Application of copula modelling to the performance assessment of reconstructed watersheds

The existence of interdependence among environmental variables has been qualitatively known for centuries. Recent studies have shown that copula modelling can provide a simple, yet powerful framework for modelling interdependence among hydrological data; however, still there are several studies which use outdated and superficial methods to perform this task. By considering the current state of knowledge, this study tries to introduce a pragmatic procedure to perform copula modelling in real-world problems. Our study uses copula modelling to find a notion for conditional quantities of the maximum annual water deficit with respect to the annual cumulative evapotranspiration. Therefore, by having an estimate for the annual cumulative evapotranspiration, the hydrological performance of the reconstructed watershed can be assessed even in nearby ungauged reconstructed watersheds with similar physical characteristics and reclamation strategy. Several competitive models are developed for joint description of these variables in a prototype reclaimed oil-sands mining site in northern Alberta, Canada. The developed joint models are compared and analyzed according to their convergence feasibility, overall accuracy, tail behaviour and a goodness-of-fit test. Our study concludes that copula modelling can be considered as a powerful option in practitioners’ toolkit. For the case under consideration, the Gumbel–Houguaard structure provides the most credible model of dependence. Moreover, our study provides some initial supports for the application of minimum distance methods for copula parameter estimation.

[1]  Arthur Charpentier,et al.  Tails of multivariate Archimedean copulas , 2009, J. Multivar. Anal..

[2]  A. Elshorbagy,et al.  A generic system dynamics model for simulating and evaluating the hydrological performance of reconstructed watersheds , 2008 .

[3]  Paul Embrechts,et al.  Copulas: A Personal View , 2009 .

[4]  C. Genest,et al.  The Joy of Copulas: Bivariate Distributions with Uniform Marginals , 1986 .

[5]  Andreas Schumann,et al.  Probability Analysis of Hydrological Loads for the Design of Flood Control Systems Using Copulas , 2010 .

[6]  Carlo De Michele,et al.  Extremes in Nature : an approach using Copulas , 2007 .

[7]  Implementation issues for total risk calculation for groups of sites , 2009 .

[8]  Rao S. Govindaraju,et al.  A copula-based joint deficit index for droughts. , 2010 .

[9]  U. Haberlandt,et al.  A space-time hybrid hourly rainfall model for derived flood frequency analysis , 2008 .

[10]  C. Genest,et al.  A Primer on Copulas for Count Data , 2007, ASTIN Bulletin.

[11]  A. Elshorbagy,et al.  Probabilistic Approach for Design and Hydrologic Performance Assessment of Reconstructed Watersheds , 2007 .

[12]  Rajib Maity,et al.  Probabilistic prediction of hydroclimatic variables with nonparametric quantification of uncertainty , 2008 .

[13]  Z. Bargaoui,et al.  Investigation with Kendall plots of infiltration index–maximum rainfall intensity relationship for regionalization , 2009 .

[14]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[15]  V. Singh,et al.  Gains from Copula-Based Multivariate Distribution Functions for Rainfall Processes , 2008 .

[16]  B. Renard,et al.  Use of a Gaussian copula for multivariate extreme value analysis: Some case studies in hydrology , 2007 .

[17]  Roger M. Cooke,et al.  Sampling algorithms for generating joint uniform distributions using the vine-copula method , 2007, Comput. Stat. Data Anal..

[18]  M. Pahlow,et al.  Planning of technical flood retention measures in large river basins under consideration of imprecise probabilities of multivariate hydrological loads , 2009 .

[19]  Two-copula based simulation for detention basin design , 2009 .

[20]  Gunky Kim,et al.  Comparison of semiparametric and parametric methods for estimating copulas , 2007, Comput. Stat. Data Anal..

[21]  A. Bárdossy,et al.  Copula based multisite model for daily precipitation simulation , 2009 .

[22]  Jean-David Fermanian,et al.  Goodness-of-fit tests for copulas , 2005 .

[23]  Jean‐François Quessy,et al.  On a new goodness‐of‐fit process for families of copulas , 2009 .

[24]  Wade T. Crow,et al.  A new data assimilation approach for improving runoff prediction using remotely-sensed soil moisture retrievals , 2008 .

[25]  Ann De Schepper,et al.  A copula test space model how to avoid the wrong copula choice , 2008, Kybernetika.

[26]  D. Dupuis.  Using Copulas in Hydrology: Benefits, Cautions, and Issues , 2007 .

[27]  T. Forshaw.  Everything you always wanted to know , 1977 .

[28]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[29]  Jun Yan,et al.  Enjoy the Joy of Copulas: With a Package copula , 2007 .

[30]  Jenq-Tzong Shiau,et al.  BIVARIATE FREQUENCY ANALYSIS OF FLOODS USING COPULAS1 , 2006 .

[31]  Clara Qualizza,et al.  Multi-criterion decision making approach to assess the performance of reconstructed watersheds , 2007 .

[32]  S. Kotz,et al.  The Meta-elliptical Distributions with Given Marginals , 2002 .

[33]  C. Genest,et al.  Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask , 2007 .

[34]  Kenneth J. Koehler,et al.  Constructing multivariate distributions with specific marginal distributions , 1995 .

[35]  Rafael Schmidt,et al.  Non‐parametric Estimation of Tail Dependence , 2006 .

[36]  Ni-Bin Chang,et al.  Copula‐based flood frequency (COFF) analysis at the confluences of river systems , 2009 .

[37]  T. Ouarda,et al.  Index flood–based multivariate regional frequency analysis , 2009 .

[38]  Jenq-Tzong Shiau,et al.  Fitting Drought Duration and Severity with Two-Dimensional Copulas , 2006 .

[39]  A. Favre,et al.  Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data , 2007 .

[40]  Walter Ruda,et al.  Target Group-Specific Design of Student Entrepreneurship Support - A German Example Focusing on Start-Up Motives and Barriers , 2009 .

[41]  F. Serinaldi.  Copula-based mixed models for bivariate rainfall data: an empirical study in regression perspective , 2009 .

[42]  Thomas Mikosch,et al.  Copulas: Tales and facts , 2006 .

[43]  O. Scaillet,et al.  Nonparametric Estimation of Copulas for Time Series , 2002 .

[44]  R. Mesiar,et al.  Copulas: an Approach How to Model the Dependence Structure of Random Vectors , 2009 .

[45]  Mohammad Karamouz,et al.  Probabilistic reservoir operation using Bayesian stochastic model and support vector machine , 2009 .

[46]  Olivier Scaillet,et al.  Testing for Equality between Two Copulas , 2006, J. Multivar. Anal..

[47]  Andrea Castelletti,et al.  Topics on System Analysis and Integrated Water Resource Management , 2006 .

[48]  Anne-Catherine Favre,et al.  Importance of Tail Dependence in Bivariate Frequency Analysis , 2007 .

[49]  Jerald F Lawless,et al.  Semiparametric estimation in copula models for bivariate sequential survival times , 2011, Biometrical journal. Biometrische Zeitschrift.

[50]  M. Lambert,et al.  Frequency analysis of rainfall and streamflow extremes accounting for seasonal and climatic partitions , 2008 .

[51]  A. Sklar,et al.  Random variables, distribution functions, and copulas---a personal look backward and forward , 1996 .

[52]  S. P. Simonovic,et al.  Managing flood risk, reliability and vulnerability , 2009 .

[53]  Luis Samaniego,et al.  Streamflow prediction in ungauged catchments using copula‐based dissimilarity measures , 2010 .

[54]  Henrik Madsen,et al.  Assessment of the risk of inland flooding in a tidal sluice regulated catchment using multi-variate statistical techniques , 2009 .

[55]  Francesco Serinaldi,et al.  Probabilistic characterization of drought properties through copulas , 2009 .

[56]  C. De Michele,et al.  A Generalized Pareto intensity‐duration model of storm rainfall exploiting 2‐Copulas , 2003 .

[57]  B. Rémillard,et al.  Goodness-of-fit tests for copulas: A review and a power study , 2006 .

[58]  Discussion of “Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask” by C. Genest and A.-C. Favre , 2008 .

[59]  F. Serinaldi.  Analysis of inter-gauge dependence by Kendall’s τK, upper tail dependence coefficient, and 2-copulas with application to rainfall fields , 2008 .

[60]  S. Simonovic,et al.  Bivariate flood frequency analysis. Part 2: a copula‐based approach with mixed marginal distributions , 2009 .

[61]  O. Scaillet,et al.  Nonparametric Estimation of Copulas for Time Series , 2002 .

[62]  R. Nelsen.  An Introduction to Copulas , 1998 .

[63]  A. McNeil,et al.  The t Copula and Related Copulas , 2005 .

[64]  Vijay P. Singh,et al.  Gumbel–Hougaard Copula for Trivariate Rainfall Frequency Analysis , 2007 .

[65]  Bruno Rémillard,et al.  Discussion of “Copulas: Tales and facts”, by Thomas Mikosch , 2006 .

[66]  B. Bobée,et al.  Multivariate hydrological frequency analysis using copulas , 2004 .

[67]  F. Serinaldi.  A multisite daily rainfall generator driven by bivariate copula-based mixed distributions , 2009 .

[68]  C. De Michele,et al.  Multivariate multiparameter extreme value models and return periods: A copula approach , 2010 .

[70]  Daniel Berg.  Copula goodness-of-fit testing: an overview and power comparison , 2009 .

[71]  W. Härdle,et al.  Statistical Tools for Finance and Insurance , 2003 .

[72]  Anne-Catherine Favre,et al.  Bayesian copula selection , 2006, Comput. Stat. Data Anal..

[73]  Saralees Nadarajah,et al.  Assessment of hydrological droughts for the Yellow River, China, using copulas , 2007 .

[74]  V. Singh,et al.  Bivariate Flood Frequency Analysis Using the Copula Method , 2006 .

[75]  C. De Michele,et al.  On the Use of Copulas in Hydrology: Theory and Practice , 2007 .

[76]  Fabrizio Durante,et al.  On the construction of multivariate extreme value models via copulas , 2009 .

[77]  A. Bárdossy.  Copula‐based geostatistical models for groundwater quality parameters , 2006 .

[78]  Hedibert Freitas Lopes,et al.  Copula, marginal distributions and model selection: a Bayesian note , 2008, Stat. Comput..