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Non-stationary time series attribution for heatwaves over Europe

Pascal Meurer, Sebastian Buschow, Svenja Szemkus, Petra Friederichs

TL;DR

This paper introduces a non-stationary Markov EVT framework to attribute complete time-series heatwaves over Europe to anthropogenic forcing. It combines a censored bivariate extreme-value likelihood with the Extremal Pattern Index (EPI) to compress spatial information and models non-stationarity via Legendre-polynomial covariates, using both constant and time-varying thresholds. By comparing ERA5 observations to CMIP6 HIST versus HIST-NAT ensembles, it derives likelihood ratios and Bayesian evidence across seasons and regions, finding decisive evidence for anthropogenic influence in central and southern Europe and substantial but regionally variable signals elsewhere. The work advances extreme-value attribution in non-stationary climates and provides a principled framework for integrating multiple climate-model realizations, with implications for risk assessment and climate-policy communication.

Abstract

The increasing occurrence of extreme weather events since the beginning of the 21st century has led to the development of new methods to attribute extreme events to anthropogenic climate change. How the extreme event is defined has a major influence on the attribution result. A frequently disregarded or evaded aspect concerns the temporal dependence and the clustering of extremes. This study presents an approach for attributing complete time series during extreme events to anthropogenic forcing. The approach is based on a non-stationary Markov process using bivariate extreme value theory to model the temporal dependence of the time series. We calculate the likelihood ratio of an observational time series from ERA5 given the distributions as estimated from CMIP6 simulations with historical natural-only and natural and anthropogenic forcing scenarios. The spatial fields are condensed by the extremal pattern index as a compact description of spatial extremes. In addition, the study examines the extent to which attribution statements about the occurrence of extreme heat events change when the effect of the mean warming is eliminated. The resulting attribution statement provides very strong evidence for the scenario with anthropogenic drivers over Europe, especially since the beginning of the 21st century. For central and southern Europe, the influence of anthropogenic greenhouse gas emissions on heatwaves could already have been proven in the 1970s using today's methods. There is no reliable signal apart from a general increase in temperature, neither in terms of the temporal dependence of extreme heat days nor in terms of the shape of the extreme value distribution.

Non-stationary time series attribution for heatwaves over Europe

TL;DR

This paper introduces a non-stationary Markov EVT framework to attribute complete time-series heatwaves over Europe to anthropogenic forcing. It combines a censored bivariate extreme-value likelihood with the Extremal Pattern Index (EPI) to compress spatial information and models non-stationarity via Legendre-polynomial covariates, using both constant and time-varying thresholds. By comparing ERA5 observations to CMIP6 HIST versus HIST-NAT ensembles, it derives likelihood ratios and Bayesian evidence across seasons and regions, finding decisive evidence for anthropogenic influence in central and southern Europe and substantial but regionally variable signals elsewhere. The work advances extreme-value attribution in non-stationary climates and provides a principled framework for integrating multiple climate-model realizations, with implications for risk assessment and climate-policy communication.

Abstract

The increasing occurrence of extreme weather events since the beginning of the 21st century has led to the development of new methods to attribute extreme events to anthropogenic climate change. How the extreme event is defined has a major influence on the attribution result. A frequently disregarded or evaded aspect concerns the temporal dependence and the clustering of extremes. This study presents an approach for attributing complete time series during extreme events to anthropogenic forcing. The approach is based on a non-stationary Markov process using bivariate extreme value theory to model the temporal dependence of the time series. We calculate the likelihood ratio of an observational time series from ERA5 given the distributions as estimated from CMIP6 simulations with historical natural-only and natural and anthropogenic forcing scenarios. The spatial fields are condensed by the extremal pattern index as a compact description of spatial extremes. In addition, the study examines the extent to which attribution statements about the occurrence of extreme heat events change when the effect of the mean warming is eliminated. The resulting attribution statement provides very strong evidence for the scenario with anthropogenic drivers over Europe, especially since the beginning of the 21st century. For central and southern Europe, the influence of anthropogenic greenhouse gas emissions on heatwaves could already have been proven in the 1970s using today's methods. There is no reliable signal apart from a general increase in temperature, neither in terms of the temporal dependence of extreme heat days nor in terms of the shape of the extreme value distribution.
Paper Structure (25 sections, 30 equations, 21 figures, 2 tables)

This paper contains 25 sections, 30 equations, 21 figures, 2 tables.

Table of Contents

  1. Data and Indices
  2. Data
  3. Extremal pattern index (EPI)
  4. Example: Summer 2025 in Europe
  5. Theory and Methods
  6. Attribution of extreme events
  7. Likelihood formulation
  8. Including non-stationarity
  9. Can extremes be attributed, and is their frequency increasing over time? - Using a constant threshold
  10. Are the extremes becoming more extreme? - Using a variable threshold
  11. Fitting procedure
  12. Attribution statements for ERA5 using ensemble simulations
  13. Model estimation with constant threshold
  14. Likelihood of ERA5 with constant threshold
  15. Model estimation with variable threshold
  16. ...and 10 more sections

Figures (21)

  • Figure 1: AR6 regions regions of (a) northern, (b) central, and (c) southern Europe as used in this study with North Africa excluded and restricted to land points only. A grid cell is considered as land point when at least $50\%$ of the cell is occupied by land.
  • Figure 2: Mean T2max anomalies in ERA5 between (a) 07 June to 5 July 2025, (b) 15 July to 01 August 2025, and (c) 07 August to 19 August 2025. Only grid points exceeding the 99%-quantile are shown. In (d) the EPI for T2max from June to August 2025 for northern (blue), central (orange) and southern (green) European regions from Fig. \ref{['fig:landmask']} is shown. The EPI is bold if it exceeds the corresponding 95% quantile (1940-2025) of the region, whereby the quantile is indicated as dotted line.
  • Figure 3: Regions of the censored likelihood model according to Eq. (\ref{['eqn:contributions']}), whereby the likelihood contribution depends on which variable is exceeding the threshold or not.
  • Figure 4: Estimated coefficients for the model with constant threshold (Sect. \ref{['sec:model_estimation_with_constant_threshold']}) in the southern European region. The coefficients (x-axis) are shown for (a) the threshold exceedance $\phi$, (b) the scale $\sigma$, (c) the shape $\xi$, and (d) the dependence $\alpha$. The estimates for the different climate models are represented as box-whiskers, those for ERA5 as green crosses. The whiskers and fliers cover the whole range of data.
  • Figure 5: Temporal evolution of (a) threshold exceedance probability, (b) scale, (c) shape, and (d) dependence parameters for the model with constant threshold in the southern European region. Mean (solid line) and standard deviation (shading) for the different climate models is plotted based on the estimates in Fig. \ref{['fig:parameter_estimates_model1']}.
  • ...and 16 more figures