Main

Arising from both natural forcing and unforced internal variability in the climate system, decadal- to multidecadal surface temperature variations will continue to be superimposed on the anthropogenic global warming trend1,2,3,4,5,6,7, affecting estimates of transient climate response to individual forcings and regional climate change8. Knowledge of multidecadal global mean surface temperature (GMST) fluctuations, as well as their relative magnitudes compared with that of the projected long-term warming trend, is thus of crucial importance for understanding future climate change9,10.

Global and regional-scale multidecadal temperature variability (MDV) was found to be under- or overestimated in climate simulations of the recent past11,12,13. Some multicentennial to millennial-scale assessments of palaeoclimate observations and simulations suggest that temperature variability at multidecadal and lower frequencies is underestimated by the Coupled Model Intercomparison Project Phase 5 (CMIP5)14 models15,16, contrasting with other evidence suggesting that models contain realistic amplitudes of MDV17,18. Global networks of instrumental observations extend back to 1850 at best19, which is too short a period to fully characterize MDV and its driving forces and evaluate the simulated MDV amplitude20. Estimating the relative contributions of various climate forcings, and assessing the ability of climate models to accurately simulate observed climate phenomena, requires a far longer perspective10.

Over the Common Era (ce, the past 2,000 yr), palaeoclimate proxy-based observations of temperature and climate forcings are available at up to sub-annual resolution and cover much of the globe21. Reconstructed GMST over the full Common Era provides context for recent warming, and offers an opportunity to benchmark the ability of climate models to simulate MDV and to study the planetary energy balance and other key aspects of the climate system. Most previous large-scale millennium-length temperature reconstructions have highlighted the extraordinary rate and magnitude of the warming in recent decades1. However, there are considerable discrepancies among the reconstructions in the amplitude, and partly the timing, of past temperature fluctuations at interannual to multicentennial timescales1,22. The extent to which these differences can be attributed to different reconstruction methodologies relative to different palaeoclimate proxy networks is unclear, although both have been shown to strongly influence temperature reconstructions23. The recently updated compilation of temperature-sensitive records in a community-vetted, quality-controlled database21 provides an opportunity to improve the reconstruction of GMST for the Common Era. Despite much innovative work in this area22,24, there have been few coordinated efforts to reconstruct global temperatures using a consistent experimental framework23.

Here, we use seven different statistical methods to reconstruct GMST over the past 2,000 yr (1–2000 ce; Table 1). The methods range from basic composite-plus-scaling (CPS) and regression-based techniques (such as principal component regression (PCR) and regularized errors in variables (M08)) frequently used in past reconstructions, to newer linear methods (optimal information extraction (OIE) and a Bayesian hierarchical model (BHM), for example) and techniques that account for nonlinear relations between proxy values and temperature (pairwise comparison; PAI) or combine information from proxy data and climate models (offline data assimilation; DA). All procedures use the same input dataset (ref. 21, Supplementary Fig. 1) and the same calibration dataset for the reconstruction target, the infilled version25 of HadCRUT419. For each method, a 1,000-member ensemble of reconstructions is generated, allowing a probabilistic assessment of uncertainties. Further details on each method, such as the various approaches used to generate the reconstruction ensembles and the ability to include low-resolution records, are provided in the Methods.

Table 1 Overview of the reconstruction methods and data-model comparison
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Temperature history of the Common Era

The temperature evolutions reconstructed by the seven methods (Fig. 1a) exhibit the major features of previous reconstructions over the Common Era1. All reconstructions show that the GMST during the first millennium of the Common Era was warmer than during the second millennium (excluding the twentieth century). All reconstructions show a significant cooling trend before 1850 ce (Methods), followed by rapid industrial-era warming. The warmest 10 yr (30 yr, 50 yr) period of the past two millennia falls within the second half of the twentieth century in 94% (89%, 84%) of ensemble members. The most pronounced difference between the reconstructions produced by the different methods is the amplitude of the pre-industrial multicentennial cooling trend, manifesting in a temperature difference of about 0.5 °C between the warmest (DA method) and the coldest (BHM method) estimates around 1600 ce.

Fig. 1: Global mean surface temperature history over the Common Era.
figure 1

a, Temperature anomalies with respect to 1961–1990 ce. The coloured lines represent 30 yr low-pass-filtered ensemble medians for the individual reconstruction methods. The grey shading shows the quantiles of all reconstruction ensemble members from all seven methods (right-hand y axis); the 2.5th and 97.5th percentiles are indicated with black dotted lines. The black curve is instrumental data25 for 1850–2017 ce. b, Same as a, but for the 30 to 200 yr bandpass-filtered ensemble; instrumental data are not shown.

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Methods that are implemented with proxy records of lower-than-annual resolution (Table 1 and Methods) yield systematically larger pre-industrial cooling trends, on average −0.23 °C kyr−1 (−0.31, −0.11) (the ensemble median (2.5th percentile, 97.5th percentile)), compared with the other methods (−0.09 °C kyr−1 (−0.27, 0.02)).

The uncertainties for all reconstruction methods increase backwards in time and are particularly large before medieval times (Supplementary Fig. 3), owing to the decreasing numbers of input proxy data (Supplementary Fig. 1) and associated changes in the observing network. The magnitude of uncertainties varies among the reconstructions depending on the method-specific approach to generating the reconstruction ensemble (Methods).

In the currently available database, many palaeoclimate archives are seasonally biased22,26,27 (Supplementary Section 3) and the relative seasonal representation varies over time. The majority of the included tree-ring records are detrended for removal of biological age effects by methods that cannot capture centennial- to multicentennial-scale temperature variability28,29. This problem probably manifests more backwards in time22, and may result in an underestimation of low-frequency variability—especially during the first millennium of the Common Era30. On the other hand, low-resolution marine records seem to overestimate the true variance31. Many of the palaeoclimate records also have a mixed and, presumably, nonlinear response to temperature and hydroclimate32,33 that is probably not stable over time34.

Although multicentennial- to millennial-scale trends differ among the GMST reconstruction methods22,35,36 (Fig. 1a), multidecadal fluctuations are more similar (Fig. 1b). Removing the centennial-scale trend reveals remarkably coherent MDV with narrower confidence ranges across reconstruction techniques (Fig. 1b), suggesting a more robust result across methods.

Multidecadal variability and response to forcing

To explore the influence of the major known climate forcings on multidecadal GMST, we compared an ensemble of 23 CMIP5 climate model simulations (Methods) with each of the temperature reconstructions. We find strong reconstruction–model agreement over the past millennium (Fig. 2) in both the timing and magnitude of multidecadal (30–200 yr bandpass-filtered) GMST variance: the median model/data variance ratio across all reconstruction and model ensemble members is 1.01 and close to one for the individual reconstruction methods (Table 1). Multidecadal data–model coherence is substantially larger than expected by chance (Table 1, Methods).

Fig. 2: MDV in reconstructions and models and volcanic forcing over the past millennium.
figure 2

Temperature anomalies with respect to 1916–1995 ce are shown. Coloured lines are the ensemble median reconstructions from the different methods subjected to 30 to 200 yr bandpass filters. The grey shading shows the model simulation percentiles. The green curve (right-hand y axis) shows the volcanic forcing52.

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Warm anomalies around 1320, 1420, 1560 and 1780 ce, cold anomalies around 1260, 1450 and 1820 ce and the periods of reduced variability during the overall cold seventeenth century and the relatively warm eleventh century are captured by both the reconstructions and simulations. These similarities between reconstructions and models suggest a dominant influence of external forcing on multidecadal GMST variability.

The data–model agreement is particularly strong between 1300 and 1800 ce. The weaker agreement before 1300 ce might be mostly explained by the reduced quality of volcanic forcing estimates used in CMIP5 during this period37; for example the 1109 ce eruption, which is followed by a clear GMST cooling (Fig. 2), is missing in the forcing datasets38,39 used for these CMIP5 simulations (Supplementary Fig. 20b). The apparent data–model mismatch in the nineteenth century is caused by the difference in response to the eruptions in 1809 and 1815 ce, which is probably overestimated by the CMIP5 model simulations40 (Supplementary Fig. 19).

To quantitatively evaluate the GMST response to external forcings in the reconstructions compared with model simulations, we apply a formal detection and attribution (D&A) analysis41,42,43. The D&A analysis uses robust linear regression of reconstructed GMST on unforced, single and cumulatively forced simulated GMST. The regression coefficients (scaling factors) may be interpreted as the extent to which reconstructed variability is explained by candidate unforced and forced responses. We choose 1300–1800 ce for the D&A analysis, as during this period the anthropogenic forcing is negligible, and the uncertainties in the forcings, simulations and reconstructions are relatively small and time-invariant compared with earlier centuries21,37,43 (Fig. 2).

For all reconstruction methods, the D&A scaling factors are significantly above zero when performing the D&A analysis on multidecadal timescales and using fully forced model simulations (Fig. 3a, left; for details see the Methods). This means that the influence of external forcing on GMST variance is detectable in the simulations. The full ensemble median of the scaling factors (0.89) is close to unity and in four of the seven reconstruction methods (CPS, PCR, PAI and BHM), the 90% range of scaling factors does encompass unity, indicating a consistent response to total external forcing in models and reconstructions. This result is robust for a single model (CESM-CAM544) and multimodel ensembles (Supplementary Fig. 7).

Fig. 3: Pre-industrial forcing response and magnitude of unforced MDV.
figure 3

a, Multidecadal detection and attribution scaling factors over 1300–1800 ce for the CESM1-CAM5 model based on differently forced runs. The grey box and whisker plot shows the D&A experiment using fully forced runs. The yellow, green and blue box and whisker plots show multivariate experiments combining runs with solar forcing only, volcanic forcing only and GHG forcing only, respectively. Circles represent the median of the individual reconstruction methods. b, Estimates of unforced natural variability based on the D&A regression residuals of the full-forcing experiment (grey), the standard deviation of the pre-industrial control simulations (black) and reconstruction and model standard deviation during the period of low external forcing variability (850–1100 ce; brown, for details see Methods). Boxes represent the interquartile range and whiskers represent the 90% range. Black points represent the individual model simulations.

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To disentangle the influence of different forcing factors, we repeat the D&A analysis by simultaneously regressing the ensemble means of three different CESM-LME model experiments (volcanic, solar and greenhouse gas (GHG) only) onto the reconstructions (see equation (3) in the Methods). Choosing combinations of model experiments that account for different forcings allows us to estimate the relative contribution associated with each forcing (Fig. 3a, right). Results show that solar forcing is not detectable at multidecadal timescales in the reconstructions (scaling factors not significantly above zero), but volcanic and GHG forcing responses are consistent in pre-industrial multidecadal reconstructions and simulations45. Volcanic forcing has scaling factors very close to one, confirming the agreement illustrated in Fig. 2. A weak relationship between solar forcing and GMST is confirmed by cross-wavelet analysis (Supplementary Fig. 16), suggesting that solar variability as currently reconstructed cannot account for GMST variability over the Common Era, although it has been detected on multidecadal timescales regionally46,47 as well as over the Northern Hemisphere on multicentennial timescales47.

The residuals of the D&A analysis can be used to quantitatively estimate the magnitude of internal unforced MDV in the reconstructions (Methods). The residuals of the total forcing D&A analysis (Fig. 3a, leftmost box and whisker plot) are shown in Fig. 3b, and they are interpretable as the unforced variability in the actual climate system after the linear response to total forcing simulations is removed. This estimate of the unforced variability in the actual climate system is consistent with the variability in pre-industrial control simulations (Fig. 3b, left two box and whisker plots): 99% of estimates based on the D&A residuals are within the 95% range of control simulations. The interquartile range of all possible paired comparisons also encompasses zero (Supplementary Fig. 8).

Differences in the estimates from the two lines of evidence may arise from both random and systematic uncertainties in both reconstructions and simulations48 (equation (3) in the Methods). Reconstruction uncertainties include underestimation of the reconstructed variance due to sparse observations, temporal smoothing, integration or seasonal bias, structural data model error and large observational errors36,49,50,51. Simulation uncertainties arise from errors in amplitude or timing of estimated radiative forcings52, incorrect timescales of the simulated responses to forcing53, and underestimation of the forced and unforced response because of an incomplete representation of processes in the actual climate system54.

To evaluate the likelihood of these biases influencing our results, we can compare the variability in reconstructions and model simulations for the Medieval ‘quiet’ period55 for external radiative forcing (850–1100 ce), which is independent from the period used for D&A (Fig. 3b, rightmost two box and whisker plots). This analysis suggests that there is no significant difference between reconstructed and simulated unforced variability (Methods). Our analysis does not confirm earlier findings of models underestimating internal MDV15,56. Instead, it suggests that current models are able to realistically simulate the relative magnitudes of the externally and internally forced GMST on multidecadal timescales.

Long-term context of the recent warming rates

To place recent warming rates into a long-term context, we calculate 51 yr running GMST trends from the unfiltered reconstructions (Fig. 4a). The different reconstruction methods agree on the magnitudes of multidecadal trends over the Common Era, as shown by the narrow uncertainty range in Fig. 4a.

Fig. 4: Multidecadal temperature trends over the Common Era.
figure 4

a, 51-year trends in reconstructions (ensemble percentiles shaded) and instrumental data25, with the temperature response to volcanic forcing52 based on an EBM. For the trends, the years on the x axis represent the end year of the 51-yr trends. b, Ensemble probability of the largest trend occurring after 1850 ce as a function of trend length.

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Energy balance model (EBM, green line in Fig. 4; see also Supplementary Figs. 14 and 15) simulations with volcanic forcing52,57 suggest a multidecadal temperature response to secular frequency and amplitude variations in the forcing (Supplementary Figs. 12 and 13), which is qualitatively consistent with reconstructions. The fractional multidecadal variance increases with increasing frequency of volcanic eruptions (Supplementary Fig. 15). Volcanic eruptions are found to coincide with strong multidecadal cooling trends in both reconstructions and model simulations58, and this is then followed by an increased probability of strong warming trends, due to the recovery from the volcanic cooling59 (Fig. 4a and Supplementary Figs. 9–13). Both the cooling and subsequent warming trends are found to occur much more frequently than would be expected by chance.

Reconstructions and models yield practically identical values for the upper range of pre-industrial temperature trends (horizontal lines in Fig. 4a), again suggesting that climate models are skilful in simulating the natural range of multidecadal GMST variability.

Of the reconstruction ensemble members, 79% have the largest 51 yr trend within the twentieth century, which includes two distinct periods of large trends. The first reflects the early twentieth century warming60, which was shown to originate from a combination of forcings including anthropogenic forcings and internal multidecadal variability of the climate system3. The second reflects the modern period of strong warming, which extends from the mid-1970s to today. The temperature trends during these two industrial-era periods are outside the range of pre-industrial variability, even more so when compared with control runs, in which strong warming trends after volcanic cooling do not occur (dashed red line in Fig. 4a). All instrumental 51 yr trends starting in 1948 ce (and ending 1998 ce) or later exceed the 99th percentile of reconstructed pre-industrial 51 yr trends.

The extraordinary rate of the industrial-era temperature increase is evident on timescales longer than approximately 20 yr (Fig. 4b), for which the probabilities of the occurrence of the largest trends greatly exceed the values expected from chance alone, using random noise predictors with realistic memory properties (Methods).

In addition to using a variety of reconstruction methods, we assessed the robustness of our analysis of multidecadal GMST variability and trends using a range of sensitivity tests, including reconstructions based on detrended calibration and different realizations of noise proxies, adjustments for variance changes back in time, and different calibration periods and proxy subsets. Our inferences on the multidecadal GMST variability for the Common Era are robust to all these permutations (Supplementary Figs. 17–20). Nevertheless, we cannot rule out biases due to errors in the individual proxy records and the unequal spatiotemporal distribution of proxy data (Supplementary Fig. 1). Warm-season-sensitive records from the Northern Hemisphere high and mid latitudes dominate the collection of proxy records21, thus our results may be biased towards this region and season, although these effects are reduced by using the global mean (Supplementary Section 3). The coherency across reconstruction methods nevertheless suggests a reduced sensitivity of the results to methodological choices compared with earlier attempts23, perhaps in large part because of the use of the largest database of temperature-sensitive proxy data yet assembled21. With respect to the forced climate simulations, independent uncertainties include: the specification of long-timescale physical processes; relatively poorly known, slowly varying forcings; and representation of grid- and sub-grid-scale processes61. That the reconstructions presented here are coherently phased and of similar amplitude to CMIP5 simulated MDV suggests improved confidence in both these sources of information.

Agreement on both the timing and amplitude of GMST variability across the reconstruction and simulation ensembles suggests that some aspect of the observed multidecadal variability is externally forced, principally by changes in the frequency and amplitude of volcanic forcing over the pre-industrial past millennium31,45,62 and anthropogenic GHGs and aerosols thereafter63 (Supplementary Fig. 7c). Furthermore, we find that estimates of the unforced variability from reconstructions and simulations span similar ranges to independent uncertainties. Our results suggest that multimodel ensemble projections for the near term produce reasonable estimates of the spread in global mean temperature change associated with both forced and unforced MDV, further justifying the use of such models as a basis for regional- to global-scale simulations64 and studies of associated societal impact, adaptation and resilience65.

Methods

Input data and preprocessing

Palaeoclimate observations

We use version 2.0.0 of the global PAGES 2k temperature proxy data collection21. For the reconstructions presented in the main text, we use the subset of records selected on the basis of regional temperature screening and to account for false discovery rates (R-FDR subset21). This screening reduces the total number of records from 692 to 257, but increases the GMST reconstruction skill for most methods and skill metrics. A comparison of reconstructions and evaluation statistics of the full versus screened networks is provided in Supplementary Fig. 17. Of the screened proxy records, 210 are annually resolved and are included by all reconstruction methods. Sub-annually resolved time series (29 records) were averaged over the April–March annual window. The 47 records with lower-than-annual resolution are only included by the M08, OIE and PAI methods (see below). Although none of the methods explicitly consider age uncertainties, the impact of age uncertainty on reconstructions can be estimated by sampling from a set of age ensembles for each of the age-uncertain records (for example, ref. 72). However, only a subset (15%) of the age-uncertain records in version 2.0.0 of the PAGES 2k database include age ensembles or the requisite geochronological data to generate new age ensembles. Consequently, full consideration of age uncertainty is outside the scope of this research; nevertheless, this probably has minimal impacts on the reconstructions, as the final proxy network mainly consists of records with very small to negligible age uncertainties (81% of records are from tree or coral archives).

Instrumental observations

We use the Cowtan and Way25 global mean temperature dataset, which is based on a kriging-interpolated version of HadCRUT419. We use an annual average based on the April–March window that represents the ‘tropical year’, which has the advantage of not interrupting the growing season of trees in both the Northern and Southern Hemispheres.

Experiment overview and settings

The time window 1850–2000 ce was used for calibration. For evaluation, we ran a second experiment calibrating over 1916–1995 ce and validating over 1881–1915 ce. Evaluation graphs are provided in the Supplementary Section 2. Results based on this shorter calibration period, which encompasses a period of significantly improved spatiotemporal coverage with instrumental and proxy data, are presented in Supplementary Fig. 17. All reconstruction methods generated an ensemble of 1,000 GMST reconstructions, which was used for uncertainty quantification.

Reconstruction methods

Details on each method are provided in the following subsections. Table 1 summarizes some key aspects and references for the methods. Methodological constraints required some minor adaptations to the experimental set-up for some techniques. These are as follows: CPS, PCR, BHM and DA include only annually and more finely resolved proxy records. CPS and PCR exclude records with more than 33% missing values and infill the remaining missing values over the calibration period. DA generates GMST from a spatially resolved temperature reconstruction. Multiple approaches to weighting the different reconstruction methods to generate a best estimate GMST time series are presented and discussed in Supplementary Section 4.

CPS

One of the most basic methods for quantitative climate index reconstruction, CPS has been applied widely for local- to global-scale reconstructions of temperature and other variables66,73,74,75,76. CPS generates reconstructions using two steps. First, the proxy records are converted into a single time series (composite) by calculating a weighted mean. Second, this composite is scaled to the mean and standard deviation of the reconstruction target over the calibration period. Here, we use a similar implementation of CPS as ref. 56: the proxies are weighted by their non-detrended correlation with the GMST reconstruction target over the calibration period. A nested approach is applied, such that each year’s CPS reconstruction is based on a calibration using all proxies that are available in that year. Several key parameters are resampled to quantify uncertainties for each member of the 1,000-member ensemble:

  • 10% of proxy records are withheld from calibration.

  • The weight of each proxy record (target correlation r) is multiplied by a factor between 1/1.5 and 1.5. In contrast to other studies (such as ref. 77), we do not perturb r with an exponent but use a factor centred at 1 to ensure that the reconstruction ensemble is centred around a reconstruction using non-perturbed weighting.

In addition, each ensemble member is perturbed with red noise that has the same standard deviation and first-order autoregression (AR(1)) coefficient as its residuals from the reconstruction target over the calibration period. We use an AR(1) model because the short length of time available for model fitting and evaluation does not allow for a reliable estimate of coefficients for higher-order and more complex models. Furthermore, CPS residuals only exhibit significant autocorrelation at lag 1 (not shown). Thus, the total uncertainty expressed by the ensemble spread is based on both parameter and calibration uncertainty78. For a detailed discussion of this ensemble approach refer to refs. 56,78.

As for many linear methods, CPS is prone to variance losses in the reconstructions and a bias towards zero. Both the bias and reconstruction variance are directly related to the correlation of the proxies with the reconstruction target22. CPS is computationally efficient and the quality (in terms of skill measures, for example) of its results is often comparable to more advanced (and often more expensive and parameter-dependent) methods (see Supplementary Figs. 2, 3 and refs. 56,78,79).

For proxy records with lower-than-annual nominal resolution, the correlations with the instrumental target have considerably reduced degrees of freedom. Furthermore many of these records are not interpolated to a value for each year in the PAGES 2k database v2.0.0, thus cannot be directly used for CPS. We therefore use only records with a median resolution (‘ResMed’ variable in the PAGES 2k proxy database) not exceeding 1. Furthermore, we only use records with a maximum of 1/3 of missing data within the calibration interval. This yields a total of 210 proxy records within the R-FDR screened subset. Missing data in the calibration window (2.23%) are infilled using DINEOF80.

PCR

This is a linear method and, along with CPS and RegEM (see below), is one of the most commonly used methods to reconstruct climate indices and fields over the Common Era22,24,75,81,82. PCR reduces the dimensionality of the proxy matrix by using only a certain number of its principal components as predictors in the regression. PCR has similar advantages and deficiencies to CPS. There is a large literature describing potential issues of the method and its parameter choices22,24,50,83,84,85,86,87.

Here, we use the nested PCR approach originally presented by ref. 67 and implemented in many reconstructions of regional to hemispheric climate56,79,88,89,90,91,92. Our implementation is similar to ref. 56; we use ordinary least squares to generate nested 1,000-member ensemble reconstructions. We use identical parameters and input data to those described for CPS above with a few exceptions: as additional parameter perturbation, principal components are truncated so that they explain between 40% and 90% of the total variance in the proxy matrix. Similar to the weighing in CPS, each proxy record is multiplied by a factor between 1/1.5 and 1.5 to perturb its weight within the PCA routine. All records are scaled to a mean of zero and unit variance over their common period of overlap before this variance adjustment.

M08

The core of the regularized errors-in-variables approach66 is the regularized expectation maximization (RegEM93) algorithm with truncated total least squares (TLS) regression94. The reconstruction by ref. 66 (hereafter M08) is the most widely used GMST reconstruction over the Common Era in the recent literature36,66,79,86,95,96, and has been much criticized22. Here we use the M08 implementation of RegEM to allow a direct comparison. Compared with other methods used herein, the most distinctive feature is a hybrid process to decompose two levels with high/low-frequency bands of the reconstructions using the RegEM method with the truncated TLS regression. Thus, the method is able to combine low- and high-frequency proxies. The instrumental and gridded April–March average temperatures25 were interpolated using RegEM to cover the whole period (1850–2000 ce) and used to calculate the regional properties of each record; for example, the correlation coefficient between the proxy record and the regional and instrumental data during the calibration period.

The uncertainty in the original M08 method was calculated using the standard deviation of instrumental data during the calibration period and the correlation coefficient between the proxy reconstruction and instrumental data during the evaluation period66. Here, we adapted the uncertainty estimation to the ensemble reconstruction by perturbing each ensemble member with red noise of the same standard deviation and AR(1) coefficient as the residuals of the reconstruction during the evaluation period56,78.

PAI

Pairwise comparison is a nonlinear method69 that allows proxy records with different temporal resolution to be combined. It has been used in a number of continental-scale temperature reconstructions69,78,97. PAI generates a nonlinear, but monotonically increasing function to reconstruct climate by performing pairwise comparisons of the proxy records. The order of successive data points in each proxy time series is compared and the relative agreement of this ordering for all proxies is then calculated. The strength of this agreement is then used to predict temporal changes in the target climatic variable. Similar to CPS, PAI generates a unitless composite on the basis of the pairwise comparisons. In contrast to CPS, which makes use of target correlations to generate the composite, the composite derived by PAI is fully independent of the reconstruction target. Therefore, careful proxy selection is important for PAI to yield skilful results. The composite is scaled to the reconstruction target over the calibration interval. Reconstruction ensembles are generated by calculating bootstraps, resampling the proxy records and adding noise to the target data. A detailed description of the method is provided in ref. 69. Here we use the same parameters as in ref. 97.

The key advantage of PAI compared to most other methods used at present is its ability to combine high- and low-resolution proxy datasets, including records with irregular temporal resolution or those that do not cover the calibration period.

OIE

The OIE method was derived from the CPS method, and developed with several versions inspired by the local (LOC) method98, a Bayesian framework (BARCAST)99, the generalized likelihood uncertainty estimation method100,101 and the ensemble reconstructions56.

We use version 2.0 of the OIE method, which was developed to reconstruct the GMST by combining previously published reconstructions (four Northern Hemisphere reconstructions with annual resolution and covering the past two millennia66,102,103,104 and a Southern Hemisphere ensemble reconstruction56 using direct and indirect regression, which generated a reconstruction through four steps. First, we matched the variance of each proxy record using the previous hemispheric reconstructions. Second, we averaged the matched records to generate a composited proxy record, which is like the CPS method to weight and average the screened records to a composited record. Third, we regressed the composite using the E-LOC method95, which set the regression coefficients as random variables within the ranges of the direct and indirect regressions, and the regression coefficients were determined using the generalized likelihood uncertainty estimation method100,101. Finally, the uncertainty is expressed as the ensemble spread, in which each ensemble member is perturbed with red noise that has the same standard deviation and AR(1) coefficient calculated from the residuals of the reconstruction in the third step56,78.

BHM

The BHM, defined in ref. 70, is based on two ingredients: a model of the GMST (T) that depends on external forcing and additive noise (equation (1)); and a model of the reduced proxy signal (RP) as linearly dependent on T, with additive noise (equation (2)). Specifically:

$$T_t = \beta _0 + \beta _1S_t + \beta _2V_t + \beta _3C_t + \sigma _T\epsilon _t$$
(1)
$${\mathrm {RP}}_t = \alpha _0 + \alpha _1T_t + \sigma _P\eta _t$$
(2)

where t is time (1–2000 ce) and

RPt is the RP series that captures the common signal of the entire proxy dataset. It is constructed through a Lasso Regression105 between proxies with similar temporal availability and the observed anomalies25 during the calibration period. This procedure is repeated over 8 250 yr nests with different temporal availability of the proxies, and the fitted values obtained for each case are combined in a single series. The number of records that were used in each nest vary between 21 (earliest nest) and 199 (latest nest).

Tt is the temperature anomalies series.

St is the time series of solar irradiance computed from SATIRE-H106,

Vt is the transformed volcanic forcing from Easy Volcanic Aerosol (EVA) dataset37,107

Ct is the transformed GHG concentrations taken from CMIP6108,

ηt and \(\epsilon _t\) are the zero-mean, unit-variance stationary stochastic processes and σP and σT are constant variance parameters.

This method proceeds by regressing a linear combination of the PAGES 2k data onto global temperature, and modelling the latter as depending on external forcings. All model parameters are assumed random, and their prior distributions are updated through a combination of Gibbs sampling and the Metropolis–Hastings algorithm. This allows us to estimate the latent variable Tt together with all the parameters of the model and thus to obtain samples of its joint posterior distribution given the set of proxy records. This procedure also allows us to obtain prediction intervals for all the parameters and the latent variable of the model. We also considered a particular case of the model for which all the external forcings Ct, Vt and St are zero. In this way, the reconstruction of the latent variable was performed using information from the proxies only. For this particular case we defined ηt as white noise and \(\epsilon _t\) an AR(1) process. When the external forcings are not null, both ηt and \(\epsilon _t\) are Gaussian white noise processes.

DA

This method fuses proxy observations and climate model output to reconstruct multiple climate fields, while leveraging climate model physics. The DA procedures used in this study were developed for the Last Millennium Reanalysis (LMR) project71,109. The DA method of the LMR reconstructs full fields rather than just global mean temperature, and it holds advantages over competing methods such as linear regression, both frequentist and Bayesian71. Because it uses precomputed output from a climate model, this approach is known as offline data assimilation. Offline assimilation enables vast improvements in computational cost with modest loss of reconstruction skill110,111. The DA code used in this paper is adapted from an intermediate version of the code being developed in the LMR project.

The DA results in this study use information from two sources: the PAGES 2k v2.0.0 proxy network21 and the NCAR Community Climate System Model version 4 (CCSM4) last millennium simulation112. The proxy records provide the temporal variations related to climate at specific locations, while the model results quantify the covariances between different fields and locations in the climate system. In other words, proxies provide temporal variations while the model covariances spread this information through space and to other climate fields in a dynamically consistent way.

The DA methodology works by first drawing 200 random years from the output of the CCSM4 last millennium simulation. This collection of climate states represents an initial guess of the climate before any assimilation takes place, and is referred to as the model prior. This prior is initially the same for every year of the reconstruction. The DA method gets no information about temporal changes from the model, and also has no knowledge of when given forcings are present in the climate system. It is therefore worthwhile to compare forced responses in the model and the DA reconstruction as we do in Figs. 2–4.

To relate proxy observations to the model states, proxy system models33 (PSMs) must be used. While the LMR framework allows for nonlinear PSMs, recent work shows that linear PSMs help safeguard against climate model biases113, which can be sizable. For tree-ring widths, which can be sensitive to temperature, moisture or both, we use a bivariate linear regression to relate ring width to temperature (HadCRUT419, spatially interpolated via kriging25) and moisture (precipitation from the Global Precipitation Climatology Centre114). For all other proxy types, we use a univariate linear regression to the temperature dataset alone.

Using these relationships, an initial estimate of each proxy value is computed from the prior. For every year, the difference between each proxy estimate and the actual proxy observation is computed, and a Kalman filter is used to update the climate state so that these differences are reduced. Because the climate system is updated using the model covariances quantified in the prior, the end result is a dynamically consistent estimate of climate-system anomalies informed by the proxy data. To be used in this assimilation, PAGES 2k v2.0.0 records need to exhibit annual (or better) resolution, as well as at least 25 yr of overlap with the instrumental datasets to calibrate the PSMs. For the primary experiment (which uses the R-FDR screened proxies), 229 records were assimilated, from 134 tree-ring chronologies, 68 corals or sclerosponges, 21 glacier ice cores, 5 lake sediments and 1 bivalve.

Model simulations used

We use the past millennium runs of the following CMIP5 model simulations, yielding an ensemble of 23 simulations: BCC-CSM115, CCSM4112, MPI_ESM_P (members 1–3)116,117,118, CSIROMk3L-1-2119, GISS-E2-R (members 121, 124 and 127)120, HadCM343, CESM_1 (members 1–13)44. Internal variability was generated from 29 CMIP5 pre-industrial control simulations of length 499−1,198 yr, giving 21,427 yr of internal variability (same models as before plus ACCESS1, BNU-ESM, CanESM2, CMCC-CM, CNRM-CM5, FGOALS (g2, s2), FIO-ESM, GFDL (2G, 2M), HadGEM2, INMCM4, IPSL-CM5A-LR, MRI-CGCM3 and NorESM1). All simulations were aggregated annually by April–March GMST and filtered in the same way as the reconstructions.

Post-processing for figures

To generate anomalies for a given reference period, we subtract the temporal average of each method’s ensemble mean over this period from all ensemble members of this method. For low- and bandpass filtering, we use butterworth filters from the R-package signal.

Pre-industrial trends

The significance of the linear pre-industrial trends of the unfiltered ensemble median reconstructions for all methods was tested using the Mann–Kendall test (n = 1,850, two-tailed) and block-bootstrapping (tsboot() function in R, 100 yr blocks) over the period 0–1850. Both tests yield significant (α = 0.05) negative trends for all reconstruction methods.

Multidecadal GMST and data–model agreement

Multidecadal GMST variability was analysed by applying a 30–200 yr bandpass butterworth filter to all data (Table 1 and Fig. 2). See Supplementary Fig. 18 for an alternative of Fig. 2 using a 20–100 yr bandwidth. To account for the very high autocorrelation strongly reducing degrees of freedom, and non-normal distributions in the 30–200 yr bandpass-filtered reconstruction and model data, we tested the significance of data–model coherence by comparison with random noise time series. We generated a red noise time series for each reconstruction and model ensemble member with the same AR(1) coefficient as the corresponding reconstruction or model data. These noise time series were then subjected to the 30–200 yr bandpass filter and correlated to each other, as was done for the reconstructions and models. The percentages of data versus model correlations that exceed the noise-based correlations are provided in Table 1. The data–model comparisons are calculated over the 1300–2000 ce period, for which uncertainties in all datasets are relatively low (see main text).

Detection and attribution

To determine the extent of forced variability in the reconstructions, a TLS regression of the fingerprint of external forcings on the reconstructions was performed (Fig. 3). A given temperature reconstruction Y(t) decomposes into a linear combination of the m fingerprints Xi(t) for the climatic response to external forcings, and the internal variability ν0(t):

$$Y(t) = \mathop {\sum}\limits_{i = 1}^m ( X_i(t) - \nu _i(t))\times \beta _i + \nu _0(t)$$
(3)

Each fingerprint Xi(t), given by a model simulation, has its own implementation of internal variability νi(t) that is decreased by averaging over an ensemble of multiple simulations.

To determine the scaling factors βi, we performed a TLS regression, accounting for the presence of noise in both the regressor and regressand. Such a regression is commonly solved using a regularized optimal detection approach, which was introduced by ref. 121, building on earlier work by ref. 41. Optimization focuses the regression on features of the forced signal that are most distinct from internal variability. Here, we choose not to optimize the signal and instead carry out a simpler non-optimized regression, following ref. 43. As described in ref. 41, the scaling factors and the noise-reduced fingerprints \(\tilde X(t)\) and reconstructions \(\tilde Y(t)\) can be calculated, using a singular-value decomposition, and defined as:

$$\tilde Y(t) = Y(t) - \nu _0(t)$$
(4)
$$\tilde X(t) = \mathop {\sum}\limits_{i = 1}^m {\left( {X_i(t) - \nu _i(t)} \right)}$$
(5)
$$\tilde Z(t) = [\tilde Y(t),\tilde X(t)]$$
(6)

To account for the uncertainty due to internal variability, we calculated a range of 1,000 scaling factors per reconstruction by superimposing random slices of internal variability sampled from pre-industrial control simulations on the columns of \(\tilde Z(t)\). Thus we obtained a distribution of 1 × 106 scaling factors per reconstruction method and fingerprint, accounting for uncertainty from both the reconstruction method and internal variability. The effect of a forcing is said to be detectable if a scaling factor to that forcing is found to be significantly greater than zero. A difference in response to forcing in the model simulations and in the reconstructions is found if the estimated 90% scaling factor confidence interval does not encompass one.

Two different analyses were carried out. The first regresses model simulations with all forcings onto the reconstructions over the time period 1300–1800 ce. This tests whether the response to external forcings is detectable (Fig. 3a, left). For this, we used the ensemble mean of the 13 CESM-LME simulations. Our results are robust to using the CESM model only or the full multimodel mean of all 23 simulations (Supplementary Fig. 7). The second analysis uses a linear combination of the fingerprints of volcanic forcing, solar forcing and GHG forcings, which were generated from the ensemble mean of CESM-LME simulations, regressed onto the reconstructions. This analysis disentangles the forced response into these three different causes (Fig. 3a, right).

The smallest eigenvalue of the singular-value decomposition can be used as an estimate of the sum-of-squares residual of the regression, which is compared, in Fig. 3b, to the model internal variability calculated as the standard deviation of the 500 yr slices of pre-industrial control simulations. As a further estimate of unforced multidecadal GMST variability, we use the 30–200 yr bandpass filtered reconstructions and model simulations during the period 850–1100 ce (brown boxplots in Fig. 3b), which reflects a quiet period of relatively little forcing activity (that is, few large volcanic eruptions and relatively constant solar forcing, see also ref. 55).

To test the consistency of estimates of the unforced variance (Fig. 3b, left, right), we applied the following procedures.We first quantified the fraction of estimates based on the D&A residuals that lie within the confidence range of the estimate from control simulations. Note that the estimates based on the D&A residuals represent the ‘true’ observed variability, so should be distributed about the true value. In contrast, each control simulation reflects one possible realization of internal variability, which may be different from the ‘real’ sample. Thus, a difference in the medians between the two estimates does not necessarily reflect an inconsistency, but can arise from internal variability alone. However, the estimates from the observed situation should be within the confidence range of estimates from the control runs. This is indeed the case for 98.7% of estimates from the D&A residuals (n1 = 7,000), which are within the 5%–95% range of control run estimates (n2 = 43). Second, we calculate all possible comparisons (subtractions) between estimates from the D&A and control runs (n = n1 × n2 = 301,000). To assess consistency of the lines of evidence, we tested whether the confidence range of these comparisons encompasses zero. Supplementary Fig. 8 (top) shows that it does, and in fact the value of zero is within the interquartile range of estimates.

The same observation is also valid for the right-hand side of Fig. 3b, indicating consistency between the estimates of unforced variability from reconstructions and simulations within the Medieval Quiet Period between 850–1100 ce (Supplementary Fig. 8, bottom). This is confirmed by a Kruskal–Wallis non-parametric test122 for differences of medians. Such non-parametric procedures may be sensitive to violation of the assumption of independence both within and between samples from populations. For the comparison of reconstructions and simulations for the Medieval Quiet Period, we expect that the largest non-independence stems from the definition of the period chosen. We therefore applied two different sensitivity tests: block-bootstrapping the period chosen, and bootstrapping the test statistic itself. Both sensitivity tests suggest that the inference is stable to sampling partly independent periods within the Medieval Quiet Period for the comparison.

Multidecadal trends

Fifty-one-year running linear GMST trends are calculated for each ensemble member of the interannual (unfiltered) reconstructions (Fig. 4). Trend values are allocated to the year at the end of each 51 yr period. The ensemble median across all 7,000 multimethod ensemble members (seven methods with a 1,000-member ensemble for each method) of each overlapping trend is displayed as black line in Fig. 4a. The upper range of natural variability in these multidecadal trends is estimated using the 97.5th percentile across all trends and ensemble members between 1–1850 ce. For the model simulations, the corresponding 97.5th percentile is calculated back to the first year, where all simulations have data (850 ce) and for the control simulations across their respective period of data availability. Instrumental trends are calculated from the reconstruction target time series between 1850–2017 ce.

For Fig. 4b, the running linear trends are calculated for lengths between n = 10 and n = 150 yr for all 7,000 ensemble members. The fraction of ensemble members for which the centre year of the largest n-year trend over the Common Era falls into the post-1850 period is displayed. AR–noise proxy reconstructions shown in Fig. 4b were generated as follows: for each proxy record, a random noise time series with the same AR spectrum as the real proxy time series was generated using the hosking.sim() function in R. Each noise time series was truncated to the same length as the corresponding real proxy record. PCR, CPS and PAI reconstructions were then generated using these noise time series as input data, but the same parameters and target data as the real proxy reconstructions. Note that these noise proxies mimic the full AR spectrum of the real proxies, but the resulting noise proxies are uncorrelated with each other. Alternative results based on cross-correlated AR(1) noise proxies confirm our findings and are presented in Supplementary Fig. 20. The random numbers (brown line) are calculated by simply dividing the number of trend segments falling entirely within 1850–2000 (for example 140 for 10 yr trends) by the number of trends within the full 2,000 yr reconstruction period (1990 for 10 yr trends). This number decreases with increasing trend length.

EBM

The temporal evolution of the Earth’s global energy balance (Fig. 4) was computed by solving the ordinary differential equation57

$$C\frac{{\mathrm {d}T}}{{\mathrm {d}t}} = (1 - \alpha )\times Q - \sigma \epsilon T^4$$
(7)

where C is the effective heat capacity of the system, set here to c = 9.01 × 108 (in J m−2 K−1) corresponding to an upper ocean box. The effective incoming radiation is reduced by the albedo (α = 0.3) and depends on the incoming solar flux Q = S0/4 + ΔF, where S0 is the solar constant (S0 = 1,365 W m−2) and ΔF is a stochastic forcing term. The longwave outgoing radiation term is given by the product of the Stefan–Boltzmann constant σB = 5.67108 W m−2 K−4, and an emissivity term \(\epsilon = 0.6\) with the fourth power of the system’s temperature.

Here C is that of a 1 m2 surface oceanic box, computed as f · ρ · c · h, where f = 0.7 is the scaling for the approximate ocean area fraction of the planet, ρ the density of water, c the specific heat capacity of water and h the depth of the box. For a fixed value of Q, the ordinary differential equation solution describes the system’s convergence to an equilibrium state. However, in a simplified experiment, volcanic perturbations can also be considered disturbances in the energy balance, modelled by a radiative forcing anomaly relative to the equilibrium state ΔF.

$$C\frac{{\mathrm {d}}T}{{\mathrm {d}}t} = (1 - \alpha )\times (Q + {{\rm{\Delta}}} F) - \sigma \epsilon T^4$$
(8)

We obtain ΔF (all = CO2 + solar + volcanic, volcanic) from the forcing reconstructions of ref. 53, translating original values values to effective radiative forcing1,123 to units of W m−2. Specifically, for the aerosol optical depth (AOD), we use −25 W m−2 per unit AOD1. We run the EBM for two different forcing combinations: volcanic only (Fig. 4 and Supplementary Fig. 14) and all = CO2 + solar + volcanic (Supplementary Fig. 14). The EBM response for both forcing scenarios shows consistent multidecadal variability (Supplementary Fig. 14). The EBM also demonstrates that with an increase of the frequency of volcanic events, the relative proportion of variance in the multidecadal to centennial band increases (Supplementary Fig. 15).