Extreme sub-daily precipitation leads to natural hazards such as urban floods, debris flows, and flash flooding. For example, in July 2025, Texas experienced a severe precipitation event, with 130 to 280 mm of rainfall recorded in under 12 h, resulting in flash floods that claimed the lives of over 130 people . Record-breaking intermittent heavy rainfall on the southern Japanese island Kyushu caused floods and landslides, resulting in over 60 deaths . The southern UK town of Boscastle saw a damaging but non-fatal flood in 2004 caused by extreme localised rainfall. In one location, 86 mm of rain was recorded in 1 h . Extreme events such as these are often discussed in relation to anthropogenic climate change . It is difficult to attribute individual extreme events directly to climate change, but they are rarely seen in the historical record and are expected to increase in frequency and magnitude in the future .

The intensification of extreme short-duration precipitation with temperature is assumed to follow the increase of atmospheric water vapour, which follows the Clausius-Clapeyron relationship, pointing to a potential intensification of 7 % °C⁻¹ in ideal atmospheric conditions . However, observational studies have found that extreme precipitation may scale with temperature differently than this theoretical value in various parts of the world . found that scaling rates over the ocean are larger than over the land, and scaling rates are especially low or negative in tropical and dry regions. Dynamic factors, such as orography, changes in the location of the intertropical convergence zone, and large-scale moisture transport mechanisms also impact the scaling rates .

Few methods are currently available that incorporate these rainfall-temperature scaling rates into predictions of rare precipitation return levels e.g.,. Among these is the TEmperature-dependent Non-Asymptotic model for eXtreme return levels (TENAX) proposed by , which requires only rainfall and temperature data and has minimal parameters. Assuming that thermodynamics is the main factor enhancing rainfall convection at short-duration , near-surface air temperature can be used as the primary covariate that explains intensification of rainfall extremes . Hence, assuming that the relation between temperature and precipitation does not change with climate change i.e., it is a physical invariant,, changes in return levels of extreme precipitation can be predicted based on the temperatures at which precipitation events occur, which can be defined according to the predictions of different climate change scenarios. So far, this framework has been tested in the Alps , Switzerland , and Germany . Its predictions were validated using hindcasts, which showed good performance in the majority of the stations.

Practically, TENAX relates the exceedance probability of high magnitudes of precipitation to the temperatures during precipitation using a non-stationary Weibull distribution as motivated by. This distribution is described by a scale parameter and by a shape parameter that determines the tail heaviness of the distribution. The tail heaviness impacts the likelihood of extremes within the distribution. For example, Fig.  shows the application of TENAX to rainfall data at a station in Japan. Allowing the shape parameter to depend on temperature results in the blue line, with larger return levels (Fig. a), while the percentiles in Fig. b show that this dependence causes rainfall rates at higher extreme percentiles to increase faster than those at lower percentiles. So far, the temperature dependence was only included in the scale parameter using an exponential dependence that approximates the Clausius-Clapeyron equation. However, observational studies have suggested that the scaling rate may vary between different percentiles of extremes, which would imply a temperature dependence in the shape parameter of the distribution . For example, found that scaling increased from approximately 5 % °C⁻¹ at the 75th percentile to 7.5 % °C⁻¹ at the 99th percentile. The implications of this dependence are substantial, as it affects the rate of increase with temperature of the rarest and most extreme events. , , and tested a linear dependence on temperature in the shape parameter and found that there was no statistically significant dependence at the stations in the Alps. Conversely, used a similar framework in the United States to find that an exponential relation well approximates the temperature dependence of the shape parameter. Temperature scaling in the shape parameter has also been tested in a generalised extreme value model, with finding that the model is a better fit for rainfall extremes when the shape parameter can vary, but the inclusion of this variation also increases uncertainty.

These contrasting results lead to the question of whether a dependence of the shape parameter on temperature should be included when using this type of model. If there is a dependence, there are three main options for dealing with the parameter that defines the dependence: it can be allowed to vary freely at a particular station, it can be set to a reasonable value based on prior knowledge of the climatology of the region, or it can be neglected (although it exists).

Here, we investigate how the temperature dependence can be estimated from observed records without introducing large errors and uncertainty. Specifically, we answer the following questions: (i) does the shape parameter exhibit a significant dependence on temperature in different regions of the world? If there is a dependence, how does its direction and magnitude vary in space?; (ii) how much of the observed variation in such dependence can be attributed to spatial variability rather than sampling uncertainty?; (iii) does including this dependence influence the time-invariance assumption behind these approaches?; and (iv) does this dependence affect the estimates of return levels? How is this affected by the length of the data record? We conclude by giving some practical recommendations for when and if a dependence of the shape parameter of the Weibull magnitude model should be included in the TENAX-like frameworks.

A full description of TENAX can be found in . Here, we summarise the basics necessary for this paper. First, independent precipitation events are identified. These are defined as the peak precipitation intensities observed over the duration of interest, in our case, 1 h, during an independent storm. To ensure independence, storms are required to be separated by a minimum of 24 dry hours. The temperature value T assigned to an event is the temperature averaged in the 24 h preceding the event. TENAX includes a temperature model g(T), which quantifies the probability of having a precipitation event at a given temperature, and a magnitude model W(x;T), which quantifies the exceedance probability of heavy precipitation magnitudes for events at a given temperature.

For the temperature model, we use a generalised normal distribution with shape parameter β=4: 1g(T)=2σΓ1βexp⁡-T-μσβ. The location parameter μ and the scale parameter σ are estimated using the maximum likelihood method.

TENAX's magnitude model is a non-stationary Weibull distribution, with temperature as the covariate. A Weibull distribution was chosen because this should approximate the tail of heavy precipitation based on atmospheric physics arguments . It is given by: 2W(x;T)=1-exp⁡-xλ(T)κ(T), where λ(T) is the scale parameter and κ(T) is the shape parameter. Since extreme precipitation is expected to scale exponentially with temperature following the Clausius-Clapeyron relationship , an exponential dependence on temperature T is used in the scale parameter: λ(T)=λ0eaT. The shape parameter affects the tail-heaviness of the Weibull distribution, and previous observational studies have suggested that this might also have a dependence on temperature . We investigate both a linear and exponential dependence in the shape parameter, namely κ(T)=κ0+bT or κ(T)=κ0ebexpT. In principle, the linear dependence raises the possibility of a negative shape parameter if temperatures change considerably. However, our analyses showed that temperature changes are not large enough for this to become an issue in real cases. More in general, the exponential dependence prevents this from happening. The parameters in the magnitude model are estimated by left-censoring the values below the 90th percentile of the events, meaning that the values below this threshold are retained for the estimation and treated as non-exceedances see, using a maximum likelihood estimator.

The combination of the magnitude model W(x;T) and the temperature model g(T) using the total probability theorem gives the cumulative distribution function of the event magnitudes: 3F(x)=∫0∞W(x;T)⋅g(T)dT. The cumulative distribution function of the annual maxima emerging from independent samples is then given by G(x)=F(x)n, where n is the average number of events per year . Since this integral doesn't have a practical closed-form solution, we use a Monte Carlo approximation with N=2×105 iterations to approximate Eq. (). Hence, 4GTENAX(x)=1N∑i=1NW(x;Ti)n, where Ti are temperatures sampled from g(T). Return levels are calculated by inverting Eq. () numerically.

Hourly precipitation data were taken from the global sub-daily rainfall (GSDR) dataset . The data lengths vary greatly. For our initial spatial analysis (see Sect. ), we used stations with at least 10 years of complete data, where a complete year is defined as one with less than 10 % missing data. When investigating hindcasts, we only used stations with at least 20 complete years. We focused our analysis on the four countries with the highest number of stations meeting the 10-year completeness criterion: the UK (913 stations), the USA (1365), Germany (696), and Japan (1207).

Hourly temperature data were obtained from the ERA5-land reanalysis dataset . The ERA5-land grid cell closest to the coordinates of each station was chosen. The majority (94 %) of stations had temperature data within the same grid cell. A few stations, such as those on small islands or very close to the coast were not directly within a land grid cell. The average distance between the coordinates of a station and the ERA5-land cell was 4 km. If there were no grid cells with data within 0.5° distance of a station, the station was not used in our analysis.

First, we calculated the parameters of the TENAX model at each station with more than 10 years of complete data using maximum likelihood estimation and a likelihood ratio test to check whether b was significantly different from 0 at the 5 % significance level. The parameters were estimated three times: for both the linear and exponential formulation of the model (“free b”) and for the case when b was set to zero (“b=0”).

The parameters of the TENAX model are estimated from a finite set of events, and as such, there is a stochastic uncertainty caused by the limited sample size. To investigate the stochastic uncertainty and its relation to b, we performed Monte Carlo simulations. For each station in a country, we simulated precipitation and temperature events equal to the number of observed events at that station. We used the average observed parameters over the country and repeated this with b=0 and with b allowed to vary. From these simulated events, we recalculated the parameters, repeating this process five times to reduce sampling uncertainty and obtain robust distributions. Five is an arbitrary number which was chosen as a balance between computation time and stability of the results. The distribution of these parameters represents the expected stochastic spread due to sampling uncertainty. For a summary of the values used to produce the Monte Carlo simulations, see Table S1 in the Supplement. To directly compare the spread of the generated and observed parameters, we calculated the ratio of the interquartile ranges.

As per TENAX's assumptions, the parameters of the magnitude model are required to be time-invariant. This is motivated by the idea that the magnitude model represents the physical relationship between temperature and extreme precipitation which is not expected to change with climate change. We investigated the effects of b on this time invariance in a hindcast setup. We split the time series at each station in half and computed the parameters of the magnitude model in each half separately. To compare each parameter between the two time periods, we calculated the root mean square difference and the Pearson's correlation coefficient in space. We also compare the magnitude model as a whole by using a likelihood ratio test to see if the model is significantly different between the two periods (see , Sect. 3.2, for more details on this test).

Finally, we used Monte Carlo simulations to understand the impact of b on the accuracy of the estimated return levels. We did this by choosing three representative values of b: -0.015, 0, and 0.015 °C⁻¹, and typical values for the other parameters: a=0.07 °C⁻¹, λ0=1 mm h⁻¹, κ0=1.1, μ=10 °C, and σ=13 °C. We created synthetic data sets of precipitation and temperature events, assuming 80 events per year for 10-, 20-, and 30-year record lengths. From the synthetic data, we estimated the parameters with three different techniques: letting b be fitted (“free”), and setting b to the value used to generate (“set”), therefore mimicking a situation in which b is known (for example, in virtue of the local climatology), and setting b=0. We then compared the return levels resulting from these different estimates with the “true” return levels obtained from the original generating parameters. This is repeated 1000 times to get a range of the possible return levels a single distribution can generate using different techniques. We calculated the “multiplicative bias in return levels” as the ratio of the recalculated return levels to the expected return levels.

We find that the shape parameter in TENAX shows a statistically significant dependence on temperature in many locations. The typical value of this dependence (negative b parameter, Fig. 1) indicates that the temperature-precipitation scaling of rarer extremes tends to be higher than that of milder extremes. Including this negative b will lead to larger estimated return levels, especially at longer return periods (Fig. ). It would be beneficial for future work to further explore in detail the physical reasons for the spatial variability of the b parameter. It could be linked to, for example, orography, elevation, or large-scale dynamical drivers. A Bayesian framework for calculating b, and the parameters of TENAX more generally, would also help to understand and reduce uncertainty.

suggest that this dependence should be exponential. We find that the difference between exponential and linear dependence is small, although the exponential dependence seems to cause larger uncertainties in the other parameters of the model. In this concern, it should be noted that use a different statistical setup, in which temperature bins are identified and several parameter sets are estimated without resorting to left censoring. The dependence of the shape parameter on temperature is then assessed ex post. found that the observed scaling in east Australia increased by about 50 % between the 75th and the 99th percentile. For the average values of the parameters in Japan, we find an increase of about 20 % between these two percentiles. In the USA, the scaling rates are smaller, but with the average values, we also find an increase of 50 % between the 75th and 99th percentile.

The apparent dependence of b on location and orography implies that it is accounting in some way for the dynamical changes associated with temperature changes. For example, in some locations, climate change is associated with more intense circulation patterns in convective precipitation and shifts from stratiform to convective regimes, and super Clausius–Clapeyron scaling is observed . Hence, using only the b parameter to represent a vast array of precipitation mechanisms in some locations may be an oversimplification.

We note that some stations in the GSDR dataset had a change in the sampling resolution of precipitation during the observation period. For example, the resolution of some records changed from 2.54 to 0.254 mm h⁻¹. This may affect the calculations of the parameters and the hindcasts. We re-run our analyses excluding all these stations, but no differences in the results could be noticed. Removing stations with less than 20 years of available observations from the hindcasts was more effective in reducing the presence of extreme outliers. This seems to indicate that the data length is more important to the quality of the results than consistency in resolution.

To match temperature values to precipitation, we used a reanalysis dataset. This may not align with the actual near-surface air temperature that one would measure using ground observations, especially on the scale of individual stations. found that there was a high correlation between the observed daily temperatures at stations and the temperatures produced by ERA5-land, especially in the countries we investigate. Although found that ERA5 was better than ERA5-land at representing extreme heat events, we chose to use ERA5-land for its finer resolution. Other products, such as regional ones, may give more accurate temperatures, but would not allow us to compare study areas across the globe. In any case, temperatures are averaged over the 24 h preceding each event, limiting the potential impact of misrepresenting the location of small-scale weather events in ERA5-land. When producing the stochastic simulations of the parameters, we used the temperature model given in Eq. (). In some locations, this may not give an accurate range of temperature values. Given infinite events, the magnitude model and temperature model are independent, so the temperature model used would not affect the parameters of the magnitude model. However, within a finite sample, the limited temperature values may affect the magnitude model.

We investigated the dependence of heavy hourly precipitation tail heaviness on temperature using the framework provided by the TENAX model for multiple stations worldwide. We found that in many of the stations the shape parameter of the distribution exhibits a significant dependence on temperature (b≠0). The sign and size of this dependence present a spatial pattern and appear to be linked to latitude, elevation, climatic conditions, and large-scale drivers. Future work could investigate these influences specifically. Monte Carlo simulations show that, aside from Germany, the variation within the countries investigated is not attributable to stochastic uncertainty alone. Including b in the magnitude model increases the variability in the other parameters of the magnitude model, but does not make a large difference to whether the magnitude model is invariant in time when performing hindcasts. Calculating b from data at a single station may lead to a large over-estimation of return levels, especially for stations with shorter record lengths. Using an exponential dependence rather than a linear one somewhat mitigates against this, but yields larger uncertainties in the remaining model parameters.

With these conclusions in mind, we recommend the following. If only short record lengths are available (between 10 and 20 years), b should not be estimated from the individual stations alone, since the uncertainty introduced may severely bias the estimation of the return levels and of the other parameters in the magnitude model, leading to inconsistent projections. At the same time, b should not be neglected completely, since it is certainly non-zero in many regions, and its exclusion leads to an underestimation of return levels. Having prior knowledge about this dependence would largely improve the estimates. In ideal conditions, b should be estimated in the context of surrounding stations, with care to ensure the values are physically reasonable. Setting b to a reasonable, well-supported, non-zero value, such as the average value within an adequately identified region, would be a viable option to reduce these uncertainties while retaining the physical meaning of this parameter. When no prior information on b is available, setting it to zero remains the recommended option.