We use 10 previously published simulations from three climate models which all simulate the period 22 to 6 ka (Fig. , Table ). Six simulations employ MPI-ESM-CR . In these simulations, GHG concentrations and orbital parameters are updated transiently. Ice sheet topographies are changed according to the GLAC-1D or ICE-6G reconstructions (see Table ). Meltwater from ice sheets is either transported into the ocean using dynamic river routing , distributed uniformly over all grid cells, or removed from the system (see Table ). MPI_Glac1D_PTK uses a parameter configuration that leads to a smaller LGM-to-Holocene temperature difference than in the other MPI-ESM simulations. Furthermore, atmospheric parameters in the “P3” simulations are slightly different from those in the “P2” simulations to correct a pre-industrial cold bias .

We further include three CCSM3 simulations from the TraCE-21ka project . In TraCE-ALL, orbital parameters, GHG concentrations, ICE-5G ice sheet topographies, and manually prescribed meltwater fluxes are adapted transiently. In TraCE-GHG, all boundary conditions except for GHG concentrations are fixed at the 22 ka state of TraCE-ALL. Similarly, only orbital parameters are changed in TraCE-ORB. Finally, we use the ALL-5G simulation from the QUEST FAMOUS last glacial cycle ensemble . Orbital parameters, GHG concentrations, and Northern Hemisphere ICE-5G ice sheet topographies are updated transiently. In contrast to the other simulations, the Antarctic ice sheet topography and land–sea mask are fixed to pre-industrial values, and the transient boundary conditions are applied with an acceleration factor of 10. No meltwater fluxes are applied in FAMOUS.

In the following, we denote the six MPI-ESM simulations and TraCE-ALL as the “main set of simulations” and TraCE-ORB, TraCE-GHG, and FAMOUS as “sensitivity experiments”. The latter three simulations either change only one boundary condition transiently or employ boundary conditions faster than they occurred in reality. Therefore, we do not expect them to cover all changes in the climate system with the same degree of realism as the other seven simulations. More information on the simulations is provided in the Supplement (Sect. S2).

The simulation ensemble has a large spread in the four components of the deglacial temperature evolution described in Sect.  (Fig. ). In the main set of simulations, the deglacial GMSAT increase is between ∼ 4 K in TraCE-ALL and ∼ 6.5 K in MPI_Glac1D_P3. With ∼ 1 K in TraCE-ORB and ∼ 3 K in TRACE-GHG and FAMOUS, the deglacial warming is lower in the three sensitivity experiments. Deglacial warming starts later in TraCE-ALL than in the MPI-ESM simulations, and the warming trend is smoother in MPI-ESM than in TraCE-ALL. Two different aspects of meltwater injection appear to play an important role in the GMSAT histories of these runs: the method of application and the progression through time. Simulations without meltwater fluxes feature weak millennial-scale fluctuations (e.g., MPI_Ice6G_P2_noMWF), and simulations with locally applied meltwater fluxes (e.g., MPI_Ice6G_P2) generate stronger GMSAT fluctuations than the simulation with global injection (MPI_Ice6G_P2_glob). Differing meltwater histories lead to an abrupt warming at ∼ 14.5 ka in TraCE-ALL but cooling events in the MPI-ESM experiments with meltwater input.

We use temperature reconstructions from the PalMod 130k marine paleoclimate data synthesis v1.1.1 , which is a compilation of published proxy records derived from marine sediment cores. V1.1.1 is an update from with 252 published (near-)surface temperature time series covering various periods of the last glacial cycle. As described in , age models are harmonized using the Bayesian age modeling algorithm BACON . For each sediment core, 1000 iterations of the age–depth model are saved in the database to quantify chronological uncertainties. The database combines temperature reconstructions from multiple proxies which are taken unchanged from the original publications. For some proxy records, reconstructions from different original publications are included in the database. We retain all records from the same sediment cores if they are based on different proxies. We average reconstructions originating from the same sediment core and proxy if all sample depths coincide. If the depths differ, we select the time series covering the longest period during the deglaciation. Reconstructions from the same proxy data but calibrated for different seasons are averaged to obtain pseudo-annual temperatures. More details on the preprocessing of the proxy records are provided in the Supplement (Sect. S3).

We select all (near-)surface temperature samples in the interval 22–6 ka from the database. Most of these records reflect surface or mixed-layer temperatures . While the used sensors occupy a range of depths, we denote all samples as sea surface temperature (SST) reconstructions in the following. To compute robust statistics, we use only time series with at least 10 samples, which cover more than 8 kyr and have a mean temporal resolution of at least 1 kyr. 74 temperature records from 50 unique sediment cores satisfy these conditions (Fig. b, Table ). Most of them are located on continental margins with the biggest clusters located in the North Atlantic and the Indo-Pacific Warm Pool. A total of 38 temperature records are reconstructed from Mg / Ca, 17 from U37k, 17 from planktonic foraminifera assemblages, 1 from TEX86, and 1 from diatom assemblages. Unlike some recent studies focusing on either assemblage-based temperature reconstructions e.g., or geochemical proxies e.g.,, we employ a multi-proxy approach using the calibrations proposed by the original authors for assemblages and geochemical proxies, respectively. We make this choice because the number of records in the database is too small to focus on specific proxy types, and proxy types tend to be regionally clustered, which makes a systematic assessment of differences between them unfeasible within our study design. For more discussion on the differences between proxy types, see , and the references therein.

As visualized in Fig. , our model–data comparison algorithm consists of four main steps which we present in the following. An enhanced description of the algorithm with computational details is provided in the Supplement (Sect. S4).

The PSM described in Sect.  requires a SNR parameter quantifying the ratio between climatic and non-climatic variations and the specification of a temporal autocorrelation structure of the additive Gaussian noise process. Previous studies only estimated SNRs and autocorrelations for a subset of our proxy types (U37k, Mg / Ca) on sub-orbital timescales . Therefore, we estimate the PSM parameters using the SST reconstruction database (see Sect. ).

To obtain these estimates, we decompose the SST records into a similar structure as Eq. (), i.e., the sum of a local mean SST signal Pspace(T) and a realization of a Gaussian noise process ε, which aggregates all deviations from the local mean SST signal. The decomposition starts by constructing clusters of SST records centered around each of the 74 SST records selected from the database. The clusters contain the records within a radius of l∈{100,200,…,1000} km around the central record (see Fig.  for an example cluster with n=3 records centered around record SO201_2_12KL). For each cluster, we compute a local mean signal by averaging over the records in the cluster (red line in Fig. a). More specifically, we interpolate nearby records to a regular temporal resolution of 100 years, center the records, and average over the resulting time series. We use the mean age model of each record and not the age ensemble members since we account for chronological uncertainties at a different step of the PSM. Using the age ensembles instead of the mean ages strongly reduces the estimated SNR and likely biases it low (not shown). Note that we average records of different temporal resolutions, which tends to underestimate high-frequency contributions to ε. However, all records have at least a millennial resolution such that the relevant millennial and orbital timescales should be less affected by the interpolation and subsequent averaging.

For the record in the center of the cluster, we compute the residual from the local mean signal (green line in Fig. b), which is treated as a realization of the Gaussian noise process (ε in Eq. ). We compute the variance ratio between the local mean signal and the residual which provides an estimate of the SNR. Due to the short time series length, the structure of the temporal autocorrelation cannot be determined from the residuals. We choose to describe ε as an autoregressive process of order one (AR1) because it is determined by only two parameters and as a compromise between a white noise process without temporal autocorrelation and power law processes with long-range autocorrelations. This AR1 process is specified by the SNR and a decorrelation length, which we estimate from the residual. We iterate this process for all 74 records if the clusters around the respective records contain at least a specified number of records. We then take the medians of the SNRs and the decorrelation lengths in all clusters to reduce the noise in the parameter estimates, which results from the predominantly small cluster sizes (most clusters contain less than five records). As the estimates can be sensitive to the construction of the clusters, we apply this procedure for cluster radii of l∈{100,200,…,1000} km and for the minimum required number of records in a cluster of n∈{2,3}.

The median SNR over all sensitivity experiments is 1.6±0.3 (1σ) and the median decorrelation length is 1289±212 years. When we decompose the SST variability of each proxy record into a signal and a noise component according to SNR = 1.6, the mean noise level across all records is 0.9±0.6 K. This estimate is consistent with an estimate of 0.6–1.3 K by in a data assimilation framework characterizing LGM-to-Holocene anomalies. Our estimate is slightly higher than the SNR of 1.0 employed in the LGM climate field reconstruction by .

We use PPEs for the following three purposes: (i) to demonstrate the main features in the simulations that are captured by the model–data comparison algorithm; (ii) to diagnose how much model–data comparison results depend on limited temporal resolution, chronological uncertainties, and the magnitude and temporal autocorrelation structure of non-climatic noise; and (iii) to investigate how sensitive results are when noise magnitude and temporal autocorrelation structure in the PSM are different from their optimal values. Note that the difference between (ii) and (iii) is that (ii) is motivated by quantifiable limitations and uncertainties of reconstructions, while (iii) specifically targets the fact that the employed PSM is just an approximation of reality and its optimal parameters are unknown.

In PPEs, the underlying climate evolution is given by a reference simulation. The temperature time series of the reference simulation at each proxy location serves as the ground truth in the PPE. For each proxy record, the PSM from Sect.  is applied to the reference simulation to generate a single realization of forward-modeled proxy time series with a randomly selected iteration of the age–depth model and one realization of the non-climatic noise process. As this realization mimics the properties of the SST reconstructions, we call it a pseudo-proxy. We simulate pseudo-proxies at the locations and with the time axes and chronological uncertainties of the 74 selected proxy records from Sect. . Following this, the algorithm from Sect.  is employed to compute the deviations between N=100 realizations of forward-modeled proxy time series derived from each simulation and the pseudo-proxies.

For (i), we use an example PPE with a subset of simulations to illustrate how simulation characteristics such as parameter configurations and the implementation of boundary conditions influence their ranking by our algorithm. We use MPI_Glac1D_P3 as reference simulation and PSM parameters given by the estimates from Sect.  (SNR = 1.6, decorrelation length = 1289 years). For the PPE, we select simulations that differ from the reference simulations in boundary conditions (MPI_Ice6G_P2_noMWF, TraCE-ALL), parameter configuration (MPI_Glac1D_PTK), and employed climate model (TraCE-ALL). Additionally, two idealized modifications of MPI_Glac1D_P3, which are shifted in time by 2 kyr in either direction (MPI_Glac1D_P3-2k, MPI_Glac1D_P3+2k), show the effects of a timing mismatch in the deglacial temperature evolution on the model–data comparison results (Fig. a).

For (ii) and (iii), we perform two sets of PPEs (Table ). In the first set we assume that the noise magnitude and type in the PSM are known but we systematically vary the noise level of the records from very high (SNR =1/4) to very low (SNR = 16) and include PPEs without additive noise process (SNR = Inf). We further vary the noise type between white noise (no autocorrelation), an AR1 process with a decorrelation length of 1 kyr, and a self-similar process following a power law distribution with exponent one (red noise). Using all 10 transient simulations as reference simulations to avoid spurious results from selecting a specific reference simulation, we perform in total 240 PPEs (8 SNRs, 3 noise types, 10 reference simulations).

In the second set, the PSM structure used for generating the forward-modeled proxy time series employed in the model–data comparison algorithm deviates from the one selected to simulate the pseudo-proxies, thus imitating the case where the PSM structure is uncertain. For each of the 10 reference simulations, we draw a realization of pseudo-proxies with AR1 noise (SNR = 2, decorrelation length = 1 kyr). For each pseudo-proxy realization, we first apply the model–data comparison algorithm with varying noise levels in the PSM (SNR = 1/4 to SNR = 16 and SNR = Inf) but the same autocorrelation structure as in the construction of the pseudo-proxies. Following this we apply the model–data algorithm with varying autocorrelation structure (white, AR1, and power-law noise) but the same noise level as in the construction of the pseudo-proxies.

Whether a certain IQD corresponds to an acceptable agreement between a simulation and a reconstruction is a subjective choice. Moreover, because the IQD uses the probability distribution of the forward-modeled proxy time series, its absolute value depends on the specification of the PSM. For example, a higher noise level results in a larger spread of the forward-modeled proxy time series created from the same simulation, such that the IQD for a high noise level will differ from the IQD for a low noise level, even if the simulated and reconstructed SST time series are the same. Therefore, we focus on the ability of the algorithm to reliably discriminate between simulations, i.e., determining whether simulation A is closer to reality than simulation B. In PPEs, we can compute the “ground truth deviation” between a simulation and the reference climate history that was used to construct the pseudo-proxies. We choose the mean absolute deviation from the reference simulation at the locations of the proxy records as ground truth deviation because the IQD reduces to the mean absolute difference in the absence of uncertainties. We then compute a reference ranking by sorting the simulations according to their ground truth deviations. Similarly, we can rank the simulations according to the IQDs between the forward-modeled proxy time series and the pseudo-proxies, which is the ranking that would be obtained in a real-world model–data comparison situation in which only the pseudo-proxies are known but not the underlying reference climate history. We call this the pseudo-proxy ranking.

Finally, we compare the reference ranking with the pseudo-proxy ranking. If the model–data comparison algorithm discriminated perfectly between simulations, the reference ranking and pseudo-proxy ranking would be identical. However, due to reconstruction uncertainties and limitations, this will not always be the case. To quantify the similarity of the two rankings, we introduce a measure called the “fraction of pairwise reversed rankings” (FPRR). This measure is based on pairwise comparisons of the rankings of simulations: if simulation A ranks higher than simulation B in the reference ranking but ranks lower in the pseudo-proxy ranking, we say that the ranking of the two simulations is reversed in the pseudo-proxy ranking, i.e., the two simulations are erroneously ranked by the model–data comparison algorithm. We assign 1 to the pairwise comparison if the ranking is reversed and 0 if it is not reversed. We compare the rankings for all pairs of simulations and define the FPRR as the mean of all pairwise comparisons. The FPRR is 0 when the pseudo-proxy and reference rankings are equal and is 1 if the two rankings are exactly reversed. The expected value for a random ranking of simulations is 0.5, which means that an FPRR below 0.5 indicates a better-than-random ranking. We focus on two aspects of the simulations' rankings: (i) the reliability of rankings, i.e., the expected probability of erroneously ranking simulations which we define as the median IQD in a set of PPEs with the same PSM parameters, and (ii) the robustness of rankings, i.e., how much the probability of an erroneous ranking depends on the reference climate history and the realization of non-climatic processes in the pseudo-proxies. Robustness is quantified by the spread of the IQD in a set of PPEs with the same PSM parameters and can be interpreted as a measure for the predictability of the reliability of model–data comparison results.

Next, we quantify the deviations between forward-modeled proxy time series derived from the 10 deglacial simulations (Sect. ) and the 74 selected SST records (Sect. ). We employ a PSM with an AR1 non-climatic noise process and vary the SNR between 1.1 and 2.2 and the decorrelation length between 865 and 1712 years (Sect. ). We study globally and regionally averaged IQDs for the Southern Hemisphere extratropics (n= 10 proxy records), the tropics (n= 44), the extratropical North Atlantic (n= 13), and the extratropical North Pacific (n= 7) (Fig. ). We select these regions based on detected inter-regional dissimilarities of the deglacial temperature evolution in an initial visual inspection of reconstructions and simulations. The averaged temporal evolution of the reconstructed temperatures and forward-modeled proxy time series at the proxy record locations is depicted for each of the four disjunct regions in Fig. . All regions contain more than five records and thus we expect the results to benefit from the spatial averaging effect found in the PPEs. Figure  shows the IQDs for all four components of the deglacial temperature evolution, simulations, and regions. An alternative visualization of the deviations, which combines magnitude and pattern deviations for a given timescale, is provided in the Supplement (Figs. S11, S12). In the next two subsections, we assess orbital- and millennial-scale variations of our main set of simulations. Finally, we analyze the model–proxy agreement of the three sensitivity experiments.

The systematic PPEs show that the reliability and robustness of simulation rankings decrease with increasing noise levels. This result is not surprising as higher noise levels make it harder to identify the underlying temperature signal. The effect can be reduced by spatially averaging results from multiple records. As we assume the non-climatic noise to be independent between records, averaging over IQDs from multiple records reduces the influence of the noise and thus effectively enhances the SNR. If modulations of the temperature signal were not independent between records in reality, the improvement from spatial averaging would be weakened.

Rankings for orbital-scale variations are more reliable and robust than for millennial-scale variations due to comparably smaller distortion by non-climatic noise. That orbital magnitude rankings tend to be more reliable and robust than orbital pattern rankings could be due to relatively subtle differences between simulations in the timing and shape of the deglacial warming trend compared to easier to identify differences in the magnitude of deglacial warming. On the other hand, we attribute more reliable and robust millennial pattern than magnitude rankings to the differing effects of non-climatic noise on these two components. Millennial patterns of simulations are often still distinguishable based on their most pronounced fluctuations that are comparatively less distorted by non-climatic noise. Meanwhile, non-climatic noise enhances the magnitude of reconstructed millennial-scale variations (in our PSM proportional to the variability of the simulation at a given location) and thus has a systematic effect on millennial magnitudes, which can further diminish the reliability of rankings.

If the assumed noise level in the model–data comparison is not strongly overestimated or underestimated (factor 4 and more), results remain reliable. Using explicitly conservative SNR values is not safeguarding from erroneous rankings as strongly overestimating noise levels reduces the reliability, whereas strongly underestimating noise levels reduces the robustness of rankings. Incorrect specifications of the temporal autocorrelation structure of non-climatic processes have a negligible effect in our PPEs. This rather unexpected result might be due to the relatively short time period of investigation (16 kyr) compared to the timescales we study. This hypothesis could be tested in future work by repeating the experiments for longer periods. Entirely neglecting existing non-climatic processes leads to less robust and reliable rankings for millennial-scale variations. On the one hand, this can be explained by non-climatic variations in reconstructions being interpreted as climate signals, such that rankings depend more on the unknown realization of non-climatic processes. On the other hand, underestimating millennial-scale variations by neglecting variability-enhancing processes can systematically distort millennial magnitude rankings. This effect is strongest for global averages.

Taken together, the PPE results suggest that the reliability and robustness of model–data comparison results can be improved the most by increasing the SNR. In contrast, reducing the uncertainty of SNR estimates or improving the specification of the temporal autocorrelation structures will barely improve rankings. A doubling of the SNR typically reduces erroneous rankings by 1–3 percentage points. Thus, incremental improvements, for example through process-based modeling of modulations of the recorded climate signal, will only have a small effect on the reliability of rankings. PPEs without non-climatic noise typically still have 5 %–10 % erroneous rankings for regionally averaged IQDs. This percentage could be reduced by more precise chronologies and higher temporal resolutions of records. Comparing global, zonal, and local estimates suggests that significantly improved reliability can also be achieved by increasing the number of proxy records and thus averaging over more records in regional averages, as long as non-climatic contributions are not strongly correlated between records.

The diversity of the simulations in terms of employed climate models and experiment protocols makes interpreting the results challenging. Comparing TraCE-ALL and the six MPI-ESM simulations, we find that none of the simulations ranks among the simulations with the smallest deviation from the reconstructions across all four components and considered regions. We confirm this visual impression from Fig.  by computing rank histograms among the main set of simulations. Rankings are computed for each proxy record and each of the four components. Averaged over all records and components, the ranks of the simulations are between 3.8 (for MPI_Glac1D_PTK and MPI_Ice6G_P2_glob) and 4.2 (for MPI_Glac1D_P3) with TraCE-ALL at an average rank of 4.0 (Fig. S13 in the Supplement). However, the ranks of TraCE-ALL concentrate strongly at 1 (highest agreement) and 7 (lowest agreement). In contrast, the rank histograms of the MPI-ESM simulations are flatter; i.e., they feature more similar occurrence rates across ranks. Thus, TraCE-ALL IQDs are more often outside than inside the range of the MPI-ESM simulations, even though it does not feature a consistently higher or lower rank. The concentration of TraCE-ALL at extreme ranks tends to hold for all four components (Figs. S14–S17). At the moment, we cannot attribute the difference in the rank score histograms to differences between either the used climate models or the employed experiment protocols. This is due to the differences in the experiment protocol between TraCE-ALL and the MPI-ESM simulations, particularly regarding the location, timing, and magnitude of freshwater injections. Nevertheless, the flatter rank histograms of the MPI-ESM simulations, despite substantial experiment protocol and parameter configuration differences among them, hint at a substantial influence from climate model differences.

Examples of regionally varying mismatches between simulations, which compensate in global averages, are found for all four components of the deglacial temperature evolution (see Sect. ). These compensations occur because simulations with higher variability than others have higher variability in almost all regions (Fig. ). Additionally, simulations tend to have similar temporal patterns at least within each hemisphere (Figs. , ). In contrast, the reconstructed variability magnitudes are the most similar to the simulations with the highest variability in some regions, but closer to those with low variability in others. Similarly, the reconstructed variability patterns vary more between and within ocean basins than in the simulations. Therefore, we attribute the absence of a simulation with consistently high agreement relative to the others to more regionally confined variability magnitudes and patterns in reconstructions than in simulations. In other words, the reconstructed spatial variability of the deglacial temperature evolution is higher than in all considered simulations. For the North Atlantic, the differences in the reconstructed deglacial temperature evolution between the Mediterranean and the Subpolar North Atlantic found in this study are consistent with a recent synthesis by .

This mismatch in the spatio-temporal variability structure could be caused by uncertainties in ice sheet reconstructions, shortcomings of the employed models, or temperature reconstruction characteristics that vary between regions. One can assess the role of systematic reconstruction deviations from mean annual SST by integrating process-based PSMs e.g., into our algorithm in future work. This could disentangle the importance of different processes occurring during the recording, archiving, and measuring of the proxy, e.g., recording season and depth preferences, confounding environmental variables, and bioturbation. Moreover, our procedure to estimate the PSM parameters requires interpolating the proxy records to a common time axis which is otherwise avoided in the model–data comparison algorithm. Developing a more sophisticated method for the parameter estimation would be beneficial for future applications of our algorithm.

The locations of proxy records are biased towards coastal regions, and, for some regions, our results rely on records clustered in small areas. This could reduce the model–data agreement if the resolution of models was insufficient for an accurate simulation of zonal temperature heterogeneity, e.g., due to coastal upwelling or deficiencies in the simulation of gyre circulations and air–sea interactions . As higher-resolution simulations of the deglaciation are currently precluded by computational limitations, including more proxy data and physically motivated downscaling of simulation output could help test this explanation. Finally, the reconstructed meltwater peaks could be too high or the models' responses to them too strong, leading to a spatially too homogeneous SST response . Insights into this potential explanation could be gained from coupled atmosphere–ocean–ice sheet simulations or replacing local meltwater input with freshwater fingerprints obtained from eddy-resolving ocean models .

The simulation with transient changes of orbital parameters only (TraCE-ORB) deviates significantly more from the reconstructions than all other simulations for orbital magnitudes, orbital patterns, and millennial magnitudes. This is due to too small magnitudes of variability in most regions and the absence of a deglacial warming trend in the Southern Hemisphere when GHG and ice sheet changes are neglected. We also find a systematically larger orbital magnitude mismatch between FAMOUS and the reconstructions compared to the main set of simulations because of weaker deglacial warming in FAMOUS. This could be explained by the acceleration in the forcing, which can delay global warming, but more simulations are needed to confirm this hypothesis.

In contrast, the neglected orbital and ice sheet forcing in TraCE-GHG does not lead to clearly higher disagreements for orbital-scale variability and millennial patterns. For millennial magnitudes, however, the absence of ice sheet forcing degrades results strongly. In particular, in the global average, all simulations with meltwater input show a better agreement with reconstructions for millennial magnitudes than those without meltwater input. The improved agreement originates mainly from a higher millennial-scale variability in the North Atlantic, where the meltwater-induced variability is the strongest. Moreover, the MPI-ESM simulation without meltwater input and TraCE-GHG have the smallest millennial pattern disagreement in the global average, which suggests that none of the employed meltwater schemes leads to a temporal pattern of millennial-scale variability that is globally consistent with the reconstructions. The uncertainties in ice sheet reconstructions currently prevent determining the reason for the millennial pattern disagreements. The contrast between higher model–proxy agreement in simulating millennial magnitudes but no improvement for millennial patterns in the fully forced simulations hints at limitations in our current understanding of the spatio-temporal structure of millennial-scale variability during the deglaciation. Addressing these challenges with designated protocols in the context of inter-model comparison projects could be a promising way forward.

Our results suggest that reproducing the patterns of a small set of proxies might be an insufficient strategy to capture the spatial structure of millennial-scale temperature patterns. For example, reproducing the patterns of a specific Atlantic meridional overturning circulation (AMOC) proxy (e.g., Pa / Th ratios at Bermuda rise), as TraCE-ALL does , will not necessarily lead to a good model–proxy agreement for millennial-scale temperature patterns across different regions. Instead, other factors, such as the magnitude of the AMOC response or the background climatic state, could have a large influence on the regional manifestations of temperature variability. Alternatively, uncertainty regarding the origins of millennial-scale variability could lead to an adequate reproduction of the pattern of AMOC variability with an incorrect mechanism, which could result in a spatially varying degree of model–proxy agreement.

A single metric is likely insufficient for fully capturing the deviations between simulations and reconstructions in an interpretable way. When combining magnitude and pattern metrics in biplots (see Figs. S11, S12), simulations with local freshwater injection perform the best in the North Atlantic for either timescale: MPI_Glac1D_P3 for orbital timescales and TraCE-ALL for millennial timescales. While strong freshwater water-induced perturbations can have an imprint on the orbital-scale signal, when the perturbations are large enough to substantially influence time averages on orbital timescales, a good model–proxy agreement for orbital timescales does not imply a good agreement for millennial timescales and vice versa in our results. Instead, we argue that a varying importance of forcings and internal feedback processes on different temporal and spatial scales substantially affects the model–proxy agreements for different components.

As the PPEs and the real-world application have shown, the pattern IQDs are sensitive to the timing of timescale-dependent temperature fluctuations. Therefore, they are only meaningful if the goal of a simulation is to reproduce a specific succession of variations observed in reconstructions. Temporal alignment cannot be expected for internally driven variations such as spontaneous millennial-scale fluctuations and in the presence of boundary conditions with large spatio-temporal uncertainties like deglacial meltwater fluxes. In these cases, the magnitude IQDs, which are insensitive to the timing of fluctuations, could be combined with a more insightful measure for temporal patterns, e.g., based on the similarity of spatial relationships in reconstructed and forward-modeled proxy time series e.g.,.

Applications of our model–data algorithm are not restricted to SST reconstructions during the last deglaciation. With new syntheses becoming available , an extension to terrestrial temperature records can be attempted. Moreover, other periods with climate transitions and changing background conditions can be assessed as long as a sufficient number of proxy records with absolute chronologies are available. Targets could, for example, be the penultimate deglaciation, the glacial inception, or the last glacial cycle. Finally, it is straightforward to adapt our algorithm for model–proxy comparison of other continuous variables such as oxygen isotopes, in particular if PSMs already exist that link the proxies to one or multiple simulated variables.