Temporal Distribution of Extreme Precipitation in Barcelona (Spain) under Multi-Fractal n-Index with Breaking Point

Abstract

Rainfall regimes are experiencing variations due to climate change, and these variations are adequately simulated by Earth System Models at a daily scale for most regions. However, there are not enough raw outputs to study extreme and sub-daily precipitation patterns on a local scale. To address this challenge, Monjo developed the n-index by characterizing the intensity and concentration of precipitation based on mono-fractal theory. In this study, we explore the use of a multi-fractal approach to establish a more accurate method of time scaling useful to study extreme precipitation events at a finer temporal resolution. This study was carried out on the reference station of Barcelona (Spain) and its surroundings in order to be representative of the Mediterranean climate. For return periods between 2 and 50 years, two variables were analyzed: the n-index and the reference intensity $I_0$. Moreover, a new parameter, the so-called “breaking point”, was designed here to describe the reference intensity $I_0$, which is predominant for low time ranges. The results showed that both parameters are dependent on the time steps and the return period, and the scores confirmed the validity of our approach. Finally, the n-index was projected under downscaled CMIP6 climate scenarios by 2100, showing a sustained increase of up to +10%.

1. Introduction

1.1. Motivation

Because of the highly non-linear behavior of the rainfall regimes, which experience important variations due to climate change [1], precipitation must be analyzed with higher time resolutions (or time steps) than the ones typically used in meteorology (between one hour and one day). To address this challenge, it is possible to simulate rainfall in a shorter time step using scaling techniques to support the analysis of future local risks such as flash floods. This theory can be applied on the basis of the fractal properties using the n-index of Monjo et al. [2,3,4].

The n-index highlights the self-similarities at the different time scales that rainfall presents [3]. In geometry, this property is known as fractality, which can be briefly explained as follows: the geometric dimension of a point is zero, while it is equal to one for a line; however, the intensity of rainfall is neither punctual nor linear, but fractional, so its dimension is bounded between zero and one [5,6,7,8]. The time scale can influence the fractality: When the geometric dimension depends on the time step, it is known as multi-fractality; on the other hand, a low dependency of the time scale is assumed under the mono-fractal hypothesis [4,7,9,10].

A study by the authors of [11] published in 2012 showed that precipitation anomalies in Sahel exhibit persistent long-range correlations for all the time lags between 4 months and 28 years, during the period of 1900 to 2010. This result states that the fluctuations of the Sahel precipitation anomalies in short time intervals are positively correlated to those in longer time intervals in a power law fashion. However, for the period 1948–2001, the behavior of this parameter became almost random. This study showed, on one hand, that power law was particularly suited in the frame of precipitation and, on the other hand, the importance of a data set that encompasses a longer time scale.

The use of fractal geometry is relevant in the theoretical framework for detecting possible changes in the intensities of future precipitation events according to the return periods. To achieve this purpose, we can fit several theoretical distribution functions such as Gumbel, Gamma, Generalized Pareto, and Generalized Extreme Values (GEV), among others [12,13]. Moreover, a power law is usually considered to accommodate the intensity-duration-frequency (IDF) curves to the maximum averaged intensity (MAI) curves [2,4,13], only dependent on two parameters: the reference intensity $I_0$ of an IDF curve (for a given partial duration $t_0$) and the n-index, which is assimilated to the fractal dimension of the precipitation intensity.

Therefore, the objective of this study was to analyze the dependence of these two parameters on the time step analyzed and the return periods considered as indicators of the multi-fractality of the precipitation. This method can be applied on any station endowed with observed data at a very low time step (i.e., a finer time resolution). Moreover, it is a key for flood modeling, such as in the Improving ClimAte Resilience of crItical Assets (ICARIA) project [14]. In particular, our technique was mainly tested in the Jardí gauge of the Fabra Observatory of Barcelona (Spain), which is considered a standard reference due to its high time resolution (5 min data since 1927) and due to the many analyses performed using its data [4,15,16]. It includes analyses of extreme rain rates [17], calibration of the pluviograph data to obtain hourly rainfall [18], the study of the convective ratio of rainfall [19] and semi-empirical approach of the maximum precipitation expected [20], as well as multi-fractal parameters to characterize local rainfall, IDF curves, and time scaling [4,16,21,22].

1.2. State of the Art

1.2.1. Maximum Averaged Intensities (MAI)

The maximum amounts of precipitation, $P_{max}\left(t\right)$, for several time intervals (t) can be ordered according to the time interval length within a particular rainfall event [2]. Thus, the maximum averaged intensities (MAI) are defined by each maximum amount divided by the time interval, as follows:

$$I\left(t\right)=\frac{P_{max}\left(t\right)}{t},$$

(1)

where $I\left(t\right)$ is the MAI and $P_{max}\left(t\right)$ is the observed maximum precipitation accumulation in a period of time t. However, Equation (1) can be theoretically approximated by the power law of Equation (2) as described by Monjo [3].

$$I\left(t\right)=I_0\left(t_0\right){\left(\frac{t_0}{t}\right)}^n,$$

(2)

where $I\left(t\right)$ and $I\left(t_0\right)$ are MAI, corresponding to times t and $t_0$, and n is a dimensionless parameter that is ranged from 0 to 1. Thus, to quantitatively classify a precipitation event, we will need three theoretical values:

• A maximum mean reference intensity of any reference $I_0\left(t_0\right)$,
• The duration of the rainfall event t,
• The variability of the intensity, according to the exponent n.

The advantage of this theoretical formula (Equation (2)) is that the classification criterion is independent of the absolute maximum intensity, so it focuses on the variability of the intensity of precipitation, represented by the n-index, whose values range between 0 and 1. That means that we can have intense precipitation that can be characterized by both constant or variable intensity at different levels. There are many advantages to using geometric indicators in water management and other related sectors. First of all, they provide a robust standardization for comparability between events at different spatial or temporal scales. In contrast, classical variables such as the total precipitation amount do not provide information on the organization or structure of the precipitation system at the finer time scales necessary to implement adaptation measures. There are multiple trans-sectorial indexes commonly used by modelers to support decision making, such as the concentration index (CI) that measures the climatic inequality of daily precipitation [13,23], the rainfall erositivity index (R), used to measure the degree of rainfall-induced soil erosion [24], the fire weather index (FWI), used worldwide and developed in Canada for forest fires [25] and the sustainability indices (SI) [26] among others, used in water management. The information provided by the n-index highlights its potential usefulness to modelers to support informed decision-making to policymakers regarding the risk involved by extreme or irregular precipitation and dry spells. The n-index has already been used in European projects to deal with extreme precipitation-related risks in hand with other indicators of the anticipation chain of dangerous events [4,14].

This n-index model is applied to the precipitation registers of a specific event, isolated in time, and tries to quantify the regularity pattern related to more convective or stratiform precipitation in the field of meteorology. However, it can also be applied to climatology to establish typical regimes, classification, and climate change analysis [4].

1.2.2. Intensity-Duration-Frequency (IDF) Curves

The most common method to analyze precipitation in the climate domain is the intensity-duration-frequency (IDF) curves [27]. The IDF curves are obtained by fitting an empirical or theoretical cumulative distribution function (e.g., Gumbel law or Generalized Extreme Value distribution, among others) and using the return periods to approach extreme events of precipitation. The IDF curves are key tools in hydrology to prevent floods with engineering solutions by using the frequency analysis method and studying the associated return period [28]. Therefore, they can also be used for precipitation. As described by Moncho et al. [2], we can adjust the IDF curves using the MAI curves. This is possible because the use of the dependence of return periods in the MAI formulation highlights the fact that the IDF curves are, by definition, a climatic representation of the temporal distribution of the MAI curves (Figure 1). Setting a specific return period p, this leads to:

$$I(t,p)=I_0(t_0,p){\left(\frac{t_0}{t}\right)}^{n\left(p\right)},$$

(3)

where $n\left(p\right)$ is now an n-index function that depends on the return period p, so it can be generalized to a multi-fractal dimension $n(t,p)$ that also depends on t, in contrast to a constant (mono-fractal) n-index [4].

Therefore, the main objective of this article was to model the two parameters, $I_0(t_0,p)$ and $n(t,p)$, according to the time step t and the return periods p to determine the eventual relation between these parameters, under the multi-fractal approach.

However, it is worth noting that there are other approaches to characterize the IDF curves. As an example, the Sherman formula [29]:

$$I\left(t\right)=\frac{a}{{(t+b)}^c}\Rightarrow I\left(t\right)=\frac{I_0{t}_0^n}{{(t+\cancel{b})}^n},$$

(4)

which depends on three parameters a, b and c. The first one, a, can be interpreted as a reference intensity ($I_0$) multiplied by a time reference ($t_0$) at power $c=n$, that is $a=I_0{t}_0^n$, while c would be the n-index and have the same interpretation. For b, this is not as obvious as the two other parameters. The dimension of b is a time indeed. We can notice that if b is null, the formula is the same as in Equation (3). We can mathematically interpret b as the breaking point between the mono-fractal approach and the multi-fractal approach, which is a key parameter in this work. Therefore, this paper proposes a novel procedure for assessing the impacts of climate change on the precipitation regimes at sub-daily scales: (1) the identification of the breaking point between the mono-fractal and the multi-fractal behavior of the n-index, (2) its application for empirical IDF curves and (3) its assessment under the context of climate change.

2. Data

Reference Observations

Although the Fabra Observatory of Barcelona is the main observatory for this study, which has the largest data set with a 5 min time resolution, three other observatories were used to test the method more robustly (Figure 2). To achieve this, there was also a collaboration with the Catalan Meteorological Service (Servei Meteorològic de Catalunya) to use their data. These complementary observatories are listed in

Table 1. Table of main characteristics of the observatories, including identification, longitude, latitude, altitude (masl), name and the specifications (time resolution and observation period) of the data used in this paper (Figure 2 left).

Id	Lon	Lat	Altitude	Name	Resol.	Period
D5	2.12	41.42	411.00	Barcelona-Obs. Fabra (main station)	5 min	1927–2016
UF	1.91	41.29	573.00	PN del Garraf - el Rascler	10 min	1996–2020
UG	2.04	41.30	3.00	Viladecans	10 min	1996–2020
VT	2.13	41.48	84.00	Cerdanyola del Vallès	10 min	1996–2020

(rows 2 to 4), and they have a maximum length of 11 years and a time resolution of 10 min.

Data from the main observatory, the Fabra Observatory of Barcelona, were obtained thanks to the collaboration with the Spanish and Catalan meteorological centers in a precedent work of the Climate Research Foundation—Fundación para la Investigación del Clima (FIClima) [4].

This station has a sample of five-minute observations from 10 June 1927 to 9 August 2016 with a gap between the 10 of October 1992 and the 13 of July 2008. However, we have a total of 7,728,191 observations, which is a robust sample and remarkable due to the date of the first observation.

3. Methods

3.1. Climate Models

In order to produce future climate projections of the daily n-index for Barcelona, it was necessary to use downscaled outputs of the Coupled Model Intercomparison Project Phase 6 (CMIP6), previously generated with the FIClima method [30] for the observatories of interest. For this purpose, climate projections (SSPE1-2.6, SSP2-4.5, SSP3-7.0 and SSP5-8.5) were obtained through statistical downscaling applied to the ten Earth System Models listed in

Table 2. Earth System Models previously downscaled with the FIClima method [30], and used for the study of future projections (1951–2100) of the n-index of Barcelona.

Name	Institution	Released Year	Resolution Lon (°) × Lat (°)	Reference
ACCESS-CM2	CSIRO-ARCCSS	2019	1.9° × 1.3°	Ziehn et al. [31]
BCC-CSM2-HR	BCC	2017	1.1° × 1.1°	Wu et al. [32]
CanESM5	CCCma	2019	2.8° × 2.8°	Swart et al. [33]
CMCC-ESM2	CMCC	2017	1.3° × 1.0°	Cherchi et al. [34]
CNRM-ESM2-1	CNRM-CERFACS	2017	1.4° × 1.4°	Voldoire et al. [35]
EC-Earth3	EC-Earth-Consortium	2019	0.7° × 0.7°	EC-Earth Con. [36]
MPI-ESM1-2-HR	MPI-M, DWD, DKRZ	2017	0.9° × 0.9°	Gutjahr et al. [37]
MRI-ESM2-0	MRI	2017	1.1° × 1.1°	Yukimoto et al. [38]
NorESM2-MM	NCC	2017	1.3° × 0.9°	Bentsen et al. [39]
UKESM1-0-LL	MOHC, NERC, NIMS, NIWA	2018	1.9° × 1.3°	Sellar et al. [40]

, simulated at a local scale for the UF, UV and UG observatories (Table 1) under the framework of the ICARIA project.

3.2. In Brief

In order to give a brief summary of the multi-fractal n-index model, Figure 3 shows an example that illustrates the self-similarities of precipitation between time scales within an event. By only analyzing the hyetograph (Figure 3a), this behavior can already be detected. The method can be summarized as follows: From the original data (aggregated with several time steps), we obtain cumulative precipitation curves ($P_{max}$) that approximately adjust to an iso-n curve, and we obtain the maximum averaged intensities (MAI) according to Equations (1) and (2) (Figure 3b,c). These intensities strongly depend on the frequency of the extreme events, so we calculate the expected values for several return periods in Section 3.3. Then, the theoretical model is validated to identify the applicability range of the model (e.g., Figure 3c shows a good performance for a duration larger than 20 min). Finally, we analyze the multi-fractal behavior of the IDF curves by focusing on the different values of the n-index according to the time step (Figure 3d). In particular, we use two semi-empirical curves to characterize the n-index at every time step and for every return period, as described by Equations (7) and (6) of Section 3.4.

3.3. Wet Spells and Return Periods

The principle of our approach is to apply a spell function to different time steps of the data set for the same station and period [4,10]. The spell function separates dry events from wet spells, defined as at least three consecutive time steps with a non-zero precipitation record. To study precipitation patterns, we focus on the MAI characteristics of wet spells, which are the parameters n-index and reference intensity $I_0$.

In fact, to obtain the temporal component of the IDF curves (Equation (3)) and then to use the downscaled climate model outputs, an approximation of these two parameters is needed. However, this equation also depends on the return period p; therefore, we need to find a threshold according to each return period, which is defined by Equation (5), so the thresholds are obtained by reversing this definition according to the extreme value theory. Thus, for a given return period p, the threshold Y is given by

$$T=F(x\le Y)=1-\frac{1}{p},$$

(5)

that is, $T\in (0,1)$ is the quantile of the empirical distribution $F(x\le Y)$ of the precipitation amount equal or lower than Y for the selected station and time period. In this study, we chose five return periods to study: 2, 5, 10, 25 and 50 years. The thresholds are detailed in

Table 3. Maximum precipitation $P_{max}$ (in mm) according to time steps t (in hours) and the return periods p (in years) on the station Jardí gauge of Fabra Observatory of the Royal Academy of Sciences and Arts of Barcelona. The expectation value of the duration d*, estimated for each return period (in days), aimed to show that the duration of an extreme event increases with the duration.

p (Years)	Time Steps t (in Hours)
	d *	1/12	1/4	1/2	1	2	3	4	6	12	24
2	5.5	12.1	20.7	27.8	36.4	40.8	42.3	47.9	49.0	59.3	66.0
5	6.5	14.1	26.6	34.3	45.5	55.0	58.2	59.3	62.2	71.7	82.6
10	7.5	17.1	32.6	37.9	50.3	69.1	70.6	70.6	76.4	87.6	96.1
25	8.4	20.2	33.3	41.8	55.5	75.6	77.1	94.1	95.1	95.1	107.8
50	8.9	20.2	37.9	46.4	64.2	84.3	91.5	100.5	104.9	105.8	122.0

* Duration of a wet spell with a step time of fifteen minutes with the thresholds associated with the two-hour return period (see Section 5.2).

.

As shown in Table 3, the number of time steps is very important to obtain a representative aspect of the evolution of these parameters. After building these thresholds and applying them to the result of the spell function, it is possible to study the extreme precipitation events. The return periods are obtained from empirical quantiles, which allows for a study of extreme rainfall events.

There are multiple methods to apply these thresholds. We focus on one of them here, namely applying thresholds on the accumulation of total precipitation of the event.

3.4. Multi-Fractal Models

3.4.1. Temporal Resolution Models for n-Index

After establishing the empirical distribution of both the n-index and the reference intensity $I_0$, we addressed their multi-fractal behavior by using regression models that adequately represent the observed relation between these features and the time steps for all return periods considered.

To study the n-index, two methods were selected: an approach with two parameters and an approach with three parameters.

The first relation considered to approximate the variability of the n-index with two parameters is based on the physical idea of $n\sim 1/log\left(t\right)$, because of its formula given by Equation (3), that is

$$n=n_{max}-\frac{k}{log(t/\min )},$$

(6)

where $t\ge 5$ min, while $n_{max}$ and k are the two parameters of the model.

Another technique was selected to analyze the multi-fractality of the n-index. In particular, it was based on the convergence of this parameter to an asymptotic value $n_{\infty }$ as the time step approaches infinity (Section 5.1), starting from an initial value $n_0$ for a time step that theoretically approaches zero (Figure 3). Therefore, we can design an empirical three-parametric model for the multi-fractal n-index of each return period as follows:

$$n=\frac{n_0}{1+at}+\frac{n_{\infty }}{1+\frac{1}{at}},$$

(7)

where t is the time step and $a>0$ is a time-scale parameter that indicates the velocity of convergence of the n-index from the initial value $n_0$ to the infinite value $n_{\infty }$.

The key idea of this approach is to quantify the three parameters $n_{\infty }$, $n_0$ and a to have a (non-linear) regression of the n-index. The method selected to approximate the three parameters is a non-linear model from R language.

3.4.2. Temporal Resolution Models for Reference Intensity

For the reference intensity, as the relation is a power law function of the time step, a relationship log-log was chosen. The formula of the curve is, according to Equation (3), outlined in Equation (8).

$$log\left(I_0(t,p)\right)=log\left(I_{00}\left(p\right)\right)+n\left(p\right)log\left(\frac{t_0}{t}\right),$$

(8)

where $I_0$ is the reference intensity research, $I_{00}$ is a reference intensity to fit the power law, t is the step time, $t_0$ is a time step reference equal to 1 (hour) here and n is the index $n\left(p\right)$ average on every event of the return period p.

We can notice that in Figure 6a, for the five-minute time step, the curve does not fit well to the data. A better curve can be found by adding more statistical parameters, but it has to include a physical sense.

As shown in Equation (4), Sherman’s formula depends on three parameters, and their physical senses are explained above. However, the parameter ‘b’ also has a meaning, which is a breaking point step between the mono-fractal and multi-fractal approach or the cascading approach.

In fact, by applying the Akaike-based non-linear minimization on the reference intensity $I_0$, it is possible to obtain an approximation of these three parameters on the station Jardí gauge of Fabra Observatory of the Royal Academy of Sciences and Arts of Barcelona. The function to minimize is

$$I_0(t,p)=I_{00}\left(p\right){\left(\frac{1+t_m}{t+t_m}\right)}^{n\left(p\right)},$$

(9)

where $I_{00}$ is a fitting parameter that represents the reference intensity at the step time $t=1$, for a breaking time $t_m$ and considering the n-index. It is a reformulation of the Sherman function.

4. Results

4.1. Multi-Fractal n-Index Models

The modeling of the multi-fractal n-index is sensitive to the number of data. Thus, for example, high return periods such as 25 or 50 years can present different behavior with respect to the lower return periods. This is found in our results where, for the 25-year return period, the curve has an unexpected shape.

This is linked to the data set because even originally, in Figure 4c, the 25-year return period had a different behavior than the four others. Taking this problem into account, we can set the choice of the accurate regression model to simulate the multi-fractal behavior of the n-index.

As mentioned previously, we can distinguish between two different sets of duration: less than an hour and more than one hour.

The regression that only takes into account the data with a time step of less than one hour is noticeably better adapted for the first time period according to Figure 5b. However, the problem of noise after one hour remains, and we notice that the curve seems to underestimate the n-index. Moreover, the shape of the curve is increasing, as expected.

The n-index is much more complicated to estimate for a time period greater than one hour according to Figure 5. The main explanation was mentioned before: the noise around a probable limit value, which depends on the data set. A proof is that we can notice, independently of the method of regression or the return period, the same pattern on every bias chart. However, despite that noise, we can conclude that the non-linear model seems to be the most appropriate way to approach the curve for the time steps over one hour.

In addition, two statistical metrics were added to estimate the most appropriate regression method: the Mean Absolute Error (MAE) and the squared Pearson correlation ($r^2$) in

Table 4. Statistical scores for the approach of n according to the two-parameter (Equation (6)) and three-parameter (Equation (7)) regression models for the different return periods. The coefficients are presented in the Appendix A.1 (Table A1).

Statistical	Return Periods p
Scores	2	5	10	25	50
	Two-parameter regression on the full period
MAE	0.0199	0.0199	0.0211	0.0230	0.0262
$r^2$	0.854	0.846	0.721	0.619	0.566
	Two-parameter regression for less than one hour
MAE	0.0311	0.0264	0.0216	0.0224	0.0255
$r^2$	0.990	0.996	0.999	0.979	0.957
	Three-parameter regression
MAE	0.0113	0.0147	0.0187	0.0247	0.0249
$r^2$	0.961	0.893	0.772	0.576	0.703

. The objective of the first one is to quantify the mean difference between the empirical curve and the regression, and the objective of the second one is to have an indicator of the correlation.

We can conclude that every regression method can be improved. Even if, considering the resulting bias, the regression for time steps lower than one hour is the most appropriate on the same period, after one hour, the value of $r^2$ decreases (because the variance remains the same as the two-parameter regression method on the full data set due to the formula and therefore the $r^2$).

The noise around a potential convergence value (determined with the non-linear method) is the reason for the low score. Considering the MAE score, the best approach would be the three-parameter method. However, the result is not clear for the high return periods using only the MAE, but $r^2$ confirmed that the three-parameter regression is better for this station.

4.2. Multi-Fractal Reference Intensity Models

The function of Equation (9), relative to the reference intensity, was easily represented (Figure 6b). The shape is still the same as the curves obtained in Figure 6a. In addition, the three-parameter approach is more accurate in representing the differences between short and long time steps.

The bias score was selected to measure the performance of the two approaches selected to model the multi-fractal behavior of the reference intensity $I_0$ (Figure 6). The main difference appears for the lowest time steps: the explanation is the parameter $t_m$. For values below the lower bound of the confidence interval, the approach cannot be the same, according to Gutierrez-Lopez [41]. This parameter is predominant for the study of precipitation at a very low time step: “Accepting that C ($t_m$ in Equation (9) of this study) is part of the duration factor of the Sherman equation, it is recommended that future studies, and even IDF curves already calculated, in very specific sites should be reviewed considering the physical meaning of parameter C.” The parameter $t_m$ is the mean duration of storms in situ, although it is not calculated in this study according to the surrounding stations. The storm with a duration shorter than $t_m$ requires to be analyzed according to multi-fractal hypothesis [4]. Storms studied with this method with a 5-min time step can be less than 0.3 h (typical value of $t_m$ according to

Table A2. Coefficient and p-value for the approaches of $I_0$ according to the two-parameter (Equation (8)) and three-parameter (Equation (9)) regression models for the different return periods (years).

Statistical	Return Periods p
Scores	2	5	10	25	50
	Two-parameter regressions on the full period (Figure 6a)
	p-value
intercepts	<$0.001$	<$0.001$	<$0.001$	<$0.001$	<$0.001$
$log\left(t\right)$	<$0.001$	<$0.001$	<$0.001$	<$0.001$	<$0.001$
	Coefficient value
$I_{00}$ [mm/h]	1.83	1.79	1.73	1.67	1.68
n-index	0.573	0.570	0.581	0.580	0.570
	Three-parameter regression (Figure 6b) *
$I_{00}$ [mm/h]	21.9 [21.3,22.4]	25.4 [24.2,26.6]	27.0 [25.5,28.5]	28.1 [27.1,29.0]	30.0 [28.4,31.5]
n	0.802 [0.737,0.866]	0.830 [0.691,0.967]	0.802 [0.659,0.945]	0.703 [0.630,0.775]	0.719 [0.605,0.834]
$t_m$ [hr]	0.331 [0.263,0.399]	0.405 [0.244,0.566]	0.323 [0.176,0.470]	0.202 [0.137,0.268]	0.249 [0.135,0.363]

* The three parameters regression is obtain by using the minimization function nlm (non-linear minimization) of R, there is therefore no p-value and the coefficients are given with confidence interval to assess the validity of the method.

). It should be noted that $t_m$ is independent of the return period, despite the large confidence interval, according to this reference and

Table A1. Coefficient and p-value for the approaches of n-index according to the two-parameter (Equation (6)) and three-parameter (Equation (7)) regression models for the different return periods (years).

Statistical	Return Periods p
Scores	2	5	10	25	50
	Two-parameter regression model on the full period (Figure 4a)
	p-value
$n_{max}$	<$0.001$	<$0.001$	<$0.001$	<$0.001$	<$0.001$
k	<$0.001$	<$0.001$	$0.002$	$0.007$	$0.012$
	Coefficient value
$n_{max}$	0.60	0.58	0.55	0.52	0.52
k	0.44	0.4	0.32	0.25	0.27
	Two-parameter regression model for less than one hour (Figure 4b)
	p-value
$n_{max}$	<$0.001$	<$0.001$	<$0.001$	<$0.001$	<$0.001$
k	0.005	<$0.001$	<$0.001$	0.010	0.022
	Coefficient value
$n_{max}$	0.54	0.53	0.52	0.53	0.50
k	0.30	0.28	0.25	0.28	0.23
	Three-parameter regression model (Figure 4c) *
$n_0$	0.342 [0.300,0.383]	0.366 [0.303,0.428]	0.394 [0.343,0.445]	0.289 [0.000,0.652]	0.398 [0.329,0.467]
$n_{\infty }$	0.563 [0.543,0.583]	0.555 [0.510,0.600]	0.552 [0.463,0.641]	0.477 [0.446,0.507]	0.567 [0.332,0.467]
a	0.026 [0.007,0.045]	0.016 [0.001,0.043]	0.006 [−0.008,0.020]	0.136 [−0.417,0.689]	0.002 [0.011,0.016]

* The three parameters of the regression model are obtained by using a non-linear minimization function so-called nlm in R language.

. That is why $t_m$ is independent of the return period in Equation (9). By comparing these two approaches with the bias statistic, we can conclude on two points:

For the first two hours, the non-linear model with the Sherman function is more accurate. This is mainly due to the third parameter $t_m$, which in a certain way includes more precision because, before this breaking point, as supported by the fractal theory, the rainfall dimension depends on the time scale (i.e., multi-fractal dimension). There is, therefore, a significant bias with the two-parameter regression. The non-linear model is more appropriate in this case, and $t_m$ is nonzero according to the confidence interval (Table A2 in Appendix A.2).

For time steps longer than two hours, both regressions are similar according to the bias. There are two explanations to conclude on this finding. The first is that, as mentioned in Section 5.2, $I_0$ converges toward a value independently of the return period and the approach period, because precipitation has a shorter duration than the time step (e.g., twenty-four hours). The second explanation is that if $t_m=0$ or b in the Sherman formula, Equations (3) and (4) are both equal. As $t_m\sim 0.2$ h, for time steps greater than two hours, there is at least one order of magnitude between these two features of the equation. Therefore, these equations are almost the same, and we can work under the mono-fractal hypothesis according to Monjo et al. [4].

However, it is worth noting that the regression with two parameters leads to an underestimation of the parameter $I_0$ and an underestimation of the global n-index. The n-index with three statistical parameters does not have the same meaning, it is still bounded between 0 and 1, but because of the presence of $t_m$, we cannot have the same conclusion as in Section 5.1 (Moncho et al. (2009) [2]).

In addition, we calculated two statistical metrics to consolidate the choice of the three-parameter regression over the two-parameter regression ones. According to

Table 5. Statistical score for the approach of $I_0$ according to the two-parameter (Equation (8)) and three-parameter (Equation (9)) models for different return periods (find the fitted parameters in Table A2).

Statistical	Return Periods p
Scores	2	5	10	25	50
	Two-parameter regression
MAE	3.43	4.32	4.50	4.28	4.80
$r^2$	0.927	0.907	0.921	0.948	0.934
	Three-parameter regression
MAE	0.458	0.872	1.12	0.782	1.13
$r^2$	0.999	0.996	0.995	0.998	0.995

, the $r^2$ is clearly higher, and the MAE is much lower for the first. This leads us to recommend the three-parameter regression, even if using the result of the other approach can be useful to obtain the optimal initial conditions in order to apply a non-linear minimization.

5. Discussion

5.1. Time Sensitivity of n-Index

The initial analysis suggested that the multi-fractal characteristics exhibited by both the n-index and the reference intensity are contingent upon the return periods; however, discernible nuances are observable for each parameter.

For the n-index, we noticed that for time steps shorter than one hour, the shape of the empirical curve looks regular, but for those greater than one hour, it seems to include unpredictable noise (see Section 4.1). The n-index values can be summarized as follows: On average, for short time intervals (e.g., less than one hour), the n-index falls within the range of 0.35 to 0.5, indicative of deep convection precipitation as outlined by Monjo et al. [3] for the Barcelona region.

The n-index is higher as the time step increases, which follows from the fact that stratiform precipitation events generally have an n-index under 0.4 at a low time scale [4] because they are mostly spatially extended and regular, but on a longer time scale, precipitation is more irregular and has a shorter relative duration (twelve-hour time step as an example). The physical origin of the n-index ranges is not the same on different time scales since there is a distinct response according to the precipitation patterns. In particular, when the time step increases, the origin of wet spells is more likely to be due to a frontal organization (due to the definition of the wet spell and their duration) even if the n-index value is characteristic of a different regime. In contrast, the finer time resolutions inform about the mesoscale or microscale processes.

The range of the n-index values is not so high, but the shape of the curve above the one-hour time step is sensitive to the data set and the station due to the precipitation regimes in this zone. As is the case here, according to Monjo et al. [3], the mean n-index in Barcelona is between 0.5 and 0.6 for sub-daily time steps (composed of data of one, three, or six hours). After one hour, we are approximately in this range.

It is relevant to remind the reader that Barcelona has multiple precipitation patterns, from unorganized convection to frontal systems, as we notice in Figure 4, due to its location near the Mediterranean Sea.

The n-index is not easy to model with a simple regression because above the one-hour time step the shape of the curve seems to include unpredictable noise (Figure 4). As highlighted in Table 4, the squared of the Pearson correlation coefficient ($r^2$) is generally less than 0.95, which indicates that the relationship is not statistically robust. To avoid this problem, a regression was considered using only the points with a time step less than one hour, plotted in Figure 4b. We see that when the r-square is above $r^2>0.95$, the relation is robust.

For the specific modeling of the n-index, despite the noise, we saw through the bias that a simple regression model below the one-hour time step provides a great result for studying the low-duration event or the long event with an inferior time step. This breaking point in the multi-fractal behavior of the n-index is an important piece of information considering the issues of this study and the IDF curves: having data on one-hour time steps allows for a lower temporal resolution according to the law.

Moreover, the third parameter $t_m$ can be endowed with a significant uncertainty ($t_m\approx 0.33\pm 0.05$ hours), which can be a problem and a significant point of contention in a way to approach precipitation with a short duration, such as the monocellular rainfall system or with short significant intensities, such as in unstable fronts. A primary study of the station using these methods to establish the IDF curve remains essential in order to obtain a better representation of a potential event. The breaking point of about $t_{bp}\sim 1$ seems to be related to the transition point given by the parameter $t_m$ such as $t_{bp}\sim 3\times t_m$.

5.2. Time Sensitivity of Reference Intensity

The results obtained for the reference intensity $I_0$ are promising; however, the statistical scores calculated in Section 4.2 hide the fact that for low time steps, the two-parameter regression is far from the empirical curve, and well-fitted for the other time steps. Nevertheless, the three-parameter regression is still better, and this means that the intensity, independently of the return period or the time steps, can be approximated with a low potential bias between the curve and the real event.

An interesting feature of the reference intensity is the convergence for the high time steps toward the same value, regardless of the return period. This is logical and the explanation is similar to the precedent note: even if the thresholds are different according to the Table 3, a three-day wet spell is composed of several different intensities; therefore, the mean intensity is almost the same independently of the return period. This can also be explained by the fact that the most extreme events are sometimes the longest (that is, similar intensity but longer effective duration), according to Table 3. At this point, the result of a fifteen-minute spell function is used with a two-hour return period, because if both have the same time step, the longest events (with the longest duration of the wet spell) are the same as the ones with the highest accumulation. Events with short return periods can have the same duration as events with long return periods. Return periods, calculated on a larger time scale basis, can be compared to the finest time scale data of accumulation to analyze the highest accumulation and not be dependent on the duration of the wet spell. The new threshold is a quantile that highlights the longest wet spells. Moreover, within a 50-year return period event, it is possible that many consecutive storms occur with variable intensities and durations, and therefore, are averaged all together in the threshold.

In any case, both the n-index and the reference intensity $I_0$ are linked to the energy of the precipitation systems, since the time structure depends on the speed of the vertical (convection) and horizontal (stratiform) fluxes. As these processes depend on each life cycle, both indicators can present sensitivity on the time step considered.

5.3. Climate Projections of Daily n-Index

To support the discussion above, future projections on the supra-daily n-index were addressed. To make this, several projections were generated for different climate change scenarios through the application of the FIClima statistical downscaling [30] to ten Earth System Model outputs mentioned in Table 2 for the UF, UG and UV observatories (Table 1).

The projections show a clear upward trend of the index over the years, increasing with the radiative forcing implied by the scenario (Figure 7). Taking advantage of the strong correlation among the n-index values at different time scales [3], we can state that an increase in the supra-daily scale index implies an increase in the sub-daily scale index, and therefore an increase in precipitation concentration at smaller time scales is also expected.

However, it should be noted that the self-similarity between supra-daily and sub-daily scales is limited for the Barcelona region [3,10]. According to the results obtained in the previous sections (Figure 4), the n-index remains approximately constant on scales equal to or larger than the hourly scale, guaranteeing the mono-fractal hypothesis at these scales. Therefore, it can be affirmed that the climate projections obtained at the daily scale can be extrapolated at least to hourly scales.

The increase in the n-index implies a rise in the irregularity of precipitation, which results in increased concentrations during rainfall events. In turn, it is consistent with the increase in energy in the climate system due to the effect of anthropogenic emissions. In particular, this can be related to the enhancement of deep convection associated with the surge in Convection Available Potential Energy (CAPE) and the rise in “capping” by Convective Inhibition (CIN) in the Mediterranean region [42]. The increase in these parameters can be related to climate change, the rise in water vapor content in the mid-level troposphere and in the reduction of the low-level relative humidity due to increased surface temperatures [42], respectively. However, the relationship between the n-index and these parameters should be studied in more depth to assess their correlation.

These future scenarios of increasing the n-index for the city of Barcelona emerge as useful information to assess the hydrological risks in the coming decades and, therefore, for well-informed decision-making. In addition, the methodology presented in this section is useful to replicate in other Mediterranean regions where, given the projected decrease in precipitation, the management of water resources becomes even more important.

6. Conclusions and Perspectives

As is well known, research on extreme precipitation presents considerable challenges in obtaining significant conclusions due to the high natural variability and the scarcity of extreme values observed. This study identifies some key points by using the geometric n-index, which is robust and does not use units to compare different time scales. The primary use of the 5-min long time series of the reference station (Fabra Observatory) ensures comparability with previous studies. In particular, our results show strong physical consistency with respect to previous studies [4].

From these results, the first main conclusion is that our approach opens up new perspectives in the study of precipitation at different time steps and identifies when they are inter-comparable up to a breaking point between the mono-fractal and multi-fractal behaviors. Thus, the transitional multi-fractal-to-mono-fractal n-index is a particularly well-suited normalized indicator that can describe the change (breaking point) in precipitation patterns even about the mesoscale-to-synoptic structures. Applying this method to a given time series allows us to obtain a precise climatology of its precipitation structures at several time steps, for both the comparison of particular rainfall events and the assessment of climatic trends of these features.

As a second main conclusion, we discovered that the reference intensity and the n-index of a specific time series are highly dependent on the duration of the wet spell, and pointed to the limits of using a defined step time to characterize a storm (an accumulation in one hour, for example). An approach dependent on the duration of the event is needed to characterize the intensity and the pattern of precipitation, consequently, the hydrological risks. The approach of the IDF curves with a mono-fractal index can be accurate, but it remains difficult to approach the constants, especially the n-index. However, the IDF curve, formulated as in Moncho et al. [2], has the advantage of being intuitive by having a physical sense and contains the dependence of both return periods and the duration of the wet spells. Therefore, this work helps to refine this approach and includes some other conclusions around this method, such as climatology over time steps and the return period of the parameters or a method to approximate these features on a station.

The approach outlined in this paper can be considered from the perspective of another method of time scales particularly fitted for extreme events. This alternative was presented by [43] in 2016 in the form of extreme events associated with El Niño. Its technique entitled “Natural Time Analysis (NTA)” allows the detection of those characteristics of the dynamics of the complex ocean-atmosphere system that could be employed for the detection of precursory signals of major ENSO maxima [43,44]. Combined with the methods presented in this study, it could be used to lead a study on extreme event detection and their projection in the future using the formulation of IDF curves presented here.

The period used during the study can be important in the context of climate change. The evolution of the parameters can be studied using the CMIP6 climate scenarios [45] regarding the precipitation patterns in the Mediterranean basin. In particular, longer dry spells are expected but alternated with more intense rainfall in shorter events. In line with this goal, other studies should be led using different data sets for different periods to study the evolution of the constant in recent years and obtain projections on parameters n and $I_0$.