the temperature anomalies is a simplifying assumption that does not account for directionality and long-range covariance relationships in the climate system. Indeed, many palaeoclimate reconstruction techniques, generally based on eigendecompositions of sample covariance or correlation matrices, are explicitly designed to exploit such covariance patterns9,44,45,46. This class of methods assumes that the characteristic spatial structures identified in the calibration interval are constant in time, but have varying amplitudes; results can be strongly dependent on the particulars of how these modes of variability are determined and used47. BARCAST, in contrast, relies on the simpler assumption of a temporally constant decorrelation length scale and represents each observation as indicative of local temperature. This local representation (equation (3)) is similar to the proxy representation in the LOC method18,19,48, although BARCAST additionally models errors in the instrumental observations and arrives at an estimate of the spatial field of temperature anomalies49,50.

Temporal stationarity is assumed through the first-order autoregressive description of interannual temperature variability (equation (1)). This parameterization lacks any representation of long-term temperature trends, or associations between temperatures and climate forcings43,51. In some sense, however, this is advantageous as the process level is agnostic regarding any changes, and inferences concerning the unprecedented nature of recent extremes are consequently derived exclusively from the data. Furthermore, this approach permits exploration of exactly where the assumption of strict temporal stationarity fails (see Fig. 3).

The assumption that the space–time covariance function is separable—that is, that it can be factored into the product of purely spatial and purely temporal elements—is also unlikely to hold in detail12. Predictive performance, however, is often not affected by incorrectly assuming a separable covariance form52. With regard to both stationarity and separability, BARCAST has been shown to perform at a level comparable to or better than other climate field reconstruction techniques22, even when the underlying assumptions are not strictly met by the data14. The validation metrics reported in Supplementary Information Section 5 also suggest that BARCAST provides an adequate statistical description of the data for the present purposes.

Analysis of results

The analysis presented here is based on 4,000 posterior draws taken from four parallel Gibbs samplers. Spatial mean values are computed by weighting the grid boxes by land area according to a 0.25° land–sea mask31. Taking the 50th percentile of the 4,000-member ensemble at each year of the spatial mean time series results in a best estimate of the time series, while taking the 5th and 95th percentiles produces 90% pointwise credible intervals20, which are used to indicate the uncertainty in the reconstruction at each year.

Pathwise 90% credible intervals are calculated by inflating the pointwise intervals such that the envelope contains 90% of the posterior time series in their entirety15,17,29. The two uncertainty intervals have different interpretations, with the pointwise intervals covering the true temperature anomaly for 90% of the years, whereas the time series of true temperature anomalies lies entirely within the pathwise envelope with 90% probability. The statement that certain recent years are warmer than all previous years with P > 0.95 follows from noting that the 90% pathwise uncertainties for these years lie entirely above those for all years in the 1400–2004 interval. Note that the statement holds at P > 0.95 because the test is one sided, as we have prior reason to believe that recent years are warmer than usual. Uncertainties for temporally smoothed time series are estimated in the same manner, after first smoothing each ensemble member using a nine-point Hanning window. Point estimates and uncertainties in the centennial-scale trends are derived by calculating the trend for each ensemble member at each position of a sliding 100-year window and then calculating both pointwise and pathwise uncertainties.

Probability estimates corresponding to specific statements, such as the probability that western Russian temperatures achieved a maximum in 2010, are obtained by calculating the proportion of ensemble members for which the statement is true. Performing such an analysis at each location leads to the maps shown in Fig. 2a. Calculating the fraction of locations for which each year is warmest or coldest, averaging results across the ensemble members and binning the years by decades results in the histograms shown Fig. 2b. Figure 2c is derived in the same manner, but from a 4,000 member ensemble obtained by running BARCAST in reduced mode using only the ice-core and log-transformed lake-varve series.

To assess recent extremes, we simulate temperature anomalies and instrument-like observations over the past 20 years using median parameter values, and in the latter case additionally record the maxima at each location as well as the five largest values in both space and time. Distributions of the simulated instrumental quantities are built up by repeating the procedure 10,000 times. When shifting the mean of these simulations to match that inferred over the past 20 years (Fig. 3), we use the simple average across the locations, ensemble members, and years within the 1992–2011 interval. Note that this mean differs slightly from the mean of the spatial average time series because of the spatial weighting inherent in the latter. Figure 3b shows the single instrumental simulation that is closest to the actual instrumental observations according to the variance of the site-wise maxima. Additional results and model diagnostics are available in Supplementary Information Sections 2 and 3 and a more detailed discussion on inferring extremes in the presence of uncertainty in Supplementary Information Section 6.

Robustness

To examine the robustness of our results to specific data types, we run BARCAST in five different reduced mode formulations using the following subsets of data types: tree-ring densities alone, ice-core series alone, lake-varve series alone, ice-core and lake-varve series together, and instrumental series alone. Results of the proxy-only analyses are then compared with each other, to the main analysis, to the instrumental-only analysis, and to the withheld instrumental observations. The time series of spatial-average temperature and centennial-scale slopes, as well as the distribution of extremes for the spatial mean, for centennial trends in the spatial mean, and as a function of space, are all compared with each other. For the spatial average, we also consider the correlations and root-mean-square error between each proxy-only analysis and the instrumental-only analysis over three different time intervals: 1850–1959, 1850–1994 and 1960–94. In general, we find that the proxy-only analyses provide consistent inferences with one another and with the instrumental-only predictions, when accounting for the uncertainties in each analysis (see Supplementary Information Sections 5.1–5.4).

An important exception is that the predictions from the tree-ring densities alone do not track the warming seen in all other data sets in the latter half of the twentieth century, a finding consistent with the so-called divergence problem33,53. To assess the robustness of our results to this divergence, we re-ran the full analysis excluding the post-1960 tree-ring-density observations and found no qualitative change in results (see Supplementary Information Section 4). The primary reason for consistency between analyses which include and exclude the post-1960 tree-ring-density observations is that instrumental data are of sufficient quality and number post-1960 so as to dominate the solution irrespective of the tree-ring data. It is also the case that the parameterization of the tree-ring-density relationship with temperature is primarily constrained by its relationship with data before 1960.

To assess the variability in the ensemble of posterior draws, we simulate the withheld instrumental observations in each proxy-only analysis and examine rank verification histograms54 for the withheld instrumental observations. To assess the coverage rates of the credible intervals, we calculate the percentage of the withheld instrumental observations that fall within the nominal 90% intervals. Results show that the ensembles of predictions of the withheld instrumental values generally have about the correct variability, and that the actual coverage rates are generally within 10% of the nominal rate. An exception is, again, for the tree-ring-density analysis during the post-1960 interval, where the coverage rate is about 15% too low and the shape of the rank verification histogram is indicative of a low bias. See Supplementary Information Section 5.5 for further details.