Extreme weather events are rare, and so they inevitably attract attention and cause socioenvironmental impacts when they occur. Explaining events that have occurred and putting them in a climate perspective constitutes a key challenge for both national weather services and the “event attribution” community.
The former have the responsibility to document historical weather events and maintain long-term statistics over their territory. This includes describing events from a synoptic perspective, quantifying their rarity (e.g., return period), and comparing with inventories of past events of the same type. The selection of documented events is generally made on the basis of fixed thresholds, so that events can be characterized by a level and duration of exceedance. As the climate warms, the thresholds for defining heatwaves (cold spells) are exceeded more (less) easily: for instance, 37 (16) of the 46 heatwaves (46 cold spells) officially reported in France by Météo-France over 1947–2022 have occurred after 1985, in the second half of the record period.1 This raises new questions for climate monitoring. Are recent episodes of threshold exceedance really extreme events in today’s climate? How rare were past events relative to their respective climate? Besides, due to possible adaptation, the impacts associated with a given temperature event are not necessarily the same today as in the past. Taking nonstationarity into account in regular climate monitoring has therefore become a necessity for weather services.
Placing extreme weather events in the perspective of climate change is precisely the aim of “event attribution” studies. This field seeks to assess how climate change has altered event probability or intensity, through quantities such as the probability ratio or the fraction of attributable risk. It has emerged since the pioneering analysis of the European heatwave of summer 2003 by Stott et al. (2004). It has become a regular activity as illustrated by the series of supplements to the annual State of the Climate report since Peterson et al. (2012)2 or the creation of the World Weather Attribution (WWA) group,3 and there is a growing interest from weather services in making it operational. Many events have been scrutinized over the past two decades, but it is often expressed that studies suffer from a “selection bias,” as in the IPCC AR6, chapter 11: “studies in the developing world are … lacking (Otto et al. 2020)” and “events that have been studied are not representative of all extreme events that occurred…” (Seneviratne et al. 2021). There are a variety of reasons for this bias (origin of the authors, quality of the data, media coverage, etc.), but so far little has been proposed to limit it and objectify event selection.
A common issue to climate monitoring and event attribution analyses is that their statements are dependent on the event definition, in particular, the spatiotemporal extent considered. Weather services need to identify the duration and extent where the event was extreme to analyze it appropriately and provide relevant information to local decision-makers. In attribution studies, using large scales generally yields a higher detection of the signal (climate change) due to the smoothing out of the noise (internal variability), and therefore larger probability ratios (Angélil et al. 2014, 2018; Cattiaux and Ribes 2018, hereafter CR18; Kirchmeier-Young et al. 2019). Event definition is often subjective in the literature, which is referred to as another type of “selection bias” (Seneviratne et al. 2021). In a previous paper (CR18), we proposed an objective definition procedure consisting of automatically selecting the spatiotemporal scale that maximizes the event rarity. This was motivated by the fact that the meteorological extremeness of an event is generally what makes it of interest and generates the greatest socioeconomic impacts. Importantly, we showed that our procedure does not artificially bias attribution statements: maximizing the rarity does not systematically maximize (or minimize) the attributable signal.
Here we show that this method can also be used to compare the rarity of different events, for example, two heatwaves of different spatiotemporal scales occurring in different years and at different locations. It can therefore serve both meteorological services to create inventories of historical events, and the event attribution community to address the issue of selection bias. Ultimately, we provide objective selections of historical heatwaves and cold waves at a national level (France) and at the global level, which constitute new information on observed extremes and can serve as databases for the validation of climate models or statistical methods.
The event probability as the ranking metric
Our study complements products of national weather services and rankings of extreme temperature events that have already been recently proposed in the literature. Russo et al. (2015) have proposed a ranking of European heatwaves on the basis of an index combining the fraction of area over which daily temperatures exceed reference (fixed) thresholds and the magnitude of the exceedances. Röthlisberger et al. (2021) and Boettcher et al. (2023) have developed a methodology to identify extreme seasonal temperatures at the gridpoint scale, also from fixed thresholds, and then form spatially coherent objects. Thompson et al. (2022) have compared the spectacular Pacific Northwest heatwave of June 2021 to the most extreme 1-day hot events ever recorded globally, defined at each region of the world as the largest daily temperature anomaly normalized with respect to local mean and variance. In a follow-up paper, Thompson et al. (2023) have also provided estimates of the return period of current records of daily maximum temperatures worldwide.
Scanning a given region: France
We first illustrate our scanning procedure at a fixed location: metropolitan France. We use the national daily thermal index provided by Météo-France over the 1947–2022 period (Ribes et al. 2022). We search for the most extreme warm and cold events in this time series (i.e., the lowest p1), exploring all years, all days of a year, and several temporal durations of n days. To keep the computation time reasonable, we restrict n to 1–8, 10, 12, 15, 20, 25, 30, 40, 50, 60, 75, 90, 120, 180, 270, and 365 days (23 values). The methodology for estimating p1 and p0 is depicted in Fig. 1 using the example of the 5-day heatwave of 15–19 June 2022. This episode is one of the three heatwaves reported by Météo-France in 2022 and the earliest one in records4 based on the reference threshold5 of 23.4°C (Fig. 1a).
(a) Daily-mean temperatures of 2022 (black). The 2022 normals (solid violet), associated ±1 and ±2 s.d. (gray shading), and 1950 normals (dashed violet). Heatwaves reported by Météo-France (i.e., with 3 days above 23.4°C) in red, with 15–19 Jun highlighted. (b) Computing calendar p1: Raw (thin gray) and detrended (w.r.t. 2022; thick black) time series of 15–19 Jun temperatures. Long-term trend (dashed colored) and its levels in 2022 (horizontal colored; factual climate) and 1950 (horizontal gray; counterfactual climate). Barcode: Detrended (factual) sample, with the fitted normal distribution (violet). Counterfactual distribution in gray. (c) Computing annual-maxima p1: as in (b), except that the time series is Tx5day and the fits are GEV distributions. (d) Calendar and (e) annual-maxima p1 as function of the time of the year (x axis) and the temporal averaging (y axis), with our selections of events indicated and ranked.
Citation: Bulletin of the American Meteorological Society 105, 1; 10.1175/BAMS-D-23-0095.1
As in CR18, the event rarity is assessed through an empirical estimation method consisting of (i) detrending the sample of observations xt to make all values representative of the climate at t1 = 2,022 (factual) or t0 = 1,950 (counterfactual) and (ii) fitting stationary parametric distributions on the detrended samples to derive p1 and p0. Our detrending procedure is described in detail in appendix A; in short, the shape of the trend is estimated from a multimodel ensemble of historical simulations and the amplitude from observations alone, with a dependency on the time of year and the temporal duration n. Two different options are then taken for the sample xt and the fitted distribution to describe the event conditionally to its calendar context or not.
The first—calendar conditioning—consists of comparing the n-day mean temperature of the event with the multiyear sample of temperatures observed on the exact same dates, and using a Gaussian distribution fit after detrending (Fig. 1b). In this case minimizing p1 is equivalent to searching for the largest normalized anomalies, i.e., departures from the mean in numbers of standard deviations (s.d.).6 For the June 2022 event we find a 5-day anomaly of 5.2 K, (2.6 s.d.), which gives a calendar p1 of 0.005; the interpretation is that an equally or hotter heatwave had a 0.5% chance to occur at these exact dates in 2022.
However, a 5-day average temperature of 25.4 K is not that exceptional in France in the climate of 2022: it is actually close to the expectance of the annual maxima of 5-day mean temperatures (hereafter Tx5day) for that year (estimate of 25.8 K, Fig. 1c). Thus, the second option—no conditioning—consists of comparing the n-day event with the sample of Txnday and using a generalized extreme value (GEV) distribution fit after detrending. In this case 1/p1 is interpretable as a formal return period. For the June 2022 event, the annual-maxima p1 is found to be 0.57, which means that a heatwave at least as hot had more than 50% chance to occur in any dates in 2022, and translates into an estimated return period of 1/p1 ∼ 1.8 years.
For both calendar and annual-maxima approaches, p0 is estimated by detrending the respective xt sample relative to t0 = 1,950 instead of t1 = 2,022 (gray distributions in Figs. 1b,c). As climate has warmed—estimates of 2.22°C for 15–19 June and 2.43°C for Tx5day—p0 is unsurprisingly smaller than p1. Finally, it must be noted that
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the formalism of Eqs. (1) and (2) and the use of annual “maxima” is well suited for hot events; to apply it to cold events one has to consider that X is the temperature multiplied by −1 and to consider annual “minima”;
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there is a debate in the community as to whether the value of the event should be included in or excluded from the sample when estimating its probability of occurrence (Philip et al. 2020; van Oldenborgh et al. 2021b). The question especially arises when the time series is stopped because the last event is particularly extreme, and we want to study it. Solutions to properly formulate stopping rules have been proposed (e.g., Barlow et al. 2020). Here we decide to systematically include the value of the event in the sample, because (i) we scan all the events of the whole period so we have no particular stopping rule and (ii) GEV distributions for annual maxima of temperature are often right bounded (negative shape parameters) and in case of a very rare event, excluding its value for the fit can lead to an estimated upper bound lower than the event effectively observed (as in Philip et al. 2022). In such a case, it would be nonsense to conclude that the event has p1 = 0 while we know that it has occurred.
The scanning procedure then consists of computing calendar and annual-maxima (annual-minima) p1 for all 27,740 days of 1947–2022 (29 Februaries are omitted) and 23 n-day temporal extents, and search for the local minima of p1—the most extreme events—within these 27,740 × 23 matrices. To do so, we select the absolute minimum of p1, then mask all overlapping calendar windows (we consider that they correspond to the same event) and iterate while there remains p1 smaller than a given threshold: we choose 0.01 for the calendar method and 0.25 for annual-maxima and annual-minima methods. In addition, we impose a minimal duration of 3 days for the calendar approach for readability and a maximal duration of 30 days for annual-maxima (annual-minima) approaches for the quality of GEV fits.
This procedure is first illustrated on the complete year 2022 (Figs. 1d,e). Three hot events are selected in the calendar approach; the June event mentioned above ranks second and is found to be the most extreme at the 3-day scale. In the annual-maxima approach, a unique 30-day event is selected that overlaps the two episodes reported by Météo-France in July and August. Its estimated return period is small (1/p1 = 10 years), reflecting the fact that exceeding the reference threshold of 23.4°C has become likely in the climate of 2022 (see also Fig. 1a). (The 180-day event from May to October has a lower p1 but the use of annual blocks and GEV is questionable for such long durations.)
Now expanding to the entire period, the calendar approach enables quick identification of abnormally hot events throughout the whole year, which is useful for routine climate monitoring and can be relevant for impacts studies (Fig. 2a). Only the heatwaves occurring near the peak of the annual cycle are retained by the annual-maxima approach (Fig. 2b). Both approaches agree that the greatest hot event for France is the heatwave of early August 2003, considered as a 10-day event (4–13 August, consistent with CR18). It corresponds to a normalized anomaly of 4.4 s.d. and an estimated return period of 410 [95 to Inf] years (Table 1, see caption for details on the confidence interval). Other major heatwaves are found in summers 1947, 1976, 1983, and 2019, with durations of, respectively, 2, 25, 25, and 1 days. This is overall consistent with the events selected by Météo-France on the basis of fixed thresholds (Fig. 2c). As for cold spells, the most extreme events are found in winters 1956, 1963, 1985, 1987, and 2012 (Figs. 2d,e and Table 1), which is also consistent with Météo-France reporting (Fig. 2f). The events of February 1956 and January 1985 have normalized anomalies of −4.4 s.d. (Table 1), i.e., are equally unusual as the August 2003 heatwave from a calendar perspective; however, their formal return periods estimated from the annual-minima approach are shorter (respectively, 170 and 110 years).
(a)–(c) Selection of the hottest events with the (a) calendar and (b) annual-maxima approaches, compared with (c) Météo-France threshold-based selection (3 days above 23.4°C; source of graphics: Les Décodeurs, Le Monde). (d)–(f) As in (a)–(c), but for cold events, with annual minima in (e) and exceedances below −2°C in (f). Rectangles and pieces of time series indicate the dates (x axis) and years (y axis) of the events, with a restricted x axis in (b), (c), (e), and (f). Rectangle colors indicate p1 with the five most extreme events highlighted. A minimal duration of 3 days and a maximal p1 of 0.01 are imposed to calendar events. A maximal duration of 30 days and a maximal p1 of 0.25 are imposed to annual-maxima and annual-minima events. The p1 values are lower in the calendar method due to the condition that the event occurs at these exact dates.
Citation: Bulletin of the American Meteorological Society 105, 1; 10.1175/BAMS-D-23-0095.1
The five hottest and coldest events selected by the annual-maxima and annual-minima approaches in France, ordered by increasing p1 (see also Fig. 2). Associated p0 and p1/p0 are indicated, as well as average temperature and normalized anomaly. Brackets indicate 90%-level confidence intervals computed by bootstrapping years in the Txnday or Tnnday samples; as random resampling removes the value of the year of interest from the N-year sample with a probability
The right panels in Fig. 2 illustrate the limitations of climate monitoring of extremes using fixed threshold approaches. On the one hand, using the same threshold for the whole year makes it impossible to describe certain events conditionally to their calendar context, e.g., late cold snaps in early spring, which have an impact on vegetation (Vautard et al. 2023). On the other hand, global warming leads to heatwave thresholds being exceeded almost every summer (we can no longer speak of “extreme” events), while cold spell thresholds are almost never reached (we can no longer speak of events at all). Our method provides complementary and useful information for national weather services on both these calendar and nonstationary aspects. Last, note that an interactive application associated to this paper enables to reproduce the analyses carried out in this section and retrace Fig. 1 for various choices of years and dates and Fig. 2 for different thresholds of p1 and duration (details in appendix C).
Scanning worldwide
To extend the scanning procedure to the global scale, we use daily-mean 2-m temperatures provided by the ERA5 dataset over the 1959–2022 period (Hersbach et al. 2020). The advantage of ERA5 is that it is a physically consistent dataset globally (model simulation) with a sufficiently fine resolution for our purpose and without missing values, which is particularly convenient for the calculation of annual maxima or minima. The main disadvantage is that it is not temporally homogenized (assimilated observations evolve in time), which can cause problems for both our detrending and p1 calculations. Nevertheless, we have verified that the main results of this section, i.e., the selections of extremes, do not suffer from obvious heterogeneity problems. To our knowledge, there exists no observational dataset of global daily temperature with time homogeneity, long history, and no missing values.
The exploration of spatial domains is performed with the hierarchical collection of economic and political regions defined by Stone (2019) within the “Weather Risk Attribution Forecast” (WRAF) framework. This collection provides successive divisions of land areas into regions of 10, 5, 2, and 0.5 million km2 (hereafter Mkm2), each division being nested in the previous one, with only a few gaps in the global coverage (for instance Europe appears only from 5 Mkm2). The set of 0.5 Mkm2 regions was also used by Thompson et al. (2022, 2023). The full range of spatial sizes covers the vast majority of event definitions in recent attribution studies, with the exception of studies at global scale or at the local scale (in this case we could refine within a 0.5 Mkm2 region). Scanning the full period at global scale thus involves considering 64 years × 365 days × 347 WRAF regions × 23 time durations ∼186 million temperature events.
First, we scan a single year to identify the most extreme events of that year, as this can be useful for annual reports of climate monitoring and event attribution. Figures 3a and 4a provide the minima of p1 at each 2 Mkm2 region for, respectively, the annual-maxima approach (heatwaves) in 2021 and the annual-minima approach (cold spells) in 2022. We have chosen these two examples because outstandingly, west Canada holds the strongest event in both cases, i.e., the 3-day heatwave of 29 June–1 July 2021—the famous “heat dome” analyzed in numerous studies (e.g., Bercos-Hickey et al. 2022; Philip et al. 2022; Schumacher et al. 2022; Thompson et al. 2022; Terray 2023; White et al. 2023)—and the 5-day cold spell of 19–23 December 2022, related to Winter Storm “Elliott.” We estimate that these two events have remarkably low values of p1, corresponding to estimated return periods of 810 years for the heatwave and 550 years for the cold spell (Table 2). There is certainly no meteorological relationship between these two events, suggesting that this region has, by chance, experienced particularly extreme temperatures over the past two years. For a closer look, the equivalent of Fig. 1 for these two events has been plotted in Figs. S1 and S2 in the online supplemental material (https://doi.org/10.1175/BAMS-D-23-0095.2). The top three heatwaves in 2021 then include a 7-day July event in Kazakhstan and a 30-day event in December 2021–January 2022 in Argentina, while a 30-day November event in Congo and a 12-day July event in Queensland complete the podium of cold spells 2022. By construction, about half of the regions have a minimum p1 above 0.5 in Figs. 3a and 4a; for a given temporal duration, the distribution of yearly minimum p1 across regions is indeed uniform (appendix B).
Scanning heatwaves worldwide. (a) For each 2 Mkm2 region, the minimum p1 of all 2021 events in the annual-maxima approach is shown. (b) Calendar p1 of a warm event occurring on 29 Jun–1 Jul 2021, i.e., coinciding with the absolute minimum of p1 in (a). (c),(d) As in (a), but for (c) 2003–22 and (d) 1959–2022.
Citation: Bulletin of the American Meteorological Society 105, 1; 10.1175/BAMS-D-23-0095.1
Scanning cold spells worldwide. (a) For each 2 Mkm2 region, the minimum p1 of all 2022 events in the annual-minima approach is shown. (b) Calendar p1 of a cold event occurring on 2022, 19–23 Dec 2022, i.e., coinciding with the absolute minimum of p1 in (a). (c),(d) As in (a), but for (c) 2003–22 and (d) 1959–2022.
Citation: Bulletin of the American Meteorological Society 105, 1; 10.1175/BAMS-D-23-0095.1
The 10 hottest and coldest worldwide events selected by the annual-maxima and annual-minima approaches for 2 Mkm2 regions, ordered by increasing p1 (see also Figs. 3 and 4). Brackets indicate 90%-level confidence intervals computed as in Table 1 with the same remark about the possibly infinite upper bound of 1/p1. Region ID and names are from Stone (2019). CEMAC = Economic and Monetary Community of Central Africa; ECO = Economic Cooperation Organization; EEA = European Economic Area; SADC = Southern Africa Development Community; USA = United States of America.
Figures 3b and 4b show the calendar p1 associated with the 2021 and 2022 west Canadian events. We find normalized calendar anomalies of, respectively, 5.0 and 3.0 s.d. over this region, which highlights the extremeness of these two events (this gives a lower p1 than in the annual-maxima approach since here it is conditioned by the time of the year). The global picture of calendar p1 can be useful for routine climate monitoring: for example, here we see that the 2021 heatwave was restricted to west Canada while the 2022 cold spell also spreads over the northwest U.S. region (2.7 s.d.), and to a lesser extent over Northwest Territories and Yukon (2.2 s.d.) and midwest United States (1.9 s.d.). This is consistent with the characteristic spatial size of atmospheric patterns responsible for summer versus winter extremes in midlatitudes. Figure 3a also shows that the 2021 Kazakhstan heatwave occurred concurrently with the west Canadian one, which could be a coincidence or result from an amplified circumglobal wave pattern. Note that for given dates, the distribution of calendar p1 across regions is expected to be uniform in our procedure, so that about half of the regions are expected to have p1 above 0.5 in such maps (appendix B).
We now extend the scan to two time periods: the last 20 years (2003–22; Figs. 3c and 4c and Table 2) over which event attribution has developed, and the whole period of study (1959–2022; Figs. 3d and 4d)—the interactive application enables to further visualize the results of any year or period (appendix C). The podium of heatwaves of the last 20 years is occupied by events well known in the scientific community (Fig. 3c): west Canada 2021, southwest Russia 2010, and south Europe 2003 (the same as in France in Fig. 2). This first one is consistent with Thompson et al. (2022) while the last two are in Russo et al. (2015). Other major events include heatwaves that have been well documented in the scientific literature (e.g., January and December 2019 in Australia and associated bushfires; e.g., van Oldenborgh et al. 2021a), and others for which no attribution study exist (e.g., South Africa 2016, South Pacific Russia 2011; see Table 2), illustrating the geographic selection bias. Over the whole period, only two events of size 2 Mkm2 (May 1998 in Chad and January 1993 in west Antarctica, Fig. 3d) appear to exceed the 2021 west Canadian heatwave, with estimated return periods of 2,200 and 1,100 years, respectively.
On the cold spell side, a 4-day event in July 1975 in central north Brazil wins the contest in the 2 Mkm2 category, with January 2014 in east Canada being the most extreme event of the last 20 years (Figs. 4c,d and Table 2). The December 2022 “Elliott” storm in west Canada ranks fourth over 2003–22 and twelfth over the whole period, which again highlights its extremeness in the climate of 2022.
Last, the hierarchical character of the collection of WRAF regions is used to refine the selection of events among spatial scales. Figure 5 shows the minimum p1 associated with the three major heatwaves identified in Fig. 3 and Table 2 on the geographical division of the 10, 5, 2, and 0.5 Mkm2 regions. The 2021 Canadian event is found to be the most extreme for a 0.5 Mkm2 region (Alberta), the 2010 Russian event for a 5 Mkm2 region (European Russia), and the 2003 European event for the 2 Mkm2 region as in Fig. 3.
Zooming over a region. (a), (from left to right) For each 10, 5, 2, and 0.5 Mkm2 region, the minimum p1 of all 2021 events in the annual-maxima approach is shown, which coincides with the 2021 Canadian heatwave listed first in Table 2 and second in Table 3. (b) As in (a), but for the 2010 Russian heatwave (second and fourth). (c) As in (a), but for the 2003 European heatwave (third and fifth). For each event, the region with the lowest p1 is explicitly named and its panel (i.e., region size) is highlighted in gray.
Citation: Bulletin of the American Meteorological Society 105, 1; 10.1175/BAMS-D-23-0095.1
More generally, when minimizing p1 both over time and the entire collection of WRAF regions for the period 2003–22, most of the events of Table 2 stand out, sometimes on parent or child domains (Table 3). Notably, two Chinese heatwaves slip into the trio of heatwaves described above: a 3-day event in 2003 and a 25-day event in 2022, with estimated return periods similar to the 2021 Canadian heatwave. These two events were reported in the media7 but seem to have received less scientific attention. Extending the scan to the entire period 1959–2022 can be done in the interactive application (appendix C): other events join the top 10, but sometimes with obvious data problems (appendix B). The top events remain the heatwave of May 1998 in Chad (at size 2 Mkm2) and the cold spell of July 1975 in Brazil (at size 0.5 Mkm2, south Pará).
As in Table 2, but for the entire collection of WRAF regions. The number of characters in the region ID indicates its size: X for 10 Mkm2, X.X for 5 Mkm2, X.X.X for 2 Mkm2, and X.X.X.X for 0.5 Mkm2. Region ID and names from Stone (2019). DR Congo = Democratic Republic of Congo; EEA = European Economic Area; USA = United States of America.
Using the collection of WRAF regions can thus be helpful to both (i) identify the most extreme events of a given period worldwide and (ii) give a rough idea of their spatial size. However, extreme events do not stop at administrative borders, and once an event has been selected for a study, its precise definition may be decided through a more systematic exploration of various spatial domains and sizes (e.g., CR18), and possibly on a more suitable dataset than ERA5 for the area of interest.
Discussion and conclusions
We have objectively identified and ranked the most extreme temperature events over a large spatiotemporal range, using the probability of event occurrence in the factual climate (p1) as a universal metric. We produce a list of the greatest hot and cold extremes over the recent past, together with their spatiotemporal scale. Applied locally, such as on a national level, this procedure can provide additional method for the climate monitoring of weather events in a warming world. Applied globally, it ensures extreme events are identified objectively, independently of the populations concerned or the media coverage. This can help the event attribution community both by providing objective lists of events to be studied in dedicated annual reports, without selection bias, and by building reference databases of extreme events on which climate models and statistical attribution methods can be confronted.
Our procedure is not without limitations, particularly because applying it in such a generalized and exhaustive way requires use of simple and fast methods. A first remark concerns the long-term trend, which is crucial when comparing extreme events of various years. As its amplitude is here estimated solely from the observations (appendix A and Fig. S3), it may contain traces of decadal internal variability, and this can locally affect p1 estimates. In regions where internal variability causes recent years to be persistently warmer than the forced response—as suspected, e.g., in France (Ribes et al. 2022)—our procedure could overestimate the trend and underestimate the rarity of recent hot events. Using a long-term trend estimated from a combination of observed and modeled data would be more reliable but for future work. Importantly, the histogram of years of selected events or the time series of yearly minimum p1 do not suggest any collective bias related to trend estimates (appendix B and Fig. S4).
A second remark concerns the estimation procedure for p1, more precisely, the fit of parametric laws onto observed samples. To compare various temporal durations and spatial domains, it is assumed here that the same, simple, distributions apply to all seasons, regions, and a wide range of time and space scales (with different parameters). For the calendar approach, we use the normal distribution, which is classical for temperature, but does not account for potential asymmetric behaviors (e.g., McKinnon and Simpson 2022). For the annual-maxima or annual-minima approaches, we use GEV distributions to fit Txnday or Tnnday samples, which is well suitable for small n (here we have limited to n = 30) but less for large n (typically n = 365, i.e., the full year). In both cases, we find pretty uniform histograms of the p1 of all events in all regions, suggesting that the estimation procedure is “collectively” correct (appendix B and Fig. S3). Ideally, one would need a family of distributions that allows a slow transition from skewed or GEV distributions for small scales to normal distributions when the law of large numbers starts to apply. This would affect the values of p1, but without changing the philosophy of our work (the idea of focusing on p1) nor even the ranking of events in a dramatic way.
Last, exploring extreme events in a generalized way over so many years and regions requires a high-quality dataset of temperature observations with temporal homogenization. Here we use ERA5, which performs well over many regions where the data assimilation and the numerical model are efficient. For France we have verified that ERA5 and the Météo-France thermal index are very close. However, there are some regions where one may question the data quality (appendix B and Fig. S4) as previously noted by Thompson et al. (2022) in their ranking of heatwaves. Repeating the scanning exercise with other datasets could make it possible to quantify how observational uncertainties reverberate in our results.
Overall, we show that p1 is an appropriate metric to identify the most extreme weather events in recent history. We have opted for a nonstationary view of climate monitoring—p1 is the probability of occurrence in the factual, evolving, climate—but minimizing p0 instead of p1 can provide a stationary view. The method can be extended to other meteorological variables, such as precipitation or surface winds, yet the estimation of p1 over various space and time scales is more delicate than for temperature. Following van der Wiel et al. (2020), it can also be adapted to select extreme events in terms of impacts, not weather, by replacing the meteorological variable (e.g., temperature) with an impact-oriented variable (e.g., heat stress). Writing p1 as a joint probability would also make it possible to consider compound events. Ultimately, the range of applications for such an exhaustive spatiotemporal scan goes beyond the area of extreme events: for example, replacing p1 with an appropriate evaluation metric could enable the identification of regions and periods where discrepancies between observations and climate models are the strongest, which could make this approach useful for a much wider community.
ClimatHD by Météo-France (www.meteofrance.fr/climat-passe-et-futur/climathd), online, accessed 15 October 2023.
BAMS Explaining Extreme Events from a Climate Perspective (www.ametsoc.org/ams/index.cfm/publications/bulletin-of-the-american-meteorological-society-bams/explaining-extreme-events-from-a-climate-perspective), online, accessed 15 October 2023.
WWA web page (www.worldweatherattribution.org), online, accessed 15 October 2023.
Weather extremes of summer 2022 by Météo-France (https://meteofrance.com/actualites-et-dossiers/actualites/changement-climatique-lete-2022-et-ses-extremes-meteorologiques), online, accessed 15 October 2023.
Definition of heatwaves by Météo-France (https://meteofrance.com/actualites-et-dossiers/comprendre-la-meteo/temperature/comment-les-climatologues-evaluent-vagues-chaleur-canicules), online, accessed 15 October 2023.
For X ∼ N(µ, σ), p1 is a monotonic function of (X − µ)/σ.
ACCESS-CM2 (3 members), ACCESS-ESM1-5 (10), AWI-CM-1-1-MR (1), BCC-CSM2-MR (1), CAMS-CSM1-0 (2), CanESM5-CanOE (3), CanESM5 (50), CAS-ESM2-0 (2), CESM2 (3), CESM2-WACCM (3), CIESM (1), CMCC-CM2-SR5 (1), CMCC-ESM2 (1), CNRM-CM6-1-HR (1), CNRM-CM6-1 (6), CNRM-ESM2-1 (5), E3SM-1-1 (1), EC-Earth3-CC (1), EC-Earth3 (3), EC-Earth3-Veg-LR (3), EC-Earth3-Veg (4), FGOALS-f3-L (1), FGOALS-g3 (4), FIO-ESM-2-0 (3), GFDL-CM4 (1), GFDL-ESM4 (1), GISS-E2-1-G (7), HadGEM3-GC31-LL (4), HadGEM3-GC31-MM (4), IITM-ESM (1), INM-CM4-8 (1), INM-CM5-0 (1), IPSL-CM6A-LR (7), KACE-1-0-G (3), KIOST-ESM (1), MCM-UA-1-0 (1), MIROC6 (49), MIROC-ES2L (9), MPI-ESM1-2-HR (2), MPI-ESM1-2-LR (10), MRI-ESM2-0 (2), NESM3 (2), NorESM2-LM (1), NorESM2-MM (1), TaiESM1 (1), UKESM1-0-LL (5).
Acknowledgments.
The authors are grateful to M. Wehner (LBNL) and two anonymous reviewers for their careful proofreading. They thank Y. Robin and M. Schneider (Météo-France) for providing the France thermal index and the associated selections of events and D. Stone (National Institute of Water and Atmospheric Research) for the WRAF regions files. They acknowledge the ERA5 dataset provided by the ECMWF and climate modeling groups involved in CMIP6 for producing and making available their simulations. J.C. particularly thanks S. Qasmi and R. Samacoïts (Météo-France) for their great support on spatial and interactive visualization with R and the shiny package (https://cran.r-project.org) and M. Goar (Le Monde).
Data availability statement.
Input data and scanning results are available on a Zenodo archive: https://doi.org/10.5281/zenodo.7966559. This includes Météo-France thermal index, ERA5 data averaged over WRAF regions, CMIP data used for the detrending, WRAF data used for the analysis, and date versus duration matrices of estimated p1 and p0 for the three methods (calendar, annual maxima, and annual minima) and all regions. R scripts for computing the scanning procedure and visualizing the results are available on GitHub: https://github.com/jlncttx/CRT23/. Figures and tables can also be reproduced on the interactive application: https://jlncttx.shinyapps.io/CRT23-app/.
Appendix A: Details on the detrending procedure
The statistical procedure used to estimate p1 involves (i) detrending the time series (making it stationary) and (ii) using standard statistical methods for stationary data. It is equivalent to empirical methods using nonstationary fits on observations (Philip et al. 2020) but we prefer to do a two-step process to ensure consistency of trends between consecutive calendar dates or n-day durations. Here we detail how we correct a (nonstationary) yearly temperature time series with respect to a reference year to make all values representative of the (stationary) climate of that year. We do that by fitting observations onto a multimodel estimate of climate change (forced response F); this is illustrated with the example of France in Fig. S3.
Two important remarks on this simple detrending procedure must be made. First, it only corrects for potential changes in the mean, not in the variance or higher moments; this is not a major problem for temperature because changes in the mean dominate the total signal, but a more sophisticated correction could be used, especially to extend the method to other variables. Second, as we allow the scaling factors βT to be different from 1, we correct for possible mismatches between observations and models on the magnitude of long-term trends; in other words, our correction βT × F has the shape of the multimodel-mean forced response but the amplitude of the observed trend.
Appendix B: Validation of the p 1 estimation procedure
Here we provide further elements of validation of the p1 estimation procedure. First, we verify that the selected events are evenly distributed throughout the scanning period, and that no clear long-term trend is visible in the yearly minimum p1 (Fig. S4). This suggests that the long-term trend is reasonably accounted for in all regions. Note that El Niño (La Niña) years coincide with peaks in the number of regions having a record heatwave (cold spell), as expected since a large number of regions are dominated by this mode of variability.
Second, we verify that the distributions of our p1 estimates are close to uniform. Figure S5 shows the histograms of p1 estimates for all events of all 2 Mkm2 regions, for various temporal durations and the three (calendar, annual-maxima, annual-minima) methods. In all cases p1 is rather uniformly distributed, suggesting that fitting normal distributions to calendar temperatures and GEV to annual maxima and annual minima is reasonable. In the calendar approach, slight departures from the uniform arise at the edges of the [0, 1] interval, which could result for asymmetric behaviors.
Finally, we show examples of problems encountered in a few cases when scanning the full period and all the regions. Importantly, there is no obvious evidence that the events listed in the main text or tables are concerned by such issues.
Figure S6a shows a time series of Tx20day in the 5 Mkm2 region of east and south Brazil. The first 5 years of the series (1959–63) appear well above the rest of the detrended series, which clearly points to a homogeneity problem. This event of October 1963 is selected by our procedure in this region and several child regions with pretty high rankings, but it is at least partly for wrong reasons.
Figure S6b illustrates another type of problem: the time series is Tn3day in Pacific Russia, for which the GEV fit gives a very negative shape parameter (ξ = −0.7). This very sharp “upper” tail—right side of the distribution since the GEV is reversed for annual minima—results in a very low p1 associated with this January 1960 event, although its value does not look that extreme. This is partly because several years have “not so low” values, and encourages one to increase block size (i.e., more than 1 year) to help estimating ξ and ensure a better GEV fit.
Appendix C: An interactive web page to further explore the results
We have deployed an R shiny application to reproduce the figures and tables of the paper for all methods, years, and other variables (daily-maximum and daily-minimum temperatures): https://jlncttx.shinyapps.io/CRT23-app/.
The “Local scan” tab enables the user to explore the France data and redo Figs. 1 and 2 and Table 1 for any selection of variable, method, and year.
The “Global scan” tab enables the user to explore results of the ERA5 scan and redo Figs. 3 and 4 and Table 2 for any selection of variable, method, year(s), and size of WRAF regions.
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