Frequency analysis of minimum flows

Revista Brasileira de Recursos Hídricos Brazilian Journal of Water Resources Versão On-line ISSN 2318-0331 RBRH, Porto Alegre, v. 23, e17, 2018 Scientific/Technical Article https://doi.org/10.1590/2318-0331.0318170080 Frequency analysis of minimum flows Análise de frequência de vazões mínimas Adelita Ramaiana Bennemann Granemann1, Miriam Rita Moro Mine1 and Eloy Kaviski1 Universidade Federal do Paraná, Curitiba, PR, Brasil 1 E-mails: (ARBG), (MRMM), (EK) Received: June 06, 2017 - Revised: December 22, 2017 - Accepted: January 19, 2018 ABSTRACT The study of minimum flows is increasingly important due to the relationship with ecosystem sustainability, the economy and its role as a sentinel of climate change. The aim of this paper is to contribute to the theoretical treatment of minimum extremes and, specifically, minimum flows. The method has two approaches: i) conventional; ii) asymptotic. In conventional analysis, the Weibull (W2) and Lognormal distributions of two parameters (LN2) were adjusted to the series of annual minimum flows and minimum averages of 7-day flows. In the asymptotic analysis approach two parent distributions, the distributions of all average daily flows, with power behavior for minimum flows, are investigated: i) Gamma; ii) LN2. The theory studied in this paper is applied to 11 gauged stations in the Iguaçu river basin with 48-year data series. It was concluded that the LN2 distribution presents the best fit according to the χ 2 test. It was found that the Gamma distribution, with respect to the minimums, tends to a power function, and consequently the W2 distribution. The parameters k, b and µ X , of the normalized annual minimum flow series are well-fitted to the LN2 distribution. According to both approaches, LN2 can be recommended for studies of minimum flows in the Iguaçu river basin. Keywords: Minimum extremes distributions; Minimum flows; Extremes asymptotic analysis. RESUMO O estudo de vazões mínimas torna-se cada vez mais importante devido à relação com a sustentabilidade de ecossistemas, da economia e como sentinela das mudanças climáticas. O objetivo deste artigo é contribuir para o tratamento teórico sobre extremos mínimos e, especificamente, vazões mínimas. O método divide-se em duas abordagens: i) convencional; ii) assintótica. Na análise convencional foram ajustadas as distribuições de Weibull (W2) e Lognormal de dois parâmetros (LN2) às séries de vazões mínimas anuais e mínimas médias de 7 dias. Na análise assintótica duas distribuições “mãe”, distribuições de todas as vazões médias diárias, com comportamento de potência para vazões mínimas, foram investigadas: i) Gama; ii) LN2. O estudo foi aplicado a 11 estações fluviométricas da bacia 2 hidrográfica do rio Iguaçu com 48 anos de dados. Concluiu-se que a distribuição LN2 apresentou melhor ajuste segundo o teste χ . Verificou-se que a distribuição Gama, com relação aos mínimos, tende a uma função de potência, e em consequência à distribuição W2. Os parâmetros k, b e µX, das vazões mínimas anuais moduladas, ajustaram-se bem à distribuição LN2. De acordo com as duas abordagens utilizadas, pode-se recomendar a LN2 para estudos de vazões mínimas na bacia hidrográfica do rio Iguaçu. Palavras-chave: Distribuições de extremos mínimos; Vazões mínimas; Análise assintótica de extremos. This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1/14 Frequency analysis of minimum flows INTRODUÇÃO FREQUENCY ANALYSIS OF MINIMUM FLOW In the extreme value theory, the analysis of maxima always attracts more attention than that of minima. This is partly because, in general, the quantiles estimated for minimum flows are not as extreme as for maximum flows. For a large number of applications, the minimum annual flow of 7 days duration and 10-year recurrence or an estimation of the flow of 95% of permanence are sufficient. These quantiles can be estimated directly from the empirical distribution, without resorting to a theoretical model adjustment. On the other hand, really severe droughts, such as occurred exceptionally during the rainy season of the 2014 hydrological year (Oct/2013 to March/2014) in the Southeast region of Brazil, where the estimated return period was over 100 years, according to Nakayama et al. (2015), require the application of theoretical statistical models that present a good fit to the hydrological variables under study. There are still many details to be discovered and clarified for the distribution function properties of minima. Increasing recognition of the importance of minimum flows for ecosystem viability, economic sustainability and as a climate change alert, makes the study of small extreme minima more and more important (GOTTSCHALK et al., 2013). In this context, the study of minimum flow is necessary, because these are the parameters for granting water use and dilution of effluents that influence the management of conflicts through use in situations of scarcity. Understanding the characteristics of minimum flow formation is crucial for the efficient development and integrated management of these water resources. It is important to raise the question: how to postulate an adequate distribution that respects the principle of parsimony (few parameters) to describe the phenomenon of interest? Bibliographical research showed the frequent use of the Lognormal distribution of two parameters(LN2) as a model for minimum flows, verified in the studies of Silvino et al. (2007), Uliana et al. (2011), Correa and Melo (2014), Victorino et al. (2014) and Finkler et al. (2015). For this reason, it is important to verify if this really is the best distribution to be considered, since its asymmetry is positive. This work proposes to analyze in detail the distributions of Weibull, Lognormal of two parameters and Gamma, with respect to the study of frequency of minimum extremes. Since most advanced statistical books deeply discuss recommended distributions to maximum extremes with real examples, but for minimum events distributions area described in a few paragraphs, merely saying that the adjustment procedures are similar The objective is to extend the theoretical treatment on minimum extremes and specifically minimum flows. Two approaches are considered: i) conventional frequency analysis; ii) asymptotic analysis of minimum extremes (Block Method). Therefore, this work was applied to 11 fluviometric stations in the Iguaçu river basin with data series of 48 years. The adjustment of distribution W2 and LN2 was verified at minimum annual flow and minimum moving average of 7 days duration. It was also analyzed whether the power behavior of the distribution of the mean daily flows (parent distribution), here postulated as Gamma and LN2, towards the minimum values (tail distribution) corresponds to a W2 distrib (...truncated)