Combining proxy information and climate model simulations reconciles these sources of information about past climates. This, in turn, strengthens our understanding of past climatic changes. The analogue or proxy surrogate reconstruction method is a computationally cheap data assimilation approach, which searches in a pool of simulated climate states the best fit to proxy data. We use the approach to reconstruct European summer mean temperature from the 13th century until present using the Euro 2k set of proxy records and a pool of global climate simulation output fields. Our focus is on quantifying the uncertainty of the reconstruction, because previous applications of the analogue method rarely provided uncertainty ranges. We show several ways of estimating reconstruction uncertainty for the analogue method, which take into account the non-climate part of the variability in each proxy record.

In general, our reconstruction agrees well at multi-decadal timescales with the Euro 2k reconstruction, which was conducted with two different statistical methods and no information from model simulations. In both methodological approaches, the decades around the year 1600 CE were the coldest. However, the approaches disagree on the warmest pre-industrial periods. The reconstructions from the analogue method also represent the local variations of the observed proxies. The diverse uncertainty estimates obtained from our analogue approaches can be locally larger or smaller than the estimates from the Euro 2k effort. Local uncertainties of the temperature reconstructions tend to be large in areas that are poorly covered by the proxy records. Uncertainties highlight the ambiguity of field-based reconstructions constrained by a limited set of proxies.

There have been numerous efforts to reconstruct regional to global surface temperature for the last 500 to 2000 years. Many of the statistical reconstruction methods essentially assume a linear relationship between proxy information and temperature data. Here, we apply a non-linear method, the analogue method, to reconstruct the mean European summer temperature over the past 750 years in annual resolution. Our main goal is to provide a perspective on estimating uncertainties for reconstructions by analogue, which only few previous applications quantified. Our approach relies on a collection of dendroclimatological proxy records and the output of paleoclimate simulations.

The core of the analogue method is the search for similar spatial
patterns in simulated temperature data compared to the proxy records.
That is, we search for simulated analogues of the climate anomalies
indicated by the set of proxies at each available date. The method
originated during the Second World War when the US Air Force catalogued
weather situations of previous decades as a means of long-range weather
forecasting. In this approach, forecasters obtain forecasts by analogy
between current observations and a past set of weather patterns

The approach allows to reconcile the spatially sparse information from
environmental and documentary proxy data with spatially complete and
dynamically consistent information from observational data or long
climate simulations in the sense of data assimilation

The analogue method is generally found to perform well, e.g., for area-averaged indices and also at the locations of the used predictors

Previous analogue search reconstructions do not usually consider the
uncertainty in the predictor data, and studies rarely provide an
uncertainty estimate for the final reconstruction. This precludes to
some extent a realistic evaluation of predictors or reconstructions.
Exceptions are the studies by

Here, we propose alternative means to estimate the uncertainty of
analogue search reconstructions based on the calibration correlation of
the proxy predictors with an appropriate observational dataset. Our
approach to estimating uncertainty ranges reduces the possibility of
producing time series of reconstructed climate. On the other hand, it
allows to provide alternative reconstructions that are compatible with
the sparse information from the proxy records. The procedure further
acknowledges the possibility that the analogue pool does not cover
certain points in the predictor space. Our proposed uncertainty
estimates originate in the uncertainty of the individual proxies,
whereas

Recent continental proxy-based reconstructions

In the following, first, we introduce our approach to the analogue
search uncertainty as well as the used proxy and model data. Then, we
discuss the results for three different approaches to an analogue
reconstruction. These are (i) using a single best analogue, (ii) using a
fixed number of good analogues, and (iii) considering all analogues
complying with the proxies within a fixed level of uncertainty. We also
consider estimates from an ensemble following the subsampling approach
of

The paradigm that past analogues may provide information for
anthropogenic climate changes is pervasive in climate science

Here, we obtain annually resolved large-scale fields of seasonal mean
summer (June, July, and August; JJA) temperature based on a pool of relevant
candidate fields and a set of local data indices as predictors for the
period of 1260 to 2003 of the Common Era (CE). The reconstruction domain is
Europe from

Reconstruction domain and locations of the
included proxies. Red squares show the proxies included in our search,
grey squares show the locations from the original Euro 2k setup which we
exclude. The grey shaded box shows the original domain of

The approach of an analogue search is usually that, for each set of
predictors, i.e., each point in time, one ranks all potential analogues
according to a criterion of similarity to the target proxy pattern. The
criterion is traditionally the Euclidean distance and only the single
pool member with the smallest Euclidean

The approach presented here differs from previous applications in some important aspects. While we also show a single best-analogue reconstruction and a reconstruction based on a fixed number of analogues, we add a reconstruction that explicitly considers the uncertainty of the proxy records in the selection of the analogue fields.

The next subsection provides details on our three different approaches. In short, first, the single best reconstruction is the common application, and our uncertainty estimates are derived from the local correlation between gridded observation data and the local proxy series. Second, a further common approach is to use a fixed number of analogues. As we want to consider the uncertainty of the local predictors, we identify for a given uncertainty level of the proxies the smallest number of valid analogues for any date in our period of interest and then provide a reconstruction for each date using this minimum number of analogues. Finally, we fix the value of the uncertainty level around the predictors and consider all valid analogues within this uncertainty level.

We consider predictors and analogues normalized by their local standard deviation to conserve the interfield relations. The final reconstructions are rescaled by a chosen standard deviation, which is, here, usually the local full-period standard deviation of one of the simulations.

Empirical reconstructions of past environmental conditions rely on proxy
data, which may be documentary notations but more often are measurements
of biological, geological, or chemical properties of the environment.
Such proxy representations of the past conditions are naturally
uncertain. A prominent source of uncertainty is that the archives
recorded signals from more than one climate or environmental variable

In the following, we describe our thinking on the uncertainty of an
analogue reconstruction. We first provide general derivations before
describing the three reconstruction approaches of (i) best analogue, (ii) fixed number of analogues, and (iii) fixed uncertainty level. Our
derivation of the uncertainty estimates relies on a number of
assumptions, which we detail in the next paragraphs. Table

Correlations provide a simple measure of the relation between
proxy observations and the climatic environment over a period when
reliable (instrumental) observations of the climatic variability exist.
We assume we can derive the uncertainty of how well a local proxy record
represents the local climate from the correlation coefficients. We
denote this uncertainty hereafter as proxy uncertainty. We use
correlations between the proxy records and the observational gridded
Climate Research Unit (CRU) data

List of mathematical expressions.

Proxies considered, their geographic position, and the correlations between the proxy records and the summer (June, July, and August; JJA) mean temperature observations from the CRU TS 3.10 data

In our present approach, we consider normalized proxy data. That is, the
variance of an individual proxy

Assuming one can interpret the squared correlation coefficient (

We can take the total variance Var

Because we consider normalized data, the total variance becomes 1:
Var

We want to use the local estimates of the proxy noise to formulate a
criterion for finding analogues in simulated field records from climate
simulations. Because we use simulated records with unit variance, we can
consider the following as a noise standard deviation:

First, we consider the case of a reconstruction from the single best
analogue. We use the normalized data and we consider two uncertainty
estimates for this reconstruction. First, we assume that we can obtain
uncertainties as the square root of the sum over the individual proxy
noise variances (Var

From our point of view, the real benefit of our derivation of
uncertainty is to use only analogues which comply with a certain
tolerance criterion. That is, a second way towards an uncertainty
estimate assumes that we can obtain a similarity criterion between proxy
data and simulation pool by considering the noise standard deviation for
an individual proxy as a local noise tolerance threshold. A candidate
field has to comply with all local thresholds to be considered a valid
analogue. We then can limit our analogue search to only those analogues
within a certain tolerance range at each location, i.e., within plus and
minus 1, 2, or 3 SD

In the following, we only consider analogues within traditional 90 %,
95 %, 99 %, and 99.9 % intervals. We consider two cases: (a) we use a
fixed number of analogues, and (b) we use a fixed noise level
SD

For a defined noise tolerance criterion, there may be at best a few
locally tolerable analogues for a certain date. For example, if we
consider a criterion of 1 SD

However, we want to provide a reconstruction at each date in the period
of 1260 to 2003 CE and want to consider a fixed number of analogues. We
find that among the tested levels, a tolerance criterion of
2.57 SD

Considering a fixed standard deviation criterion, the number of valid
analogues can become large for individual years. For example, the
largest number of analogues for a single year for 1 standard
deviation is 2105 in our approach. We regard this still a subjectively
reasonable maximal number. Thus, we choose a 1 SD

We later show the results for these reconstructions in comparison to the single best-analogue reconstruction. For ensembles of analogues, uncertainty estimates are the full range of the ensemble and an uncertainty envelope based on the intra-ensemble variance.

As a side note, we could also use the individual local values for all proxies to construct a maximally tolerated Euclidean distance. The obvious caveat of this approach is that the analogues may locally lie outside the tolerance range of some of the proxy records, although the Euclidean distance is smaller than the maximally tolerated value. On the other hand, the criterion that the analogue should lie within each individual proxy tolerance may exclude the overall best analogue according to the minimal Euclidean distance. We consider this downside acceptable and only consider these. Furthermore, we do not weight the analogues, e.g., according to their distance, because our approach of explicitly considering the uncertainty in the proxies already accounts for the mismatch between the proxies and candidate pool.

Recently,

More specifically, our set of eight proxies (see the next section) allows for 70 combinations of four proxies. We exclude those combinations without any information in northern Europe. Thereby, we obtain 65 combinations of four proxies. In addition, we choose 100 sets of simulated candidate fields. Each set includes 4824 candidate fields. We then produce 100 reconstructions for each of the 65 combinations of proxies. That is, our ensemble has in total 6500 reconstructions. We use the same 100 sets of candidate fields for all 65 combinations of proxies. For each date and each reconstruction, we only consider the single best field according to the Euclidean distance.

We compare our reconstructions to the European data by

The target of our application of the analogue method is a representation
of European temperature in summer (JJA), equivalent
to the original Euro 2k reconstruction by the

Since neither the Albanian nor the Slovakian proxy records provided by
the

We describe results for the period of 1260 to 2003 CE, although two of the Euro 2k proxy series extend back to the year 138 BC, and the analogue approach is suited to use variable numbers of proxies. The latest start date of any of the used eight proxy indices is the year 1260 CE, and thus all eight records cover the period of 1260 to 2003 CE. We decide against using uneven numbers of proxies and against extending the reconstruction further back to ease the comparison of the results and our different uncertainty estimates.

Thanks to the PMIP3 effort

We opt here for a single model ensemble predating the PMIP3 effort but
compliant with its protocol, i.e., the millennium simulations with the
COSMOS setup of the Max-Planck-Institute Earth System Model (MPI-ESM) by

We use data centered on the full period of 1260 to 2003 CE, and the data are
normalized with the standard deviation over the same period.

The best-analogue reconstruction relative to
its full-period mean:

Simulations in our pool of analogue candidates: ID, forcing components, data reference. We consider for all eight simulations the period of 800 to 2005 CE, i.e., 1206 simulated years. Forcings are stratospheric sulfate aerosols from volcanic eruptions (V), variations of total solar irradiance (large amplitude: S, small amplitude: s), changes in Earth's orbit (O), land use change (L), greenhouse gases (G); note, only methane and nitrous oxide were prescribed; the carbon dioxide concentration was calculated interactively. For details, see data references and

Figures

Further information about the single best-analogue reconstruction:

The analogue reconstruction shows rather small centennial variations as
does the Euro 2k reconstruction (Fig.

Figure

Considering uncertainties for the reconstructions, Fig.

The noise-variance-based envelope for the best-analogue reconstruction is generally wider than the uncertainty of the Euro 2k reconstruction, while the mean squared error (MSE)-based analogue uncertainty is usually narrower. The MSE-based uncertainty is also generally narrower than the noise-based uncertainty but can become occasionally very wide. The latter widening reflects that the best analogues may fit poorly to the proxy records. The MSE-based uncertainty estimates become particularly wide in the late 20th century, highlighting that the single best analogues found for this period do not match the proxy data well. The best-analogue reconstruction is generally within the 2 standard deviation uncertainty of the Euro 2k reconstruction. Similarly, the noise-based uncertainty estimate for the analogue reconstruction usually includes the Euro 2k data.

Both uncertainty measures for the analogue reconstruction describe different but not mutually exclusive parts of the uncertainty of the reconstruction. The variance-based envelope estimates the reconstruction uncertainty based on the local agreement between proxies and observations over the period when instrumental data are available. Thus, it is unlikely that the uncertainty of the reconstruction at any time is smaller than this estimate because we can assume that the quality of the proxies is best in the recent period. The proxy-based noise uncertainty estimate includes local information but extrapolates this over the period without instrumental data. On the other hand, the mean square error captures the misfit between the uncertain proxies and the final reconstruction product. Where it is smaller than the variance-based estimate, we would call it unrealistic. When it exceeds this estimate, it is preferable.

Normalized proxy values (squares) for proxies
included and the values of the best analogue for selected years (lines).
Proxy locations on the

Both measures of reconstruction uncertainty rely on the level of
agreement between reconstructed and observed data. In the following, we
particularly look at the agreement between the reconstructed data and
the proxy data as it enters the MSE-based uncertainty estimate. Figure

A slightly disconcerting feature is visible for, e.g., the year 1947,
where the analogue appears to underestimate the intra-location
variability. Figure

Figure

We do not investigate the differences in intra-location variability in detail. There are a number of explanations on which we only very shortly touch here. First, the noisy proxy series may overestimate the true intra-location variability. Second, our selected simulations may be spatially too smooth. This, thirdly, might be due to the low resolution, and simulations with higher resolutions might help then. Fourth, the chosen distance measure may result in such a feature depending on the characteristics of the simulation pool, which however should usually not be the case. Including a more diverse set of simulations may be the simplest way to investigate this in future applications.

Differences between local grid-point series
for the single best analogue and the proxy series as swarm plots.
Numbers above the

Figure

In summarizing, the general agreement between the Euro 2k and the
analogue reconstruction as seen in Fig.

Analogue reconstruction values at the
locations of the included proxies. Shown are the normalized proxies in
red, the median of 39 analogue values in black, and the full range of the
39 local analogues in blue. The

As described above, we also consider a reconstruction based on a set of
good analogues. One could base such a selection on an arbitrary number
of, e.g., 10 analogues. However, we base our choice of the number of
analogues on our considerations in Sect.

Figure

The good agreement between the proxies included in our analogue search
and our reconstructed local series extends beyond correlations. The
range of reconstructed values usually is narrow for these proxies.
However, there are also obvious mismatches, e.g., 16th century warmth in
the Austrian Alps and, more frequently, individual very cold excursions,
which are not matched in the analogues (Fig.

Figure

The analogue reconstruction for the 39 best
analogues.

The median of the fixed-number analogue ensemble is shown in Fig.

Although the uncertainty of the regional average for central Europe
shows a wide uncertainty for the 39 analogues, the full domain
reconstruction has a rather narrow uncertainty range. The full ensemble
range and a 2 standard deviation uncertainty based on the variance of
the ensemble are nearly indistinguishable in Fig.

The distribution of the uncertainty estimates of the 39-analogue median
is narrower than for the single best analogue, and the distribution also
has smaller values than for the two estimates for the single best
analogue. However, in this case, the variability of the fixed number of
analogues does not encompass the full range of potential analogues
compliant with a specific uncertainty level. Again, we note that as long
as an uncertainty estimate is smaller than the proxy noise-based
estimate as seen in Fig.

Interannual differences between the single best-analogue reconstruction
and the median of the 39-analogue reconstruction appear to be of similar
size as the interannual differences between the Euro 2k reconstruction
and the 39-analogue median (not shown). The smoothed representations
align quite well for the two different analogue approaches. On the other
hand, there are some systematic differences between the 39-analogue
median and the Euro 2k reconstruction in the smoothed version
particularly in the 14th and 16th centuries and starting from approximately the
year 1850. We generally assume that such systematic differences are due
to differing sensitivities between the regression-based approach of the
Euro 2k reconstruction and our analogue search. However, considering the
mid-16th century, the work by

Differences between the two analogue approaches do not show such systematic differences except maybe for the early 20th century. Both analogue approaches appear to overestimate the warming trend since the early 19th century. This is more pronounced in the single best reconstruction compared to the median of the 39 analogues, for which we already noted the reduced variability.

The use of a fixed number of analogues in the previous section implies
that we consider for each date a different level of proxy uncertainty
according to our considerations in Sect.

Analogue reconstruction based on an
1 SD

Figure

As indicated before, if one chooses smaller uncertainty intervals around
the proxy values, it becomes more likely that the method fails to
identify suitable analogues. This becomes obvious when considering the
smoothed estimates. This way of constraining the analogue space quite
frequently fails to provide any analogue at all. Small ticks at the
time axes of Fig.

Another period without suitable analogues occurs after the year 2000 CE.
Considering the results of

Figure

Analogue fields for three reconstructed cases with different numbers of analogues; color bars are temperature anomalies in Kelvin relative to the full period. From left to right, 1459 CE with 6 analogues, 1424 CE with 24 analogues, and 1827 CE with 817 analogues. From top to bottom are the median, local minimum, and local maximum. Black dots signal the proxy locations in the top row.

Until now, we concentrated on time series. Figure

It is surprising that, e.g., the proxies anchor the year 1827 in Turkey
only within a range of up to 8 K for the more than 800 analogues.
Even central Scandinavia may be rather cold or rather warm, although it
should be constrained by three proxy records. Indeed, the best analogue
for that year is close to the proxies (compare Fig.

The 24 analogues for the year 1424 have a tendency towards warm values, but again warm and cold conditions are found within a 1 standard deviation interval around our proxy anchors for southeastern and southwestern Europe. On the other hand, the six analogues available for the year 1459 mostly give slightly cold conditions over wide parts of the domain and especially for continental Europe.

Figure

We noted for Fig.

The fact that the fixed uncertainty analogue search commonly fails in finding suitable analogues obviously reduces its value if we are interested in complete reconstruction series. However, such deficiencies also provide valuable information about how well our pool of analogues represents the variability recorded by the proxies within a certain interval of confidence. Furthermore, the occasionally large numbers of potential analogues together with their potentially locally wide range are a note of caution that field reconstructions may be of limited value locally even if the area mean is a valid representation of past mean climates.

The local range of analogues over the
reconstruction period:

Warmest and coldest periods help to characterize the reconstructions. Note that the start date in 1260 CE prevents an assessment of the Medieval Climate Anomaly for the best-analogue data. Similarly, the occasional failure of the method to find analogues for a fixed noise level complicates any attempt to identify coldest centuries. That is, the validity of any identified period is limited, and thus the exercise is of reduced value for the fixed noise level approach.

For the period from 1260 to 1850 CE, the Euro 2k reconstruction and the best-analogue reconstruction both have the warmest 100-year period from 1353 to 1452 CE (results not shown). Considering the full period until 2003, the last hundred years were warmest. The coldest 100-year period was from 1549 to 1648 CE according to the best-analogue reconstruction but from 1579 to 1678 CE in the Euro 2k record. Estimates of coldest decades and 30-year periods fall within this coldest century and overlap between both reconstructions. Estimates for shorter warmest periods disagree more.

The coldest and warmest periods are very similar in the 39-analogue reconstruction compared to the best-analogue version. Again, coldest conditions on decadal, 30-year, and centennial timescales occur mainly in the 17th century (not shown). This holds for the median as well as the coldest and warmest analogue estimates for the periods.

As mentioned, the validity of any identified coldest century is limited for the fixed 1 noise standard deviation ensemble. However, the coldest decades and 30-year periods again are in the early 17th century as for our other approaches. We find the warmest periods usually centered about the early 15th century for the period before 1850 CE, which compares well with the Euro 2k reconstruction. However, considering only the warmest estimates of the envelope, the warmest decade occurs in the second half of the 18th century, which is more in line with the estimates of our other analogue approaches.

We now consider the response to volcanic forcing, as volcanoes are
considered to be the most important external forcing over the
pre-industrial period. They are also the best constrained past climate
forcing for the last 500 to 2000 years

Ensemble of subsampled reconstructions.

Individual eruptions show usually some cooling, though it may be quite
small for the best-analogue reconstruction (not shown). Noteworthy is
the lack of a clear response for, e.g., the Kuwae eruption, which took
place in 1458 CE according to

Regarding the reconstruction from a fixed number of analogues, we again find summer cooling following some events, but others barely leave a signal in the European area mean data (not shown). For spatial fields, similarly, there is not a distinct signal of post-eruption summer cooling. The range of analogues even allows for some regional warming.

The lack of appropriate analogues also hampers evaluating the response to well-dated tropical volcanic eruptions for the fixed 1 noise standard deviation approach. For example, there are no analogues available for the year without summer 1816 CE. Otherwise, the common feature is again that some eruptions appear to have resulted in European summer cooling, while there is no identifiable imprint for other eruptions in our European area mean data (not shown). Comparing spatial fields for this reconstruction, anomalies are more homogeneous but also smaller than for the reconstruction from 39 good analogues (not shown). While we find cooling, the wide range of the analogues also allows for notable warming for some eruptions.

Recently,

Individual reconstructions and the median of the sub-ensembles differ strongly from one another and also may display strong differences to our single best-analogue reconstruction using all data. However, the overall median and the single best analogue from all data agree well in their smoothed representation. Differences are most visible in the 14th to 16th centuries, the early 19th century, and the middle of the 20th century. The range of the subsampling ensemble is slightly larger than most of our discussed uncertainty estimates but is still generally smaller than the assumed 2 standard deviation uncertainty based on the proxy noise.

The subsampling uses only four out of eight available proxies for our domain and their coverage may be very uneven. Nevertheless, even sub-selecting the proxies appears to validly constrain the candidate pool with respect to the regional mean although with notable uncertainty. We do not provide further evaluation of the subsampling ensemble. In view of the results of our previous analyses, we presume that four proxies may indeed provide a constraint on the area mean but will fail to do so locally.

Figure

We note that the uncertainty distribution for the subsampling-based
ensemble range is centered at larger values compared to most of our other
estimates using the full set of proxies. Including less proxies in the
search is a weaker constraint on the candidate pool compared to using
all proxies, and therefore the range of potential analogues also likely
widens. The wide range of the root mean squared error (RMSE)-based estimate for
the single best analogue is mainly due to the large errors in the late
20th century as already seen in Fig.

Comparison of uncertainty estimates: from top to bottom, histograms in bins of 0.01 K for Euro 2k 2 standard deviation uncertainty (red), range of subsampling ensemble (black), range of fixed 1 noise standard deviation ensemble (dark orange), range of 39 analogues (light orange), assumed 2 standard deviation interval for the MSE-based uncertainty estimate for the single best analogue (dark blue), ensemble-variance-based 2 standard deviation interval for the 1 noise standard deviation (light blue), ensemble-variance-based 2 standard deviation intervals for the 39 analogues (yellow). We only show one-sided estimates when the estimates are symmetric. The green line throughout the panel marks the uncertainty estimate based on the proxy noise.

Neither the estimates of the fixed number of analogues nor the fixed 1
standard deviation interval likely represent the full range of
uncertainty. For most dates, the fixed number of analogues represents
only part of all valid analogues according to our assumptions on local
uncertainty, and the fixed 1 standard deviation interval is by
construction a rather narrow estimate. The assumed 2 standard
deviation uncertainty estimates based on the proxy noise are generally
larger than estimates from all other approaches as seen in the green
line in Fig.

Comparison of local grid-point analogue data for the fixed 1 standard deviation approach with an arbitrary selection of regionally representative data from BEST. Location, station name, and correlation over available station data are at the top of the panels. Grey and black indicate the interannual and smoothed analogue medians. Red and blue lines are the interannual station data and their smoothed estimate. The

Station data allow to evaluate our reconstruction against sources of
information independent of the proxies or other reconstructions. The
Berkeley Earth project

Correlations are often of notable strength between the reconstructed
median data close to locations of the long instrumental records with the
regionally representative data series from the BEST project

Comparing the data series, however, indicates notable shortcomings of
the reconstruction median. The reconstruction median often overestimates
the recent warming trend and the median shows notably less variability
than the BEST series. The underestimation of the variability on the
other hand occasionally leads to an underestimation of the most recent
warm anomalies. There are also cases where both series appear to agree
quite well over the period when both are available. Examples are the
central England temperature and Montdidier. Figure

Earlier proxy surrogate reconstructions from the analogue method usually considered the single best match or a small set of best fits to reconstruct past climate states compliant with limited local proxy information. Testing the analogue method against a prior reconstruction for the European domain shows that the method indeed allows to reconstruct past climate variability comparably to more common approaches. It appears even to appropriately capture the intra-proxy variability and the proxy variability over time. This holds for different implementations of the method using either a single best or multiple good analogues.

Our focus, however, is on the uncertainty of reconstructions by analogue
search. The method traditionally neglects the uncertainty of the final
estimate. An exception considering the Common Era of the last 2000 years
is the study by

We describe alternative approaches of obtaining uncertainty estimates
for analogue reconstructions, which do not require reducing the
available information from proxies and simulations. These estimates rely
ultimately on the assumption that the calibration correlation between a
proxy record and climate observations gives us information about how
well the proxies represent the climate. We use these correlations to
construct an estimate of the uncertainty of the area average
reconstruction based on these proxies. The square root of the sum over
the Var

We further construct two types of reconstruction ensembles based on our estimate of the local proxy uncertainty. For these ensembles, we provide two uncertainty estimates, which are their full range and an estimate based on the variance of the ensemble members. Ensemble envelopes reflect the mean uncertainty, whereas estimates based on the proxy noise generally are local uncertainties.

The uncertainty estimates from the subsampling using degraded information and the range of an ensemble from a fixed 1 standard deviation proxy noise uncertainty are similar to the uncertainty estimates from an earlier reconstruction of European summer temperature. However, our uncertainty estimates vary more than these earlier estimates. Most other estimates have a tendency to be smaller than these earlier reconstruction uncertainties although reconstructions are comparable. The time-constant estimate based on the proxy noise is larger than prior and present uncertainty estimates except for cases where the uncertainties clearly reflect that the reconstruction is a bad match to the anchoring proxy information.

We note that problems arise if we use a fixed uncertainty interval
around the proxies. In this case, we are not able to obtain good
analogues for some dates. Our approach is particularly unlikely to find
valid analogues in the fixed uncertainty level setup for years of strong
observed cooling, e.g., due to strong volcanic eruptions. This is a
fundamental shortcoming of an analogue search that considers uncertainty
in the way we do in this case. Our other estimates as well as the
approach by

We only consider complete proxy records starting at the same date with
the same temporal resolution. However, the analogue method does not rely
on these assumptions. It easily compensates for missing values and data
with different resolutions.

While we focused on the temperature fields, it is easy to additionally
reconstruct other variables that are compatible with the temperature
proxy records, since the climate models do not only simulate surface
temperature but the full climate/weather situations

We have to note that the reconstruction neglects possible information
about the past climate forcing trajectory. This has implications for
dynamical inferences, which may be misleading. While one can account for
this by including the forcing reconstruction in the anchoring dataset,
this reduces the pool of potential analogues. Furthermore, all results
depend on the consistency and quality of the pool of analogues, i.e., the
simulations and the underlying sophisticated climate models. An
interesting extension of our approach can be to preprocess the
simulation pool data by using proxy forward models

Applications of the analogue method commonly only focus on the best analogue. The failure to find any analogue and the occurrence of multiple good analogues raise the issues of extrapolation and interpolation of the analogue pool and the analogue ensemble. Interpolation of analogues may be of interest for obtaining one optimal representation for the reconstruction. More crucially, extrapolation is one solution to obtain reconstructions for situations, e.g., extremes, which are not included in the pool of potential analogues. Extrapolation of the current pool may be possible by generating synthetic analogues. Data science methods may be available to do this.

Proxy surrogate reconstructions from the analogue method often neglect that the proxies and, in turn, the reconstruction are uncertain estimates. Here, we suggest uncertainty estimates for single best-analogue reconstructions as well as analogue reconstructions from multiple good analogues. We are primarily interested in the case where we only consider analogues which fall within a certain uncertainty interval of the original proxies.

We compare reconstructions and uncertainty estimates to a previously published reconstruction. This evaluation suggests that the analogue approaches capture the variability as well as a composite-plus-scaling approach. The analogue reconstructions also appear to capture the intra-proxy variability and the proxy variability. Similarly, our results suggest that our approach compares well to independent data.

If we only use analogues that comply with the proxies within a certain uncertainty interval, the problem arises that there may be no compliant candidates in the pool of simulated fields. Generally, the uncertainties and the evaluation of the local range of reconstructions suggest that the proxies only loosely constrain the reconstructions.

Upscaling the local proxies to obtain larger-scale climate information holds many opportunities to infer information about past climate states. However, one has to add relevant estimates of uncertainty to provide meaningful information.

This Appendix provides a number of additional figures to assist the comparison of our reconstructions and our uncertainty estimates to previously published work.

Figure

The best-analogue reconstruction as in
Fig.

Differences between the CRU TS data and the Euro 2k reconstruction plotted against the differences between the CRU TS data and the single best-analogue reconstruction.

Figure

Finally, Figs.

Comparison of the reconstructions of

Comparison of local grid-point analogue data for the fixed number of analogues approach with an arbitrary selection of regionally representative data from BEST. Location, station name, and correlation over available station data are at the top of the panels. Grey and black indicate the interannual and smoothed analogue medians. Red and blue lines are the interannual station data and their smoothed estimate. The

Comparison of local grid-point analogue data for the single best analogue with an arbitrary selection of regionally representative data from BEST. Location, station name, and correlation over available station data are at the top of the panels. Grey and black indicate the interannual and smoothed analogue medians. Red and blue lines are the interannual station data and their smoothed estimate. The

This paper uses a number of external software packages. These include
the Climate Data Operators (CDOs;

The simulation data are available from the World Data Center for Climate
(WDCC) at

OB devised the analyses, performed them, and wrote the first draft. OB and EZ discussed the results and revised the manuscript.

The authors declare that they have no conflict of interest.

Funding in the projects PRIME2 and PALMOD (

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. ZO133/6-2) and the Bundesministerium für Bildung und Forschung (grant no. 01LP1509A).The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Jürg Luterbacher and reviewed by three anonymous referees.