How Extreme Can Solar Flares Get? A Statistical View
| Nugget | |
|---|---|
| Number: | 525 |
| 1st Author: | Lapo CECCARELLI |
| 2nd Author: | Daniela CASTRO-CAMILO |
| Published: | May 4, 2026 |
| Next Nugget: | TBD |
| Previous Nugget: | Observations of Slow Elemental Abundance Decay in Association to CME |
Introduction
Quantifying the upper limits of solar flare activity is central to understanding space weather risk. In this work, we analyse the lengthy record of soft X-ray (1-8 Å) fluxes from GOES (1975-2022, see Figure 1 and Ref. [1]) through the lens of Extreme Value Theory (EVT), focusing explicitly on the statistical structure of the tail of the flare-intensity distribution function. Note that a recent Nugget dealt with this subject from a different observational point of view.
Extreme Value Modelling Framework
We adopt two complementary EVT approaches. First, a block maxima framework, modeling weekly maxima using the Generalised Extreme Value (GEV) distribution. Second, a peaks-over-threshold (POT) approach, modeling the exceedances above a high quantile (a specified threshold) using the Generalised Pareto Distribution (GPD). These approaches target the same tail behaviour but rely on different asymptotic regimes, providing a useful consistency check.
The GPD distribution for exceedances y = X - u > 0 is defined as
where μ is the real-valued "location parameter", σ > 0 is the "scale parameter", and ξ is the "shape parameter" governing the behaviour of the tail of the distribution, i.e. the greatest values.
The GPD distribution for exceedances y = X - u > 0 is defined as:
where σu > 0 is the scale parameter above threshold u and ξ is the same shape parameter controlling tail "heaviness".
A key quantity in both formulations is the shape parameter, which governs tail heaviness. Our estimates consistently suggest a positive shape parameter, indicating heavy-tailed behaviour and supporting the plausibility of very large events beyond the observed range.
Diagnostics and Model Assessment
We assess model adequacy using standard EVT diagnostics. "Quantile-quantile" plots (Figure 2) show good agreement between empirical and fitted distributions in the tail region. "Dependence diagnostics" further highlight the role of temporal clustering, suggesting that extreme flares may not occur independently, an aspect that is partially accounted for through the weekly aggregation in the GEV framework.
Return Levels and Risk Quantification
"Return level" estimates derived from both GEV and GPD models (Figure 3 shows results for the GEV model) provide a quantitative interpretation of extreme flare magnitudes. Extrapolation indicates that Carrington-like events correspond to return periods on the order of a century, while more extreme scenarios remain within the support of the fitted heavy-tailed models. Importantly, uncertainty increases rapidly with the return period, as reflected in widening confidence intervals. This highlights the intrinsic difficulty of inference in the far tail, even with several decades of satellite data.
Implications for Solar Physics
From a statistical perspective, the results reinforce that assumptions about the nature of the tail are critical when extrapolating beyond observed data. The consistency we find between GEV and GPD approaches strengthens confidence in the inferred tail behaviour, while also emphasising the limits of data-driven extrapolation.
Limitations and Future Directions
While the GEV and GPD models provide principled and asymptotically justified frameworks for modelling extremes, their adequacy is limited in settings where events are irregularly spaced and potentially dependent, as is the case for solar flares. "Point-process" approaches, such as the Hawkes self-exciting processes, offer a natural way to capture temporal clustering and triggering mechanisms. Combining such models with EVT for the flare magnitudes represents a promising direction for more realistic and physically informed modelling of extreme solar activity