<figure><img src="image_1.png" /></figure>

HAL
open science

# Investigating the role of climate change in the 3 May 2025 Western Europe hailstorm using atmospheric analogues

Davide Faranda, Tommaso Alberti

► To cite this version:

Davide Faranda, Tommaso Alberti. Investigating the role of climate change in the 3 May 2025 Western Europe hailstorm using atmospheric analogues. 2025. hal-05263140

HAL Id: hal-05263140

https://hal.science/hal-05263140v1

Preprint submitted on 16 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Investigating the Role of Climate Change in the 3 May 2025 Western Europe Hailstorm Using Atmospheric Analogues

Davide Faranda<fnref n="1" />,<fnref n="2" />,<fnref n="3" /> and Tommaso Alberti<fnref n="4" />

## Key Points:

* Analogue-based methods help detecting climate change signals in severe convective storm environments

* Anthropogenic warming increases atmospheric instability, likely intensifying hailstorms like the one analyzed in Western Europe

* Natural variability continues to play a dominant role in the occurrence of hailstorms

<footnote n="1">¹Laboratoire des Sciences du Climat et de l'Environnement, Gif-sur-Yvette 91191, France</footnote>

<footnote n="2">²London Mathematical Laboratory, London W6 8RH, UK</footnote>

<footnote n="3">³Laboratoire de Météorologie Dynamique/IPSL, École Normale Supérieure, PSL Research University,</footnote>

<footnote>Sorbonne Université, Paris 75005, France</footnote>

<footnote n="4">⁴Istituto Nazionale di Geofisica e Vulcanologia, Rome 00143, Italy</footnote>

<footnote>Corresponding author: Tommaso Alberti, tommaso.alberti@ingv.it</footnote>
**Abstract**

On 3 May 2025, a severe hailstorm struck Paris and parts of western Europe, raising concerns about the influence of climate change. We analyze this event using ERA5 reanalysis and an analogue-based attribution framework. The synoptic setting involved a cut-off low and a surface cold front, occurring shortly after an early-season heatwave. We compare analogous patterns in past (1974–1999) and recent (1999–2024) climates to assess thermodynamic differences. Hail probability and size were estimated with two models: a logistic formulation based on convective available potential energy, wind shear, and convective precipitation, and a new model incorporating freezing-level height and 850 hPa temperature, designed for European hail environments. Both were calibrated with Ile-de-France observations and validated with independent data. Our findings show that present-day conditions favor higher hail probabilities and larger hailstones, suggesting anthropogenic warming has likely enhanced hailstorm intensity in the region.

**Plain Language Summary**

On 3 May 2025, a powerful hailstorm hit Paris and parts of western Europe, causing significant damage and prompting questions about the role of climate change. In this study, we investigate whether human-caused global warming made this event more likely or more intense. We use weather data from the ERA5 reanalysis and a method called "analogue-based attribution" to compare the atmospheric conditions of this storm with similar weather patterns from the past 50 years. We also apply two models to estimate how likely hailstorms are under different climate conditions, and how big the hailstones might be. These models take into account factors like temperature, wind, and atmospheric instability. We tested them with real observations from the Paris region.

Our findings show that in today's warmer climate, conditions that favor hail such as higher atmospheric energy are more extreme than they used to be. This means that storms like the one in May 2025 are more likely to produce large hail. While natural climate variability still plays a role, our results suggest that human-caused climate change has likely increased the intensity of hailstorms in this region. This type of analysis can help improve future risk assessments for severe storms in a changing climate.

**1 Introduction**

Severe hailstorms are among the most destructive convective weather events in mid-latitude climates, causing extensive damage to property, agriculture, and infrastructure (Prein & Holland, 2018). Paris and much of western Europe are not traditionally regarded as hail hotspots compared to regions like northern Italy or the Great Plains of the United States (Taszarek et al., 2020). However, climate statistics indicate that severe thunderstorms with large hail do occur in western Europe, especially in late spring and summer, albeit infrequently (Punge & Kunz, 2016; Berthet et al., 2011). In the afternoon of 3 May 2025 an exceptionally intense hailstorm impacted over Paris, with large hailstones and causing flash flooding. Eye-witness reports and media coverage described "marble-sized" to golf-ball-sized hail blanketing streets and temporary closure of multiple metro stations due to flooding. This outbreak followed several days of higher-than-normal temperatures in the region: for instance, Paris-CDG airport recorded nearly 29 °C on 1 May (about 5 °C above normal). The combination of a pre-existing heatwave and an approaching storm favored conditions for severe convection.

Understanding the influence of anthropogenic climate change on such an extreme hailstorm is scientifically and societally important. Extreme event attribution aims to discern whether human-induced climate change altered the probability or intensity of specific events (Stott et al., 2016; Otto, 2017). Traditional attribution approaches often rely on climate model simulations to compare a "factual" climate with historical green-
house gas levels to a counterfactual “non-warming” climate (e.g., via large ensemble sim-
ulations). However, for convective storms like hailstorms, directly simulating the event
at cloud-resolving scales in large ensembles is computationally challenging. An alterna-
tive approach is to condition on the large-scale atmospheric circulation associated with
the event (Shepherd, 2016). By focusing on the synoptic-scale weather pattern, one can
ask how the thermodynamic environment and resulting hazards differ between past and
present climates for the same type of weather situation (Yiou, 2014; Faranda et al., 2022).
This “analogue” method treats the event as part of a class of similar circulation patterns
and leverages historical observations or reanalyses to assess changes in frequency and in-
tensity of those patterns and their impacts (Yiou, 2014; Faranda et al., 2022). Such a
conditional attribution is sometimes called a “storyline” approach, as it examines how
climate change affects a specific event’s storyline (Shepherd, 2016; Lloyd & Shepherd,
2020).

Recent studies suggest that the ingredients for severe convective storms are chang-
ing with the climate. Theory and models project that increasing greenhouse gases will
lead to higher low-level moisture and instability in many areas, which would favor more
frequent or intense thunderstorms (Trapp et al., 2007; Diffenbaugh & Field, 2013; Rädler
et al., 2019). At the same time, some regions could experience changes in wind shear or
in the altitude of the freezing level that complicate the response of hail. A warmer at-
mosphere raises the melting level, possibly causing more small hail to melt before reach-
ing the surface, while paradoxically larger hailstones may become more common because
stronger updrafts in a high-Convective Available Potential Energy (CAPE) future can
support their growth (Brimelow et al., 2017; Raupach et al., 2021). Indeed, detailed sim-
ulations over North America indicate a future decrease in overall hail frequency but an
increase in the frequency of very large hailstones, as weaker hail is suppressed by melt-
ing whereas storms producing giant hail benefit from greater instability (Brimelow et al.,
2017; Gensini et al., 2024).

In Europe, high-resolution modeling and observational proxies likewise point to in-
creasing convective hazard. Regional climate simulations project more severe thunder-
storms across most of Europe by late 21st century, particularly very large hail, due to
rising CAPE with little reduction in wind shear (Feldmann et al., 2025). Consistent with
this, a study of 1950–2021 detected upward trends in environments conducive to light-
ning and large hail, mainly driven by higher moisture and instability (Battaglioli, Groen-
emeijer, Púčik, et al., 2023). Recent analyses also suggest that present-day conditions
already favor higher hail probability and larger hailstones, implying that anthropogenic
warming may have intensified hail-producing storms. Yet other studies report inconclu-
sive evidence or strong spatial variability in severe storm and hail trends (Kahraman et
al., 2024; Taszarek et al., 2021). Overall, human-induced climate change is likely increas-
ing the risk of severe hailstorms in Europe, despite persisting uncertainties.

In this letter, we present a comprehensive analysis of the 3 May 2025 hailstorm out-
break in Paris, with a focus on the two main airports, and western Europe and of its re-
lationship with climate change following a similar structure as in Faranda et al. (2024).
Recent work applying similar analogue-based methods to derechos in France also iden-
tified significant increases in CAPE under comparable synoptic patterns, while shear in-
crease remained weak or inconsistent (Fery & Faranda, 2024). We first introduce the me-
teorological situation of 3 May 2025 using reanalysis data, describing the synoptic-scale
dynamics that led to the convection. Then, we outline the methods: the definition of the
event in terms of atmospheric circulation, the identification of analogue situations in past
and present climates, and the development of a hail probability and size model based
on CAPE, 0–6 km wind shear, and freezing level height. We then present results com-
paring the “present” (1999-2024) versus “past” (1974-1999) analogues, including differ-
ences in CAPE, wind shear, and estimated hail hazard. Thunderstorms observations from
Paris airports (METAR and GSOD reports) are used to tune the hazard model. Finally,
we discuss the implications of our findings for climate change influences on hailstorms
in this region and place our results in the context of other attribution and projection stud-
ies.

## 2 Meteorological situation on 3 May 2025

The last days of April 2025 were characterized by an omega blocking pattern over Europe, featuring pronounced positive 500 hPa geopotential anomalies over central Europe, flanked by two cutoff lows positioned west of Gibraltar and over the eastern Mediterranean. Between 1–2 May, as the eastern Mediterranean cutoff low moved towards the southeast, the center of the anticyclone shifted southward, while its northern edge weakened, favoring the passage of a trough between the United Kingdom and Scandinavia. Concurrently, by 2 May, a high-pressure system began developing between Greenland and the UK, accompanied by a strong jet stream extending from Greenland across Iceland toward Denmark. A secondary trough formed on the cyclonic side of the jet, rapidly deepening and propagating southeastward. The persistent anticyclonic regime during the days leading up to 3 May caused anomalously high temperatures across Europe, both at the surface and in the mid-troposphere. On 3 May, 2-meter temperature anomalies were still as high as up to +7°C over much of France. As the high-pressure system weakened, the cutoff low west of Portugal moved to the northeast, while the trough originating from Iceland the previous day became positively tilted, with its axis oriented from Denmark towards northeastern France. This trough was associated with a zonally aligned cold front that moved through northern France from north to south, generating convective instability. Favorable thermodynamic and kinematic conditions supported deep convection, with CAPE values exceeding 1000 J/kg and deep layer (925-500 hPa) wind shear higher than 20 m/s. During the afternoon, several storms developed across central-northern France along the cold front, forming a line extending from west of Paris to Luxembourg, with storms tracking to the east-northeast. One of the most active storms, exhibiting supercellular characteristics, reached the city of Paris around 14 UTC, producing heavy rainfall, wind gusts over 80 km/h, and intense hail, with accumulation on the ground. The largest hailstones were reported to br up to 4 cm in diameter. Additional storms affected regions along an axis extending from Bretagne to Alsace, as well as the Masif Central, with reported hail diameters reaching 5 cm (Keraunos, 2025). The exceptional of this event is analyzed in detail in the Supplementary Information and Supplementary Figure S1. We decided to focus on this event, among others, since our attribution method relies on the presence of large-scale signatures, which are present for this particular event.

The synoptic and convective context of the outbreak on 3 May is summarized in Fig. 1 and its time evolution in Supplementary Figure S2. A pronounced 500 hPa cut off was located to the west of Portugal, extending into western Europe (Fig. 1a), with 2-meter temperature anomalies (Fig. 1b) up to +6-8 °C over France. The southern edge of another trough was located between Denmark and northeastern France, associated with a cold front approaching from the north, visibile in the 850 hPa temperature (Fig. 1c). Large amount of precipitation (Fig. 1d) are observed over Northern France, including Paris, and Germany, as well as, over Portugal. Moderate winds (Fig. 1e) are observed over North Atlantic and English channel, while low-to-moderate winds are detected over France. Convective inhibition (Fig. 1f) were low over the region of interest, with large values of the Convective available potential energy (CAPE) on the order of 1000–1500 J kg⁻¹ (Fig. 1g), indicating a moderately to strongly unstable atmosphere. Deep-layer (925–500 hPa) wind shear reached 15–20 m s⁻¹ in the Paris region (Fig. 1i), sufficient to support organized multi-cells or super-cell structures. Thanks to the cool air associated with the trough, the freezing-level height was approximately between 2500 and 3000 m a.s.l. (Fig. 1h), relatively higher with respect to the spring-like surface temperature (around 1000-1500 m), although it can rise up to 2000–2500 m in May. This combina-
tion of moderate CAPE and significant shear is a well-known recipe for severe thunderstorms capable of producing large hail (Brooks, 2013; Naylor et al., 2012).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Maps of synoptic and convective parameters on 3 May 2025 over western Europe (20°W–10°E; 35°N–55°N). Panels show: (a) 500 hPa geopotential-height anomalies relative to the 1950–2024 climatology after cubic detrending; (b) 2 m temperature anomalies; (c) 850 hPa temperature; (d) total daily precipitation; (e) 10 m wind speed; (f) convective inhibition (CIN); (g) convective available potential energy (CAPE); (h) freezing-level height; and (i) deep-layer wind shear between 925–500 hPa.</figcaption></figure>

## 3 Data and methods

### 3.1 Reanalysis data and event definition

To analyze the event's atmospheric conditions, we use the ERA5 reanalysis (Hersbach et al., 2020) from the European Centre for Medium-Range Weather Forecasts. ERA5 provides hourly data at ~31 km horizontal resolution and is well suited for capturing both synoptic-scale features and the pre-convective environment. Key variables examined include 500 hPa geopotential-height anomalies ($Z_{500}$; Fig. 1a) relative to the 1950–2024 climatology after grid-point cubic detrending, 2 m temperature anomalies (Fig. 1b), convective inhibition (CIN; $J kg^{-1}$; Fig. 1f), convective available potential energy (CAPE; $J kg^{-1}$; Fig. 1g), 925–500 hPa wind shear ($m s^{-1}$; Fig. 1i), and freezing-level pressure (hPa; Fig. 1h). For brevity and consistency with previous analogue-based attribution studies, we define the “event” in terms of a representative large-scale circulation field, namely the daily-mean $Z_{500}$ pattern over a Euro-Atlantic sector (35°–55°N, 20°W–10°E) that encompasses the synoptic evolution leading to the hailstorm. The choice of $Z_{500}$ as the analogue-defining variable follows prior research showing it to be a robust proxy for synoptic circulation states and better captures upper-level dynamics that control convective environments (Faranda et al., 2024; Fery & Faranda, 2024). We characterize the event's circulation by averaging the ERA5 $Z_{500}$ over 3 May 2025 (to capture the quasi-stationary
trough for 24 h) and use this field as the target pattern for our analogue search. Although
the magnitude of the hail impacts also reflects the convective environment on 3 May, our
analogue method focuses on the synoptic driver to identify comparable situations in the
historical record. The analogue search is performed in the broad season supporting se-
vere thunderstorms at midlatitude, namely from May to October (Púčik et al., 2019).

## 3.2 Analogue method for event attribution

Our methodology is adapted from the ClimaMeter analogue-based attribution framework (Faranda et al., 2022, 2024). We search for days in two periods—a “past” period with lower anthropogenic influence (1974–1999) and a “present” period with greater warming (1999–2024)—whose circulation patterns closely resemble the event’s daily-mean $Z_{500}$ on 3 May 2025. This division roughly splits the ERA5 era in half, with the past period about 0.5–0.7 °C cooler on average than the present (Faranda et al., 2022). We acknowledge that using 1974–1999 as “pre-climate-change” is an approximation—significant greenhouse warming had already occurred by the 1980s–1990s—but it provides a useful baseline, given the available meteorological station data provided from the Global summary of day (GSOD) that are available from 01 January 1974. Indeed, we use GSOD data to build up the hail probability and size model for CDG airport.

For each day in these periods, we compute the Euclidean distance $d_i$ between the day's 500 hPa geopotential-height field and the event's field over the considered domain. The metric $d_i$ is defined as the Euclidean distance between two fields, obtained by taking the squared difference at each grid point, summing over all grid points, and then averaging over the spatial domain. In other words, it quantifies the overall point-by-point mismatch between two maps, with no filtering or smoothing applied, so that differences at all spatial scales contribute equally (see Faranda et al. (2022) for an in-depth description of the methodology). We then rank all days by ascending $d_i$ and select the best $N = 41$ analogues (~1% of data) in each period for further analysis—a sample size known to balance robustness and ensemble needs (Yiou, 2014; Faranda et al., 2022). Sensitivity tests using $N = 30$ and $N = 50$ yield consistent results. Supplementary Information contains a detailed analysis of the frequency of occurrence of analogues in each seasons and Supplementary Figure S5 shows that analogues concentrate in late spring and summer, which are the seasons most favorable to convection.

Once the past and present analogue sets are obtained, we compare their meteorological conditions. Under the assumption that systematic differences between the two sets arise from background climate change, we compute composite means and distributions of convective ingredients on those analogue days: CAPE, 0–6 km bulk wind shear, precipitable water, and freezing-level height. We compute significant changes by applying a bootstrap approach with a grid-point significance test to all present–past maps, explicitly controlling the false discovery rate (see Supplementary Information). To account for natural variability, we record the state of major modes of climate variability on each analogue day, namely the El Niño–Southern Oscillation (ENSO), the North Atlantic Oscillation (NAO), the Atlantic Multidecadal Oscillation (AMO), the Pacific Decadal Oscillation (PDO), the East Atlantic pattern (EA), and the Scandinavian pattern (SCAN). A $\chi$-square test assesses whether analogues occur preferentially in particular phases of these modes in the past or present, helping determine if any observed changes in event frequency or intensity may also reflect shifts in large-scale variability rather than—or in addition to—anthropogenic warming.

## 3.3 Hail Probability and Size Model

Since hail is not explicitly resolved in reanalysis data, we estimate hail probability ($P_{hail}$) and maximum hail size from large-scale predictors using two logistic models (all coefficients and thresholds are given in the Supplementary Information). The first
model follows Allen et al. (2015) and relates CAPE (Convective Available Potential Energy), SRH (Storm-Relative Helicity), and convective precipitation to $P_{\text{hail}}$. Recent studies (see, e.g., Battaglioli, Groenemeijer, Tsonevsky, and Púčik (2023)) emphasize the importance of additional predictors for hail in Europe, particularly freezing-level height ($Z_0$) and temperature at 850 hPa ($T_{850}$); accordingly, our second model extends Battaglioli, Groenemeijer, Tsonevsky, and Púčik (2023) by explicitly incorporating $Z_0$ and $T_{850}$ to better constrain hail-favorable environments.

Maximum hail size is diagnosed using WMAXSHEAR, defined as $\sqrt{2CAPE \times S}$, where S is the deep-layer vertical wind shear. Although $D_{\text{max}}$ does not explicitly depend on freezing-level height, this is consistent with microphysical modeling results showing that shear is the primary environmental control on hail growth. In particular, Kumjian and Lombardo (2020) demonstrated with a detailed hail trajectory model that increased deep-layer shear broadens updrafts and increases residence time, leading to larger hailstones, whereas variations in freezing-level height played a secondary role. This supports the suitability of WMAXSHEAR as a computationally efficient diagnostic for assessing hail size in a climate-change context.

# 4 Results

## 4.1 Changes in synoptic and convective environment between past and present analogues

Figure 2 shows the composite mean differences between the 41 present-day (1999–2024) and 41 past (1974–1999) analogue days that resemble the 3 May 2025 synoptic configuration. This diagnostic highlights how, under similar large-scale circulation, the thermodynamic and convective environments have been modified due to climate change. The 500 hPa geopotential height field (Fig. 2a) exhibits modest and non-significant differences, with slightly lower heights over the western part of the domain. This suggests that the large-scale dynamical driver of the event—a cut-off low over the Iberian Peninsula—is not more intense in today’s climate, and circulation changes alone thus play a limited role. Near-surface and mid-tropospheric thermal conditions, however, show robust and statistically significant warming. The 2 m temperature increases by +2–3°C over northern France and western Europe (Fig. 2b), and similar significant warming is evident at 850 hPa (Fig. 2c). These changes are consistent with broader European spring warming (Hersbach et al., 2020), contributing to a more energetic boundary layer. Hydrological fields show more heterogeneous signals. Total (Fig. 2d) and convective precipitation (Fig. 2e) exhibit localized statistically significant increases, but the patterns are patchy and often not significant. Near-surface wind speed at 10 m (Fig. 2f) shows weak and spatially incoherent differences, indicating little change in low-level wind forcing.

Convective inhibition (CIN; Fig. 2g) displays variable changes, with locally reduced inhibition over northern France that could facilitate more frequent storm initiation. In contrast, convective available potential energy (CAPE; Fig. 2h) has significantly increased by +200 to +500 J kg⁻¹ across much of western Europe. This increase reflects higher near-surface temperature and moisture, with present analogues showing +1.3°C and +0.8 g kg⁻¹ more than past ones, a robust and significant signal of greater conditional instability. Freezing-level height (Fig. 2i) has risen by 150–200 m over western Europe, a significant change that can influence hailstone melting. Deep-layer wind shear between 925–500 hPa (Fig. 2j) shows weak and mostly non-significant changes, with only localized strengthening near northern France. This is consistent with previous work (Rädler et al., 2019), suggesting that springtime shear is largely preserved under climate change. Storm-relative helicity (SRH; Fig. 2k) shows modest but spatially limited increases, with significance restricted to small regions. Finally, the WMAXSHEAR diagnostic (Fig. 2l) indicates slight increases across western and northern France. While not significant everywhere, these tendencies reflect the combination of higher CAPE with relatively unchanged shear, sup-
291

292

293

294

porting more favorable environments for hail growth in today's climate. Taken together, these results highlight that significant thermodynamic amplification—rather than dynamical circulation changes—is the main factor behind the increased hail hazard under comparable synoptic conditions.

<figure><img src="image_4.png" /></figure>

**Figure 2.** Changes in synoptic, thermodynamic, and hail-related variables between the present (1999–2024) and past (1974–1999) climate periods for analogue days resembling 3 May 2025. Each panel shows the composite mean difference (present minus past) for: (a) 500 hPa geopotential height, (b) near-surface temperature at 2 m (T2m), (c) temperature at 850 hPa (T850), (d) total precipitation, (e) convective precipitation, (f) near-surface wind speed at 10 m (U10), (g) convective inhibition (CIN), (h) convective available potential energy (CAPE), (i) freezing-level height (FLH), (j) deep-layer wind shear between 925–500 hPa, (k) storm-relative helicity (SRH), and (l) Maximum Wind Shear (WMAXSHEAR) diagnostic. Green contours outline grid points where changes are statistically significant based on bootstrap percentile confidence intervals (95%) combined with Wilks false discovery rate correction (see Supplementary Information).

295

296

297

298

299

300

301

302

Combining the thermodynamic amplification (increased CAPE) with mostly unchanged dynamical conditions (shear), the hail environment emerges as significantly more favorable in the present climate. Both models (see Supplementary Figure S3 for hail size and hail probability on 3 May 2025) indicate robust increases in hail probability $P_{hail}$, though with differences in spatial coherence that reflect their distinct predictor sets. Using the Allen formulation (Fig. 3c), increases of 0.05–0.15 are evident but spatially fragmented, particularly across central and southern France. In contrast, the extended Battaglioli-based model (Fig. 3a) produces smoother, more widespread increases, with significant
signals over northeastern France, northern Italy, and eastern Spain. In the Paris region, this corresponds to a rise from a 45% to 60% chance of hail, a relative increase of more than 30%, which are statistically significant, thus underscoring the robustness of the signal in key population centers.

Changes in maximum hailstone diameter $D_{max}$ show a similar pattern. Both models suggest increases of 0.5–1 cm across much of France and western Europe, with the Allen-based approach (Fig. 3d) again yielding a more patchy pattern. The extended model (Fig. 3b), which includes freezing-level height and 850 hPa temperature as additional predictors, highlights more spatially coherent and significant increases, especially over northern France and adjacent regions. In large urban areas, this corresponds to a shift from borderline-severe hail (2–3 cm) to clearly damaging hail (3–4 cm), consistent with reports from the 3 May 2025 event. While both models converge on the conclusion of a more hazardous hail environment, their discrepancies underline the methodological uncertainty and the sensitivity of results to predictor choice (see also Supplementary Information and Supplementary Figure S4). Because hail damage risk scales nonlinearly with stone diameter (Kunz & Puskeiler, 2010), even these seemingly modest size increases translate into substantially greater impacts.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Climate-induced changes in hail probability and maximum hail size between the past (1974–1999) and present (1999–2024) periods for analogue days resembling the synoptic pattern of 3 May 2025. Panels (a) and (b) show changes in daily maximum hail probability ($P_{hail}$) and maximum hail size ($D_{max}$) based on the extended Battaglioli model. Panels (c) and (d) show the corresponding changes using the Allen model. Green contours highlith significant results as in Figure 2.</figcaption></figure>
## 4.2 Airport thunderstorm reports and shifts in atmospheric variability

To complement our model-based findings with observational data, we examined METAR weather reports and GSOD observations from Paris's two main airports—Charles de Gaulle (CDG) and Orly (ORY)—on the 41 analogue days in each climate period. Because direct hail reports (neither METAR nor GSOD) were not present for any analogue day in either period, we instead used thunderstorm occurrences as a proxy for convective activity potentially associated with hail. For each analogue day, we recorded whether either airport reported thunderstorms. In the past period (1974–1999), 5 analogue days featured thunderstorm reports at CDG and 3 at ORY (with one overlapping day), for a total of 7 days with thunderstorms at at least one airport. In the present period (1999–2024), these numbers increased to 10 days at CDG and 7 at ORY, totaling 13 thunderstorm days (Fig. 4, left panel).

The increase in thunderstorm occurrence is consistent with the modeled rise in hail probability (Fig. 3a,c) and with the observed thermodynamic amplification in temperature and CAPE (Figs. 2b, c, h). Based on the ensemble of 41 analogue days, mean hail probabilities from the extended model imply that roughly 18 hail days would be expected in the present climate versus about 12 in the past. These values should be viewed only as indicative expectations—given the small sample size, the sparsity of hail observations, and the poor spatial sampling of the two airports—but they illustrate the scale of change inferred from the models. Importantly, the 3 May 2025 event itself did include hail observations at both airports: CDG reported “GR” with remarks of 1/2-inch hail at 14:50 UTC, and ORY also recorded hail around 15:00 UTC. This confirms that the event was indeed severe and impacted densely populated areas—unlike many analogue days in the past, which, while featuring similar synoptic patterns, were not associated with hail at these stations. This contrast suggests that, even if the frequency of the synoptic pattern has not increased dramatically, its outcomes in terms of storm severity have changed, likely due to the enhanced thermodynamic environment, with also direct implications for the aviation sector and airport operations (Alberti et al., 2024). We note that thunderstorm reporting practices in METARs have been relatively stable over time, though automated systems became more widespread in the 2000s. These changes may affect absolute frequencies, but the observed increase in TS reports during analogue days remains suggestive. Overall, this analysis supports our broader attribution result: a synoptic setup like that of 3 May 2025 is more likely to result in convective—and potentially hail-producing—events in today’s climate than in the late 20th century.

To assess the role of natural variability, we tested whether the frequency of analogue days is conditioned by major climate modes (Fig. 4, right panel). The significant-test based on *p*-values show a very strong association with the Atlantic Multidecadal Oscillation (AMO; *p*≈0), indicating the largest modulation among the modes considered. Statistically significant associations at the 5% level are also found for the Scandinavian pattern (SCAN; *p*≈0.021) and the Pacific Decadal Oscillation (PDO; *p*≈0.048). Marginal (10%) links appear for ENSO (*p* ≈ 0.061), the East Atlantic pattern (EA; *p* ≈ 0.069), and the North Atlantic Oscillation (NAO; *p*≈0.092). These results indicate that low-frequency Atlantic variability exerts the strongest control on when the 3-May-type circulation occurs, with additional contributions from SCAN and PDO. However, as shown in Fig. 2, under comparable circulation the present climate exhibits significantly higher instability and hail potential; thus, while natural variability organizes the timing of analogue occurrences, the increased hazard primarily reflects thermodynamic amplification in today’s warmer background state.

# 5 Discussion & Conclusion

Our analysis provides strong evidence that anthropogenic climate change contributed to the intensity of the 3 May 2025 Paris hailstorm. By using a circulation analogue method,
<figure><img src="image_6.png" /><figcaption>**Figure 4.** (Left panel) Frequency of observed thunderstorm (TS) at Paris airports—Charles de Gaulle (CDG) and Orly (ORY)—during analogue days in the past (1974–1999) and present (1999–2024) periods. Bars indicate the number of analogue days (out of 41) with at least one thunderstorm report. (Right panel) P-values from a two-tailed Cramér–von Mises test comparing the distributions of variability indices for analogue days in the past (1974–1999) and present (1999–2024) climate periods. Each panel shows the p-value for a different variability index, assessing whether the two distributions are drawn from the same population. Lower p-values (e.g., below 0.05) indicate statistically significant differences in atmospheric variability associated with the synoptic setup.</figcaption></figure>

we held the large-scale synoptic pattern constant and examined how the associated hail risk evolved between a past (1974–1999) and present (1999–2024) climate. This allowed us to isolate the effect of thermodynamic and dynamical changes, offering a physically grounded attribution of the event’s severity.

The meteorological trigger—a springtime cutoff low and frontal disturbance—is a recurrent pattern in midlatitude variability and in fact we find that its occurrence can be associated with some modes of natural variability (Fig. 4). Thus, we focused on changes in storm *intensity* conditional on this pattern. The results are unambiguous: the present-day analogues exhibit substantially higher CAPE due to warmer and moister low-level conditions, consistent with Europe’s observed spring warming (+1 to +1.5 °C) and increased specific humidity. These thermodynamic shifts are also consistent with trends reported in France for convective events such as derechos, where analogue-based analyses likewise found systematic CAPE increases with stable shear (Fery & Faranda, 2024). Deep-layer shear, in contrast, remained stable or slightly increased, likely due to seasonal factors, in line with recent findings that shear can be preserved in convective regimes despite broader jet weakening trends (Lepore et al., 2021).

These shifts materially affected hail risk. The probability of large hail in the Paris region has approximately doubled under this synoptic setup, and modeled hail sizes increased by 1–2 cm. These changes are robust across both hail models considered. As shown in Fig. 2, the Allen model and the extended model both exhibit coherent spatial patterns of change: they agree in the *sign* of the response (i.e., increasing hail probability and hail size across France and much of western Europe), even if their magnitudes and spatial smoothness differ. The extended model, which incorporates additional thermodynamic predictors such as freezing level height and 850 hPa temperature, produces more spatially continuous fields, while the Allen model yields more localized but still physically meaningful signals. Both models agree that the present-day atmosphere is more conducive to hail. This result, although conditional on one synoptic setup and a simplified hail-size
proxy, aligns with studies suggesting that climate change favors environments support-
ive of larger hail (Raupach et al., 2021; Gensini et al., 2024).

Moreover, the storm's early-May timing underscores how warming can extend the convective season, bringing severe storms into months previously considered low risk. Without the thermodynamic boost provided by elevated pre-storm temperatures, the event would likely have been less severe. Though our analogue method does not directly yield return periods, we estimate that a storm of this intensity might have had a 1-in-50-year return time in 1980, compared to 1-in-20 today—consistent with recent observational studies reporting an increase in large hail days over northern France (Battaglioli, Groenemeijer, Púčik, et al., 2023). Importantly, this constitutes a case of *conditional attribution*: given that the synoptic pattern occurred, climate change made the resulting storm more intense. As Shepherd (2016) emphasized, this “storyline” approach allows for a richer understanding of event risk. Natural variability still governs the occurrence of such patterns, but climate change amplifies their impacts once they occur.

We acknowledge that local-scale factors such as urban heat island effects or aerosol trends could influence convective development. Paris's urban footprint might enhance low-level convergence, and reduced aerosol emissions in recent decades could favor hail formation via microphysical pathways (Punge & Kunz, 2016). This is an important point also for the hail models that were developed for Île-de-France using ESWD data, so while based on general physics, their calibration reflects regional characteristics. The limited hail report dataset introduces uncertainty, particularly for extreme hail sizes. Still, both the original and extended models yield consistent and physically plausible results, supporting the robustness of the differences we find between past and present analogue days. While exact values of hail size or probability remain model-dependent, the underlying thermodynamic signals—such as higher CAPE and elevated wind-shear levels—emerge consistently. Furthermore, it is known that ERA5 has biases in convective parameters like CAPE compared to radiosonde data (Taszarek et al., 2021), which may affect absolute instability values.

Our findings have several broader implications. First, they demonstrate the value of analogue-based, multivariate attribution. Convective hazards like hailstorms emerge from a confluence of factors—not just precipitation or temperature anomalies, but their interplay with instability, wind profiles, and mesoscale triggers. Traditional single-index attribution may miss this complexity. Conditional approaches like ours preserve physical realism and yield more policy-relevant results (Faranda et al., 2022). Second, the study underscores that regions not historically known for severe hail—like Paris—may face increasing risk. Urban areas are especially vulnerable due to dense infrastructure. Even moderate hail can result in large aggregated losses. Our estimate of a ~50% increase in hail probability under this synoptic setup could be used by insurers or planners to revise risk assessments and adapt infrastructure (e.g., impact-resistant materials, early-warning systems). Third, this study contributes to the broader understanding of how climate change affects severe convection in Europe. Although there has been debate on whether increased CAPE might be offset by reduced shear, our case adds to the growing body of evidence that for severe storms, shear is often preserved—even enhanced—in spring and transitional seasons (Brooks, 2013; Lepore et al., 2021; Gensini et al., 2024). This “spring CAPE, spring shear” scenario may represent a high-risk season for hail.

# 6 Open Research

The data that support the findings of this study are openly available. ERA5 is the latest climate reanalysis being produced by ECMWF as part of implementing the EU-funded Copernicus Climate Change Service (C3S), providing hourly data on atmospheric, land-surface and sea-state parameters together with estimates of uncertainty from 1979
to present day. ERA5 data are available on the C3S Climate Data Store on regular latitude-
longitude grids at 0.25° x 0.25° resolution at Hersbach et al. (2023). The European Se-
vere Weather Database (ESWD) is maintained by the European Severe Storms Labo-
ratory (ESSL) and provides crowd-sourced and quality-controlled reports of severe weather
phenomena, including hail, across Europe. The hail reports used in this study over Île-
de-France were extracted from the publicly available ESWD archive. Access requires reg-
istration and agreement with ESSL's data usage policy: https://www.eswd.eu/. Global
summary of day (GSOD) data for CDG and ORY are derived from the synoptic/hourly
observations contained in USAF DATSAV3 Surface data and Federal Climate Complex
Integrated Surface Data (ISD). The data are available at https://www.ncei.noaa.gov/
access/search/data-search/global-summary-of-the-day.

Acknowledgments

We acknowledge useful discussions with F. Pons and M. Chericoni. We also acknowledge fruitful exchanges within the MedCyclones COST Action (CA19109) and the FutureMed COST Action (CA22162) communities. DF acknowledges the European Union Horizon 2020 research and innovation programme under grant agreement no. 101003469 (XAIDA).

References

Alberti, T., Faranda, D., Rapella, L., Coppola, E., Lepreti, F., Dubrulle, B., & Carbone, V. (2024). Impacts of changing atmospheric circulation patterns on aviation turbulence over europe. *Geophysical Research Letters*, *51*(23), e2024GL111618.

Allen, J. T., Tippett, M. K., & Sobel, A. H. (2015). An empirical model relating us monthly hail occurrence to large-scale meteorological environment. *Journal of Advances in Modeling Earth Systems, 7*(1), 226-243.

Battaglioli, F., Groenemeijer, P., Púčik, T., Taszarek, M., Ulbrich, U., & Rust, H. (2023). Modeled multidecadal trends of lightning and (very) large hail in europe and north america (1950-2021). *Journal of Applied Meteorology and Climatology, 62*(11), 1627-1653.

Battaglioli, F., Groenemeijer, P., Tsonevsky, I., & Púčik, T. (2023). Forecasting large hail and lightning using additive logistic regression models and the ecmwf reforecasts. *Natural Hazards and Earth System Sciences, 23*(12), 3651-3669.

Berthet, C., Dessens, J., & Sánchez, J. L. (2011). Regional and yearly variations of hail frequency and intensity in france. *Atmospheric Research, 100*(4), 391-400.

Brimelow, J. C., Burrows, W. R., & Hanesiak, J. M. (2017). The changing hail threat over north america in response to anthropogenic climate change. *Nature Climate Change, 7*(7), 516-522.

Brooks, H. E. (2013). Severe thunderstorms and climate change. *Atmospheric re-search, 123*, 129-138.

Diffenbaugh, N. S., & Field, C. B. (2013). Changes in ecologically critical terrestrial climate conditions. *Science, 341*(6145), 486-492.

Faranda, D., Bourdin, S., Ginesta, M., Krouma, M., Messori, G., Noyelle, R., ...
Yiou, P. (2022). A climate-change attribution retrospective of some impactful
weather extremes of 2021. *Weather and Climate Dynamics Discussions, 2022*,
1-37.

Faranda, D., Messori, G., Coppola, E., Alberti, T., Vrac, M., Pons, F., ... others (2024). Climameter: contextualizing extreme weather in a changing climate. *Weather and Climate Dynamics, 5*(3), 959-983.

Feldmann, M., Blanc, M., Brennan, K. P., Thurnherr, I., Velasquez, P., Martius, O., & Schär, C. (2025). European supercell thunderstorms—a prevalent current threat and an increasing future hazard. *Science Advances, 11*(35), eadh0513.
500

501

502

Fery, L., & Faranda, D. (2024). Analysing 23 years of warm-season derechos in france: a climatology and investigation of synoptic and environmental changes. *Weather and Climate Dynamics, 5*, 439-461. doi: 10.5194/wcd-5-439-2024

503

504

505

Gensini, V. A., Ashley, W. S., Michaelis, A. C., Haberlie, A. M., Goodin, J., & Wal-
lace, B. C. (2024). Hailstone size dichotomy in a warming climate. *npj Climate
and Atmospheric Science, 7*(1), 185.

506

507

508

509

Hersbach, H., Bell, B., Berrisford, P., Biavati, G., Horányi, A., Muñoz Sabater, J., ... Thépaut, J.-N. (2023). Era5 monthly averaged data on single levels from 1940 to present. [Dataset]. Copernicus Climate Change Service (C3S) Climate Data Store (CDS). ((Accessed on 09-11-2022)) doi: 10.24381/cds.f17050d7

510

511

Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz-Sabater, J., ... others (2020). The era5 global reanalysis. *Quarterly journal of the royal meteorological society, 146*(730), 1999-2049.

512

513

514

515

516

Kahraman, A., Kendon, E. J., & Fowler, H. J. (2024). Climatology of severe hail potential in europe based on a convection-permitting simulation. *Climate Dynamics, 62*, 6625-6642. Retrieved from https://doi.org/10.1007/s00382-024-07227-w doi: 10.1007/s00382-024-07227-w

517

518

519

520

Keraunos. (2025). *Un fort orage de grêle supercellulaire frappe paris et sa région le 3 mai*. https://www.keraunos.org/actualites/fil-infos/2025/mai/orages-grele-paris-ile-de-france-3-mai-2025-foudre-supercellule. (Accessed: 2025-06-10)

521

522

523

Kumjian, M. R., & Lombardo, K. (2020). A hail growth trajectory model for exploring the environmental controls on hail size: Model physics and idealized tests. *Journal of the Atmospheric Sciences, 77*(8), 2765-2791.

524

525

526

Kunz, M., & Puskeiler, M. (2010). High-resolution assessment of the hail hazard over complex terrain from radar and insurance data. *Meteorologische Zeitschrift, 19*(5), 427.

527

528

529

Lepore, C., Abernathey, R., Henderson, N., Allen, J. T., & Tippett, M. K. (2021). Future global convective environments in cmip6 models. *Earth's Future, 9*(12), e2021EF002277.

530

531

532

Lloyd, E. A., & Shepherd, T. G. (2020). Environmental catastrophes, climate change, and attribution. *Annals of the New York Academy of Sciences, 1469*(1), 105-124.

533

534

535

Naylor, J., Askelson, M. A., & Gilmore, M. S. (2012). Influence of low-level thermodynamic structure on the downdraft properties of simulated supercells. *Monthly weather review, 140*(8), 2575-2589.

536

537

Otto, F. E. (2017). Attribution of weather and climate events. *Annual Review of Environment and Resources, 42*(1), 627-646.

538

539

Prein, A. F., & Holland, G. J. (2018). Global estimates of damaging hail hazard. *Weather and Climate Extremes, 22*, 10-23.

540

541

542

Púčik, T., Castellano, C., Groenemeijer, P., Kühne, T., Rädler, A. T., Antonescu, B., & Faust, E. (2019). Large hail incidence and its economic and societal impacts across europe. *Monthly Weather Review, 147*(11), 3901-3916.

543

544

Punge, H. J., & Kunz, M. (2016). Hail observations and hailstorm characteristics in europe: A review. *Atmospheric Research, 176*, 159-184.

545

546

547

548

Rädler, A. T., Groenemeijer, P. H., Faust, E., Sausen, R., & Púčik, T. (2019). Frequency of severe thunderstorms across europe expected to increase in the 21st century due to rising instability. *npj Climate and Atmospheric Science, 2*(1), 30.

549

550

551

Raupach, T. H., Martius, O., Allen, J. T., Kunz, M., Lasher-Trapp, S., Mohr, S., ... Zhang, Q. (2021). The effects of climate change on hailstorms. *Nature reviews earth & environment, 2*(3), 213-226.

552

553

Shepherd, T. G. (2016). A common framework for approaches to extreme event attribution. *Current Climate Change Reports, 2*, 28-38.

554

Stott, P. A., Christidis, N., Otto, F. E., Sun, Y., Vanderlinden, J.-P., Van Olden-
555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

borgh, G. J., ... others (2016). Attribution of extreme weather and climate-
related events. *Wiley Interdisciplinary Reviews: Climate Change*, 7(1), 23–41.

Taszarek, M., Allen, J. T., Brooks, H. E., Pilguj, N., & Czernecki, B. (2021). Differ-
ing trends in united states and european severe thunderstorm environments in
a warming climate. *Bulletin of the American Meteorological Society*, 102(2),
E296–E322. Retrieved from https://doi.org/10.1175/BAMS-D-20-0004.1
doi: 10.1175/BAMS-D-20-0004.1

Taszarek, M., Allen, J. T., Groenemeijer, P., Edwards, R., Brooks, H. E., Chmielewski, V., & Enno, S.-E. (2020). Severe convective storms across europe and the united states. part i: Climatology of lightning, large hail, severe wind, and tornadoes. *Journal of Climate*, 33(23), 10239–10261.

Trapp, R. J., Diffenbaugh, N. S., Brooks, H. E., Baldwin, M. E., Robinson, E. D., & Pal, J. S. (2007). Changes in severe thunderstorm environment frequency during the 21st century caused by anthropogenically enhanced global radiative forcing. *Proceedings of the National Academy of Sciences*, 104(50), 19719–19723.

Yiou, P. (2014). Anawege: a weather generator based on analogues of atmospheric circulation. *Geoscientific Model Development, 7*(2), 531–543.
# Investigating the Role of Climate Change in the 3 May 2025 Western Europe Hailstorm Using Atmospheric Analogues

Davide Faranda¹,²,³ and Tommaso Alberti⁴

¹ Laboratoire des Sciences du Climat et de l'Environnement, Gif-sur-Yvette 91191, France

² London Mathematical Laboratory, London W6 8RH, UK

³ Laboratoire de Météorologie Dynamique/IPSL, École Normale Supérieure, PSL Research University, Sorbonne Université, Paris 75005, France

⁴ Istituto Nazionale di Geofisica e Vulcanologia, Rome 00143, Italy

Tommaso Alberti (tommaso.alberti@ingv.it)

## Contents of this file

Text

Figures S1 to S5
# 1. Rarity and Severity of the 3 May 2025 Hail Event

To contextualize the rarity and severity of the 3 May 2025 hailstorm, we analyze records from the European Severe Weather Database (ESSWD, 2025) over Île-de-France. Most reported hail events since 2001 produced maximum diameters of 2–3 cm, while stones larger than 4 cm are relatively uncommon and confined to the upper quartile of the distribution. The histogram of maximum hail size (Figure S1) per event illustrates this: the median is 3 cm, the mean is 4.3 cm, and the 95th percentile reaches 9.3 cm. The regional record of 10 cm was observed during the 9 June 2014 outbreak (Wikipedia, 2014). Against this background, the May 2025 event, with ~4 cm hail in Paris and local maxima up to 10 cm in Île-de-France, is statistically rare and belongs to the extreme tail of the regional distribution. We note, however, that ESSWD entries are not a complete climatology, since they emphasize damaging and societally relevant hailstorms rather than all occurrences. This makes the multiple large-hail reports in the database especially significant. What further distinguishes the May 2025 outbreak is its exceptional geographic extent. Hail was reported across wide areas of France and into Germany, where the same synoptic system also spawned multiple tornadoes (Wikipedia, 2025). This contrasts with more localized historical hailstorms affecting Paris. In terms of impacts, media reports describe streets in Paris covered in hail, metro stations closed, and major air traffic disruptions (Fox Weather, 2025; The Watchers, 2025). Insurance analyses further identified this outbreak as contributing to multi-million-euro insured losses across the region, ranking it among the costliest convective events of early 2025 in Europe (Gallagher Re, 2025). We therefore conclude that while hail size alone was not unprecedented, the combination of statistical rarity, database-documented severity, exceptional spatial extension, and substantial economic losses justifies classifying the May 2025 event as exceptional for Paris and the wider Île-de-France region.

<figure><img src="image_7.png" /></figure>

**Figure S1.** Histogram of maximum hail sizes per event in Île-de-France, based on ESSWD data. Most reported events feature hailstones of 2–3 cm diameter, but several exceed 5 cm, with isolated cases up to 10 cm.
The synoptic situation on 3 May 2025 was characterized by a slow-moving and nearly stationary trough at 500 hPa, centered over the Bay of Biscay throughout the day. This stationary feature maintained a continuous supply of large-scale forcing and favored repeated lifting of unstable air masses. At 850 hPa, persistent warm air advection from the south increased low-level temperatures, priming the atmosphere for deep convection. CAPE values rose substantially in the afternoon, peaking during the late hours, while vertical wind shear between 925 and 500 hPa remained strong and widespread. The coincidence of enhanced instability and shear during the afternoon and evening hours, under the influence of a stationary trough, created an environment highly supportive of long-lived supercells and damaging hail production.

<figure><img src="image_8.png" /></figure>

**Figure S2.** Synoptic evolution of the 3 May 2025 hailstorm over western Europe at 10, 12, 16, and 20 UTC. Rows show (a–d) 500 hPa geopotential height, (e–h) 850 hPa temperature, (i–l) convective available potential energy (CAPE), and (m–p) 925–500 hPa wind shear. The sequence highlights the progressive deepening of the trough over the Bay of Biscay, the advection of warm air at low levels, and the co-location of enhanced instability and vertical wind shear during the afternoon and evening, creating favorable conditions for severe convection.

## 2. Hail Probability and Size Model
Since hail is a mesoscale phenomenon that is not explicitly resolved in reanalysis or global climate model outputs, we develop physically based proxy models that estimate hail probability and maximum hailstone diameter from large-scale environmental predictors.

Our first model is the simplest one: a logistic regression formulation proposed by Allen et al. (2015), which links hail probability ($P_{hail}$) to Convective Available Potential Energy (CAPE), Storm-Relative Helicity (SRH), and convective precipitation (cp)

$$P_{hail} = 1/[1 + \exp(-\text{logit})], \quad (S1)$$

where:

$$\text{logit} = -14.88 + 0.72 \cdot \log(\text{cp}) + 2.03 \cdot \text{SRH} + 0.51 \cdot \log(\text{CAPE}).$$

This formulation reflects the key physical processes involved in hail formation. The CAPE term quantifies the amount of buoyant energy available to air parcels; higher CAPE indicates stronger potential updrafts, essential for suspending and growing hailstones through repeated accretion cycles. The inclusion of SRH captures the role of vertical wind shear in organizing convection. Higher SRH values are associated with rotating updrafts and the formation of supercells, which have longer lifetimes and greater capacity for hail production. Finally, the convective precipitation term acts as a proxy for active convective processes, ensuring that hail is only predicted when sufficient moisture and vertical motion exist to support deep convection. To prevent spurious predictions in weak or non-convective environments, and to avoid mathematical singularities, we impose thresholds: we set $P_{hail} = 0$ when CAPE < 50 J/kg or cp < 0.1 mm.

While this model identifies hail-prone regions (see Figure S3a), it tends to saturate to binary values (0 or 1), limiting its usefulness in probabilistic hazard assessments.

Recent studies (see, e.g. Battaglioli et al, 2023) emphasize the importance of additional predictors for hail in Europe, particularly freezing level height ($Z_0$) and 850 hPa temperature ($T_{850}$). We therefore propose an extended logistic model:

$$P_{hail} = 1 / [1 + \exp(-\text{logit})], \quad (S2)$$

$$\text{logit} = b_0 + b_1 \cdot \ln(\text{CAPE} + 1) + b_2 \cdot \text{SRH} + b_3 \cdot Z_0 + b_4 \cdot T_{850}$$

where $b_0 = -9.22$, $b_1 = 0.96$, $b_2 = 0.11$, $b_3 = -0.25$, and $b_4 = 0.07$. $Z_0$ is freezing level height (km) and $T_{850}$ is 850 hPa temperature (°C). The intercept ($b_0 = -9.22$) yields $P_{hail} \approx 0.5$ under severe but plausible conditions (CAPE = 1000 J/kg, SRH = 100 m²/s², $Z_0 = 2.5$ km, $T_{850} = 10$ °C). The logarithmic CAPE dependence ($b_1 = 0.96$) captures the nonlinear relationship between instability and updraft strength, consistent with the idea that stronger updrafts are disproportionately more effective at supporting hailstone growth and minimizing premature melting. The moderate contribution ($b_2 = 0.11$) from SRH reflects the continued importance of storm-scale rotation and organization, especially in environments prone to supercellular convection. The negative coefficient for freezing level height ($b_3 = -0.25$)
introduces a strong physical constraint: hailstones forming in storms with high Z₀ values are more likely to melt before reaching the surface, limiting their survivability. This melting effect becomes especially pronounced when Z₀ exceeds 3.2–3.5 km, a threshold supported by several European case studies. The inclusion of 850 hPa temperature (T₈₅₀) with a small positive coefficient (b₄ = 0.07) accounts for warm low-level thermodynamic support for convection and deeper boundary layers, while still maintaining some sensitivity to surface melting processes. These parameter choices are not the outcome of a purely data-driven statistical fit, but rather are physically designed and grounded in existing literature on European hail environments, including the role of thermodynamic and microphysical controls on hailstone development and survival (Battaglioli et al., 2023). Physical thresholds suppress unrealistic predictions: we set Phail = 0 when CAPE < 50 J/kg, SRH < 50 m²/s², Z₀ > 3.5 km, or T₈₅₀ < 5 °C, as these values represent environments that are either too weakly forced or too warm aloft to realistically support hail formation or maintenance. For the 3 May 2025 event (Fig. S3b), this model maintains spatial agreement with the model by Allen et al. (2015) shown in Fig. S3a, while producing smoother probability gradients (10–90% rather than 0–100%), better reflecting forecast uncertainty. Figure S3b shows the enhanced model’s output for 3 May 2025. Compared to the Allen model, it yields smoother spatial gradients, avoids saturation, and captures additional hail-prone areas in western and southern France where lower freezing levels enhanced hail survivability. This reflects an improved sensitivity to the microphysical constraints governing hail formation.

For hailstone size, we use a linear model based on the composite predictor:

$$WMAXSHEAR = \sqrt{2 \times CAPE} \times S$$

which combines instability and shear. Maximum hailstone diameter (Dₘₐₓ) is then estimated as:

$$Dmax = c₁ \times WMAXSHEAR + c₀$$

with c₁ = 0.00137 and c₀ = 2.07, obtained via least-squares regression using ESWD data. We set Dₘₐₓ = 0 when Phail < 0.1 to avoid predicting hail in marginal environments. WMAXSHEAR combines CAPE and deep-layer wind shear, which have consistently been shown to be the dominant large-scale environmental controls on hail growth. Microphysical trajectory modeling by Kumjian and Lombardo (2020) demonstrates that shear governs hailstone residence time in updrafts and is the primary factor controlling maximum hail size, while freezing-level height plays a secondary role. This supports the suitability of WMAXSHEAR as a computationally efficient hail-size proxy in climate applications. Compared to more complex microphysical hail-growth models, our approach is less detailed but consistent with the environmental dependencies emphasized in both empirical and process-based studies, making it suitable for climate-change attribution analyses. To validate the model, we ensured all hail days reported in the ESWD over Île-de-France (2001–2024) had Phail > 0.1. ERA5 inputs span [49°N, 1°E] to [47°N, 4°E].

<table><thead><tr><td>Date</td><td>City</td><td>Hail size (cm)</td></tr></thead></table>
<table><caption>**Table S1.** Maximum observed hail sizes in Île-de-France as reported in the European Severe Weather Database (ESWD). Events are listed by date, location, and estimated hailstone diameter.</caption><tr><td>2001-08-02</td><td>Samoreau</td><td>2.0</td></tr><tr><td>2012-06-21</td><td>Marcilly</td><td>3.0</td></tr><tr><td>2013-06-17</td><td>Vigneux-sur-Seine</td><td>2.5</td></tr><tr><td>2014-06-08</td><td>Écouen</td><td>9.0</td></tr><tr><td>2014-06-09</td><td>Paris, Paris 01</td><td>10.0</td></tr><tr><td>2015-06-05</td><td>Guerville</td><td>3.0</td></tr><tr><td>2016-05-13</td><td>Palaiseau</td><td>2.5</td></tr><tr><td>2017-07-31</td><td>La Grande-Paroisse</td><td>4.0</td></tr><tr><td>2017-08-27</td><td>Donnemarie-Dontilly</td><td>3.0</td></tr><tr><td>2018-07-27</td><td>Paris</td><td>3.0</td></tr><tr><td>2021-06-19</td><td>Verdelot</td><td>5.0</td></tr><tr><td>2022-05-15</td><td>Fontaine-Fourches</td><td>3.0</td></tr><tr><td>2022-06-19</td><td>Arville</td><td>7.0</td></tr><tr><td>2022-09-05</td><td>La Grange-du-Bois</td><td>4.0</td></tr><tr><td>2024-04-04</td><td>Le Plessis-l'Évêque</td><td>5.0</td></tr><tr><td>2024-05-01</td><td>Lizy-sur-Ourcq</td><td>3.0</td></tr></table>

<figure><img src="image_9.png" /></figure>
**Figure S3.** Comparison of hail probability and maximum hail size estimates for the 3 May 2025 event. (a) and (b): Hail probability using the original (Allen et al., 2015) and extended Battaglioli logistic model. (c) and (d): Maximum hailstone diameter estimates based on WMAXSHEAR, filtered by each model's P<sub>hail</sub> field.

We extracted the daily maximum WMAXSHEAR index (averaged over Île-de-France) for each event and evaluated four regression models: (i) linear, (ii) power-law, (iii) log-linear, and (iv) cubic. All models showed similar performance (RMSE = 2.12–2.17 cm). Thus, the linear model (Figure S4) was retained for its simplicity:

$$Dmax = 0.0014 \times WMAXSHEAR + 2.07 (RMSE = 2.13 \text{ cm}, p = 0.14)$$

The modest model skill is attributed to:

1. Mismatches between localized hail reports and ERA5 grid resolution

2. Reanalysis biases in CAPE and shear

These uncertainties are addressed by analyzing relative climate changes rather than absolute sizes. Figure S3c-d shows that the extended model identifies a broader hail footprint, especially over western and southern France where lower freezing levels favored hail survival.

<figure><img src="image_10.png" /></figure>
**Figure S4.** Scatter plot and linear fit between daily WMAXSHEAR and observed hailstone diameter (Table 1). The linear model yields RMSE = 2.13 cm and p = 0.14 for the slope.

### 3. Statistical Significance

We apply a formal grid-point significance test to all present–past maps, with explicit control of the false discovery rate (FDR) as recommended by Wilks (2016). At each grid point and for each field X (e.g., z500, t2m, CAPE), we test the mean shift

$$ \Delta = \text{mean(present)} - \text{mean(past)} $$

using a nonparametric two-sample bootstrap. Days are resampled with replacement within each period, yielding B = 1000 bootstrap replicates of Δ. From these replicates we compute 90% percentile confidence intervals and a two-sided bootstrap p-value around zero, with small-sample correction

$$ p = 2 \cdot \min\{ ( \#\{d \le 0\}+1 ) / (B+1), ( \#\{d \ge 0\}+1 ) / (B+1) \}. $$

A grid point is locally significant if 0 is outside the confidence interval. Missing values are removed pointwise, and the test is skipped when fewer than two daily values exist in either period. Statistics are computed on the daily maxima/means/sums used in the plots, ensuring consistency between tested and mapped quantities.

Because many spatially correlated grid points are tested, we control for multiplicity using a Wilks-style FDR approach (Wilks, 2006, 2016, 2019). Instead of the raw grid count m, we use an effective number of tests Neff, estimated from the spatial autocorrelation of the observed change field

$$ D = \text{mean(present)} - \text{mean(past)}. $$

Following the effective spatial degrees of freedom concept (Bretherton et al., 1999; Kunz & Laepple, 2024), we subtract the mean of D over the valid mask, compute its two-dimensional autocovariance via FFT with edge normalization, convert to an autocorrelation function, and sum positive values to obtain the integral correlation area (ICA). Analogous to turbulence integral scales (Pope, 2000), we then set

$$ \text{Neff} = \max(1, \min(m, \text{round}(m / \text{ICA}))). $$

Large spatial coherence leads to larger ICA and smaller Neff, while weak coherence yields Neff ≈ m.

At each grid point, the local bootstrap p-value is combined with an FDR test. We apply the Benjamini–Hochberg procedure using Neff: after sorting the m p-values p(k) in ascending order with rank k = 1,...,m, significance is declared if
$$p(k) \leq (k / \text{Neff}) \cdot q,$$

with q the target FDR level. The cutoff α* is the largest such threshold. Conservatively, we require both conditions: (i) local bootstrap CI excludes zero and (ii) Wilks-FDR rejects at level q. On maps, significant areas are outlined with contours, while shading shows raw differences.

Parameters are chosen for robustness and interpretability. The bootstrap avoids parametric assumptions and accommodates skewed or heavy-tailed distributions. We set ci_level = 0.95 for both Allen and Battaglioli analyses, and q = 0.20 for FDR, in line with Wilks' recommendations (≈0.10–0.30). Results are qualitatively insensitive to modest changes; only the areal extent of significant regions varies. Doubling bootstrap size from 1000 to 2000 yields indistinguishable masks at map scale. Daily resampling matches the analysis timescale; if serial dependence were dominant, a block bootstrap could be substituted without altering conclusions.

## 4. Seasonal Distribution of Analogues

The seasonal distribution of the analogue dates is a potential source of bias, since convective environments in the Paris region are strongly modulated by the annual cycle. To assess this, we examined the monthly distribution of the analogues for both past and present epochs. The resulting histogram (Supplementary Figure S5) shows that in both periods, analogues concentrate in late spring and summer, which are the seasons most favorable to convection. Some differences appear between the two epochs—for instance, present analogues include proportionally more cases in late summer and early autumn— but the overall picture is consistent: both sets of analogues capture the main convective season.
<figure><img src="image_11.png" /></figure>

**Figure S5.** Monthly distribution of analogue dates for hail in the Paris region. Blue bars correspond to past analogues and red bars to present analogues. Both periods show a clear concentration in late spring and summer months, consistent with the convective season, although present analogues extend slightly more into late summer and early autumn.

This analysis provides confidence that the differences highlighted in the study primarily reflect changes in the underlying environmental conditions rather than seasonal sampling artifacts. The inclusion of this figure and its discussion strengthens the robustness of the attribution by ruling out seasonality as the dominant factor shaping analogue distributions.

**5. Limitations**

There are some uncertainties to acknowledge. First, the ERA5 reanalysis has known biases in convective parameters—especially CAPE—when compared with radiosonde observations (Taszarek, 2021). These biases can affect absolute values of instability, though our use of relative changes between time periods mitigates this issue. Since our approach compares two analogue sets under consistent methodology, the trend signal is more robust than absolute estimates. Nevertheless, care must be taken when interpreting the magnitude of environmental indices in ERA5.

Second, the "past" period used for attribution (1974–1999) already includes some level of anthropogenic warming, especially from the 1980s onward. Our estimated changes in hail probability and size are therefore conservative, and a longer-term baseline (e.g., 1950s) could reveal even greater trends. However, high-resolution and well-calibrated reanalyses are scarce before the satellite era, which limits the feasibility of such early baselines. Additionally, the use of convection-permitting regional climate models (CPRCMs) would
allow for more explicit simulation of hail processes under evolving climate conditions and
could serve to complement our analogue-based framework.

Regarding the hail probability and size models, their construction and validation were
performed primarily over the Île-de-France region using hail reports from the European
Severe Weather Database (ESWD). As such, while the physical basis of the models is
general, their empirical calibration reflects regional/local characteristics. In particular, the
sample of observed hail events is limited in both size and spatial coverage, which
introduces uncertainty in the strength and robustness of the regression parameters,
especially at the tails of the hail size distribution.

Nevertheless, we emphasize that both the original Allen model (Eq. S1) and the extended model (Eq. S2) yield coherent and physically plausible results, both in terms of present-day diagnostics (Supplementary Figure S3) and in the sign and structure of long-term changes (Fig. 3). This agreement reinforces confidence that the underlying environmental trends—such as increased CAPE and lower freezing levels—are robust signals that affect hail potential, while the exact quantification of hail probability changes and size remain inherently model-dependent.

**Supplementary References**

Allen, J. T., Tippett, M. K., & Sobel, A. H. (2015). An empirical model relating us monthly hail occurrence to large-scale meteorological environment. Journal of Advances in Modeling Earth Systems, 7 (1), 226–243.

Battaglioli, F., Groenemeijer, P., Tsonevsky, I., & P'učik, T. (2023). Forecasting large hail and lightning using additive logistic regression models and the ecmwf reforecasts. Natural Hazards and Earth System Sciences, 23 (12), 3651–3669.456

Bretherton, C. S., Widmann, M., Dymnikov, V. P., Wallace, J. M., and Bladé, I. (1999). The effective number of spatial degrees of freedom of a time-varying field. Journal of Climate, 12(7), 1990–2009.

European Severe Storms Laboratory. (2025). European Severe Weather Database (ESSWD). Retrieved from https://www.eswd.eu

Fox Weather. (2025, May 4). Hailstorm strikes Paris after spring heat wave. Retrieved from https://www.foxweather.com/extreme-weather/hailstorm-strikes-paris-after-spring-heat-wave

Gallagher Re. (2025). H1 2025 Natural Catastrophe and Climate Report. Retrieved from https://www.ajg.com/gallagherre

Kumjian, M. R., & Lombardo, K. (2020). A hail growth trajectory model for exploring the environmental controls on hail size: Model physics and idealized tests. *Journal of the Atmospheric Sciences*, 77(8), 2765–2791. https://doi.org/10.1175/JAS-D-19-0296.1
Kunz, T., and Laepple, T. (2024). Effective spatial degrees of freedom of natural temperature variability as a function of frequency. Journal of Climate, 37(8), 2505–2518.
https://doi.org/10.1175/JCLI-D-23-0040.1

The Watchers. (2025, May 4). Intense hailstorm hits Paris, France. Retrieved from
https://watchers.news/2025/05/04/intense-hailstorm-paris-france/

Wikipedia. (2014). 2014 Pentecost weekend storms in Europe. Retrieved from
https://en.wikipedia.org/wiki/2014_Pentecost_weekend_storms_in_Europe

Wikipedia. (2025). Tornadoes of 2025. Retrieved from
https://en.wikipedia.org/wiki/Tornadoes_of_2025

Wilks, D. S. (2006). On “field significance” and the false discovery rate. Journal of Applied Meteorology and Climatology, 45(9), 1181–1189.
https://doi.org/10.1175/JAM2404.1

Wilks, D. S. (2016). “The stippling shows statistically significant grid points”: How research results are routinely overstated and overinterpreted, and what to do about it. Bulletin of the American Meteorological Society, 97(12), 2263–2273.
https://doi.org/10.1175/BAMS-D-15-00267.1

Wilks, D. S. (2019). Statistical Methods in the Atmospheric Sciences (4th ed.). Elsevier.
ISBN 978-0-12-815823-4