Our interest is in temperature. Specifically, we concentrate on means of seasonal or annual temperature at the surface. We consider proxies for which the literature previously reports a sensitivity to temperature in the form of a calibration relation. We search for analogues within fields of simulated surface temperature. To do the comparison, we consider the model variable “surface temperature” over the European–North Atlantic domain shown in Fig. . The reconstruction also uses these fields.

Theoretically, the variable or variables to be reconstructed can be different from the variable or multiple variables represented by the paleo-observational predictors. Indeed, we here assume that it is possible to reconstruct annual temperatures from proxy records with diverse seasonal attributions.

Using temperature in a multi-proxy comparison requires a number of assumptions. First, we assume that the proxy recorders indeed were temperature sensitive. More importantly, here, we assume that all the different recorders, aquatic or otherwise, represent temperature at the surface. This is an assumption of convenience in view of potential habitat biases of the proxy records .

Section  gives details on our selected proxies. In short, we choose 17 proxies at locations in the European–North Atlantic sector (Fig. ) from the compilation of . These are from a variety of different proxy systems. We take these as published by either or the original publications. Therefore, calibrations and uncertainty estimates have diverse origins. Considering the proxy ages and their uncertainties, we adopt those as published.

Optimally, one would aim for maximal consistency in the comparison. Consistency among parameters and calibration ensures a relation among the proxy predictors, which, one can assume, increases the chance that the proxy records lead to a selection of physically meaningful analogues. In this case, the proxies can effectively anchor the analogue selection. We here assume that all chosen proxy types reliably represent the target of interest and a multi-proxy approach is viable.

The analogue method allows searching for analogues at dates when there is information. One can pool the predictor dates into consistent intervals of, for example, 500 years, and search for analogues for these 500-year pools. One can follow the example of and interpolate the proxy records to consistent time steps using the age models for the individual records. We choose here a different approach; we identify all the years for which at least one of the chosen proxy records includes a dated value. We perform analogue searches for these dates according to our considerations on uncertainty, which we describe in the following (Sect. ).

Each data point of a proxy series potentially represents a time interval of a specific length, and the comparison should consider this temporal resolution. That is, if one data point represents a 50-year accumulation and another data point represents a 500-year accumulation, the procedure ideally accounts for these differences. We decide to use typical resolutions instead of individual resolution estimates to simplify the procedure and allow a computationally more efficient analogue search for data- and time-uncertain proxies. Indeed, it is not necessarily the case that a proxy-record publication includes the information to estimate the pointwise temporal resolution. Considering information provided by on their proxies, we conclude for our chosen subset of these (compare Table ) that the proxies have at best centennial average resolutions. While there are proxies with higher and lower resolutions, a reasonable estimate for the overall average resolution is centennial.

Therefore, we decide to compare the proxy estimates to 101-year averages of the model simulation output. That is, we compare them to 101-year mean values, which we obtain by using a 101-year moving mean on the simulation output time series that is closest to the proxy location.

In one test case, we do not preprocess the simulation output but use the annually resolved values of the output for the comparison. For this specific test, we also include the simulation data from the FAMOUS-HadCM3 simulations for the Quantifying and Understanding the Earth System (QUEST) project compareand Sect.  for which output data are only representative of decadal forcing conditions. We refer to these as QUEST FAMOUS simulations. We do two more tests with differing resolutions that use 51- and 501-year averages, respectively.

We test the whole approach by using pseudo-proxies. We construct the pseudo-proxies following the ensemble approach of . Their approach takes simulated grid-point data and transforms them in multiple steps into a pseudo-proxy record. The steps follow the framework of a proxy system model including a sensor model, an archive model, and an observation model see. first add a noise estimate for environmental non-temperature influences at the sensor stage. This stage also includes adding a bias term due to changing insolation. Next, the archive stage primarily represents a smoothing of this record, which is meant to reflect effects of, e.g., bioturbation. The measurement stage adds another noise term. After sampling this record at a specific number of dates, the procedure, finally, also adds an error term for the proxy data reflecting effective dating uncertainties. We set this term to zero in the script of because we do not aim to transfer dating uncertainty to the data uncertainty. Pseudo-proxy locations are simulation data grid points close to the real proxy locations. Figure  shows the 17 pseudo-proxy and proxy locations and allows us to identify their slight offsets due to the discrete character of the simulation data. The pseudo-proxy generation smooths each record to mimic the temporal filtering effects of the real environmental archive. The smoothing length is randomly chosen but temporally uniform for each record. The search for analogues again uses 101-year mean estimates from the simulation pool (compare the paragraph above).

Simulations potentially differ in their modern-day climate mean compare, e.g.,. Using anomalies can circumvent this issue. However, there is not a clear path in our application towards computing them equivalently in simulations and proxy data. If such a path exists, one can consider simulation output as anomalies to the climatology over the 20th century or over the full simulation period or over the longest period common to all simulations. For example, construct the anomaly record for a data series by subtracting the temporal mean calculated over the full period of the record of interest. Their period of interest backs this decision. The proxy records of suggest an overall rather stable climate in the North Atlantic during Marine Isotope Stage 3, although a number of Dansgaard–Oeschger (DO) events occurred during this period. We presume that using anomalies allows us to include a wider range of simulations and analogue candidates for each date.

However, in the present case, the period of interest includes mainly the last 15 kyr. Thus, it spans part of the deglaciation from the Last Glacial Maximum to the Holocene optimum. Our selection of simulations can only piecewise cover that period of interest, which complicates the construction of a surface temperature candidate pool. Indeed, the most recent dates differ among the proxy records, and thus there is no simple procedure to provide anomalies relative to a consistent modern climate. Additionally, using anomalies may introduce climatic inconsistencies if we are interested in climate variables other than temperature. For these reasons, we decide that we cannot reasonably use anomalies. Instead, we try to find analogues for the local proxy reconstructions in their absolute temperature units without subtracting any climatology.

We are interested in millennial timescales from the last deglaciation until the recent past. On these timescales, uncertainty affects our proxy predictors in two ways. First, we have to consider the age or dating uncertainty. Second, the measured proxy data and the temperatures inferred from them are affected by various sources of uncertainty compare, e.g.,and their references.

Previous applications of the analogue method usually did not consider proxies with considerable age uncertainties except for . consider the age uncertainty by shifting the date of each proxy by ±500 years. Thereby, they obtain an ensemble of 214 reconstructions from which they calculate confidence intervals for their final reconstruction. They do not separately consider the uncertainty of the proxy/reconstruction value. For details, see .

Uncertainty of proxies in time and date is commonly expressed as central value and a given uncertainty range. These ranges may be given as plus and minus standard deviations around the central value, e.g., ∼±1.64 or ∼±2.58 standard deviation ranges, or as percentage confidence intervals, e.g., 90 % or 99 % intervals. Such usage of a percentage view can be extended to pairwise expressions of the uncertainty. This is central to our approach.

That is, we choose a different approach (Fig. ) compared to, e.g., . We interpret each data point in a proxy series together with its dating as a data point in the two dimensional space spanned by temperature and time. Each proxy data point is located on this two-dimensional temperature–time plane and each point is surrounded by uncertainty ranges along both dimensions. We can utilize the uncertainty ranges in the two dimensions as our “area of tolerance”, in which the analogue candidate simulation fields should be located to be considered as good analogues. That is, the area of tolerance defines our criterion for selection of good and valid analogues. If a candidate field is within the tolerance areas at all considered proxy locations, it is thought to be a valid analogue. We can define tolerance ranges for different levels of pairwise proxy–time uncertainty. In equivalence to common expressions of uncertainty or confidence intervals, these can be formulated as pairwise 90 % or 99 % intervals. These choices of intervals yield increasingly larger areas of tolerance. For the real proxies, the data uncertainties that enter the computation of the tolerance area follow our simplifying assumptions detailed in Sect. , while the dating uncertainties are taken from the compilation published by . For the pseudo-proxy setup, the pseudo-proxy algorithm of provides estimates for data and dating uncertainty.

To define these areas of tolerance, we still have to define their shape. Our interest is in finding analogues that agree with the proxy data but also account for these uncertainties. Then, we could take the uncertainty estimates of temperature and time to construct a two-dimensional uniform estimate in the form of a rectangle of tolerance. Analogue candidates would be valid analogues if they fall locally within these rectangles. If they fall outside of the rectangle, they would not be considered valid analogues. Although the uncertainties in temperature and time are commonly taken to be Gaussian, the rectangular approach is the best one if we consider the uncertainties of date and temperature isolated from each other. Then, our tolerance for the temperature data has the same structure at the border of our temporal tolerance range as it has at the central estimate for the date. However, in our application, we do not see both tolerance ranges in isolation. We assume that our tolerance range is a two-dimensional pairwise construct in time and temperature. Then, our tolerance construct takes the shape of a two-dimensional Gaussian. This implies that our tolerance areas are ellipses. Such ellipses can be computed dependent on an assumed pairwise confidence level or coverage or in our interpretation tolerance range. We refer to these as percentage levels.

According to our view of tolerance ranges as tolerance ellipses, we accept fewer analogues for dates far away from the median proxy age estimate. For these dates, analogue candidates need to be numerically very close to the proxy. In contrast, we accept more analogues close to the central age estimate of the proxy and tolerate that they may more strongly differ from the numerical central estimate of the proxy. We acknowledge that it may seem counterintuitive that we reduce the range in data uncertainty at dates far away from our central best estimate of temperature and date. This originates from our assumption that the pair of data and date stems from a two-dimensional distribution that is centered on our best estimate. Thereby, the likelihood of a valid pair of data and date reduces further away from our best estimate according to the assumptions on the distribution.

As we have estimates of the uncertainties of the data point, we can construct and visualize the ellipses of tolerance around each data point under the assumption of two-dimensional Gaussian tolerance areas. We use the R package ellipse whose default ellipse function follows by implementing the ellipse equation as 1(x,y)=(α⋅σx⋅cos⁡(θ+d/2)+μx,α⋅σy⋅cos⁡(θ-d/2)+μy), where x is our time dimension and y is our temperature dimension. Furthermore, α is the tolerance level of interest (i.e., the percentage levels mentioned above) transformed to a t-test statistic as implemented by , σi are the 1 standard deviation levels of the uncertainties in the x and y directions, μi are the best estimates of the values in the x and y directions, i.e., date and data, and θ∈[0,2π]. cos⁡(d)=ρ is the correlation between temperature and time uncertainties. However, we do not consider potential non-zero covariances between dating uncertainty and proxy uncertainty. For simplicity, we also do not take account of the likely correlations between subsequent tuples of data and date.

A two-dimensional tolerance ellipse represents tolerance levels for two-dimensional normal distributed data. However, as in the simple case of a tolerance rectangle, our interest is only in the ellipse as a binary decision criterion to consider the data included in the ellipse and to neglect the data outside of the ellipse. That is, we use the ellipse as an area of tolerance to identify valid analogues from our analogue candidate simulation field pool. The ellipses provide the maximal acceptable distance for simulated data to be considered as an analogue (Fig. b). That is, the ellipses are not meant to represent the uncertainty ranges in the value of the proxies. They are rather meant to define a limit beyond which an analogue candidate is not considered anymore. Essentially, the ellipses define a weighting scheme (although with binary weights) based on the proxy and dating uncertainties.

The ellipses are defined from points in the proxy–time space (see Fig. b). We construct ellipses for those data points for which a published record provides ages. Our tolerance range for a specific date as well as the tolerance envelope for the full proxy record follows from the superposition of the tolerance ellipses from successive data points (see panels of Fig.  and later Figs.  and ). This envelope generally provides for each date upper and lower limits of values that the analogue candidates need to fall between. However, the envelope may also result in the impossibility to define analogues for specific locations or even for all locations for specific dates.

That is, the superposition of ellipses constructs a tolerance envelope (Fig. a), which we use to identify valid analogues from our candidate pool. The ellipses around the data points mark the limit of their pointwise two-dimensional area of influence in our search for spatially resolved analogues. Their superposition is essential for identifying those simulated data to be considered as analogues. If the tolerance ranges for multiple data points in a record overlap for a given year, we simply take their maximal ranges. Simulated data that fall outside the tolerance ranges are rejected. Thus, for a selected date, candidate analogues have to fall within the tolerance range at all considered locations to be valid analogues.

Because we provide reconstructions only for those years for which one of the chosen proxy records includes a dated value, and because our tolerance estimates are essentially pointwise, the envelope may not be one continuous envelope over the full period of interest. Furthermore, because we use the envelopes as a decision criterion, it can happen that the method fails to find any valid analogues for given years.

Our pointwise estimates are compliant with the initial uncertainty of the proxies, and our final reconstruction uncertainties are an expression of this initial confidence in the local data. This is in contrast to , who provide an ensemble of reconstructions. Their uncertainty estimate measures the uncertainty of the initial reconstruction relative to shifted ages. That is, the two different applications of the analogue method consider different things in their uncertainty estimates. The reconstruction uncertainty in our approach originates from the selected analogues.

The ellipses of tolerance allow in theory to produce reconstructions for each year included in the dating uncertainty. That is, if a proxy series has a value dated to the year 500 BP with a dating uncertainty of σ=50 years, and if we decide to consider dates within ±2σ, then we can search for analogues from 600 to 400 BP. However, we decide to only reconstruct values at those dates at which at least one proxy is dated. That is, if only this hypothetical proxy has a dated value between 600 and 400 BP and it only has this one dated value, we perform the reconstruction only for the year 500 BP. Our assumption is that this maximizes the link between the reconstruction and the underlying proxies. Thus, if we increase the width of the tolerance envelope, we usually do not obtain reconstructed values at more dates but only increase the probability to find a valid analogue at a given date. As we show later, there are exceptions, when the wider tolerance envelopes lead to the inclusion of more proxies at specific dates so that the search becomes more constrained and finding an analogue becomes less likely.

In other applications of the analogue method, the choice of a valid analogue usually relies on a distance metric. This is commonly the Euclidean distance compare, although use an unweighted root mean square error (RMSE) as distance metric between their proxies and the analogue candidates from their simulation pool. Based on such a distance, one can select the best fit, a small number of good fits, e.g., the 10 analogues with the smallest distance, or a composite or interpolation of a small number of good fits.

Here, we deviate from this and decide neither on a fixed number of analogues nor a defined metric. Candidates in our pool are valid analogues if they are within the tolerance range (compare Sect. ) at all considered locations for a selected date. That is, as described above, we have an envelope of tolerance values for specific years and each proxy record. For our standard approach, a candidate is a valid analogue for a date if it falls within the ellipse of tolerance for all proxies. We also mention tests where an analogue is valid if it is outside the ellipses at one location, at two locations, or at 25 % of the locations. We consider only a small set of potential ellipses. These use 90 % and 99.9 % percentage levels for the pseudo-proxy approach and either 99 % or 99.99 % percentage levels for the various proxy setups.

We additionally show one instance of a reconstruction using just one best analogue. For this test, we choose the analogue with the smallest Euclidean distance to our proxy values. As we deal with proxy records that are irregularly spaced in time, we have to find a way to select dates for which to do a single best analogue reconstruction and get the proxy values for these dates. To do so, we consider the proxy values valid at all dates within a given range around their dating. We identify the range of these values and take the midpoint of that range as the proxy value for this date. We consider values within a 90 % or ∼1.64 standard deviation dating uncertainty around the dating. We compute these based on 1 standard deviation inferred from the originally published dating uncertainty.

In short, our reconstruction is based on the following workflow. We have a set of sparse proxy predictors and a pool of simulated fields. As our proxies are not only sparse in space and uncertain in their values but also irregular and uncertain in time, we have to decide (a) when to compare them, (b) in which resolution to compare them, and (c) how to consider the uncertainties in time and value. Therefore, we decide to (i) compare the proxies and simulated data for all dates when one proxy is dated, (ii) compare the proxies to 101 moving means of the simulated data, and (iii) take the proxy data values as valid within an ellipse of tolerance around the dated value in time and temperature space. Then analogue candidate simulation fields are valid analogues if they are within these tolerance ranges around all proxy records included in the search.

We concentrate on a European–North Atlantic domain (Fig. ). There, we choose 17 locations with proxy-inferred temperature records from the collection of see also Tables  and . Nine of these series use alkenone U37K′ but the set also includes temperatures derived from foraminifera Mg/Ca (two records), pollen (two), chironomids (two), TEX86 (one), and foraminiferal assemblages (one) (compare Table  and Fig. ). For the various proxy types, see, e.g., and their references or and their references for U37K′, and their references or and their references for foraminiferal Mg/Ca, and their references or and their references for TEX86, and for the specific pollen records, for the specific chironomid records, and for the specific record using foraminiferal assemblages.

We do not include all records from within the chosen domain. We do not consider additional seasonal attributions for the foraminifera assemblage data of compare also . We further excluded the alkenone unsaturation ratios of see also as well after initial tests due to concerns about the potential influence of sea ice in simulations. Indeed, we find (not shown) that including this record puts very strong constraints on the analogue candidates and can reduce the chance of finding valid analogues. We exclude two more records because they are co-located with other proxies. That is, we do not use the stacked radiolarian assemblage records of see also because the upper part of the record is from the same upper core as the U37K′ data of see also . Similarly, we decide ad hoc to use the U37K′ data of see also instead of that of see also , which are basically co-located. We use the data of in one alternative proxy setup. Table  and Fig.  provide details on our different proxy setups. All in all, we consider nine different setups of proxy networks, which we name E01 to E09. In a pseudo-proxy setup, we use a network of locations equivalent to E01 and therefore name this pseudo-proxy setup P01.

We consider the seasonal attributions of individual proxy records in our search for analogues. We generally take the attributions and the calibrations for the records as published by but also check the references provided by them. Seasonal attributions are diverse for the various proxy records. The majority is either for summer season (seven) or annual (eight) according to . We compare the proxies to the simulation output season that is close to the seasonal attribution as given by or the original publication. For simplicity's sake, we only consider the modern meteorological seasons DJF (December to February), MAM (March to May), JJA (June to August), and SON (September to November) as well as the calendar annual simulation means (compare Table ). We do ignore possible calendar effects e.g.,.

Regarding proxy uncertainty, we decided to assume an uncertainty of σ=1 K for all proxies as we were not able to infer full uncertainties for every temperature reconstruction either from or from the original publications. This reduces the uncertainty for some records and potentially increases the uncertainty for others. We regard this to be a reasonable simplification.

We performed reconstruction exercises for various proxy setups. We concentrate on the full set of proxies mentioned above (see Fig.  and first 17 lines of Table ). Figure b visualizes how many of these 17 proxies are available for the dates for which we aim to reconstruct temperature. The figure shows this for two different assumptions on uncertainty (red and grey lines; see Sect. ).

Figure  and Table  give a first impression of setups for additional reconstructions. We shortly describe the results for these alternative setups in our results section below. Most notably, among these alternative tests are setups that use only U37K′ proxies (Fig. ). The difference between the two U37K′ setups is that E03 uses the GeoB 5901-2 record instead of the M39-008 record (compare Table  and Fig. ).

We use pseudo-proxies calculated following to test our approach. provide pseudo-proxies based on simulated annual mean temperature and for a global selection of grid points from the TraCE-21ka simulation . Here, we calculate the pseudo-proxies for annual average data and for the chosen European–North Atlantic domain only. The approach also provides randomly chosen pseudo-age uncertainties. Following and their repository , these are based on assumptions on the smoothing of the pseudo-proxies and a Gaussian term.

Here, the pseudo-proxy computation uses QUEST FAMOUS simulation data . Specifically, we use the simulation ALL-5G (see Tables  and ). For details on this and the other QUEST FAMOUS simulations, please see . The FAMOUS-HadCM3 simulations for QUEST use accelerated forcings compare. That is, the last glacial cycle of approximately 120 000 years of climate forcing was simulated in approximately 12 000 simulation years. Thus, the annual simulation data are only representative of 10 years of climate evolution. The data are available in monthly resolution for the full simulation period for air temperature at 1.5 m height and as snapshots every 10 simulation years for surface temperature. We use the simulation year annual means of the air temperature data for the construction of the pseudo-proxies. The FAMOUS-HadCM3 simulations use a very-low-resolution atmospheric model with a 5∘ latitude by 7.5∘ longitude grid. Therefore, we use the Climate Data Operators (CDO) application from the Max Planck Institute for Meteorology (https://code.mpimet.mpg.de/projects/cdo/, last access: 18 August 2020) to remap the data to a 0.5 by 0.5∘ grid and use this for the pseudo-proxy calculations. In these remapped data, we follow and use grid-point data close to proxy locations used in the realistic setup.

We modify the pseudo-proxy script of to account for the reduced temporal resolution of the available QUEST FAMOUS data. This primarily means considering the default parameter settings that are given in time units. It also includes ad hoc scaling of the randomly chosen dating uncertainty to approximate the distribution of the observed dating uncertainties. The latter modification also avoids individual data points extending their influence too far along the time dimension.

The 17 pseudo-proxy locations are close to the realistic proxy locations (compare Fig. ). Figure a visualizes the number of pseudo-proxy locations with data against the dates at which we try to reconstruct values. The pseudo-proxy records are shown in Fig. . The figure also visualizes our assumptions on the uncertainty of the pseudo-proxies in terms of the tolerance envelope (compare Sect. ).

Table  provides a general overview of the various simulations in our pool of candidates. Tables  to give additional information. We only consider previously published simulations. These stem from a variety of projects and were performed with a variety of models. The projects are TraCE-21ka (“Simulation of Transient Climate Evolution over the last 21 000 years”) , the Paleoclimate Modelling Intercomparison Project phase III PMIP3;, the CESM Last Millennium Ensemble Project , the Max Planck Institute Community Simulations of the last millennium , and Quaternary QUEST e.g.,. We include the QUEST FAMOUS simulations only for a test case and exclude them for the main discussions due to their specific characteristics (compare Sect. ).

We use simulations for various different time periods to increase the candidate pool. We assume that simulation climatologies can differ over a relatively wide range e.g.,. Simulations from the TraCE-21ka and the QUEST projects are transient over periods covering approximately the last 22 kyr and the last glacial cycle, respectively. Otherwise, the simulations are transient over the last millennium or time slices for the mid-Holocene and the Last Glacial Maximum. Additionally we also include pre-industrial control simulations. Such a multi-model and multi-time-period candidate pool effectively follows suggestions of . We note that considering simulations for the last millennium as candidate for the Last Glacial Maximum can introduce climatological inconsistencies if the method identifies these fields as analogues.

We remap all simulation output to a 0.5 by 0.5∘ grid for the construction of pseudo-proxies and for the search for analogues. The motivation is that thereby fewer proxies are close to the same grid point. However, resulting differences between grid points are likely small. We use the original resolution for the final regional average reconstructions and the evaluation of field data. Local grid-point evaluations are done against the remapped files.

Table  introduces a number of additional proxy setups (E02 to E09). These use different subselections of proxies from our initial selection. Further, most of them test sparser sets of locations around central Europe (compare Table ). Figure  provides additional information about which records are included in the different setups and their proxy types. Here, we shortly present the results.

Experiment E01 is our main setup. It was described in the previous section. It uses the 17 chosen proxy locations, which we also use for the pseudo-proxy setup. Setups E02 and E03 are based only on alkenone U37K′ records and E03 replaces M39-008 by GeoB 5901-2, as both are co-located. E04 to E09 include different numbers of other proxy types instead of U37K′. Figure  shows the availability of proxies for the different setups. Figure  presents the proxy data and assumed uncertainties including the GeoB 5901-2 record. For more information, see Table  and Fig. .

Figure  shows the reconstruction results for the proxy setups E01 to E09. All panels plot the reconstructions using the 99 % and the 99.99 % tolerance envelopes. Panel (a) adds for our main setup a reconstruction where we consider interannual data for the simulations and include the QUEST FAMOUS-HadCM3 simulations. The panel also includes the results of testing an analogue approach where only the single best analogue is considered at each date.

The panels of Fig.  highlight that loosening the tolerance constraint and thereby widening the tolerance envelope leads to valid analogues for notably more dates as well as a wider range of valid analogues. We also obtain analogues at more dates if we keep the tolerance envelope at the lower level but do not preprocess the simulation output to 101-year moving means (Fig. a, black lines). This inclusion of interannual data increases the number of analogues throughout the reconstruction period. This variation of the experiment also uses more simulation data by including the QUEST FAMOUS data, but this only affects the reconstruction success in the 15th millennium BP in this setup. We performed further tests with different averaging periods of 51 and 501 years, respectively, while keeping the narrow tolerance envelope (not shown). Increasing the averaging period to 501 years reduces the number of valid analogues and the number of dates with any valid analogues. Reducing the averaging period to 51 years allows us to find a few valid analogues in the 15th and 16th millennia BP. In this setting, the approach also finds more valid analogues in recent millennia.

Generally, the method appears to provide more complete reconstructions among our proxy setups for those that only include U37K′ records (Fig. b, c). That is, such consistent sets of proxies provide a more continuous reconstruction for both local tolerance assumptions. Nevertheless, we fail to obtain valid analogues, i.e., reconstructed values at the end of the deglaciation. While results are quite similar over much of the period between both reconstruction attempts (E02 and E03), the second setup allows a wider and potentially colder range in the period before ∼ 12 kyr BP.

Further panels of Fig.  add different setups. Panel (d) complements the U37K′ proxies by one foraminiferal assemblage record. Panels (e) and (f) also test different setups dominated by U37K′ but including other proxies. Panels (g) to (i) use different small setups of proxies around the European area.

Multi-archive setups with fewer proxies give generally wider ranges of possible analogues. Otherwise, all setups tend to be in a comparable range regarding their median and their range considering the last 10 millennia. Differences between all setups are largest in the 14th millennium BP due to a larger range for some reconstructions.

Both multi-proxy setups in panels (e) and (f) fail to provide analogues before the deglaciation for the narrower tolerance assumption. The setups in panels (g) and (i) are notably warmer in the 14th millennium BP compared to results in panel (h) but also compared to other setups. This holds for both tolerance envelopes. A common difference is the inclusion of M39-008 while excluding the U37K′ D13882 record (compare Table  and Fig. ). The latter record is thought to represent summer temperatures off the west coast of Portugal, while the former is meant to represent annual temperatures in the Gulf of Cádiz. We note that panels (e) and (f) also are warmer compared to other setups in the 14th millennium BP for the wider tolerance range. These also include M39-008 and exclude D13882. Please note that the Supplement of refers to D13882 as D13822.

Generally, we find that the reconstructions from different setups differ in their ability to reconstruct climate for specific periods. Indeed, different setups may provide notably different climates, particularly for the early part of the time period of interest. Particular proxies appear to shift the results for the earlier part of our reconstruction between a warmer and a colder deglacial estimate. It is beyond the scope of this paper to disentangle the reasons for this. All setups provide rather constant reconstruction ranges.

As noted, Fig. a adds a single best analogue reconstruction, where the reconstructed value for a given date is the analogue candidate with the smallest Euclidean distance to the proxy values for that date. During the past approximately four millennia, as well as during the period from 8000 to 10 000 years BP, the single best estimate is included in the ranges of our other reconstruction efforts. Indeed it is close to the test with interannual data throughout the Common Era of the last 2000 years.

In the period between 4000 and 8000 years BP, when other approaches give very narrow ranges due to few valid analogues, there are cases when the result from the single best analogue setup differs notably from the other efforts. However, it is still within the range of results from the other experiments for earlier and later periods. Such deviations from the tolerance area approach are reasonable since our construction of the proxy values for the single best analogue search can provide a notably different proxy state compared to the tolerance envelopes constructed for our standard approach. Another potential explanation is that the analogue that minimizes the overall distance may be outside of one or even multiple tolerance ranges. Finally, we already mentioned that changing a tolerance level may change the number of proxy locations included in a search. For example, widening a tolerance level may result in inclusion of more proxy locations for specific dates. The construction of the proxy values for the single best search similarly changes the underlying multi-dimensional proxy vector. Indeed, an inspection of the data indicates that, in our test case, the found analogue does fall outside the tolerance ranges at least at one location.

We also note that the single best analogue approach allows us to obtain estimates when the other approaches fail between 10 000 to 14 000 years BP. Comparably to our other reconstruction attempts, the single best analogue reconstruction shows only little variability. Noteworthy are the reconstructed values in the 15th millennium BP where the single best analogue represents a Holocene-level warm climate and not a deglacial climate.

We consider two more modifications of our approach. Figure  shows, first, results using a rectangular tolerance region, and, secondly, reconstructions for tests where an analogue candidate is valid although it falls outside the tolerance region at one, two, or 25 % of the locations. The rectangular setup has minimal influence on the reconstruction but gives more homogeneous ranges of valid analogues and succeeds on slightly more dates in finding valid analogues. Relaxing the tolerance criteria results in very wide ranges in the early part of the study period. Due to few available proxies in that period, the criterion to fit 75 % of locations is stricter than the criterion that allows the analogue search to fail at two locations. The resulting reconstructions still have little variability. They also either give a wide and nearly constant range of potential values or a very narrow range.