Copula-based stochastic simulation of hydrological data applied to Nile River flows Taesam Lee and Jose D. Salas ABSTRACT Modelling a multivariate distribution is a classical issue in statistics. Copula functions offer a useful solution to this issue by modelling the multivariate distribution as a function of its marginal distributions. They have been used in various problems in hydrology and water management such as flood frequency analysis and drought or rainfall intensity-duration frequency analysis. However, to the knowledge of the author, they have not been applied for stochastic simulation of hydrologic data. In this study we explore the applicability of the copula concept for stochastic streamflow simulation. Parametric and non-parametric functions are applied for fitting the distribution of the original observed data and the serial dependence structure is then modelled with alternative copula functions. The pros and cons of different copula models are investigated by comparing the statistics of the generated data. Two major features of the copula models include: (1) portraying the heteroscedasticity embedded in the serial correlation of the observed data and (2) the flexibility of applicable marginal distributions. The suggested copula models are applied to simulate synthetic annual streamflow data of the Nile River. The results showed that the benefits of using these copula models are somewhat marginal with respect to the well-known modelling procedures. Taesam Lee (corresponding author) Engineering Research Institute, Department of Civil Engineering, Gyeongsang National University, 501 Jinju-daero, Jinju-si, Gyeongsangnam-do, 660-701 Republic of Korea E-mail: tae3lee@gnu.ac.kr Jose D. Salas Department of Civil and Environmental Engineering, B208 Engineering Building, Colorado State University Fort Collins, Colorado 80523-1372, USA Key words | copula, drought, nonparametric, stochastic simulation, streamflow, time series INTRODUCTION A copula is a multivariate distribution function with stan- dard uniform marginals. Copulas allow the modelling of any multivariate distribution from its marginal distributions and its copula function. Because of this flexibility, copulas have become popular in several fields such as economics (e.g. Chen & Fan a, b; Gagliardini & Gourieroux ; Chen et al. ; Lee & Long ) and hydrology for flood and drought analysis (e.g. De Michele & Salvadori ; Salvadori & De Michele ; Shiau ; De Michele et al. ; Shiau et al. ; Laux et al. ; Serinaldi et al. ; Shiau & Modarres ). An application of copulas for time series modelling and generation of annual stream- flows is presented here. Time series generation is useful for analyzing water resources systems such as for determining the required storage capacity of a reservoir and estimating the severity of drought, etc. Conventional time series generation models used in hydrol- ogy and water resources management such as autoregressive moving average (ARMA) (Salas et al. ; Salas & Obeysekera ; Salas ; Brockwell & Davis ; Salas et al. ) and fractional Gaussian noise (Mandelbrot & Wallis ; Mandel- brot ; Koutsoyiannis ) have some drawbacks such as the assumption of Gaussian marginal distribution. However, hydrologic data such as precipitation and streamflow usually have skewed distributions and sometimes show bi- or multi- modal characteristics (Lall & Sharma ; Sharma et al. ; Prairie et al. ) where there is more than one generat- ing mechanism (e.g. runoff resulting from convective rainfall and snowmelt). Data transformation methods such as logarith- mic, Box-Cox, gamma and power transformations are common ways of tackling with the problem of skewness. However, there still exist some limitations that may affect the reproduction of 318 © IWA Publishing 2011 Hydrology Research | 42.4 | 2011 doi: 10.2166/nh.2011.085 Downloaded from http://iwaponline.com/hr/article-pdf/42/4/318/371177/318.pdf by guest on 14 June 2022