1 Introduction

Recent changes in the climate may be linked to an increase in the frequency and intensity of extreme precipitation events worldwide, causing socioeconomic losses and environmental impacts (Yilmaz and Perera 2015; Tabari 2020; Fowler et al. 2021; Silva et al. 2021). Such events can exacerbate natural hazards, including landslides and flash floods, in areas with reduced vegetation cover (Merz et al. 2014; Yilmaz et al. 2014; Yilmaz 2017; Dalagnol et al. 2021; Silva Cruz et al. 2022).

In this context, maximum flow analyses are crucial for hydraulic projects such as dam spillways, urban and agricultural drainage, and water-related soil erosion control (Zalina et al. 2002; Cunderlik and Ouarda 2006; Alam et al. 2018; Morabbi et al. 2022). When historical flow data are unavailable, studies of intense rainfall using pluviographic data can provide an alternative. Some of these studies use intensity-duration-frequency (IDF) relationships to estimate rainfall duration on a subdaily scale (usually from 5 to 1440 min) (Beskow et al. 2015; Fadhel et al. 2017; Yilmaz et al. 2017; Costa et al. 2020).

Brazil’s lack of subdaily rainfall data poses a significant challenge to designing engineering structures that can mitigate the socioenvironmental impacts of extreme rainfall (Diez-Sierra and del Jesus 2019; Santos et al. 2019a; Santos et al. 2019b; Costa et al. 2020; Lima et al. 2021). Acquiring long subdaily data remains challenging, particularly for field observations in developing countries such as Brazil. To address this issue, rainfall disaggregation methods, such as disaggregation coefficients, are needed to estimate subdaily values from daily data. Disaggregation coefficients, which are based on the relationships observed between hourly rainfall data with varying durations and the annual maximum daily rainfall in a specific area, are widely employed in Brazil due to their straightforward implementation (Koutsoyiannis 2003; Sane et al. 2018; Silva Neto et al. 2017; Abreu et al. 2022).

The use of methodologies, such as disaggregation coefficients, to estimate rainfall at smaller temporal intervals has emerged as a commonly used technique (Caldeira et al. 2015; Martins et al. 2019; Passos et al. 2021). Disaggregation coefficients allow the calculation of precipitation totals based on the characteristics of hourly rainfall rates in a particular region and enable the assessment of the return probability of extreme rainfall events in shorter periods (Pui et al. 2012; Kunkel et al. 2013). Previous studies in Brazil have employed CETESB’s disaggregation coefficients (1986), which are averaged from coefficients from several rainfall stations throughout the country; thus, their applicability to specific regions is potentially limited.

The widely used technique for generating IDF curves at subdaily intervals across Brazil is based on the disaggregation coefficients developed by CETESB (1986), as demonstrated in studies by Silveira (2000) for Rio Grande do Sul State, Ferreira et al. (2005) for São Paulo State, Oliveira et al. (2008) for Goiás State, and Passos and Mendes (2018) for the Balsas Municipality in Maranhão State. However, these coefficients may lead to overestimation or underestimation of maximum hourly rainfall due to their generation throughout Brazil, as reported by Back and Wildner (2021) and Silva Neto et al. (2021). Regional coefficients, however, closely matching local rainfall characteristics, atmospheric dynamics, and associated weather systems are essential.

Thus, this study aimed to analyze the frequency and intensity of extreme rainfall in the Itacaiúnas River basin (IRW) on daily and hourly scales. Additionally, the study evaluated methodological procedures to estimate heavy (R95p) and extreme (R99p) rainfall in greater detail and accuracy. This study uses the nomenclature proposed by Frich et al. (2002), where heavy rainfall is defined as records above 95% of the quantile and extreme rainfall as records that exceed 99% of the quantiles. In addition, the study innovates by adapting the rainfall disaggregation coefficients proposed by CETESB (1986) for Brazil to regional conditions.

The IRW is in the Brazilian Amazon’s “Deforestation Arch” in Pará State. Extensive deforestation has occurred since the 1970s, with natural cover replaced by pasture, crop, and mining areas (Souza-Filho et al. 2018; Cavalcante et al. 2019b; Silva Júnior et al. 2019; Silva Júnior et al. 2022; Lima et al. 2022). This intense human intervention in the natural land cover has increased the vulnerability of the IRW to the adverse socioenvironmental effects of extreme rainfall, thereby necessitating further investigation.

This paper’s organization is as follows: Section "Materials and methods" presents the study area and the methods used, including data collection, an analysis of homogeneous groups, the probability distribution of extreme rainfall, and IDF based on mean disaggregation coefficients for Brazil and estimated with in situ gauge stations. Section "Results and discussion" presents the results of these analyses, including rainfall regionalization (homogeneous groups), evaluation of FDP fitting to the annual maximum daily rainfall, and comparison of estimates of maximum hourly rain performed by the disaggregation coefficients calculated in this research to those defined by CETESB (1986).

2 Materials and methods

2.1 Study area

The IRW is situated in the southeastern region of Pará in the eastern Brazilian Amazon, covering an area of approximately 41,300 km2, bounded by latitude 05°10′ to 07°15′S and longitude 48°37′ to 51°25′W (see Fig. 1). It is characterized by the presence of the Serra dos Carajás, a plateau with altitudes ranging between 400 and 900 m, contrasting with the adjacent regions’ lower elevations (see Fig. 1A). The Itacaiúnas River is a tributary of the Tocantins River’s left bank after the latter’s confluence with the Araguaia River (Pontes et al. 2019).

Fig. 1
figure 1

The geographic location of the Itacaiúnas River watershed (IRW), southeastern Pará State (PA), Brazil, and the spatial distribution of meteorological stations and rainfall stations (in A). Climogram for the Marabá (PA) meteorological station from 1976 to 2016 (in B)

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The basin is primarily used for pasture (51.1%) and mining (0.3%); natural forest covers 48.1% of it, and the remainder is occupied by water bodies (0.3%) and savanna (0.2%) (Nunes et al. 2019). Furthermore, increasing urbanization and industrialization in the cities in the region, especially in Marabá and Parauapebas, two of the largest cities in Pará State (IBGE 2021), have led to a rise in water demand.

The climate of the IRW is classified as “Aw,” tropical with a summer rainy season and dry winter, according to the Köppen climate classification (1936) adopted for the Brazilian territory by Álvares et al. (2013). The climate is characterized by humid equatorial conditions with rainfall concentrated primarily in the summer and high temperatures throughout the year. The climogram of the locality of Marabá (PA), used as a reference to characterize the area between 1976 and 2016, shows an average annual temperature of 27.1 °C and average annual rainfall of 1833.4 mm (Fig. 1B). Approximately 83% of the annual rainfall occurs between October and May, which is the rainy season, while from June to September, accumulated rainfall is less than 17% of the annual volume (Tavares et al. 2018; Cavalcante et al. 2019a). The highest rainfall records are observed between December and April, with rainfall totals exceeding 200 mm, contrasting with the period from June to September, with an average rainfall of less than 50 mm. The annual thermal amplitude is low, characteristic of areas near the equator. In Marabá, the thermal averages are above 26 °C for all months of the year, with higher values above 28 °C in the drier months (August and September) and below 27 °C in the wettest months (December and April) (Fig. 1B).

2.2 Methodological procedures

The flowchart in Fig. 2 describes the methodological procedures used, where methodologies differ according to the objective of the analysis and the spatiotemporal availability of hourly and daily data.

Fig. 2
figure 2

Flow chart for methodological procedures applied for extreme rainfall analysis in the IRW

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2.2.1 Obtaining rainfall data

Daily rainfall data were collected from six Brazilian National Water Agency (ANA 2021) stations and one conventional meteorological station of the Brazilian National Institute of Meteorology (INMET 2021), covering the period 1988 to 2018. The data had a maximum of 10.58% missing values (station ID 02—Eldorado dos Carajás) and a minimum of 0.13% (station ID 01—Marabá). Missing data were filled in using the PERSIANN-CDR satellite product, following the recommendations of Xavier et al. (2021) for the Mearim River basin. In addition, complete hourly rainfall data during the selected period for 2016 to 2018 were obtained from weather stations managed by the Vale Institute of Technology (ITV 2021) and the Marabá (INMET) station.

2.2.2 Cluster analysis for identifying the homogeneous rainfall groups

The study employed a clustering technique to identify homogeneous rainfall groups in the IRW. The squared Euclidean distance, which measures the geometric distance between two observations in multidimensional space, was used as a proximity measure. The Ward method (Ward 1963) was applied to daily data series to group meteorological and rainfall stations with similar rainfall characteristics. The dendrogram was generated using Statistica 10 software. The physiographic factors, such as relief, rainfall regime, and proximity space between the rainfall stations, were used to delimit homogeneous groups, a technique used in previous studies (Teodoro et al. 2016; Shiau and Lin 2016; Terassi et al. 2020; Brasil Neto et al. 2021; Zerouali et al. 2022).

Descriptive statistics were computed to determine the maximum value, lower quartile (LQ), higher quartile (HQ), outliers, and percentiles (95th and 99th) of the data set (Yang et al. 2017). Outliers were identified using Eq. 1, which involves adding 1.5 times the interquartile range (HQ-LQ) to the average of the daily rainfall.

$$Outlier= average+1.5\ \left( HQ- LQ\right)$$
(1)

The nomenclature suggested by Frich et al. (2002) was adopted to define heavy rainfall, using the acronym R95p for daily records equal to or greater than the 95th quantile and R99p for intense daily rainfall equal to or greater than the 99th quantile.

2.2.3 Determining probability distributions and associated metrics

Annual maximum daily rainfall data for the period 1988–2018 from the ANA and INMET meteorological stations were fitted to 11 probability distribution functions (PDFs): Fréchet, gamma, gamma 3 parameters, generalized extreme value (GEV), Gumbel, log gamma 3 parameters, log-normal, log normal 3 parameters, normal, Weibull, and Weibull 3 parameters. These PDFs are widely used in the literature for maximum daily rainfall estimation in various regions (Koutsoyiannis et al. 1998; Katz 2010; Rulfová et al. 2016; Yuan et al. 2018; Xavier et al. 2019a; Lima et al. 2021).

The Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests were employed to assess the goodness of fit of each PDF (Fischer et al. 2012; Ye et al. 2018; Xavier et al. 2019b; Moccia et al. 2021). The Akaike information criterion (AIC), Bayesian information criterion (BIC), and log-likelihood function (LLF) validation metrics were used to compare and select the best-fitting PDF (Coles and Dixon 1999; Svensson et al. 2007; Svensson and Jones 2010). The PDFs with the lowest AIC and BIC values were considered to best describe and model the maximum daily annual rainfall estimates. The Friedman nonparametric test was utilized to categorize the PDFs based on their ordering within a categorized set (Cahill 2003; Kim et al. 2017; Navares and Aznarte 2020).

$$AIC=-2 loglog\ (ML)+2k$$
(2)
$$BIC=-2\mathit{\log}\ (ML)+ klog\ (n)$$
(3)

where log (ML) is the maximized log-likelihood function under the proposed model and k is the number of parameters in a given model.

2.2.4 IDFs and disaggregation coefficients

IDF curves are a widely used tool in hydraulic engineering for estimating the “project design rainfall,” a hypothetical rainfall event used to design hydraulic structures based on the maximum daily rainfall records. This study calculated disaggregation coefficients for ITV and INMET meteorological stations using the hourly rainfall average and its proportional relationship to other hourly intervals (01, 06, 08, 10, and 12 h). Table 1

Table 1 Identification (ID), geographic location (latitude and longitude), altitude, analysis period, and data record failures (%) of the rainfall stations and meteorological stations* located on the IRW and its surroundings
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The frequency of a rainfall event is commonly associated with its return period (RP), which represents the time interval in which a rainfall value can be equal to or exceeded (Papalexiou et al. 2013). The longer the RP of a precipitation value is, the lower the probability of that event being equaled or surpassed, whereas the shorter the RP is, the greater the probability of rainfall being equaled or exceeded. This concept has been well established in the literature (Mohymont et al. 2004; Ghiaei et al. 2018).

The creation of the IDF curves followed these steps:

  1. i

    Definition of annual daily maximums for the conventional (ITV and INMET) data series (1988 to 2018)

  2. ii

    Determination of the maximum daily rainfall for the RPs (2, 5, 10, 25, 50, and 100 years) according to the 11 probability distributions (Section "Determining probability distributions and associated metrics")

  3. iii

    Delimitation of homogeneous groups of daily rainfall considering the 12 meteorological stations (INMET)

  4. iv

    Calculation of the disaggregation coefficients from the hourly rainfall of the network of ITV and INMET stations (Table 2)

  5. v

    Disaggregation of maximum daily rainfall from the disaggregation coefficients of CETESB (1986) and calculated by the present research

  6. vi

    Generation of IDF (mm/h) curves from the disaggregation coefficients calculated by the hourly data acquired from the ITV and INMET meteorological stations

Table 2 Disaggregation coefficients for the homogeneous groups (HGs) in the IRW, calculated for Brazil (CETESB 1986)
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The study estimated the disaggregation coefficients of rainfall intensity for two meteorological stations (ITV and INMET) and compared them with the mean values for Brazil (CETESB) across different hourly durations (1 h, 6 h, 8 h, 10 h, and 12 h) out of 24 h. The coefficients were calculated by evaluating the direct proportionality between the maximum hourly rainfall and the maximum daily rainfall at the selected stations. Although subhourly coefficients were available for CETESB, hourly time steps of ITV and INMET stations limited their estimation for the IRW.

3 Results and discussion

3.1 Analysis of homogeneous groups and rainfall intensity

3.1.1 Daily rainfall characterization and cluster analysis

Cluster analysis revealed the presence of two homogeneous groups (HGs) that differed primarily from December to May (Fig. 3). The monthly averages in HG-I were greater than 300 mm in February (313 mm) and March (333.9 mm), while HG-II reported a maximum of 247.4 and 257.7 mm during these same months. The average annual rainfall was estimated at 1826.7 mm and 1559.0 mm for HG-I and HG-II, respectively.

Fig. 3
figure 3

Dendrogram (Ward’s method—in A) and the spatial distribution of homogeneous groups (HGs) and hypsometry (m) for daily rainfall in the IRW (in B). Monthly rainfall distribution (mm) in homogeneous groups (HGs) of the IRW (in C), according to daily records (1988–2018)

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The analysis of daily rainfall data showed that HG-I exhibited higher levels of heavy (R95p) and intense (R99p) rainfall between February and March, with thresholds greater than 40 mm and 70 mm in December, respectively. The maximum daily rainfall for HG-I was recorded in October (225.3 mm), December (182.0 mm), and April (175.3 mm), with rainfall above 150 mm typically observed from October to December, February, April, and May. Outliers were detected between January and April, with values exceeding 20 mm and a maximum above 30 mm between February and March (Table 3).

Table 3 Descriptive daily rainfall (mm) statistics for HG-I and HG-II in the IRW
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For HG-II, heavy and intense rains were also identified between February and March, with R95p thresholds greater than 40 mm and R99p records exceeding 70 mm in December. The maximum daily values were observed between December and May (> 140 mm), especially in January (170.0 mm) and April (165.5 mm). Outliers exceeding 15 mm were detected between January and April, with the maximum in February (23.6 mm) and March (23.3 mm) (Table 3).

The rainfall regime in the studied region is mainly influenced by the southern position of the Intertropical Convergence Zone (ITCZ) during the Southern Hemisphere summer and the South Atlantic Convergence Zone (SACZ), which generates rainfall (Nobre et al. 1991; Marengo 2005). Variability in the convection center of the Lower Amazon (Hastenrath and Geischar 1993; Carvalho et al. 2002) also impacts rainfall amounts. The role of frontal systems in organizing mesoscale convective systems that produce abundant rainfall, particularly in the rainy season, is noteworthy (Falck et al. 2015; Serrão et al. 2021). Instability lines, which usually form along Brazil’s northern coast and propagate inland every 2 days, are also significant, with more or less frequency between April (October) and June (November) (Cohen et al. 1995; Alcântara et al. 2011; Almeida et al. 2017).

3.1.2 Hourly rainfall characterization and cluster analysis

The cluster analysis conducted on hourly rainfall data in the IRW identified four HGs (Fig. 4), which differed from the grouping obtained with daily rainfall data. These differences were more pronounced between the wet and dry months. The clustering was critical in establishing and adapting the disaggregation coefficients, which were then applied to the daily time series between 1988 and 2018.

Fig. 4
figure 4

Dendrogram (Ward’s method—in A) and the spatial distribution of the homogeneous groups (HGs) and hypsometry (m) for hourly rainfall in the IRW (in B). Monthly rainfall distribution (mm) in homogeneous groups (HGs) in the IRW (in C), according to hourly rainfall records (2016–2018)

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HG-I represents the wettest sector of the IRW, in the E and NE, with an average annual rainfall of 1877.2 mm. HG-II, situated in the central and SE sectors, has an average annual rainfall of 1238.9 mm. HG-IV, in the N and NE sectors of the IRW, has an average annual rainfall of 1587.2 mm. In contrast, HG-III, found in the W and SE sectors of the watershed, has an average annual rainfall of 1554.0 mm (Fig. 4).

From November to April, HG-I exhibits average monthly rainfall exceeding 200 mm and reaching values above 300 mm in February and March. HG-II and HG-III have average monthly rainfall exceeding 200 mm between January and March, with HG-III having maximum rainfall exceeding 300 mm between February and March. During the rainy season, HG-II has significantly lower averages than the other homogeneous groups, with values below 150 mm between November and January and higher rainfall between February and March (> 200 mm). The driest period is from June to September, with averages below 60 mm, and in July, the average rainfall does not exceed 10 mm in any homogeneous group (Fig. 4).

3.2 Probability distributions, IDF curves, and return periods

3.2.1 Probability distributions of heavy and intense rainfall

The observed data adjustments generated by the PDFs were ineffective based on the results of the KS and AD tests. For ID03 (HG-II) station values, more robust adjustments were identified for PDFs gamma 3, normal, and Weibull (95% significance), while gamma and log-normal presented 90% significance (Tables 4 and 5  - Fig. S1). AIC values lower than 300 were identified for all PDFs in ID 01, ID 06, and ID 03, except for Fréchet and log gamma 3 in the latter. BIC values lower than 300 were obtained for all PDFs in ID 06, except for Weibull in ID 01. The log-likelihood function indicated values lower than − 145 in all ID 06 (HG-II) PDFs and, except for normal and Weibull, in ID 01 (HG-I). However, none of the PDFs showed goodness of fit for the observed data in ID 06 (HG-I), ID 04 (HG-II), and ID 07 (HG-II) when considering all the metrics and tests described above (Tables 4 and 5—Fig. S2).

Table 4 Descriptive level (p value) for the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) test, log-likelihood (loglik), Akaike information criteria (AIC), Bayesian information criteria (BIC), and estimated annual daily rainfall intensity for 95 (R95p) and 99 percentiles (R99p) for probability distribution functions to estimate the maximum daily rainfall at rain gauges and the weather station for HG-I in the IRW, eastern Amazon, Brazil
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Table 5 Descriptive level (p value) for the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) test, log-likelihood (LLK), Akaike information criteria (AIC), Bayesian information criteria (BIC), and estimated annual daily rainfall intensity for 95 (R95p) and 99 percentiles (R99p) for probability distribution functions to estimate the maximum daily rainfall at rain gauges and the weather station for HG-II in the IRW, eastern Amazon, Brazil
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The use of PDFs for analyzing historical heavy and intense daily rainfall series indicated that the Fréchet distribution had the highest values, ranging from 140.1 to 162 mm for R95p and 179.1 to 219.7 mm for R99p. However, the Gumbel PDF produced the highest values in ID 02, with 173.2 mm for R95p and 221.0 mm for R99p. In contrast, the normal distribution yielded the lowest values for R95p and intense R99p rains at most rainfall stations, ranging between 121.9 and 154.3 mm and 147.9 and 176.2 mm, respectively (Tables 4 and 5 - Fig. S3). The KS and AD tests revealed that the PDFs had better performance in ID 02 and ID 06, while poorer performances were observed in ID 04 and ID 05 for the KS test and in ID 01 and ID 02 for the AD test (Fig. S1).

The AIC metric indicated that the Gumbel (300.8) and gamma (301.1) probability density functions (PDFs) had better fits to the time series of annual maximum daily rainfall. In contrast, the BIC metric revealed that the Gumbel (303.6) and gamma (303.9) PDFs provided better adjustments. The Weibull (304.4) and normal (303.9) PDFs had lower adjustments for the AIC metric, while the Log gamma 3 (307.4) and Weibull (307.3) distributions produced the worst results for the BIC metric (Table 6 - Fig. S2).

Table 6 The mean value for each metric related to the model’s goodness of fit (loglik, AIC, and BIC) and the corresponding compact letter display of the Friedman multiple comparison post hoc tests at a 5% significance level. According to the Friedman test, the metrics for distributions followed by the same letter are not different
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The log-likelihood function revealed that GEV (− 147.6), Weibull 3 (− 147.6), log-normal 3 (− 147.7), and gamma 3 (− 147.9) are the most suitable PDFs for estimating the maximum annual rainfall in the IRW. However, the Weibull (− 150.2) and normal (− 150.0) PDFs showed the greatest misfits and are thus not recommended for use in the IRW (Table 6 - Fig. S2).

The Gumbel and gamma distributions better fit the maximum annual rainfall series in the IRW, as indicated by the metrics. However, the GEV and Weibull 3 PDFs best fit according to the log-likelihood function. The normal and Weibull PDFs significantly underestimated the intense rains in the IRW, and they showed the worst adjustments to the annual maximum daily rainfall in the IRW in all the evaluated metrics.

Beskow et al. (2015) conducted a study on Rio Grande do Sul in southern Brazil and identified the kappa probability distribution as the most appropriate for estimating maximum rainfall, with GEV and Gumbel distributions also suitable. Blain et al. (2021) investigated maximum annual rainfall data for the state of São Paulo in southern Brazil and found that the GEV and generalized logistic distributions (GLOs) accurately modeled annual maximum daily rainfall. Similarly, Lima et al. (2021) indicated that the Gumbel, GEV, and log-normal distributions were suitable for representing annual maximum daily rainfall in Rio de Janeiro in southern Brazil; similar results were obtained in this research.

Abreu et al. (2018) conducted statistical tests using KS and AD to evaluate extreme daily rainfall in the southwest region of Minas Gerais (southern Brazil). They found that the Gumbel and GEV probability distributions were better suited for estimating rainfall extremes in this area. In the Brazilian Amazon, Santos et al. (2015) showed that the GEV and Pareto distributions had a good fit for estimating seasonal maximum daily rainfall in different subbasins, with the GEV distribution being the most suitable. In the southeast region of Pará, where the IRW is located, the Gumbel distribution estimated maximum daily rainfall values between 106.5 and 299.6 mm during the austral autumn (March to May).

Ximenes et al. (2021) analyzed the monthly average rainfall series for the Brazilian northeast region (NEB) and found that the gamma and Weibull distributions fit well. The log-normal and generalized Pareto distributions also presented satisfactory results in some regions and certain months. However, these results differ from the estimates of maximum daily rainfall for the IRW.

3.2.2 IDF curves and return periods of extreme rainfall

A comparison of the disaggregation coefficients showed that CETESB's values from 1986 underestimated hourly rainfall at HG-I compared to those obtained in this study. The maximum rainfall intensity for a 100-year return period event in 60 min was 125.0 mm/h using the coefficient established in this research, while it was 117.6 mm/h using CETESB’s (1986) coefficient in ID 02. This pattern of underestimation is consistent across different rainfall durations (Fig. 5). In contrast, for HG-II and IV, CETESB’s (1986) coefficients overestimate the intensity of estimated rains for various durations and return periods. The differences between the rainfall intensity values generated by the two methodologies were smaller in HG-IV than in HG-II. The results demonstrate that the application of CETESB’s (1986) DC overestimates the rainfall intensity of different durations and return periods in this sector of the IRW. For HG-III, the maximum estimated rainfall lasting 60, 360, 480, and 1440 min had higher intensity than CETESB’s (1986) DC, while the maximum rain lasting 720 and 600 min was more intense in CETESB (1986) than in the DC proposed in this study (Fig. 5).

Fig. 5
figure 5

Scatter graph of maximum hourly rainfall and different return periods estimated from the disaggregation coefficients developed by CETESB (1986) and calculated by this research and the maximum daily rainfall (mm) for different return periods estimated at rain gauges at the weather station in the IRW. The return period of 2 (in a), 5 (in b), 10 (in c), 25 (in d), 50 (in e), and 100 years (in f)

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The study area’s adapted coefficients yielded maximum rainfall intensities of 125.0 and 117.6 mm/h for a 100-year RP in ID 02 and ID 05 (HG-I), respectively, while ID 06 and ID 04 had the lowest rainfall intensity values of 63.1 and 64.7 mm/h, respectively, for the same duration (Table 7). The ID 02 rainfall station (HG-I) recorded the highest rainfall intensity totals for 1440 min, with values ranging between 7.7 and 9.8 mm/h for RPs equal to or greater than 25 years, whereas for RPs of 2 to 10 years, ID 01 (HG-IV) exhibited the highest daily rainfall, ranging from 4.0 to 6.2 mm/h (Table 7).

Table 7 Intensity-duration-frequency (IDF) (mm h−1) for different return periods (RPs) at rain gauges and weather stations in the homogeneous groups (HGs) in the IRW
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For RPs equal to or greater than 10 years, ID 06 (HG-III) is expected to have less intense rainfall intensities, ranging from 4.9 to 7.0 mm/h over 1440 min. Consequently, the expected 1440-min rainfall intensity for a 100-year RP in ID 02 (9.8 mm/h) is 40% greater than that estimated for ID 06 (7.0 mm/h). The lowest 1440-min intensities for RPs of 2 and 5 years are anticipated at ID 07 (HG-III), with values of 2.9 mm/h and 4.2 mm/h, respectively (Table 7).

Silva Neto et al. (2020) analyzed the state of Tocantins and reported rainfall intensities lasting 1440 min ranging from 7 mm/h (RP10y) to 12 mm/h (100-year RP). For rainfall durations of 720 min, the intensities were 14, 20, and 23 mm/h for RPs equal to 10, 50, and 100 years, respectively. The extreme N and NW regions of Tocantins, adjacent to the IRW, experienced the highest intense rainfall due to the prevalence of the Continental Equatorial air mass, which facilitates convective rainfall of short duration and high intensity. Santos et al. (2015) identified the Brazilian Amazon as the rainiest region in Brazil, with a maximum daily intensity estimated to be between 219.5 mm (west) and 430.5 mm (east) for a 100-year RP.

4 Conclusions

The IRW sectors with the highest rainfall totals were identified through cluster analysis. HG-I in the SE and NE watershed sectors had the highest rainfall, followed by HG-II in the S and SW sectors. Rainfall is concentrated (> 90%) in both homogeneous rainfall groups from October to May, with the highest totals in February and March. HG-I (February to April) experienced the most intense daily rainfall for the thresholds of R95p and R99p. The statistical tests and metrics indicated that the rainfall time series adhered well to the PDFs. The Gumbel and gamma distributions fit more closely to the series of maximum annual rainfall in the IRW, as indicated by the AIC and BIC metrics. The LLF indicated the best fit for the GEV and Weibull 3 PDFs. The normal and Weibull distributions mostly underestimated the intense rains in the IRW, as the PDFs were less adjusted to the IRW rain time series.

Calculating disaggregation coefficients as a function of homogeneous groups and patterns allows for evaluating and estimating rainfall totals at other stations by adapting to the dynamics of regional rainfall. However, CETESB’s disaggregation coefficients (CETESB 1986) underestimate the maximum hourly rainfall for HG-I and overestimate the maximum rainfall for HG-II and IV for different RPs. IDF curves demonstrate values exceeding 115.0 mm/h for a maximum rainfall of 60 min and RP of 100 years in HG-I. Conversely, the lowest value (~ 65 mm/h) for maximum hourly rainfall for the RP of 100 years was recorded in HG-II. This study is expected to contribute to redimensioning agricultural, urban, and road drainage hydraulic works by indicating essential parameters that may prevent or minimize the socioenvironmental impacts of floods and landslides. Future studies should investigate the occurrence of intense rainfall in different climate scenarios, considering the Intergovernmental Panel on Climate Change (IPCC) reports, and examine how the IDF curves would change under these scenarios.

For public managers, we recommend maintaining and improving an active pluviograph station network to measure and understand subdaily extreme events, which is of great relevance to understanding watersheds that have high erosivity levels and dense drainage networks. This information will subsidize the management of risks and vulnerabilities in complex watersheds of interaction between the natural attributes of the landscape and different land use and types of occupation.