The aim of this study is to estimate the return period of maximum daily precipitation for each season of the year in different subregions of the Brazilian Amazon. For this, the extreme value theory was used, through the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD). The GEV distribution and GPD were applied in precipitation series from homogeneous regions of the Brazilian Amazon. The GEV and GPD goodness of fit were evaluated by the application of the Kolmogorov–Smirnov (KS) test, which compares the cumulative empirical distributions with theoretical ones. The KS test results indicate that the tested distributions have a good fit, particularly the GEV distribution. Thus, they are adequate to study the seasonal maximum daily precipitation. The results indicate that extremes of more intense rainfall are expected during the rainy or transition seasons of each subregion. Using the GEV distribution (GPD), a daily total of 146.1 (201.6), 143.1 (209.5), and 109.4 (152.4) mm is expected at least once a year in the south, at the Atlantic coast in the Amazon catchment, and in the northwest of the Brazilian Amazon, respectively.
In tropical regions, particularly in the case of the Amazon region, the spatiotemporal variation of meteorological attributes, especially rainfall, is related to the performance of meteorological phenomena at different scales, modulated by ocean–atmosphere mechanisms, which produce total rainfall above and/or below the climatological average.
In the Brazilian Amazon, most of the annual rainfall occurs between the austral summer and austral autumn seasons. The highest values found in the austral summer occur in the southern Amazon, oriented from northwest to southeast, because of the action of the South Atlantic convergence zone (SACZ; Carvalho et al. 2004; Grimm 2011; de Oliveira Vieira et al. 2013; de Quadro et al. 2012). In this season, higher rainfall is observed in the central part of the Amazon, which may be associated with condensation of moist air transported by the trade winds, to the east of the Andes Mountains (Nobre et al. 1991; Da Rocha et al. 2009).
During the austral autumn, precipitation is reduced over the southern Amazon region. However, the highest values of rainfall associated with displacement to the south of the intertropical convergence zone (ITCZ), which is the main regulatory system of rainfall variability in eastern Amazonia, are recorded to the northwest and on the coast. In most parts of the Amazon, the drought season is throughout the austral winter, because it is established as the ITCZ moves to its position farther north (Fu et al. 2001; de Souza and da Rocha 2006; Chen et al. 2008; de Souza et al. 2009; Moura and Vitorino 2012). During the austral spring the convective activity associated with the SACZ in the southern and southeastern Amazon begins, starting the seasonal cycle of rainfall in these areas.
The main mechanisms of tropical ocean–atmosphere circulation that can affect rainfall anomalies in this region are El Niño–Southern Oscillation (ENSO) over the Pacific Ocean and the interhemispheric meridional gradient of sea surface temperatures (SST) anomalies over the Atlantic Ocean (Nobre and Shukla 1996; Souza et al. 2000; Liebmann and Marengo 2001), which act in different phases favoring or disfavoring convective activity in tropical areas (de Souza et al. 2005).
The largest floods recorded in the Amazon occurred in 1954, 1989, 1999, 2009, 2011, and 2012 (Marengo et al. 2013b). The main causes of these floods were La Niña and/or anomalously warm ocean waters in the tropical South Atlantic (Vale et al. 2011; Sena et al. 2012; Marengo et al. 2013a,b; Satyamurty et al. 2013; Espinoza et al. 2013). Because of SST anomalies in the tropical South Atlantic, the ITCZ stays in the south for a longer time, when compared to its mean position, leading to extreme rainfall in Amazonia (Marengo et al. 2012b, 2013a).
According to Gloor et al. (2013), since 1990 there has been an intensification of the hydrological cycle in the Amazon basin, with an increase in runoff during the rainy season and occasional severe droughts. Brito et al. (2014) studied different categories of extreme precipitation events in the Amazon analyzing the frequency, intensity, and contribution to the climatology of accumulated precipitation between 1998 and 2013 and found that extreme precipitation produced more rain in the last 7 years, reaching its peak during 2011 and 2012.
In the Amazon, various activities of the productive sector, particularly those related to agriculture, industry, hydropower generation, distribution of energy, and so on, are affected by extreme precipitation events, making the population vulnerable to variability in the climate system. Intense and prolonged rainfall may have negative consequences, primarily for the population occupying the shores of rivers, because when there is an elevation of the water level, in general, there are floods (river waters rise to the height of its banks, without overflowing) and/or inundations (river waters overflow). During 2014, two Brazilian states (Acre and Rondônia) declared a state of emergency because of floods caused by heavy rainfall in the headwaters of its rivers.
In this sense, the probabilistic prediction of the occurrence of extreme precipitation events is of vital importance for the planning of activities exposed to its adverse effects. One way to model these events is to use the extreme value theory (EVT), through the generalized extreme value (GEV) distribution, which includes the distributions of Gumbel, Fréchet, and Weibull, and the generalized Pareto distribution (GPD), as the exponential, Pareto, and the beta. Thus, this study aims to estimate the return period of such events through the GEV and GPD, considering the seasonal maxima as extremes, and to indicate the regions and the period (season) with more serious occurrences in the Brazilian Amazon.
2. Material and methods
The daily rainfall dataset was obtained from the National Water Agency (Agência Nacional de Águas) and the Bank of Meteorological Data for Education and Research (Banco de Dados Meteorológicos para Ensino e Pesquisa) of the National Institute of Meteorology (Instituto Nacional de Meteorologia). The stations were selected following the recommendations of the World Meteorological Organization (WMO 1989) for the period from 1983 to 2012. In this document, it is recommended to 1) discard the month that shows any missing daily value and 2) exclude from the climatological normal the monthly data that present three or more consecutive gaps or that have more than five alternate months missing. The initial set consisted of 1129 rain gauges, but following the WMO recommendations, 305 remained.
The return period of extreme precipitation events in the Brazilian Amazon was obtained for homogeneous rainfall regions, determined by Santos et al. (2015). These authors used Ward’s hierarchical clustering method and, as a similarity measure, the Euclidean distance. Six subregions of homogeneous rainfall were identified (Fig. 1): two subregions in the southern Brazilian Amazon and four subregions up north (two in the coastal area and two in the northwest portion). According to Santos et al. (2015), these subregions are sufficient to represent the rainfall in the Brazilian Amazon. These six subregions feature different precipitation patterns and intensities.
In the present study, synthetic series of precipitation were used, which consist of the daily maximum values of each subregion. The synthetic series were analyzed considering the EVT, which is a branch of theoretical probability that studies the stochastic behavior of the extremes associated with a distribution function F, which is normally unknown. Its main goal is to estimate the upper tail of a probability distribution of a set of independent observations that are equally distributed.
To ensure the independence of the time series of daily precipitation, values were placed in a disorderly manner, so that the daily maximum did not occur on consecutive days. To test the hypothesis of the independence of the data, the nonparametric test of sequences of adherence to the normal distribution was used, called a run test, which checks whether the elements of the series are independent of each other. A 5% significance level for the test was adopted. According to Sharma et al. (1999), the implementation of this assumption ensures the achievement of satisfactory statistical inferences from probabilistic models of extreme values.
The goodness of fit of the distributions was checked through the nonparametric test of normal distribution adherence, the Kolmogorov–Smirnov (KS) test, with a 5% significance level. In this test, the null hypothesis is and the alternative hypothesis is . The test statistic is obtained by is the theoretical cumulative distribution function, and G(x) is the empirical cumulative distribution function, to n random observations with a cumulative distribution function. This test represents the upper extreme limit of differences between absolute values of the empirical and theoretical cumulative distribution considered in the test (Lucio 2004). The null hypothesis is rejected if the value is greater than the tabulated one. This is necessary to determine if the exact probability of the test is lower than the significance level.
The EVT uses the GEV distribution and the GPD to model the rainfall extremes. The estimate of distribution parameters from the GEV distribution and from the GPD were made by the maximum likelihood method (Smith 1985). In this theory, the return period (or the average recurrence interval) corresponds to the probability p of a return level that has a 100% chance of being exceeded in given year. The concepts of return level and return period are commonly used to convey information about the likelihood of rare events such as floods. A return level with a return period (years) of T = 1/p is a high threshold (e.g., maximum annual rainfall) whose probability of exceedance is p.
1) GEV distribution
The GEV distribution combines three asymptotic forms of extreme value distributions—Gumbel, Weibull, and Fréchet (Fisher and Tippett 1928)—in a unique form, defined according to Jenkinson (1955) as follows:
where is the location parameter with , is a scale parameter with , and is the shape parameter with .
The extreme value distributions of Weibull and Fréchet correspond to the particular cases of (1a) in where and , respectively. When , the function assumes a form (1b), which represents Gumbel distribution.
For the quantile of the GEV distribution, the cumulated probability is given by , which results in (Palutikof et al. 1999):
In the GEV distribution, the sample is divided in subperiods (blocks) that may be monthly, seasonal, annual, etc. From each block, a maximum or minimum value is extracted to compose a set of extreme data, according to the block maximum methodology, or annual maxima (Gumbel; Maraun et al. 2009; Sugahara et al. 2009).
In the present paper, for the evaluation of each GEV goodness of fit, the seasonal maxima were considered as extremes values, through the block maxima method. Thus, the final database used consists of seasonal maximum precipitation observations for the four seasons of the year.
Pickands (1975) has shown that the asymptotic distribution of excesses of a random variance above a threshold value may be approximated by GPD. As for GEV, the GPD may be understood as a family of distributions that, depending on the parameter value of the form, includes particular cases, defined as
where is the selected threshold, or in other words, the values of are the exceedances. For , the GPD is an exponential distribution, for Pareto and for beta.
where is equal to , where is the total number of exceedances over and is the number of years of the registry.
In the GPD, the datasets were determined according to the picks over threshold methodology, which only considers the values above the established threshold (Sugahara et al. 2009). The threshold indicates the minimum value of the extremes selected for each subregion and calculated from the quantiles of complete series.
There were some problems in choosing the threshold because very high thresholds increase uncertainty in the sample (variance) associated with the estimated quantile. At the same time, very low thresholds tend to increase the quantile bias. Thus, it is expected that an optimal threshold is found to minimize both the bias and the variance (An and Pandey 2005).
In this work, testing the quantiles above 95%, the best goodness of fit of the GPD was found using quantile 99% as the threshold. Then we selected the 1% of data located in the upper end of each distribution, corresponding to 28 observations of extremes in each subregion and season.
3. Results and discussion
a. General aspects of extreme events
In Fig. 2, each box plot represents the first quartile, the median, and the third quartile of the precipitation extremes used in the GEV distribution and the GPD. The whiskers extend from the box to the minimum or maximum values unless there are outliers. The whiskers only extend to values that are not outliers. The individual circles represent the outliers.
In subregions of southern Amazonia (R1 and R2) the largest extreme rainfall was registered in the austral summer (December–February; Fig. 2a), which is the rainy season of R1 and R2. These subregions are influenced by the monsoon system in South America (Marengo et al. 2012a), which modulates the formation of the SACZ (Zhou and Lau 1998; Carvalho et al. 2004), and whose spatial and temporal variability have a critical role in the distribution of precipitation extremes (Carvalho et al. 2002, 2004; Grimm 2011; de Oliveira Vieira et al. 2013; de Quadro et al. 2012).
In the subregions R1 and R2, the extremes observed in the austral summer (Fig. 2a) in the GEV distribution are between 139.3 and 285.3 mm (R1) and between 123.0 and 224.9 mm (R2). In GPD, the observed values were higher, between 165.0 and 285.3 mm for R1 and between 151.0 and 224.9 mm for R2. However, the highest recorded value (299.6 mm) in R2 did not occur in the austral summer, but in the austral autumn (March–May; Fig. 2b), in which extremes were recorded between 106.5 and 299.6 mm (GEV) and between 139.4 and 299.6 mm (GPD). In Fig. 2a, one may observe that the medians found in the GPD were higher, with values of 183.7 and 162.5 mm for R1 and R2, respectively. In these subregions, the lower extremes are found during the austral winter (June–August; Fig. 2c), with medians of 92.40 (GEV) and 99.4 mm (GPD) for R1 and 82.5 (GEV) and 84.80 mm (GPD) for R2. Despite the less intense extremes compared to other seasons of the year, in the austral winter the precipitation extremes are greater than 140 mm of rainfall. R1 presented a maximum of 148.70 mm, and R2 presented a maximum of 233.70 mm.
In coastal subregions (R3 and R4), the more intense precipitation extremes are found in the austral autumn (Fig. 2b), the rainy season of these subregions, associated with the ITCZ (de Souza et al. 2005; de Souza and da Rocha 2006). They are also associated with the coastal squall lines (CSLs), which are more frequent during the austral winter and austral autumn (Cohen et al. 1995; Alcântara et al. 2011). In this season, 50% of the extremes (median) are above 175.2 (GEV) and 192.6 mm (GPD) in R3 and 154.0 (GEV) and 164.0 mm (GPD) in R4. The maxima were 277.0 and 225 mm for R3 and R4, respectively. In R3, which is closer to the coast, the lower rainfall extremes were recorded in the austral spring (September–November; Fig. 2d), where CSLs are less frequent (Alcântara et al. 2011), with medians of 103.3 (GEV) and 119.5 mm (GPD). In R4, lower extremes were found in the austral winter (Fig. 2c), because of the displacement of the ITCZ to the north (de Souza et al. 2005; Broccoli et al. 2006), with medians of 99.0 (GEV) and 105.1 mm (GPD).
R5 and R6 are located in the northwestern part of the Amazon. R5 does not present a well-defined dry season (Santos et al. 2015). However, according to Fig. 2, the extremes of R5 are less intense compared to the other subregions, with the exception of R6. In R5, precipitation extremes are found a little higher in the austral autumn (Fig. 2b), with medians of 135.2 (GEV) and 140.3 mm (GPD). The maximum precipitation in the northwestern part of the Amazon can be explained in terms of the condensation of moist air transported by the trade winds and lifted because of the influence of the Andes (Nobre et al. 1991; Garreaud and Wallace 1997; Da Rocha et al. 2009). R6 consists of stations in the state of Roraima that are in the Northern Hemisphere and thus show the climatic characteristics of the Northern Hemisphere. The highest rainfall in R6 was recorded during the austral winter (Fig. 2c), with medians of 119.0 (GEV) and 128.3 mm (GPD). Therefore, precipitation extremes in R6 are not as high compared to other subregions.
b. Extreme distributions via EVT
The extreme precipitation events observed in Fig. 2 were modeled through the GEV distribution and GPD. Before estimating parameters of the distributions, it was found that, after being disordered, the time series of all the subregions became independent.
The parameters of the distributions obtained by the maximum likelihood estimation are shown in the figures of the return periods. In the GEV, estimates of the shape parameter are between −0.5 and 0.5 and can therefore be applied to the method according to the suggestion of Smith (1985). In the austral autumn (Fig. 3), only two types of distributions were observed, Fréchet and Gumbel . In austral summer (Fig. 4), austral winter (Fig. 5), and austral spring (Fig. 6), the three distributions—Fréchet , Gumbel , and Weibull —were found.
The KS test was conducted to check the goodness of fit of the distributions for the 5% significance level. In this study, as there are 30 observations of seasonal maximum precipitation in the GEV, the critical value used in the test is 0.24. In the GPD, as there are 28 observations of seasonal maximum exceeding the adopted threshold 99% quantile, the critical value used in the test is 0.26. Table 1 shows the test results, indicating that the settings of the GPD were accepted with a 5% significance level, with few exceptions. The settings of the R6 in the austral summer and austral spring and of the R2 in the austral autumn and austral winter were not accepted with a 5% significance. In GEV, the goodness of fit was accepted with a 5% significance level in all subregions and in all seasons. Thus, we suggest that the goodness of fit of the distributions to the studied series is suitable.
It is expected that the higher rainfall extremes occur during the austral summer (Fig. 4) and austral autumn (Fig. 3) in all subregions. However, for some return levels (5 and 10 years) of R6, it is expected that the higher rainfall levels occur during the austral winter (Fig. 5). These results indicate that when the heaviest rainfall does not occur in the rainy season in the subregion, it occurs in a period of transition.
The results of the estimated model are in accordance with observed extreme events (Fig. 2). Extremes of more intense rainfall in the austral summer and/or austral autumn are expected in all subregions, except R6. During the austral summer (Fig. 4) and austral autumn (Fig. 3), a daily rainfall of 146.1, 128.8, 143.1, 134.2, and 109.4 mm is expected at least one day per year. A daily rainfall of 234.2, 195.9, 231.4, 201.0, and 169.1 mm is expected at least once every 10 years. Finally, every 100 years, it is expected that there will be at least one day when a total of 430.5, 295.6, 297.4, 264.6, and 219.5 mm of rainfall occurs in R1, R2, R3, R4, and R5, respectively. R6 is the region in which less intense precipitation extremes are expected, being more likely in the austral autumn (Fig. 3) and austral winter (Fig. 5), when a daily rainfall of 92.1, 157.9, and 249.3 mm is expected every year, every 10 years, and every 100 years, respectively.
The fact that the most intense events occur during the austral summer and austral autumn is in agreement with Marengo et al. (2012b), who analyzed the extremes of 1989, 1999, and 2009, and shows an amount of rainfall above what is climatologically normal during the austral summer. The authors found that the rains were above normal from November 2008 to April 2009. In 1989 and 1999, the precipitation anomalies persisted during the austral autumn and austral winter.
In the GPD, all estimates of the shape parameter are greater than −0.5 and less than 0.5, except in R5 in the austral winter (Fig. 9, described in greater detail below) and R2 in the austral spring (Fig. 10, described in greater detail below); therefore, the GPD may be applied. According to Smith (1985), the regularity conditions for estimation by the maximum likelihood estimation are not necessarily satisfied when . In these cases, the maximum likelihood estimators exist but do not satisfy the conditions of regularities. When , the maximum likelihood estimators do not exist.
Similarly to the GEV, it was found that larger extremes of precipitation were observed with the GPD in the rainy season or during the transition period of each subregion. In R1, R2, R3, R4, and R5, it is expected that the most intense events are during the austral summer and austral autumn, and during austral winter and austral spring for R6. During the austral summer (Fig. 7) and austral autumn (Fig. 8), rainfall of 201.6, 171.9, 209.5, 178.4, and 152.4 mm is expected at least one day per year. Every 10 years daily totals of 279.6, 269.6, 261.8, 234.9, and 198.6 mm are expected, and at least once every 100 years daily precipitation totals of 380.7, 459.3, 296.7, 310.5, and 244.9 mm are expected in R1, R2, R3, R4, and R5, respectively. However, for the return period of 100 years, in R1 (southern Amazonia) a higher daily total (395.0 mm) is expected in the austral spring (Fig. 10). In R6, the most intense extremes are expected during the austral winter (Fig. 9) and austral spring (Fig. 10), with a daily total of 141.9, 188.5, and 288.1 mm per 1, 10, and 100 years, respectively.
Overall, it was observed that the estimates of the return period using the GPD are larger than those found in the GEV. However, in some cases, as in R1 for the return period of 100 years, in the austral summer a daily total of 430.5 mm with the GEV and 380.7 mm for the GPD is expected. The smaller GEV estimates are explained by the fact that the maximum value in this distribution is to be extracted by subperiods (monthly, annual, etc.); thus, the maximum value of a subperiod can be smaller than the lower end of the other values. In this study the maximum of precipitation per season was obtained, but the maximum of a given season may not necessarily be an extreme event.
This work consists of fitting the GEV distribution and the GPD to the daily precipitation data in the subregions of the Brazilian Amazon, with the goal of estimating the return period of the maximum seasonal precipitation.
The goodness of fit of the distributions was assessed using the nonparametric KS test. The GEV presented the best goodness of fit, despite having the disadvantage of selecting the extremes by subperiods. Thus, a maximum within a period may not necessarily be an extreme event.
All estimates achieved by the GEV were satisfactory. In GPD, few cases were not significant at 5%. Therefore, the GEV distribution and the GPD are suitable to study the maximum precipitation, with the GEV distribution being the most appropriate. In the GPD, as some results found by the maximum likelihood estimation were not necessarily satisfied, it would be appropriate to estimate the parameters of this distribution using other methods in an attempt to obtain better goodness of fit.
The results indicate that the extremes of more intense rainfall are expected in the austral summer and/or austral autumn in all subregions, except R6. This region consists of stations in the state of Roraima, all from the Northern Hemisphere, with climatic characteristics of the same hemisphere. Therefore, in R6 the occurrence of intense precipitation events during the austral winter is expected. These results are in agreement with the extremes recorded in the six subregions of the Brazilian Amazon, in the period from 1983 to 2012, where the extremes of more intense rainfall occur during the rainy season of each subregion.
The highest daily precipitation amounts are expected in the southern region (R1 and R2) and at the Atlantic coastal region of the Amazon catchment (R3 and R4) during the austral summer or austral autumn. Using the GEV distribution, it is expected that there will be a daily rainfall total of 146.1 and 143.1 mm at least once a year and a daily total of 234.2 and 231.2 mm at least once every 10 years in the south and at the coast of the Amazon, respectively. In the GPD, it is expected that there will be a daily rainfall total of 201.6 and 209.5 mm at least once a year and a daily total of 279.6 and 261.8 mm at least once every 10 years in the southern and coastal areas of the Amazon, respectively. These results may contribute to better strategic planning, since possessing this information allows people to take measures that minimize the disruption caused by the floods in these regions.