How Frequent Is an Extraordinary Episode of Precipitation? Spatially Integrated Frequency in the Júcar–Turia System (Spain)

Abstract

An extraordinary episode is a torrential rainfall event that produces significant societal impacts, which poses a major natural hazard in the western Mediterranean, particularly along the Valencia coast. This study evaluates the feasibility and added value of an explicitly spatial approach for estimating return periods of extraordinary precipitation in the Júcar and Turia basins, moving beyond traditional point-based or micro-catchment analyses. Our methodology consists of progressive spatial aggregation of time series within a basin to better estimate return periods of exceeding specific catastrophic rainfall thresholds. This technique allows us to compare 10 min rainfall data of a reference station (e.g., Turís, València, 29 October 2024 catastrophe) with long-term annual maxima from 98 stations. Temporal structure is characterized using the fractal–intermittency n-index, while tail behavior is modeled using several extreme-value distributions (Gumbel, GEV, Weibull, Gamma, and Pareto) and guided by empirical errors. Results show that $n\approx 0.3$–0.4 is consistent for extreme rainfall, while return periods systematically decrease as stations are added, stabilizing with about 15–20 stations, once the relevant spatial heterogeneity is sampled. Specifically, the probability of exceeding extraordinary thresholds is between 3 and 10 times higher for the areal than the point approach, so recurrence of a catastrophe would be once a few decades rather than centuries. Overall, the results demonstrate that spatially integrated return-period estimation is operational, physically consistent, and better suited for basin-scale risk assessment than purely point-based approaches, providing a relevant baseline for interpreting recent catastrophic events in the context of ongoing climatic warming in the Mediterranean region.

1. Introduction

1.1. Context

Extreme precipitation constitutes one of the primary natural hazards in the coastal regions of the western Mediterranean. In particular, the Júcar–Turia system (Spain) is among the most exposed areas in Europe to torrential rainfall episodes, with historical records frequently approaching or exceeding 500 mm in less than 24 h [1,2]. This hazard arises from quasi-stationary mesoscale convective systems (MCS) as a result of the interaction of: (i) a massive supply of warm, moist air from the Mediterranean Sea, (ii) an upper-level flux divergency due to the proximity of an isolated depression (DANA or cut-off low), and (iii) a complex orography that facilitates an efficient humidity convergence. The confluence of all these factors is extremely rare (i.e., extraordinary) at the point scale, although it has repeatedly triggered catastrophic events across different sectors of the Júcar–Turia system, mainly in October. Notable examples include the floods of 1957 (València), 1982 (Tous), 1987 (Oliva) and the episode of 29 October 2024, which reached 771.8 mm in 24 h at Turís [3], the second-highest 24 h rainfall total recorded for Spain, surpassed only by the 817.0 mm observed in Oliva on 3 November 1987 [4]. Such phenomena lead to devastating flash floods, substantial economic losses, and high mortality rates, underscoring the increasing systemic risk in the region. A detailed synoptic-scale analysis of the October 2024 Valencia event is provided by Campos et al. [5], who document the large-scale atmospheric configuration and mesoscale processes responsible for the persistence and severity of the episode, offering a complementary physical perspective to the statistical approach adopted in this study.

In the context of climate change, the intensification of these events has been widely documented [3,6,7,8,9]. Both regional scenarios and observational studies indicate an increasing intensity of rainfall up to +20% in areas that are already highly vulnerable, such as the Valencian coastline [3,10]. Therefore, the study of the temporal structure and statistical recurrence of these episodes is essential for risk management, hydrological planning, and territorial adaptation.

Numerous studies have highlighted the role of large-scale atmospheric dynamics in modulating the frequency and persistence of extreme precipitation events in the Mediterranean region [11,12,13,14,15]. In particular, the break of the Rossby waves along the jet stream disrupts zonal flow and promotes the development of quasi-stationary circulation patterns, often leading to cut-off lows (DANAs). These systems enhance low-level moisture transport and maintain persistent convective activity over affected regions. Moreover, Rossby wave breaking can induce persistent synoptic-scale configurations that simultaneously promote drought conditions in some areas while triggering extreme rainfall and flooding in others, a behavior that is especially relevant for the western Mediterranean during autumn [16,17]. Under ongoing climate warming, such circulation regimes are projected to become more frequent or longer-lasting, increasing the likelihood of spatially extensive, high-impact precipitation extremes [18,19,20,21]. Against this physical background, an open key question remains on how often these events are expected on the scale of a river basin. The present work aims to address that question.

1.2. Estimation of Episode Recurrence

Numerous studies have addressed the characterization of these phenomena from multiple perspectives. At the synoptic scale, atmospheric patterns associated with extreme events are classified through statistical circulation analysis, confirming the decisive role of blocking configurations and retrograde flows [22,23,24]. On the regional scale, studies such as González-Hidalgo et al. [2] have cataloged a very large number of extraordinary precipitation episodes in the Iberian Peninsula since the beginning of the twentieth century, while Ruiz et al. [25] investigated palaeofloods in the Júcar–Turia system through geomorphological analyses of alluvial fans and fluvial plains, with at least 10 major floods in five centuries (1571, 1590, 1632, 1776, 1779, 1805, 1864, 1897, 1923, 1949 and 1957). These reconstructions reveal that the actual frequency of catastrophic events far exceeds estimates based solely on instrumental meteorological records [26,27].

Despite this extensive body of work, estimating the recurrence of extreme precipitation events remains a major methodological challenge, particularly when the hazard of interest is spatially extended.

An important limitation repeatedly identified in the hydrological literature is the excessive reliance on the classical micro-catchment or point-based approach of Intensity–Duration–Frequency (IDF) curves [28]. Here, micro-catchments are understood as basins of small spatial extent (typically tens to <100 km²), with short concentration times (hours) and rapid hydrological responses strongly controlled by local orography and drainage structure. These units exhibit pronounced spatial and temporal variability in rainfall distribution and runoff generation: the same convective event may produce markedly different maxima at stations separated by only a few kilometers. This spatial heterogeneity is a pattern widely documented in recent studies [1,29].

The point-based (micro-catchment) approach—which summarizes hazards using a single representative value per basin—fails to capture this behavior and, critically, the spatial probability of exceedance, i.e., the likelihood that a given threshold is exceeded in at least one subregion of the domain. Recent studies show that extreme precipitation events often display complex spatial dependence structures whose extent and intensity evolve with climate forcing [30], and that neglecting spatial variability leads to systematic underestimation of flood hazard [29]. Moreover, the spatial dependence of extreme rainfall can substantially modify recurrence intervals in a basin, making the analysis of a single station insufficient for regional hazard assessment [31].

Therefore, since basin- or regional-scale hazard depends fundamentally on the joint spatial behavior of extremes, it is necessary to complement (or replace) purely point-based estimates (e.g., Gumbel fitted to a single station or micro-catchment) with explicitly spatial methodologies that integrate information from multiple stations and model the spatial dependence of extremes. The continued use of point-based representations assumes a homogeneous, station-equivalent behavior that is incompatible with the spatial complexity of extreme precipitation processes under current and future climate conditions [32]. As a consequence, the erroneous idea has become widespread that extreme events such as those mentioned above have return periods of hundreds or thousands of years, whereas empirical, geological, and statistical records indicate that they recur several times per century [25,27].

1.3. Estimation of Rainfall Concentration

In addition, recent studies highlight the need to use new statistical indicators that capture the temporal variability of precipitation beyond simple accumulations. Among them, the fractal intermittency n-index proposed by Monjo [33] stands out as a dimensionless measure that describes the structure of precipitation events according to their degree of concentration and intensity across time scales. Specifically, the maximum precipitation amount ($P_t$) expected in a duration of t hours is given by

$$P_t=P_{1h}{t}^{1-n}⟹P_{\mathrm{tot}}=P_{1h}{t_{\mathrm{ef}}}^{1-n}$$

(1)

where $P_{1h}$ is the maximum precipitation recorded in 1 h, and the effective duration $t_{\mathrm{ef}}$ is defined when the maximum amount ($P_t$) is equal to the total amount ($P_{\mathrm{tot}}$). Equation (1) can be derived from the Maximum Average Intensity (MAI) relationship implemented in empirical IDF curves with power-law behavior [28]. The n-index is a dimensionless indicator that describes precipitation variability on different temporal scales. Its value ranges between 0 and 1, where values close to 0 indicate more uniform–continuous rainfall (typical stratiform), whereas values close to 1 indicate highly concentrated–intermittent rainfall (typical convection), and $n=0.5$ denotes an efficient balance between intensity and duration as well as between advection and convection currents (i.e., organized deep convection) [33,34].

The n-index approach has been applied in both global and regional analyses and has proven useful for complementing the hydrological modeling of intense episodes [1,35,36,37,38,39]. Its application is key in climate modeling, allowing a better understanding of rainfall patterns and their impact on water resource management and the prediction of extreme events.

Against this background, the present work aims to bridge physical characterization and statistical recurrence by combining indicators of rainfall concentration with spatially integrated return-period analysis.

The analysis begins with a detailed examination of the event that occurred on 29 October 2024 in the Júcar–Turia system (Figure 1), one of the most devastating in recent decades, which particularly affected the municipality of Turís (Valencia Province, Spain).

Through the use of the fractal–intermittency n-index and the estimation of empirical and theoretical return periods through fitting to different probability distributions, the representativeness and frequency of this episode are evaluated within the context of the last decades. Unlike previous studies, the approach adopted here integrates multiple stations within each water basin, allowing the analysis of annual maxima and their aggregated spatial behavior. In addition, a sensitivity analysis is introduced as a function of the number of stations to assess how return periods and fitting errors vary as greater spatial coverage is incorporated.

The article is structured as follows: Section 2 presents the data used and describes the methodological framework, including the calculation of spatially integrated return periods with sensitivity analysis and the n-index; Section 3 presents the results obtained; and Section 4 discusses their climatic, hydrological, and territorial interpretation, and draws general conclusions to outline future research directions.

2. Materials and Methods

2.1. Observed Data

A total of 98 time series of daily precipitation (1981–2024), recorded in stations located in the Júcar–Turia system, were collected from the Spanish Meteorology Agency (AEMET) database: 58 stations correspond to the Júcar basin and 40 are located in the Turia basin (Figure 1). To analyze the extreme precipitation episode that occurred on 29 October 2024 and severely affected Valencian territory, 10-minute precipitation data recorded by an automatic weather station located in the municipality of Turís were also used (records from 28 October 2024 00:00 h to 31 October 2024 00:00 h, local time).

2.2. Point and Spatially Integrated Frequency

Point and spatially integrated return periods were estimated for several reference thresholds of extraordinary episodes, including the 2024 case. Point return periods ($\rho$) represent the expected average time between precipitation events p exceeding a prescribed threshold P (e.g., pluvial amount or intensity), based on the empirical (or fitted) exceedance probability $\Pi (p\le P)$ of a single time series ($N=1$), that is,

$$\rho \equiv {\displaystyle \frac{1}{\Pi (p>P)}}={\displaystyle \frac{1}{1-\Pi (p\le P)}}.$$

(2)

In contrast, spatially integrated periods also represent return periods $\rho$ but using $N>1$ time series (from stations located nearby), so the probability of exceeding thresholds is significantly higher. Both indicators are used to assess the probability of occurrence of extreme events over a given time span. The longer the return period, the less frequent and more intense the associated event.

The time series used for all subsequent calculations consist of the annual maximum values extracted from a set of N time series (subsequently adding stations); that is, the maximum of the annual maxima is considered to build a synthetic annual time series that integrates all the N stations considered. The resulting synthetic series represents an annual spatial extreme defined under an OR-type framework, i.e., an extreme event is said to occur in a given year whenever at least one station within the domain exceeds the prescribed threshold. This interpretation naturally leads to a basin-scale hazard indicator rather than a point-based extreme [40]. The annual maxima (AM) approach was adopted because it provides a consistent temporal basis for spatial aggregation across heterogeneous station networks and allows a direct interpretation of regional extremes under this OR-type definition. Peaks-over-threshold (POT) approaches were not considered, as they would require additional declustering assumptions and threshold selections that become non-trivial in a spatially integrated context and could compromise the comparability of return periods across different station subsets [41]. Distribution fitting was performed using families commonly employed in extreme-value theory (EVT) and hydrological frequency analysis, including Weibull, Gumbel, Generalized Extreme Value (GEV), Gamma, and Pareto distributions [42]. These distributions provide flexible representations of upper-tail behavior relevant for extreme precipitation, whereas central-limit distributions such as the normal distribution are unsuitable due to their rapidly decaying tails.

Extreme-value analysis requires suitable regularity conditions on tail behavior and dependence, which may involve local stationarity or explicit modeling of non-stationarity and spatial dependence. These conditions are compatible with the present spatially integrated construction, since the synthetic series is defined as the maximum of annual block maxima across a fixed number N of stations. Specifically, for each station i we compute the annual block 1-day maximum precipitation $P_{1,y,i}={max}_{d\in y}{p}_{i,d}$ among the days d of each year y, and then define the N-regional maximum $P_{1,y;N}={max}_{1\le i\le N}{P}_{1,y,i}$. Although this involves a second maximization step, standard EVT results ensure that if each $P_{1,y,i}$ belongs to an attraction domain of an extreme-value distribution, then $P_{1,y;N}$ also exhibits valid limiting behavior. Conceptually, $P_{1,y;N}$ represents the annual regional maximum for a set of N stations and corresponds to the block-maximum analog of an OR-type spatial extreme, since $P_{1,y;N}>p$ whenever at least one station exceeds p. This provides a clear statistical basis for treating $P_{1,y;N}$ as a synthetic y-annual series suitable for basin-scale frequency and areal IDF analyses.

2.3. Areal IDF Curves

Consistently with spatially integrated periods (Section 2.2), areal IDF curves denote intensity–duration–frequency relationships derived for a spatial domain (e.g., a basin), where the precipitation threshold is exceeded by at least one station within the domain, thus representing a basin-scale rather than a point-based IDF. These resulting curves were compared to the empirical power law found by Moncho and Caselles [28]:

$$P_{t,\rho ;N}=P_{t_0,{\rho }_0;N_0}{\left({\displaystyle \frac{\rho N}{{\rho }_0{N}_0}}\right)}^m{\left({\displaystyle \frac{t}{t_0}}\right)}^{1-n}⟹\rho \sim 1/N,$$

(3)

where n is the n-index (Equation (1)), m is an empirical scaling parameter, and $P_{t,\rho ;N}$ and $P_{t_0,{\rho }_0;N_0}$ are, respectively, the maximum precipitation amounts recorded in a small area by N and $N_0$ stations, as expected for a return period $\rho$ and ${\rho }_0$, within an episode of duration t and $t_0$. If $N=N_0=1$, Equation (3) provides a classical empirical IDF curve as $I_{t,\rho }=P_{t,\rho }/t$. On the other hand, for $N\ge N_0>1$ statistically independent stations, Equation (3) represents an empirical relationship between the spatially integrated return period $\rho$ and the maximum precipitation amount $P_{t,\rho ;N}$. To analyze the effect of N on the return period $\rho$, we set the curves of Equation (3) for constant precipitation thresholds $P_{t,\rho ;N}=P_{t_0,{\rho }_0;N_0}$, for example, 200 or 500 mm. Here, we assumed that the return period changes as follows:

$$\rho ={\rho }_{min}+({\rho }_{max}-{\rho }_{min}){\left({\displaystyle \frac{1}{N}}\right)}^{\lambda \left(N\right)},$$

(4)

where ${\rho }_{max}$ is the return period considering only the most extreme station, while ${\rho }_{min}$ is the lowest return period, found when all the stations are used, and $\lambda \left(N\right)$ is an independence parameter: Statistically, $\lambda =1$ means that the stations are purely independent (i.e., recovering the $\rho \sim 1/N$ relationship of Equation (3) with ${\rho }_{min}=0$), while $\lambda =0$ implies that all stations are purely dependent and do not contribute to new extremes. For this study, we considered a reduction of $\lambda$ from 1 to 0 as $N/N_0$ increases with respect to a certain $N_0$, specifically $\lambda \left(N\right)\approx exp(-N/N_0)$.

2.4. Sensitivity Analysis of Episode Recurrence

The fit of the theoretical cumulative distributions was compared with the empirical cumulative function to evaluate which model best reproduces the observed behavior across different ranges (e.g., fitted versus extrapolated). To differentiate between well-sampled and extreme cases, two disjunct ranges of return periods were considered: the fitting range (40 values) and the validation range (3 most extreme values). The fitting range includes return periods where multiple observations are available, allowing for robust statistical fitting. In contrast, the validation range corresponds to very large return periods, often represented by only one or two observations. These extreme cases are exceptionally rare, leading to return periods that are extremely high and potentially distant from the true values due to the lack of additional extreme observations. Evaluating the validation range separately helps quantify the uncertainty associated with extrapolating beyond the observed data.

The accuracy of the fitted distributions was quantified using the mean absolute error (MAE) and the root mean square error (RMSE), calculated between the modeled and empirical values of the spatially integrated return periods. Given a set of K observations (i.e., K annual maxima and their return periods), the errors are defined as:

$$\mathrm{MAE}={\displaystyle \frac{1}{K}}\sum _{i=1}^K\left|{\rho }_i^{\mathrm{model}}-{\rho }_i^{\mathrm{empirical}}\right|$$

(5)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{K}}\sum _{i=1}^K{\left({\rho }_i^{\mathrm{model}}-{\rho }_i^{\mathrm{empirical}}\right)}^2}$$

(6)

where ${\rho }_i^{\mathrm{model}}$ and ${\rho }_i^{\mathrm{empirical}}$ are the modeled and observed return periods, respectively. These metrics were calculated separately for the fitting range and the validation range, allowing assessment of model performance in well-sampled versus extreme, poorly sampled conditions. For each station subset and basin, all candidate distribution families were systematically fitted, and model selection was performed automatically by minimizing the error metrics (MAE and RMSE, Equations (5) and (6)), evaluated separately over the fitting and validation ranges. This procedure ensures that the choice of the theoretical distribution is objective, reproducible, and not based on subjective or a posteriori criteria.

The spatial-integration analysis was then conducted by computing the evolution of the return period as stations are progressively added to the analysis, in order to capture a spatial evolution rather than a purely temporal one. Calculations were performed on maximum series derived from different station subsets, with blocks of five stations. For each subset, the different fitting methods were compared in order to select the one with the lowest error.

2.5. Rainfall-Concentration Comparison

To complete the time structure of the obtained IDF curves, a comparison of different extreme precipitation episodes was made together with the data for the 29 October 2024 episode using the fractal–intermittency n-index (Equation (1)), which is assumed to be independent of the return periods (Equation (3)). It is calculated from the multifractal analysis of the precipitation series, using a multiplicative cascade approach to assess variability at different scales [33]. The analysis of rainfall concentration was carried out mainly using the n-index.r script and functions_spell.r scripts [43] to compute the n-index and display the results of the case study. For the estimation of return periods, both fitdistrplus v1.2-4 and evd v2.3-7.1 libraries were applied [44,45].

For general data processing and analysis, RStudio 2025.09.0+387 (“Cucumberleaf Sunflower” release) was used, an integrated development environment for the R programming language that facilitates data manipulation, statistical model development, and result visualization, providing advanced tools for the exploration and analysis of precipitation time series.

3. Results

3.1. Findings on Episode Recurrence

For the Júcar basin (Figure 2a), the best fit is obtained using either the Gumbel or the GEV distribution, although it should be noted that for very high precipitation values none of the methods is able to adequately reproduce the observed maximum. In contrast, for the Turia basin (Figure 2b), the Pareto distribution appears to provide the best fit to the empirical data, except for the highest precipitation values (validation range), where Weibull or GEV show a better performance. To more precisely assess which fit performs best, fitting errors were calculated with respect to the empirical distribution (

Table 1. Errors of return periods (years) for fitted distributions (top) and for their validation ranges (bottom) for all the stattions of the Júcar and Turia basins.

		Júcar Basin		Turia Basin
Range	Distribution	MAE	RMSE	MAE	RMSE
Fitting	Weibull	0.80	1.67	0.65	1.31
	Gumbel	0.26 a	0.56	0.59	1.01
	GEV	0.28	0.82	0.84	1.68
	Gamma	0.44	0.98	0.65	1.04
	Pareto	0.72	1.62	0.28	0.68
Validation	Weibull prediction	15,400	22,000	2.5	2.5
	Gumbel prediction	3800	5300	18	20
	GEV prediction	133	188	4.1	5.3
	Gamma prediction	10,500	14,900	13.6	15.0
	Pareto prediction	240	330	190	270

^a Bold figures indicate the best results.

).

After computing the errors, Pareto is identified as the best fit for the Turia basin, except for the values used for validation, for which Weibull provides the best performance. For the Júcar basin, Gumbel exhibits the lowest error across most of the distribution, except for the most extreme values, where GEV better defines the return periods.

Although the analysis was carried out using the best-performing method for each individual case (and even for each subset of stations), overall, GEV produced the lowest average error. Finally, for stations located within the Júcar basin, a GEV fit was applied to all subsets due to the small differences in error in the estimation of the return period compared to the large differences observed in return periods for extreme values. In contrast, for the Turia basin, GEV was the dominant method for the first two station subsets, whereas Weibull was applied to the remaining subsets.

The results showed that the evolution of the return period follows a power-law decay curve with a final stabilization (Figure 3). This structure is more stable for the Turia basin, as there is less variation between extreme values, while for the Júcar basin the evolution for the highest thresholds (e.g., 400 and 500 mm) is more irregular, as the incorporation of many stations (>30) with less extreme rainfall reduces the severity. It should also be noted that the evolution of the fitting errors is proportional to the number of stations considered: the greater the number of stations, the larger the uncertainty in the return periods.

For both basins, the five or ten stations that exhibit the highest maxima are those that logically introduce the largest reduction in the estimates of the spatially integrated return period, with the station that records the highest precipitation maximum in each basin standing out most clearly.

Table 2. Estimated parameters (${\rho }_{min}$, ${\rho }_{max}$, $N_0$) with their standard errors for the Júcar and Turia basins according to Equation (4).

	Threshold [${\mathit{P}}_{1\mathit{day},\mathit{\rho }}$]	${\mathit{\rho }}_{min}$ (Years)	${\mathit{\rho }}_{max}$ (Years)	${\mathit{N}}_0$	${\mathit{\rho }}_{max}/{\mathit{\rho }}_{min}$
Júcar	200 mm	$1.81\pm 0.13$	$11.58\pm 0.36$	$9.30\pm 1.21$	$6.40\pm 0.50$
	300 mm	$5.23\pm 0.37$	$28.94\pm 1.05$	$9.45\pm 1.45$	$5.53\pm 0.44$
	400 mm	$14.14\pm 0.94$	$56.26\pm 2.70$	$8.54\pm 1.95$	$3.98\pm 0.32$
	500 mm	$33.20\pm 2.50$	$94.28\pm 7.50$	$6.91\pm 3.24$	$2.84\pm 0.28$
Turia	120 mm	$1.00\pm 0.03$	$7.20\pm 0.09$	$2.77\pm 0.31$	$7.20\pm 0.22$
	150 mm	$1.19\pm 0.07$	$13.12\pm 0.19$	$3.27\pm 0.34$	$11.03\pm 0.65$
	180 mm	$2.02\pm 0.11$	$21.90\pm 0.30$	$3.51\pm 0.32$	$10.84\pm 0.60$
	210 mm	$5.34\pm 0.15$	$34.12\pm 0.41$	$3.04\pm 0.30$	$6.39\pm 0.19$

confirms that a critical number of stations is $N_0\approx 9$ for Júcar and $N_0\approx 3$ for Turia, which provides most of the reduction in the return period (from ${\rho }_{max}$ to ${\rho }_{min}$). Specifically, the probability of exceeding extreme thresholds in Júcar basin is between 3 and 6 times higher for the areal IDF curves ($N>N_0>1$) than for the point IDF estimates ($N=1$), as the quotient ${\rho }_{max}/{\rho }_{min}$ shows (Table 2). The factor is even higher for the Turia basin, with an increasing of the probability up to 6–10 times more (although with lower precipitation amounts).

Moreover, if $N=N_0$ is fixed, Equation (3) provides a representative areal IDF curve such as

$$P_{1\mathrm{day},\rho }=P_{1\mathrm{day},1\mathrm{year}}{\left({\displaystyle \frac{{\rho }_{min}}{1\mathrm{year}}}\right)}^m$$

(7)

with scaling parameter $m=0.31\pm 0.02$ for Júcar and $m=0.30\pm 0.09$ for Turia basin, which reproduces the thresholds ($P_{1\mathrm{day},\rho }$, Table 2) from $P_{1\mathrm{day},1\mathrm{year}}=171\pm 8$ mm and $P_{1\mathrm{day},1\mathrm{year}}=134\pm 10$ mm, respectively.

3.2. Findings on Rainfall Concentration

For the 2024 catastrophic event, two major precipitation sub-events were identified, with the second clearly dominating the overall episode; consequently, our analysis concentrates on this latter segment. The first sub-event exhibited an n-index of approximately 0.65 over a duration of about 1–2 h, whereas the second peak showed an n-index near 0.27, with a structure composed of several sub-peaks of roughly $n\sim 0.5$. For this main sub-event, an effective duration of 4.94 h was estimated by noting that the maximum hourly precipitation reached 170.2 mm and that the maximum accumulated rainfall within the segment associated with the dominant peak amounted 620.6 mm (Figure 4).

The effective duration was also computed considering the entire precipitation time series, while maintaining an n-index of $0.27$, as this value best represents the entire series. In this case, the expected duration is 7.3 h. If n is replaced by 0.5 to evaluate the duration that would be required in an “efficient” case for the same total amount of precipitation, an effective duration of 13.3 h is obtained.

To compare with other cases, the n-index and effective duration were estimated according to Equation (1) and are shown in

Table 3. Comparison of the 2024 episode features (Equation (1)) with recent historical episodes.

Date	Station	Basin	${\mathit{P}}_{1\mathit{h}}$ (mm)	n	${\mathit{P}}_{\mathbf{tot}}$ (mm)	${\mathit{t}}_{\mathbf{ef}}$ (Hours)
20 October 1982	Casas del Baró	Júcar	140	0.37	975 *	21.8
03 November 1987	Gandia	Serpis	154	0.42	1000 *	26.3
03 November 1987	Oliva	Serpis	150	0.41	817	17.7
22 October 2000	Carlet	Júcar	60	0.35	532	28.7
12 October 2007	Alcalalí	Xaló–Gorgos	90	0.35	440	11.5
23 September 2008	Sueca	Júcar	142	0.14	350	2.9
29 October 2024	Turís	Turia-Jucar	180	0.27	772	7.3

* Estimation according to proxy data and testimonies.

. The n-index of Turís during the 2024 DANA episode is the second-lowest record for extreme precipitation after the Sueca event of 23 September 2008 (326 mm in 3 h and 144 mm in 1 h), with $n=0.14\pm 0.01$ for an effective duration of 2.9 h. The difference can be characterized by the spatial scale of the storms. The anchored convective cell of the 2008 episode (horizontal scales of 5–10 km) was four times smaller compared to the quasi-stationary MCS of the 2024 DANA episode (storm focus size of about 20–40 km).

Assuming stationarity (i.e., omitting climate change effects) and taking into account the parameters of Equation (7) and Table 3 in Equation (3), total amount (772 mm) of the 29 October 2024 episode in Turís is given by the following curve:

$$772\mathrm{mm}=(171\pm 8)\mathrm{mm}{\left({\displaystyle \frac{{\rho }_{min}}{1\mathrm{year}}}\right)}^{0.31}{\left({\displaystyle \frac{t_{ef}}{7.3\mathrm{hours}}}\right)}^{1-0.27}$$

(8)

where the effective duration of the 1-day accumulation was $t_{ef}=7.3$ hours. Therefore, the return period of this catastrophe is obtained from Equation (8) as ${\rho }_{min}=130\pm 40$ years for the Júcar basin, while the threshold of 500 mm has only $33\pm 2$ years of return period (Table 2).

4. Discussions and Conclusions

4.1. Spatially Integrated Return Periods

This work was conceived as a feasibility assessment of an explicitly spatial approach to estimate return periods of extreme precipitation, progressively incorporating stations within each basin (Júcar and Turia). The results confirm that the approach is operational and consistent with probabilistic reasoning: as the number of stations increases, the return period (when at least one station exceeds a given threshold within the domain) decreases up to a certain spatial sample size of about 15–20 stations. From a practical perspective, this result indicates that basin-scale hazard cannot be reliably inferred from a small number of isolated rain gauges, as a limited spatial sampling systematically underestimates the likelihood of catastrophic rainfall affecting at least part of the basin. Moreover, beyond that sample, the spatially integrated return period stabilizes into a plateau (i.e., a minimum return period ${\rho }_{min}$), because newly added stations no longer contribute annual maxima higher than those previously observed. This decreasing power law (e.g., Equation (4)) is particularly clear in the Turia basin and somewhat more irregular in the Júcar basin for high thresholds, suggesting differences in internal heterogeneity between domains. In particular, the consideration of areal IDF curves instead of the classical point approach leads to a significant reduction of the return period of about a factor of 3–6 for amounts higher than 200 mm.

Overall, the evidence supports the idea that a highly localized anomaly should not inflate regional return periods when the objective is basin-scale hazard and when the metric is correctly defined in spatial terms, so with areal IDF curves. In operational terms, this implies that events often perceived as “exceptional” at a single station may correspond to much more recurrent basin-scale situations when spatial coverage is properly accounted for. Moreover, the value of the plateau ${\rho }_{min}\sim$ 33–130 years for catastrophic thresholds (e.g., 500–700 mm/day) is consistent with the observed palaeofloods in the Júcar–Turia system: at least 10 major floods in five centuries [25].

The observed uncertainty of the estimates is also consistent with the spatial variability and the aggregation effects. As stations are added, estimation errors initially increase: expanded spatial coverage introduces heterogeneity (different local regimes), and spatial dependence among series reduces the effective sample size, so uncertainty does not decrease mechanically with the number of stations. Once the regional tail becomes “well sampled”, both return periods and errors tend to stabilize. This stabilization provides a practical criterion for identifying when the spatial information is sufficient to support robust risk assessments at the basin scale. This interpretation is consistent with the reported pattern that “the probability is proportional to the number of stations”, as well as with the plateau observed in the return period–number of stations relationship for different precipitation thresholds.

4.2. Selection of Theoretical Distributions

Regarding tail modeling, the study documents differences by basin and range. In the Júcar basin, Gumbel exhibits the lowest mean error across the distribution, although GEV better represents the extremes, which is why GEV is used for the spatial station sets in this basin. In the Turia basin, Pareto provides the best overall fit except in the validation range, where Weibull improves performance; for block aggregation, GEV dominates in the first two station sets, while Weibull is applied to the subsequent ones. On the aggregated scale, the GEV yields the lowest mean error, but a case-by-case criterion is adopted, prioritizing the best-performing fit for each station set. Although this flexibility increases methodological complexity, it ensures that return-period estimates remain physically meaningful and avoids systematic biases associated with the rigid application of a single distribution family. This result reinforces two key ideas: (i) the need for adaptive selection of the distribution family depending on the domain and the range considered (observed versus extrapolated), and (ii) the importance of reporting uncertainty bands specific to the fitted model used in each range.

Finally, several limitations and avenues for improvement, already implicit in the study design, should be acknowledged: (a) the use of annual maxima facilitates comparability but may underutilise POT (peaks-over-threshold) information and seasonality; (b) spatial dependence among stations suggests the use of resampling techniques or spatial extreme-value frameworks to better quantify uncertainty; and (c) the order in which stations are incorporated could be explored through sensitivity analyses (e.g., starting with stations exhibiting the highest maxima or based on climatic proximity criteria). Nevertheless, the primary objective is fulfilled: the proposed spatial method is viable, behaves as expected, and produces stable and traceable return period estimates once the relevant climatological heterogeneity of each basin is adequately covered. In this sense, the annual maximum approach proves sufficient for first-order basin-scale hazard estimation when combined with explicit spatial aggregation. In any case, the annual maximum approach is enough for the areal IDF analysis. More sophisticated options such as three- and four-parametric distributions would be more adequate for daily precipitation modeling from entire time series [42].

4.3. Episode Time Structure

The use of the fractal–intermittency n-index provides physical–statistical context to the analysed episodes (e.g., the 2024 DANA event in Turís), allowing comparisons of temporal efficiency and intermittency against historical cases (such as the 2008 event), and linking temporal structure to accumulated severity. From a hazard perspective, this temporal characterization helps explain why events with similar daily totals may result in very different impacts, depending on their degree of concentration and effective duration. The IDF combination of n-index, annual maxima, and tail fitting offers a consistent interpretation of hazard: episodes with values close to $n\approx 0.3$–$0.5$ (more efficient rainfall) can concentrate large amounts within relatively short effective windows, which is highly relevant for threshold definition and for the interpretation of regional maxima employed in spatial return period estimation.

Therefore, although the n-index is independent of the spatial return periods [46], it provides complementary physical insight into the structure of extreme events dominating the spatial maxima. Events characterized by intermediate n-index values exhibit an efficient concentration of rainfall, which increases the likelihood that at least one station within the domain records an extreme annual maximum. As a result, such efficient events are more prone to dominate the OR-type spatial maxima and, consequently, to control the highest spatially integrated return periods. In this sense, the n-index helps to interpret why certain events, such as October 2024, emerge as regional extremes beyond what would be expected from point-based frequency analysis alone.

4.4. Final Recommendation

From an operational perspective, the study supports three recommendations: (i) Explicitly define the spatially integrated return period (e.g., “at least one station”, k-of-N, or areal average) according to the management question, as values are not interchangeable across definitions. (ii) When aggregating stations in blocks (here, groups of five), jointly assess return periods and errors, and stop aggregation when changes in return periods fall below a practical threshold (e.g., <5%) and the error reaches its plateau. (iii) Accompany each return period estimate with its associated uncertainty and with the fitting method selected for that specific station set and range, in line with the distribution-specific error evidence for each basin. (iv) Analyze the dependence of the spatially integrated return period and the number of stations considered using an empirical relationship (e.g., Equation (4)), and finally obtain a better estimate of the annual maxima ($P_{1day,1year}$, Equation (7)) and its corresponding IDF curve (e.g., Equation (8)).