Modeling the spatial dependence of floods using the Fisher copula

Abstract. Floods often affect not only a single location, but also a whole region. Flood frequency analysis should therefore be undertaken at a regional scale which requires the considerations of the dependence of events at different locations. This dependence is often neglected even though its consideration is essential to derive reliable flood estimates. A model used in regional multivariate frequency analysis should ideally consider the dependence of events at multiple sites which might show dependence in the lower and/or upper tail of the distribution. We here seek to propose a simple model that on the one hand considers this dependence with respect to the network structure of the region and on the other hand allows for the simulation of stochastic event sets at both gauged and ungauged locations. The new Fisher copula model is used for representing the spatial dependence of flood events in the nested Thur catchment in Switzerland. Flood event samples generated for the gauged stations using the Fisher copula are compared to samples generated by other dependence models allowing for modeling of multivariate data including elliptical copulas, R-vine copulas, and max-stable models. The comparison of the dependence structures of the generated samples shows that the Fisher copula is a suitable model for capturing the spatial dependence in the data. We therefore use the copula in a way such that it can be used in an interpolation context to simulate event sets comprising gauged and ungauged locations. The spatial event sets generated using the Fisher copula well capture the general dependence structure in the data and the upper tail dependence, which is of particular interest when looking at extreme flood events and when extrapolating to higher return periods. The Fisher copula was for a medium-sized catchment found to be a suitable model for the stochastic simulation of flood event sets at multiple gauged and ungauged locations.

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

[2]  A. Davison,et al.  Statistical Modeling of Spatial Extremes , 2012, 1208.3378.

[3]  J. Segers.  MAX-STABLE MODELS FOR MULTIVARIATE EXTREMES , 2012, 1204.0332.

[4]  Claudia Czado,et al.  Selecting and estimating regular vine copulae and application to financial returns , 2012, Comput. Stat. Data Anal..

[5]  Anthony C. Davison,et al.  Extremes on river networks , 2015, 1501.02663.

[6]  T. Ghizzoni,et al.  Multivariate skew-t approach to the design of accumulation risk scenarios for the flooding hazard , 2010 .

[7]  T. Ouarda,et al.  Delineation of homogenous regions using hydrological variables predicted by projection pursuit regression , 2016 .

[8]  Aloke Phatak,et al.  A max-stable process model for rainfall extremes at different accumulation durations , 2016 .

[9]  Renzo Rosso,et al.  Extremes in Nature , 2007 .

[10]  Philippe Naveau,et al.  Modeling jointly low, moderate, and heavy rainfall intensities without a threshold selection , 2016 .

[11]  J. Seibert,et al.  Flood‐type classification in mountainous catchments using crisp and fuzzy decision trees , 2015 .

[12]  Francesco Serinaldi,et al.  Irreversibility and complex network behavior of stream flow fluctuations , 2016 .

[13]  Eric P. Smith,et al.  An Introduction to Statistical Modeling of Extreme Values , 2002, Technometrics.

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

[15]  Extensive spatio-temporal assessment of flood events by application of pair-copulas , 2015 .

[16]  Jonathan A. Tawn,et al.  A conditional approach for multivariate extreme values (with discussion) , 2004 .

[17]  A. Bárdossy,et al.  Upper tail dependence in rainfall extremes: would we know it if we saw it? , 2015, Stochastic Environmental Research and Risk Assessment.

[18]  Francesco Serinaldi,et al.  Can we tell more than we can know? The limits of bivariate drought analyses in the United States , 2016, Stochastic Environmental Research and Risk Assessment.

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

[20]  Jean-François Quessy,et al.  On the family of multivariate chi-square copulas , 2016, J. Multivar. Anal..

[21]  Francesco Serinaldi,et al.  An uncertain journey around the tails of multivariate hydrological distributions , 2013 .

[22]  Harry Joe,et al.  Multivariate extreme value copulas with factor and tree dependence structures , 2018 .

[23]  Laurens de Haan,et al.  Stationary max-stable fields associated to negative definite functions. , 2008, 0806.2780.

[24]  Felice D'Alessandro,et al.  Practical guidelines for multivariate analysis and design in coastal and off-shore engineering , 2014 .

[25]  Gianfausto Salvadori,et al.  Frequency analysis via copulas: Theoretical aspects and applications to hydrological events , 2004 .

[26]  J. Blanchet,et al.  Trend in the Co‐Occurrence of Extreme Daily Rainfall in West Africa Since 1950 , 2018 .

[27]  Roberto Rudari,et al.  Multisite flooding hazard assessment in the Upper Mississippi River , 2010 .

[28]  Andreas Schumann,et al.  Rainfall Generators for Application in Flood Studies , 2011 .

[29]  Zhuo Wang,et al.  Incorporating spatial dependence in regional frequency analysis , 2014, Water resources research.

[30]  P. Naveau,et al.  Variograms for spatial max-stable random fields , 2006 .

[31]  A conditional approach to modelling multivariate extreme values (with discussion). , 2004 .

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

[33]  Jonathan A. Tawn,et al.  Statistics of Multivariate Extremes , 1990 .

[34]  C. Sempi,et al.  Principles of Copula Theory , 2015 .

[35]  James E. Helmreich.  Regression Modeling Strategies with Applications to Linear Models, Logistic and Ordinal Regression and Survival Analysis (2nd Edition) , 2016 .

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

[37]  A. Bárdossy,et al.  Geostatistical interpolation using copulas , 2008 .

[38]  A. Favre,et al.  Stochastic generation of multi-site daily precipitation focusing on extreme events , 2018 .

[39]  Mathieu Ribatet,et al.  Extreme value copulas and max-stable processes , 2013 .

[40]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[41]  A. Frigessi,et al.  Pair-copula constructions of multiple dependence , 2009 .

[42]  Mathieu Vrac,et al.  Multivariate statistical modelling of compound events via pair-copula constructions: analysis of floods in Ravenna (Italy) , 2017 .

[43]  D. Viviroli,et al.  Continuous simulation for flood estimation in ungauged mesoscale catchments of Switzerland - Part II: Parameter regionalisation and flood estimation results , 2009 .

[44]  E. Steyerberg,et al.  [Regression modeling strategies]. , 2011, Revista espanola de cardiologia.

[45]  Anne-Catherine Favre,et al.  The new family of Fisher copulas to model upper tail dependence and radial asymmetry: Properties and application to high‐dimensional rainfall data , 2018 .

[46]  J. R. Wallis,et al.  Regional frequency analysis , 1997 .

[47]  Narayanaswamy Balakrishnan,et al.  A General Purpose Approximate Goodness-of-Fit Test , 1995 .

[48]  Christian Genest,et al.  On the multivariate probability integral transformation , 2001 .

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

[50]  Delbert Dueck,et al.  Clustering by Passing Messages Between Data Points , 2007, Science.

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

[52]  R. Furrer,et al.  Synthetic design hydrographs for ungauged catchments: a comparison of regionalization methods , 2018, Stochastic Environmental Research and Risk Assessment.

[53]  B. Gräler.  Modelling skewed spatial random fields through the spatial vine copula , 2014 .

[54]  Yannick Malevergne,et al.  Testing the Gaussian copula hypothesis for financial assets dependences , 2001, cond-mat/0111310.

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

[56]  Fabrizio Durante,et al.  A multivariate copula‐based framework for dealing with hazard scenarios and failure probabilities , 2016 .

[57]  J. Blanchet,et al.  A regional model for extreme rainfall based on weather patterns subsampling , 2016 .

[58]  Caroline Keef,et al.  Spatial dependence in extreme river flows and precipitation for Great Britain. , 2009 .

[59]  S. Padoan.  Max‐Stable Processes , 2013 .

[60]  Felice D'Alessandro,et al.  Practical guidelines for the multivariate assessment of the structural risk in coastal and off-shore engineering , 2015 .

[61]  Patrick J. F. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 2003 .

[62]  Rob Lamb,et al.  Estimating the probability of widespread flood events , 2013 .

[63]  Keith Beven,et al.  Probabilistic flood risk mapping including spatial dependence , 2013 .

[64]  Hongxiang Yan,et al.  A regional Bayesian hierarchical model for flood frequency analysis , 2015, Stochastic Environmental Research and Risk Assessment.

[65]  András Bárdossy,et al.  Calibration of hydrological model parameters for ungauged catchments , 2006 .

[66]  Anthony C. Davison,et al.  Threshold modeling of extreme spatial rainfall , 2013 .

[67]  Fabrizio Durante,et al.  Clustering of concurrent flood risks via Hazard Scenarios , 2018 .

[68]  Ying Sun,et al.  OF SPATIAL EXTREMES : MEASURING SPATIAL DEPENDENCE AND MODELING SPATIAL EFFECTS , 2012 .

[69]  Frank E. Harrell,et al.  Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis , 2001 .

[70]  Zhenzhou Lu,et al.  A new method for model validation with multivariate output , 2018, Reliab. Eng. Syst. Saf..

[71]  Bruno Merz,et al.  Handling uncertainty in bivariate quantile estimation – An application to flood hazard analysis in the Mekong Delta , 2015 .

[72]  H. Joe.  Dependence Modeling with Copulas , 2014 .

[73]  J. Seibert,et al.  Influence of hydro-meteorological data spatial aggregation on streamflow modelling , 2016 .

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

[75]  Francesco Serinaldi,et al.  A Blueprint for Full Collective Flood Risk Estimation: Demonstration for European River Flooding , 2016, Risk analysis : an official publication of the Society for Risk Analysis.

[76]  J. R. Wallis,et al.  Regional Frequency Analysis: An Approach Based on L-Moments , 1997 .

[77]  Hannes Kazianka,et al.  Spatial Interpolation Using Copula-Based Geostatistical Models , 2010 .

[78]  R. Furrer,et al.  Uncertainty Assessment of Synthetic Design Hydrographs for Gauged and Ungauged Catchments , 2018 .

[79]  Andrea Petroselli,et al.  Catchment compatibility via copulas: A non-parametric study of the dependence structures of hydrological responses , 2016 .