Abstract
Worsening temperature extremes are among the most severe impacts of human-induced climate change. These extremes are often defined as rare events that exceed a specific percentile threshold within the distribution of daily maximum temperature. The percentile-based approach is chosen to follow regional and seasonal temperature variations so that extremes can occur globally and in all seasons, and frequently uses a running seasonal window to increase the sample size for the threshold calculation. Here, we show that running seasonal windows as used in many studies in recent years introduce a time-, region-, and dataset-depended bias that can lead to a striking underestimation of the expected extreme frequency. We reveal that this bias arises from artificially mixing the mean seasonal cycle into the extreme threshold and propose a simple solution that essentially eliminates it. We then use the corrected extreme frequency as reference to show that the bias also leads to an overestimation of future heatwave changes by as much as 30% in some regions. Based on these results we stress that running seasonal windows should not be used without correction for estimating extremes and their impacts.
Similar content being viewed by others
Introduction
Percentile-based temperature extremes are defined as events of a certain rarity and are frequently used to derive downstream impact metrics such as heatwave properties, particularly in a warming climate. The aim of percentile-based extreme definitions is to account for spatial [e.g.,1,2] and seasonal [e.g.,3,4,5] variations in temperature distributions, so that extremes can occur across the globe and throughout the year. In addition, percentile-based extremes are intended to offset differences between datasets such as models, reanalyses, and observations [e.g., 2,6].
The rarity of the extreme is set by calculating exceedances of an appropriate percentile threshold, typically on a daily basis. For example, on average 10% of days are expected to have a maximum temperature exceeding the 90th percentile when evaluated in the same period as the threshold is calculated. This base period is frequently chosen as 1961–1990 as recommended by the Expert Team on Climate Change Detection and Indices (ETCCDI; http://etccdi.pacificclimate.org/list_27_indices.shtml;7).
The period length of 30 years is a compromise between the length of the time series on the one hand and the number of samples available for the percentile calculation on the other hand. In particular observational datasets often do not provide high-quality data for long time periods, and longer periods may contain strong forced warming, which is often undesirable. Short time periods, in turn, limit the number of samples on which the percentile threshold is based, which can lead to considerable biases8. Therefore, individual studies have also used longer periods5,9,10,11,12.
The ETCCDI recommends a 5-day running window across the seasonal cycle to counter the limited number of samples. Combined with the recommended 30-year period, this results in a sample size of 150 values per calendar day for the percentile calculation. However, many recent heatwave studies do not follow this recommendation and use longer windows of 15 days4,5,9,10,12,13,14,15,16,17,18,19 or even 31 days6,20,21,22,23,24,25,26,27,28. Notably, the widely used Heat Wave Magnitude Index daily (HWMId) is defined based on a 31-day running window29. The decision to deviate from the ETCCDI recommendation and use a longer window is often not explicitly motivated in the literature. However, it can be assumed that the intention is to increase the sample size beyond 150 values in order to avoid biases in the frequency of exceedances8 as well as large variations in the extreme threshold between adjacent calendar days16.
In the following, we reveal serious pitfalls when using such long-running windows. We show that they lead to large and systematic biases in the frequency of temperature extremes and discuss how these biases undermine commonly accepted assumptions about the properties of percentile-based extreme definitions. We demonstrate a simple solution that essentially eliminates the bias and uses the corrected frequencies as a reference to investigate the effect of the bias on estimates of extreme changes under warming.
Results
Systematic biases in percentile-based temperature extremes
We define a bias in the frequency of daily maximum temperature extremes as a deviation from the theoretically expected extreme frequency. As an example: for the 90th percentile the expected frequency is 10% extreme days on average when calculating threshold and extremes in the same period (see methods for details). However, we find that in the ERA5 dataset, the 1961–1990 average, global average, daily maximum temperature extreme frequency based on the 90th percentile using a 31-day running window (TX90p31w) is only 9%, a relative bias of −10%. Regionally the bias can be considerably larger and exceed −30% as shown in Fig. 1a for ERA5 and in Fig. S1 in the supplementary information for the CMIP6 multi-model mean.
In individual months, the bias can be even larger, exceeding −75%. This almost excludes the occurrence of extremes altogether as shown in Fig. 1c for the example of a grid cell with a strong bias in the North Atlantic. This striking underestimation of threshold exceedances, particularly in the transition seasons, may seem counter-intuitive: The moving window used in the percentile calculation is symmetric around each calendar day—it encompasses both seasonally colder as well as seasonally warmer values. One might expect these to average out and lead to the desired smooth and well-defined extreme threshold, which is exceeded for ~10% of days throughout the seasonal cycle and across the globe. But this expectation does not hold as the threshold is, by design, an extreme percentile and not an average. It is, therefore, dominated by the seasonally warmer days in the window, making it exceedingly unlikely for the considered central day to exceed the threshold if (1) the seasonal gradient is strong and (2) the day-to-day variability is low.
The strongest bias is, hence, found in regions and seasons with a strong seasonal gradient but weak day-to-day variability, as showcased in Fig. 1c and Fig. S2. While many of the strongest biases, therefore, occur over oceans there are also several land regions with considerable bias such as India and the western US (Fig. S2a, c). Across most of Europe the strong seasonal cycle is offset by strong day-to-day variability, and only a weak bias remains (Fig. S2e). In regions with weak seasonal variations, such as most of the tropics and the Southern Ocean (Fig. S2i), the bias is consequently negligible. In the following, we will refer to this bias as running window bias.
To further investigate and contextualise this running window bias, we build on a study by Zhang et al.8 based on synthetic data. As detailed in the methods, we include an additional seasonal cycle of varying amplitude into their auto-correlated white noise data. Figure 2a shows the frequency bias in absence of a seasonal cycle reproducing their results. For the in-base case, the same random sample is used to calculate the threshold and the exceedances; for the out-of-base case, exceedances are based on but the same threshold but a new random sample. The bias for the 5-day window is equivalent to Zhang et al.8, while we use a slightly different second window size of 31 days (as opposed to their 25 days) to be consistent with the rest of our work.
The difference between the out-of-base and in-base and bias is shown in Fig. 2b. The temporal inhomogeneity introduced by this difference is widely recognised in the community and can be corrected following the recommendation by the ETCCDI and Zhang et al.8. For the 5 day window, the bias (Fig. 2a) and the difference (Fig. 2b) grow strongly with increasing percentile and vary periodically due to the the limited number of samples, as discussed in detail by Zhang et al.8. Based on this result alone, it is understandable why a longer window seems preferable, with the bias in Fig. 2a,b remaining much lower for the 31-day running window, especially for high percentiles.
Crucially, however, this advantage of the longer window does not hold in the presence of even a moderately strong seasonal cycle (equivalent to the median in ERA5) as shown in Fig. 2c. While the bias for the 5-day window hardly changes compared to the case without a seasonal cycle, the 31-day running window leads to heavily increased bias. In fact, the running window bias introduced by the seasonally running window in combination with a seasonal cycle is larger than the bias due to the limited number of samples in Fig. 2a and larger than the difference in Fig. 2b for all but the highest percentiles. The evolution of this running window bias across the seasonal cycle is shown in Fig. S3 for a range of different window sizes.
Pitfalls in the interpretation of temperature extremes
The presence of the running window bias undermines the main assumption about percentile-based extreme definitions made (more or less explicitly) by many studies, namely that they allow extremes equally across the seasonal cycle [e.g.,4,9,14]. In the following, we discuss three additional pitfalls in the interpretation of extremes and connect them to further assumptions about percentile-based extremes made in the literature: (1) spatial inhomogeneities, (2) artificial dataset differences, and (3) spurious change signals.
(1) An important argument for the use of percentile-based indices is that they are based on thresholds following the local temperature distribution, thus making extremes and their changes comparable between regions [e.g.,15,30]. However, for a fair comparison, the probability for extremes should be the same across regions, which is undermined by the spatial variations of the bias (see Fig. 1a). As a measure for the unequal treatment of regions with low and high bias in a comparison we define spatial inhomogeneity as the difference between the spatial 5th and 95th percentiles of grid cell biases. Figure 3 shows the spatial inhomogeneities for different combinations of percentiles and window sizes. They generally worsen with increasing percentile and window size, for TX90p31w, the inhomogeneity reaches ~25%. Note that for high percentiles and small window sizes, the mean bias greatly increases due to a lack of samples, so these cases are also problematic even though they have little or no spatial inhomogeneities (see ref. 8 for a detailed discussion). For TX90p5w (recommended by the ETCCDI) the global mean bias is almost zero, which is also reflected in a small inhomogeneity of ~3%, showing that this setup allows a fair comparison between regions. We note, however, that for such short window sizes the limited number of samples also leads to a bias (Fig. 2a) and to strong day-to-day variations in the extreme threshold, which might also pose a problem. Finally, we also show the 50th percentile in Fig. 3 to demonstrate that central estimates are not affected by the running window bias.
(2) Another important argument for percentile-based indices is that they provide an implicit bias correction of models compared to observations [e.g., refs. 2, 6]. The underlying extreme frequencies are, hence, assumed to be the same when analysing derived heatwave properties, such as duration, area, or cumulative heat. This means that any model-observation differences in these derived metrics are interpreted as higher-order differences. However, differences in these metrics can also result from a diverging bias structure leading to artificial dataset differences. Figure 4a shows the CMIP6 multi-model mean artificial difference to ERA5 and reveals several regions where model-observation differences exceed ± 10% robustly across most of the 26 models used. To illustrate the extent of possible differences, we compare a grid cell in the Amazon region between ERA5 and the CanESM5 model in Fig. 4c. The differences in the extreme frequency exceed 70% in April, due to a wrong representation of the mean seasonal cycle in CanESM5. Such a difference could easily be misinterpreted as a wrong representation of extremes in CanESM5 compared to ERA5 but is, in fact, founded in a wrong mean seasonal cycle interacting with the running window.
(3) A final argument for percentile-based temperature extreme definitions is that they account for the warming trend when using shifting baselines [e.g., refs. 20, 21]. This is intended to allow, for example, the investigation of non-linear changes in heatwave properties under climate change and, generally, follows an interpretation of extremes as events, which are rare by definition, even in a warming world [e.g., refs. 31, 32]. However, the running window bias can shift between time periods and translate into spurious change signals. Figure 5a shows such spurious change signals as the difference between two periods, for which separate thresholds are calculated: historical (1961–1990) and future (2071–2100) using the high emission scenario SSP3-7.0. Figure 5d shows a grid cell in the Arabian Sea for the CanESM5 model as an example. The decrease in the amplitude of the mean seasonal cycle in the future leads to a greatly reduced running window bias and, therefore, a striking increase in extreme day frequency by >30% in the annual mean. For the month of September, where extremes were basically absent in the historical period, the frequency even increases by a factor of 15.
A simple solution to eliminate the running window bias
To essentially eliminate the running bias even when using a 31-day window, we suggest to account for the seasonal variations first, to avoid mixing them into the extreme threshold. A simple way of doing so is to remove the mean seasonal cycle before calculating the threshold and subsequent extremes (see methods for details). In fact, this has been proposed already as early as 1999 by Folland et al.33 and Jones et al.34 and later been reiterated by Zhang et al.8, but it has not been applied in the recent literature to the best of our knowledge. One reason for this may be that these earlier studies did not discuss the potential for biased results due to the seasonal cycle we show here.
Figure 1b, d clearly reveals the benefit of removing the mean seasonal cycle before calculating extreme frequencies. The global mean frequency bias is reduced from −10% to −0.5% and from −33% to 0% for the example grid cell in the North Atlantic (see Fig. S2 for the other example grid cells). As a result, the spatial inhomogeneity is also vastly reduced from 24% to 3% (Fig. 3), and hardly any systematic bias pattern remains.
Exceptions are regions in the Southern Ocean and Arctic Ocean where the correction leads to a slight increase in bias. In the Southern Ocean the upper end of the temperature distribution is mostly constant throughout the year, basically following the 0 ∘C isotherm (Fig. S2i) when the seasonal cycle is still included. In the Arctic Ocean the same is true during northern hemispheric summer (Fig. S2g). At the same time, these regions also exhibit a strong seasonality in the day-to-day variability, with the amplitude being considerably lower during summer. This leads to an increase in the seasonality of the 90th percentile threshold when the mean seasonal cycle is removed and, hence, to a slight increase in the running window bias. However, the bias in these regions is small overall, with the highest amplitudes staying below 10% in both the uncorrected and corrected cases.
Artificial dataset differences are also greatly reduced when the mean seasonal cycle is first removed (Fig. 4b). The fraction of grid cells with robust differences (stippling in Fig. 4a,c) decreases from 37% to 2% when the correction is applied. Figure 4d shows the example of the grid cell in the Amazon that removes the mean seasonal cycle before calculating the threshold accounts for offsets between the datasets. The same holds true for differences between time periods, when using multiple base periods as shown in Fig. 5. The spurious change signal showcased for the grid cell in the Arabian Sea is essentially eliminated, and the only robust differences remaining are at high latitudes, connected to the changes in day-to-day variability discussed above.
Impact of the bias on heatwave changes
Now we use the corrected extreme frequencies as a reference to show the impact of the running window bias on climate change signals. We focus on two metrics: (1) annual mean extreme frequency, which corresponds to the rest of our study and (2) summer heatwaves. The second metric is calculated from the extreme frequencies by restricting them to the respective extended summer season in both hemispheres and then applying a 3-day persistence criterion following, for example, Lyon et al.16 and Perkins–Kirkpatrick and Lewis17. As change metric we show the CMIP6 multi-model mean ratio between 2071–2100 and 1961–1990 frequencies following Fischer and Schär4 (see methods for details). Extremes in the future are calculated using a fixed 1961–1990 threshold in this section.
For the change in the extreme frequency, we can set an upper limit to the ratio based on theoretical considerations: if we expect a historical extreme frequency of about 10%, then a tenfold increase should be the maximum, corresponding to every day being extreme in the future. However, the historical extreme frequency can be considerably lower due to the running window bias, as we have shown. At the same time the extreme frequency greatly increases in the future when using a fixed baseline, leading to a decrease in the bias everywhere on the globe (Fig. S6). Figure 6 (top row) shows changes exceeding a factor of 13 for the biased case, while our correction leads to increases being mostly limited to the theoretical maximum. A direct comparison of extreme changes between biased and corrected cases is shown in Fig. 6c and reveals that the bias leads to an overestimation of extreme changes in most regions. The overall pattern is similar to the historical bias shown in Fig. 1a, which consists of our expectations based on the general decrease of bias in the future.
Finally, we repeat the same steps as above for the case of extended summer heatwaves, which is a frequently used metric in more impact-focused studies. For this case, no a priori theoretical maximum for the increase exists since the 3-day persistence criterion leads to spatially varying heatwave frequencies in the historical period (ranging between 3% to 7%; see Fig. S7a, d). Figure 6d shows the resulting bias in the heatwave ratio with ocean grid cells masked to highlight changes over land. The Arabian Peninsula is a particular hot spot of overestimation which can exceed 30% here. This is founded in a strong historical bias combined with a strong reduction in bias in the future as heatwave frequency approaches 100% in this region (Fig. S6).
Discussion
We have demonstrated that the percentile-based temperature extreme frequency can exhibit considerable deviations from the expected frequency due to a bias introduced by the use of running seasonal windows. This running window bias can vary with season, region, time period, and dataset undermining generally accepted properties of percentile-based extreme definitions. The bias changes in the future and, therefore, has implications also for climate changes studies. We also note that, while we have demonstrated the effect on hot extremes here, the same considerations apply equally to cold extremes.
One way to avoid the running bias is to limit the size of the running window to 5 days as recommended by the Expert Team on Climate Change Detection and Indices (ETCCDI), since such a window length causes only a negligible running window bias. However, the limited sample size in this case can also lead to a bias and to the sampling of day-to-day variability into the threshold, which may also pose a problem.
To allow the use of longer-running windows without bias, we recommend removing the mean seasonal cycle before calculating the extreme threshold. This essentially eliminates the bias as shown for the annual extreme frequency. Drawing on the corrected frequencies as reference, we show that the running window bias also affects summer heatwaves based on biased extreme frequencies. The effect on other derived extreme metrics and heatwave properties remains to be investigated.
To conclude, we strongly warn against the use of long-running windows without correction when calculating extreme thresholds. The use of such a biased method is never advisable, even though the impacts on derived metrics might not always be strong or immediately apparent. Many studies, naturally, adopt methods from previous work, so avoiding the use of biased methods also helps to prevent their spread. Crucially, the same biased method may have a negligible effect in one setting and a large impact in another.
Methods
ERA5 data
We draw on data from the fifth generation of the European Centre for Medium-Range Weather Forecasts (ECMWF) Retrospective Analysis (ERA535;). We use hourly instantaneous 2 m surface air temperature with a native resolution of 0.25° × 0.25°, re-sampled to the daily maximum value and a spatial resolution of 2.5° × 2.5° using conservative remapping and restricted to the 30-year period 1961–1990. To simplify seasonal cycle-related calculations, we exclude February 29th from leap years so that each of the 30 years has 365 days.
CMIP6 data
To complement ERA5, we also use daily maximum 2 m surface air temperature from 26 models from the Coupled Model Intercomparison Project Phase 6 (CMIP636;) as listed in table S1 in the supplementary information. These models represent an “ensemble of opportunity” of all models that are available from the ETH Zurich CMIP6 next-generation archive37 and provide daily maximum temperature for the historical period and the Shared Socio-economic Pathway 3-7.0 (SSP3-7.038;) in the future. While we acknowledge that this pool of models does not constitute a representative or exhausting sample of the full diversity of models in the CMIP6 archive [e.g., refs. 39, 40] our aim here is mainly to showcase the potential for biases in the calculation of extremes rather than an in-depth model evaluation and we, therefore, chose this pragmatic approach. The pre-processing applied to the CMIP6 models is identical to ERA5, except that we use the period 2071–2100 in addition.
Calculation of percentile-based extreme thresholds
Extreme thresholds are defined for each dataset and grid cell separately by calculating the p-th percentile over 30 years and the ±d days surrounding each day of the year. Temperature extremes for a given day are then defined as exceedances of the respective percentile value centred on the same day of the year. In our study, we focus on the 90th percentile calculated using a 31-day window (termed TX90p31w) but also investigate a range of other combinations of percentiles and window sizes to identify their interaction with the seasonal cycle.
The calculation of percentiles is done empirically on the pooled data from the 30 years and ±d days, so on 30 × 31 = 930 values for TX90p31w. Modern programming languages offer a range of choices to calculate the percentile and the choice of method can influence the results, in particular for high percentiles, which are exceeded by only a few values. For the main part of the manuscript we use the method “linear”, which is the default setting in Python’s NumPy package (https://numpy.org/doc/stable/reference/generated/numpy.percentile.html), in R (https://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html), and in Julia (https://docs.julialang.org/en/v1/stdlib/Statistics/). It can, therefore, be assumed to be a frequently employed setting for the calculation of percentile-based thresholds. For our comparison to the results from Zhang et al.8 in Fig. 2 we, instead, use NumPy’s “weibull” method to be consistent with their approach. Biases similar to Fig. 2 but using “linear” are shown in Fig. S8 in the supplementary information.
Calculation of extreme frequency, bias, and difference
The extreme frequency f (p, w) is the fraction of days exceeding the extreme threshold in a given month or the entire year in percent:
with p indicating the percentile and w the window size.
The running window bias \({f}^{{\prime} }(p,w)\) is defined as the extreme frequency minus the expected extreme frequency \({f}_{\exp }(p)=100-p\) (so 10% for the 90th percentile, as an example) divided by the expected frequency in percent:
In Fig. 6 we use the corrected frequency as \({f}_{\exp }\).
Equivalently, the difference Δf between two datasets or periods is calculated relative to the reference extreme frequency fref (p, w) following:
With the reference being ERA5 in Fig. 4 and the historical period in Fig. 5.
Removing the mean seasonal cycle
The mean seasonal cycle is calculated over 30 years on a day-of-the-year basis (and, hence, without any running window) for each grid cell separately. We choose this approach here for its simplicity, an alternative option would be to use a running window in the calculation for the mean seasonal cycle to increase the sample size. This is not problematic as central estimates are not affected by the running window bias (see Fig. 3).
For the historical 1961–1990 period the mean seasonal cycle is always calculated in the same period. For the future two options exist: (1) for Fig. 5 the mean seasonal cycle for the 2071–2100 period is used, mirroring the percentile calculation with multiple baselines; (2) for Fig. 6 the historical 1961–1990 seasonal cycle is used.
Calculation of summer heatwaves and climate change signals
In a first step extremes are constrained to the extended summer season: May-September in the northern and November-March in the southern hemisphere. Then heatwaves are defined at a grid cell basis as at least three consecutive extreme days. Finally, to select land grid cells we apply a common land-sea mask to the regridded data, based on Natural Earth (https://www.naturalearthdata.com/) and implemented in Python’s regionmask package (https://regionmask.readthedocs.io).
Climate change signals between the periods 1961–1990 and 2071–2100 (using the first period as fixed baseline for both cases) in the last section are calculated as extreme frequency ratio r following Fischer and Schär4:
For such a comparison of frequencies between in-base and out-of-base periods (Fig. 6) Zhang et al.8 recommend the use of a cross-validation approach to avoid an artificial frequency jump between the periods. Their cross-validated extreme frequency is calculated as mean over 29 folds for each of the in-base years (see ref. 8 for details). This means the output is no longer a time series of individual days exceeding the threshold or not, but an average extreme frequency. This prohibits the calculation of heatwaves from the extreme frequency by applying the 3-day persistence criterion. Therefore, we, here, directly use the extreme frequency to calculate heatwaves in the base-period as well as outside, again, following Fischer and Schär4. In this context, we note that the jump for the 90th percentile using a 31-day window is very small at only ~3% as can be seen from Fig. S8. In addition, we focus on a comparison between the uncorrected and corrected frequencies in Fig. 6, which are both affected equally by the jump.
Creation of synthetic time series
For the creation of the synthetic time series, we combine auto-correlated white noise emulating day-to-day variability and a sine-function emulating the seasonal cycle. The white noise is defined to have a mean of zero, a standard deviation of one, and a lag 1-day auto-correlation of 0.8. The lag 1-day auto-correlation of 0.8 is used to be consistent with the work by Zhang et al.8 and because it is reasonably close to the median lag 1-day auto-correlation of the de-seasonalized day-to-day variability in ERA5, which is 0.76 in the period 1961–1990.
The amplitude of the sine function is set to three distinct values so that the ratio of the standard deviation over the seasonal cycle and the standard deviation over the white noise (which is one by definition) represents three cases: 0 (no seasonal cycle), 1.8 (same as the spatial median in ERA5), and 3 (corresponding to the 90th percentile in ERA5).
The ratio of standard deviations Δs in ERA5 is calculated for each grid cell separately following:
where the enumerator is the standard deviation of the mean seasonal cycle (calculated on a day-of-the-year basis with a window size of 1 and the denominator is the mean over standard deviations calculated from the de-seasonalized day-to-day variability for each year separately.
Following Zhang et al.8 we produce a synthetic 30-year interval and calculate the percentile threshold, exceedance, and bias from it (in-base case). For the out-of-base case we produce a separate 30-year interval which is evaluated against the threshold calculated from the first 30 years. This process is repeated 5000 times, and the presented biases are the average over all iterations.
Data availability
All raw data used in this study are freely available for research applications. ERA5 hourly 2 m temperature: https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysis-era5-single-levels?tab=overview; CMIP6 daily maximum 2 m surface air temperature: https://esgf-node.llnl.gov/; An example dataset providing extreme frequencies for ERA5 using a 31-day window for the 90th percentile is available here: https://doi.org/10.5281/zenodo.10639317.
Code availability
The code to re-create the figures in the main manuscript is available at: https://github.com/lukasbrunner/running_window_bias.
References
Zhang, X. et al. Indices for monitoring changes in extremes based on daily temperature and precipitation data. WIREs Clim. Change 2, 851–870 (2011).
Schoetter, R., Cattiaux, J. & Douville, H. Changes of western European heat wave characteristics projected by the CMIP5 ensemble. Clim. Dyn. 45, 1601–1616 (2015).
Tank, A. M. G. K. & Können, G. P. Trends in indices of daily temperature and precipitation extremes in Europe, 1946–99. J. Clim. 16, 3665–3680 (2003).
Fischer, E. M. & Schär, C. Consistent geographical patterns of changes in high-impact European heatwaves. Nat. Geosci. 3, 398–403 (2010).
Hirsch, A. L., Ridder, N. N., Perkins-Kirkpatrick, S. E. & Ukkola, A. CMIP6 multimodel evaluation of present-day heatwave attributes. Geophys. Res. Lett. 48, e2021GL095161 (2021).
Freychet, N., Hegerl, G., Mitchell, D. & Collins, M. Future changes in the frequency of temperature extremes may be underestimated in tropical and subtropical regions. Commun. Earth Environ. 2, 1–8 (2021).
Klein Tank, A. M., Zwiers, F. W. & Zhang, X. Guidelines on analysis of extremes in a changing climate in support of informed decisions for adaptation https://www.ecad.eu/documents/WCDMP_72_TD_1500_en_1.pdf (2009).
Zhang, X., Hegerl, G., Zwiers, F. W. & Kenyon, J. Avoiding inhomogeneity in percentile-based indices of temperature extremes. J. Clim. 18, 1641–1651 (2005).
Della-Marta, P. M., Haylock, M. R., Luterbacher, J. & Wanner, H. Doubled length of western European summer heat waves since 1880. J. Geophys. Res. Atmos. 112, 1–11 (2007).
Perkins, S. E., Alexander, L. V. & Nairn, J. R. Increasing frequency, intensity and duration of observed global heatwaves and warm spells. Geophys. Res. Lett. 39, 20714 (2012).
Stefanon, M., Dandrea, F. & Drobinski, P. Heatwave classification over Europe and the Mediterranean region. Environ. Res. Lett. 7, 014023 (2012).
Engdaw, M. M., Ballinger, A. P., Hegerl, G. C. & Steiner, A. K. Changes in temperature and heat waves over Africa using observational and reanalysis data sets. Int. J. Climatol. 42, 1165–1180 (2022).
Della-Marta, P. M. et al. Summer heat waves over western Europe 1880-2003, their relationship to large-scale forcings and predictability. Clim. Dyn. 29, 251–275 (2007).
Perkins, S. E. & Alexander, L. V. On the Measurement of Heat Waves. J. Clim. 26, 4500–4517 (2013).
Spinoni, J. et al. Heat and cold waves trends in the Carpathian Region from 1961 to 2010. Int. J. Climatol. 35, 4197–4209 (2015).
Lyon, B., Barnston, A. G., Coffel, E. & Horton, R. M. Projected increase in the spatial extent of contiguous US summer heat waves and associated attributes. Environ. Res. Lett. 14, 114029 (2019).
Perkins-Kirkpatrick, S. E. & Lewis, S. C. Increasing trends in regional heatwaves. Nat. Commun. 11, 3357 (2020).
Reddy, P. J., Perkins-Kirkpatrick, S. E. & Sharples, J. J. Interactive influence of ENSO and IOD on contiguous heatwaves in Australia. Environ. Res. Lett. 17, 014004 (2021).
Wu, S. et al. Season-dependent heatwave mechanisms: a study of southern China. Weather Clim. Extrem. 42, 100603 (2023).
Vogel, M. M., Zscheischler, J., Fischer, E. M. & Seneviratne, S. I. Development of future heatwaves for different hazard thresholds. J. Geophys. Res.: Atmos. 125, e2019JD032070 (2020).
Schielicke, L. & Pfahl, S. European heatwaves in present and future climate simulations: a Lagrangian analysis. Weather Clim. Dyn. 3, 1439–1459 (2022).
Russo, E. & Domeisen, D. I. V. Increasing intensity of extreme heatwaves: the crucial role of metrics. Geophys. Res. Lett. 50, e2023GL103540 (2023).
Dosio, A., Mentaschi, L., Fischer, E. M. & Wyser, K. Extreme heat waves under 1.5 °C and 2 °C global warming. Environ. Res. Lett. 13, 054006 (2018).
Russo, S., Marchese, A. F., Sillmann, J. & Immé, G. When will unusual heat waves become normal in a warming Africa? Environ. Res. Lett. 11, 054016 (2016).
Ceccherini, G., Russo, S., Ameztoy, I., Patricia Romero, C. & Carmona-Moreno, C. Magnitude and frequency of heat and cold waves in recent decades: the case of South America. Nat. Hazards Earth Syst. Sci. 16, 821–831 (2016).
Brunner, L., Schaller, N., Anstey, J., Sillmann, J. & Steiner, A. K. Dependence of present and future European temperature extremes on the location of atmospheric blocking. Geophys. Res. Lett. 45, 6311–6320 (2018).
Spensberger, C. et al. Dynamics of concurrent and sequential central european and scandinavian heatwaves. Q.J.R. Meteorol. Soc. 146, 2998–3013 (2020).
Aadhar, S. & Mishra, V. The 2022 mega heatwave in South Asia in the observed and projected future climate. Environ. Res. Lett. 18, 104011 (2023).
Russo, S., Sillmann, J. & Fischer, E. M. Top ten European heatwaves since 1950 and their occurrence in the coming decades. Environ. Res. Lett. 10, 124003 (2015).
Perkins-Kirkpatrick, S. E. & Gibson, P. B. Changes in regional heatwave characteristics as a function of increasing global temperature. Sci. Rep. 7, 12256 (2017).
Röthlisberger, M. et al. A new framework for identifying and investigating seasonal climate extremes. J. Clim. 34, 7761–7782 (2021).
Amaya, D. J. et al. Marine heatwaves need clear definitions so coastal communities can adapt. Nature 616, 29–32 (2023).
Folland, C. et al. Workshop on indices and indicators for climate extremes, Asheville, NC, USA, 3-6 June 1997, Breakout Group C: Temperature indices for climate extremes. Clim. Change 42, 31–43 (1999).
Jones, P. et al. The use of indices to identify changes in climatic extremes. Clim. Change 42, 131–149 (1999).
Hersbach, H. et al. The ERA5 global reanalysis. Q. J. R. Meteorol. Soc. 146, 1999–2049 (2020).
Eyring, V. et al. Overview of the coupled model intercomparison project phase 6 (CMIP6) experimental design and organization. Geosci. Model Dev. 9, 1937–1958 (2016).
Brunner, L., Hauser, M., Lorenz, R. & Beyerle, U. The ETH zurich CMIP6 next generation archive : technical documentation (2020).
Meinshausen, M. et al. The shared socio-economic pathway (SSP) greenhouse gas concentrations and their extensions to 2500. Geosci. Model Dev. 13, 3571–3605 (2020).
Brunner, L. et al. Reduced global warming from CMIP6 projections when weighting models by performance and independence. Earth Syst. Dyn. 11, 995–1012 (2020).
Merrifield, A. L., Brunner, L., Lorenz, R., Humphrey, V. & Knutti, R. Climate model selection by independence, performance, and spread (ClimSIPS v1.0.1) for regional applications. Geosci. Model Dev. 16, 4715–4747 (2023).
Acknowledgements
The authors thank Erich M. Fischer and Gabriele C. Hegerl for helpful discussions on the manuscript, and Urs Beyerle for downloading the CMIP6 data for the ETH next-generation archive.
Author information
Authors and Affiliations
Contributions
This work was conceptualised and written by L.B. with contributions from A.V. Data acquisition, analysis, and visualisation by L.B.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Brunner, L., Voigt, A. Pitfalls in diagnosing temperature extremes. Nat Commun 15, 2087 (2024). https://doi.org/10.1038/s41467-024-46349-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467-024-46349-x
- Springer Nature Limited