A coupled system based on Differential Evolution for the determination of Rainfall intensity equations

Scientific/Technical Article A coupled system based on Differential Evolution for the determination of Rainfall intensity equations Um sistema acoplado baseado em Evolução Diferencial para determinação de equações de chuvas intensas Guilherme José Cunha Gomes1  Eurípedes do Amaral Vargas Júnior2  1 Departamento de Estradas de Rodagem do Espírito Santo, Vitória, ES, Brasil 2 Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brasil ABSTRACT Rainfall intensity equations are fundamental in hydrological studies of road design, which require a project rainfall definition to estimate the project flow and the subsequent design of the hydraulic structure. This paper develops an integrated framework for rainfall intensity equations analyses from global optimization via Differential Evolution. The code was specially developed to facilitate the Gumbel model adjustment in the frequency analysis of annual series, as well as the intensity-duration-frequency model fit, without prior knowledge about the parameters of both models. The developed system was evaluated by using Markov chain Monte Carlo simulation, that search efficiently the model parameter space in pursuit of posterior samples and the posterior prediction uncertainty for both models. The results indicate that simulations are shown to be in good agreement with the measured flow and precipitation data. The optimal parameters obtained with the developed framework agreed with the maximum a-posteriori value of the Monte Carlo simulations. The paper illustrates explicitly the benefits of the method using real-world precipitation data collected for a hydrologic study of a highway design. Keywords:  Precipitation; Rainfall intensity equations; Global optimization; Differential Evolution; Highway design RESUMO As equações de chuvas intensas são fundamentais em estudos hidrológicos de projetos rodoviários, os quais requerem a definição da chuva de projeto para estimativa da vazão de projeto e o posterior dimensionamento da estrutura hidráulica. Neste trabalho, desenvolveu-se um sistema computacional para a obtenção da equação de chuvas intensas, a partir de otimização global via Evolução Diferencial. O sistema foi especialmente projetado para facilitar o ajuste do modelo de Gumbel na análise de frequência de séries anuais, bem como o ajuste do modelo da curva intensidade-duração-frequência, sem que o usuário necessite conhecimento prévio sobre os parâmetros de ambos os modelos. Simulações Monte Carlo via cadeia de Markov foram realizadas para avaliar a metodologia desenvolvida através da distribuição posterior dos parâmetros e da incerteza preditiva de ambos os modelos. Os resultados obtidos indicaram bons ajustes para os modelos de Gumbel e da equação de chuvas intensas com o uso da otimização global via Evolução Diferencial, uma vez que os parâmetros ótimos obtidos pelo sistema concordaram com o valor máximo a-posteriori oriundo das simulações Monte Carlo. Um estudo de caso envolvendo a obtenção da equação de chuvas em um estudo hidrológico de projeto rodoviário ilustram a usabilidade e aplicabilidade do sistema e da metodologia desenvolvida. Palavras-chave:  Precipitação; Equação de chuvas intensas; Otimização global; Evolução Diferencial; Projetos rodoviários INTRODUCTION Rainfall controls a very wide range of processes on the Earth´s surface. However, the behavior of exceptional rainfall is hard to determine, and is generally handled in a probabilistic way. These rains can be determined through rainfall intensity equations, which are broadly utilized for dimensioning drainage structures with different purposes, including agricultural ( VIVEKANANDAN, 2017 ) and urban drainage systems ( SABÓIA et al., 2017 ), channels ( WRIGHT; SMITH; BAECK, 2014 ), and engineering structures in road projects ( KANG et al., 2009 ). A precise characterization of rainfall intensity equations is therefore vital for the appropriate sizing of different drainage structures. Different probabilistic models (e.g. normal, log-normal, and Gumbel) can be used to obtain an event (rainfall or discharge) that can be exceeded, on average, once at every specified period of time (usually years). Such models, non-linear in their parameters, consist in the probability cumulative functions of each statistical distribution. From the historical observations of the event (e.g. annual maximum daily precipitations), the statistical model is mixed with the observed events through the estimation of parameters that provide better adjustment of the employed model to the observed rainfall data. Among the statistical distributions, the Gumbel model is widely used for estimating precipitation or discharge data regarding a certain return period ( Tr ). A detailed description of the adjustment of statistical distributions to data on rainfall events is provided by Tucci (1993) , including different techniques for estimating the parameters of the models. After this optimization process is completed, the maximum daily precipitations are determined for different values of Tr . The rainfall intensity equation, also called intensity-duration-frequency (IDF) curve, provides estimates of rainfall intensity concerning different rainfall durations and different values of Tr . The IDF relations, obtained from the disaggregation of the annual maximum daily precipitations, constitute one of the elements necessary for calculating the design discharge, i.e. the discharge employed for the sizing of drainage structures. Different models for the IDF curves have been proposed in the literature ( KOUTSOYIANNIS; KOZONIS; MANETAS, 1998 ) and, like the probability cumulative functions of the statistical distributions, they require techniques for estimating the parameters of adjustment between calibration data and models. Different techniques are available for estimating the parameters of the Gumbel model. These techniques include the method of moments ( TUCCI, 1993 ; MADSEN et al., 2002 ; MOHYMONT; DEMARÉE; FAKA, 2004 ), maximum likelihood estimation ( CHOW; MAIDMENT; MAYS, 1988 ; FADHEL et al., 2017 ), approximate Bayesian computations ( DEMIRHAN, 2017 ), among others. For the IDF equations, the search for the model parameters can be performed through linear regression with the linearization of the model ( SILVA et al., 2012 ), besides iterative methods or non-linear regressions, such as the Gauss-Newton ( GARCIA et al., 2011 ) and Levenberg-Marquardt models ( PENNER; LIMA, 2016 ). These last techniques are also called methods of local optimization, in which, in order to start the optimization process, there must be an initial estimate of the parameters that needs to be relatively close to the optimal values. Regarding models that are non-linear in their parameters, such as the Gumbel model and the IDF curve, it is widely established in the literature ( VRUGT, 2016 ) that methods of glo (...truncated)