1. INTRODUCTION

Excessive rainfall can cause agricultural losses, soil erosion, and flooding. In addition to causing material damage, these events represent a major risk to life, especially in urban areas. Thus, knowledge about extreme rainfall in a given location has great application in urban and agricultural planning, besides being used in environmental risk analysis, water infrastructure projects, irrigation, and dimensioning of engineering drawings (^{Deng et al., 2017}; ^{Coelho Filho et al., 2017}).

To characterize extreme rainfall, one must know the intensity, duration of the event, and the frequency of occurrence, which can be represented by intensity-duration-frequency (IDF) curves and equations (^{Silveira, 2016}). These equations have great application in the hydrological dimensioning of urban drainage structures, in hydrological models for flow estimation, and in agricultural drainage and soil conservation (^{Marra et al., 2017}; ^{Ouali and Cannon, 2018}).

Intensity-duration-frequency equations are determined using traditional methods based on data from rainfall stations (^{Martins et al., 2017}; ^{Tfwala et al., 2017}). In the absence or scarcity of long data series, information from rainfall stations is gathered, and the maximum 1-day rainfall is clipped to shorter rainfall events, thus allowing the fitting of equations (^{Penner and Lima, 2016}; ^{Rangel and Hartwig, 2017}; ^{Dias and Penner, 2019}). Some studies assess the possibility of using satellite or radar observations to obtain IDF equations (^{Marra et al., 2017}).

^{Mirzaei et al. (2014)} claim that is important to assess the uncertainties related to extreme rainfall estimates and to propagate those uncertainties into design decisions and risk assessment, and point out that uncertainty in depth-duration-frequency curves is usually disregarded in the view of difficulties associated in assigning a value to it. ^{Mirzaei et al. (2015)} investigated uncertainties incorporated in the distribution function of the series of annual maximum daily rainfall.

There are different statistical distributions of extreme events that can be used to fit a set of hydrological data. Among them is Gumbel, which is the most used for fitting data in studies of extreme rainfall (^{Gonçalves et al., 2019}; ^{Petrucci and Oliveira, 2019}). Another distribution that has been shown to be quite adequate to represent extreme natural events such as heavy rainfall is the Generalized Extreme Value (GEV) distribution (^{Quadros et al., 2011}; ^{Tfwala et al., 2017}). ^{Olofintoye et al. (2009)} point out that many statistical distributions can be applied to describe extreme annual rainfall events in a given location. However, choosing the appropriate model is one of the biggest problems in engineering practice, as there is no general agreement on which distribution or distributions to use in the analysis of extreme rainfall frequency. The selection of the appropriate model depends mainly on the characteristics of the available rainfall data for a particular station. That is why it is necessary to evaluate several distributions to find the model that allows obtaining the best extreme rainfall estimates.

In Alagoas State there is a lack of information on IDF equations (^{Dias and Penner, 2019}). Thus, the present study analyzes the maximum annual rainfall and determines IDF equations in 44 rainfall stations distributed throughout Alagoas State, Brazil.

2. MATERIALS AND METHODS

Alagoas State is bathed by the Atlantic Ocean and borders the states of Pernambuco, Sergipe, and Bahia. The state is located between latitudes 8°54’57” S and 9°19’50” S and longitudes 35°09’08” W and 38°13’38” W. The relief is divided into three major types, starting from east to west through the coastal plain, followed by the tablelands region, and the plateau that corresponds to most of the Alagoas territory. According to ^{Barros et al. (2012)}, the annual rainfall averages in Alagoas State vary from 2,000 mm, on the coast, to 400 mm in the hinterlands (Sertão). These values gradually decrease from east to west.

Climatic variation throughout the state is quite significant. According to the Köppen climate classification, Alagoas State is divided into four zones. There is the occurrence of humid tropical climate (Ams) and semi humid tropical climate (As) in the most coastal region of the state, which corresponds to the forest zone, the most humid part of the Agreste region, and the coast. Such climates are characterized by abundant rains throughout the year and a well-defined dry season. To the west of Alagoas, in the Agreste and Sertão, climatic classification comprises the driest types, with a hot semi arid climate (BSsh), in which evaporation exceeds rainfall. There is also the presence of As climate in a small area northwest of the state (^{Barros et al., 2012}).

The study included rainfall data from 44 rainfall stations located in Alagoas State, belonging to the Hydrological Network of the National Water Agency (ANA). We selected stations that presented at least 20 years of data. Table 1 contains the stations used, their respective coordinates, historical series, and climate data. Figure 1 shows the spatial distribution of these stations.

Figure 1. Location map of the rainfall stations used. 

To analyze the historical series of extreme rainfall data, we used the Gumbel distribution (GD) (Type I distribution of extremes) and the Generalized Extreme Value (GEV) distribution, whose probability density functions (PDFs) are given, respectively, by Equations 1 and 2:

Where x is the maximum annual daily rainfall; α, β, and k are parameters of the probability distribution (^{De Alcântara et al., 2019}).

The parameters of the Gumbel distribution were estimated using the Moments (MM), Maximum Likelihood (MLM), and L-moments (MML) methods (^{Naghettini and Pinto, 2007}), in addition to the method proposed by Chow (CH) (^{Back and Bonetti, 2014}). The parameters of the GEV distribution were adjusted by the Moments (MM) and L-moments (MML) methods (^{Naghettini and Pinto, 2007}).

Following ^{De Alcântara et al. (2019)}, two tests were used to analyze the fitting to the distribution: Kolmogorov-Smirnov (KS) and Anderson-Darling (AD), considering the ranking of distributions and the respective methods of estimating parameters. The distribution with the lowest sum of the ranks of the two tests was selected.

Using the selected distribution for each data series, the values of maximum daily rainfall with return periods of 2, 5, 10, 15, 25, 50, and 100 years were determined. The breakdown of daily rainfall into shorter duration rainfall followed the methodology proposed by ^{CETESB (1986)}, estimating maximum rainfall intensities for 5, 10, 15, 20, 25, 30, 60, 360, 720, and 1440 minutes.

With the values obtained from maximum rainfall intensities for different durations and return times, we determined the parameters of the Equation 3 that expresses IDF:

Where: I is the average maximum rainfall intensity (mm h^-1); K, m, b, n are the coefficients to be fitted; T is the return period (years); t is the rainfall duration (minutes).

To fit the Equation 4, we minimized the standard error (RMSE), expressed by:

With the IDF equations obtained for each station, we determined rainfall intensity for different durations (15, 30, and 60 minutes, and maximum daily rainfall with a 10-year return period). To represent the spatial distribution of extreme rainfall, data were interpolated in ArcGIS 10.3 software using the ordinary kriging method with spherical model.

3. RESULTS AND DISCUSSION

For all data series, neither GD nor GEV distributions were rejected by the KS and AD adhesion tests. Although all distributions were adequate, the one with the best adherence was chosen (Table 2). In general, the GEV distribution showed better results, with this distribution being chosen for approximately 80% of stations. The GEV distribution obtained by the L-moments method was highlighted with the best fitting in 32 (73%) historical series. The Gumbel distribution is widely used for maximum annual rainfall (^{Ottero et al., 2018}; ^{Mistry and Suryanarayana, 2019}). Notwithstanding, there are studies indicating that the GEV distribution has been shown to be superior to the Gumbel distribution (^{Beskow et al., 2015}; ^{Namitha and Vinothkumar, 2019}).

Table 2 also presents the values of the coefficients of the IDF equation fitted for each station, the standard error values, and the coefficients of determination (R²). For all stations, correlation coefficients greater than 0.991 and RMSE values ranging from 1.94 to 6.82 mm h^-1 were obtained. ^{Sabino et al. (2020)} evaluated the fitting of the IDF equation for 14 rainfall stations in Mato Grosso State. The authors obtained a correlation coefficient (R²) ranging from 0.8665 to 0.9596, and RMSE ranging from 8.40 to 15.69 mm h^-1. These data show the good quality of the fitting of IDF equations for Alagoas State.

The K coefficient values ranged from 268.5 to 1107.4, and the m coefficient values ranged from 0.092 to 0.324. Moreover, b values approached 9.19 for all rainfall stations, and the n coefficient values were equal to 0.706. Other studies have already reported values practically constant for parameters b and n in fitting the coefficients of the IDF equation (^{Caldeira et al., 2015}; ^{Souza et al., 2012}). ^{Sabino et al. (2020)} fitted IDF equations for 14 rainfall stations in Mato Grosso State and also observed a higher coefficient of variation in coefficients K and b.

The K parameter is directly proportional to the rainfall intensity. The places where the highest values of this parameter were found coincide with the regions with the highest rainfall values, corresponding to the eastern/coastal region of tropical climate. In turn, the lowest K values are observed in the interior of the state, since in this region there is a dry climate. Therefore, there are coincidences with the characteristics of the Köppen climate classification, as already noted by ^{Silva and Oliveira (2017)}.

The fitting of IDF equations allowed estimating rainfall intensities for 15, 30, and 60 minutes with a 10-year return time, in addition to the maximum 1-day rainfall, using kriging to interpolate the data (Figure 2). The highest intensities occur on the coast, decreasing from east to west. Knowledge of IDF relationships, especially in places where hydrological monitoring is scarce, is an important tool for urban, agricultural, and environmental planning. Several engineering areas demand information about extreme rainfall, such as power generation, dams, civil construction, and urban drainage.

Figure 2. Rainfall intensity (mm h^-1) for different durations with a 10-year return period, and maximum 1-day rainfall. 

There is a marked spatial variation in maximum rainfall intensity in Alagoas State. Knowledge of this variation in rainfall intensity is important for planning water resource management actions as well as for soil conservation and engineering projects. The 5-minute rainfall intensity is used in the dimensioning of gutters to capture rainwater (^{Back and Bonetti, 2014}). For soil conservation and gradient terracing, it is common to use the 15-minute rainfall intensity and a 10-year return period (^{De Maria et al., 2016}). For level terraces, the maximum 1-day rainfall intensity and a 10-year return period is recommended. These values can be obtained from the IDF equations established for the rainfall stations in Alagoas (Table 2). The maximum 30-minute rainfall intensity is used as an indicator of the rainfall erosive potential. Therefore, Figure 2 indicates the locations with the greatest rainfall erosive potential in Alagoas State.

4. CONCLUSIONS

Alagoas’ climate is quite varied, with tropical climate to the east and dry climate to the west. The highest averages of maximum annual rainfall coincide with the regions of tropical climate.

The series of maximum annual rainfall showed good fitting to Gumbel and GEV distributions, all of which were approved by the Kolmogorov-Smirnov and Anderson-Darling adhesion tests.

The GEV distribution with parameters obtained by the L-moments method was considered the best in 73% of rainfall stations.

The estimated IDF equations showed a good fit, with determination coefficients above 0.991. These equations allow estimating rainfall intensity from 5 minutes to 24 hours with a return period of 2 to 100 years, and standard error of 6.822 mm h^-1.

There is a marked spatial variation in maximum rainfall intensity in Alagoas State, showing the need for hydrological studies addressing each climatic region of the state.