Advancing research on compound weather and climate events via large ensemble model simulations

Abstract

Societally relevant weather impacts typically result from compound events, which are rare combinations of weather and climate drivers. Focussing on four event types arising from different combinations of climate variables across space and time, here we illustrate that robust analyses of compound events — such as frequency and uncertainty analysis under present-day and future conditions, event attribution to climate change, and exploration of low-probability-high-impact events — require data with very large sample size. In particular, the required sample is much larger than that needed for analyses of univariate extremes. We demonstrate that Single Model Initial-condition Large Ensemble (SMILE) simulations from multiple climate models, which provide hundreds to thousands of years of weather conditions, are crucial for advancing our assessments of compound events and constructing robust model projections. Combining SMILEs with an improved physical understanding of compound events will ultimately provide practitioners and stakeholders with the best available information on climate risks.

Full text

Article https://doi.org/10.1038/s41467-023-37847-5
Advancing research on compound weather
and climate events via large ensemble model
simulations
Emanuele Bevacqua
1
, Laura Suarez-Gutierrez
2,3,4
, Aglaé Jézéquel
5
,
Flavio Lehner
6,7,8
,MathieuVrac
9
,PascalYiou
9
& Jakob Zscheischler
1
Societally relevant weather impacts typically result from compound events,
which are rare combinations of weather and climate drivers. Focussing on four
event types arising from different combinations of climate variables across
spaceandtime,hereweillustratethat robust analyses of compound
events —such as frequency and uncertainty analysis under present-day and
future conditions, event attribution to climate change, and exploration of low-
probability-high-impact events —require data with very large sample size. In
particular, the required sample is much larger than that needed for analyses of
univariate extremes. We demonstrate that Single Model Initial-condition Large
Ensemble (SMILE) simulations from multiple climate models, which provide
hundreds to thousands of years of weather conditions, are crucial for advan-
cing our assessments of compound events and constructing robust model
projections. Combining SMILEs with an improved physical understanding of
compound events will ultimately provide practitioners and stakeholders with
the best available information on climate risks.
Most environmental impacts result from combinations of multiple
weather and climate drivers that are referred to as compound
events1–3. Given the complexity of the climate system, such events can
occur in multiple ways, for example involving extreme conditions in
different variables happening simultaneously or across space and
time4. For instance, in 2014/2015, Rio de Janeiro (Brazil) was hit by
concurrent extremely hot and dry summer conditions, which led to
wide-ranging socio-economic repercussions that exceeded the
impacts that would have been caused by heat and dryness in isolation5.
The hot-dry weather favoured widespread wildfires as well as water
scarcity that impacted coffee production5. Furthermore, excess fatal-
ities due to a severe dengue fever outbreak were linked to the increase
in water storage tanks installed by the population to mitigate the
drought5. In a similar fashion, impacts from hurricanes typically arise
from concurrent hazards via precipitation-driven flooding and wind-
driven hazards such as storm surges, as happened when Hurricane Ida
led to destruction in Louisiana (United States) in 2021, causing about
30 fatalities6. Extreme impacts can also arise from sequences of
weather events that increase system vulnerability or lead to dis-
proportionate damages, as exemplified by ecosystem7and
agricultural8impacts from the two European drought years in 2018 and
2019, and by the critically reduced water storages in South Africa due
to a sequence of three dry winters in 2015-20179,10.Furthermore,
combinations of extremes occurring simultaneously (or almost
simultaneously) across multiple regions can cause extreme impacts to
connected global systems11. For example, simultaneous soil moisture
Received: 24 June 2022
Accepted: 31 March 2023
Check for updates
1
Department of Computational Hydrosystems, Helmholtz Centr e for Environmental Research—UFZ, Leipzig, Germany.
2
Max Planck Institute for Meteorology,
Hamburg, Germany.
3
Institute for Atmospheric and Climate Science, ETH Zurich, Zurich, Switzerland.
4
Institut Pierre-Simon Laplace, CNRS, Paris, France.
5
LMD/IPSL, ENS, Université PSL, École Polytechnique, Institut Polytechnique de Paris, Sorbonne Université, CNRS, Ecole des Ponts, Marne-la-Vallée, France.
6
Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY, USA.
7
Climate and Global Dynamics Laboratory, National Center for
Atmospheric Research, Boulder, CO, USA.
8
Polar Bears International, Bozeman, MT, USA.
9
Laboratoire des Sciences du Climat et de l’Environnement, IPSL,
CEA-CNRS-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France. e-mail: emanuele.bevacqua@ufz.de
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droughts across multiple breadbaskets can reduce global food pro-
duction and threaten food security12–15.
Compound events are thuscharacterized by complex multivariate
relationships. Recognizing that a univariate perspective on hazards
may severely underestimate risk16 led to a rapidly growing body of
literature on different types of compound events3,4,17, including lit-
erature on interconnected, interacting, and cascading risks18. However,
the limited temporal length of observations and routinely used model
simulations often limits the possibility of identifying and studying the
sparse multidimensional constellations of climate variables constitut-
ing compound events, an issue related to the curse of
dimensionality1,19–21. Combining routinely used single-member simu-
lations of multiple climate models, such as the standard multi-model
ensembles from the coupled model intercomparison project (CMIP)22,
offers larger sample sizes but leads to confounding irreducible
uncertainties arising from internal climate variability (the inherent
chaotic nature of climate) and from potentially reducible uncertainties
arising from structural differences across models23–26. Robustly dis-
tinguishing between these uncertainties is important, however, as it
can ultimately help reduce uncertainties from structural model
differences25,26.
In this perspective, we argue thatusing output from climate Single
Model Initial-condition Large Ensembles24 (SMILEs) should become
standard for studying and projecting compound events. A SMILE
consists of many simulations (i.e. ensemble members) from a single
climate model based on the same model physics and under the same
external forcings, but each starting from slightly different initial states.
Thus, each realization inthe ensemble evolves differently solely due to
internal climate variability. Combining the multiple members of a
SMILE ensures therefore a much better sampling of the data sparse
regions associated with compound events compared to traditional
model simulations27. Using a combination of different SMILEs then
allows for identifying model differences.
Here we propose six important topics in compound event
research and illustrate associated challenges. Each of the following six
sections provides information on the importance of one of the six
topics and demonstrates, based on our analyses and the literature,
both the challenge of addressing the topic and how using SMILEs can
be of help in this context. Specifically, we demonstrate that SMILEs are
important to (i) robustly assess compound event frequencies and
associated uncertainties; (ii) quantify the extent to which anthro-
pogenic climate change has contributed to observed compound
events; (iii) provide robust quantification of historical and future
changes in compound events; (iv) quantify and explore uncertainties
in future projections, including distinguishing between uncertainties
arising from internal climate variability and structural model differ-
ences; (v) identify most dangerous low-likelihood compound events,
advancing their physical understanding, and exploring associated
risks; and (vi) evaluate the skill of advanced statistical models devel-
oped to improve the assessments of compound events. Via analyses
related to four different compound event types, we illustrate how
these topics can be addressed using SMILEs. The considered com-
pound event types are compound hot-dry warm-season events, con-
current precipitation and wind extremes, multi-year soil moisture
droughts, and concurrent soil moisture droughts across global soy-
bean breadbaskets. See Table 1for a comprehensive list of the six
research topics and the associated analyses related to different com-
pound events.
Results
Estimating compound event likelihoods and associated
uncertainties
An important challenge of climate studies is assessing the likelihood of
extreme events26. This information ultimately serves decision-makers,
international development agencies, and insurances to develop
preparedness for climate impacts3. Employing the approach of Bev-
acqua et al.26 and focussing on the historical (1950–1980) frequencies
of compound hot-dry warm seasons (f
HD
), concurrent daily precipita-
tionandwindextremes(f
PW
), and multi-year (i.e., consecutive annual)
droughts (f
MYD
), we illustrate that using a large sample size is essential
for robustly estimating compound event probabilities. Using
3100 years of data from 31-year periods of the 100 ensemble members
of the MPI-GE climate model (40 members of the CESM1-CAM5 model
for f
PW
) allows for robustly identifying hotspot areas where compound
event frequencies are higher than what would be expected when
assuming that the compound drivers are statistically independent (see
unstippled regions in Fig. 1a–c).
Importantly, if such frequencies were estimated based on much
smaller sample sizes contained in typically available model simulations
and observations, they would be highly uncertain and could ultimately
lead to misleading risk estimates. For example, based on 31 years of
data from a single ensemble member, someone interested in com-
pound hot-dry event risk over Southern Asia could obtain regionally-
averaged f
HD
ranging from 2% to 6% (centred 90% range across the
ensemble members), i.e. differing by a factor of 3 (Fig. 1d, g; note that
frequencies at the local scale can even range from about 0% to 10%).
That means, depending on the data sample selected at random, a wide
rangeofpossiblerisksmaybeestimated.Similarly,f
PW
estimates could
differ by a factor of 2 on average over Portugal, while f
MYD
by a factor of
12 over Central NorthAmerica. Hence, based on a short sample of data,
in principle, a low compound eventrisk may beestimated over regions
that are, instead, at high risk. This sampling uncertainty stems fr om the
fact that 31-year periods are, in this case, highly insufficient to sample
internal climate variability, resulting in a wide range of possible fre-
quency estimates across single ensemble members26. Similarly, this
sampling uncertainty is also inherent in estimates derived from short
observational records. SMILEs allow users to obtain a robust model-
based estimate of the compound event probability (i.e., the frequency
computed from many years of data obtained by pooling multiple
ensemble members) and allow for a quantification of how uncertain
compound event frequency estimates are based on a given sample
size. The latter can be estimated as the range of frequencies across
single ensemble members. The differences in compound event fre-
quencies across different ensemble members are likely to partially
arise from fluctuations in climate modes of variability that influence
regional weather over decadal scales. For example, Atlantic Multi-
decadal Variability modulates hot and dry conditions in Southern
Asia28 and droughts in Central North America29. SMILE-based analyses
can help to assess the influence of the modes of variability on the
compound events12.
It is important to note that the uncertainties in compound event
estimates are substantially larger than for univariate extremes inde-
pendently of the considered sample size, highligh ting that employing a
large sample size is much more important for compound than for
univariate events. This is exemplified by the relative uncertainty in the
compound frequency, which is defined here as the ratio of twice the
standard deviation to the mean of the frequency across ensemble
members26. Based on 31 years of data, this relative uncertainty is 2.4
and 1.1 on average over land for compound hot-dry events and uni-
variate hot events, respectively (see continuous lines in Fig. 2a). To
reach a relative uncertainty of 1.1 for compound events, 132 years of
data would be required (see arrow in Fig. 2a). The results further
highlight that the relative uncertainty in the compound frequency is
higher at locations wherethe coupling between the compound drivers
is weaker (compare dashed and dotted lines in Fig. 2a), that is, when
the statistical dependencies make the compound event rarer, there-
fore more difficult to sample.
The larger uncertainties for compound than for univariate events
holds for very different event types, including compound precipitation
andwindextremes(Fig.2b), despite the fact that the daily scale of
Article https://doi.org/10.1038/s41467-023-37847-5
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Table 1 | The six research topics and the associated analyses related to different types of compound events
Research topic Analysis Type of compound event Data Figure
(i) Robustly assess compound event frequencies and associated
uncertainties
Uncertainty in the historical frequency of
compound events
Compound hot-dry event (focus on
Southern Asia)
MPI-GE (100 ensemble members) 1
- - Concurrent precipitation and wind extremes
(focus on Portugal)
MPI-GE (100) -
- - Multi-year drought (focus on Central North
America)
CESM1-CAM5 (40) -
- Dependence of the uncertainty in the frequency of
compound and unviariate events on the
sample size
Compound hot-dry event (global scale) MPI-GE (100) 2
- - Concurrent precipitation and wind extremes
(global scale)
MPI-GE (100) -
- - Multi-year drought (global scale) CESM1-CAM5 (40) -
(ii) Quantify the extent to which anthropogenic climate change
has contributed to observed compound events
Required sample size for attribution of compound
events to anthropogenic climate change
Generic bivariate compound event Synthetic data from bivariate normal
distribution
3
(iii) Provide robust quantification of historical and future chan-
ges in compound events
Effect of internal climate variability on changes in
dependencies
Dependence between precipitation and
temperature (global scale, with focus on
Central Europe)
MPI-GE (100) 4
- Effect of internal climate variability on changes in
compound event frequencies
Simultaneous soil moisture drought over
main soybean regions worldwide
MPI-GE (100) 5
(iv) Quantify and explore uncertainties in future projections,
including distinguishing between uncertainties arising from
internal climate variability and structural model differences
Partitioning uncertainty in the future frequency of
compound events (internal climate variability vs
model differences)
Compound hot-dry event (global scale) CESM1-CAM5 (40), CSIRO-Mk3-6-0 (30),
CanESM2 (50), EC-EARTH (16), GFDL-CM3
(20), GFDL-ESM2M (30), and MPI-GE (100)
6
- High- and low-risk climate storylines Compound hot-dry event (South Africa) - -
(v) Identify most dangerous low-likelihood compound events,
advancing their physical understanding, and exploring
associated risks
Identification of extreme event-based storylines Concurrent precipitation and wind extremes
(Portugal)
CESM1-CAM5 (40) 7
- - Simultaneous soil moisture drought over
main soybean regions worldwide
MPI-GE (100) -
(vi) Evaluate the skill of advanced statistical models developed
to improve the assessments of compound events
Evaluation of statistical models via a perfect-model
approach
Simultaneous soil moisture drought over
main soybean regions worldwide
MPI-GE (100) 8
Article https://doi.org/10.1038/s41467-023-37847-5
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precipitation-wind events yields a much larger sample size (compared
to compound hot-dry warm-season events). In line with the curse of
dimensionality, when we consider multi-year droughts, the relative
uncertainties increase with the dimensionality of the considered
compound event (Fig. 2c). Hence, employing large sample sizes is even
more important for estimating the occurrence of higher-dimensional
compound events such as multi-year floods, co-occurring fire
drivers30,31, concurrent droughts in multiple regions13, and spatially
extended floods32. While arguably our examples are somewhat
extreme by focusing on a sample size as small as 31 years, the issues
remain for larger sample sizes (Fig. 2). Furthermore, we note that it is
not uncommon in the literature that probabilities of compound events
are estimated based on such small sample sizes33–38, including prob-
abilities of high-dimensional events13,20,30,39. Shorter periods than
31 years are often used for projections at fixed global warming levels,
e.g. in reports of the Intergovernmental Panel on Climate Change3.
Compound event attribution
In the aftermath of a high-impact weather event, the extent to which
anthropogenic climate change has contributed to the event is regularly
questioned40. Addressing this attribution question is important to gain
a better understanding of how climate change has affected and might
affect extreme events and their impacts41.Onewaytoapproachthe
question is to frame it in terms of a probability ratio PR, defined as the
ratio of the probability of an extreme event occurring under current
conditions (factual world) to the probability that the event occurs in a
world without anthropogenic climate change (counterfactual world)42.
Such probabilities can be estimated using climate model simulations,
for example by comparing runs simulating both worlds43,orbyusing
approaches based on non-stationary extreme value theory, where the
occurrence of extreme events is fitted against a covariate such as
global mean temperature44. Values of PR > 1 imply that climate change
contributed to a certain event occurrence; for instance, PR = 2 implies
that climate change made a certain event class (typically defined as all
events exceeding a critical threshold) two times more likely. So far,
attribution studies have mainly treated events as univariate, however—
becausemany climate-related impacts are caused by a combination of
multiple drivers—studies on compound event attribution are
emerging43–46.
As attribution studies typically focus on very extreme events, they
are often based on large ensemble climate model simulations. For
example, the weather@home experiment, a large ensemble of simu-
lations of the regional atmosphere-only climate model HadRM3P47 has
been widely used by the attribution community. Other studies have
relied on the multi-model CMIP5 and CMIP6 ensembles to gather
enough data to isolate any climate change forced signal from internal
climate variability noise, though the ability to disentangle internal
variability from model differences ultimately depends on the number
Fig. 1 | Historical (1950–1980) frequency of compound events and associated
uncertainties. a Ensemble mean frequency of compound hot-dry events (f
HD
;
during the warm season) based on the MPI-GE model. Stippling indicates areas
where the compound event ensemble-mean frequency issmaller than expected in
a referencecase that assumesindependencebetween the compound drivers(here,
independence between average temperature and precipitation during the warm
season). d,gf
HD
of the ensemble members associated with the 5th and 95th per-
centilesamongst all the members, respectively, of the f
HD
averaged over the region
in the green box. b,e,hAs in panels a,d,g, but for concurrent daily precipitation
and wind extremes (f
PW
) and based on the CESM1-CAM5 model. c,f,iAs in panels
a,d,g, but for the frequency of three consecutive annual soil moisture droughts
(f
3YD
). Univariate extremes were defined via percentile-based thresholds, i.e. hot
and dryseasons occurring every10 years on average,precipitation and windevents
occurring twice a year, and soil moisture droughts every 5 years (see Methods).
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of runs available for each model24,48. The required ensemble size to
estimate internal variability, which is needed to provide robust attri-
bution assessments, depends on several factors, such as the spatio-
temporal scale of the event, the variable, and the amplitude of the
forced trend49.
Here, wedemonstratethat for compound event attribution,larger
sample sizes are required than for univariate event attribution, in
particular if the drivers are not strongly correlated and have trends of
similar m agnitude. We build on the synthetic data approach developed
by Zscheischler and Lehner (2022)43. For the sake of simplicity, we
focus on events that increase in probability with climate change
(Fig. 3a). That is, we calculate the required sample size for the attri-
bution of concurrent high extremes of climate variables that show
positive trends due to anthropogenic climate change such as coastal
flooding followed by a heatwave50.Wedefine the required sample size
as the number that guarantees correctly identifying PR > 1 in the pre-
sence ofa forced trend, assuming no trendin the dependence between
drivers43 (see Methods). In general, very large sample sizes are needed
to disentangle small climate change driven trends from internal cli-
mate variability (Fig. 3b). We find that when trends in the drivers are
comparable, attributing compound events requires a larger sample
size than that needed for attributing any of the contributingunivariate
events (see stippling in Fig. 3b). In particular, a substantially larger
sample size for compound than for univariate events is needed for
smaller forced trends (Fig. 3c). For example, the sample size needed
under identical trends of 0.4 ⋅σ
i
in the drivers differs by about 330 (i.e.
by a factor 2) between the bivariate (under zero correlation) and the
univariate cases (Fig. 3c). Furthermore, larger sample sizes are
required when the dependence between the compound event drivers
is small or negative (Fig. 3c), i.e.—in general—when the dependence
makes the compound event rare. For example, the sample size needed
under identical trends of 0.4 ⋅σ
i
in the drivers differs by about 1200 (a
factor 3.5) between the bivariate case under −0.5 and +0.5 correlation
(Fig. 3c).Whenthetrendsinthecompoundvariablesdifferstronglyin
magnitude, attributing the compound event requires a sample size in
between what is needed for attributing the univariate events in isola-
tion (missing stippling in Fig. 3b). That is, for example, attributing a
compound hot-dry event to anthropogenic warming trends requires a
larger sample size than that needed for attributing the heat event in
isolation, though smaller than for attributing the drought event alone
given that trends in droughts are much weaker than trends in heat-
waves in most regions3.
Detecting changes in compound event characteristics
Internal climate variability arising from the chaotic nature of the cli-
mate system can obscure anthropogenic climate change signals within
Fig. 3 | Required sample size for attribution of compound and univariate
events. a Idealized bivariate distribution of two compound drivers (X,Y)under
historical conditions (green), and under present conditions (red) obtained via
shifting the mean of the historical drivers by forced trends Trend
i
equal to one
standard deviation (σ
i
) of the historical distribution. Shading in the top-right
depicts compound events, i.e. concurrent high extremes. bMinimum sample size
required for attribution of concurrent high extremes (>90th percentile) of syn-
thetic data simulated from a bivariate standard Gaussian distribution (with con-
stant zerocorrelation)in the presenceof positive forcedtrends Trend
X
and Trend
Y
in the two drivers (expressed in units of standard deviations). Grey on the bottom
bottom-left indicates a required sample size larger than the values in the palette.
The large orange dotindicatesthe combination of trends displayed ina.Stippling
indicates combinations of Trend
Y
and Trend
X
for which attributing the compound
event requires a larger sample size than that needed for attributing any of the
underlying univariate events (elsewhere compound attribution requires a sample
size in between that needed for the two univariate events in isolation). cSample
size required for attributing any of the two univariate events (orange), and the
compound event (green) assuming equal forced trends for both drivers and dif-
ferent constant correlation (ρ) between the drivers (see legend).
Fig. 2 | Dependence of the uncertainty due to internalclimate variability on the
sample size. a Relative uncertainty due to internal climate variability in the fre-
quency of univariate hot events f
H
(orange) and compound hot-dry events f
HD
(solidgreen). Dashedand dotted greenlines show therelative uncertaintyin f
HD
for
grid points with the 35% strongest and weakest coupling, respectively (see Meth-
ods). Relative uncertainty is defined as the ratio of twice the standard deviation to
the mean of the frequency facross ensemble members averaged over global
landmasses (see Methods). bAs in a, but for the frequency of wind extremes (f
W
,
orange)and concurrent precipitation and windextremes (f
PW
,green).cAs in a,but
for the frequency of a single-year soil moisture drought (f
1YD
, orange), two (f
2YD
,
green) and three (f
3YD
, purple) consecutive annual soil moisture droughts,
respectively. The climate models and the definitions of the extremes are the same
as in Fig. 1.
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short time periods51. Hence, especially for low-probability events,
robust sampling of internal climate variability under different global
warming states is crucial for assessing anthropogenically-forced
changes robustly51. For compound events, identifying
anthropogenically-forced changes requires, in addition to considering
changes in the individual compound event drivers, to also quantify
changes in their coupling, i.e. their statistical dependence32,52–54.How-
ever, quantifying any potential anthropogenic signals in the depen-
dence, and therefore understanding the processes driving such
signals, has proven to be particularly challenging so far due to the
limitedsamplesizeofconsidereddatasets
33,34,38,52,54.Forexample,for
compound coastal flooding—which is often driven by concurrent wind
and precipitation extremes54—Wahl et al.52 identified an increase in the
dependencies between the drivers in the historical period over the
United States’coast, but it was not possible to attribute this increase to
anthropogenic climate change. Bevacqua et al.54 suggested that inter-
nal climate variability may dominate such dependency changes even
under high greenhouse gas emissions and that SMILEs may therefore
be necessary for identifying any potential forced changes. The SMILE
experimental design allows for a quantification of the forced response
(including changes arising from mean and variability55 of individual
compound drivers, as well as from their dependence), by averaging
changes across ensemble members, hence filtering out spurious
trendsdue to internalclimate variability. Once SMILE simulationsallow
us to disentangle such changes, further analysis can be carried out to
identify the physical mechanisms driving these changes.
Focussing on compound hot-dry events, previous studies based
on single-member simulations highlighted disagreement among
model runs on the sign of the changes in the statistical dependence
between temperature and precipitation53, ultimately causing large
uncertainties in impact assessments. In particular, changes in this
dependence are expected to modulate the heat sensitivity of crops as
temperatures rise, thereby affecting crop production56. We illustrate
that such changes in the dependence can be robustly estimated using
the large ensemble simulations of the MPI-GE model as an example
(Fig. 4a). Highly misleading results in the strength and sign of the
changes may be obtained based on short time periods (31 years) from a
single model simulation (note that this sample size is similar to that
used in previous studies33–35,38). This is due to the large uncertainty due
to internal climate variability in the dependence change (Fig. 4b,
compare with Fig. 4a). Thus, using model output from only one model
simulation does not even guarantee to identify the correct sign in the
projected change. For example over Central Europe, one may identify
either a positive or negative change (Fig. 4c, d) despite the clear
underlying negative forced change (Fig. 4a).
Similar considerations apply to the overall changes of com-
pound event occurrence (not just changes in statistical dependence
of drivers), as exemplified for trends in spatially compounding
events. A previous study, by comparing two recent adjacent 20-year
observational periods, indicated an increase in the probability of
multiple breadbasket failures for wheat, maize, and soybean, and a
decrease for rice13. However, model simulations (MPI-GE model)
indicate a large variability in the frequency of a similarly defined
5-dimensional compound event, i.e. concurrent annual soil moisture
droughts across the five main soybean breadbaskets worldwide
(Fig. 5a), when estimated based on just31 years of data (derived from
a single ensemble member) (see blue bars in Fig. 5b). As a result, even
two consecutive 31-year periods under the same historical conditions
could exhibit a large increaseor decrease in this frequency solely due
to internal variability (grey bars in Fig. 5c). This indicates the chal-
lenge of attributing detected changes in spatially compounding
events to anthropogenic climate change without employing large
sample sizes. Notably, the potential for detecting both positive and
negative changes persists even when considering projections in a
world 2 °C warmer than pre-industrial conditions (brown bars in
Fig. 5c, and Fig. 5b), in which case forced changes are expected to be
much larger than in the historical period. Here, large ensemble
simulations allow for robustly identifying a future forced response.
For instance, by merging one-hundred ensemble members of
31 years, the change in the probability of Nconcurrent regions under
droughts (lines overlaid to brown bars in Fig. 5c), is statistically sig-
nificant (95% confidence level) up to N=4.
Fig. 4 | Internal climate variability can obscureforced changes in intervariable
dependence. a Ensemble mean (MPI-GE model) change in Spearman correlation
(Δρ
S
) between summertime mean temperature and mean precipitation in a 2 °C
warmer world relative to preindustrial conditions. bUncertainty in Δρ
S
due to
internal climate variability (twice the standard deviation of Δρ
S
across ensemble
members, i.e. 2 ⋅U
IV
; see Methods). c,dΔρ
S
of the ensemble members associated
with the5th and 95th percentiles amongst allthe members, respectively, rankedby
Δρ
S
averaged over Central Europe.
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Uncertainty sources in projections and plausible climate story-
lines for impact modelling
Quantifying and understanding uncertainties in climate projections
and identifying the potentially wide range of plausible climate futures
is highly relevant for policymakers25,57,58. It is important to distinguish
between uncertainty arising from internal climate variability, which is
irreducible because of the inherently chaotic nature of the climate
system, and uncertainty from structural differences amongst climate
models, which could be reduced through, for instance, model
improvements or emergent-constraints59. Uncertainty partitioning
based on multi-model single-ensemble simulations23 is expected to be
biased, especially at regional scales and for variables with large
variability60, which includes compound event frequencies. Here, we
illustrate how multiple SMILEs can be used to partition uncertainties in
compound event projections following recently proposed
approaches25,26 (see Methods).
For compound hot-dry events, we demonstrate thatcontributions
from internal climate variability and structural model differences to
uncertainties in compound event projections depend on the warming
level, with the two sources broadly contributing comparably in a 2°C
warmer world (Fig. 6a) and model differences dominating in a 3 °C
warmer world (see purple dominating in Fig. 6b). Uncertainty in cli-
mate projections (i.e. in the climate over a period of, for instance,
30 years) can be communicated via climate storylines, which are dis-
tinct plausible future climates61 that can be derived from large
ensemble climate model simulations26,62,63. For example, one storyline
could be derived from a model simulation that exhibits an increase in
precipitation over a region of interest, while another from a simulation
that exhibits a precipitation decrease. Note thatclimate storylines may
also be used as input to impact models to allow for an efficient
exploration ofthe full rangeof plausible future climates and associated
impacts (see final discussion).
Distinct climate storylines derived from different ensemble
members of a single SMILE26,62 allow for communicating “certain”
irreducible uncertainty stemming from internal climate variability. In
fact, the robust sampling of internal variability offered by the large
number of ensemble members of a SMILE allows for identifying very
extreme, albeit plausible future scenarios that may lead to the largest
impacts26,61,62. An illustration of climate storylines with different com-
pound hot-dry event frequencies in a 2 °C warmer world over Southern
Africa is provided in Fig. 6d, g. When uncertainties from model dif-
ferences dominate, as it happens in a 3 °C warmer world in this
example, climate storylines derived from different models can be used
to communicate such uncertainties transparently (Fig. 6e, h). In prin-
ciple, such uncertainty from model differences could be reduced by
improving the representation of the processes driving compound
event changes. For example, in Southern Africa, improving the
representation of processes dominating precipitation uncertainties, in
particular related to shifts in theconvective region and the structure of
sea surface temperature in the surrounding oceans64, is essential to
reduce uncertainties in future compound risk26.
Event-based storylines
The largest socioeconomic impacts are often caused by individualvery
rare events65,66. Estimating the occurrence probabilities of such rare
events is very challenging, but a better understanding of the events’
dynamics is critical for risk analyses. This can be done using an event-
based storyline67, which represents a physically self-consistent unfold-
ing of a plausible individual event. The concept of event-based story-
line differs from that of climate storyline introduced above, as the
former refers to a plausible individual event (e.g., a very intense storm
or crop failure and its consequences), and the latter to a plausible
future climate over a long period (e.g. 30 years). The concept of event-
based storylines has been employed to study several types of rare high-
impact events over recent years68–72. Event-based storylines can be
used to improve our understanding of the physical drivers of very
extreme eventsand, through collaborationsbetween climate scientists
and impact modellers, the event’s socio-economic consequences66.
However, identifying rare high-impact events that typically consist of
combinations of multiple climatic drivers from short samples of data is
difficult73,74. For example, based on datasets with limited sample size,
identifying and studying the potential for combinations of physical
drivers that lead to record-shattering heat events can be difficult73,and
therefore the threats of such rare but plausible extreme events could
be neglected.
In this context, the large sample of atmospheric conditions
obtained by pooling multiple ensemble members of a SMILE can be
used to identify very rare and potentially high-impact events that are
plausible but have not yet occurred in the observational record (so-
called unseen events73,75,76), which is particularly useful for compound
events as we illustrate with two analyses. To quantify rarity in a multi-
dimensional space we rely on a copula-based approach77,78 by selecting
events that maximise the copula describing the distribution of the
compound event drivers (see Methods).
The first analysis concerns compound precipitation and wind
extremes over Portugal. Using all available simulations from multiple
ensembles of the model CESM1-CAM5, we find that much more
extreme events are possible compared to the most extreme event
identified in a single simulation of that model (Fig. 7a). Specifically,
based on the first ensemble member we identify as the most extreme
event an extratropical cyclone with a central pressure below 990 hPa,
which causes wind extremes along the Portuguese coast, but virtually
no heavy precipitation (Fig. 7b). Based on all simulations the most
extreme event is a deep cyclone with a core pressure below 970 hPa,
Fig. 5 | Internal climate variability can obscure forced changes in compound
events. a The five soybean regions. bHistorical (blue) and future (orange) fre-
quency of annual soil moisture drought (defined as events occurring every 4 years
on average)occurring in Nsimultaneous (x-axis)soybean regions(based on MPI-GE
model). Coloured bars indicate the centred 95th percentile range and the mean
amongst the ensemble members (historical probabilities based on individual
membersare shown throughbackground lines). cProjected change in probabilities
of simultaneous droughts in a 2 °C warmer world relative to preindustrial condi-
tions (brown) and variationsbetween two historical periods due to internal climate
variability (grey; 95% range of changes obtained from randomly paired historical
ensemble members).
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which causes extreme wind speeds and daily precipitation above
100mm in Portugal (Fig. 7c). Such an eventmay cause a combination of
inland flooding and landslides2, coastal storm surges54, uprooted trees,
anddownedpowerlines
79. In a second analysis, we search for con-
current droughts in five key soybean production regions using data
from the model MPI-GE. The most extreme event based on all pooled
ensemble members is associated with droughts that are characterized
by (sometimes substantially) lower soil moisture over all regions than
based on the first ensemble member alone (Fig. 7d). Here, in line with a
recentSMILE-based analysis12 that revealedthe contribution of oceanic
modes of variability on concurrent droughts across multiple bread-
basket regions, further analyses could help identify plausible oceanic
precursors of worst-case spatially compounding droughts. In general,
the simulated worst-case events should be considered plausible only if
the driving processes are in line with basic physical principles80.
Developing and evaluating statistical tools for compound events
analysis
Properties of compound events are often studied via statistical
modelling2,16,20,81,82. For example, statistical weather generators, which
are used to simulate realistic weather conditions, can—in principle—
allow sampling unobserved extreme events and estimating the like-
lihood of very rare events, reducing issues related to the limited
sample size of observations77. To meet the challenge of studying dif-
ferent aspects of compound events, new statistical methodologies of
different levels of complexity are being developed2,20,83–86. However,
novel statistical tools need to be tested and evaluated, and the limited
sample size of observational records is a constraining factor in this
regard.
Building on a perfect-model approach (introduced below), SMILEs
offer a powerful testbed for new and existing methodologies24,75.This
can be particularly useful for assessing the skills of novel methods in
compound event research, which can be very complex given the need
for modelling multiple inter-variable relationships and multivariate
extremes87,88. In a perfect-model approach, the climate represented by
the large ensemble simulations of one climate model is used as a
testbed. The statistical model is calibrated to the data of a single
ensemble member, representing the pseudo-observations that are
limited in sample size. Then, the skill of the statistical model in
representing the statistics of interest in the underlying climate can be
Fig. 6 | Model differences dominate internal climate variability at higher
warming levels. a,bUncertainty in the frequency of compound-hot-dry events
(f
HD
) due to model-to-model differences (U
MD
) relative to the sum of U
MD
and the
uncertainty due to internal climate variability (U
IV
)inaworld2°C(a)and3°C(b)
warmer than pre-industrial conditions (expressed in percentage; uncertainty is
dominated by model-to-model differences for values above 50% and by internal
climate variability otherwise). c,fAverage of the future f
HD
over Southern Africa
(green box in d) according to individual ensemble members of different climate
models in a 2°C and 3 °C warmer world, respectively (larger symbols indicate
ensemble mean of individual models). d,gIn a 2 °C warmer world, climate story-
lines of compound hot-dry events resulting from internal climate variability, i.e.
future f
HD
from the ensemble members associated with the lowest and highest
values averaged over Southern Africa amongst all the members of the MPI-GE
model (shown in the box in c). e,hIn a 3°C warmer world, storylines resulting in
the lowest and highest risk model, i.e. ensemble mean of the future f
HD
associated
with the two models outof seven (shown in the boxes in f)showingthelowestand
highest values averaged over Southern Africa. Compound hot-dry events were
defined as in Fig. 1.
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evaluated against the climate derived from all other ensemble
members.
For example, one could be interested in evaluating a multivariate
statistical model that is fitted to 31 years of soil moisture observations
across the five soybean breadbasket regions with the goal to estimate
the probability of simultaneous droughts in the five regions. We
illustrate using the perfect-model approach (Methods) that a statistical
model based on pair-copula-constructions, a statistical tool that is
widely used for high-dimensional compound events13,20,89,under-
estimates the probability of two regions under drought in favour of a
possible overestimation of four regions under droughts (Fig. 8;the
correct probability shown by the black line in the middle of the blue
bar is outside of the sampling uncertainty in grey of the statistical
model). The statistical model does also not represent the statistics of
the data used for calibration (the red lines do not always fall inside the
grey range), which in principle may be due to structural biases in the
chosen statistical model, indicating the challenge of developing mul-
tivariate statistical models for compound events based on limited
sample sizes and the advantage offered by the large sample size of
SMILEs. While it seems unwise to estimate high-dimensional prob-
abilities from limited sample sizes, this is not uncommon in the
literature13. Finally, note that the uncertainty in the frequency of con-
current droughts arising from internal climate variability (blue bars;
the range of frequencies that can be measured in a given 31-year per-
iod) is conceptually different from the sampling uncertainty from the
statistical model (grey bars; an uncertainty around the frequency
averaged across multiple 31-year periods), hence—based on theory—a
match between these uncertainties should not be expected. Finally, we
note that a user may employ a perfect model approach to test different
types of statistical models and eventually choose the most appro-
priate one.
Discussion
While large ensemble simulations are known to be important to study
very extreme univariate events24,73,90,91,wedemonstratedthatsuch
simulations are even more relevant for compound event analysis. In
principle, when studying mild extremes under stationary climate
conditions, pooling data from observations or a historical single
ensemble member could increase sample size and partially reduce
uncertainties. However, for studying very extreme compound events
and non-stationary climate conditions, very large sample sizes from
multiple ensemble members are essential14,26,32,92–95. Furthermore, cli-
mate models are an essential tool for investigating future changes in
compound events. We recommend using SMILEs to avoid estimating
potentially incorrect compound event frequencies and associated
changes. SMILEs are especially important when the correlations
between the compound drivers areweak and for veryhigh dimensional
compound events, that is, when identifying compound event char-
acteristics becomes particularly difficult based on short sample sizes.
As we illustrated with several analyses, the large sample sizes of
SMILEs allow for tackling the challenge of robustly estimating statis-
tical features of compound events. However, we note that—as for vir-
tually all of the research related to climate and weather—research gaps
do not arise merely from statistical challenges. It is important to
advance our understanding of the physical processes behind com-
pound events and their changes via in-depth analyses that combine
Fig. 7 | Identifying extreme event-based storylines. a Pairs of wintertime
(December-February, 1950–1980) daily-mean precipitation and wind values aver-
aged overPortugal (box in b,c) based on data of thefirst ensemble member(violet)
and based on all pooled ensemble members (orange) of the CESM1-CAM5 model.
Large dots indicate the most extreme compound events in the twodatasets, i.e. the
selected event-based storylines (Methods). b,cPrecipitation (shading) and sea
level pressure (isolines) associated with the selected extreme events based on one
(b) and all ensemble members (c). Stippling indicates locations exhibiting extreme
precipitation (blue), extreme wind (green), and concurrent extremes (magenta),
where extremes are defined as values exceeding the local 10-year return levels—
based on December–February data. dAnomalies (%, with respect to the mean;
1950–1980) of annual regionally averaged soil moisture over the five soybean
breadbasket regions according to the most extreme compound event storyline
based on data from the first ensemble member (violet) and based on all pooled
ensemble members (orange) of the MPI-GE model.
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models and long observational datasets. The large sample size of
SMILEs can help advance the physical understanding of compound
events. For example, a SMILE-based analysis revealed that a relatively
small percentage increase in precipitation intensities caused by a
combination of dynamic and thermodynamic atmospheric changes
will drive a disproportionately larger increase in the spatial extent of
precipitation extremes32. Similarly, SMILEs helped reveal that the
concurrence of El Niño with the cold phase of the Tropical North
Atlantic can substantially enhance the probability of concurrent
droughts over global breadbasket regions12.Suchclimate-model-based
analyses should be carried out side by side with a climate model eva-
luation of the primary processes responsible for compound events
(see discussion below), e.g. the climate modes of variability.
SMILEs can also help advance our understanding of very extreme
events. For example, climate model simulations helped uncover the
physical drivers of record-shattering hot events73,96. Given that climate
models can have biases, very extreme simulated compound events
should be considered plausible only if the identified driving processes
are consistent with basic physical processes80.Modelsthatareunable
to capture events as extreme as those observed even with the
improved sampling provided by SMILEs are likely affected by biases in
their representation of real-world processes.
SMILEs further allow for robustly quantifying forced changes in
compound event features, including the dependencies between
compound drivers. In fact, large changes may appear when analysing
data with limited sample size, although no forced trend may exist. This
effect, which also exists for univariate quantities51, is much enhanced
for compound events. Once forced changes are identified, they can be
linked to their driving physical processes. For example, SMILEs can
help disentangle the physical mechanisms behind dependence chan-
ges, e.g. it could be studied whether the strengthening of land-
atmosphere feedbacks97 strengthens temperature-precipitation
dependencies53, whether projected changes in storm tracks54,98 influ-
ence the dependence between precipitation and wind extremes, and
how changes in dependence scale with global warming.
While SMILEs offer many advantages compared to single-member
climate model simulations, a series of considerations are worthwhile.
SMILEs have already helped substantially to advance our under-
standing of climate variability24; however, climate models can have
biases that should be considered carefully. That means climate model
skills need to be evaluated43,86,99. Assuming that any difference between
model- and observation-based estimates is only due to biases can be
misleading as, especially for multivariate relationships, large differ-
ences may arise from internal climate variability43,54. That is, climate
models with a few or even one single member may fail to capture
observed behaviours not because of model biases but simply because
of insufficient sampling of internal climate variability. The improved
sampling of internal climate variability in SMILEs allows for better
model evaluation100—specifically, it allows for avoiding rejecting
models for the wrong reason. In general, we highlight that a purely
statistical assessment of the model skills in representing the ultimate
quantity of interest (e.g., frequency of extreme events) should always
be complemented by a process-based approach that evaluates the
model representation of the driving processes of extreme events72,80,101
and therefore the physical realism of the simulated events. Here, we
argue that the latter, process-based oriented evaluation is particularly
relevant for compound events. In fact, when observations are very
short in sample size, while SMILE-based model evaluation can avoid
rejecting models for the wrong reason, it may be very difficult to dis-
entangle model biases from noise arising from internal climate varia-
bility in highly uncertain statistical quantities such as the frequency of
compound events.
Models that pass a process-based climate model evaluation may
still have residual biases102. Hence, bias adjustment or downscaling
methods are operationally used for impact assessments102.Forcom-
pound events, multivariate downscaling methods103 or bias
adjustments104–107—which adjust both the distributions of the indivi-
dual variables of interest and their statistical coupling—should be
considered104,108. Correcting SMILEs can be challenging80,104; however,
there are recent developments that provide bias-adjusted large
ensembles80,109 (e.g., the CanLEADv1110), as well as dynamically down-
scaled SMILEs27 (i.e., SMILE from regional climate models that should
better represent local processes). Building on a perfect-model
approach24,75, SMILEs could also be used to assess the skill of multi-
variate bias-adjustment methods104.
In general, running impact models for many SMILEs can be chal-
lenging due to the computational costs of some impact models and
the need of bias adjusting climate models111.Inthiscontext,storylines
derived from SMILEs can be used to explore and communicate a wide
range of plausible future climates over a period of, for instance,
30 years (via climate storylines, Fig. 6), as well as the consequences of
individual worst-case events (via event-based storylines, Fig. 7). The
selection of such storylines should be based on a metric thattakes into
account the combinations of drivers that cause impacts to a given
sector of interest72, hence the choice of the metric will depend on the
sector and should be guided by impact modellers. For example,
exploring impacts associated with a year characterised by widespread
droughts over breadbasket regions (event-based storyline) may allow
testing different adaptation strategies related to, for instance, irriga-
tion, therefore supporting decision-making1,2,66,68, whereas climate
storylines derived based on frequencies of cold spells occurring
together with low winds might be explored for the renewable energy
sector112. Overall, employing a limited set of storylines as input in
impact models can allow for efficient exploration of a substantial range
of possible impacts, therefore supporting climate risk assessments65,72.
Given the cost of dynamically downscaling full SMILEs, downscaling
single events or ensemble members can provide an effective approach
for ultimately driving impact models with high-resolution climate
data72.
In our illustrations, we employed seven CMIP5 generation global
SMILEs24 (each providing 16-100 ensemble members, see Table 1).
However, new generation CMIP6 outputsthat include more SMILEs are
now becoming available (see https://www.cesm.ucar.edu/projects/
community-projects/MMLEA/ for a current overview). The sub-
stantial reduction of uncertainties in compound event frequencies
with the increase in the sample size (Fig. 2) indicates that the many
Fig. 8 | Evaluating statistical modelling of compound events using large
ensemble simulations. Blue bars as in Fig. 5b. Red line,the historicalprobability of
annual soil moisture drought in Nsimultaneous regions based on data of the first
ensemble member. Grey bar, sampling uncertainty (centred 95% confidence
interval) in the historical probability estimated through a statistical model fitted to
data of the first ensemble member (the mean prediction of the statistical model is
shown by the black line).
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CMIP6 models that are now available with several ensemble members
(16 models with at least 10 members), will offer new opportunities for
studying different types of compound events. Considering such a
variety of models will enable a more comprehensive study of uncer-
tainties from model-to-model differences in climate risk assessments.
Furthermore, new SMILEs from regional climate models have also
started to increase in number and can allow for comprehensive
regional analyses by combining regional and global SMILEs27.The
increase in computational resources should eventually enable the
generation of high-resolution SMILEs at the global scale. We would like
to note that alternatives to SMILEs exist. For example, climate
emulators113,114 simulate large sample sizes at a low-computational cost;
however, despite some exceptions88, they are typically not yet tailored
to compound events. Furthermore, combining our understanding of
physical processes with intelligent use of methods can also help
reduce required sample sizes and thus render single-member model
simulations more useful. For instance, using approaches termed
dynamical adjustments allows estimating the forced response of cli-
mate change on various quantities of interest under certain assump-
tions with comparably small sample sizes115,116.
Based on a number of illustrative analyses (Table 1), we conclude
that assessing compound events and their changes based on datasets
of limited sample size may be misleading. Observational data, parti-
cularly when not characterised by a very small sample size, remain
paramount for analysing extreme events. Nevertheless, climate mod-
els are essential tools for studying the dynamics of the most extreme
events and for estimating future changes. Hence, given the increasing
availability of SMILEs, we argue they should be preferred over single-
ensemble model simulations to provide stakeholders with well-
informed information on future climate risks associated with com-
pound events.
Methods
Data
We used seven SMILEs: CESM1-CAM5117 (including 40 ensemble
members), CSIRO-Mk3-6-0118 (30), CanESM290 (50), EC-EARTH119 (16),
GFDL-CM3120 (20), GFDL-ESM2M121 (30), and MPI-GE122 (100), providing
data for the period 1950-2099 (based on the RCP8.5 emission
scenario123 after 2005). We considered the period 1950–1980 as the
historical baseline and periods of the same length in a world 2 °C (or
3°C,in Fig. 6) warmer than pre-industrial conditions in 1870–190026.
Compound events
Given a variable and a model, extremes were defined as values
exceeding a percentile-based threshold of the variable’s distribution
obtained by pooling together data of the historical period from all
available ensemble members of the model. Compound event fre-
quencies were computed empirically via counting26,53,94.
Compound hot-dry events were defined as high mean tem-
perature (above the 90th percentile) and low mean precipitation
(below the 10th percentile) values over the warm season (the three
consecutive months with highest mean temperature during
1950–1980)26. In Fig. 4, to quantify historical and future Spearman
correlation between mean temperature and precipitation, for each
ensemble member and grid point, we first removed linear trends in
the two variables53 (detrending different ensemble members
independently).
To study multi-year droughts, we defined droughts as annual
average soil moisture over the total column below the 20th percentile
(model MPI-GE). To assess concurrent droughts across soybean
growing regions, we used the five soybean regions from Gaupp et al.13;
here, a drought was defined when the regionally averaged annual soil
moisture was below the 25th percentile. For droughts and compound
hot-dry events, data were bilinearly interpolated to a 2.5° spatial grid
before all calculations.
For concurrent wind and precipitation extremes, we considered
historical daily data of accumulated precipitation and mean wind
speed on the native model grid (model CESM1-CAM5). Frequencies of
concurrent daily extremes (f
PW
) are based on precipitation and wind
extremes defined as values occurring twice a year on average (i.e.,
values above their 100 ⋅(1–2/365)th percentiles).
Overall, we note that considering more extreme thresholds to
define compound events (e.g., the 16th percentile to define a soil
moisture drought) would result in higher relative uncertainties in the
frequency of compound events than those shown in Fig. 2.
Ensemble mean and uncertainty quantification
Each of the seven SMILEs consists of several simulations starting from
different initial conditions, resulting in multiple ensemble members
that spana range of plausible climates and associated distributions for
anyquantityofinterestX. Given a SMILE, we obtained robustestimates
of Xas the average of Xfrom individual ensemble members (ensemble
mean). Furthermore, when Xis a projected change, e.g., the change in
the frequency of compound events (Δf),theforcedresponseoffin the
considered SMILE is the ensemble mean of Δf.
For a given SMILE, following Maher et al. (2021)25 and Bevacqua
et al. (2022)26,wequantified the uncertainty in the statistical quantity X
in a single realisation due to internal climate variability as twice the
standard deviation of Xfrom individual ensemble members of a given
SMILE model (2 ⋅U
IV
). In Fig. 6a, b, where we consider multiple SMILE
models to compute such an uncertainty due to internal climate
variability, we obtain an average of U
IV
across models as
UIV =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Nmod P
Nmod
s=1
σ2ðXsÞ
s,whereNmod = 7 is the number of models, and
σ(X
s
) is the standard deviation of Xfrom individual ensemble members
of the model s.InFig.6a, b, we show the uncertainty in the future f
HD
due to model-to-model differences (U
MD
) relative to the sum of U
MD
and U
IV
. To this end, we quantified U
MD
based on the standard devia-
tions of the ensemble-mean of f
HD
in the seven SMILEs26.
Dependence of uncertainty on sample size
In Fig. 2, following the procedure of Bevacqua et al.26,wecomputedthe
ratio of twice the standard deviation to the mean of f(i.e., 2 UIV=f)
across N
ens
= 12 ensemble members (for compound precipitation-wind
extremes we use N
ens
=30)ofvariousfixed sample size Nyears.TheN
ens
members were obtained via sampling data from pooled concatenated
historical ensemble members of the SMILE considered to analyse the
compound event of interest. When assessing (1) compound hot-dry
events and (2) precipitation-wind extremes, we randomly sampled
calendar years54 from the concatenated data. For assessing (3)
droughts over, e.g., two consecutive years, we first modified each
ensemble member simulation by creatinga dataset with pairs of all two
consecutive soil moisture annual values (we disregarded the first year
of each ensemble member for which no previous soil moisture value
exists). Then, we concatenated these modified ensemble members.
Finally, we sampled the N
ens
ensemble members taking pairs from the
pooled concatenated data. To obtain dashed and dotted lines in
Fig. 2a, we first computed the Spearman correlation between tem-
perature and precipitation (based on all pooled ensemble members)
and then selected the grid points with the 35% strongest and weakest
coupling (i.e., strongest and weakest negative correlation coefficient),
respectively.
Required sample size for attribution
Extending the approach of Zscheischler and Lehner (2022)43,basedon
synthetic data of two variables Xand Ywith a given Pearson correlation
cor
XY
, we quantified—in the presence of hypothetical combinations of
anthropogenically forced positive trends Trend
X
and Trend
Y
in the two
variables—the minimum sample size required for attribution of
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univariate high extremes and concurrent high extremes. We repeated
the experiment three times, for different values of the correlations
cor
XY
(−0.5, 0, and +0.5). Note that we assume no trends in the com-
pound drivers’dependency43. Each experiment was carried out in
three steps:
(1) Based on a bivariate standard Gaussian distribution with a given
correlation cor
XY
,wesimulatedNpairs (X,Y) both for a reference
period representing hypothetical preindustrial conditions, and
for a hypothetical present-day period under forced trends in the
variables (via modifying the means of the distribution of the two
variables by Trend
X
and Trend
Y
). To identify the sample size N
required for attribution, such simulations were conducted for a
set of Nranging from 5 to 3000 and repeated N
bootstrap
= 5000
times each.
(2) For each of the N
bootstrap
simulations associated with each sample
size N, extreme events of individual variables were defined as
values above the 90th percentile of the reference period. The
probability ratio PR = p
1
/p
0
was computed, where p
0
is the
empirically estimated probability of an extreme event occurring
in the reference period simulations and p
1
is that probability for
the present-dayperiod. PRs werecomputed for thetwo univariate
extreme events and concurrent extremes.
(3) We identified the minimum sample size Nfor which the attribu-
tion is robust,defined as a sample size for which the 95% con-
fidence interval of PR is greater than one in 90% of the N
bootstrap
simulations, where the confidence interval was estimated follow-
ing Zscheischler and Lehner (2022)42,43.
Event-based storylines
Compound events are found in a multidimensional space, for which, in
contrast to a univariate space, there is no unique natural ordering,
therefore identifying the most extreme compound events requires
employing a metric whose extreme values are associated with mean-
ingful events from an impact perspective. Here, we use a copula-based
approach to select the most extreme events77,78. To identify extreme
compound wintertime (December-February) precipitation and wind
events, we first defined, for all pooled members of CESM1-CAM5, the
variables X
1
and X
2
as the spatially-weighted average of daily pre-
cipitation and wind, respectively, over Portugal (Fig. 7). Secondly,
throughtheR-packagecopula
124, we computed the bivariate empirical
copula C of these (X
1
,X
2
), that is the empirical joint cumulative dis-
tribution function125 of (U
1
=F
1
(X
1
), U
2
=F
2
(X
2
)), where F
i
is the empiri-
cal cumulative distribution function of X
i
.Thepairs(X
1
,X
2
) associated
with the highest copula values (and therefore the highest probability
Pr(X
1
≤x
1
,X
2
≤x
2
)) are then selected as the most extreme events. To
identify extreme compound droughts across soybean breadbasket
regions, we employed the same conceptual approach, but extended it
to five dimensions and used data from the model MPI-GE. That is, we
considered five variables (X
1
,X
2
,...,X
5
)thatweredefined as the
negative values (given the interest in their extremely low values) of the
breadbasket-regionally averaged annual soil moisture and computed
the associated five-dimensional empirical copula C. Finally, the com-
binations (X
1
,…,X
5
) associated with the highest copula values were
selected as the most extreme events.
Perfect-model approach for evaluating a statistical model of
spatially compounding droughts
For Fig. 8, we used the breadbasket regionally averaged annual soil
moisture data for the historical period. We tested how a multivariate
statistical model fitted to pseudo-observations with a limited sample
size (here, 31 years from the first member) represents the compound
event statistics of the underlying climate defined by pooling all avail-
able ensemble members (31 ⋅100 years).
Following the modelling of Bevacqua et al.20, the multivariate
statistical model consists of a five-dimensional probability density
function (pdf) of the variables (X
1
,X
2
,...,X
5
) (the soil moisture in the
five breadbasket regions). The pdf is decomposed into the five
marginal distributions of X
i
and the copula density c, where c
accounts for the dependence amongst the variables X
i
regardless of
their marginal distributions and was modelled via pair copula con-
structions (PCCs)20. Firstly, marginal distributions were fitted
through a kernel density estimate89 (via the stats R-package126, func-
tion density). Then, we fitted pair-copula families and selected the
best PCC structure via the function RVineStructureSelect (based on
AIC) of the VineCopula R-package127. Copulas were fitted to the vari-
ables U
i
=F
i
(X
i
), where F
i
is the empirical cumulative distribution
function of X
i
.
The resulting statistical model, i.e. the fitted 5-dimensional pdf, is
associated with a unique probability of simultaneous droughts across
N-regions, which is, however, affected by sampling uncertainty
because the pdf is fitted to a sample of finite size. We estimated the
sampling uncertainty via standard parametric bootstrap similar to
previous studies20,21,39, in four steps. (1) We simulated
N
bootstrap
=900 samples of (X
1
,X
2
,...,X
5
) with the same length as the
original data (i.e., 31 years). This was achieved by simulating the vari-
ables (u
1
,u
2
,...,u
5
) and transforming them to (x
1
,x
2
,...,x
5
) via the
inverse of the marginal kernel densities. (2) We fitted a pdf model to
each of the N
bootstrap
samples via the same procedure outlined above.
(3) From each of these N
bootstrap
models, we simulated a sample of
(X
1
,X
2
,...,X
5
) with a length of 1000 times the original, such as to
robustly obtain N
bootstrap
probabilities of simultaneous droughts across
N-regions associated with each pdf. (4) Finally, we computed the
centred 95% confidence interval of such N
bootstrap
probabilities, i.e. the
final sampling uncertainty in the probability of simultaneous droughts
across N-regions (shown via grey boxplots in Fig. 8along with an
overlaid black line indicating the mean of the N
bootstrap
probabilities).
Finally, we note that a user may be interested in carrying out the
analyses considering different ensemble members of a given SMILE as
pseudo-observations (or different SMILEs to carry the overall analysis).
Data availability
The model data used in the study are available online at https://www.
earthsystemgrid.org/dataset/ucar.cgd.ccsm4.CLIVAR_LE.html (for
CanESM2, CESM-LE, CSIRO-Mk3-6-0, GFDL-ESM2M, and GFDL-CM3)
and at https://esgf-data.dkrz.de/projects/mpi-ge/ (for MPI-GE). The
HadCRUT5 data set can be found at https://www.metoffice.gov.uk/
hadobs/hadcrut5/. All maps were obtained by using the oce
R-package128.
Code availability
All custom codes are direct implementations of standard methods and
techniques, described in detail in Methods. The code used to prepare
the figures in this paper is available from the corresponding author
upon reasonable request.
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Acknowledgements
We acknowledge the participants of the workshop “Large ensemble
simulations for compound event research" held in October 2021 at École
normale supérieure Paris. The workshop was funded by the European
COST Action DAMOCLES (CA17109). This project has received funding
from the European Union’s Horizon 2020 research and innovation pro-
gramme under grant agreement No 101003469. L.S.G. received funding
from the German Ministry of Education and Research (BMBF) under the
ClimXtreme project DecHeat (Grant number 01LP1901F) and from the
European Union’s Horizon Europe Framework Programme under the
Article https://doi.org/10.1038/s41467-023-37847-5
Nature Communications | (2023) 14:2145 15
Content courtesy of Springer Nature, terms of use apply. Rights reserved

Marie Skłodowska-Curie grant agreement No 101064940. F.L. was
supported by the US DOE Office of Science, Office of Biological &
Environmental Research (BER), Regional and Global Model Analysis
(RGMA) component of the Earth and Environmental System Modeling
Program under Award Number DE-SC0022070 and NSF IA 1947282.
M.V. acknowledges support from the COESION project funded by the
French National programme LEFE (Les Enveloppes Fluides et l’Envir-
onnement). P.Y. was supported by the French ANR (grant agreement
ANR-20-CE01-0008-01, SAMPRACE). J.Z. acknowledges the Helmholtz
Initiative and Networking Fund (Young Investigator Group COM-
POUNDX, Grant Agreement VH-NG-1537). We thank the US CLIVAR
Working Group on Large Ensembles and NSF AGS-0856145 Amend-
ment 87 for supporting the Multi-Model Large Ensemble Archive. We
also thank the modelling centers for providing the simulations.
Author contributions
E.B. initiated the study and carried out the analyses. E.B. wrote the
manuscript with contributions from J.Z., L.S.-G., and A.J. E.B. designed
the study with contributions from J.Z. All authors (E.B, A.J., L.S., F.L.,
M.V., P.Y., and J.Z.) discussed the results and reviewed the manuscript.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Competing interests
The authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to
Emanuele Bevacqua.
Peer review information Nature Communications thanks Jennifer Hel-
geson, Mohammad Reza Najafiand the other, anonymous, reviewer(s)
for their contribution to the peer review of this work.
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