Many climate-related disasters often result from a combination of several climate phenomena, also referred to as “compound events’’ (CEs). By interacting with each other, these phenomena can lead to huge environmental and societal impacts, at a scale potentially far greater than any of these climate events could have caused separately. Marginal and dependence properties of the climate phenomena forming the CEs are key statistical properties characterising their probabilities of occurrence. In this study, we propose a new methodology to assess the time of emergence of CE probabilities, which is critical for mitigation strategies and adaptation planning. Using copula theory, we separate and quantify the contribution of marginal and dependence properties to the overall probability changes of multivariate hazards leading to CEs. It provides a better understanding of how the statistical properties of variables leading to CEs evolve and contribute to the change in their occurrences. For illustrative purposes, the methodology is applied over a 13-member multi-model ensemble (CMIP6) to two case studies: compound wind and precipitation extremes over the region of Brittany (France), and frost events occurring during the growing season preconditioned by warm temperatures (growing-period frost) over central France. For compound wind and precipitation extremes, results show that probabilities emerge before the end of the 21st century for six models of the CMIP6 ensemble considered. For growing-period frosts, significant changes of probability are detected for 11 models. Yet, the contribution of marginal and dependence properties to these changes in probabilities can be very different from one climate hazard to another, and from one model to another. Depending on the CE, some models place strong importance on both marginal properties and dependence properties for probability changes. These results highlight the importance of considering changes in both marginal and dependence properties, as well as their inter-model variability, for future risk assessments related to CEs.

In September 2017, heavy rainfall and storm surge associated with Hurricane Irma resulted in record-breaking floods in Jacksonville, Florida. In 2019, Australia experienced high temperatures and prolonged dry conditions, which resulted in one of the worst bush fire seasons in its recorded history. In April 2021 and 2022, Central Europe experienced consecutive days of frost events following a warm early spring, which caused severe damage to agricultural yields. These recent climate events are some examples of so-called compound events (CEs), i.e. high-impact climate events that result from interactions of several climate hazards. These climate hazards are not necessarily extremes themselves, but their simultaneous or successive occurrences can generate strong impacts

From a statistical point of view, CEs are characterised by the statistical features of the variables forming the CEs, i.e. their marginal properties (e.g. mean and variance) and dependence structures. These key statistical properties can be affected by future climate change

In this paper, we propose a new methodology to assess the ToE of CE probabilities. We also develop a copula-based multivariate framework, which allows for an adequate description of the contribution of the changes in marginal and dependence properties to the evolution of multivariate hazard probabilities. This CE analysis is applied to two case studies. Please note that the goal of the paper is not to provide precise results of ToE in these two case studies, but rather to introduce the conceptual framework and raise awareness among climate scientists on the potential emergence of CE probabilities, as well as the contributions of statistical properties to probability changes. We first analyse compound wind and precipitation extremes over the coastal region of Brittany (France). This bivariate CE, i.e. composed of co-occurring climate hazards over the same region and time, has been analysed in several studies

The rest of this paper is organised as follows: Sect.

One ensemble of 13 global climate models (GCMs) following the CMIP6 protocol

List of CMIP6 simulations used in this study, their run, approximate horizontal resolution and references.

For compounding wind and precipitation extremes, we use the spatial mean of daily wind speed maxima and the spatial sum of daily precipitation time series during winter (December, January and February) over the region of Brittany, France (

For growing-period frost events, data are extracted over central France (

For illustrative purposes, Fig.

Our aim is to design a statistical method to assess the ToE of CE probabilities, that is to detect from which period changes of probability are statistically significant relative to a baseline period. Probabilities of CEs can be computed with copulas. Copulas are functions that make it possible to describe the dependence structure between random variables separately from their marginal distributions, which greatly simplifies calculations involving multivariate distributions

The concept of time of emergence (ToE) has been developed to assess the significance of climate changes relative to background variability. Comparing changes of climate signal relative to natural variability is particularly relevant as human societies and ecosystems are inherently adapted to the local background level of variability, and major impacts arise most likely when changes emerge from it

In this study, we use copula modelling to compute CE probabilities. We first consider two random variables

Bivariate exceedance probability refers to the probability that both random variables exceed a certain value

Let us now consider the realisations

changes in the marginal properties of

and changes in the dependence structure (i.e. in the copulas) between

Then, do exceedance probability values change significantly between reference and future periods? And if so, how much of this change is due to changing marginal properties, and how much is due to changing dependence structure? Attributing probability changes to changes of marginal and dependence properties has already been introduced by

Illustration of the influence of marginal and dependence properties on bivariate exceedance probabilities for an artificial distribution of two contributing variables

To assess how much marginal and dependence properties contribute to exceedance probabilities change between reference and future period, we use the four probabilities derived above to decompose the overall probability change. We first define

The methodology described above to assess ToE of CE probabilities and marginal and dependence contributions to these changes is applied to the 13 CMIP6 models by considering successively all 30-year sliding windows spanning the period 1871–2100. The methodology is applied to each climate model individually (“Indiv-Ensemble” version). In particular for contributions and ToE, multi-model median estimates are derived to summarise the information given by all the models. The Indiv-Ensemble version makes it possible to analyse the modelling of hazards separately and to assess the uncertainty in ToE arising from the inter-model differences. We also applied the methodology in the “Full-Ensemble” version, which consists of pooling the contributing variables of the 13 climate models together to derive unique ToE estimates and contribution values accounting for the global uncertainty in climate modelling. However, the details of the “Full-Ensemble” version and its results are not discussed in the main article but are given in Sects. S1–S5 in the Supplement. A summary of the successive steps of our methodology for the Indiv-Ensemble version is provided in the form of a flowchart in Fig.

Flowchart for the computations of time of emergence and contributions for the Indiv-Ensemble version.

In this section, results are presented for compound wind and precipitation extremes during winter in Brittany. Please note that, for this section as well as for the rest of the study, the period 1871–1900 is considered as the baseline period for natural variability to evaluate ToE and contributions. To focus on wind and precipitation extremes, we applied our methodology to points of high values. For each model, we selected points where, concurrently, wind and precipitation values exceed the individual 90th percentiles (denoted

To illustrate our methodology, we first explain the results obtained for compound wind and precipitation extremes and a single bivariate exceedance threshold before extending the results to several bivariate thresholds. We evaluate the probabilities of exceeding the 80th percentiles of the bivariate points belonging to

Before computing any probability, Fig.

Change of winter (December to February) bivariate wind and precipitation extremes distributions in Brittany based on CNRM-CM6 simulations due to

Time series of exceedance probabilities over all sliding windows for the bivariate threshold (

The evolution of the bivariate FAR

The results for ToE and contributions have so far been presented for the probability of events exceeding the 80th percentiles of selected points belonging to

CNRM-CM6

We now present the results obtained for ToE and contributions for the Indiv-Ensemble version for a single exceedance threshold. The methodology, previously illustrated with the CNRM-CM6 simulations, is now applied to each of the 13 models. Among the 13 models of the ensemble, only one model (INMCM-5.0) had more than 5 % of goodness-of-fit tests over all sliding windows, thus rejecting the hypothesis that the copula is a good fit, and hence was excluded from the analysis (see Appendix

We first present the results obtained for probabilities of exceeding the 80th percentiles of selected points of high values of wind and precipitation for the 1871–1900 reference period. Figure

The evolution of bivariate FAR, relative differences and contributions time series with respect to the reference period, as well as their decomposition in terms of marginal, dependence and interaction terms, is displayed in Fig.

Figure

As previously done in Sect.

Time of emergence (at 68 % confidence level) matrices of compound wind and precipitation extremes for the Indiv-Ensemble version due to changes of

Median contribution of marginal, dependence and interactions terms is displayed in Fig.

We now apply our methodology to analyse a second type of CE: growing-period frost. Contrary to compound wind and precipitation extremes, for which we were interested in exceedance probabilities (i.e. both contributing variables exceedance thresholds), we are interested here in the probability of growing-period frost, i.e. the probability of having a GDD value exceeding a threshold of 200 (GDD

We now present the results for the growing-period frost. As previously done, only one model (CMCC-ESM2) is excluded from the ensemble since it presents more than 5 % of goodness-of-fit tests, thus rejecting the hypothesis that fitted copulas are a good fit (see Appendix

Figure

The evolution of bivariate FAR, relative differences and contributions time series for the Indiv-Ensemble version and their decomposition in terms of marginal, dependence and interaction terms are shown in Fig.

Same as Fig.

In this study, we have presented a new methodology to assess the ToE of compound hazards probabilities. Using a copula-based multivariate framework, we also propose to quantify the contributions of marginal and dependence properties to probability changes of hazards leading to CEs. The methodology has been applied to analyse two different climate hazards with potentially high impacts, using a 13-member multi-model ensemble (CMIP6): compounding wind and precipitation extremes in Brittany and growing-period frost events over central France. For each hazard, the methodology has been applied to individual climate models to derive ToE of probabilities and contributions of statistical properties of each model separately. It enables us to estimate the uncertainty in ToE values and contributions to multivariate hazards probability changes arising from inter-model difference.

Results for compounding wind and precipitation extremes over Brittany show that occurrence probabilities of such events are likely to increase and potentially emerge before the end of the 21st century. However, the reason for these increased probabilities can be different depending on climate models: while for some models, probability changes are mainly driven by marginal changes only, other models give a strong importance to both marginal properties and dependence properties. This results in having a mixed importance (

Concerning growing-period frost events over central France, a large majority of models agree on the emergence of probabilities of such events. They also agree on the dominant contribution of marginal properties changes, while the contribution of dependence properties is mostly negligible.

By analysing two different case studies, our results highlight that the importance of marginal and dependence properties to probability changes can differ from one compound hazard to another, and from one climate model to another. It thus stresses the importance of considering both marginal and dependence properties carefully, as well as their inter-model variability, to analyse the future evolution of multivariate hazards leading to CEs.

In this study, the emergence of probabilities of multivariate hazards has been investigated with respect to the 30-year baseline period 1871–1900. This period can be considered as representative of the beginning of the industrial era

Moreover, in this study, the ToE of probability signals is defined as the year or time period for which the probability signal

In addition, changes in marginal properties of the different variables and their contributions to probability changes have been assessed together, i.e. without separating the changes and contributions from wind and precipitation, nor those from GDD and minimum temperature. Thus, it does not allow us to quantify by how much changes in individual variables drive probability changes. Some studies already concluded on the importance of individual variables in the change of occurrence of multivariate hazards

In this study, we demonstrated our conceptual framework using simulations from an ensemble of 13 GCMs. While using GCMs permitted us to illustrate our methodology and draw general conclusions when analysing changes of CE probabilities, the resolution of such climate models is often considered too coarse for a realistic representation of climate variables at a regional scale, such as for precipitation and wind

This study shows that both univariate and multivariate properties can be essential in determining CE properties. However, despite substantial improvements in climate modelling, climate simulations often remain biased compared to observations or reanalyses in terms of both univariate and multivariate properties

It should be noted that uncertainty in probabilities of multivariate hazards has been assessed by considering uncertainty in both statistical fitting procedures and model-to-model differences. However, uncertainty arising from internal climate variability, i.e. from the inherent chaotic nature of the climate system, has not been investigated. Assessing and analysing these uncertainties is, however, key to better characterising them and thus providing useful information for policy-makers

It is also important to note that the role of physical drivers of multivariate hazards has not been investigated in this study. Indeed, recent studies highlight the importance of large-scale climate modes

As mentioned in Sect.

Confidence intervals of bivariate exceedance probabilities are estimated by combining the confidence intervals from the fitted parameters for both marginal distributions and copulas. For both marginal distributions and copulas, the fitted parameters and their 68 % (or. 95 %) confidence intervals are estimated using MLE (as described in Appendix

For the fitting of the marginal distributions, we considered the Akaike information criterion (AIC) to select the best families among Gaussian, generalised extreme value and generalised Pareto distributions. The marginal distributions of wind speed and precipitation beyond the selection thresholds were modelled by generalised Pareto distributions. For growing-period frost events, the marginal distributions of the GDD indices were modelled using Gaussian distributions. We modelled the negative of the minimal temperatures using GEV distributions and transformed back.

For fitting of the copulas, marginal distributions are transformed into uniform distribution using normalised ranks

Custom codes developed for the analyses are publicly available at

CMIP6 climate model data can be downloaded through the Earth System Grid Federation portals. Instructions to access the data are available here:

The supplement related to this article is available online at:

MV had the initial idea of the study. MV and BF designed the experiments and protocols. BF made all the computations and figures. BF and MV made the analyses and interpretations. BF wrote the first complete draft of the manuscript, with input, corrections and additional writing contributions from MV.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modelling groups (listed in Table

The authors acknowledge support from the EUR IPSL Climate Graduate School project managed by the ANR under the “Investissements d'avenir” programme with the reference ANR-11-IDEX-0004-17-EURE-0006, the European Union’s Horizon 2020 research and innovation programme via the “XAIDA” project (grant no. 101003469), as well as from the “COESION” project funded by the French National programme LEFE (Les Enveloppes Fluides et l’Environnement).

This paper was edited by Joaquim G. Pinto and reviewed by Jakob Zscheischler and one anonymous referee.