Regional Frequency Analysis Using L-Moments for Determining Daily Rainfall Probability Distribution Function and Estimating the Annual Wastewater Discharges

Abstract

The spatial distribution of precipitation is one of the major unknowns in hydrological modeling since meteorological stations do not adequately cover the territory, and their records are often short. In addition, regulations are increasingly restricting the amount of wastewater that can be discharged each year. Therefore, understanding the annual behavior of rainfall events is becoming increasingly important. This paper presents Rainfall Frequency Analysis (RainFA), a software package that applies a methodology for data curation and frequency analysis of precipitation series based on the evaluation of the L-moments for regionalization and cluster classification. This methodology is tested in the city of Palma (Spain), identifying a single homogeneous cluster integrated by 7 (out of 11) stations, with homogeneity values less than 0.6 for precipitation values greater than or equal to 0.4 mm. In the evaluation of the prediction capacity, the selected cluster of 7 stations performed in the first quartile of the 120 possible combinations of 7 stations, both for the detection of the occurrence of rainfall—in terms of Probability of Detection (POD), False Alarm Ratio (FAR), Critical Success Index (CSI) and Bias Score (BS) statistics—and for the accuracy of rainfall—according to Root Mean Square Error (RMSE), Nash–Sutcliffe Efficiency coefficient (NSE) and Percent Bias (PBIAS). The cluster was also excellent for predicting different rainfall ranges, resulting in the best combination for both light—i.e., [1, 5) mm—and moderate—i.e., [5, 20) mm—rainfall prediction. The Generalized Pareto gave the best probability distribution function for the selected region, and it was used to simulate daily rainfall and system discharges over annual periods using Monte Carlo techniques. The derived discharge values were consistent with observations for 2023, with an average discharge of about 700,000 m³ of wastewater. RainFA is an easy-to-use and open-source software programmed using Python that can be applied anywhere in the world.

1. Introduction

The Mediterranean climate is characterized by complex spatial and seasonal variability, with significant interannual fluctuations in precipitation levels [1]. Furthermore, in the context of anthropogenic climate change, there is a consensus that the frequency and intensity of heavy precipitation events have increased since the 1950s and will continue to increase [2]. In urban contexts, the occurrence of precipitation events, coupled with the intensification of urbanization activities, underscores the necessity to enhance the capacity of urban drainage systems. It is, therefore, evident that the statistical characterization of precipitation constitutes an indispensable component of the implementation of reliable hydrological and hydraulic models [3].

Frequency analysis of rainfall from weather station records depends on both the quality of the data and the length of the historical series. Despite the importance of developing in situ frequency analysis [4,5] both for extreme events [6,7] or drought periods [8], the lack of sufficiently long records limits the robustness of the required statistical data treatment [9]. Neighboring climate observatories are often used for quality control and gap filling, requiring long data series with few missing records and significant time overlap with the data series to be completed [10].

A notable improvement in accuracy can be achieved through regional frequency analysis [4,6,11], using data from the neighboring observatories to complete the information. In terms of statistical analysis, the L-moments method is particularly well-suited to the task, offering a number of advantages over alternative techniques [6,9,11,12,13,14]. These advantages include [4]: (1) their robustness in the presence of outliers, (2) a more effective discrimination of probability density functions (PDF), (3) a less biased estimation method, and (4) successful implementation experiences worldwide. The implementation of such techniques for hydrological applications has been widely distributed as an R package under an open-source license [3,4,10,12] and, to the best of our knowledge, it is not common practice to find such analysis using Python libraries.

This paper presents a new software package coded in Python for Rainfall Frequency Analysis (RainFA) [15] and applies its procedures to the case study of the city of Palma (Spain), where the insufficient capacity of its sewerage system determines the number of discharges into the Mediterranean Sea, affecting the water quality and increasing the possibility of beach closures, which ultimately affects the tourism sector [16,17]. Starting with subdaily rainfall data from 11 stations for the period 2015–2021, RainFA is used to homogenize to daily and curate the registers and, after a statistical analysis and clustering process, to determine a homogeneous group of 7 stations that can be considered to be a region. Next, a PDF is fitted for both the daily rainfall and the number of rainy days per year. The resulting runoff generates pollution vectors that have a significant impact on fresh and estuary waters; consequently, efforts to minimize their impact are required, in accordance with European regulations [18,19] and their respective state transpositions [20]. In response to these challenges, RainFA implements a methodology to quantify the annual discharges into the Mediterranean Sea, which will improve the capabilities of the future digital twin of the sewerage system that EMAYA is currently developing in the area [16].

2. Materials and Methods

2.1. RainFA

RainFA develops a methodology for data curation and frequency analysis of rainfall series, programmed in Python, that allows (see Figure 1): (1) preliminary data analysis, by means of (a) homogenizing the time-step; (b) performing data quality control—i.e., detecting anomalous values based on user-defined thresholds as well as through statistical analysis to identify outliers, performing homogeneity tests, and using double-mass curves and HDR boxplots between tuples of data series; (2) clustering, supported by the Silhouette width and Mantel statistics, as well as the Ward’s dendrogram and evaluating the Discordancy and Homogeneity measures using the L-moments method; (3) IDW interpolation, for geospatial and temporal distribution of precipitation within the homogeneous regions; (4) frequency analysis, to adjust the best-fitting PDF function for the daily precipitation determination; and (5) wastewater discharge estimation, according to a simplified methodology based on the Monte Carlo simulations of N years of daily precipitation events as described in Section 2.7. This methodology has been applied to the time series (2015–2021) of 11 weather stations and the urban drainage system of the metropolitan area of Palma (Spain).

2.2. Preliminary Data Analysis

Data continuity and stationarity are two of the most important concerns associated with historical series analysis [4,12]. In terms of continuity and data quality, a set of control tests are planned, including user-defined homogenization of the temporal discretization and rainfall thresholds, descriptive statistics—e.g., boxplots and outliers—and double-mass graphs. In addition, the non-parametric Mann–Kendall [21,22] trend test is applied for stationarity.

2.2.1. Temporal Homogenization

First of all, RainFA adjusts the sample size of the data using the resample function of the Pandas library of Python [23]. To do this, the time-step is determined with the least common multiple of the different series, and the sum is used to complete the series in cases where the original frequency is smaller.

2.2.2. Daily Quality Control

After the time homogenization, a series of different tests and procedures for quality control are proposed to ease the detection of unusual values that should be discarded from the data series. A daily timestamp has been selected for the data curating procedure in accordance with [3,10,12,24] and the different methods are described in the following paragraphs.

User-Defined Thresholds

A user-defined threshold option has been implemented within the RainFA software package (v1.0.1) to allow a manual series cleaning procedure to discard some values that may influence the following statistical analysis. Default values are set in Not a Number (NaN) for negative numbers and will require a completeness procedure in further steps of the method. Additionally, an equivalent to 40 mm for 10 min cumulative rainfall has been adopted for the definition of the upper threshold.

Outlier Detection and Bivariate HDR Boxplots

Rainfall time series are cataloged as functional data to provide measurements in a temporal continuum that can be plotted on curves or surfaces for their analysis [25]. According to [26], a side benefit of some of the functional data analysis techniques is the identification of outliers that may not be obvious from a standard statistical analysis or a graphical representation of the original data. An early approach to this representation was John Tukey’s box and whiskers plot (or boxplot) that represents, in a compact form, the distribution summary statistics [27]. RainFA introduces boxplots of each data series to identify the potential outliers of the dataset, after the definition of a lower threshold that would be used to trigger rainfall. The default considered value is 0.2 mm, as it corresponds to a common error in a telemetry rain gauge device.

In addition, ref. [28] introduced the concept of 2D dimensions to the boxplot analysis, relying on a density estimate of the selected variable to compare between two different data series. To implement this comparison, ref. [29] presented a method of calculating and plotting Highest Density Regions (HDR), which are often considered the most appropriate subset to use to summarize a probability distribution. According to [26], the Bivariate HDR boxplot is constructed using a bivariate kernel density estimate $\widehat{f}\left(z\right)$, which is defined as:

$$\begin{array}{c}\widehat{f}\left(z\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{K}_{h_i}(z-Z_i),\end{array}$$

(1)

where z correspond with the data points of the time series, $Z_i$ represents a set of bivariate points, $K_{h_i}$ is the kernel function, and $h_i$ is the bandwidth for dimension i.

Next, from the kernel density estimates, an HDR is defined as:

$$\begin{array}{c}{R}_{\alpha }=\left\{z:\widehat{f}\left(z\right)\ge f_{\alpha }\right\},\end{array}$$

(2)

where $f_{\alpha }$ is such that ${\int }_{R_{\alpha }}\widehat{f}\left(z\right)dz=1-\alpha$, which corresponds to a region with probability coverage $1-\alpha$, meaning that all points within the region have a higher density estimate than any point outside its boundaries. RainFA considers an outer 90% highest density region for the outer contour, which defines the outlier consideration.

Double-Mass Curves

Double-mass curves were presented as a tool for evaluating the consistency of hydrological data—e.g., streamflow, rainfall, sediment series, etc.—from a single station by comparing it with a pattern composed of several nearby stations [30]. The principle of the double-mass curve is based on the fact that the cumulative graph of one variable against the accumulation of another variable over the same period is represented as a straight line as long as the two quantities are proportional, with the slope of this line being the constant of proportionality between them.

A change in the slope of the curve could mean a variation in the constant of proportionality or perhaps that there is no relationship. For this reason, it is common practice in hydrology to plot the double-mass curves of each individual time series against the average registers of all stations.

RainFA considers both the Bivariate HDR boxplots—as mentioned in the previous sub-section—and the double-mass curves to study the rainfall patterns and the existing proportionality between the different stations, expecting to detect potential anomalies and the cluster relationships between stations.

Trend Analysis

Non-stationarity of hydroclimatic series is one of the main concerns in the analysis of historical data and has been particularly considered in the regional frequency analysis of rainfall and flood series [4,12]. As noted in [31,32,33], the Mann–Kendall test is a robust procedure to the influence of extremes and suitable for use with skewed data because it does not require that the data be normally distributed, only that they be independent [31,34].

The null hypothesis,$H_0$, assumes that the data ($x_1,...,x_n$) are samples of n independent and equally distributed random variables [4,31]. The alternative hypothesis, $H_1$, of a two-sided test assumes that the distributions of $x_j$ and $x_k$ are not identical for all $j,k\le n$ with $j\ne k$ [31]. First, the statistic S, which has mean zero and is asymptotically normal [31,33], is computed with the following equations:

$$\begin{array}{c}S=\sum _{k=1}^{n-1}\sum _{j=k+1}^{n}sgn(x_j-x_k),\end{array}$$

(3)

$$\begin{array}{c}sgn(x_j-x_k)=\left\{\begin{array}{cc}+1& \mathrm{if}sgn(x_j-x_k)>0\\ 0& \mathrm{if}sgn(x_j-x_k)=0\\ -1& \mathrm{if}sgn(x_j-x_k)<0\end{array}\right.\end{array}$$

(4)

The variance of S is determined with the following equation [32], which takes into account that ties might be present:

$$\begin{array}{c}Var\left(S\right)={\displaystyle \frac{1}{18}}\left[n(n-1)(2n+5)-\sum _{p=1}^q{t}_p(t_p-1)(2t_p+5)\right],\end{array}$$

(5)

where q corresponds with the number of tied groups and $t_p$ is the number of data values in the $p^{th}$ group.

And finally, statistic S and its variance—i.e., $Var\left(S\right)$—are used to determine the test statistic Z as:

$$\begin{array}{c}Z=\left\{\begin{array}{ccc}{\displaystyle \frac{S-1}{\sqrt{Var\left(S\right)}}}& \mathrm{if}& S>0\\ 0& \mathrm{if}& S=0\\ {\displaystyle \frac{S+1}{\sqrt{Var\left(S\right)}}}& \mathrm{if}& S<0\end{array}\right.\end{array}$$

(6)

The Z value evaluates the presence of a statistically significant trend, where a positive trend corresponds to a positive Z value and a negative trend corresponds to a negative Z value. Since Z has a normal distribution, the goodness of the $H_0$ hypothesis associated with a $\alpha$-confidence level is determined by the expression $\left|Z\right|\le Z_{1-{\displaystyle \frac{\alpha }{2}}}$, corresponding the $Z_{1-{\displaystyle \frac{\alpha }{2}}}$ with 1.96 for a confidence level of 95%. RainFA uses the existing pyMannKendall software package to apply the Mann–Kendall trend test [35].

2.3. Identification of Candidate Homogeneous Regions

2.3.1. Cluster Overview

Cluster analysis is used to identify the candidate homogeneous regions by comparing different site characteristics, either related to spatial or meteorological conditions, which are analyzed using multivariate statistics and classified according to their similarity [11,12,36]. RainFA considers the spatial characteristics, which are UTM coordinates for latitude and longitude, together with the elevation in meters. It also includes three rainfall characteristics from the AEMET gauging stations: the maximum rainfall in 24 h per year, the average annual rainfall, and the average number of days with daily rainfall above 1 mm [37]. If rainfall characteristics are not available for the site, they can be extracted from the rain gauge data series (site statistics).

As recommended in [12], RainFA rescales the features using the Min-Max normalization methodology, which involves determining the minimum and maximum values of the different parameters and the features are ranged between 0—used for the minimum value—and 1—set for the maximum value—according to the aforementioned methodology. Then, the optimal number of clusters is identified using the Silhouette width and Mantel curve comparison methodologies [38], while the homogeneous regions are determined by applying Ward’s dendrogram [39]. Finally, the homogeneity of the selected regions is evaluated using the statistics derived from the L-moments [11]: (1) the Discordancy measure ($D_i$), which is a measure of the dissimilarity of a given site from the rest of the sites within a group; and (2) the Heterogeneity measure (H), or H-statistic, which compares the dispersion of the L-moments from what would be expected for a homogeneous region.

2.3.2. Cluster Optimization

The optimal number of clusters is determined by measuring the degree to which an object—in this case, a gauging station—belongs to its cluster compared to its membership in the next closest cluster [38]. According to [40], the Silhouette width, $s\left(i\right)$, is defined as:

$$\begin{array}{c}s\left(i\right)={\displaystyle \frac{b\left(i\right)-a\left(i\right)}{max\left\{a\left(i\right),b\left(i\right)\right\}}},\end{array}$$

(7)

where $a\left(i\right)$ corresponds to the average dissimilarity of object i to all other objects of cluster A and $b\left(i\right)$ is determined as

$$\begin{array}{c}b\left(i\right)=\underset{C\ne A}{min}\left(d(i,C)\right),\end{array}$$

(8)

where C represents a cluster different from A and $d(i,C)$ corresponds to the average dissimilarity of object i to all other objects in cluster C. Thus, $b\left(i\right)$ corresponds to the closest cluster to A and is denoted as cluster B. $s\left(i\right)$ varies between −1 and 1, which means that i is “well clustered” when $s\left(i\right)$ is close to 1 and, on the contrary, when $s\left(i\right)$ is close to −1, the object i is clearly “misclassified”, and it is on average much closer to cluster B than to cluster A. An intermediate situation is reached when $s\left(i\right)$ is close to 0, and it is not clear whether the object i should be assigned to cluster A or B [40].

The Silhouette width extends this cluster classification to all the objects in the dataset and determines the average of the clustering procedure, allowing the optimal number of groups to be determined. The Silhouette width is recognized as one of the strongest indices for data clustering [41].

In addition, RainFA implements Mantel’s correlation coefficient test, which examines two distance matrices from the same samples. In this case, the initial distance matrix and the distribution into clusters matrix are compared, and, as a result, a correlation matrix is determined using the Pearson r coefficient for calculations [38]. The higher the correlation between the two matrices, the better the Mantel correlation.

2.3.3. Ward’s Dendrogram

The minimum variance hierarchical clustering algorithm, or Ward’s dendrogram [39], is recommended for the identification of homogeneous regions in a clustering process such as that applied by the RainFA package for the evaluation of rainfall data in a specific geographical context [11]. Ward’s dendrogram algorithm uses the W function to minimize the sum of the squares of the deviations of the object vectors from the centroid of their respective clusters [42]:

$$\begin{array}{c}W=\sum _{k=1}^K\sum _{j=1}^m\sum _{i=1}^{N_k}{\left(y_{ij}^k-y_{·j}^k\right)}^2,\end{array}$$

(9)

where K is the number of clusters, m corresponds with the dimension of the attribute vector, $N_k$ represents the number of feature vectors in cluster k, $y_{ij}^k$ denotes the rescaled value of the attribute j in the feature vector i assigned to cluster k, while $y_{·j}^k$ is the mean value of the feature j for cluster k.

The algorithm starts with single station groups where the centroids coincide with the feature vectors and W is zero. At each step of the algorithm, a group fusion is performed, and the smallest increment in the function W is the fusion criterion. Ward’s dendrogram results are very useful for identifying homogeneous regions in regionalization processes [11,39]. However, it does not allow for the reallocation of stations that may have been poorly classified in the early stages of the analysis [42].

2.3.4. L-Moments Homogeneity Measure

RainFA evaluates the homogeneity of the regions using two statistics based on L-moments and described in [6,11,43]: (1) the Discordancy ($D_i$) of a station i with respect to the rest of the stations in a group; and (2) the Heterogeneity ($H_i$) measure, which checks whether the variation in L-moments between the sites of a region is similar to what would be expected for a homogeneous region [4,8].

The Discordancy ($D_i$) is defined according to:

$$\begin{array}{c}{D}_i={\displaystyle \frac{1}{3}}\left[{(u_i-\overline{u})}^T(u_i-\overline{u})S^{-1}\right],\end{array}$$

(10)

where $u_i$ is the vector of L-moment ratios corresponding to the site i—and includes the L-variation (L-CV), the L-skewness (${\tau }_3$) and the L-kurtosis ${\tau }_4-,\overline{u}$ is the vector of average L-moments ratios in the selected cluster and S is the sample covariance matrix of the L-moments of all sites in the cluster. In general terms, sites with $D_i\ge 3$ are considered discordant [11,12].

The second statistic, the Heterogeneity ($H_i$), requires the Monte Carlo simulations of regions with equal L-moments and length registers, which may allow the determination of homogeneous regions to compare with [11,44]. This is carried out by first fitting a four-parameter kappa distribution function that is adjusted to the previously mentioned L-moments ratios, i.e., L-variation (L-CV), L-skewness (${\tau }_3$) and L-kurtosis (${\tau }_4$); and then the computed function is used to carry out the Monte Carlo simulations [4,11].

The Heterogeneity (based on the L-variation ratio [6,43]) is then calculated using:

$$\begin{array}{c}H={\displaystyle \frac{V-{\mu }_v}{{\sigma }_v}},\end{array}$$

(11)

where ${\mu }_v$ and ${\sigma }_v$ are the mean and standard deviation of the simulated values, and the V-statistic is determined from the observed regional data as:

$$\begin{array}{c}V={\left\{{\displaystyle \frac{{\sum }_{i=1}^N{n}_i{[t^{\left(i\right)}-t^R]}^2}{{\sum }_{i=1}^N{n}_i}}\right\}}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(12)

where i refers to a station with a registration length of $n_i$, $t^i$ corresponds to the L-variation ratio, N is the total number of stations of the homogeneous region and $t^R$ represents the average L-variation ratio of the region, which is determined as:

$$\begin{array}{c}{t}^R={\displaystyle \frac{{\sum }_{i=1}^N{n}_i·t^{\left(i\right)}}{{\sum }_{i=1}^N{n}_i}},\end{array}$$

(13)

According to the original criterion [11], the region can be considered “acceptably homogeneous” if $H<1$, “possibly heterogeneous” if $1\le H<2$, and “definitely heterogeneous” if $H\ge 2$.

2.4. Generalized IDW Interpolation for Homogeneous Regions

The Inverse Distance Weighted (IDW) is a widely used interpolation method for both image and spatial data contexts [45]. The main characteristic of this method is that all the points on the Earth’s surface are considered to be interdependent, based on a relationship that is inversely proportional to the distance to each point [46]. By generalizing this relationship in the selected cluster regions, it is possible to determine an influence weight parameter for each station and thus to compound the variable to be simulated, which in this case is rainfall. Since rainfall varies over time, it is advisable to first determine the weights and then apply them to the time series, using the following procedure in a Geographical Information System (GIS) program: (1) represent the stations according to their coordinates; (2) create a new field called Weights in the station point layer; (3) assign the value 1 to station i and 0 to the rest of the stations; (4) apply the IDW interpolation method to calculate the weighted raster map of station i; (5) repeat the steps (3) and (4) for all the stations; (6) combine the calculated weighted raster maps into a multiband raster map, integrating the weights of all the stations into a single file.

2.5. Probability Distribution Functions Selection for Homogeneous Regions

The regional probability distribution function is selected according to the goodness-of-fit statistic $Z^{Dist}$, described in [11] and applied in [4,12,47], as an alternative of different statistical tests, such as Anderson-Darling (AD), Kolmogorov–Smirnov (KS) or Chi-Squared (CS), as described and applied in [9,44,48]. The different candidate distributions are fitted to the regional average L-moments, and its L-kurtosis (${\tau }_4$) is determined. As indicated in [4] and its sources, and in accordance with the determination of the Heterogeneity measure, the simulated length records of each site correspond to the observed data. The $Z^{Dist}$ is then calculated as:

$$\begin{array}{c}{Z}^{Dist}={\displaystyle \frac{{\tau }_4^{Dist}-t_4^R+B_4}{{\sigma }_4}},\end{array}$$

(14)

where ${\tau }_4^{Dist}$ is the L-kurtosis of the fitted distribution and $t_4^R$ the regional average L-kurtosis, while ${\sigma }_4$ and $B_4$ are the standard deviation and bias of $t_4^R$, which can be obtained by repeated simulation of a homogeneous region whose sites have the same fitted frequency distribution and record lengths as those in the observed data. [11] declares the fit to be adequate if $Z^{Dist}$ is close to zero, which is a reasonable criterion: $\left|Z^{Dist}\right|\le 1.64$.

2.6. Evaluation Criteria

2.6.1. Detection of Occurrence of Rainfall

The ability of each precipitation product to detect the occurrence of rainfall events is assessed using four categorical indices (see

Table 1. Precipitation occurrence performance metrics.

Statistic	Equation	Optimal Value
POD	${\displaystyle \frac{N_{11}}{N_{11}+N_{01}}}$	1
FAR	${\displaystyle \frac{N_{10}}{N_{11}+N_{10}}}$	0
CSI	${\displaystyle \frac{N_{11}}{N_{11}+N_{01}+N_{10}}}$	1
BS	${\displaystyle \frac{N_{11}+N_{10}}{N_{11}+N_{01}}}$	1

): (1) probability of detection (POD); (2) false alarm ratio (FAR); (3) critical success index (CSI); and (4) bias score (BS). POD indicates the ability of precipitation products to accurately capture the actual occurrence of precipitation. FAR assesses the proportion of false alarms detected by rainfall products. The ability of the data to comprehensively detect true precipitation events is measured by CSI. CSI is an accurate and balanced detection metric and is based on POD and FAR. BS is the ratio of the estimated rainfall to observed rainfall.

In Table 1, $N_{11}$ denotes the observed rainfall that is correctly detected, $N_{01}$ represents the observed rainfall that has not been detected, and $N_{10}$ is the precipitation that is detected but not observed.

2.6.2. Accuracy of the Simulations

Three quantitative performance metrics are employed to measure the accuracy of the precipitation datasets in terms of daily rainfall (see

Table 2. Precipitation accuracy performance metrics.

Statistic	Equation	Value Range
RMSE	$\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(O_i-S_i)}^2}{n}}}$	$[0,+\infty )$
NSE	$1-{\displaystyle \frac{{\sum }_{i=1}^n{(O_i-S_i)}^2}{{\sum }_{i=1}^n{(O_i-\overline{O})}^2}}$	$(-\infty ,1]$
PBIAS	${\displaystyle \frac{{\sum }_{i=1}^n(O_i-S_i)·100}{{\sum }_{i=1}^n{O}_i}}$	$(-\infty ,+\infty )$

): (1) Root Mean Square Error (RMSE); (2) Nash–Sutcliffe Efficiency coefficient (NSE); and (3) Percent Bias (PBIAS). The RMSE explains the deviation between two data sets, i.e., the degree of dispersion between observed and estimated precipitation data. The NSE indicates how well the correlation of observed and simulated data fits the 1:1 line when plotted together. In contrast, PBIAS indicates the degree to which the observed value is overestimated or underestimated as a percentage.

In Table 2, $O_i$ and $S_i$ are the observed and simulated data, respectively, i is the data index, and n is the total number of measurements, while $\overline{O}$ is the mean of the observed data.

In addition, the ability of the simulated rainfall to predict different daily rainfall intensities is evaluated according to the thresholds [49,50] shown in

Table 3. Classification of rainfall events based on its daily rainfall intensity. Modified for daily values from the World Meteorological Organization (WMO).

Type of Event	Daily Rainfall Intensity (mm/day)
No rain	<1
Light rain	[1, 5)
Moderate rain	[5, 20)
Heavy rain	[20, 40)
Violent rain	≥40

.

2.7. Annual Sewage Discharge Estimation

The Royal Decree 665/2023 [20] and the proposal of adaptation of the Urban Wastewater Treatment Directive [18] establish criteria for the design of sewerage infrastructure, limiting the system’s outflows and direct discharges into the sea. The calculations of the former are based on daily rainfall, while the latter considers annual flows. To evaluate them, RainFA adjusts the daily rainfall probability function for the calculated region and determines both (1) the expected number of annual sewer system discharges and (2) the total discharge volume. This approach is based on the simulation of all rainfall events occurring during the N years simulated using the Monte Carlo method.

The probability distribution function for the annual number of precipitation events can be fitted as a normal distribution, denoted as $N({\mu }_V,{\sigma }_V^2)$, where V represents an array with the total number of significant precipitation events (greater than 0.4 mm) in one year in the selected homogeneous region:

$$\begin{array}{c}{\mu }_V={\displaystyle \frac{{\sum }_{i=1}^n{V}_i}{n}},\end{array}$$

(15)

$$\begin{array}{c}{\sigma }_V^2={\displaystyle \frac{{\sum }_{i=1}^n{(V_i-{\mu }_V)}^2}{n}}.\end{array}$$

(16)

n precipitation events (n = 1000) can be simulated for both probability distribution functions. For each year i, n different daily precipitation events $P_{ij}$ are simulated with the selected probability distribution function (PDF) of the homogeneous region. The goodness of fit of the distribution is checked with the Shapiro–Wilk test [51].

The other variables that are needed to estimate the annual average discharge are:

•

Effective Area (EA): non-permeable area in the watershed.

•

Rainfall Inflow (${\mathit{RI}}_{\mathit{ij}}$): total rainfall entering the system for the event j of the year i. $R{I}_{ij}=EA\times P_{ij}$.

•

Attenuation and Treatment Capacity (${\mathit{C}}_{\mathit{A}+\mathit{T}}$): defined as the sum of the volume of the sewer system and attenuation tanks ($C_A$), and the capacity volume of the treatment plant ($C_T$). The parameter $C_T$ depends on the duration of the rainfall, as shown in Figure 2.

•

Dry Volume (${\mathit{V}}_{\mathit{dry}}$): average daily volume of wastewater during dry days.

•

Discharge event (${\mathit{D}}_{\mathit{ij}}$): discharge for the event $P_{ij}$ when the attenuation and treatment capacity is exceeded.

$$\begin{array}{c}{D}_{ij}=\left\{\begin{array}{ccc}R{I}_{ij}+V_{dry}-C_{A+T}& \mathrm{if}& R{I}_{ij}+V_{dry}-C_{A+T}>0\\ 0& \mathrm{if}& R{I}_{ij}+V_{dry}-C_{A+T}\le 0\end{array}\right.,\end{array}$$

(17)

Figure 3 shows the complete procedure for estimating the annual sewage discharge. It should be noted that the methodology does not take into account the duration of the rainfall, so it is recommended to use the time of concentration in the catchment as the duration of all rainfall events to obtain a conservative estimate of the discharge to the environment.

Figure 2. Example of hydrogram showing the treated water in the wastewater treatment plant and the discharged volume without being treated.

Figure 2. Example of hydrogram showing the treated water in the wastewater treatment plant and the discharged volume without being treated.

Finally, the defined methodology allows the estimation of the Annual Average Discharge ($\overline{AAD}$), which can then be evaluated for compliance with future European discharge limits as [18]:

$$\begin{array}{c}\overline{AAD}=\sum _{i=1}^N\sum _{j=1}^n\left(D_{ij}>0\right),\end{array}$$

(18)

Figure 3. Example of a hydrogram showing the treated water in the wastewater treatment plant and the discharged volume without being treated.

Figure 3. Example of a hydrogram showing the treated water in the wastewater treatment plant and the discharged volume without being treated.

3. Description of the Study Area and Data Sources

3.1. Case Study

The city of Palma is located in the southwest of the island of Mallorca, Spain, with an area of 208.63 ${\mathrm{km}}^2$ and a population of 420,000 inhabitants. The metropolitan area, which discharges its wastewater into the municipal sewerage system, includes three other municipalities—Esporles, Bunyola, and Marratxí—so that the total area associated with the study area is 382.8 ${\mathrm{km}}^2$ with 471,000 inhabitants.

Specifically, the study area is in the lower depression and estuary of the Sa Riera, na Bàrbara, and Gros gullies. Geomorphologically, the basin is bounded by the Serra de Tramuntana to the north and the Serra de na Burguesa to the west. It is also bordered by the lower hills near Pòrtol, known as Serra de Son Mayol to the northeast and Costes des Xorrigo slopes to the east [52] (Figure 4).

Annual rainfall in the area has averaged 449 mm over the period 1980–2010, with a rainy season in autumn—i.e., from September to November—accounting for 52.2% of annual rainfall [53]. Furthermore, the interannual variability of rainfall is considerable, with pluviometric irregularity indices reaching 3.49 at the Portopí station (B228) for the 1980–2010 data series [54].

In addition, the maximum daily rainfall series shows a high spatial variability. The barrier effect of the Tramuntana mountain range is transformed into a reduction of the maximum daily rainfall in the area of Palma. In particular, for a 10-year return period, precipitation levels can vary considerably, from 130 mm/day in the municipality of Bunyola—in the northern part of the study area—to 70 mm/day in the coastal area of the city of Palma [55]. Specifically, the area surrounding Sant Jordi is characterized by the lowest maximum daily rainfall records due to the sheltering effect of the Serra de Son Mayol to the north and the Costes des Xorrigo to the east, which prevents the penetration of the north and east winds, respectively [56].

3.2. Data Sources

This study uses data from 11 rain gauges collected between 2015 and 2021. Three of these rain gauges are managed by AEMET [37]: (1) Palma Portopí (B228); (2) Palma Universitat Illes Balears (B236); and (3) Palma Aeropuerto (B278). The remaining 8 stations are operated by the non-profit organization BalearsMeteo [57], with which EMAYA has an agreement for access to the data recorded by their stations in the municipalities where they both operate.

Table 4. Availability of information per rain gauge station.

Station Name	Series Start Date-Time	Series End Date-Time
El Pil·lari	31/12/2015 18:50	18/05/2022 23:00
La Bonanova	12/02/2015 11:30	18/05/2022 10:30
Pòrtol	08/10/2016 10:20	11/05/2022 00:00
Son Rapinya	21/11/2018 14:50	11/05/2022 00:00
Pont d’Inca	23/01/2009 20:00	19/05/2022 12:00
Puntiró	14/01/2017 16:40	26/06/2022 22:30
Secar de la Real	05/02/2021 21:10	23/06/2022 19:50
Son Ferriol	03/05/2021 23:40	22/06/2022 15:10
Portopí (B228)	01/01/2015 00:10	01/07/2022 00:00
UIB (B236)	01/01/2015 00:10	01/07/2022 00:00
Aeropuerto (B278)	01/01/2015 00:10	01/07/2022 00:00

shows the date-time range for each station.

In the present case study, the different meteorological stations have recorded every 1, 5, and 10 min, so the 10 min time-step was chosen for homogenization, with the sum of 2 or 10 values being used for resampling, as appropriate. The daily rainfall was used for clustering and homogeneity evaluation.

Finally, the period chosen to assess the accuracy of the simulated rainfall datasets was from 2021 to 2022. All rain gauges were operational during this period.

4. Results

4.1. Data Curation and Stationarity

After the temporal homogenization to a 10-min time-step, four different data quality control tests were performed: (1) user-defined thresholds; (2) boxplots, both the traditional Box and Whisker and the Bivariate HDR boxplot; (3) double-mass curves; and (4) the Mann–Kendall trend test.

No values exceeded the 40 mm threshold defined for the 10-min time-step. Next, individual boxplots were constructed for all the stations (see Figure 5), both for precipitation values greater than 0.2 mm and values exceeding 2 mm, showing consistent outlier densities and similar ranges across the different stations, except for the Son Ferriol station, which had no daily records greater than 2 mm. Furthermore, both the double-mass curves (see Figure 6) and the Bivariate HDR boxplots (see Appendix A) show that the Son Ferriol station has no correspondence with the mean values of all the stations or with the rest of the stations in tuples. In addition, as shown in Table 4, there are only about 13 months of records. As a result of both findings, the time series of Son Ferriol was excluded from the rest of the study.

From the analysis of the double-mass curves (Figure 6), a fairly uniform behavior is observed among all stations, albeit with certain peculiarities: B228, B236, El Pil·lari, and Pont d’Inca align with the overall station average; meanwhile, La Bonanova, Secar de la Real, and Pòrtol exhibit rainfall levels that exceed the average; finally, B278, Puntiró, and Son Rapinya present rainfall levels that are below the average. Furthermore, when looking at the results of the Bivariate HDR boxplot (see Appendix A), additional insights emerge both from the shape and the average of the individual density plots of each station, and especially from the common density regions, where the higher the correspondence between the values of the axes and the shape of their 50% and 90% HDR regions, i.e., with a smaller deviation from the 1:1 line, the greater the similarity between the stations.

The stationarity of the data is tested using the non-parametric Mann–Kendall test [21,22], the results of which are summarized in

Table 5. Results of trends in daily rainfall using the Mann–Kendall Test.

Station	s	Var(S)	Z	Trend
B228	−962	2.72 $\times {10}^6$	−0.58	No
B236	−3942	4.01 $\times {10}^6$	−1.97	Decreasing
B278	−1413	1.66 $\times {10}^6$	−1.09	No
El Pil·lari	−2213	1.38 $\times {10}^6$	−1.88	No
La Bonanova	−3161	2.56 $\times {10}^6$	−1.98	Decreasing
Pont d’Inca	7737	1.07 $\times {10}^7$	2.36	Increasing
Pòrtol	−2297	1.64 $\times {10}^6$	−0.90	No
Puntiró	−1156	1.02 $\times {10}^6$	−2.27	Decreasing
Secar de la Real	91	3.56 $\times {10}^4$	0.48	No
Son Rapinya	−835	2.47 $\times {10}^5$	−1.68	No

. Four out of ten stations show a certain trend, i.e., Pont d’Inca and Puntiró, and two other stations can be considered in the threshold area so that the region as a whole could be considered to be stationary.

4.2. Clustering and Homogeneity

According to the Mantel Score and the Silhouette Width graphs, between two and three cluster regions could be defined in the selected stations (Figure 7). This result is consistent with the results of Ward’s dendrogram (Figure 8) findings, as there is a group of six stations that could be associated with two different station tuples. However, an alternative interpretation of Ward’s dendrogram result is that there is a single cluster with four possible exclusions.

According to the above results, the only region to be studied would include six rainfall stations located in the floodplain of the Sa Riera, na Bàrbara and Gros torrents. In addition, the station B278 shows rainfall patterns very similar to those of this group, so it could be included, also in line with the Heterogeneity results that will be presented subsequently in this section. The selected region has a maximum distance from the sea of just over five kilometers, an average altitude of around 40 m, and an average annual rainfall of 400 (±50) mm/year. Furthermore, the region geomorphologically corresponds to a semicircular depression open to the sea in the Palma watershed [52].

Next, the L-moments homogeneity measure [11] statistics, i.e., Discordancy and Heterogeneity, are applied. First, for the Discordancy measure, all the stations within the selected region agree with the statistical data of the region.

Table 6. Results of the Discordancy Test.

No.	Name	Di	Critical Value	Classification
2	La Bonanova	1.08	1.92	Non-discordant
4	Son Rapinya	0.60	1.92	Non-discordant
5	Pont d’Inca	0.86	1.92	Non-discordant
7	Secar de la Real	1.55	1.92	Non-discordant
9	B228	0.76	1.92	Non-discordant
10	B236	1.20	1.92	Non-discordant
11	B278	0.95	1.92	Non-discordant

shows the results of applying the Discordancy test to the data.

The results of the Heterogeneity test (see

Table 7. Heterogeneity results.

No.	Name	All	Ward Group	W. Group*
1	El Pil·lari	x	-	-
2	La Bonanova	x	x	x
3	Pòrtol	x	-	-
4	Son Rapinya	x	x	x
5	Pont d’Inca	x	x	x
6	Puntiró	x	-	-
7	Secar de la Real	x	x	x
9	B228	x	x	x
10	B236	x	x	x
11	B278	x	-	x
Heterogeneity for different minimum daily precipitation
Pd ≥ 0.2 mm		9.74	3.61	3.69
Pd ≥ 0.4 mm		2.72	0.79	0.56
Pd ≥ 1 mm		−0.04	0.97	0.59
Pd ≥ 2 mm		−0.66	0.37	0.14

) show that the extended cluster (W. group*), including the B278 station, has an acceptable measure of homogeneity (H < 1) for daily rainfall greater than 0.4 mm. The lack of homogeneity in daily rainfall data, including the ’No rain’ consideration (i.e., <1 mm), can be attributed to wind effects, dew, installation errors, or other types of error [58]. Consequently, precipitation inference calculations and PDF selection are performed for the acceptably homogeneous region ($P_d\ge 0.4$ mm).

Moreover, for rainfall events greater than 1 mm, all the studied combinations could be considered to be homogeneous regions. Further investigation of these results, including the study of different combinations of stations, the addition of new stations with completely different characteristics within the island of Mallorca, and the clustering capacity of the Heterogeneity measure, would be beneficial.

4.3. Evaluation of the Precipitation Inference Model Using the IDW Method in the Selected Homogeneous Region

All the indices referred to in Table 1, Table 2 and Table 3 (see Section 2.6.1 and Section 2.6.2) are calculated not only for the selected homogeneous region but also for the 120 possible combinations of seven out of the ten available stations. This allows for the determination of whether the selected homogeneous region performs well in comparison to the other possible combinations.

4.3.1. Detection of Occurrence of Rainfall

The values of our categorical indices for the selected homogeneous region are found to be highly satisfactory, both in absolute terms and in relation to the other possible combinations of seven stations (see

Table 8. Precipitation occurrence performance results.

Statistic	Value	Performance Rank
POD	0.95	25°
FAR	0.15	13°
CSI	0.81	12°
BS	1.11	19°

and Appendix B for the complete set of results). The POD, FAR, and CSI obtained values are significantly higher than the thresholds typically considered acceptable for this type of meteorological inference. Furthermore, it is noteworthy that the combination of the homogeneous regions presents the sixth-best performance in terms of rainfall detection and false alarm avoidance, as quantified by the CSI index. The BS index suggests that the model may have a slight tendency to overestimate the probability of rainfall.

4.3.2. Accuracy of Rainfall

The NSE index shows favorable values in both absolute and relative terms, as summarized in

Table 9. Precipitation accuracy performance results.

Statistic	Value	Performance Rank
RMSE	2.86 mm	25°
NSE	0.86	14°
PBIAS	−2.49%	58°

—the complete results are presented in Appendix D. Furthermore, the PBIAS values show favorable results, indicating a minimal degree of overestimation of the proposed inference model. Unfortunately, the RMSE for this combination is about 70% of the mean observed rainfall. Nevertheless, this suboptimal performance in absolute, though not in relative terms, could be improved by incorporating more robust inference methods, such as the Inverse Distance Weighting (IDW) method or Krigging [36,59].

Figure 9 shows that the model performs suboptimally in absolute terms, especially for rainfall below 10 mm. However, the results of the homogeneous regions combination are comparable to the best of the combinations. It is also noteworthy that the model shows a slight tendency to underestimate, except for moderate daily rainfall (Pd < 10 mm), where it clearly overestimates. It is, therefore, essential to carry out a cluster analysis before implementing any inference method to ensure that the most appropriate approach is taken for each type of rainfall. This is particularly important in regions such as the case study presented here, where summer convective rainfall can be highly localized.

The ability of the simulated precipitation to predict different ranges of daily rainfall according to WMO criteria is a remarkable achievement, both in absolute and relative terms, as shown in

Table 10. Critical Success Index (CSI) results and ranking for different types of daily rainfall events.

Type of Event	Daily Rainfall Intensity (mm/day)	CSI Value	Rank
No rain	<1 mm	0.84	5°
Light rain	[1, 5) mm	0.51	1°
Moderate rain	[5, 20) mm	0.62	1°
Heavy rain	[20, 40) mm	0.44	26°
Violent rain	≥40 mm	0.7	15°

. The complete results are presented in Appendix C.

4.4. PDF Selection

A number of candidate distributions are tested, including lognormal (GLO), Generalized Extreme Value (GEV), Generalized Pareto (GPA), and the Pearson type III (PE3), with the aim of identifying the optimal PDF for the case study homogeneous region. The goodness-of-fit statistic $Z^{Dist}$ is used to evaluate the quality of fit for the different probability functions. The results, which are summarized in

Table 11. Goodness-of-fit results.

PDF	${\mathit{Z}}^{\mathit{Dist}}<1.64$
GLO	1.54
GEV	2.97
GPA	1.06
PE3	3.62

and

Table 12. Parameters of the different PDFs evaluated.

PDF	Parameters
GLO	k (shape): −1.103, $\alpha$ (scale): 4.708, $\xi$ (location): 3.944
GEV	k (shape): −0.462, $\alpha$ (scale): 3.821, $\xi$ (location): 2.894
GPA	k (shape): −0.392, $\alpha$ (scale): 4.326, $\xi$ (location): 0.4
PE3	$\gamma$ (shape): 0.321, $\sigma$ (scale): 11.91, $\mu$ (location): 0.120

, indicate that the GPA PDF provides the best fit (Figure 10).

Furthermore, the PDF selection was applied at the different rainfall station locations, showing that the GPA PDF also provides the best local fit, according to the $Z^{Dist}$ statistic (see

Table 13. Goodness-of-fit results at local scale.

Name	${\mathit{Z}}^{\mathit{Dist}}$
	GEV	GLO	GPA	PE3
La Bonanova	0.95	0.61	0.46	1.38
Son Rapinya	0.69	0.24	0.01	1.22
Pont d’Inca	1.32	0.84	0.57	1.69
Secar de la Real	0.39	0.04	0.06	1.21
B228	0.92	0.52	0.38	1.68
B236	1.59	1.22	0.64	0.85
B278	1.01	0.57	0.42	1.44

). The graphical results, represented in Appendix A, range from 13.85 mm to 19.23 mm of cumulative rainfall for the 10-year return period or 0.9 marked quantile, indicating clear variability in volume among the stations despite them being part of a homogeneous group.

4.5. Sewage Discharge Estimation

The sewerage system of the city of Palma is evaluated using the discharge estimation methodology presented in Section 2.7. The effective ‘impervious’ area (EA) is estimated at 605 ha, which is considered a high figure for an almost completed separate sewerage system, but includes both (1) roofs that are incorrectly connected to the sewerage network and (2) an old mixed network in the city center that is challenging to remove. The latter accounts for 25% of the total contribution.

The urban wastewater treatment plant (known as EDAR II) receives a daily inflow of 78,000 m³ ($V_{dry}$) of wastewater from its catchment area. In addition, the treatment capacity of this plant, combined with the support of the other urban treatment plant (known as EDAR I), is 90,000 m³ ($C_T$). This estimate takes into account that the treatment plants can operate at a maximum capacity of 6000 m³/h during a 2-h rainfall event (corresponding to the time of concentration of the tributary area). Furthermore, in terms of the attenuation capacity, a stormwater tank with a capacity of 40,000 m³ has recently been added; this, together with the capacity of the sewerage network, including both the collection system and the treatment plants, calculated at 30,000 m³, gives a total for attenuation and treatment capacity ($C_{A+T}$) of 160,000 m³.

The wastewater discharged into the environment has an approximate mean value of 715,000 ${\mathrm{m}}^3\pm 33.000$ with a 95% confidence interval (see Figure 11), which is in line with the annual discharge estimate of 700,000 m³ calculated by EMAYA using its level and flow sensors at the discharge points for the year 2023.

As can be seen in Figure 11, the volume distribution of annual discharges is considerably scattered due to the high dispersion obtained previously for the annual rainfall with the n = 1000 years of simulations using the Monte Carlo method. These results are due to the fact that the proposed methodology does not take into account the interdependence of the two statistical variables—i.e., the number of rainy days, with a mean of $56.35\pm 0.88$ with a 95% confidence interval; and the daily rainfall selected in Section 4.4 (generalized Pareto). The data show that years with a high number of episodes have a lower rainfall per episode, suggesting that the expected dispersion should not be as high as observed.

5. Discussion

As can be seen in the results, after the first stage of the data curation process, the time series from the Son Ferriol station were discarded from the evaluation both for the short length of their records and for their inconsistency with the magnitude of the precipitation records of the other stations. Next, from the analysis of the double-mass curves, and especially from the Bivariate HDR boxplots (see Appendix A, it is possible to detect relationships between stations that are consistent with the results of the clustering process. In this sense, for example, there is a clear alignment with the 1:1 line and density distribution in the Bivariate HDR boxplots of La Bonanova and B228 or Son Rapinya and Secar de la Real—and even with these last two with Pont d’Inca—as shown in the clustering hierarchy of Ward’s dendrogram (represented in Figure 8).

The two scores used to determine the number of clusters suggested the creation of either two or three distinct groups. According to Ward’s dendrogram mentioned above, this results in two groups of two stations and an additional group of six stations. Geographically, B278 and El Pil·lari stations are located on the periphery of the study area, in a drained marshland close to the Mediterranean Sea. On the other hand, Pòrtol and Puntiró stations are in a hilly area called Serra de Son Mayol, which could explain the differences in rainfall patterns with the rest of the stations (see Figure 4). Finally, the remaining stations are in the floodplain of three ephemeral streams that flow into the Bay of Palma. Notwithstanding this final conclusion, the revision of the annual rainfall, the average altitude, and the proximity to the main cluster group suggested the inclusion of B278 in the main cluster, separating it from El Pil·lari station.

Regarding the PDF selection method, different statistical methods, such as KS and AD, were introduced in Section 2.5. However, the RainFA package implements the $Z^{Dist}$ statistic as described in [11] to select the most appropriate PDF. According to this, the GPA function was selected as the best fit. It is important to note that the present case has been applied to study the sewerage capacity, which is usually designed for a return period of between 10 and 25 years, which is in line with the quantity of input data. If the objective were different and required a higher return period, additional goodness-of-fit methods would need to be applied and compared to determine the best PDF selection.

From the evaluation of the 120 possible combinations of seven out of ten available stations, the selected cluster performed outstandingly, demonstrating the ability of the present methodology to carry out regional analysis of the hydrology of regions with poor meteorological data availability.

Finally, the results and methodology for calculating the expected mean discharge values are considered acceptable and provide a reliable order of magnitude of the runoff produced in the sewer system. In any case, future studies will investigate the relationship between the rainfall intensity per event and the average annual rainfall, together with the duration of the rainfall, which could refine the results of the discharges to the environment.

The proposed methodology makes it possible to estimate the resources required for attenuation and treatment within the sewerage system, as well as for the reduction of impervious surfaces with inadequate connection to the sewerage network. The aim is to ensure compliance with existing and future European regulations [18,19].

Figure 12 illustrates possible scenarios for achieving the requirement of discharges of less than 1% of the total volume of water treated by the system, as previously required by the proposed European Directive (it should be noted that this percentage has since been increased to 2%). [19]. It can be seen that this would require a total of up to 260,000 m³ of attenuation and treatment capacity if the current old mixed network remains connected. Conversely, a reduction of only 100 hectares would require only 230,000 m³, which would mean increasing the volume of stormwater tanks by 70,000 m³, while maintaining the current treatment capacity of the wastewater treatment plants.

In the present study, RainFA has been applied to a data curation process for 10-min time-step series to determine a homogeneous region in terms of daily rainfall. However, the software is prepared to and the method has been widely applied to different types of data (either historic or future projections under climate change scenarios), such as monthly, daily, or annual maximum rainfall in 24 h, streamflows or temperature [1,4,9,11,12,13,14,36,48,60,61,62], among others.

6. Conclusions

The spatial and temporal distribution of rainfall data is an important factor in the simulation of hydrological processes. The present study develops a comprehensive methodology for Rainfall Frequency Analysis using a Python software package called RainFA. It incorporates a number of key techniques, such as time scale homogenization, data curation, cluster analysis, L-moment homogenization measure, spatial distribution for series completion, and PDF fitting, for a complete Rainfall Frequency Analysis that could be applied worldwide. In addition, the package includes a tool for estimating sewer discharge from Monte Carlo rainfall simulations generated from the selected PDF distribution and the characteristics of the existing sewer network.

The RainFA software has been applied to the city of Palma (Spain) and has successfully identified a region with acceptable homogeneity for daily rainfall above 0.4 mm, with stations covering almost the entire catchment area of the EMAYA drainage system. Furthermore, the results show a satisfactory performance in terms of the detection of rainfall events, both in absolute and relative terms. However, the rainfall accuracy shows suboptimal performance in detecting the total amount of daily rainfall. Nevertheless, the results for our homogeneous region could be improved using more robust inference techniques. In addition, a methodology for simulating annual rainfall events and estimating annual wastewater discharges is proposed and tested to meet future European requirements [19]. This new methodology and the complete application procedure implemented in RainFA and applied to the context of the metropolitan area of Palma complements and develops the calculation procedure regulated in Spain by [20], providing a toolbox for the determination of the discharge capacity of urban drainage systems. Finally, in the case of Palma, it has allowed EMAYA to analyze its attenuation and treatment needs and to evaluate the possible measures that need to be implemented in order to comply with the objectives of the EU regulation.