To consider trends in extremes, the GEV distribution can be made non-stationary by allowing at least one GEV parameter to vary with time: μ(t), σ(t) and/or ξ(t). Owing to the difficulty of estimating the tail parameter ξ, it is assumed stationary, i.e. ξ(t)=ξ0. Three non-stationary models are considered: M1(θ1)θ1=(μ0,μ1,σ0,ξ0)μ(t)=μ0+μ1tσ(t)=σ0ξ0M2(θ2)θ2=(μ0,σ0,σ1,ξ0)μ(t)=μ0σ(t)=σ0+σ1tξ0M3(θ3)θ3=(μ0,μ1,σ0,σ1,ξ0)μ(t)=μ0+μ1tσ(t)=σ0+σ1tξ0

Furthermore, based on previous studies that showed trends in extremes in France emerging in the mid-80s , we consider the case of a trend starting at year t+=1985, leading to the three models M1∗, M2∗, and M3∗. For example, in M1∗: μ(t)=μ0if t≤1985,μ0+μ1(t-1985)if t≥1985.

Figure  illustrates the case of a non stationary GEV with trends in μ and σ starting in 1985 (corresponding to M3∗), with the corresponding GEV densities and associated return level plots for several years.

This gives in total six non-stationary models, in addition to the stationary model denoted (M0), which corresponds to μ(t)=μ0, σ(t)=σ0, and ξ(t)=ξ0. Although the starting year t+=1985 will be hold fixed below, considering starting year a few years before or after does not change the main results of this study (not shown).

For each model, the GEV parameters θ are estimated by maximum likelihood method for each spatial location separately. At each spatial point, we thus have seven estimated models M0, M1, M2, M3, M1∗, M2∗, and M3∗ that we temporarily rename M0 to M6. For the hourly series (1990–2022), because all observations start after 1985, the breakpoint models M∗ are equivalent to the standard linear models M, and the selection procedure defaults to comparing the four models M0, M1, M2 and M3. We apply for each non-stationary model Mj, j≥1, a likelihood ratio test to compare the goodness of fit of Mj to that of the stationary model M0, accounting for their respective number of parameters to avoid overfitting. The test is based on computing the statistics Λj=2(ℓj-ℓ0), where ℓ are the maximum likelihood values. Under the null hypothesis that the maxima are stationary, the statistics Λj follows a χ2 with k degress of freedom, with k equal the difference in the number of parameters between Mj and M0.

Let pj be the corresponding p-value. If pj ≤ 10 %, the model Mj is considered to perform better than M0, otherwise M0 is selected. If several models Mj are preferred to M0, we apply the following procedure: (1) if either M3 or M3∗ are preferred to M0, we select among these two models that with the smallest p-value among p3 and p3∗; (2) otherwise, we select among M1, M1∗, M2, M2∗ that with the smallest p-value among p1, p1∗, p2, p2∗.

This two-step process prioritizes models with simultaneous temporal effects on μ and σ when statistically justified, ensuring complexity is introduced only when it provides plus-value. In the rest of the article, only the selected model is considered at each location (grid point or station). However, for the evaluation of 10-year return levels as a climatological metric, we consistently rely on the stationary model (M0).

For a non-stationary GEV with parameters μ(t), σ(t) and ξ0, the T-year return level is still given by Eq. () but it is now a function RLT(t) of the year t. In this study, we focus on the 10-year return level (RL10) as the primary metric for extreme precipitation. Our daily series (63 years) and hourly series (33 years) provide sufficient data for its robust estimation. As a sensitivity check, trends in RL2 and RL5 show the same regional patterns as RL10 (daily: r = 0.86–0.99 among significant stations; hourly: r = 0.79–0.98), confirming that the detected signals are robust and not driven by GEV fitting uncertainty.

Owing to the linear form of μ(t) and σ(t) in the considered non-stationary models, return levels are linear functions of time in models M1, M2, and M3, and linear only after 1985 in the breakpoint models M1∗, M2∗, and M3∗. In the stationary model M0, return levels are of course constant over time. For the non-stationary models, the relative trend in the T-year return level (%) over the most recent 30-year period (between 1992 and 2022) is computed as RelTrend=RLT(2022)-RLT(1992)RLT(1992).

It is obviously 0 if the stationary model has been selected.

Noting that the T-year return level function can be written as RLT(t)=RLT,0+RLT,1t for models M1, M2, M3 and as RLT(t)=RLT,0if t≤1985,RLT,0+RLT,1(t-1985)if t>1985, for models M1∗, M2∗, M3∗, confidence intervals for the trend coefficient RLT,1 are obtained by profiling the likelihood with respect to RLT,1 . If the 90 % confidence interval of RLT,1 does not contain 0, the T-year return level trend is significant at level 10 %.

At the hydrological year scale, the stationary model is selected for about two-thirds of the stations. For the remaining one-third, significant changes in the GEV distributions are observed, with nearly equal proportions of linear trend model (M) and breakpoint models (M∗). AROME model exhibit similar proportions. At the seasonal scale, the only notable difference occurs in winter, where the linear model is slightly more frequent than the breakpoint model (22 % vs. 13 %). Before analyzing trends in daily return levels, we verified the consistency of GEV-estimated trends in the mean against those obtained via a simple linear regression on observed annual maxima. This preliminary check shows a very strong spatial and quantitative agreement (Pearson correlation r = 0.86 for absolute trends, r = 0.88 for relative trends), confirming the robustness of the statistical fits.

At the daily scale, station-based trends in the 10-year return level (Fig. ) show spatially coherent but region-dependent signals across France over 1959–2022. For the hydrological year (YEAR), we see a clear reinforcement along the Rhône Valley (+5 % to > +30 %) and in the southern Alps (+20 % to +30 %). Elsewhere, trends are weak, of mixed sign, and often not significant, supporting a seasonal breakdown. Autumn (OND) shows a similar pattern in trends as the annual scale, with the exception of the Paris area that shows a decrease (-5 % to -20 %). In winter (JFM), France is cut in three with a diagonal of decrease from the Mediterranean coast to the Atlantic coast (-10 % to -40 %) and mainly increases elsewhere. Spring (AMJ) is characterized by a more spatially uniform positive trend over much of the country, with local increases exceeding +35 %. Summer (JAS), shows predominantly negative trends in the southern half of France, apart in the Rhône valley, with an accentuated decrease along the Mediterranean coast (up to -40 %) and mainly positive trends in the northern half.

AROME reproduces the broad spatial organization of these daily-scale patterns but generally underestimates their amplitudes (negative bias). For the hydrological year, the model partly captures the significant increase in the Southern Alps, albeit with weaker intensities, and does not reproduces the Rhône Valley reinforcement observed in the stations. Autumn is the season when trend patterns are best represented (r=0.40). AROME correctly identifies the positive signal in the Rhône valley and the negative trends in the Paris area, although it barely sees the decrease in the northern Alps. In winter, the model barely retrieves the diagonal of decrease from the Mediterranean to the Atlantic coast and it misses the decrease over the Alps. All the largest trends are anyway less marked. In spring, AROME produces weak and poorly organized signals and fails to reproduce the widespread positive trends measured by the stations. In summer, the model correctly identifies a general negative tendency over the souther half of France but does not see the increase over the Rhône valley and the northern half of France. Overall, AROME moderately captures the large-scale signs and structure of daily return-level trends but tends to underestimate both their intensity and spatial coherence.

Similarly to the daily scale, we verified the consistency of GEV hourly trends by comparing GEV-estimated trends in the mean to linear regression trends of observed annual maxima. Despite the shorter record length and the higher noise level of sub-daily extremes, we find a robust agreement (Pearson correlation r = 0.72 for absolute trends, r = 0.70 for relative trends).

At the hourly scale (Fig. ), station-based trends in the 10-year return level over 1990–2022 are markedly more heterogeneous, noisy, and locally more extreme than their daily counterparts (Fig. ), with significant trends ranging [-50%,+100%] versus [-30%,+30%] at the daily scale. Most of the significant trends are highly positive. Though noisier than at daily scale, these trends are robust in the sense that they are not produced by a single year. Removing the overall maxima of each station and fitting the GEV distributions on these new series gives slightly lower trends for the positive trends and slightly larger trends for the negative ones (Fig. ) – which was expected since the largest value is omitted – but the magitudes and patterns remain close. Over the hydrological year, no clear spatial pattern of trends is found apart in few isolated stations where trends reach ± 50 % and occasionally exceed +100 %. At the hydrological year scale, the stationary model M0 is selected for 74.2 % of the stations, and standard linear trend models (M) for 25.8 % (compared to 75.1 % and 24.9 % for the AROME simulations, respectively). Seasonally, the signals remain highly contrasted. In autumn, a pronounced positive pattern emerges along the Mediterranean coast, locally approaching +100 %. In winter, no clear spatial pattern of trends is found apart at some isolated stations spread over western France, the eastern Pyrenees and the Rhône valley, where strong increases of up to +100 % are found. Spring is the only season when a quite coherent pattern is found. Stations exhibit widespread and very strong increases across much of the country, frequently above +60 % and locally approaching +100 %. In summer, spatial variability reaches its maximum, with trends spanning the full range from -100 % to +100 %, particularly in the Rhône Valley. These patterns reflect both the strong local variability of convective extremes and the limited length of the hourly record, which jointly hinder the emergence of a clear regional-scale climate signal.

AROME only partially reproduces these hourly-scale patterns. Across all seasons, the model simulates weak and spatially inconsistent signals and fails to capture the most pronounced positive trends seen in the station data. For the hydrological year, spatial correlation between AROME and stations remains low (r ≈ 0.12), and seasonal correlations are close to zero for several seasons. Mean biases are substantial and of varying sign, with values reaching ME = -84.7 % in spring and ME = -28.4 % in summer. While AROME occasionally recovers the sign of the observed trends in some regions, it generally does not reproduce their fine-scale spatial organization or amplitude.

The monthly analysis (Fig. ) further highlights the specificity of convective regimes and the strong temporal concentration of some signals. Figure a compares the monthly median GEV relative trends at Météo-France stations and the corresponding AROME grid points (side-by-side subpanels). At the daily scale, station median trends remain moderate throughout the year, ranging from -16.27 % in September to +14.05 % in April/May. In contrast, station-based hourly GEV trends show far larger values concentrated in specific convective windows, with monthly medians reaching +43.28 % in February, +28.26 % in March, +101.86 % in June, +53.88 % in October, and +30.09 % in December. AROME reproduces the positive sign of these monthly trend peaks but systematically underestimates their magnitude. For instance, in June, AROME simulates a median hourly GEV trend of +17.70 % (against +101.86 % for stations), and in February, it simulates +11.46 % (against +43.28 % for stations).

This systematic underestimation is reflected in the monthly mean errors (ME) between the stations and the corresponding AROME grid points shown in Fig. b, with large negative values at the hourly scale during convective months (e.g., ME = -75.6 % in February, ME = -182.0 % in March, and ME = -136.1 % in June), whereas daily mean errors remain much smaller (e.g., ME = -0.9 % in January and ME = 3.1 % in March).

Finally, Fig. c shows that spatial correlation (r) is consistently higher at the daily scale (peaking at r = 0.62 in December) than at the hourly scale. At the hourly scale, spatial correlations are extremely low or close to zero during the convective months (e.g., r = 0.08 in June, r = 0.07 in October, r = 0.04 in March, and r = 0.02 in May), and virtually zero in July and September (r ≈ 0.00), reflecting the high local spatial variability of hourly trends. February stands out as the only month showing moderate spatial agreement (r = 0.45). The detailed monthly spatial patterns of hourly GEV trends are presented in Appendix C (Fig. ).

It should be noted that temperature trends (cf. Fig. ) with the AROME model forced by ERA5 are about one-third weaker than observed trends. This implies a dampening of the Clausius–Clapeyron-related component of extreme precipitation trends, which is expected to influence the magnitude of the trends we can detect.

The weaker simulated temperature trend suggests an internal dampening of the warming signal in France compared to observations. Since aerosols and greenhouse gases evolve realistically in this simulation (Sect. 2.2) and ERA5 reproduces observed temperature trends well , the underestimation likely stems from the model's internal sensitivity to radiative forcings, the absence of interactive vegetation, and/or the lack of temporal evolution of the land use/land cover map.

First, station precipitation measurements are frequently affected by undercatch errors, which are particularly pronounced in cold and mountainous regions with strong winds. Since undercatch typically results in an underestimation of actual precipitation – especially for high-intensity, short-duration convective events–, correcting these errors would increase the observed extreme values. Consequently, the model's underestimation of hourly extremes is likely even larger than what we report here. Part of this underestimation may also stem from the 2.5 km horizontal resolution of AROME: showed that increasing the resolution to 1.3 km results in a better representation of maximum precipitation intensities, with less pronounced underestimation. Furthermore, station time series can also exhibit some inhomogeneities (non-climatic shifts in the time series caused by changes in station location, environment, or sensor types over decades), which could introduce local artificial trend components, although the data used here have been quality-controlled by Météo-France.

Second, raingauges provide measurement over short periods, particularly the hourly data that are available since the 1990s only. This is particularly problematic when studying extremes within a GEV framework, as only one value (the maximum value) is used per year. In some cases, isolated outliers can significantly influence estimates of the return level . For example, in Fig. , some individual points stand out from their surroundings, which is probably due to sampling bias and not to differences in climate. To make the estimation more robust, other solutions include taking into account threshold exceedances within a generalized Pareto distribution (GPD) framework, or using a distribution modeling all precipitation values, such as the recent extended generalized Pareto distribution or the Simplified Metastasis Extreme Value (SMEV) distribution , or using a neighborhood or region-of-influence approach to reduce sampling bias by using more data. All these methods have their advantages and disadvantages. Our general advice is not to interpret trends at an individual point, but to look for consistent trends across regions, to avoid over-interpretation.

Third, the stations are not evenly distributed across the country. In particular, hourly stations are lacking in the Alps, the Pyrenees and, to a lesser extent, the Massif Central. This naturally has an impact on our evaluation measures (correlation and bias) and prevents us from fully evaluating AROME in mountainous regions.

A limitation of the evaluation is that it compares point-scale raingauge measurements with 2.5 km × 2.5 km AROME grid-box averages. This mismatch in spatial support is relatively modest for daily accumulations, but it becomes a major limitation at the hourly scale: a single convective core may affect only part of a grid cell, so that local station extremes can be much higher (or lower) than the corresponding AROME value even when the event is realistically simulated at the grid-box scale. A high-resolution gridded observational product would therefore be better suited to evaluating hourly extremes than individual stations. In this respect, Météo-France’s COMEPHORE product – a 1 km, hourly radar-gauge precipitation reanalysis over metropolitan France – would be particularly valuable . It would have been interesting to include COMEPHORE in this study, but the reanalyses only begin in 1997 and the method for calculating the rainfall fields changed in 2007, which could affect trend estimation.

Another limitation stems from the modeling framework. We have applied a non-stationary GEV framework to estimate trends in extremes. This requires more parameters to be estimated and additional assumptions to be made about the shape of the trends. Here, we have assumed that the distribution of parameters, and therefore return levels, evolve linearly over time. This is obviously a simplification, as (i) there is no physical reason for the evolution to be linear, (ii) there is no direct physical dependence between return levels and time. The linearity assumption could be relaxed by considering non-linear transfer functions (e.g. power functions), with the risk of over-fitting and increased uncertainty if more parameters have to be estimated. Here, as in many other studies based on GEV distributions, we prefer to use a rigid (linear) relationship to increase robustness. Previous studies analyzing the relationship between temperature and precipitation lead us to believe that temperature might be a more appropriate covariate. However, which temperature to choose and at which scale (e.g. local temperature at the time of the event, annual region-mean temperature, sea-surface temperature) is a matter of debate in itself , which is why we chose to keep time as the simplest covariate. Temperature change over the recent past is anyway close to linear in time, so we believe that the magnitude of the trends would be barely affected.

The daily trends derived from Météo-France stations confirm and refine previously documented spatial structures of extreme-precipitation evolution in France. The significant increases in the Rhône Valley and the southern Alps (+5 % to +30 %) are fully consistent with earlier regional analyses reporting an intensification of daily extremes in southeastern France . The marked increases detected in northern France (locally up to +35 %) also agree with national-scale projections indicating a northward-shifted reinforcement of daily extremes under strong warming scenarios . The weak or non-significant trends observed across large parts of western and central France likewise corroborate the high spatial variability highlighted in the IPCC report .

Overall, AROME reproduces the main spatial structures and magnitudes reasonably well when compared with the station-based climatology. The model correctly identifies regions of positive change (southern Alps, Rhône Valley, parts of northern France) and regions with weak or negative signals (western France, the Paris area), indicating an overall ability to capture the pattern of daily return-level evolution. However, AROME systematically underestimates the amplitude of the trends. This reduced magnitude is consistent with the underestimation of temperature trends in AROME that is about one-third weaker than observed trends (Fig. ).

Unlike the daily scale, no national mapping of hourly 10-year return-level trends has previously been available for France. The present work therefore provides the first nationwide characterization of hourly extreme-precipitation trends, offering new insight into their spatial organisation.

Over 1990–2022, trend magnitudes in hourly extremes frequently reach ± 50 % and can exceed +100 %, with strong seasonal dependence. Coherent monthly signals emerge nonetheless: February trends in the Rhône Valley, March trends along the western Mediterranean arc, November increases in the eastern Mediterranean arc, and widespread June increases across most of France. These patterns, though noisier than daily trends, are robust in the sense that they are not produced by a single year (Fig. ). This indicates the start of an emerging trend in hourly extremes even within the constraints of a relatively short observational period.

AROME reproduces part of the seasonal structure of these hourly signals – concurring on the months associated with enhanced extreme activity (Fig. a) – but fails to capture their spatial organization and magnitude. The model systematically underestimates the magnitude of positive hourly GEV trends, leading to a much larger underestimation of relative trends at hourly than at daily scale (e.g., ME = -136.1 % vs. -1.4 % in June, see Fig. a and b). Furthermore, hourly spatial correlations remain close to zero for most convective months (e.g., r = 0.08 in June and r = 0.04 in March, see Fig. c), reflecting shifts in the simulated convective precipitation zones, especially in mountain regions (such as the Massif Central and the Alps) where summer convective activity dominates. Part of this trend magnitude underestimation is consistent with the underestimation of temperature trends in AROME (Fig. ).

Overall, the hourly results highlight two major conclusions: (i) the station-derived hourly trends presented in this study constitute a new contribution to the climatology of French precipitation extremes, revealing structured yet highly localized patterns; (ii) AROME currently shows limited skill in representing both the magnitude and spatial heterogeneity of hourly return-level trends, even when significant signals are isolated, underscoring the challenges of simulating sub-daily extremes.