1 Introduction

The design of hydraulic structures and flood management are dependent on adequate maximum streamflow estimates. Such estimates can be obtained through concepts related to deterministic or statistical hydrology. Among the methodologies associated with deterministic hydrology, rainfall-runoff models stand out. In statistical hydrology, in addition to statistical-type hydrological models, flood frequency analysis (FFA) has been investigated.

FFA relates the frequency of occurrence of extreme streamflows to their respective magnitudes, also known as “Design Floods”. Several authors have approached the FFA by means of the most varied methods (Rahman et al. 2013; Aydoğan et al. 2016; Lam et al. 2016); among them, at-site FFA and regional FFA (RFFA) have caught researchers’ attention. At-site FFA can be applied to define design flood as reference for assessing RFFA accuracy and for rainfall-runoff hydrologic models (Noto and La Loggia 2009; Rahman et al. 2013).

The selection of a probability distribution function (PDF) and its adequate parameters estimation are key factors in studies concerning extreme events, such as at-site FFA and RFFA (Rahman et al. 2013). Numerous studies worldwide have addressed the probabilistic modeling of maximum streamflows. Nevertheless, Rahman et al. (2014) highlighted that due to the inherent complexity, little attention has been given to PDFs with more than three parameters.

According to Beskow et al. (2015), simplified PDFs, such as 2 and 3 parameters Log-Normal (LN2 and LN3, respectively), and Gumbel (or Extreme Value Type 1 - EV1), are the most commonly used in Brazil. Using the Extreme Value Type 1 (EV1) PDF, Mello et al. (2010) constructed statistical confidence intervals for maximum annual streamflow to the 6 largest basins of the Upper Grande River, in Minas Gerais, southeastern Brazil. Silvino et al. (2007) verified the adjustment of different PDFs (Normal, LN2, Exponential, Gamma and Weibull) to maximum streamflow series of a sub-basin of the Paraguay River, in Mato Grosso State, middle-west Brazil. Despite the satisfactory results of traditional 2-parameter distributions found in the aforementioned studies, the use of multiparameter distributions may lead to more appropriate fits between simulated and observed data (Rahman et al. 2014; Cassalho et al. 2017).

Contrary to observed in Brazil regarding the application of traditional distributions, the use of multiparameter distributions is widespread in some countries. Lam et al. (2016) stated that multiparameter PDFs, especially Log Pearson type III, Generalized Extreme Values (GEV) and Generalized Pareto (GPA), have their application widely diffused for at-site FFA in Australia. Similarly, various RFFA studies around the world employed the Generalized Logistic (GLO), GEV, GPA, LN3 and Pearson type III (PE3) distributions, with parameters estimated by the L-moments (LMO), as proposed by Hosking and Wallis (1997), such as in the Middle Ganga Plains, India (Kumar et al. 2003), on the island of Sicily, Italy (Noto and La Loggia 2009), and in the Çoruh basin, Turkey (Aydoğan et al. 2016).

In addition to the multiparameter distributions traditionally used in FFA, many authors propose the use of the 4-parameter Kappa (KAP) and the 5-parameter Wakeby (WAK), which have a greater number of parameters, thereby tending to a better frequency modeling (Rahman et al. 2013; Beskow et al. 2015; Cassalho et al. 2017). In Brazil, the study conducted by Beskow et al. (2015) is one of the precursors in the use of multiparameter distributions in hydrological frequency analysis; however, its focus was the analysis of heavy rainfall. These researchers found superior performance of KAP and GEV distributions in terms of the number of historical series better adjusted, in comparison to LN2 and EV1 distributions, when conducting probabilistic modeling of extreme rainfall series for 342 rain gauges in the Rio Grande do Sul State, Brazil. Hosking and Wallis (1997) pointed out that some of the advantages of the mimic-everything distribution (WAK) over other distributions with fewer parameters are: i) a better adjustment to the series; ii) the separation effects that other distributions do not have; iii) better performance in the presence of outliers due to its heavy upper tail; and iv) the existence of an easy-to-apply quantile estimation function. The performance of WAK in FFA has been discussed in some countries worldwide (Kumar et al. 2003; Hussain and Pasha 2009; Rahman et al. 2014). However, its use in Brazil is still incipient, making the present study likely the first one to apply this PDF in a FFA in the country. Thus, this pioneer study represents a significant improvement in water resources management in Brazil, where the use of more advanced statistical procedures has not followed the historical efforts towards hydrological monitoring.

Naghettini (2017) highlights that randomness, independence, homogeneity, and stationarity are fundamental assumptions for hydrological frequency analysis, which are often considered true for most FFA studies (Beskow et al. 2015; Aydoğan et al. 2016; Lam et al. 2016). Among them, stationarity is the most discussed in recent literature (Ishak et al. 2013, and Wang et al. 2015). Ward et al. (2014a), conducting a global assessment of the relationship between El Niño-Southern Oscillation (ENSO) and annual floods from simulated MAS series for 11,558 basins, found that ENSO has significantly influenced more than a third of the Earth’s land surface. Considering return periods (RPs) from 2 to 1000 years, Ward et al. (2014b) found significant anomalies in flood volumes during El Niño and La Niña years in Brazil. Flantua et al. (2016) demonstrated that other major ocean-atmospheric mechanisms (e.g. Inter-decadal Pacific Oscillation, Southern Annular Model, Atlantic Multidecadal Oscillation, tropical North Atlantic, and tropical South Atlantic) exert influence on hydrological anomalies. Furthermore, anthropogenic actions may also influence MAS (Ishak et al. 2013). It should be highlighted that the aforementioned common FFA assumptions might result in misrepresentation of design flood estimates, thus limiting its applicability.

In this context, the objectives of this study are to: i) evaluate the performance of multiparameter distributions, still not used in the probabilistic modeling of maximum streamflow in Brazil, comparing them to the traditional 2-parameter distributions; ii) evaluate the gain obtained in estimation of at-site design floods by using multiparameter distributions. All the methodology and hypotheses were verified taking as reference over a hundred basins located in southern Brazil.

2 Material and Methods

2.1 Study Area and Database

Rio Grande do Sul State (Fig. 1a, b) is located in extreme southern Brazil between the coordinates 27° and 34°S and 49° and 58°W, bordering with Santa Catarina State to the northeast, Uruguay to the southeast, Argentina to the northwest and the Atlantic Ocean to the east. Its altitude varies between 0 and approximately 1600 m above sea level. Rio Grande do Sul is the fifth most populous state in the country, with an estimated population of 11,286,500 inhabitants, in a total area of 281,738 km2 (IBGE 2016). According to Köppen’s classification, the climate falls within the classes Cfa and Cfb (Sparovek et al. 2007).

Fig. 1
figure 1

Location of the Rio Grande do Sul State in Brazil and South America (a); streamflow gauging stations used in this study, drainage network, and elevation (b); initial year of the MAS (c); final year of the MAS (d); total length of the MAS (e); and percentage of missing data between initial year and final year for each MAS series (f)

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Taking as reference the classification method adopted by National Water Agency (ANA), Brazil is split into 8 hydrological regions. Among them, Uruguay and South Atlantic regions have drainage areas within Rio Grande do Sul State. The Maximum Annual Streamflow (MAS) series employed in this study were coded according to the method proposed by ANA (2009). By this method, the first and second numbers given to a MAS series are derived from its hydrological region and sub-basin, respectively. The Uruguay hydrological region (first number equal to 7) is subdivided into 9 sub-basins, of which 8 are located in Rio Grande do Sul. Similarly, the South Atlantic region (8) has 4 sub-basins in the state.

All the 123 MAS series were obtained from the ANA’s hydrological database (Fig. 1b). These series met the criterion of a minimum limit of 10 years of data, which has been adopted for frequency analysis of extreme hydrological variables in poorly monitored regions (Beskow et al. 2015; Caldeira et al. 2015; Cassalho et al. 2017). Other 10 MAS series were excluded from the analysis as they belong to watersheds whose drainage areas are partially outside the geographic limits of Rio Grande do Sul (Fig. 1b).

2.2 Temporal Trend Detection

The rank-based and nonparametric Mann-Kendall test (Mann 1945; Kendall 1975) is the most widely used for detecting monotonic trends in historical series (Villarini et al. 2009; Wang et al. 2015). Under stationarity condition, it is assumed that the hydrological variable of interest does not present significant temporal variations, which can occur in the form of trends (upward or downward), shifts and periodicities (i.e. nonmonotonic trends) (Villarini et al. 2009). According to Wang et al. (2015), the null hypothesis H0 states that the data (x1, …, xn) are samples of n equally distributed, independent random variables (Eq. 1).

$$ \mathrm{Z}=\left\{\begin{array}{c}\hfill \left(S-1\right)/\sqrt{\mathrm{Var}\ \left(\mathrm{S}\right)}\hfill \\ {}\hfill 0\hfill \\ {}\hfill \left(S+1\right)/\sqrt{\mathrm{Var}\ \left(\mathrm{S}\right)}\hfill \end{array}\kern2.5em \right.{\displaystyle \begin{array}{c}\hfill \mathrm{if}\hfill \\ {}\hfill \mathrm{if}\hfill \\ {}\hfill \mathrm{if}\hfill \end{array}}\kern2.5em {\displaystyle \begin{array}{c}\hfill \mathrm{S}>0\hfill \\ {}\hfill \mathrm{S}=0\hfill \\ {}\hfill \mathrm{S}<0\hfill \end{array}} $$
(1)
$$ \mathrm{S}=\sum_{\mathrm{k}-1}^{\mathrm{n}-1}\sum_{\mathrm{j}=\mathrm{k}+1}^{\mathrm{n}}\operatorname{sgn}\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_{\mathrm{k}}\right),\operatorname{sgn}\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_{\mathrm{k}}\right)=\left\{\begin{array}{c}\hfill +1\hfill \\ {}\hfill 0\hfill \\ {}\hfill -1\hfill \end{array}\kern2.5em \begin{array}{c}\hfill \mathrm{if}\hfill \\ {}\hfill \mathrm{if}\hfill \\ {}\hfill \mathrm{if}\hfill \end{array}\kern2.5em \begin{array}{c}\hfill \left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_{\mathrm{k}}\right)>0\hfill \\ {}\hfill \left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_{\mathrm{k}}\right)=0\hfill \\ {}\hfill \left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_{\mathrm{k}}\right)<0\hfill \end{array}\ \right. $$
(2)

Where Z is the standardized test statistics, x corresponds to x univariate time series, n is the length of the historical series, and j and k are time indices associated with individual values. The statistical S can be considered normally distributed when n ≥ 8. The hypothesis H0 that there is no trend in the series should be accepted if the absolute value of z is less than or equal to Z1 − α/2.

2.3 PDF Fitting by the L-Moments Method

Statistical moments are responsible for describing the form of probability distributions. Among the most common moments are the mean (μ), standard deviation (σ) or variance (σ2), skewness (γ) and kurtosis (к). More satisfactory PDFs’ parameter estimates can be obtained by L-moments (LMO), which is a derivation of probability weighted moments (PWM) (Hosking and Wallis 1997; Hussain and Pasha 2009). The PWMs, defined by Greenwood et al. (1979), are expressed by:

$$ {\mathrm{M}}_{\mathrm{p},\mathrm{r},\mathrm{s}}=\mathrm{E}\left\{{\mathrm{x}}^{\mathrm{p}}{\left[\mathrm{F}\left(\mathrm{x}\right)\right]}^{\mathrm{r}}{\left[1-\mathrm{F}\left(\mathrm{x}\right)\right]}^{\mathrm{s}}\right\} $$
(3)

Where x is the variable to be parsed, F(x) is the respective cumulative distribution function (CDF) and p, r, and s are integer numbers that represent the PWM’s order. In its simplest form (r = s = 0 and p > 0), the PWM Mp , 0 , 0 expresses the conventional moment of pth order (Greenwood et al. 1979). The PWM βr = M 1 , r , o is particularly useful in the characterization of probability distributions (Hosking and Wallis 1997; Hussain and Pasha 2009). As x(F) is the inverse form of the CDF F(x), used in the estimation of quantiles, the PWM β r can be rewritten as (Hosking and Wallis 1997):

$$ {\upbeta}_{\mathrm{r}}=\underset{0}{\overset{1}{\int }}\mathrm{x}\left(\mathrm{F}\right){\mathrm{F}}^{\mathrm{r}}\mathrm{dF} $$
(4)

PWMs can be used in parameter estimation, however, they are not as easily interpreted as L-moments (Hosking and Wallis 1997). These authors further state that the L-moment λ r + 1 of any order (r + 1), can be obtained by a linear combination with the PWMs:

$$ {\uplambda}_{\mathrm{r}+1}=\sum_{\mathrm{k}=0}^{\mathrm{r}}{\mathrm{P}}_{\mathrm{r},\mathrm{k}}^{\ast }{\upbeta}_{\mathrm{k}} $$
(5)
$$ {\mathrm{P}}_{\mathrm{r},\mathrm{k}}^{\ast }=\frac{{\left(-1\right)}^{\mathrm{r}-\mathrm{k}}\left(\mathrm{r}+\mathrm{k}\right)!}{{\left(\mathrm{k}!\right)}^2\left(\mathrm{r}-\mathrm{k}\right)!} $$
(6)

The first four L-moments (λ1, λ2, λ3 and λ4), referring to the location, scale, skewness, and kurtosis, respectively, can be obtained by (Hosking and Wallis 1997):

$$ {\uplambda}_1={\upbeta}_0 $$
(7)
$$ {\uplambda}_2=2{\upbeta}_1-{\upbeta}_0 $$
(8)
$$ {\uplambda}_3=6{\upbeta}_2-6{\upbeta}_1+{\upbeta}_0 $$
(9)
$$ {\uplambda}_4=20{\upbeta}_3-30{\upbeta}_2+12{\upbeta}_1-{\upbeta}_0 $$
(10)

On dividing the highest order L-moments by λ2 (scale measure), the dimensionless versions of the L-moments, the L-moment ratios (τr), are obtained. It is worth noting that the coefficient of L-variation (L-CV=τ) is calculated by taking the ratio between the scale moment and that of position (Hosking and Wallis 1997).

$$ {\uptau}_{\mathrm{r}}=\frac{\uplambda_{\mathrm{r}}}{\uplambda_2},\mathrm{r}\ge 3 $$
(11)
$$ \uptau =\frac{\uplambda_2}{\uplambda_1} $$
(12)

The L-moments λ 1 and λ 2 , the coefficient of L-variation τ, and the L-skewness τ 3 and L-kurtosis τ 4 are the most useful quantities in the description of probability distributions (Rao and Hamed 2000).

2.4 Probability Distribution Functions (PDFs)

The MAS series were adjusted to 2-parameter (EV1, LN2 and Gamma) and multiparameter (GEV, LN3, PE3, GLO, GPA, KAP and WAK) distributions, used in the probabilistic modeling of maximum streamflow in at-site FFA (Rahman et al. 2013; Rahman et al. 2014; Lam et al. 2016).

The PDF of the Generalized Extreme Value (GEV) and Extreme Values (EV1) is expressed by:

$$ \mathrm{f}\left(\mathrm{x}\right)={\upalpha}^{-1}{\mathrm{e}}^{-\left(1-\mathrm{k}\right)\mathrm{y}-{\mathrm{e}}^{-\mathrm{y}}},\kern1.25em \mathrm{y}=\left\{\begin{array}{c}\hfill -{\mathrm{k}}^{-1}\log \left[1-\mathrm{k}\left(\mathrm{x}-\upxi \right)/\upalpha \right],\hfill \\ {}\hfill \left(\mathrm{x}-\upxi \right)/\upalpha, \hfill \end{array}\right.\kern1.75em {\displaystyle \begin{array}{c}\hfill \mathrm{k}\ne 0\hfill \\ {}\hfill \mathrm{k}=0\hfill \end{array}} $$
(13)

Where, x is the variable under study, α, ξ and k are scale, position and shape parameters, respectively. In case of k = 0, this PDF corresponds to EV1 distribution and if k > 0, it represents the 3-parameter GEV.

The 2-parameter Log-Normal (LN2) PDF of the logarithmic normally distributed variable x is expressed by:

$$ \mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}\upsigma}_{\mathrm{y}}\sqrt{2\uppi}}\exp \left\{\frac{{\left[-\log \mathrm{x}-{\upmu}_{\mathrm{y}}\right]}^2}{2{\upsigma}_{\mathrm{y}}^2}\right\} $$
(14)

Where, μ y and σ y are the mean and standard deviation of the natural logarithms of the variable x.

The 3-parameter Log-Normal distribution (LN3) differs from LN2 because x is subtracted by a value α in the former, which represents a lower bound, characterizing a coefficient of asymmetry (Rao and Hamed 2000):

$$ \mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\left(\mathrm{x}-\upalpha \right){\upsigma}_{\mathrm{y}}\sqrt{2\uppi}}\exp \left\{-\frac{1}{2{\upsigma}_{\mathrm{y}}^2}{\left[\log \left(\mathrm{x}-\upalpha \right)-{\upmu}_{\mathrm{y}}\right]}^2\right\} $$
(15)

Where μ y and \( {\sigma}_y^2 \) are also position and scale parameters, respectively, except that they correspond to the mean and variance of the logarithm of (x − α) (Rao and Hamed 2000).

The 2-parameter Gamma distribution is a variation of PE3, for which the position parameter ξ is zero (Rao and Hamed 2000; Naghettini and Silva 2017), thus, its PDF is written as:

$$ \mathrm{f}\left(\mathrm{x}\right)=\frac{{\left(\mathrm{x}-\upxi \right)}^{\upalpha -1}{\mathrm{e}}^{-\left(\mathrm{x}-\upxi \right)/\upbeta}}{\upbeta^{\upalpha}\Gamma \left(\upalpha \right)} $$
(16)

Where x is the variable under study (0 < x < ∞), β is the scale parameter, α is the shape parameter, and Г(.) represents the gamma function. However, Hosking and Wallis (1997) proposed an alternative parameterization for PE3, which allows the use of this PDF in situations where the skewness of the observed data is negative. The conventional moments of the distribution, μ, σ and γ are given by:

$$ \upalpha =\frac{4}{\upgamma^2},\kern0.5em \upbeta =\frac{1}{2}\upsigma \left|\upgamma \right|,\kern1.75em \upxi =\upmu -\frac{2\upsigma}{\upgamma} $$
(17)

Generalized Logistic (GLO) distribution, with position (ξ), scale (α) and shape (k ≠ 0) parameters, is defined by Hosking and Wallis (1997) as:

$$ \mathrm{f}\left(\mathrm{x}\right)=\frac{\upalpha^{-1}{\mathrm{e}}^{-\left(1-\mathrm{k}\right)\mathrm{y}}}{{\left(1+{\mathrm{e}}^{-\mathrm{y}}\right)}^2},\kern1.25em \mathrm{y}=-{\mathrm{k}}^{-1}\log \left[\frac{1-\mathrm{k}\left(\mathrm{x}-\upxi \right)}{\upalpha}\right] $$
(18)

The Generalized Pareto (GPA) distribution also has parameters of position (ξ), scale (α) and shape (k = 0), and its PDF is defined as:

$$ \mathrm{f}\left(\mathrm{x}\right)={\upalpha}^{-1}{\mathrm{e}}^{-\left(1-\mathrm{k}\right)\mathrm{y}},\kern1.25em \mathrm{y}=-{\mathrm{k}}^{-1}\log \left[1-\mathrm{k}\left(\mathrm{x}-\upxi \right)/\upalpha \right] $$
(19)

Following recommendations of Hosking and Wallis (1997), PDFs with a greater number of parameters were also considered, as they can capture the behavior of a wider variety of frequency distributions. The 4-parameter Kappa distribution (KAP) (20) and its respective CDF (21) are described below:

$$ \mathrm{f}\left(\mathrm{x}\right)={\upalpha}^{-1}{\left[1-\mathrm{k}\left(\mathrm{x}-\upxi \right)/\upalpha \right]}^{1/\mathrm{k}-1}{\left[\mathrm{F}\left(\mathrm{x}\right)\right]}^{1-\mathrm{h}} $$
(20)
$$ \mathrm{F}\left(\mathrm{x}\right)={\left\{1-\mathrm{h}{\left[1-\mathrm{k}\left(\mathrm{x}-\upxi \right)/\upalpha \right]}^{1/\mathrm{k}}\right\}}^{1/\mathrm{h}} $$
(21)

Where x is the analyzed variable, ξ is the position parameter, α is the scale parameter and k and h are characteristic parameters. Depending on k and h values, KAP may take the form of different distributions, e.g. GEV (h = 0 e k ≠ 0), GLO (h =  − 1 e k ≠ 0), GPA (h = 1 e k ≠ 0), Exponential (h = 1 e k = 0), EV1 (h = 0 e k = 0), Logistic (h =  − 1 e k = 0), Normal (h = 1 e k = 1) and Inverse Exponential (h = 0 e k = 1) (Hosking and Wallis 1997; Beskow et al. 2015).

Differently from the aforementioned PDFs, WAK cannot be defined analytically because there is no explicit expression for its PDF f(x) and CDF F(x); nevertheless, it is possible to define an expression for the inverse of its CDF (Hosking and Wallis 1997; Rahman et al. 2014; Naghettini and Silva 2017). Thus, the quantile estimation equation for the Wakeby distribution (WAK) is given by:

$$ \mathrm{x}\left(\mathrm{F}\right)=\upxi +\frac{\upalpha}{\upbeta}\left[1-{\left(1-\mathrm{F}\right)}^{\upbeta}\right]-\frac{\upgamma}{\updelta}\left[1-{\left(1-\mathrm{F}\right)}^{-\updelta}\right] $$
(22)

Where ξ is the position parameter, a and γ represent scale parameters and β and δ correspond to shape parameters. When a and γ are equal to zero, the distribution represents GPA.

The parameters of each PDF were obtained based on the L-moments using the System of Hydrological Data Acquisition and Analysis (SYHDA) software, which was also used in other studies (Beskow et al. 2015; Caldeira et al. 2015; Cassalho et al. 2017).

2.5 Goodness-of-Fit

The Anderson-Darling test (Eq. 23) has been widely used to indicate the PDF that best fits the MAS series under analysis (Heo et al. 2013; Beskow et al. 2015), as it assigns greater weight to distribution tails, which is a desirable characteristic in the probabilistic modeling of extreme events. According to Heo et al. (2013), the greater weight attributed to the distribution tails is due to the greater emphasis on the difference between empirical distribution function (EDF) and CDF (Eq. 23).

$$ {\mathrm{A}}^2=\mathrm{n}\underset{-\infty }{\overset{+\infty }{\int }}\frac{{\left[{\mathrm{F}}_{\mathrm{n}}\left(\mathrm{x}\right)-\mathrm{F}\left(\mathrm{x}\right)\right]}^2}{\mathrm{F}\left(\mathrm{x}\right)\left[1-\mathrm{F}\left(\mathrm{x}\right)\right]}\mathrm{dF}\left(\mathrm{x}\right) $$
(23)

Where A2 corresponds to the test result, x is the studied variable, Fn(x) the EDF, F(x) the CDF and n the series length. Heo et al. (2013) reported that Eq. (23) can be rewritten for computational reasons, as shown (Eq. 24):

$$ {\mathrm{A}}^2=-\mathrm{n}-\frac{1}{\mathrm{n}}\sum_{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{i}-1\right)\log \mathrm{F}\left({\mathrm{x}}_{\mathrm{i}}\right)-\frac{1}{\mathrm{n}}\sum_{\mathrm{i}=1}^{\mathrm{n}}\left(2\mathrm{n}-2\mathrm{i}+1\right)\log \mathrm{F}\left({\mathrm{x}}_{\mathrm{i}}\right) $$
(24)

With the aid of the SYHDA, AD test was applied to evaluate the performance of all PDFs presented in relation to each MAS series. It is worth mentioning that the performance of all PDFs is discussed in terms of best fit, in other words, MAS series better adjusted by a particular PDF, and adequate fit, corresponding to MAS series adequately adjusted by a particular PDF for a 5% significance level according to the AD test (Beskow et al. 2015).

3 Results and Discussion

3.1 PDF Fitting

Based on the non-parametric Mann-Kendall test, at a significance level of 5%, only 7 out of 113 series (Fig. 2) presented significant monotonic trend, thus, they were not used for the sequence of this study.

Fig. 2
figure 2

PDF more indicated for each MAS series according to AD test and identification of the non-stationary series (monotonic trends)

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The remaining 106 series were then adjusted to the 10 PDFs evaluated in the present study. The PDFs with the best fit to the historical series, according to AD test at a significance level of 5%, are presented in Fig. 2. As a complement, the main results for each PDF are summarized in Table 1.

Table 1 Performance of the considered PDFs and descriptive statistics of their respective parameters
Full size table

As one can note in Fig. 2 and Table 1, the 2-parameter PDFs, commonly used in the probabilistic modeling of extreme events in Brazil, presented fewer best fits and adequate fits, when compared to multiparameter PDFs. Although an adequate adjustment was found for more than 70 series, the EV1 and LN2 PDFs had no best fit in any of the 106 series evaluated. The 2-parameter Gamma, 3-parameter LN3 and GLO distributions were the best PDFs to only 1 series. Rahman et al. (2013) also observed this behavior for 2-parameter distributions, especially in the case of LN2 and Gamma PDFs, when analyzing 15 PDFs. Similarly, Beskow et al. (2015) found that for 342 maximum daily rainfall series, EV1 and LN2 presented the best fit in only 27 and 17 series, respectively. It should be mentioned that GLO distribution had the lowest number of adjusted series and was considered as the most appropriate PDF for only 1 series. Despite the unsatisfactory at-site performance for Rio Grande do Sul State, several authors have reported acceptable performance of GLO distribution in regional studies (Kumar et al. 2003; Heo et al. 2013).

The GPA was suitable to approximately half of the analyzed series (Table 1). Even though this PDF was considered the most appropriate to 12 series, 11 of them were better fitted by a special case of the WAK distribution (γ = 0 and δ = 0) where it mimics the behavior of GPA. Table 1 summarizes the GPA’s parameters (ξ, α and β) found for the special case of the WAK. In the first RFFA study conducted in Pakistan with basis on the L-moments, Hussain and Pasha (2009) observed that GPA was one of the best distributions for the description of regional behavior of maximum streamflows. In an at-site FFA, Rahman et al. (2013) found that GPA had a lower number of adequate fits, based on AD test, only when compared to GEV and LP3 distributions. It should be mentioned that, in the aforementioned study, when considered other less restrictive goodness-of-fit tests, GPA was the most indicated PDF for a greater number of series.

The PE3 distribution was adequately fitted to more series than the above mentioned PDFs, corresponding to over 70% of the series. However, PE3 resulted in the best fit for only 8 of the 106 series. On analyzing hydrologically homogeneous regions in the Çoruh basin, Turkey, Aydoğan et al. (2016) reported that this PDF outperformed the other five 3-parameter PDFs.

The GEV distribution has been extensively used in FFA, and it is often cited as the most robust distribution in the probabilistic modeling of extreme events (Kumar et al. 2003; Noto and La Loggia 2009; Rahman et al. 2013; Beskow et al. 2015). Beskow et al. (2015), studying the frequency analysis of heavy rainfall in Rio Grande do Sul, Brazil, stated that 97% of the series were adequately adjusted to GEV. The present study corroborates the findings of Beskow et al. (2015) as more than 96% of the series were adequately adjusted by GEV. In the present study, GEV provided better adjustment to shorter series, contrasting with other PDFs that resulted in better adjustment to longer average length series. This characteristic is important for regions where hydrological monitoring frequently lacks long historical series.

The KAP and WAK were the most indicated PDFs for over two-thirds of the 106 series. After GEV, KAP was the distribution with the greatest number of series adjusted (~80% of the series). Beskow et al. (2015) found that, although GEV distribution has adjusted to the greatest number of series, KAP was the most appropriate in the probabilistic modeling of heavy rainfall in the state of Rio Grande do Sul, Brazil. In the present study, WAK was the distribution with best fit to approximately 45% of the total series. However, fewer series were adequately adjusted to WAK in comparison with other PDFs that generated the best fit to a lower number of series (e.g. EV1, LN2, LN3, Gamma). On conducting a case study in 91 watersheds in eastern Australia, Rahman et al. (2014) also observed a good performance of WAK. However, these authors stated that this PDF was more sensitive to the series length, therefore, its use should be restricted to long historical series (at least 40 years), going along with the findings of the present study where WAK was better adjusted to series with an average length of 39 years. The use of the KAP and WAK distributions is still scarce in Brazil, especially in studies involving MAS. In this context, the present study stands out as probably the first regarding MAS to address such multiparameter distributions in FFA in the country.

3.2 Assessing Quantile Estimate Errors

Given the high variability of the PDFs’ parameters and the distributions that produced the best fit, Beskow et al. (2015) emphasized the importance of analyzing the influence of the PDF choice on estimating quantiles associated with predefined RPs. Therefore, for illustrative purposes, the estimates of MAS quantiles by 2-parameter distributions widely used in Brazil (EV1 and LN2) and PDFs whose best results were obtained (GEV, KAP and WAK) are presented in Fig. 3a. In the following analyses, RPs of 20, 50 and 100 years, widely used in hydrological engineering projects, were established. It is worthwhile to mention that for the 4 cases presented in Fig. 3a, the series met the monotonic trend criterion, in accordance with Mann-Kendall test, and could therefore be used in the estimates. In order to illustrate the inexistence of monotonic trends and the possible presence of nonmonotonic trends, Fig. 3b depicts the time series of the four selected sites.

Fig. 3
figure 3

Annual maximum streamflow quantiles for 4 randomly selected basins in Rio Grande do Sul State (highlighted in Fig. 2) in which the best PDF is indicated between parentheses beside the respective station code (a); and the respective time series (b)

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It can be inferred from Fig. 3a that LN2 progressively overestimated maximum streamflow magnitudes as RP increased. In the case of station 88575000, LN2 overestimated annual maximum streamflow by more than 50% when associated with a 100-year RP. Although on a smaller scale, the same behavior was observed for the quantiles derived from EV1 distribution, which reinforces the need for more complex PDFs. The multiparameter PDFs presented similar estimates, especially for RPs of 20 and 50 years. However, for the 100-year RP, GEV tended to differentiate from the best fit distributions, i.e. KAP (76500000) and WAK (74370000, 86160000, and 88575000). This fact reflects the characteristic of distributions with greater number of parameters (e.g. KAP and WAK) to better adapt to the extreme occurrence frequencies (Rahman et al. 2014; Naghettini and Silva 2017), thus leading to the best fit in the tails of the distributions and, consequently, to more adequate estimates for events associated with high RPs (i.e. more complex hydraulic structures).

The trend analysis of the time series is represented in Fig. 3b, where no monotonic trend can be visually identified for any of the 4 randomly selected sites. The lack of a proper continuous time series illustrated in Fig. 1 (e, f) and presented in Fig. 3b, commonly experienced in developing countries, diminishes the quantile estimates’ reliability. This reinforces the need for a robust statistical hydrology approach as proposed in the present study.

Table 2 presents the maximum, minimum and average Relative Absolute Error (RAE) generated by the PDFs in relation to the 3 distributions that resulted in the greatest number of best fits. Therefore, the 13 series that were better fitted by GEV are compared to the estimates of the other 9 PDFs. The same was done for the 23 MAS series that obtained a better adjustment by KAP and the 47 series fitted by WAK distribution.

Table 2 Minimum, maximum and mean Relative Absolute Error (RAE in %) in the maximum streamflow estimates for fixed RPs considering the distributions with acceptable fit by the AD test in relation to the PDFs that had the best adjustment
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The findings presented graphically in Fig. 3a are also reported in Table 2, but considering all the series under analysis. The RAEs associated with the use of LN2 reached 168.4% in relation to the best estimate, which were obtained by KAP. In a study carried out by Beskow et al. (2015), intended for modeling of extreme rainfall events in the same state as the present study, the authors found errors even greater than those presented in Table 2. Still in agreement with these authors, despite not having had the highest maximum RAEs, 2-parameter distributions such as EV1 presented mean RAEs of more than 10%, reflecting a systematic lower precision in the estimates.

For the 13 series in which GEV was the most appropriate PDF, KAP was that with the lowest average RAE for all RPs. With respect to the 23 series better represented by KAP, the lowest mean RAE for the most critical RPs (50 and 100 years) were obtained by WAK, not exceeding 3% of RAE in the estimates. Similarly, for the 47 series better modeled by WAK, KAP presented the lowest errors. It can be stated that the multiparameter distributions, especially KAP and WAK, presented more precise estimates for maximum quantiles, once both result in estimates having low average RAE.

Despite the importance and applicability of the results for water resources management and engineering, one must have in mind the proposed approach’s limitations. One of the main limitations comes from the short MAS series length found in some studied basins (Fig. 1e), associated with high percentage of missing years (Fig. 1f). Watt et al. (1988) recommend the use of the fitted quantile functions for RP of up to four times longer than the series length, which should be at least 10 years long. The extrapolation of such limit may cause some of the large errors depicted in Fig. 3a. Furthermore, the period covered by some series (Fig. 1c, d) might be coincident with periods of nonmonotonic trend, which were not detected by the Mann-Kendall test (Fig. 3b). Thus, the design flood estimates in such cases may not reflect properly what is expected for future events.

4 Conclusions

Based on the results presented, it can be concluded that: i) the capability of multiparameter distributions (i.e. KAP and WAK) in statistically modeling the observed MAS series in terms of AD test and RAE was superior to that of the traditional 2-parameter distributions, such that 2-parameter PDFs application would result in substantially high absolute errors for quantile estimates; ii) GEV was the PDF with the greatest number of MAS series adjusted according to the AD test; however, KAP and WAK distributions resulted in the best fit for a greater number of series, thus reinforcing the gain in the design flood estimates due to the superiority of PDFs with more parameters; and (iii) GEV presented a better fit to shorter series when compared to the other PDFs, which is an important characteristic for basins where long historical series are not frequently available. Due to the limitation imposed by the stationarity hypothesis, especially associated with nonmonotonic trend, which was not verified in the present study, it is recommended that future studies are intended for evaluating the influence of natural and anthropogenic causes on PDFs’ parameters.