Main

Palaeoclimatologists have long sought to refine estimates of temperature changes during the LGM, as both a benchmark for climate models and a constraint on Earth’s climate sensitivity. In the 1970s, the Climate Long-Range Investigation, Mapping and Prediction (CLIMAP) project collated assemblages of foraminifera, radiolarians and coccolithophores and used transfer functions to create maps of seasonal sea surface temperatures (SSTs) for the LGM1. Along with geological constraints on sea level and ice sheet extent, these maps were used as boundary conditions for pioneering atmospheric global climate model (GCM) simulations—the first Paleoclimate Modelling Intercomparison Project (PMIP)5. Three decades later, the Multiproxy Approach for the Reconstruction of the Glacial Ocean Surface (MARGO) project remapped the LGM oceans using foraminiferal, radiolarian, diatom and dinoflagellate transfer functions and two geochemical proxies—the unsaturation index of alkenones (\({U}_{37}^{{K}^{{\prime} }}\)) and the Mg/Ca ratio of planktic foraminifera2. This product has served as a touchstone for model–data comparison in the second and third phases of PMIP (PMIP2 and PMIP3)6 as well as calculations of climate sensitivity7.

In spite of this extensive work, estimates of global cooling during the LGM remain poorly constrained due to proxy uncertainties and methodological limitations. Microfossils occasionally present ‘no-analogue’ assemblages; that is, groups of species that are not observed today and therefore are difficult to interpret. In the LGM in particular, no-analogue assemblages appear in north Atlantic dinocysts1 and tropical Pacific foraminifera8, and have cast doubt on the CLIMAP and MARGO inference of relatively mild LGM cooling in the tropics and subtropics9,10,11. Likewise, geochemical proxies are subject to seasonal biases and sensitivity to non-thermal controls, all of which affect calculated SSTs12,13. Beyond proxy uncertainties, the data from the LGM present a methodological challenge in that they are not evenly distributed in space; the SST data cluster near coasts, where sediment accumulation rates are high. This heterogeneous sample distribution complicates the calculation of both regional and global average values. Furthermore, the translation of changes in SST to global mean surface air temperature (GMST)—the quantity needed for calculations of climate sensitivity—requires the use of an uncertain scaling factor14. As a result of these uncertainties, estimates of the change in LGM GMST (ΔGMST) range from −1.7 °C to −8.0 °C (refs. 7,14,15,16,17,18,19,20), yielding poorly bounded estimates of climate sensitivity of 1–6 °C per doubling of CO2 (ref. 21).

Here, we infer the magnitude and spatial pattern of LGM cooling using geochemical SST proxies, Bayesian proxy forward models, isotope-enabled climate model simulations and offline data assimilation. The SST proxy observations are assimilated with new simulations conducted with the isotope-enabled Community Earth System Model (iCESM)22. The resulting estimates of ΔGMST are combined with published constraints on radiative forcing to produce probabilistic estimates of climate sensitivity based on the LGM climate state.

Paleoclimate data assimilation

Our data collection consists of 956 LGM (23–19 kyr ago, ka) and 879 late Holocene (4–0 ka) SST proxies (Fig. 1). The proxy values are averages over the chosen time intervals and in many cases come from the same location, such that the total numbers of unique locations for each timeslice are 636 and 664, respectively (see Methods for further details). For the purposes of this study, the late Holocene average is interpreted as representative of the pre-industrial (PI) climate state, and is the benchmark from which we compute LGM cooling (see Methods). Distinct from previous work, this study focuses exclusively on geochemical proxies for SST; specifically, \({U}_{37}^{{K}^{{\prime} }}\), the TetraEther indeX of 86 carbons (TEX86), δ18O and Mg/Ca. We previously developed Bayesian models for each of these proxy systems12,13,23,24, which enables us to propagate proxy uncertainties and seasonal bias into the data assimilation. Although including marine assemblage data would improve spatial coverage, the outstanding no-analogue problems and lack of comparable Bayesian models prevent us from using these data in the framework presented here. Likewise, we do not assimilate terrestrial temperature estimates, because many of these are also based on pollen or other microfossil assemblages.

Fig. 1: Locations of geochemical SST proxies used for the LGM climate reconstruction.
figure 1

a, Proxy sites for the LGM (956). b, Proxy sites for the late Holocene (LH; 879). Proxies are colour-coded by type and the number of each is shown in parentheses.

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To circumvent problems associated with spatial representation and averaging, we use an offline data assimilation technique25 (see Methods) to blend information from proxies with full-field dynamical constraints from iCESM. The assimilation begins with an ensemble ‘prior’ of possible climate states taken from the model; in our case, these are 50 yr average states from simulations of the glacial state (21 and 18 ka) and the late Holocene (3 ka and PI) (see Methods). The water-isotope-enabled model simulations facilitate the direct assimilation of δ18O data (comprising 60% of our collection; Fig. 1), without the need to rely on empirical relationships between seawater δ18O and salinity derived from present-day observations. At locations where there are proxy data, values from the ensemble prior are translated into proxy units using our Bayesian forward models to calculate the ‘innovation’—the difference between the observed proxy value and the computed value from the model ensemble. The innovation is weighted by the Kalman gain, which considers both the covariance of the proxy with the rest of the climate fields and the uncertainties in the proxy observation and the model ensemble. The weighted innovation is then added to the prior ensemble to produce an ensemble of assimilated (analysis) climate states (see Methods for a mathematical description). For each time interval (LGM and late Holocene), we conducted 25 assimilation experiments with a 40-member model ensemble in which we withheld 25% of the proxy data at random to calculate verification statistics (see Methods). These collectively yield a total of 1,000 ensemble realizations of LGM and late-Holocene climates.

Spatial patterns in LGM cooling

The assimilated SST field shows distinctive spatial patterns in LGM cooling (Fig. 2a), with changes in SST (ΔSST) in excess of −8 °C in the north Atlantic, north Pacific and the Pacific sector of the Southern Ocean; enhanced cooling in eastern boundary upwelling zones; and reduced cooling in the western boundary regions. Many of these patterns are broadly consistent with CLIMAP and MARGO; however, an important difference is that there is no evidence for warming in the subtropical gyres, a feature associated with assemblage data1,2 (Fig. 2a). Strong cooling near 40° N in the Pacific reflects a southward shift in the subpolar gyre; this agrees well with independent model simulations of the LGM and a recent regional synthesis of δ18O data26. In the Indian Ocean, our reconstructed cooling pattern closely resembles the proxies and reflects the impact of the exposed Sunda and Sahul shelves27. Previous investigations of cooling in the glacial tropical Pacific offer conflicting conclusions: some suggest enhanced cooling in the eastern equatorial Pacific (EEP; as we observe here)8, whereas others suggest greater cooling in the warm pool28. Analysis of our proxy collection (separate from the assimilated product) indicates that there is no significant difference in the magnitude of cooling between the warm pool and EEP (−0.2 ± 1.0 °C, 2σ, see Methods). This could reflect a limitation of the proxy network, which is biased towards the coasts (Fig. 1). The stronger cooling in the EEP in our assimilated product thus reflects the CESM prior and possibly the influence of proxies that are teleconnected to the EEP, such as those situated along the California margin.

Fig. 2: Global changes in temperature during the LGM derived from palaeoclimate data assimilation.
figure 2

a, LGM–late Holocene ΔSST. b, LGM–late Holocene ΔSAT. c, LGM ΔGSST. d, LGM ΔGMST. In c and d, dots represent the median values and bars the 95% CI for values derived from the data, the data assimilation (DA) and the model prior (Model).

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The covariance of SST with surface air temperature (SAT) allows us to recover the latter directly from the assimilation algorithm, rather than having to scale from one to the other14. Over land, we observe the expected large cooling over the Northern Hemisphere ice sheets, but noticeably little cooling in Alaska and western Beringia (Fig. 2b). These results agree with observations that these locations remained unglaciated during the LGM29 and experienced minimal cooling or even a slight warming30, probably due to dynamical changes induced by the Laurentide Ice Sheet31,32.

Our reconstruction provides updated estimates of tropical cooling. The glacial change in SAT across the tropics (30° S–30° N) is −3.9 °C (−4.2 to −3.7 °C; 95% confidence interval, CI). This estimate is greater than the PMIP2 and PMIP3 multi-model range of −3.2 to −1.6 °C (ref. 33), but is not as large as calculations based on snowlines of tropical glaciers34 and noble gases35 (approximately −5 °C). Our reconstruction indicates that the change in tropical SSTs was −3.5 °C (−3.8 to −3.3 °C, 95% CI). This value is larger than the spatial mean computed from the tropical SST proxies on their own (−2.6 °C, −3.0 to −2.3 °C, 95% CI), partly due to the enhanced cooling throughout the east-central tropical Pacific in the assimilated field (Fig. 2a). However, the magnitude of both the proxy- and data assimilation-inferred tropical SST cooling is far greater than CLIMAP or MARGO estimates (−0.8 °C and −1.5 °C, respectively) and closer to the estimates of Ballantyne et al.10 (−2.7 °C) and Lea et al.36 (−2.8 °C).

Validation with δ18O of precipitation

To assess the reliability of our reconstruction, we included the δ18O of precipitation (δ18Op) in our model prior to compare the assimilated value to independent ice core and speleothem proxy data (see Methods). Overall, our reconstruction explains 67% of the variance in observed Δδ18Op (Fig. 3a). This is a marked improvement over the prior (which explains only 35% of the variance; Extended Data Fig. 1) and suggests that our reconstruction provides a reasonable estimate of global LGM climate. The improvement comes mainly from the ice core sites, which show a better match to observations after the assimilation. A notable feature captured by our reconstruction is the difference in Δδ18Op between ice core sites in west and east Antarctica (Fig. 3b); the latter region is warmer (Fig. 2b) and experiences less isotopic depletion. At face value, a warmer east Antarctica contradicts previous work: at Epica Dome C, ice core δ18O is interpreted to indicate a change in SAT (ΔSAT) of approximately −8 °C (ref. 37), whereas our assimilated product indicates a more modest 5 °C of cooling. However, the former estimate assumes that the δ18O–SAT relationship remains constant in time37. Isotope-enabled modelling experiments have shown that the δ18O–SAT slope in Antarctica may have been different during the LGM, and strongly depends on changes in Southern Ocean SSTs38.

Fig. 3: Validation of the data assimilation with independent δ18Op data.
figure 3

a, Observed changes in ice core (Antarctica and Greenland) and speleothem LGM–late Holocene δ18Op compared with predicted changes from the data assimilation ensemble. Dots indicate median values, error bars represent the 95% CI. The R2 value is shown in the lower right corner. b, Map of the median changes in LGM–late Holocene δ18Op from the data assimilation, overlain with ice core and speleothem observations (dots). Speleothem δ18O values have been converted from δ18O of calcite or aragonite to δ18Op (in ‰ Vienna standard mean ocean water, VSMOW) before plotting (see Methods).

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Global temperature change

As the data assimilation technique yields spatially complete fields, we can compute both global SST (GSST) and GMST change during the LGM without needing to consider missing values or use a scaling factor14. Our calculated change in GSST (ΔGSST) is −3.1 °C (−3.4 to −2.9 °C, 95% CI) (Fig. 2c). This is more tightly constrained than the model prior, which spans −4 to −2.7 °C (Fig. 2c), reflecting the influence of the data. The assimilated ΔGSST is slightly larger than the proxy data suggest (−2.9 °C, −3.0 to −2.7 °C, 95% CI, Fig. 2c), a difference that arises from biases due to incomplete and uneven proxy data sampling. The ΔGSST from data assimilation agrees well with estimates of glacial cooling based on a subset of SST proxies spanning multiple glacial–interglacial cycles (−3.1 °C)20, but is much larger than CLIMAP (−1.2 °C) and MARGO (−1.9 °C) estimates, as well as previous LGM ocean state estimates that used the MARGO dataset (−2.2 to −2.0 °C)39,40.

The change in GMST in the assimilated product is −6.1 °C (−6.5 to −5.7 °C, 95% CI) (Fig. 2d). This result agrees with a number of previous studies, including those that estimated LGM cooling from a restricted network of proxies spanning multiple glacial–interglacial cycles14,20, noble gas measurements in ice cores19 and changes in tropical SSTs15,16 (Fig. 4a). However, our calculated ΔGMST does not overlap with the estimates that used the MARGO product7,18 (−4 to −2 °C), or an average of time-continuous marine and terrestrial temperature proxies17 (Fig. 4a).

Fig. 4: LGM global temperature change and climate sensitivity derived from data assimilation.
figure 4

a, Estimates of ΔGMST from previous studies (vertical bars represent the 95% CI, dots show the median) compared with the data assimilation result (the height of the horizontal bar represents the 95% CI). SC11, ref. 7; SH12, ref. 17; AH13, ref. 18; FT19, ref. 20; SVD06, ref. 15; HO10, ref. 16; SN16, ref. 14; BE18, ref. 19. b, LGM-constrained ECS determined using the ΔGMST from data assimilation. Dots indicate median values and coloured bars indicate the 95% CI associated with ΔGMST and RGHG, RICE and RAE. The lower bar shows the range of ECS with an ice-sheet efficacy of 0.7.

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One caveat of our results is that they are based on assimilation with an ensemble prior from a single climate model. As discussed above, the structure of the temperature fields in CESM directly shapes our assimilated fields. In addition, CESM1.2 simulates a larger LGM cooling (6.7 °C) than the PMIP2 and PMIP3 models (5.9 to 3.1 °C)21, although CESM ECS in our configuration is well within the multiple model range of the Coupled Model Intercomparison Project Phase 541 (see Methods). An assimilation with a model with different covariance structures and a more moderate global cooling could yield a different result. Future work should thus test our results by assimilating data across different models, preferably with water isotope capabilities. This said, several lines of evidence suggest that our inferred median value of approximately −6 °C is realistic. First, as already noted, our result agrees well with a number of previous studies (Fig. 4a). Second, our data-only GMST estimate has a median value of −5.6 °C and the assimilated estimate falls entirely within its error bounds (Fig. 2d). Finally, an assessment of bias in the model prior, based on rank histograms of withheld proxies, suggests minimal bias in the mean (Extended Data Fig. 2). However, the uncertainties associated with our ΔGMST estimate could be too narrow due to a lack of structural variability in the model prior (see Methods and Extended Data Fig. 2).

Equilbrium climate sensitivity

Our GMST results can be used to provide updated constraints on ECS. To calculate an ECS that approximates the classical ‘Charney’ definition, we must consider, in addition to GHG forcing (ΔRGHG), the slow feedback processes that affect LGM climate, which (following ref. 42) are treated here as radiative forcings. These include albedo changes associated with the expanded land ice and lowered sea level (ΔRICE) and increases in mineral dust aerosols (ΔRAE). Although vegetation changes could also impact LGM climate, we do not consider these here because biome reconstructions are poorly defined outside of the northern high latitudes43 and little is known about the associated radiative forcing44 and efficacy (effectiveness in changing surface temperature). Likewise, we do not consider non-dust aerosol emissions, as these are not well constrained for the LGM.

We estimate ΔRGHG, ΔRICE and ΔRAE from published syntheses and propagate uncertainties associated with these into the calculations of ECS (see Methods). As these estimates derive from simplified expressions or multi-model estimates, they are relatively independent from CESM. We show results with and without ΔRAE for comparison with previous work (Fig. 4b). Without dust aerosols, ECS is 3.8 °C (2.8–4.7 °C, 95% CI); with dust aerosols, ECS is 3.4 °C (2.4–4.5 °C, 95% CI). With our ΔGMST from data assimilation, global temperature change is no longer the primary source of uncertainty; it accounts for about 25% of the 95% CI in each estimate (Fig. 4b). Instead, most of the uncertainty comes from the forcings (Fig. 4b). ΔRGHG and the forcing associated with a doubling of CO2 can be directly estimated from ice core GHG concentrations and simplified equations (−2.48 W m−2 and 3.80 W m−2, respectively)45 but still carry uncertainties and account for about 10% of the 95% CI. ΔRICE is estimated from CESM and available PMIP2 and PMIP3 simulations (see Methods) and its value varies between −2.6 and −5.2 W m−2, thus accounting for ~40% of the 95% CI. Finally, although it is well known that the LGM atmosphere was dustier46, the magnitude of ΔRAE also varies widely across models (0 to −2 W m−2)46 and contributes ~25% of the 95% CI.

It has recently been argued that ice sheets are not as ‘effective’ as GHGs in terms of global radiative forcing, because their impact might be concentrated at high latitudes47,48,49. We explore this possibility by presenting a distribution of ECS with a reduced ice sheet efficacy (ε) of 0.7 (rather than 1.0 as used above; see Methods) (Fig. 4b). Under this scenario, the median ECS rises to 4.0 °C with a 95% CI of 2.8–5.3 °C. However, these values should be interpreted with caution because ε is an under-constrained quantity.

Despite the broad distributions that uncertainties in the radiative forcings yield, we can use these new calculations to make probabilistic statements concerning ECS. Given the uncertainty space explored here, the LGM data suggest that ECS is virtually certain (>99% probability) to be above 2.2 °C and below 5.5 °C, with the latter only plausible under a condition of low ε. Assuming ε = 1, ECS is very likely (90% probability) to be between 2.5 °C and 4.3 °C. These are tighter constraints on LGM-constrained ECS than those stated in Assessment Report 5 of the Intergovernmental Panel on Climate Change (IPCC) (1–6 °C per doubling of CO2)21, and are in excellent agreement with the traditional consensus range of 2–4.5 °C (refs. 3,4). Unlike previous work, we find little evidence that ECS estimates based on the LGM climate are abnormally low7,44. ECS is unlikely to remain constant across climate states; palaeoclimate and modelling evidence suggests that it scales with background temperature, with lower values during cold climates and higher values during warm states50. Taking this into consideration, our LGM results place a strong constraint on the minimum ECS in the climate system, which is almost certainly greater than 2 °C, and is more likely to lie between 3 and 4 °C.

Methods

SST proxy data collection

We compiled a total of 956 and 879 average proxy values from the LGM and late Holocene (LH) time periods, respectively. In many cases, multiple types of proxy data are available from the same location. Using a cutoff criterion for ‘same location’ of 0.1° in latitude or longitude, our proxy collection contains 636 and 664 unique locations for the LGM and LH, respectively. Following MARGO, the LGM was defined as 23–19 ka (ref. 2). The LH was defined as 4–0 ka to match the interval of time averaged for the LGM. This choice is consistent with previous LGM–LH comparisons12,51. For the purposes of this study, the LH is considered representative of pre-industrial conditions. Although climate has certainly changed within the past 4 kyr in response to shorter-term forcings such as volcanic eruptions and solar irradiance, we posit that this assumption is reasonable in the context of the large, slow climate changes associated with orbitally driven glacial–interglacial cycles. This differs from CLIMAP and MARGO, in which LGM cooling was calculated relative to twentieth-century observations1,2. However, historical observations carry their own distinctive type of uncertainty52 and also include the signature of anthropogenic global warming, so are not ideal in terms of isolating the LGM climatic response. Furthermore, using proxy estimates of the pre-industrial climate, rather than observational estimates, offers an ‘apples-to-apples’ comparison with the LGM.

The proxy data consist of both continuous time series that pass through the LGM and the LH, and data from ‘timeslice’ studies that focused on the average LGM climate state. The latter derive in part from the MARGO collection, with additional data from more recent studies. In many cases, LGM slice data were not accompanied by corresponding LH data. To provide matching data for these cases, we searched the core top datasets for each proxy system for a nearby value, with a cutoff radius of 1,000 km. This provided roughly similar spatial coverage between the LGM and LH target slices (Fig. 1). Core tops that were used as LH data were subsequently removed from the calibration datasets, and the parameters for the Bayesian forward models were recalculated without these data, to avoid circularity. However, as the number of core tops used in the data synthesis is small (100 or less) relative to the size of the core top datasets (~1,000) the effect on the model parameters was negligible.

The time series consist of 260 core sites with radiocarbon-based age models. Since we are only reconstructing the mean LGM and LH states, we did not apply any criteria regarding sampling resolution. Before taking time averages from these proxy data, we recalibrated all age models using the Marine13 radiocarbon curve53 and the BACON age modelling software (translated into Python as snakebacon and available at https://github.com/brews/snakebacon) to ensure consistent treatment. Not all of the core sites extend through the LGM and/or the LH, so not all of these sites are represented in the analysis. However, many of these sites typically contain more than one proxy for SST, thus the total number of data points from the time series is 335 for the LGM and 273 for the LH.

Model simulations

We employ iCESM22, which is based on CESM version 1.254. iCESM has the capability to explicitly simulate the transport and transformation of water isotopes in hydrological processes in the atmosphere, land, ocean and sea ice22. iCESM reproduces instrumental records of both physical climate and water isotopes reasonably well22,54.

We conducted iCESM1.2 simulations of the LH and the LGM, consisting of timeslices of the PI, 3, 18 and 21 ka. These simulations had a horizontal resolution of 1.9° × 2.5° (latitude × longitude) for the atmosphere and land, and a nominal 1° displaced-pole Greenland grid for the ocean and sea ice. The PI simulation used standard climatic forcings at ad 1850, and the slice simulations of 3, 18 and 21 ka used boundary conditions of GHGs, Earth orbits and ice sheets following the PMIP4 protocol55. Specifically, CO2, CH4 and N2O concentrations were 284.7 ppm, 791.6 ppb and 275.68 ppb for the PI; 275 ppm, 580 ppb and 270 ppb for 3 ka; 190 ppm, 370 ppb and 245 ppb for 18 ka; and 190 ppm, 375 ppb, and 200 ppb for 21 ka. Changes in surface elevation, albedo and land ocean distribution associated with the ice sheets were derived from the ICE-6G reconstruction56. For the 18 and 21 ka simulations, global volume mean seawater δ18O was set to 1.05‰ in accordance with ice-volume effects. A value of 0.05‰ was used for the PI and 3 ka simulations. The simulations were run with a prescribed satellite phenology in the land model. This choice was based on the overall poorer simulation of vegetation processes with a prognostic phenology, which could potentially be more problematic for palaeoclimate57. The LH (3 ka and PI) and the LGM slices (21 and 18 ka) were initialized from equilibrated CESM1.2 simulations with water isotopes27,58 and extended for 900 years. All of the slices reached quasi-equilibrium in both the physical climate and the water isotopes with a top-of-atmosphere energy imbalance of less than 0.09 W m−2.

Model ECS was estimated to be 3.6 °C and 2.8 °C in the PI and LGM simulations, respectively. The model ECS was calculated as the ΔGMST between a baseline simulation (PI or LGM) and a simulation with twice the corresponding atmospheric CO2 amount in a slab ocean model (SOM) configuration. Ocean dynamics were inactive in SOM simulations and their effects were approximated using prescribed mixed-layer depth and heat transport convergence that were derived from the fully coupled PI or LGM simulation. The PI ECS of 3.6 °C is in the middle of the multiple-model range (2.1–4.7 °C) in the Coupled Model Intercomparison Project phase 541, but slightly smaller than the published values of 4.0–4.2 °C in CESM1.259,60. The difference is probably due to the prescribed satellite phenology in our configuration of the land model, which excludes vegetation feedbacks61.

We also made use of available iCESM1.3 simulations of the LGM and PI58. iCESM1.3 differs from iCESM1.2 primarily in the gravity wave scheme, along with a few bug fixes in the cloud microphysics and radiation62. The iCESM1.3 PI and LGM simulations have lengths of 400 and 1,000 years, respectively.

Data assimilation

The data assimilation technique used here is the ensemble square-root Kalman filter63; specifically, an offline method that has previously been used to reconstruct climate during the past millennium25,64,65,66. This data assimilation method solves the following equation to compute an ensemble of climate states (Xanalysis) from prior states taken from a climate model (Xprior) and proxy data observations (y):

$${X}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}={X}_{{\rm{p}}{\rm{r}}{\rm{i}}{\rm{o}}{\rm{r}}}+K(y-{Y}_{{\rm{e}}})$$
(1)

Xprior is an ensemble of climate states taken from iCESM, of dimension N state vector elements (SST, SAT and so on collapsed into a vector) by M ensemble members (40, in our case). In a typical data assimilation application, the length of time represented by the ensemble would equal the length of time represented by the data; for example, annual data would be used to update an annual prior. However, in our case the data represent average conditions across 4,000 years. As we cannot run iCESM for 100,000 years or more, we must use a different time average for our model prior. Experimentation with time averaging the model states revealed that once the average exceeded the interannual timescale, the patterns in the assimilated fields, as well as GSST and GMST, were largely insensitive to the length of the average (Extended Data Fig. 3). Thus we chose 50 years—the longest time average that we could use while still retaining enough ensemble members for the assimilation technique (40 members).

y − Ye is the innovation: the difference between the column vector y (dimensions P × 1) of observed proxy values for each location P and the matrix Ye (dimensions P × M) of proxy values calculated from the model prior at the same locations. The innovation thus takes place in the native proxy units. Ye is calculated by feeding model variables from the prior ensemble (SST, SSS, δ18O of seawater) into previously published Bayesian forward models for each proxy type12,13,23,24 that convert those variables into an estimate in proxy units. These models take the general form:

$$\begin{array}{c}{\rm{P}}{\rm{r}}{\rm{o}}{\rm{x}}{\rm{y}}\sim {\mathscr{N}}(\mu ,{\sigma }^{2})\\ \mu =f({\rm{S}}{\rm{S}}{\rm{T}},{X}_{i}),\end{array}$$
(2)

where σ2 is the residual error variance associated with the function that describes μ, Xi denotes i additional X climatic influences that may contribute to the proxy, and ~ indicates ‘has the distribution of’. For complete details of each model, we refer the reader to the original publications12,13,23,24. Briefly, the \({U}_{37}^{{K}^{{\prime} }}\) and TEX86 forward models are only a function of SST (monthly values for \({U}_{37}^{{K}^{{\prime} }}\), to account for seasonal responses in the North Atlantic, North Pacific and Mediterranean regions12; annual for TEX86). The model for δ18O of planktic foraminifera (δ18Oc) requires monthly SST and the annual δ18O composition of seawater24. δ18Oc is computed for the optimal growing season using the species-specific hierarchical model described in ref. 24. The model for Mg/Ca of planktic foraminifera requires monthly SST and sea surface salinity (SSS), as well as surface water pH, bottom water calcite saturation state (Ω) and the cleaning method used in the laboratory13. The latter was recorded as part of the data collection effort, but for pH and Ω we must make assumptions, as iCESM does not simulate the ocean carbonate system. In the absence of good information regarding spatial changes in Ω, we assume that it is the same as today for each given site location, with values drawn from GLODAPv2 (ref. 67). For pH, we use modern estimates from GLODAPv2 for the LH timeslice, and then for the LGM we add 0.13 units to the modern values to account for the global increase in pH due to lowered CO2, following refs. 13,68. As with δ18Oc, Mg/Ca is forward-modelled for the optimal growing season using the species-specific hierarchical model described in ref. 13. In this manner, the seasonal preferences of foraminifera are explicitly accounted for in the assimilation.

K is the Kalman gain (of dimension N × P), which weights the innovation according to the covariance of the forward-modelled proxy value with the rest of the climate state, as well as the uncertainties of the model ensemble and the proxy observations. Following ref. 63, K is defined as:

$$K={W}_{{\rm{l}}{\rm{o}}{\rm{c}}}\circ [{X}_{{\rm{p}}{\rm{r}}{\rm{i}}{\rm{o}}{\rm{r}}}^{{\prime} }{Y}_{{\rm{e}}}^{{\prime} {\rm{T}}}/(n-1)]\times {[{Y}_{{\rm{l}}{\rm{o}}{\rm{c}}}\circ [{Y}_{{\rm{e}}}^{{\prime} }{Y}_{{\rm{e}}}^{{\prime} {\rm{T}}}/(n-1)]+R]}^{-1},$$
(3)

where \({X}_{{\rm{prior}}}^{{\prime} }\) and \({Y}_{{\rm{e}}}^{{\prime} }\) are the matrices of deviations, for example, \({X}_{{\rm{prior}}}^{{\prime} }={X}_{{\rm{prior}}}-{\bar{X}}_{{\rm{prior}}}\), R is a diagonal matrix of the uncertainty (as a variance) of each proxy observation, and Wloc and Yloc are weights that apply a covariance localization, a distance-weighted filter that limits the influence of each proxy in space69. Covariance localization is applied to minimize spurious relationships from producing artefacts at large distances away from the proxy location. Following ref. 25, we use the Gaspari–Cohn fifth-order polynomial70 with a specified cutoff radius (outside of which all covariance is eliminated) of 12,000 km (the rationale behind this value is discussed below). Wloc contains the localization weights of each N state vector element relative to each proxy estimate in Ye and is of dimension N × P. Yloc contains the localization weights of each proxy estimate in Ye relative to the other estimates in Ye and thus is of dimension P × P. n is the number of ensemble members, and division by (n − 1) is applied to obtain an unbiased estimate. ◦ denotes element-wise multiplication.

Following ref. 63, the update equation (equation (1)) is solved by decomposing the problem into an update of the mean value of the prior state (\({\bar{X}}_{{\rm{prior}}}\)) and the deviations from the mean (\({X}_{{\rm{prior}}}^{{\prime} }\)):

$${\bar{X}}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}={\bar{X}}_{{\rm{p}}{\rm{r}}{\rm{i}}{\rm{o}}{\rm{r}}}+K(y-\bar{{Y}_{{\rm{e}}}}),$$
(4)
$${X}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}^{{\prime} }={X}_{{\rm{p}}{\rm{r}}{\rm{i}}{\rm{o}}{\rm{r}}}^{{\prime} }-\mathop{K}\limits^{ \sim }{Y}_{{\rm{e}}}^{{\prime} },$$
(5)

where \(\tilde{K}\) is defined as:

$$\begin{array}{c}\mathop{K}\limits^{ \sim }={W}_{{\rm{l}}{\rm{o}}{\rm{c}}}\circ [{X}_{{\rm{p}}{\rm{r}}{\rm{i}}{\rm{o}}{\rm{r}}}^{{\prime} }{Y}_{{\rm{e}}}^{{\prime} {\rm{T}}}/(n-1)]\,{[{(\sqrt{{Y}_{{\rm{l}}{\rm{o}}{\rm{c}}}\circ [{Y}_{{\rm{e}}}^{{\prime} }{Y}_{{\rm{e}}}^{{\prime} {\rm{T}}}/(n-1)]+R})}^{-1}]}^{{\rm{T}}}\\ \,\times {[\sqrt{{Y}_{{\rm{l}}{\rm{o}}{\rm{c}}}\circ [{Y}_{{\rm{e}}}^{{\prime} }{Y}_{{\rm{e}}}^{{\prime} {\rm{T}}}/(n-1)]+R}+\sqrt{R}]}^{-1}.\end{array}$$
(6)

The full assimilated ensemble is then recovered through:

$${X}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}^{}={\bar{X}}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}+{X}_{{\rm{a}}{\rm{n}}{\rm{a}}{\rm{l}}{\rm{y}}{\rm{s}}{\rm{i}}{\rm{s}}}^{{\prime} }.$$
(7)

The assimilated ensemble has dimensions N × M, like the prior model ensemble.

The diagonal proxy error matrix R and the cutoff radius for the covariance localization weights are user-defined, so we used validation metrics based on withheld SST proxies and independent proxies for the oxygen isotopic composition of precipitation (δ18Op) to guide our choices. The most conservative choice for R is to assign the σ2 error terms (equation (2)) associated with the Bayesian forward models for each proxy to the diagonal of the matrix. As the forward models are trained on a globally distributed set of data, these terms represent the error in the proxy system on a global level and so we denote these as Rg. In temperature space, these translate to 1.5–4 °C 1σ errors. However, these are probably too high for an individual proxy location (otherwise, one would expect that the proxies would not be able to detect LGM cooling at all). Rather, at a single site, proxy uncertainty is expected to lie somewhere between analytical precision and the global error. Unfortunately, this can only be directly observed by analysing parallel sediment cores, which is not commonly done in palaeoceanography. Thus, to experimentally determine an optimal value of R, we systematically reduced it from Rg to Rg/500 and analysed validation statistics. Values of SST, SSS and δ18O of seawater from the assimilated ensemble were then forward-modelled to predict the withheld proxy values. We calculate both the coefficient of efficacy (CE)71 and the root mean square error (RMSE) between the observed and mean of the predicted proxy values. These were calculated in normalized units to account for the different ranges of absolute proxy values between \({U}_{37}^{{K}^{{\prime} }}\), TEX86, δ18O and Mg/Ca.

In addition, we calculated the R2 between observed Δδ18Op (LGM – PI) derived from ice cores and speleothems and predicted Δδ18Op from the assimilation. Model prior δ18Op in the assimilation was calculated as an annual mean weighted by the monthly precipitation amounts. Ice core Δδ18Op values were taken from the compilation in ref. 72 (their table 1). Speleothem Δδ18Op values were computed from the SISAL database, version 1b73. We first searched the SISAL database for sites that contained both LH (4–0.2 ka, to exclude anthropogenically influenced values) and LGM (23–19 ka) data and recorded the mean δ18O of calcite or aragonite. We then converted these average δ18O values to dripwater δ18O (considered analogous to δ18Op) following the recommendations of ref. 74 (their equations 1–3). This conversion accounts for the influence of temperature on fractionation as well as kinetic effects, and converts from the VPDB to the VSMOW scale. These calculations require an estimate of cave temperature; for this we use the SAT value from our data assimilation at the grid cells closest to the speleothem locations.

Extended Data Table 1 shows the validation results from scaling Rg. For both the LGM and LH, validation CE and RMSE are relatively insensitive to the choice of R, although there is slight improvement up to Rg/5. This is in part because the validation is being calculated across the globe and the spatial variation in proxy values is always large; indeed, even the prior produces decent prediction of the withheld proxies, particularly for the LH (Extended Data Table 1). More useful information can be gleaned from the comparison with independent δ18Op data. Here, we observe a substantial increase in R2 from the case of Rg to Rg/5 from 0.50 to 0.67. This suggests that an increase in proxy precision drives the assimilated product closer to a state that agrees with the δ18Op proxies. However, it is also clear that if the proxy uncertainty is set too low (Rg/500), the R2 drops back down (0.59). Thus, there appears to be an ideal value of R somewhere between Rg/5 and Rg/100. We choose the lowest scaling value of Rg/5—the most conservative choice. Rg/5 translates to ~0.6–1.3 °C of 1σ uncertainty, depending on the proxy. This is greater than analytical error (~0.3 °C 1σ) and strikes us as a reasonable estimate of site-specific error.

To find an optimal cutoff radius for the covariance localization, we experimented with values from ∞ (for example, no localization) to 6,000 km and analysed the same validation statistics as above. As was the case with varying R, validation CE and RMSE are not very sensitive to localization (Extended Data Table 2). However, comparison with δ18Op shows that the best skill is achieved with a cut-off radius of 12,000 km (Extended Data Table 2), so we use this value for our assimilation.

Each time interval (LGM and LH) is carried out as a separate ensemble assimilation consisting of 25 iterations, in which we withhold 25% of the proxies at random for validation. Across all iterations, the validation CE is 0.94 and 0.92 for the LH and LGM slices respectively, and the validation RMSE is 0.24 and 0.29 in normalized proxy units, respectively. Figure 3 shows the independent validation with the δ18Op proxies. The relatively good fit (R2 = 0.67) is a substantial improvement over the model prior (Extended Data Fig. 1). The improvement mainly comes from the ice core sites, which have an R2 of 0 in the prior case and an R2 of 0.18 in the assimilated product. However, these values are skewed by a negative outlier, the Camp Century ice core. Excluding this outlier, the R2 values are 0 and 0.41 in the prior and assimilated product, respectively, indicating substantial improvement. In contrast, the speleothem sites show less difference between the prior and posterior, with an R2 of 0.12 and 0.13, respectively. The speleothem group also has an outlier—Sofular Cave. Excluding this site, the R2 is 0.24 and 0.32 in the prior and assimilated product, respectively, indicating some improvement. The limited change at the speleothem sites may reflect the fact that prior variability in mean annual δ18Op is small at these site locations.

Influence of different proxy types

The advantage of using a multi-proxy approach is that structural biases associated with each proxy type counterbalance each other, resulting in a more robust estimate of LGM cooling. To investigate how each proxy type influences the assimilated result, we conducted experiments in which only δ18O, \({U}_{37}^{{K}^{{\prime} }}\) and Mg/Ca data were assimilated, respectively (TEX86 was not investigated as there are only 20 sites available). Extended Data Fig. 4 shows the assimilated ΔGSST and ΔGMST estimates for LGM cooling for each proxy type, alongside maps of proxy locations. Median values for ΔGSST are −3.0 °C, −3.1 °C and −3.3 °C for the \({U}_{37}^{{K}^{{\prime} }}\), Mg/Ca and δ18O experiments respectively, all of which are very close to the value using all proxy types (−3.1 °C). Error bounds for \({U}_{37}^{{K}^{{\prime} }}\) and Mg/Ca are larger and most probably reflect the smaller number of total data points assimilated. Median values for GMST are −6.1 °C, −6.6 °C and −6.3 °C for the \({U}_{37}^{{K}^{{\prime} }}\), Mg/Ca and δ18O experiments respectively. The \({U}_{37}^{{K}^{{\prime} }}\) and δ18O experiments yield results close to the value using all proxy types (−6.1 °C). Assimilation with Mg/Ca yields a slightly cooler result, which might reflect the spatial distribution of the proxies (that is, a concentration of data in the North Atlantic) or an imperfect estimation of non-thermal controls. Although temperature dominates foraminiferal Mg/Ca, salinity, pH and dissolution also exert substantial influence. These secondary factors are accounted for in the forward model (BAYMAG13), but we must make simple assumptions about the change in pH and bottom water saturation (see the description in Data assimilation above) for the LGM case that could deviate from actual conditions at the respective core sites.

Assessment of bias in the model prior

As a formal combination of model results and data, our data assimilation product and its uncertainties are inherently influenced by bias in the model’s ability to accurately reproduce the true LGM and LH climate states. However, we do not know what these true states were (especially the LGM), so we can only indirectly assess whether the model prior is ‘biased’ in this sense. To explore potential bias, we calculated proxy verification rank histograms75 for the LGM and LH time slices respectively (Extended Data Fig. 2). Rank histograms are simply counts of the rank of each observation (the proxy data withheld for validation) relative to each of the 40 ensemble members from the assimilated product (forward-modelled into proxy units). If a proxy ranks below all of the ensemble members, it gets a rank of 1; if it ranks above all the ensemble members, it ranks as 41. As the proxy observations themselves are not error-free, following ref. 75 we add our inferred proxy uncertainty of Rg/5 to the ensembles before calculating ranks.

A perfectly ideal model prior would produce a flat rank histogram, which indicates that the ensemble is drawn from the same distribution as the proxy; however, this is rarely the case as all models carry some structural biases relative to observations. Both the symmetry and the shape of the rank histogram can be used to intuit the nature of the bias75. In our case, we observe that our rank histograms are close to symmetrical (particularly for the LGM; Extended Data Fig. 2), which suggests that our model prior does not have too much bias in the mean (otherwise we would expect the proxies to systematically rank higher or lower than the ensemble). However, our histograms have a distinctive U shape (Extended Data Fig. 2), indicating that a number of proxies rank lower or higher than the assimilated ensembles. A U shape can arise if the error on the observations is underestimated; however, our validation against Δδ18Op data suggests that our estimate of this value (Rg/5) is robust. A U shape can also arise if there are structural biases in the proxies that are not accounted for in our forward models. However, we found that the U shape persists across all of our four proxy types, and that there are no geographical patterns in the locations of high and low ranks (not shown). These observations are encouraging in that they suggest that there is not a systematic error in a single proxy system. Nevertheless, the forward models are fit globally and thus do not account for errors that may arise due to localized non-thermal influences (such as, for example, nutrient availability) so we cannot completely rule out the possibility that structural proxy errors contribute to the shape of the rank histogram.

The most likely explanation for the U shape is that our model prior is underdispersive, or rather, lacks variability relative to what the proxies consider to be the ‘true state’. Experimentation with inflation of the model prior variance, and a range of time-averaging (5–50 years) did not substantially reduce the U-shaped rank histograms (not shown), which suggests it is not so much the fact that the 50-yr ensemble members lack variance, but rather that CESM maintains relatively consistent covariance patterns across the pre-industrial and glacial states, respectively. The structure of the SST, SSS and seawater isotope fields, compared to modern observations, are not perfect, with known biases22,76,77. This would cause proxies to consistently rank lower or higher in certain locations.

The implications of an underdispersive prior are mainly that our estimate of uncertainties—both in the assimilated fields and the estimates of global mean temperature—may be too small. As we do not know the true LGM state, we cannot estimate how overconfident our assimilated results are. We need model priors from multiple other isotope-enabled climate models to assess this more thoroughly, an experiment that must be left to future work, when such simulations become available.

Proxy-only estimates of LGM cooling

To provide a point of comparison for the data assimilation results, we computed global and tropical ΔSST and ΔGMST from the proxy data in isolation. The proxy data were calibrated to SST using a suite of Bayesian prediction models12,13,23,24 producing 1,000-member ensemble estimates for each data point. To approximate the proxy observational uncertainty used in the data assimilation (see discussion above), data were sorted along the ensemble dimension and normally distributed site-level error (\({\mathscr{N}}(0,0.5)\,^\circ {\rm{C}}\)) was added back to the ensemble. GSST was computed following the method of ref. 60—data were first binned and averaged into latitudinal bands, then latitudinal averages were used to calculate an area-weighted global average. As the results are sensitive to the size of the bin60, we computed an ensemble of GSST across bin sizes of 2.5° to 15° (at 2.5° intervals). GSST was scaled to GMST using the method of ref. 14, in which scaling factors (determined from PMIP LGM simulations) were drawn from a uniform distribution spanning values between 1.5 and 2.3. Results for ΔGSST and ΔGMST are shown in Fig. 2c, d.

Analysis of tropical Pacific cooling

As stated in the main text, we analysed LGM cooling across the tropical Pacific in the proxy data alone to compare with the data assimilation result. The proxy data were calibrated to SST as described in the previous subsection, and then the Pacific zonal gradient was computed for both the LGM and LH as the difference between the average SST in the western Pacific (10° S–10° N, 130–170° E) and the eastern Pacific (5° S–5° N, 75–140° W). We then computed the LGM – LH difference in the zonal gradient, yielding a median value of −0.2 °C. These calculations were conducted for all 1,000 ensemble members, yielding an uncertainty of 1.0 °C (2σ).

Climate sensitivity calculations

We calculate ECS as:

$${\rm{E}}{\rm{C}}{\rm{S}}=\frac{\Delta {\rm{G}}{\rm{M}}{\rm{S}}{\rm{T}}}{\Delta R}\times {F}_{2\times {{\rm{C}}{\rm{O}}}_{2}},$$
(8)

where ΔGMST is taken from the data assimilation, \({F}_{2\times {{\rm{CO}}}_{2}}\) is the forcing associated with the doubling of CO2 from the pre-industrial state and ΔR is the total change in radiative forcing, including the slow feedbacks that affect the LGM climate state. In Fig. 4, we present one solution that includes GHG and ice sheet forcing (\(\Delta R=\Delta {R}_{{\rm{GHG}}}+\Delta {R}_{{\rm{ICE}}}\)) and another that also includes aerosol forcing from mineral dust (\(\Delta R=\Delta {R}_{{\rm{GHG}}}+\Delta {R}_{{\rm{ICE}}}+\Delta {R}_{{\rm{AE}}}\)). We estimate the mean value of ΔRGHG and \({F}_{2\times {{\rm{CO}}}_{2}}\)to be −2.48 W m−2 and 3.80 W m−2, respectively, using published equations45. Average GHG concentrations for the LGM and LH are used for the calculation of ΔRGHG. Following the assessment in successive IPCC reports, we assume a 10% (90% CI) uncertainty for \({F}_{2\times {{\rm{CO}}}_{2}}\) (ref. 45) (1σ = 0.23). The 1σ uncertainty in ΔRGHG, which is dependent on \({F}_{2\times {{\rm{CO}}}_{2}}\), is:

$$\sqrt{{\left(\frac{\Delta {R}_{{\rm{G}}{\rm{H}}{\rm{G}}}}{{F}_{2\times {{\rm{C}}{\rm{O}}}_{2}}}\right)}^{2}\times {0.23}^{2}}=0.15\,{\rm{W}}\,{{\rm{m}}}^{-2}.$$
(9)

ΔRICE accounts for the radiative forcing from surface albedo changes associated with the LGM ice sheets and exposed land due to lowered sea level. We calculate ΔRICE using an approximate partial radiative perturbation method78 in CESM and also make use of additional published results in 11 PMIP2 and PMIP3 models6,79. The resulting ΔRICE has a multi-model ensemble mean of −3.64 W m−2 and a range from −2.59 to −5.20 W m−2 (Extended Data Table 3). ΔRAE was obtained from a published compilation of the top of atmosphere instantaneous direct radiative forcing of LGM dust in nine modelling studies46. ΔRAE has a large spread, ranging from 0 to −2 W m−2.

We also calculate a third solution for ECS that assumes a lower ε in which ΔRICE is multiplied by 0.7, a value consistent with previous work48,49. This value of ε is used merely to show the impact of a smaller-than-unity efficacy on ECS and should not be considered a definitive number. In ref. 48 it was demonstrated that there is a wide range of possible values for ε; however, some of the derived range comes from models of intermediate complexity47 that have substantially different physics than the GCMs in PMIP2 and PMIP3 that we use to derive the values of ΔRICE (Extended Data Table 3). Thus we prefer to show results for a single value of ε as an example rather than sampling from the full range in ref. 48.

To propagate uncertainties into the final calculations of ECS, we used a Monte Carlo approach, sampling the full 1,000-member ensemble of ΔGMST from our assimilated product, and combining these with 10,000 samples of each distribution of ΔR as well as \({F}_{2\times {{\rm{CO}}}_{2}}\). ΔRGHG and \({F}_{2\times {{\rm{CO}}}_{2}}\) were assumed to have normal distributions. Since their uncertainties are dependent, \({F}_{2\times {{\rm{CO}}}_{2}}\) was sampled first, then the distribution of ΔRGHG was derived by multiplying the resulting Monte Carlo samples by the ratio of the mean value of ΔRGHG to \({F}_{2\times {{\rm{CO}}}_{2}}\) (−2.48/3.8 = −0.65). ΔRICE and ΔRAE were treated as empirical random distributions because the limited number of samples (derived from modelling experiments) prevents us from knowing the shape of these distributions.

Total ΔR for the case in which only GHG and ice sheet forcing is considered is −6.0 W m−2 (−7.9 to −5.0 W m−2, 95%CI). Total ΔR including dust aerosol forcing is −6.8 W m−2 (−9.6 to −5.2 W m−2, 95%CI). Total ΔR for the case with reduced ice sheet efficacy is −5.7 W m−2 (−8.2 to −4.4 W m−2, 95%CI).