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HYDROLOGICAL REGIONALIZATION OF THE ANNUAL MAXIMUM STREAMFLOWS OF THE UPPER AND MIDDLE PARAOPEBA RIVER – MG USING THE INDEX-FLOOD TECHNIQUE

ABSTRACT

Estimating peak streamflows in a basin is an essential issue in hydrology, especially for projects and management of hydraulic infrastructures such as dams, spillways, and bridges. In this context, this article presents a regionalization of annual maximum streamflows for the upper and middle Paraopeba River region in Minas Gerais, proposing a regional equation to calculate maximum streamflows using the index-flood technique by the generalized extreme value (GEV) distribution. GEV distribution parameters were estimated using the L-moments method, and the adjustment of the statistical model was performed by applying the Anderson-Darling test. The results of the hydrological regionalization were compared with those obtained by the local frequency analysis, performed for the considered fluviometric stations, observing absolute deviations from 11.91 to 58.78%.

design floods; L-moments; frequency analysis; return periods

INTRODUCTION

The 20th century was characterized by intense changes in land use regarding agricultural practices, urbanization, and forest management, and these changes can cause alterations in hydrological and ecological systems, affecting the rainfall-runoff ratio and contributing to the risk of flooding ( O’Brien & Burn, 2014O’Brien NL , Burn DH ( 2014 ) A nonstationary index-flood technique for estimating extreme quantiles for annual maximum streamflow . Journal of Hydrology . DOI: http://dx.doi.org/10.1016/j.jhydrol.2014.09.041
http://dx.doi.org/10.1016/j.jhydrol.2014...
; Markert et al., 2018)Markert K , Schmidt C , Griffin R , Flores A , Poortinga A , Saah , D ( 2018 ) Historical and operational monitoring of surface sediments in the lower Mekong basin using landsat and Google Earth engine cloud computing . Remote Sensing 10 : 909 . DOI: https://doi.org/10.3390/rs10060909
https://doi.org/10.3390/rs10060909...
.

The estimation of the frequency of flooding is fundamental both for an approach to temporal variability ( Bocchiola et al., 2003Bocchiola D , De Michele C , Rosso R ( 2003 ) Review of recent advances in index flood estimation . Hydrology and Earth System Sciences 7 ( 3 ): 283 - 296 . ) and for environmental planning, and, according to Formetta et al. (2018)Formetta G , Prosdocimi I , Stewart E , Bell V ( 2018 ) Estimating the index flood with continuous hydrological models: an application in Great Britain . Hydrology Research 49 ( 1 ): 123 - 133 . DOI: https://doi.org/10.2166/nh.2017.251
https://doi.org/10.2166/nh.2017.251...
, the hydrological records of a place of interest are often not available or are too short to allow reliable statistical analysis. Moreover, the observed data series in some situations, mainly in the Brazilian scenario, are rarely long enough to provide reliable estimates of flood quantiles ( Wu et al., 2019aWu Y , Xue L , Liu Y , Ren L ( 2019a ) Uncertainty assessment of extreme flood estimation in the Dongting lake basin, China . Hydrology Research 50 ( 4 ): 1162 - 1176 . DOI: https://doi.org/10.2166/nh.2019.088
https://doi.org/10.2166/nh.2019.088...
), leading to a large number of uncertainties in the estimates of hydrological variables in engineering projects ( Wu et al., 2019bWu Y , Xue L , Liu Y ( 2019b ) Local and regional flood frequency analysis based on hierarchical Bayesian model in Dongting lake basin, China . Water Science and Engineering 12 ( 4 ): 253 - 262 . DOI: https://doi.org/10.1016/j.wse.2019.12.001
https://doi.org/10.1016/j.wse.2019.12.00...
).

Thus, hydrological regionalization has been used as an approach to overcome this limitation, consisting of a technique that allows the transfer of data between points where there is an operational fluviometric station to other points without monitoring ( Lelis et al., 2020Lelis LCS , Nascimento JG , Duarte SN , Pacheco AB , Bosquilia RWD ( 2020 ) Assessment of hydrological regionalization methodologies for the upper Jaguari river basin . Journal of South American Earth Sciences 97 : 102402 DOI: https://doi.org/10.1016/j.jsames.2019.102402
https://doi.org/10.1016/j.jsames.2019.10...
), provided that this region is considered homogeneous.

In this sense, a large number of models and regionalization methods based on the concept that information on the hydrological response can be transferred between hydrologically homogeneous basins have been proposed in the literature, such as SMAP ( Lopes et al., 1982Lopes JEG , Braga BPF , Conejo JGL ( 1982 ) SMAP: a simplified hydrologic model . In: Singh VP Applied modeling in catchment hydrology . Water Resources Publications . p. 167 - 176 . ), SWAT ( Arnold et al., 1998Arnold JGR , Srinivasan R , Muttiah RS , Williams JR ( 1998 ) Large‐area hydrologic modeling and assessment: Part I. Model development . Journal of the American Water Resources Association 34 ( 1 ): 73 ‐ 89 . DOI: https://doi.org/10.1111/j.1752-1688.1998.tb05961.x
https://doi.org/10.1111/j.1752-1688.1998...
), traditional method ( Eletrobras, 1985ELETROBRAS - Centrais Elétricas Brasileiras ( 1985 ) Metodologia para regionalização de vazões . Rio de Janeiro , v. 1 . ), the method proposed by Chaves et al. (2002)Chaves HML , Rosa JWC , Vadas RG , Oliveira RVT ( 2002 ) Regionalização de vazões mínimas em bacias através de interpolação em sistemas de informações geográfica . Revista Brasileira de Recursos Hídricos 7 ( 3 ): 43 - 51 . , differing in terms of the type of information that is transferred, transfer method, and intrinsic properties of the basin used to justify the similarity ( Pagliero et al., 2019Pagliero L , Bouraoui F , Diels J , Willems P , McIntyre N ( 2019 ) Investigating regionalization technique for large-scale hydrological modelling . Journal of Hydrology 570 : 220 - 235 . DOI: https://doi.org/10.1016/j.jhydrol.2018.12.071
https://doi.org/10.1016/j.jhydrol.2018.1...
).

According to Vogel (2005)Vogel RM ( 2005 ) Regional Calibration of Watershed Models . In: Singh VP , Frevert DF . Watershed models . Boca Raton , CRC Press , p. 47 - 71 . , one of the essential issues in hydrology is the transfer of hydrological data to regions without monitoring and, in this sense, hydrological regionalization can be used to increase the reliability of the estimated quantiles for an already monitored point, as well as for estimating quantiles in locations lacking an efficient hydrometric monitoring network.

In addition, basin dynamics can be incorporated into statistical models to obtain reliable estimates. According to Bocchiolla et al. (2003), the methodology for statistical forecasting of floods by regionalization started with Darlymple in the 1960s, with the index-flood method, which requires a two-step procedure. The first step refers to the identification of a homogeneous region where a maximum probability model can accommodate the normalized streamflow, while the second step involves the search for the appropriate index-flood estimator for the specific basin.

Maximum streamflow events can be characterized as a hydrological phenomenon, presenting great temporal and spatial variability due to climate variations on a global and regional scale, as pointed out by Naghettini (2017)Naghettini M ( 2017 ) Fundamentals of statistical hydrology . Cham , Springer . 660 p. DOI: https://doi.org/10.1007/978-3-319-43561-9
https://doi.org/10.1007/978-3-319-43561-...
. This hydrological variable is related to stochastic processes, considered a random variable described by probability distributions and, therefore, allowing statistical hydrology, whose purpose is to provide adequate means to interpret these characteristics of hydrological processes and extract more accurate information about the studied events.

Considering the different options for streamflow regionalization methodologies and aiming to contribute to users, technicians, and managers of water resources in the state of Minas Gerais in the choice of methods for estimating streamflows associated with a return period in the upper and middle Paraopeba River basin, this study presents the regionalization of annual maximum streamflows, proposing a regional equation to calculate these streamflows using the index-flood technique by the generalized extreme values (GEV) distribution, with its parameters estimated by the L-moments method (LMM), as presented by Hosking & Wallis (1997)Hosking JRM , Wallis JR ( 1997 ) Regional frequency analysis: an approach based on L-moments . New York , Cambridge University Press . 224 p. .

MATERIAL AND METHODS

Study area

The Paraopeba River basin is a sub-basin of the São Francisco River, located in the central region of Minas Gerais, in the range of coordinates −20°51′ S and −18°35′ S latitude and −45°11′ W and −43°38′ W longitude, with a drainage area of approximately 13,640 km 2 , covering 48 municipalities and over 1,318,000 inhabitants. Its main rivers are the Paraopeba, Águas Claras, Macaúbas, Betim, Camapuã, and Manso. The Paraopeba River rises in the city of Cristiano Otoni and has a total length of 428 km, and its mouth is located at the Três Marias Hydroelectric Power Plant dam ( Teixeira et al., 2021Teixeira DBS , Veloso MF , Ferreira FLV , Gleriani JM , Amaral CH ( 2021 ) Spectro-temporal analysis of the Paraopeba river water after the tailings dam burst of the Córrego do Feijão mine in Brumadinho, Brazil . Environmental Monitoring and Assessment 193 : 435 . DOI: https://doi.org/10.1007/s10661-021-09218-4
https://doi.org/10.1007/s10661-021-09218...
).

The upper and middle Paraopeba River region in the present study is considered to be the one upstream of the control section of the Alberto Flores fluviometric station (40740000), with a drainage area of 4,120 km 2 . Seven fluviometric stations were used to regionalize the maximum streamflows in the study area, which are shown in Figure 1 .

FIGURE 1
Upper and middle Paraopeba river basin and the considered monitoring sites.

The application of hydrological regionalization requires that the region under analysis be considered homogeneous, which is described by Naghettini & Pinto (2017)Naghettini M , Pinto EJA ( 2017 ) Regional Frequency Analysis of Hydrologic Variables . In: Naghettini M . Fundamentals of Statistical Hydrology . Cham , Springer . p. 441 - 495 . DOI: https://doi.org/10.1007/978-3-319-43561-9_10
https://doi.org/10.1007/978-3-319-43561-...
as those in which there is sufficient evidence that the different samples of a group have the same frequency distribution. In this sense, CPRM (2001)CPRM – Companhia de Pesquisa de Recursos Mineirais (Serviço Geológico do Brasil) ( 2001 ) Regionalização de vazões sub-bacias 40 e 41, Convênio 015/2000 ANEEL – 013/CPRM/2000, Relatório Final . Belo Horizonte , CPRM . 137 p. mentions that homogeneous regions can be defined from geographic characteristics and the similarity of the trend of individual frequency curves.

Table 1 shows the fluviometric stations and their main attributes. The stations are operated by the Geological Survey of Brazil – CPRM and are part of the National Hydrometeorology Network managed by the National Water and Sanitation Agency (ANA).

TABLE 1
Main attributes of the fluviometric stations.

Although Dalrymple (1960)Dalrymple T ( 1960 ) Flood-frequency analysis, Manual of hydrology: Part.3. Flood-flow Techniques, Geological Survey Water Supply Paper 1543-A . Washington , USGPO . 80 p. has shown that the used series must contain common periods of data for the application of regionalization, Hosking & Wallis (1997)Hosking JRM , Wallis JR ( 1997 ) Regional frequency analysis: an approach based on L-moments . New York , Cambridge University Press . 224 p. argue that it is not necessary to use periods in common if the series are homogeneous and representative of the variable under analysis (in this case, annual maximum streamflows), as considered in the present study.

Subsequently, each element X ij of the series, where i is the order number of the element at station j , is nondimensionalized through the relationship between the element and the nondimensionalization factor μ j of station j , forming the series of dimensionless elements X ij / μ j .

Definition of the regional frequency curve

The calculation of regional parameters of the distribution adopted for the homogeneous region allows the estimation of the dimensionless curve of regional quantiles. According to Hosking & Wallis (1997)Hosking JRM , Wallis JR ( 1997 ) Regional frequency analysis: an approach based on L-moments . New York , Cambridge University Press . 224 p. , the parameters of the regional curve of dimensionless quantiles, denoted by x ( F ; θ 1 , …, θ p ), can be obtained by weighting the local parameters θ^k(j) , k = 1, …, p , estimated for each station j by the respective sample sizes.

Thus, the regional parameters can be obtained by the means of the local parameters of the distribution adopted for the homogeneous region, weighted by the respective lengths of the series n j of dimensionless values of each station in the region, according to [eq. (1)]:

θ ^ k j = j = 1 N n j θ k j j = 1 N n j (1)

The L-moments method (LMM), which consists of equating the first k sample L-moments to their corresponding population moments by applying L-statistics, was applied to estimate the local parameters of the probability distribution, as described by Hosking & Wallis (1997)Hosking JRM , Wallis JR ( 1997 ) Regional frequency analysis: an approach based on L-moments . New York , Cambridge University Press . 224 p. .

The generalized extreme value (GEV) distribution was applied in this study, as this distribution model is derived from the classical theory of extreme values (Gumbel, 1958) and has theoretical justifications for its use for modeling maximum values of hydrological data, as reported by Naghettini & Pinto (2017)Naghettini M , Pinto EJA ( 2017 ) Regional Frequency Analysis of Hydrologic Variables . In: Naghettini M . Fundamentals of Statistical Hydrology . Cham , Springer . p. 441 - 495 . DOI: https://doi.org/10.1007/978-3-319-43561-9_10
https://doi.org/10.1007/978-3-319-43561-...
.

The mathematical equation of the inverse function of the cumulative probability function of the GEV distribution model is given by:

x ( T ) = β + α κ 1 ln 1 1 T κ (2)

Where:

α , β , and κ are the parameters of scale, position, and shape, respectively, which describe the GEV distribution. The chi-square (χ 2 ), Kolmogorov-Smirnov (KS), and Anderson-Darling (AD) tests were applied to verify the fit of this distribution model to the sample data, as presented in Sabino et al. (2021)Sabino M , Souza AP , Uliana EM , Almeida FT , Lisboa L , Zolin CA ( 2021 ) Distribuições de probabilidade para intensidade de precipitação máxima no estado de Mato Grosso . Revista Brasileira de Climatologia 29 : 321 - 340 . DOI: https://doi.org/10.5380/abclima
https://doi.org/10.5380/abclima...
.

Regression analysis

According to Costa (2017)Costa V ( 2017 ) Correlation and Regression . In: Naghettini M ( ed ) Fundamentals of statistical hydrology . Cham , Springer , p. 391 - 440 . DOI: https://doi.org/10.1007/978-3-319-43561-9_9
https://doi.org/10.1007/978-3-319-43561-...
, a regression analysis aims to explain the spatial variation of the nondimensionalization factor μ j of each station j based on basin characteristics, such as drainage area, annual precipitation, and channel slope.

There are several regression models, such as potential, exponential, and logarithmic, among others. The drainage area of a basin is known to be correlated with the maximum streamflows observed at that site. According to Singh (1992)Singh VP ( 1992 ) Elementary hydrology . Englewood Cliffs , Prentice Hall . 973 p. , a potential model that relates the maximum streamflows Q max and the drainage area of basin A is given by:

Q max = k A n (3)

The k and n coefficients can be estimated through statistical analysis of peak streamflow data ( Q max ), observed in basins of different drainage areas. The model of [eq. (3)] was adopted in the present study, which was also applied to some homogeneous regions delimited by CPRM (2001)CPRM – Companhia de Pesquisa de Recursos Mineirais (Serviço Geológico do Brasil) ( 2001 ) Regionalização de vazões sub-bacias 40 e 41, Convênio 015/2000 ANEEL – 013/CPRM/2000, Relatório Final . Belo Horizonte , CPRM . 137 p. .

The coefficients of correlation r and determination r 2 were calculated and the Student’s t-test (with a 5% significance level) was applied to verify the linearity relationship between the variables considered in the regression.

Estimation of an event associated with any return period

The dimensionless quantile associated with a given return period ( X / μ ) T is estimated from the regional dimensionless curve. The event X T is calculated for a given return period T using regression to estimate the nondimensionalization factor μ j , valid for the homogeneous region, that is:

X T = ( X / μ ) T μ j (4)

According to Pires (1994)Pires CLF ( 1994 ) Análise de frequência - revisão metodológica . In: CPRM . A água em revista . Belo Horizonte , CPRM , p. 13 - 22 . , the index-flood method, in the context of flood streamflow regionalization, has an advantage over other methods because the regression is performed with a measure of central tendency, as the mean values tend to have less sampling variability.

RESULTS AND DISCUSSION

Verification of regional homogeneity

A regional analysis of data consistency was performed based on usual techniques of data consistency analysis and auxiliary statistics, in which the results indicated that the samples presented no discrepant statistical characteristics. In this sense, the final definition of homogeneity was performed using the routines described by Hosking (1991)Hosking JRM ( 1991 ) Fortran routines for use with the method of L-moments. Version 2 . Yorktown Heights, New York . IBM Research Report RC 17097 , allowing considering that the region under analysis is characterized as homogeneous. Moreover, the alignment of dimensionless empirical distributions is a positive indicator that the region is homogeneous ( Figure 2 ). The coefficients of variation of the series of annual maximum streamflows of the stations were calculated to complement this analysis ( Table 2 ).

FIGURE 2
Dimensionless frequency curve for the analyzed stations.

TABLE 2
Local statistics of the series of annual maximum streamflows of the stations.

Importantly, this methodology for the definition of hydrologically homogeneous regions was also used by Moreira & Silva (2014)Moreira MC , Silva DD ( 2014 ) Análise de métodos para estimativa das vazões da bacia do rio Paraopeba . Revista Brasileira de Recursos Hídricos 19 ( 2 ): 313 - 324 . , who analyzed the methods for estimating the minimum reference streamflows in the Paraopeba River basin and identified two homogeneous regions, with the region corresponding to the upper Paraopeba River called Region 1 in that study, and also by Dantas et al. (2014)Dantas CEO , Cirilo JA , Neto AR , Silva ER ( 2014 ) Caracterização da formação de cheias na bacia do rio Una em Pernambuco: análise estatística regional . Revista Brasileira de Recursos Hídricos 19 ( 4 ): 239 - 248 . in the characterization of flood formation in the Una River basin in Pernambuco.

Adherence tests were applied after verifying the regional homogeneity to examine the goodness of fit provided by the GEV distribution for the data series at a 5% significance level. The KS, χ 2 , and AD tests indicated the applicability of GEV for the selected stations. Among these tests, KS is the least rigorous in rejecting hypotheses of adherence to the series of annual maximums, as its application assumes that the distribution to be tested is previously known ( Back & Bonfante, 2021Back AJ , Bonfante FM ( 2021 ) Evaluation of generalized extreme value and Gumbel distributions for estimating maximum daily rainfall . Revista Brasileira de Ciências Ambientais 55 ( 4 ): 654 - 664 . DOI: https://doi.org/10.5327/Z217694781015
https://doi.org/10.5327/Z217694781015...
). Still, the KS test is widely used many times as the only adherence test ( Ottero et al., 2018)Ottero CR , Chargel LT , Hora MAGM ( 2018 ) Análise de frequência dos dados pluviométricos observados em 2011 a 2013 na região serrana do Rio de Janeiro . Revista Brasileira de Meteorologia 33 ( 1 ): 131 - 139 . DOI: http://dx.doi.org/10.1590/0102-7786331007
http://dx.doi.org/10.1590/0102-778633100...
and to select the best distribution ( Back & Cadorin, 2020)Back AJ , Cadorin SB ( 2020 ) Extreme rainfall and intensity-duration-frequency equations for the state of Acre, Brazil . Brazilian Journal of Environmental Sciences 55 ( 2 ): 159 - 170 . DOI: https://doi.org/10.5327/Z2176-947820200597 .
https://doi.org/10.5327/Z2176-9478202005...
.

As shown by Naghettini & Pinto (2007)Naghettini M , Pinto EJA ( 2007 ) Hidrologia estatística . Belo Horizonte , CPRM . 552 p. , the LMM method was used to estimate the local parameters of the selected stations, while the regional parameters were obtained by the means of the local parameters of the distribution adopted for the homogeneous region, weighted by the respective series lengths. The parameters estimated by this method were considered good, following the results by Franco et al. (2014)Franco CS , Marques RFPV , Oliveira AS , Oliveira LFC ( 2014 ) Distribuição de probabilidades para precipitação máxima diária na bacia hidrográfica do Rio Verde, Minas Gerais . Revista Brasileira de Engenharia Agrícola e Ambiental 18 ( 7 ): 735 - 741 . and Junqueira Júnior et al. (2015)Junqueira Júnior JA , Mello CR , Alves GJ ( 2015 ) Eventos extremos de precipitação no alto rio Grande, MG: Análise probabilística . Revista Brasileira de Engenharia Agrícola e Ambiental 19 ( 4 ): 301 - 308 . , who evaluated the precision of fit of GEV and other probability distributions for Minas Gerais and found that the best fits were obtained by LMM and MLM.

According to Alam et al. (2018)Alam A , Emura K , Farnham C , Yuan J ( 2018 ) Best-fit probability distributions and return periods for maximum monthly rainfall in Bangladesh . Climate 6 ( 9 ). DOI: 10.3390/cli6010009 , who evaluated probability distributions applied to monthly maximum rainfall events in Bangladesh, the LMM developed by Hosking in 1990 is more robust than conventional methods, in addition to having greater sensitivity in the estimation of parameters.

GEV distribution application in the seven analyzed stations allowed obtaining the dimensionless quantiles for the respective return periods considered for different engineering applications. Morais et al. (2020)Morais MAV , Viola MR , Mello CR , Rodrigues JAM , Oliveira VA ( 2020 ) Regionalization of reference streamflows for the Araguaia river basin in Brazil . Semina: Ciências Agrárias 41 ( 3 ): 829 - 846 . DOI: https://doi.org/10.5433/1679-0359.2020v41n3p829
https://doi.org/10.5433/1679-0359.2020v4...
analyzed the efficiency of 10 probabilistic models in a regionalization study in the Araguaia River basin and identified GEV as the only one that had a good fit for the maximum streamflow series. This robustness was also observed by Cassalho et al (2018)Cassalho F , Beskow S , Mello CR , Moura MM ( 2018 ) Regional flood frequency analysis using L-moments for geographically defined regions: An assessment in Brazil . Journal of Flood Risk Management , e12453 . DOI: https://doi.org/10.1111/jfr3.12453
https://doi.org/10.1111/jfr3.12453...
when evaluating six probability distributions in the Mirim-São Gonçalo basin in Rio Grande do Sul and Rodrigues et al. (2021)Rodrigues JAM , Viola MR , Mello CR , Morais MAV ( 2021 ) Hydrological regionalization of streamflows for the Tocantins river basin in Brazilian cerrado biome . Ambiente & Água 16 ( 6 ) e2716 . DOI: https://doi.org/10.4136/1980-993X
https://doi.org/10.4136/1980-993X...
, who regionalized reference streamflows in the Cerrado biome in Tocantins.

The results obtained for the regional parameters of scale ( α ), position ( β ), and shape ( κ ) were 0.334, 0.784, and −0.046, respectively. The inverse function of the regional GEV distribution was rewritten with the parameters estimated for the homogeneous region, according to [eq. (5)], presented below.

Q max Q med max ( T ) = 0 , 784 + 0 , 334 0 , 046 1 ln 1 1 T 0 , 046 (5)

Table 3 shows the dimensionless quantiles associated with different return periods.

TABLE 3
Nondimensionalized regional quantiles.

Once the inverse function and the nondimensionalized quantiles were obtained, we proceeded with the regression application between the means of the annual maximum streamflows and the drainage areas of the analyzed stations to obtain the estimate of the index-flood. Table 4 shows the values of the drainage areas of the stations and their respective annual maximum mean streamflows, together with the logarithmic transformation of these variables.

TABLE 4
Fluviometric stations and values of areas and means of annual maximum streamflows and logarithmic transformation.

The linearization of [eq. (3)] allowed obtaining the parameters a = ln ( k ) and b by applying simple linear regression, as shown in the equation below:

ln ( Q ) = ln ( k ) + b ln ( A ) (6)

Figure 3 shows the dispersion diagram relating the drainage area and the mean annual streamflows (after logarithmic transformation), where the approximately linear relationship between the two variables can be seen in a log-log graph, suggesting the applicability of a potential model, as shown in [eq. (3)].

FIGURE 3
Linearity between the means of annual maximum streamflows and the drainage area.

The linear relationship was also evaluated by applying the Student’s t-test at a 5% significance level. The relationship between the analyzed variables was considered linear with 95% confidence. Table 5 shows the results of the linear regression application.

TABLE 5
Result of coefficients obtained by linear regression.

Pruski et al. (2016)Pruski FF , Rodriguez RG , Pruski PL , Nunes AA , Rego FS ( 2016 ) Extrapolation of regionalization equations for long-term average flow . Engenharia Agrícola 36 ( 5 ): 830 - 838 . DOI: https://doi.org/10.1590/1809-4430-Eng.Agric.v36n5p830-838/2016
https://doi.org/10.1590/1809-4430-Eng.Ag...
regionalized the long-term mean streamflow (Q ltm ) for the São Francisco River sub-basins and obtained models with r 2 equal to 0.898 and 0.893, using the drainage area and the mean annual precipitation as explanatory variables for each model, respectively. Morais et al. (2020)Morais MAV , Viola MR , Mello CR , Rodrigues JAM , Oliveira VA ( 2020 ) Regionalization of reference streamflows for the Araguaia river basin in Brazil . Semina: Ciências Agrárias 41 ( 3 ): 829 - 846 . DOI: https://doi.org/10.5433/1679-0359.2020v41n3p829
https://doi.org/10.5433/1679-0359.2020v4...
found a similar result with r 2 ranging from 0.83 to 0.96. In this context, the models proposed for the upper Paraopeba River are parsimonious and can be considered effective and efficient to estimate the flood streamflow.

Therefore, [eq. (6)] can be rewritten as shown below:

ln ( Q ) = 1 , 5526 + 0 , 9136 ln ( A ) (7)

Table 6 shows the result of calculating the residuals of linear regression after estimating coefficients a and b , while Figure 4 shows the residuals of the regression carried out around the predicted variable (mean of annual maximum streamflows) by the regression model. The variance can be considered approximately constant, indicating the fulfillment of the hypothesis of homoscedasticity.

TABLE 6
Result of the regression residuals.

FIGURE 4
Variation of residuals around the mean of the annual maximum streamflows obtained by the regression model.

Equation 7 describes the simple linear regression of the transformed variables, but the equation written in potential form is used to estimate the index-flood, that is:

Q med-max = 0 , 2117 × A 0 , 9136 (8)

Where:

Qmean-max is the mean of the annual maximum streamflows in m 3 s 1 , and

A is the basin drainage area in km 2 .

A comparison was performed between the values of the means of the observed annual maximum streamflows relative to those estimated according to the application of [eq. (8)], as shown in Table 7 , together with the calculation of the respective percentage deviations ( PD ), with a mean value of 2.98%.

TABLE 7
Comparison of streamflows estimated by linear regression relative to the mean values of the observed maximum streamflows.

Importantly, the flood data quality has to be verified before any application of hydrological analysis because some factors associated with flood data cannot be fully controlled due to their nature. Therefore, the local factors that influence the index-flood may not be adequately represented in the regression models, as pointed out by Muhammad & Lu (2020)Muhammad M , Lu Z ( 2020 ) Estimating the UK Index Flood: na improved spatial flooding analysis . Environmental Modelling & Assessment 25 : 731 - 748 . DOI: https://doi.org/10.1007/s10666-020-09713-x
https://doi.org/10.1007/s10666-020-09713...
, and the regionalization equation should be used with caution.

Thus, the calculation of the nondimensionalization factor Q mean-max , obtained by linear regression, can be substituted in [eq. (5)], that is:

Q max ( T ) = 0 , 2117 × A 0 , 9136 × 0 , 784 + 0 , 334 0 , 046 1 ln 1 1 T 0 , 046 (9)

Thus, the estimation of quantiles associated with different return periods for unmonitored locations, located within the considered homogeneous region (upper Paraopeba), can be performed by applying [eq. (9)]. This regional equation for calculating maximum streamflows is valid for drainage areas between 130 and 4,120 km 2 .

Figure 5a shows the estimate of maximum streamflow quantiles associated with different return periods, obtained as a function of the application of [eq. (9)], considering the respective drainage areas of the stations used in the study. Additionally, a local frequency analysis was performed for the stations, whose results are shown in Figure 5b . The GEV distribution was also applied with its local parameters estimated by the LMM method to estimate maximum streamflow point quantiles.

FIGURE 5
Results of quantile regionalization by regional (a) and local (b) frequency analysis for the analyzed fluviometric stations.

Figure 6 shows the percentage deviations between the maximum streamflow quantiles using regional and local frequency analysis.

FIGURE 6
Percentage deviations between the quantiles estimated by applying regional and local frequency analysis.

The absolute deviations calculated by estimating the maximum streamflow using the regional equation varied between 11.91% (for T = 2 years) and 58.78% (for T = 100 years) compared to that obtained by the local frequency analysis for each return period. However, the mean of the percentage deviations obtained for all monitoring points (return periods from 2 to 100 years) was 5.40% (mean of the obtained deviations). Zhao et al. (2021)Zhao G , Bates P , Neal J , Pang B ( 2021 ) Design flood estimation for global river networks based on machine learning models . Hydrology and Earth System Sciences 25 ( 11 ): 5981 - 5999 . DOI: https://doi.org/10.5194/hess-25-5981-2021
https://doi.org/10.5194/hess-25-5981-202...
developed a global model for flood estimation and found deviations of up to 70% for a return period equal to 100 years. It demonstrates the challenge of estimating the flood rates for a given location even for locations provided with monitoring.

CONCLUSIONS

This study proposed a regional equation for the estimation of maximum streamflow quantiles in the upper and middle Paraopeba River basin region, in Minas Gerais. The region was considered homogeneous since the dimensionless frequency curves for the fluviometric stations showed the same trend, in addition to similar coefficients of variation.

The linearity of the relationship between the mean of the annual maximum streamflows and the respective drainage areas of the stations was observed, with a correlation coefficient of 0.975. The linear regression performed to obtain the parameters of the potential model of the relationship between the mentioned variables resulted in a coefficient of determination r 2 of 0.951. The application of Equation 9 depends only on the value of the drainage area, which must be between 130 and 4,120 km 2 , which are the limits of the contribution area of the stations considered in the regionalization.

The mean deviations between the streamflow quantiles obtained by the application of the regional equation relative to those by the application of the local frequency analysis were calculated for each return period. Absolute deviations between 11.91 and 58.78% were observed, which are in the same range of variation obtained in similar studies of the literature.

An overall mean deviation (mean of the obtained deviations) of 5.40%, considering all monitoring stations and return periods from 2 to 100 years, was observed despite the values mentioned above. The application of other distribution models is recommended for future investigations so that the reduction of these percentage deviations can be verified.

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Edited by

Area Editor: Fernando António Leal Pacheco

Publication Dates

  • Publication in this collection
    09 Sept 2022
  • Date of issue
    2022

History

  • Received
    14 Mar 2022
  • Accepted
    24 July 2022
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