Scaling Properties of Rainfall as a Basis for Intensity–Duration–Frequency Relationships and Their Spatial Distribution in Catalunya, NE Spain

Abstract

The spatial distribution of rainfall intensity–duration–frequency (IDF) values, essential for hydrological applications, were estimated for Catalunya, Spain. From a larger database managed by the Meteorological Service of Catalunya and after rigorous quality control, 163 high-quality daily series spanning from 1942 to 2016, with an average length of 39.8 years and approximately one station per 200 km², were selected. A monofractal downscaling methodology was applied to derive rainfall intensities for sub-daily durations using the intensities from a reference 24 h duration as the basis, followed by spatial interpolations on a 1 km × 1 km grid. The scaling parameter values have been found to be higher in the northwestern mountainous areas, influenced by Atlantic climate, and lower in the central–western driest zones. A general negative gradient was observed toward the coastline, reflecting the increasing influence of the Mediterranean Sea. The IDF results are presented as spatial distribution maps, providing intensity–frequency estimates for durations between one hour and one day, and return periods between 2 and 200 years, with an estimated uncertainty below 12% for the 200-year return period, and lower for shorter return periods. These findings highlight the need to capture rainfall spatial variations for urban planning, flood control, and climate resilience efforts.

1. Introduction

The relationship between rainfall intensity, its duration, and its frequency at a given location is of paramount importance, not only in the field of meteorology but also in hydrology and civil engineering [1,2]. These variables’ relationship is commonly represented through intensity–duration–frequency (IDF) curves, which serve as a critical tool for understanding and modeling precipitation patterns. IDF curves illustrate the variation in rainfall intensity with respect to different durations and return periods, the latter representing the likelihood of such an event occurring over a given number of years or the annual frequency of exceedance. IDF curves are widely used in the design and management of water-related infrastructure such as drainage systems, dams, and stormwater facilities [3]. The accurate estimation of these relationships is essential for creating infrastructure resilient to extreme rainfall events, thereby reducing the risk of flooding and structural failures. The importance of these curves extends to urban planning, as they ensure the capacity of stormwater systems aligns with the demands of expected rainfall scenarios.

The foundation of modern IDF analysis was laid by pioneering studies [4,5], which introduced the first empirical mathematical relationships to link rainfall intensity, duration, and frequency. Over the decades, these initial formulations have been enhanced through the incorporation of probabilistic models and regional adaptations [6,7,8,9,10,11,12,13,14], providing deeper insights into precipitation behavior and its variability. Some limitations, such as the challenges related to short-duration applications [15,16,17], the assumptions of stationarity [18,19,20], and the omission of precipitation in the form of snow, which can significantly influence hydrological responses in certain regions [21], have also been discussed in the literature. A comprehensive compilation of IDF studies, tracing their evolution from the early foundational work to the near-present advancements can be found in [22].

The IDF curves of a location, $I(t,T)$ can be formulated by means of a generalized $I(t,T)=a\left(T\right)/b\left(t\right)$, where $I$ represents the rainfall intensity, $t$ the duration, and $T$ the return period. The functions $a\left(T\right)$ and $b\left(t\right)$ are mutually independent. Function $a\left(T\right)$ has often been determined empirically, sometimes represented as a linear logarithmic function [8,23,24,25]. A particularly noteworthy approach [10] proposed deriving this function based on theoretical consistency with probability distribution functions suitable for modeling maximum rainfall intensity in the location of interest. Regarding the $b\left(t\right)$ function, the fractal property of scale invariance in rainfall series [26,27,28] offers a particularly convenient framework for its characterization. This property allows $b\left(t\right)$ to be expressed as a power law, with its exponent serving as a simple scaling parameter to be determined. This parameter has been shown to correlate with the local rainfall regime [29,30,31,32], generally aligning with total precipitation but also reflecting the irregularity of the rainfall pattern to which the proportion of convective rainfall contributes.

Beyond serving as a simple descriptor of rainfall pattern characteristics, the primary practical application of the scaling parameter is in temporal downscaling. For example, it can be used to estimate sub-daily rainfall amounts from daily data [16,33,34,35]. This could be particularly useful for temporally downscaling precipitation forecasts from numerical weather prediction models, which typically provide daily outputs. An example of this is the use of simple scaling analysis to derive hourly rainfall from daily projected rainfall series for the 21st century in Barcelona, Spain [36]. In that work, the expected future extreme rainfall was found to increase more for hourly than for daily durations, aligning with studies that identify the increasing frequency and intensity of extreme rainfall events as defining characteristics of altered precipitation patterns in Europe due to climate change [37,38].

Global warming and the potential intensification of the hydrological cycle have raised concerns about changes in extreme precipitation patterns, with significant implications for hydrology and flood risk management [39]. Short-duration extreme rainfall is expected to increase more notably than longer-duration events [40], heightening the risk of flash floods [41] and posing challenges for infrastructure systems. This underscores the need for adaptive infrastructure and flood resilience strategies, as climate change is expected to amplify the frequency and intensity of extreme rainfall events. In some regions, heavy rainfall events are anticipated to become more frequent despite a decrease in total annual precipitation. The Mediterranean basin, and especially the Iberian Peninsula, exemplifies this paradox, with observations of reduced average rainfall in recent decades, yet an increase in extreme precipitation events [42,43,44]. These shifts suggest an evolving precipitation regime, marked by greater irregularity and intensity, which can be examined through scaling analyses. For example, in Andalusia (southern Iberian Peninsula, Spain), despite the methodological challenges posed by its diverse climate, ranging from extremely arid areas to notably wetter regions, an overall increase in rainfall irregularity has been identified [34], possibly due to the climate variations in the areas with Mediterranean influence. Research in Catalunya (northeastern Iberian Peninsula, Spain) [32] identified a temporal trend of increasing rainfall irregularity, reflected by a declining scaling parameter, over much of the 20th and early 21st centuries. Similarly, studies of monthly rainfall data from this region point to a growing irregularity in rainfall patterns, which may correlate with the substantial rise in atmospheric CO₂ concentrations [45].

The present work focuses on representing expected rainfall amounts across the region of Catalunya for a range of frequencies and durations. By constructing spatial maps for different rainfall durations (including hourly and daily scales), we aim to visualize the distribution patterns and gradients that underlie regional precipitation dynamics. These findings not only support infrastructure planning but also contribute to a deeper understanding of regional hydrological behavior in response to meteorological drivers. The maps depicting the spatial distribution, at high temporal and spatial resolution, of intensity–duration–frequency values derived using the methodology outlined in this study, along with the interpolated data for each municipality, are available on the website of the Meteorological Service of Catalunya [46].

2. Materials and Methods

2.1. Rainfall Data

The present analysis utilized rainfall data from meteorological stations managed by the Meteorological Service of Catalunya (Servei Meteorològic de Catalunya, SMC), all of which underwent rigorous quality control [47]. Data series were initially assessed for absolute quality using a customized index designed to identify and address common issues such as digitization errors and encoding inaccuracies. Subsequently, daily rainfall values were subjected to a relative comparison based on factors including proximity, elevation differences, and correlations in measurements. To ensure data reliability, only high-quality series identified in the initial assessment were included in the relative comparison stage. For more detailed information, please refer to [47]. Based on the outcomes of quality control and homogenization checks, including the third generation of the Applied Caussinus—Mestre Algorithm for homogenizing Networks of Temperature series (ACMANT3) for identifying homogeneity breaks in monthly precipitation series [48], a selection of rainfall series was made. ACMANT3 is a software package comprising six programs with a total of 174 sub-routines. Specifically, the ACMANP3month executable was employed [49], enabling the automatic detection and correction of precipitation data at a monthly resolution. The procedure was chosen for its proven performance, complete automation, and specific design for precipitation data.

From the series pre-selected based on the previously mentioned quality and completeness criteria, the final set was determined after passing both the ACMANT homogenization test and the Mann–Kendall rank test. These tests ensured that no detectable trends were present in the monthly precipitation series. This process aimed to ensure spatial uniformity, and preference was given to the highest-quality or longest-duration series. Some series from neighboring regions to Catalunya, provided by the State Meteorological Agency of Spain, AEMet, were included to enhance the accuracy of subsequent spatial analyses. For clarity, it is specified that the station series from the surrounding regions underwent the same quality controls as those from Catalunya. A total of 163 daily rainfall series, each with a minimum of 25 years of data recorded between 1942 and 2016, from the stations shown in Figure 1, were ultimately selected.

To extract the annual maximum 24 h rainfall values from daily series, a subseries of these maxima was created. However, most historical rainfall data have been recorded with a fixed daily start time (e.g., 8:00 a.m. local time), which affects the measurement of true maximum values over unrestricted 24 h intervals [50,51,52,53,54,55,56,57]. To address this, it was recommended to apply a multiplicative correction factor, which adjusts the recorded maxima to approximate values that would be obtained using a sliding 24 h window. Rather than applying the standardized global correction factor of 1.13 proposed by Hershfield [50], we used a region-specific seasonal correction factor calculated for Catalunya, as detailed in [57] and [48] (pp. 68–88), with spring showing an average value of 1.161, summer 1.093, autumn 1.124, and winter 1.139.

To establish the relationship between rainfall intensity and frequency across different durations in a later subsection, annual maximum rainfall series for durations of 48 h, 72 h, and up to 15 days are required; therefore, the same correction procedure proposed in [57] was applied to the aggregated series according to their respective durations. As a result, annual maximum rainfall series for 24, 48, 72, …, and 360 h were generated for each of the 163 selected locations, covering the available years within the 1942–2016 period.

2.2. Statistical Analysis of the Daily Maximum Intensity Series

The process of deriving IDF relationships begins with determining the statistical distribution of the expected maximum rainfall intensity for a reference duration, which in this case is 24 h. This step is essential for calculating intensity values corresponding to each return period of interest, effectively linking intensity to frequency for various recurrence intervals. The annual maximum rainfall series for daily durations for each of the 163 selected locations were initially analyzed using the Generalized Extreme Value (GEV) distribution, where applicable. To address potential challenges and limitations in the statistical treatment of the series, particularly those arising from sampling constraints that could impact the reliability of long-term predictions, alternative statistical functions were employed. The extreme value theorem states that sequences of independent and identically distributed random variables converge to a GEV distribution under suitable conditions [58]. The GEV distribution encompasses three families of extreme value distributions: Gumbel (type I), Fréchet (type II), and Weibull (type III), making it a flexible tool for modeling extremes. For extreme rainfall analysis, the GEV framework provides a robust method to describe the probability of maximum rainfall intensity over given durations.

Nevertheless, the GEV distribution was not always the most applicable option. For some maximum rainfall series, fitting the data to the GEV model resulted in a positive shape parameter. This parameter suggests a minimal increase in expected intensity for very long return periods, which likely does not represent the physical characteristics of rainfall. Instead, it may reflect a sampling limitation, such as the absence of rare, extreme rainfall events in the dataset. It has been suggested that the shape parameter might be influenced by the dominant precipitation mechanisms [59]. Despite these interesting insights, the present study found no discernible relationship between the spatial distribution of the GEV’s shape parameter and any geographic characteristic or rainfall pattern.

When the GEV distribution was not the most applicable option, alternative statistical distributions such as Gumbel (GEV type I), Generalized Normal (GNO), Generalized Logistic (GLO), Generalized Pareto (GPA), and Pearson Type III (PE3) were used to fit the data.

The probability distribution fitting was carried out using the R programming language along with the lmom [60] and lmomRFA [61] packages (versions 3.2 and 3.3). The latter was specifically employed to assess the goodness of fit. L-moments, originally introduced by Hosking [62], are particularly well suited for analyzing maxima series due to their robustness against outliers. These moments were used to evaluate the statistical properties of the annual maximum rainfall series, with a focus on the L-skewness and L-kurtosis parameters. These parameters are instrumental in selecting an appropriate probability distribution function to fit the dataset effectively.

After fitting the annual maximum rainfall series for 24 h durations to an appropriate probability distribution at each location, the maximum rainfall intensities for several return periods were calculated. The selected return periods for estimating the expected maximum intensity were 2, 5, 10, 20, 50, 100, 200, and 500 years. Using these data, spatial interpolation was performed for each selected return period to derive the intensity–frequency relationship for precipitation at high spatial resolution. The interpolation was carried out using ArcGIS software (ArcGIS Pro 2.1) [63], which applied geographically weighted regression (GWR) with altitude as an independent variable to model spatially varying relationships, followed by kriging to correct for residuals. After GWR, the residuals (unexplained spatial variability) were interpolated using kriging, effectively combining the strengths of local regression and spatial autocorrelation modeling. This two-step approach improved the accuracy of predictions by accounting for both the local influences of altitude and residual spatial patterns.

2.3. Simple Scaling Methodology

Annual maximum rainfall intensity is a variable known to exhibit scaling relationships of a statistical nature across different durations [27,28]. These relationships connect the probability distributions of rainfall intensity for a specific duration $t$, denoted as $I_t$, to those of another duration $\lambda t$, expressed as $I_{\lambda t}$, through a scaling factor formulated as a power law of the scale parameter λ. This relationship is mathematically defined in Equation (1), where $K\left(q\right)$ serves as the scaling function. In the case of monofractals, or simple scaling, $K\left(q\right)$ exhibits linear behavior, taking the form $\beta q$, where β represents the scaling parameter mentioned in the Introduction Section. For multifractal systems, $K\left(q\right)$ assumes a non-linear form [27,29,64].

$$I_{\lambda t}\stackrel{dist}{\overbrace{=}}{\lambda }^{K\left(q\right)} I_t$$

(1)

The symbol $\stackrel{dist}{\overbrace{=}}$ in Equation (1) indicates a probabilistic equivalence between the two distributions, one on each side of the equality [65]. Consequently, their statistical moments, quantiles, and other statistical characteristics maintain equivalence under this relationship. For statistical moments of order $q$, the rainfall intensity for a duration $t$, $〈I_t^q〉$, defined by Equation (2), satisfies the simple scaling relationship expressed in Equation (3).

$$〈I_{\lambda t}^q〉={\displaystyle \frac{\sum _{i=1}^n{I}_{t_i}^q}{n}}$$

(2)

$$〈I_{\lambda t}^q〉={\lambda }^{\beta q} 〈I_t^q〉$$

(3)

The annual maximum rainfall series for 24, 48, 72, …, and 360 h, previously generated for each of the 163 selected stations, are used to determine the scaling parameter $\beta$ at each location by performing a linear regression of the logarithmic values of the statistical moments with the logarithm of $t$. The straight lines obtained, with a slope of $\beta q$, confirm scale invariance.

By definition, $\beta$ values are constrained to exceed the limit of −1, corresponding to a scenario where a single intense rainfall event occurs in isolation within the dataset, such as a solitary daily peak of amount $P_1$, surrounded by dry periods. Aggregating these data over $n$ days results in $I_n=I_1/n$, yielding $\beta =-1$ for $q=1$. Conversely, for uniformly distributed rainfall over consecutive days, where intensities remain constant across durations, $\beta =0$. In practical observations, daily rainfall series typically yield $\beta$ values ranging from approximately −0.5 to −1. For longer durations, such as monthly aggregates, smoother distributions result in higher $\beta$ values [45].

The scaling parameter $\beta$ correlates with the regularity of rainfall patterns. Regions with consistent rainfall generally exhibit higher $\beta$ values, while areas characterized by irregular precipitation, such as arid zones with occasional intense rainfall episodes, display values closer to the lower limit of −1 [29,30,31,32].

The scaling analysis conducted in this study was applied to the 163 selected rainfall series by calculating the q-order statistical moments of rainfall intensities for the aggregated series spanning 1 to 15 days. A linear regression was then performed between the logarithmic values of these moments and the logarithm of the duration $t$ for each value of q in order to demonstrate scale invariance and determine $\beta$ at each station. This process enabled the interpolation of $\beta$ across Catalunya using kriging methods, facilitating comparisons with the region’s diverse climatic zones.

2.4. Downscaling for Sub-Daily Intensity Estimations

The equality of the quantiles of the probability distribution given in Equation (1) implies that these quantiles are also governed by the same scaling relationship. Specifically, this scaling relationship also applies to quantities with certain characteristics, such as rainfall intensity for a specific return period, since the return period is determined by the statistical function that describes the variable. If the statistical function remains the same, the intensity values at two different timescales, adjusted appropriately by the scaling factor, will be identical for the same frequency. Therefore, this applies to the extreme rainfall intensity $I(t,T)$, with return period $T$ and duration $t$, which is linked to the IDF curves, and the rainfall intensity with the same frequency but for a different duration. Building on this, once the scaling exponent $\beta$ has been determined, daily rainfall intensity values for a given return period $T$, $I(24, T)$ can be used to estimate the IDF values for sub-daily durations $t$, $I(t, T)$. This process assumes that the simple scaling relationship holds for sub-daily durations, which should be regarded as an approximation. The equation used to obtain these values, derived from Equation (3), is shown in Equation (4).

$$I\left(t, T\right)={\left({\displaystyle \frac{t}{24}}\right)}^{\beta }I(24, T)$$

(4)

Expression (4) will be crucial to downscale the 24 h rainfall distribution maps for several return periods, calculated from the statistical analysis of the daily maximum intensity series, to maps for sub-daily durations and the same return periods.

3. Results

3.1. Statistical Analysis of the Daily Maximum Intensity Series

The 163 annual maximum daily rainfall series were statistically analyzed, with the GEV distribution being chosen unless the data had a significantly better fit to another distribution, or the results yielded high intensities that seem questionable at high return periods. It is important to point out that the subjective component of these replacements has been minimized as much as possible. Recent studies have highlighted the potential negative consequences of subjective choices in such cases [66]. Therefore, replacements were made only when the expected values provided by the GEV distribution were close to the probable maximum precipitation (PMP) values estimated in the area [67], while also considering that the affected series were among the shortest in the dataset. In such cases, the statistical distribution that provided the best fit among all those considered was selected. Out of the 163 series, 100 (61%) were satisfactorily fitted by the GEV distribution. For the remaining cases, alternative distributions were used: Gumbel (GEV type I) for forty-four series, GNO for seven, GLO for five, GPA for four, and PE3 for three, in order to provide reliable results [48]. The 44 series for which the Gumbel distribution (GEV type I) was preferred were those where the fit to the distribution resulted in a positive shape parameter (Weibull or GEV type III), as its heavy tail implies only a very small increase in the expected intensity for sufficiently long return periods. Figure 2 shows the L-moments ratio diagram for the fittings performed on our series.

After fitting the annual maximum rainfall series for 24 h durations at each location, the maximum rainfall intensities corresponding to the return periods of 2, 5, 10, 20, 50, 100, 200, and 500 years were calculated, some of which are shown in Table S1. From these 163 values for every return period, a spatial interpolation was performed at high spatial resolution (1 km × 1 km). Figure 3 presents results for the return periods of 2, 20, 100, and 200 years, with maps simplified by grouping areas with uniform shading into zones defined by 10 mm or 20 mm intervals.

There is a general similarity between the spatial patterns of daily maximum rainfall and annual rainfall depth maps. The most prominent feature is the lower maxima observed in the western–central region, corresponding to the driest areas of Catalunya. However, the effect of changing the meteorological scale from monthly to daily reveals that in areas where the predominant rainfall type occurs on shorter timescales (typically convective rainfall), discrepancies in certain regions emerge. Thus, the highest expected daily rainfall is generally concentrated in the southern and northeastern parts of the region, with comparable values in both areas. In the southern part, the location of maximum values shifts depending on the return period: for lower return periods, the maximum rainfall is found at the southern coastal end, while for higher return periods, the maximum values move 30 to 40 km westward, toward mountainous areas reaching altitudes of 1400 m.a.s.l. A relative maximum in the eastern part of the central region is significant for high return periods, and the high expected intensities in this area are not isolated but are linked to intense rainfall in the mountainous northeast. In contrast, in the northwestern corner of Catalunya, a region with high total annual rainfall, lower daily maxima are observed. In this area there is a predominance of continuous rain with limited convective activity, resulting in significant total rainfall amounts but without sharp peaks over shorter time periods.

3.2. Simple Scaling Analysis

The scaling analysis outlined in Section 2.3 was conducted on the corrected annual maximum rainfall series for durations ranging from 1 to 15 days at each of the 163 study locations. The statistical moments of order $q$ ($q=0.5, 1, 1.5, 2, 2.5, 3$) were calculated for these aggregated series. To examine scale invariance and estimate the scaling exponent $\beta$, linear regressions were performed between the logarithms of the statistical moments and the logarithm of the duration $t$ for each $q$. Figure 4 illustrates the log–log plots of statistical moments versus the duration for two selected stations, Vielha and Lleida, as well as the straight lines fitted by linear regression, with slopes $K\left(q\right)=\beta q$, confirming scale invariance. These stations were chosen to highlight contrasting climates in Catalunya, reflected in their differing empirical $\beta$ values. Vielha, in the northwesternmost area of Catalunya, has a climate influenced by the Atlantic, characterized by regular precipitation and high annual totals (around 920 mm/year), yielding a scaling exponent of $\beta =-0.71$. Conversely, Lleida, in central–western Catalunya, experiences a dry continental Mediterranean climate with low (340 mm/year on average), irregular annual rainfall and seasonal peaks in spring and autumn, resulting in $\beta =-0.83$.

Another interesting set of metrics to assess and compare rainfall variability in both locations includes the number of wet days per year, which is 125 in Vielha and 82 in Lleida. The Simple Daily Intensity Index (SDII), that is, the mean daily precipitation over wet days, is 7.4 mm/day in Vielha compared to 4.1 mm/day in Lleida. Additionally, the rainfall disparity [68] provides a mathematical approach to quantify rainfall irregularity at monthly and annual scales. It serves as an alternative or complement to other indices that rely on ratios or differences between consecutive monthly or annual rainfall amounts, as well as simpler metrics like the well-known coefficient of variation, which has acknowledged limitations, such as its dependence on the mean and the length of the series. The rainfall disparity between consecutive months [69] is approximately 0.7 in Vielha and around 1.3 in Lleida [70].

Monofractal or simple scaling behavior was observed consistently across all analyzed locations. The scaling parameter $\beta$ ranged from −0.89 to −0.65, with the most extreme values observed at stations located outside Catalunya (Table S1). Despite being outside the region, these stations were instrumental in refining the spatial distribution of this parameter. The interpolation was carried out using simple kriging with a hole-effect variogram model, which incorporates anisotropy by accounting for differing rates of change in the parameter across directions. This approach enhances the accuracy of spatial representation [71]. Figure 5 shows the spatial distribution of $\beta$, grouped into zones corresponding to 0.1 intervals of the parameter range. The spatial analysis reveals an expected pattern that corresponds with the climate characteristics of different areas: higher values, averaging −0.75, are primarily concentrated in the northwest, corresponding to a mountainous region influenced by Atlantic conditions, particularly at its most northwestern corner where Vielha is located. Conversely, lower values, averaging −0.81, are observed in the southwestern region, aligning with Catalunya’s driest areas, including Lleida.

The spatially interpolated results were used to derive the scaling exponent at grid points with a resolution of 1 km × 1 km. This approach enables the intensity–frequency results to be interpolated and extracted at the same grid points for various selected return periods. Consequently, the intensity–duration–frequency relationships can be calculated at high spatial resolution for any desired duration without requiring additional interpolations.

3.3. Downscaling the Intensity–Frequency for 24 h to Sub-Daily Durations

IDF relationships were derived by applying the downscaling Equation (4) to the high-resolution 1 km × 1 km grids of the maximum expected 24 h rainfall intensity for each considered return period (Figure 3), using the scaling exponent $\beta$, also calculated at the same spatial resolution in the previous section (Table S1). Figure 6, Figure 7 and Figure 8 present a selection of the resulting intensity–duration–frequency maps, illustrating the outcomes for sub-daily durations of 12 h, 6 h, and 1 h. In these figures, the high-resolution rainfall intensity maps were simplified by grouping areas with uniform shading into zones defined by 10 mm or 20 mm intervals, using an equal-interval classification. This approach highlights regional trends in expected rainfall intensity while reducing the visual complexity of the original data. The original high-resolution rainfall intensity maps, along with interpolated data for each municipality, are available on the SMC’s website [46].

Since the spatial discrepancies observed in the southern and northwestern areas of Catalunya, when compared with monthly and annual rainfall maps, have been attributed to the predominance of convective rainfall at shorter timescales, these differences are expected to persist in the same zones as the scale is reduced to sub-daily durations. However, this persistence cannot be confirmed until a sufficiently long sub-daily dataset becomes available to conduct the analysis for each sub-daily duration. Until then, the downscaling method used in the present study provides sub-daily maps with the same spatial distribution as those for 24 h, which can be considered a reasonable approximation. While the same spatial distribution patterns are obtained for the sub-daily durations, the method provides specific values adjusted for each duration, and an expansion of the value range is observed as the return period increases.

The validation of these results was carried out in two ways. First, the results for a 24 h duration were compared with those from [72], using similar datasets and methodologies. The comparison showed similar outcomes, with spatial differences in maximum expected rainfall generally being small. Over 90% of the region had differences in less than 20 mm/day for 2-year return periods, and up to 40 mm/day for 100-year return periods. The weighted interpolation in this study, which accounts for altitude, improved spatial resolution, revealing higher expected intensities in mountain peaks compared to lower areas. Second, downscaled 1 h rainfall results were validated using hourly data from 105 automatic stations evenly distributed across Catalunya, which have been operational for over 15 years and are carefully quality controlled. Frequency analyses and kriging were applied to the hourly records, and the comparison with the present results revealed small differences in most parts of the region. Over 95% of the area showed differences in less than 20 mm/h for 2-year return periods, a frequency for which the comparison is most reliable due to the shorter available series.

Two error analysis procedures were carried out in order to quantify the uncertainty of the results provided. The first analysis is related to series length, since we have a small percentage of series that could be considered short for the study conducted, and this affects results at the daily scale. The second analysis is associated with the downscaling process and affects results at the sub-daily scale.

Our dataset comprises 163 high-quality daily rainfall series, with an average length of 39.8 years. Out of the 163 series, 37 (23%) have a length between 25 and 30 years, and 62 (38%) have a length between 31 and 40 years. Given the potential limitations of these shorter series, we performed an error analysis of the results at the daily scale based on series length. To do this, we selected three 75-year-long series, divided them into 20 random subsets of 25 years each, and fitted all of them with GEV distributions. We compared the intensity values for each frequency across the 60 subseries and the 75-year-long series. The differences were 18.3% for a 200-year return period, 14.3% for a 100-year return period, and smaller for shorter return periods. Given the even distribution of shorter series across the region, the error due to series length at any point can be estimated as less than 11% for the 200-year return period map and less than 9% for the 100-year return period map. For a 500-year return period, the uncertainty increases to almost 15%.

The second analysis focuses on the uncertainty associated with the scaling parameter determination in the downscaling process, which affects results at the sub-daily scale. This uncertainty arises from the calculation of $K\left(q\right)$ through the linear regression of intensity moments across different durations. The R² values for all the regressions have a mean of 0.998, and the error for the scaling parameter (β) was estimated at ±0.015. This uncertainty allows for the estimation, via error propagation, of an additional uncertainty for the downscaled rainfall amounts, which was found to be less than 5%.

Combining the two independent errors gives a total uncertainty for the sub-daily maps, calculated as the root sum of squares of the individual uncertainties: nearly 15% for the 500-year return period, under 12% for 200 years, and lower for shorter return periods.

The maps presented illustrate the spatial distribution of rainfall intensities across various return periods, offering valuable insights for assessing flood risk and infrastructure resilience. Comparing these results with established risk thresholds allows for evaluating the potential of rainfall to exceed critical limits in different areas. For example, the rainfall intensity threshold of 40 mm/h (

Table 1. Percentage of Catalunya exceeding hourly and daily rainfall thresholds associated with weather warnings.

Return Period (Years)	Intensity Threshold 40 mm/h	Accumulation Threshold 100 mm/24 h	Accumulation Threshold 200 mm/24 h
2	19.1%	1.2%	0.0%
5	72.0%	36.0%	0.0%
10	90.5%	65.4%	0.0%
20	97.5%	81.8%	3.6%
50	100%	90.9%	15.4%
100	100%	96.2%	30.6%
200	100%	98.7%	47.3%
500	100%	100%	65.8%

), commonly used by insurance companies to assess risk and identified by the SMC as representative of heavy rainfall intensity [73], is exceeded in much of the region for medium return periods, such as 5 or 10 years. This highlights critical areas prone to more frequent flooding events.

In contrast, the 24 h rainfall accumulation thresholds, defined by the SMC as 100 mm (low-level risk) and 200 mm (high-level risk), are exceeded less frequently. The 100 mm threshold is expected to be surpassed in over 80% of the territory only for return periods of at least 20 years, indicating a higher long-term risk in certain regions. Meanwhile, the 200 mm threshold is exceeded in roughly half of the territory only for very long return periods, beyond 200 years, emphasizing its rarity and association with extreme, high-risk events.

4. Discussion

The spatial distribution of the maximum expected 24 h rainfall intensity identified in this study closely resembles the patterns observed in monthly and annual rainfall depth maps [48,72,74]. The most notable feature is the lower maxima in the western areas, which correspond to the driest zones of Catalunya. However, discrepancies in certain regions suggest the influence of factors beyond total annual rainfall, such as the proportion of convective rainfall, which becomes more apparent when transitioning from annual to daily scales. For instance, the northwestern corner of Catalunya, despite being a region with high annual rainfall totals, exhibits low daily and sub-daily maxima due to the prevalence of continuous, low-convection rainfall, which generates high cumulative totals without significant short-term peaks. In contrast, the highest maximum daily rainfall intensities are observed in both the northeastern and southwestern extremes of the territory, areas where annual maxima are not particularly prominent. This finding aligns closely with the results reported by [72] for Catalunya, despite differences in datasets and methodologies. The incorporation of weighted interpolation, accounting for altitude, in the present study enhances spatial resolution, capturing more distinct variations between peaks and valleys in mountainous areas. For example, slightly higher expected intensities are found at higher altitudes compared to their lower-altitude surroundings.

Comparing the spatial differences in maximum expected daily rainfall with the results of [72], differences remain relatively small: over 90% of Catalunya exhibits differences below 20 mm/day for 2-year return periods, dropping to 78% for 10-year periods and rising to up to 40 mm/day for 70% of the territory over 100-year return periods. Overall, the present study yields slightly higher expected intensities, with notable differences in regions exhibiting the most pronounced rainfall extremes. These differences are likely attributable to the predominant use (61%) of the GEV distribution to fit the maximum intensity series in this study, while only the Gumbel distribution (GEV type I) was used in [72], a distribution that has been reported to have, among other disadvantages, limited flexibility in modeling extreme event tails [75].

Utilizing scaling relationships and the monofractal approach for downscaling daily rainfall totals has proven to be a satisfactory methodology for estimating sub-daily rainfall maxima, aligning well with the findings reported by other researchers [29,30,31,32,33,34,35,36]. This approach is particularly valuable in the absence of sufficiently long sub-daily rainfall data, as automatic weather stations currently face limitations due to the relatively short duration of their sub-daily time series. As longer sub-daily datasets become available in the future, they will allow for the more reliable construction of IDF relationships and provide the opportunity to confirm whether spatial patterns observed in the southern and northwestern areas of Catalunya, attributed to the predominance of convective rainfall, persist at sub-daily scales.

Until such datasets become available, the downscaling method used in this study results in sub-daily maps having the same spatial distribution as those for 24 h, a reasonable approximation given that the observed discrepancies in 24 h maps, driven by convective rainfall, are expected to persist at shorter timescales.

The spatial distribution of the scaling parameter β across Catalunya aligns closely with the region’s climatic zones, reflecting variations in precipitation regimes and generally corresponding to the mean annual precipitation pattern. Higher β values, averaging around −0.75, are observed in wetter regions characterized by more regular rainfall, whereas lower values, averaging approximately −0.81, are found in drier areas. However, discrepancies identified in study [32] highlight the influence of specific precipitation events, particularly the contribution of convective rainfall and the proportion of annual rainfall concentrated on the maximum rainfall day. These findings suggest that β may also serve as an indicator of rainfall irregularity.

The results for Catalunya are consistent with findings from other studies. In [30], scaling exponents between −0.77 and −0.83 were reported for Catalunya and similar patterns were noted across Spain, with values ranging from −0.55 to −0.66 in wetter regions and [−0.84] to −0.92 in drier areas. These variations were linked to differences in precipitation regimes and convective rainfall proportions. All the cited values were calculated from daily rainfall series. If the temporal scale changes, the range of scaling exponents also varies. For instance, using monthly rainfall series, Ref. [45] found higher β values for locations in Catalunya compared to those derived in this work. This difference arises because coarser temporal aggregations lead to smoother datasets, and monthly series are inherently more regular than daily ones, resulting in less variability and higher scaling parameter values. On a broader scale, Ref. [31] identified six climatological regions across Canada and the United States, each exhibiting distinct scaling behaviors driven by local topography, coastal effects, and atmospheric circulation. Similarly, Ref. [29] demonstrated that β depends on climate type, with higher values for Melbourne (−0.65, temperate climate) and lower values for Warmbaths, South Africa (−0.76, semi-arid climate).

Further support for the relationship between scaling properties and precipitation comes from studies in Spain. For example, Ref. [76] analyzed the fractal dimension of rainfall and its link to rainfall concentration and self-similarity. In Andalusia, Ref. [34] observed the influence of Mediterranean disturbances and convective rainfall on β, particularly in drier areas with irregular rainfall patterns. Together, these studies underscore the dependence of β on local climatic conditions, precipitation characteristics, and geographical features, providing a broader context for interpreting the results obtained for Catalunya.

The downscaling of maximum expected rainfall intensities to 1 h periods has made it possible to assess the areas where heavy rainfall intensity risk thresholds, such as 40 mm/h, may be exceeded, as well as the frequency of such events. This threshold, which is representative of heavy rainfall intensity [73], is surpassed across much of the region for medium return periods (e.g., 5 or 10 years), indicating areas that are particularly vulnerable to more frequent flooding. Insurance companies, particularly those covering weather-related damages such as floods, rely on specific rainfall thresholds to assess risk [77]. These thresholds are essential for determining flood insurance claims and reflect the intensity and duration of rainfall that can lead to significant surface water flooding.

5. Conclusions

Monofractal temporal downscaling has been found to be an effective methodology for estimating sub-daily rainfall maxima from daily rainfall data, particularly in the context of limited sub-daily observations. Until datasets from automatic weather stations become more extensive and provide longer time series, the downscaling approach remains a practical and reliable alternative for deriving critical rainfall estimates.

The findings of this study contribute to a more detailed understanding of regional variability in rainfall and flood risk, offering essential insights for urban planning, flood control measures, and insurance policy adjustments. The observed relationship between the scaling parameter β and climatic conditions highlights its potential as an indicator of rainfall irregularity and a valuable metric for hydrological studies across diverse climatic zones.

These results emphasize the importance of capturing spatial and temporal variations in rainfall for developing targeted adaptation strategies. Municipalities and insurance companies, in particular, can leverage this knowledge to mitigate flood damage, optimize resource allocation, and enhance resilience to extreme weather events. Moreover, the insights into scaling behaviors and rainfall distributions provide a baseline for assessing the potential impacts of climate change on extreme rainfall patterns.

By establishing a foundation for both immediate practical applications and future research, this study advances hydrological modeling and disaster risk reduction efforts, addressing the challenges posed by growing climatic uncertainties.