Data collected during three measurement campaigns with devices operated as part of the Hydrology Meteorology and Complexity laboratory TARANIS observatory (exTreme and multi-scAle RAiNdrop parIS observatory) of the Fresnel Platform of École nationale des ponts et chaussées (https://hmco.enpc.fr/portfolio-archive/fresnel-platform/, last access: 24 July 2026) are used. Summary information for each measurement campaign can be found in Table , and locations in Fig. .

The first campaign, called ENPC-Campus, takes place on the roof of the Carnot building located on the campus of ENPC. In this paper, we use the rainfall data (only the rain rate in mm h⁻¹) measured with the help of a Parsivel² disdrometer manufactured by OTT, and temperature obtained with the help of a sensor by Campbell Scientific. Both devices provide data with 30 s time steps. The corresponding data base, references presenting the devices, description of the campaign, as well as complete samples of data can be found in . The rainfall and temperature time series used in this paper for the second part of ENPC-Campus campaign are displayed in Fig. as an illustration of the studied data in this paper.

From November 2016 to September 2017 the instruments were moved to SIRTA (Site Instrumenté de Recherche par Télédétection Atmosphérique) on the Ecole Polytechnique campus for a joint intensive measurement campaign over the Ile-de-France region, where Paris is located. The site is about 38 km away from ENPC campus towards south west of Paris. This campaign is denoted SIRTA in this paper, and is much shorter than the other two campaigns, meaning that the associated results are less robust.

The last measurement campaign used in this paper took place at a wind farm operated by Boralex and located at Pays d'Othe (name of the campaign), approximately 120 km south east of Paris in a slightly rolling area. As for the other campaigns, a Parsivel² disdrometer with 30 s time steps provided rainfall data. Temperature data were collected with the help of a mini meteorological station manufactured by Thies Clima and operated with a sampling rate of 1 Hz. Temperature data was upscaled to 30 s time steps to match the resolution of the rainfall data. The devices were installed on a meteorological mast at a height of 45 m. Complete description of the campaign and samples of data can be found in .

The whole rainfall and temperature time series used in this paper, from each measurement campaign, can be accessed in .

Measurements of rainfall with the help of optical disdrometers are subject to instrumental limitations, notably calibration issues and influence of wind (see for a review). Observed deviations can go up to roughly 10 % , and some correction scheme (when wind data is available) have recently been suggested . Addressing these issues and the influence of the associated uncertainty on the multifractal analysis carried out would be an interesting topic, but it remains outside the scope of this paper. Here, we can just mention that the portion of the rainfall time series with heaviest events are less affected by wind induced biases because they are associated with a greater portion of large drops which are less affected by these biases. Hence we expect a limited influence of these biases on UM analysis, which should be confirmed by dedicated analysis.

In a first step, analyses are carried out up to synoptic scale, which corresponds to the typical duration of a large scale meteorological situation with typical temporal extent of up to 10 d and spatial extent of up to 1000 km. This scaling regime is called “large scales” in the rest of the paper.

More precisely, for each measurement campaign, the whole time series of rain rate and temperature are split into successive samples of 215 time steps, which corresponds to roughly 11.4 d. The process is initiated at the beginning of the available data without accounting for potential effects of diurnal variations. The paper focuses on rainfall, so potential snowfall should be removed in order to avoid introducing potential biases. In order to achieve this, all samples for which some “rainfall” was recorded while the temperature for the same time steps was below 4 °C were simply removed from the studied set of samples. Hail is very rare in these locations, and filters for potential hail were not implemented. The total number of samples per measurement campaign can be found in Table S1 of supplementary material. The total length of the studied samples corresponds to roughly 5.4, 0.7 and 2.0 years for the ENPC campus, SIRTA and Pays d'Othe measurement campaigns respectively.

Analyses were also implemented at event scale. To achieve this, rainfall events were selected by considering that a rainfall event is a rainy period of time during which more than 1 mm is collected and that is separated by more than 15 min of dry time steps before and after. More precisely, in order to identify them, a rainy time step found after a dry period of at least 15 min is considered as the start of a potential event. The end of the latter is the beginning of the next 15 min dry period. If the collected amount of rainfall is greater than 1 mm in between, then it is considered as a rainfall event. The number of events per measurement campaign can be found in Table S2 of supplementary material. As for the large scale analysis, events for which some “rainfall” was recorded while the temperature for the same time steps was below 4 °C were simply discarded (see supplementary material for numbers of events kept in analysis), in order to avoid potential biases associated with snowfall.

The selected rainfall events do not all have the same duration. Hence, a sample length is set for further joint analysis of the events. For technical reasons (see next section), it must correspond to a power of 2 of number of time steps. Longer sample lengths enable the study of rainfall across a wider range of scales, getting more robust results; but they impose to discard shorter events. Shorter sample lengths enable to keep a maximum of rainfall events; but limit the robustness of the results, being obtained across a more limited range of scales. As a trade-off, a sample length of 128 time steps corresponding to 64 min (∼ 1 h) was used. Then, in order to study the maximum possible data, the process illustrated in Fig. was implemented for each event: (i) the maximum number of sub-events, i.e. non overlapping samples of size 128, was computed. It is equal to the integer part of the number of time steps of the event divided by 128 (ii) In order to study as much rainfall as possible, the portion of length equal to the product of the number of samples times 128 with highest cumulative depth was found within the time series of the event. Since disdrometer measurements were originally given at drop scale, there is a unique maximum. (iii) This selected portion was finally split into sample(s).

In this paper, variability and ultimately extremes of the rainfall times series across various ranges of scales, were quantified in the framework of Universal Multifractals (UM). Here, only the key elements are reminded and interested readers are referred to a recent review by and references therein for more details.

To introduce the framework, let us consider a conservative (i.e. its average does not change with scales) 1D time series ϵλ at a resolution λ. It is defined as the ratio between the outer scale (Tout) and observation scale (Tobs), i.e. λ=Tout/Tobs. For multifractal 1D time series, the moment of order q of the series exhibits a power law relation with regard to the resolution: 1ϵλq≈λK(q) where K(q) is the scaling moment function. The notation 〈.〉 denotes the average over the time series. It can be shown that, in an equivalent way, the probability of exceeding a scale dependent threshold (λγ) defined with the help a scale-invariant singularity γ, also scales with the resolution as: 2Pr(ϵλ≥λγ)≈λ-c(γ) where c(γ) is the codimension function . The values of multifractal 1D time series behave as λγ. The singularity (γ) remains the same across scales while the value of the time series (λγ) changes with scales. The functions K(q) and c(γ) fully characterize the variability across scales of the time series ϵλ and are linked through a Legendre transform . This notably means that a singularity can be associated uniquely to each moment and vice-versa. Equations () and () also mean that multifractal properties are statistical properties which are valid on average over numerous samples.

In the specific framework of UM , which are a limit behaviour of all multiplicative cascades processes (see discussion in for more details on this debates), K(q) and c(γ) are characterized with the help of only two parameters with physical interpretation:

C1, the mean intermittency co-dimension, which measures the clustering of the (average) intensity at smaller and smaller scales. C1=0 for a homogeneous field;

Greater values of α and C1 correspond to stronger extremes. For UM, we have: 3K(q)=C1α-1(qα-q).

A Trace Moment (TM) analysis basically consists in checking the scaling behaviour of the time series and estimating K(q) by plotting Eq. () in log-log. To achieve this, the time series is upscaled from its maximum resolution Λ by averaging over adjacent time steps, then raised to various powers q, and finally the ensemble average (over various samples independently upscaled) is performed to obtain an estimate of the empirical moments and their scaling behaviour. UM parameters are estimated with the help of the Double Trace Moment technique which is an extension of TM tailored for UM .

Let us now consider a non-conservative 1D time series, denoted ϕλ, whose average changes with scales, i.e. we have 〈ϕλ〉≠1. In that case, it is usually assumed that it can be written as (with an equality in probability distribution): 4ϕλ=dϵλλ-H where ϵλ is a conservative time series (〈ϵλ〉=1) of moment scaling function Kc(q) (the sub-index “c” refers to the conservativity of ϵλ), and H the non-conservation parameter. Kc(q) only depends on UM parameters C1 and α. H characterizes the scale dependence of the average behaviour, and is equal to zero for a conservative time series. H should not be confused with the Hurst exponent: although both quantify long-range correlations for values greater than zero, the latter lacks a simple general expression for multifractal processes. It is notably not unique and needs to be generalized defining one for each studied statistical moment. H corresponds to the order of the fractional integration (if H>0) or differentiation (if H<0) needed to retrieve the studied process from a conservative one. More details about the link with the Hurst exponent can be found in Appendix A4 of , which is dedicated to this topic.

H is related to spectral analysis. Indeed, for scaling time series, the power spectrum follows a power law with regard to wave number: 5E(k)≈k-β and the spectral slope β is related to H with the help of the following formula : 6β=1+2H-Kc(2)

TM and DTM techniques should theoretically be implemented on a conservative time series ϵλ. However if H is roughly smaller than 0.3–0.4, it can be implemented directly on ϕλ, without substantially biasing the estimates of α and C1. In case of greater H, ϵλ should be used in the TM and DTM techniques. Retrieving ϵλ from ϕλ theoretically requires a fractional integration of order H (equivalent to a multiplication by kH in the Fourier space). A common approximation, which provides reliable results, consists in taking ϵΛ as the absolute value of the fluctuations of ϕΛ at the maximum resolution and renormalizing it .

The insight one can get of a statistical process is limited by the size of the studied sample. For multifractal processes, this will result in a maximum singularity γs and corresponding moment order qs beyond which the values of the statistical estimates of respectively the codimension and scaling moment functions are not considered as reliable e.g.,.

More precisely, let's consider Ns independent samples with a resolution λ. In a d-dimensional space, there are λd values per sample (d=1 for the time series studied in this paper). The maximum singularity (γs) that one can expect to observe in the available samples is defined by: 7NsλdPrϵλ≥λγs≈1

Introducing the notion of sampling dimension ds: Ns=λds (dS=0 for a single sample as it will be the case here), it yields: 8c(γs)=d+ds which enables to estimate γs. For γ>γs one expects that c(γ)=+∞, which means that the estimates of c(γ) will not be reliable. As a consequence of the Legendre transform, the empirical estimates of K(q) become linear for q>qs=c′(γs): K(q)=γS(q-qs)+K(qs). γs quantifies in a scale-invariant way the extremes that can be expected within a time series. The corresponding physical quantity at a given resolution λ is expressed as λγs. It is especially useful to quantify how extremes evolve when α and C1 exhibit different trends. Indeed, γs combines the influence of both UM parameters into a single one. For example, analysed rainfall output of climate models over France and found decreasing trend for α and increasing one for C1. Relying on the notion of maximum observable singularity which combines the effects of both parameters, they showed that rainfall extremes are expected to increase over France in the context of climate change. used it to discuss the concept of maximum probable rainfall. relied on this tool to quantify the impact of rainfall space-time variability on the usefulness of nature-based solutions in urban environment.

In this subsection, we implemented multifractal analyses on large scales, i.e. up to synoptic scale (see Sect. 2.2). In a first step, an ensemble analysis was carried out, that is to say all samples were upscaled independently and used to compute average statistical moments in Eq. () or spectra in Eq. (). Such analysis is used to study general features of scaling.

Let us illustrate the results with the ENPC-Campus campaign. The outcome of spectral analysis, i.e. Eq. () in log-log, is displayed in Fig. a. A good scaling behaviour on scales ranging from roughly 30 min to 11 d is found, and smaller scales are investigated in next the subsection. The spectral slope β is smaller than 1, meaning that across this range of scales the studied time series is conservative and multifractal analysis can be implemented directly on the time series. Trace Moment analysis (i.e. Eq. in log-log) outcome is displayed in Fig. b. The coefficients of determination of the linear regressions are all greater than 0.99 for q>0.5 and we use the one for q=1.5 as a metric. Results confirm the good scaling behaviour across this range of scales.

UM parameters estimated with the help of DTM technique are reported in Table . In the DTM technique, moments ranging from 1.3 to 1.7 (centered around the commonly used value of q=1.5) are tested enabling to provide an uncertainty range on the estimates of UM parameters. It corresponds to the uncertainty arising from the studied ensemble and not to the uncertainty arising from sample variability which is visible in the individual sample analysis below. The uncertainty on the DTM estimates of UM parameters tends to be larger when there are less samples. It remains in all cases smaller than 2.5 % and is very small compared to the differences between sites (and samples in upcoming analysis), so it will not be discussed further. UM parameters values are typical for this range of scales e.g.,. The empirical scaling moment function K(q) derived from TM analysis and the theoretical one plotted using Eq. () and DTM estimates of UM parameters are in excellent agreement as it can be seen visually in Fig. .c. The discrepancies noticeable for q<0.5 are explained by a multifractal phase transition associated with the numerous zeros which are in the time series (see for more details on this phenomenon). Very small values of H (last column of Table ), i.e. smaller than 0.1, corresponding to almost conservative time series, are retrieved for this range of scales.

Same good scaling behaviour was observed for this range of scales for the two other measurement campaigns. Similar UM parameters were retrieved with only limited variations of α and C1 (see Table ) according to the campaign.

In a second step, in order to investigate how UM features are changing with temperature, the average temperature <T> for each sample was computed from the available data. Even if the temperature also exhibits some variability, only the average temperature was considered. Joint multifractal analyses could be envisaged in future studies e.g.. Then, two types of analyses were carried out:

Individual sample analysis.

In the first type of analysis, a UM analysis was implemented on each sample individually using the same range of scales that was identified in the ensemble analysis. Individual samples with bad scaling, i.e. with r2<0.9 for q=1.5 were discarded (see supplementary material for numbers of samples kept in the analysis) from this analysis. Scaling being an average behaviour, it is expected that some samples do not exhibit a good scaling behaviour. And they were incorporated in the ensemble analysis before and the binned one after. Yet, adding their assessed parameters in the individual analysis would not be relevant because they are not reliable so they could bias results. Scatter plots of retrieved UM parameters vs. <T> are displayed in Fig. for ENPC-Campus campaign. Significant scattering is observed for all parameters. Potential overall trends were identified with the help of a linear regression, which corresponds to the simplest potential relationship. It is displayed through the red line on the plots. The quality of the linear regression was quantified with the help of the Pearson coefficient of correlation, denoted r. It ranges from -1 (exact linear relationship with negative slope) to 1 (exact linear relationship with positive slope). Values close to 0 correspond to lower levels of correlation. As a complement, Pearson p-value was computed in order to test the null hypothesis that the distribution of the underlying samples are uncorrelated. A threshold of 0.05 was used to reject the null hypothesis. In addition, Spearman rank correlation was also computed along with the corresponding p-value. Assessed slopes,Pearson coefficients of correlation r and Spearman rank correlation, along with associated levels of confidence are displayed in Table .

The |r| coefficients are low, which is expected given the observed scattering. Similar comments can be drawn from the Spearman rank correlation. It means that the retrieved trends are only valid on average over numerous events, and they are not all statistically significant. For example, for H none of the observed trends for individual campaigns pass the statistical test of significance. Also, in the case of the SIRTA campaign, there are not enough samples to get statistically reliable trends. For the other two campaigns, we observed a decreasing trend for α and an increasing trend for C1. Hence the consequences on extremes were not obvious and γs was needed to combine the effects of both. It appears that γs exhibits an increasing trend with <T>. It means that stronger variability and extremes are retrieved on studied samples with increasing temperature. A slightly decreasing trend is found for β and very slightly decreasing one for H. This suggests that at large scales, the long range correlation of the field tends to decrease with increasing temperature, i.e. that successive rainy periods are less correlated and more independent from one another.

Keeping in mind that scaling is an average feature valid on numerous samples, a binned analysis was performed. It is not just the average values for all the events of a given bin that is considered, but the output of the ensemble analysis over all the samples of the bin. This enables to assess more robust scaling estimates which yields more reliable trends, even if some precision is lost with regard to temperature. More precisely, in the second type of analysis, the temperature range from 2 to 24 °C was split into bins of 2 °C width. Then, an ensemble analysis of all the samples whose average temperature is within a given bin was performed. Given that UM features from the various campaigns were very similar and in order to increase the number of samples to obtain more reliable results, all the available samples from all the campaigns were used. Results are displayed in Fig. in red for the large scales studied in this subsection. First of all, the scaling (r2) is very good for all temperature bins. The total number of samples per bin is not the same (see Table S3 in Supplement), so a weighted linear regression (weights being the number of samples available) was performed to account for this. A larger data set would enable to explore whether this trend is real and associated with changes of precipitation type with temperature. The same trends that were obtained with the previous analysis are retrieved, i.e. α, β and H decrease with <T>, while C1 and γs increase. The obtained trends exhibit much higher coefficients of correlation than in the previous analysis, as expected when aggregating across samples, though it does not independently strengthen the individual-sample evidence.

In this subsection, analyses were carried out at event scale (see Sect. 2.3). The same analysis as for longer samples across large scales were carried out, with ensemble analysis first, to identify scaling behaviour, and then individual sample analysis in two formats.

Results are illustrated with the data from ENPC-Campus campaign. Spectral analysis (Fig. .a) shows that data exhibits a very good scaling behaviour on the whole range of scales from 30 s to 1 h. The spectral slope β is of 1.73 meaning that on this range of scales, the studied time series is not conservative, i.e. there are some strong long range correlations. Hence the analysis is done on the conservative part (see Sect. 3.1). Figure b shows TM analysis. The good scaling behaviour on the whole range of studied scales (30 s – 64 min) is confirmed, with coefficients of determination all greater than 0.99 for q>0.5.

UM parameters obtained via DTM analysis are in Table . These values, around 1.7–1.8 for α and 0.2 for C1, are consistent with those commonly reported in the literature for this range of scales e.g.,. Here we confirm them at even higher resolutions, which were not available in the previous studies. Similarly to the large scale analysis of previous section, the uncertainty on UM parameters estimates is very small compared to the differences observed between measurement campaigns and in between samples as we will see. Hence this will not be further accounted for. As for the results across large scales, there is a very good agreement between the empirical scaling moment function K(q) and the theoretical one. Some differences become visible for q greater than ≈2.3–2.5 which is slightly smaller than the expected value of qs equal to 2.7 in this case.

Very similar qualitative and quantitative behaviour (see Table ) were retrieved on this range of scales for the two other measurement campaigns.

As for the large scales in the previous section, the average temperature <T> over the whole event was assessed for each event, and the same two types of UM analyses on individual events (the sample(s) of a same event are analyzed together) and ensemble of events binned by temperature intervals were carried out.

As before and for the same reasons, the individual events with bad scaling, i.e. with r2<0.9 for q=1.5, were discarded (see supplementary material for numbers of samples kept in analysis). Scatter plots of retrieved UM parameters vs. <T> are displayed in Fig. for ENPC-Campus campaign. Stronger scattering than for the large scales is retrieved. Similarly potential overall trends were computed with the help of linear regressions. Slopes and r and p-value are reported in Table for all measurement campaigns.

In general similar results but with less pronounced trends (smaller |r|) are retrieved for UM parameters with a slightly decreasing α, an increasing C1 and an increasing γs. It means that variability and extremes also tend to increase with temperature over this range of scales. Contrarily to what is observed at large scales, β and H are increasing with temperature, corresponding to a greater non conservativeness of the time series. This suggests that with stronger temperature, the short range correlation within a given rainfall event tends to increase. It should be noted that opposing trends are observed for the dependency of H with regard to temperature for large scales (decreasing trend) and event scale (increasing trend). This suggests that warming is associated with greater within-event temporal coherence but reduced inter-event persistence, a finding that merits further investigation, to understand better which physical processes are involved in each case.

Results for the binned analysis are displayed in Fig. in black. As in the case of the large scales, they basically yield consistent trends with regard to the one discussed in the previous paragraph, with stronger correlations, as expected given the aggregation.

Until now, all the analyses carried out account for events all around the year. In order to explore potential seasonal sensitivity, the same analysis was implemented considering the events for each season separately. The events/samples for which start month are December, January and February were labelled as winter. Similarly, the others were classified in spring (March, April, May), summer (June, July, August) and autumn (September, October, November). Slopes, as well as indicators of the quality of the linear regression of the value of the studied parameters vs. <T> at both large and event scales for all measurement campaigns are displayed in Fig. . During winter, for large scales, <T> for all samples ranged from 7.3 to 11 °C with an average value of 9.0 °C. Those numbers are equal to 8.5, 19 and 14 for spring, 17, 24 and 21 for summer and 6.9, 18 and 13 for autumn.

The “All” column corresponds to the results on average considering all the events of the whole years, which were discussed in the previous two sections. With regard to results according to seasons, it should first be mentioned that they are less statistically reliable, and this is notably due to the fact that the number of samples/events was lower (roughly divided by 4) than when all events were considered at once. Yet, it seems that some trends are visible. For example for γs at large scales, it appears that there is a slight decrease of it with increasing temperature (i.e. a negative slope is found), while the trend discussed in the previous subsection is valid for the other seasons. This could suggest that the temperature signal across all seasons on γs is partly affected by seasonal influence, i.e. that γs covaries with temperature associated regime shifts. This effect is less pronounced at event scale. β also seems to behave differently in winter for large scales (except for the SIRTA campaign for which there is a limited number of samples available). Longer time series would be needed to further confirm these preliminary observations.

In addition to these analyses, sensitivity to rainfall intensity and temperature were tested. More precisely, the same analysis considering only the events or samples with average rain rate belonging to the 10 % upper percentile was carried out. Using the available data, similar results were found, not enabling to detect some differences. An analysis splitting the events between those with average temperature below and above 12 °C also yielded similar results for both subset, not enabling to detect some differences. Studies on more events and other geographical areas would be needed to further investigate this, because other work found a dependency of the increase rate with temperature e.g., Indeed, some authors reported more pronounced trends when considering high or low temperature range or only the upper percentiles of rainfall extremes. Longer time series, enabling the study of more samples/events would be needed to implement this methodology to check whether similar effects are also found in this scale-invariant framework. Further analyses with longer time series and various climate types, would be interesting to carry out, to check for example how the extremes behave depending on rainfall types, i.e. stratiform vs. convective situations for example e.g.,.

The analyses carried out did not aim at determining whether a super-CC or sub-CC is observed, since they focused on variability in general. Yet, given that rainfall extremes can be derived from some multifractal features, it is possible to relate the current results to previous findings available in the literature. More precisely, as discussed in the introduction, numerous studies have reported a scale dependence of the increase of rainfall extremes with temperature, i.e. that the increase is stronger in percentage for shorter durations . Results presented here, provide a framework to interpret this.

In this study we find in general an increase of the scale invariant concept of maximum observable singularity γs of rainfall time series with temperature. γs being related to the extreme rainfall (greater γs corresponds to stronger extremes), one can note that, at least for small scales, it is consistent with previous findings from the literature reporting an increase of rainfall extremes with temperature. It is even possible to go a step further and explore how this increase is expected to change with observation scale. Indeed, the greater rainfall rate (i.e. an indicator of extreme rainfall) that one can expect to observe in a sample at resolution λ behaves as λγs. We remind that λ is the resolution, i.e. the ratio between the outer scale and the observation scale, and that it increases with shorter duration. Hence, an increase of the scale invariant γs with temperature results in greater increase of extreme rainfall in percentage at higher resolutions, i.e. with shorter observation scales. Indeed, this percentage of increase %incr can be written as: 9%incr=100×λγs(T1)-γs(T2)-1 for a change from temperature T1 to T2. Hence the change with temperature in the scale invariant parameter γs provides a framework to explain changes in increase of rainfall extremes with temperature according to scale (mainly from daily to hourly) which were reported in previous studies. Indeed, with γs(T1)-γs(T2) fixed in Eq. () (i.e. the increase of γs with temperature), when λ increases (i.e. shorter durations are considered), the percentage of increase of the expected rainfall extreme rises. Let us consider an illustration with a temperature shift of 2 °C for the large scales. It corresponds to a shift of γs equal to 0.0077×2 (Table ). Tout here is 11 d as shown in the multifractal analysis. At daily scale Tobs is one day. This yields to λ=11/1 which gives an increase of the maximum expected rainfall of roughly 4 %. At hourly scale Tobs is one hour, leading to λ=11/(1/24) and an increase of roughly 9 % on expected maximum rainfall. This shows that relying on a scale invariant framework enables to grasp an understanding valid across a given range of observation scales. This constitutes a novelty with regard to previous studies. It should be mentioned that found a faster strengthening of extremes for shorter duration with climate change using multifractal framework, which is a result consistent with what is reported here, using a different approach.