Elsevier

Advances in Water Resources

Volume 97, November 2016, Pages 233-240
Advances in Water Resources

Probabilistic modelling of flood events using the entropy copula

https://doi.org/10.1016/j.advwatres.2016.09.016Get rights and content

Highlights

  • Theentropy copulawas usedto simulate the flood events.

  • The entropycopulawas compared to Gaussian copula and Archimedean copula method in statistical flood simulation.

  • Gibbs sampling technique was used for Archimedean copula and entropy copula to extend to three-dimension.

  • The entropy copula shows superior in multivariable simulationfor the study sites.

Abstract

The estimation of flood frequency is vital for the flood control strategies and hydraulic structure design. Generating synthetic flood events according to statistical properties of observations is one of plausible methods to analyze the flood frequency. Due to the statistical dependence among the flood event variables (i.e. the flood peak, volume and duration), a multidimensional joint probability estimation is required. Recently, the copula method is widely used for multivariable dependent structure construction, however, the copula family should be chosen before application and the choice process is sometimes rather subjective. The entropy copula, a new copula family, employed in this research proposed a way to avoid the relatively subjective process by combining the theories of copula and entropy. The analysis shows the effectiveness of the entropy copula for probabilistic modelling the flood events of two hydrological gauges, and a comparison of accuracy with the popular copulas was made. The Gibbs sampling technique was applied for trivariate flood events simulation in order to mitigate the calculation difficulties of extending to three dimension directly. The simulation results indicate that the entropy copula is a simple and effective copula family for trivariate flood simulation.

Introduction

In the context of hydrology, the return periods of flood events are always used to provide the important information for hydraulic structure design, flood protection strategies and water resources management. The univariate return periods are traditionally estimated by fitting a probability distribution function to the historical observations. However, for a flood event, which is customarily characterized by peak (P), volume (V) and duration (D), a univariate probability distribution analysis is apparently not enough as these three random variables are not mutually independent due to their internal physical connections. For a system with two or more variables, the return period of outcomes is not equal to the forcing return period of a particular variate (Hawkes et al., 2002). In the case of flood frequency estimation, merely analyzing the flood peak or flood volume frequency will lead to an underestimation or overestimation of risk (De Michele et al., 2005; Yue and Rasmussen, 2002). Therefore, a multivariate statistical analysis is required for a more complete flood frequency analysis (Grimaldi and Serinaldi, 2006). One effective approach of carrying out the multivariate frequency analysis is to generating the synthetic flood events according to the statistical properties of observations.

The most significant issue of multivariate probability analysis is the construction of dependence structure for the involved correlated random variables. Recently, copula functions developed by Sklar (1959), has received tremendous attention in extreme hydrological events simulation for its efficiency and flexibility in marginal distribution selection. An application of copula in the field of Hydrology was firstly proposed by De Michele and Salvadori (2003), the authors construct the dependency structure of storm duration and intensity using bivariate Frank copula. Favre et al. (2004) proposed an approach based on copulas (Farlie-Gumbel-Morgenstern copula, Frank copula and Clayton copula), and applied it to bivariate (P, V) frequency analysis. Grimaldi and Serinaldi (2006) built a trivariate joint distribution of flood event variables (P, V, D) using asymmetric Archimedean copula functions. Zhang and Singh (2007) used the Gumbel copula to analyze the trivariate flood frequency. A simple copula family, Gaussian copula, was employed to analyze the multivariate extreme value in hydrology by Renard and Lang (2007). Salvadori and De Michele (2015) present a multivariate frequency analysis for droughts by using Archimedean copulas and survival copula. For more applications and discussions about the copulas in hydrology, the reader is directed to (Salvadori and De Michele, 2007; Dupuis, 2007), and the references therein.

A problem in the application of copula method comes from the choice of a copula family for the specified data set, which is a rather challenging, or even impossible, task when the true copula is not in the set of copulas under the null composite hypothesis (Chu, 2011). Normally, the selection of a copula function is followed by two rules, the fitting quality of the alternative possible copulas and the difficulties in calculation.

The Principle Of Maximum Entropy (POME) was first expounded by Jaynes, 1957a, Jaynes, 1957b, proposed a criterion to choose the most suitable probability distribution function based on the rationale, that the desired probability distribution has maximum uncertainty, subject to representative and known information (Zhao and Lin, 2011). Since the work by Amorocho and Espildora (1973) and Sonuga, 1972, Sonuga, 1976 in the early 1970 s, there has been a proliferation of applications of POME in hydrological sciences. An extensive review on the applications of POME in hydrology and water resources has been provided by Singh (October 2011).

The copula is an efficient and accurate tool to obtain a suitable multivariable distribution with different marginal distributions (De Michele et al., 2007), however a copula family should be determined at first. While the POME approach can produce a probability distribution with least bias from limited information in a more objective way. A new copula family, entropy copula, combine the advantages of these two approaches was then proposed to avoid the copula family selection process for probabilistically modelling a multivariate event. The fundamental of entropy copula is that the copula function with the maximum entropy is the most suitable multivariate distribution function, and it has been successfully applied in the field of economy (Chu, 2011; Zhao and Lin, 2011) and hydrology (Hao and Singh, 2013; AghaKouchak, 2014).

In this study, the entropy copula is used to model the dependence structure of trivariate flood events and stochastically simulate the flood events. The Lutaizi hydrological gauge located on the Huai River and Quzhou hydrological gauge located on the Qujiang River serve as two case studies to illustrate the entropy copula estimation and comparisons between the entropy copula and other copula families was also made. The study sites and required variates are introduced in Section 2. In Section 3, the marginal distributions are fitted and the dependency structures are constructed by Gaussian copula, Archimedean copulas and entropy copula. Results section compares these copula performances, following by the discussion and conclusion of the work in Section 5.

Section snippets

Study site and data

The probabilistic modelling methods in this paper concern the trivariate analysis of P, V and D in the stem stream of Huai River and Qujiang River. The Huai River is notoriously vulnerable to flooding in China. The flood events of Huai River were abstracted from the observations of Lutaizi hydrological gauge (period: 1954–1998, location: 116°38′E, 32°34′N), which is located in the midstream of the river with the plain basin of 88,630 km2. Flood events of Qujiang River were collected in Quzhou

Marginal distributions

For the univariate analyses, the Generalized Pareto (GP) distribution, Generaliz Value (GEV) distribution, Gamma distribution, exponential distribution and Pearson Type III (P-III) distribution had been selected as the possible distribution functions. In this paper, the location parameters of GP distributions for the variates were selected based on the Normalized Root Mean Square Error (NRMSE) and the principle that the threshold should be as small as possible to retain the largest sample (Li

Results

The best fitted marginal distributions were found by the NRMSE formulation (Eq. (2)) for the two study sites. These marginal distributions were later used to simulate the multivariable flood events by applying the Gaussian copula, Archimedean copula and entropy copula. Finally, the NRMSE information was again used as goodness-of-fit (GoF) statistic to compare the simulated and empirical multivariate probability distributions for numerically illustrating the simulation quality.

Summary and conclusion

A maximum entropy theory based method was used to construct the copula function for the statistical simulation of trivariate flood events (P, V, D). A comparison with the widely used Gaussian copula and Archimedean copula was made by applying these methods to two hydrological gauges in China. The main advantage of the entropy copula is that no assumption is made about the copula family. By numerically searching for the maximum copula entropy, the copula function can be obtained objectively,

Acknowledgements

This research was funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Natural Science Foundation of Jiangsu Province, China (No. BK20160470) and Natural Science Research Program of the Universities in Jiangsu Province (No.15KJD170003). The authors thank Assistant Professor Amir AghaKouchak (Center for Hydrometeorology and Remote Sensing, Department of Civil and Environmental Engineering, University of California) for his code of parameters

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