On some aspects of peaks-over-threshold modeling of floods under nonstationarity using climate covariates

This paper discusses some aspects of flood frequency analysis using the peaks-over-threshold model with Poisson arrivals and generalized Pareto (GP) distributed peak magnitudes under nonstationarity, using climate covariates. The discussion topics were motivated by a case study on the influence of El Niño–Southern Oscillation on the flood regime in the Itajaí river basin, in Southern Brazil. The Niño3.4 (DJF) index is used as a covariate in nonstationary estimates of the Poisson and GP distributions scale parameters. Prior to the positing of parametric dependence functions, a preliminary data-driven analysis was carried out using nonparametric regression models to estimate the dependence of the parameters on the covariate. Model fits were evaluated using asymptotic likelihood ratio tests, AIC, and Q–Q plots. Results show statistically significant and complex dependence relationships with the covariate on both nonstationary parameters. The nonstationary flood hazard measure design life level (DLL) was used to compare the relative performances of stationary and nonstationary models in quantifying flood hazard over the period of records. Uncertainty analyses were carried out in every step of the application using the delta method.

[1]  Holger Rootzén,et al.  Design Life Level: Quantifying risk in a changing climate , 2013 .

[2]  Dörte Jakob,et al.  Nonstationarity in Extremes and Engineering Design , 2013 .

[3]  O comportamento da precipitacao na bacia do Itajai , 2000 .

[4]  Francesco Serinaldi,et al.  Dismissing return periods! , 2015, Stochastic Environmental Research and Risk Assessment.

[5]  Luc Neppel,et al.  Non-stationary frequency analysis of heavy rainfall events in southern France , 2013 .

[6]  Francesco Serinaldi,et al.  Stationarity is undead: Uncertainty dominates the distribution of extremes , 2015 .

[7]  Walter R. Lynn,et al.  Mathematical Model for Flood Risk Evaluation , 1964 .

[8]  Francesco Pauli,et al.  Penalized likelihood inference in extreme value analyses , 2001 .

[9]  M. Lang,et al.  Statistical analysis of extreme events in a non-stationary context via a Bayesian framework: case study with peak-over-threshold data , 2006 .

[10]  Richard W. Katz,et al.  Statistical Methods for Nonstationary Extremes , 2013 .

[11]  T. Yee,et al.  Vector generalized linear and additive extreme value models , 2007 .

[12]  Irving I. Gringorten,et al.  A plotting rule for extreme probability paper , 1963 .

[13]  Mauro Naghettini,et al.  On peaks-over-threshold modeling of floods with zero-inflated Poisson arrivals under stationarity and nonstationarity , 2014, Stochastic Environmental Research and Risk Assessment.

[14]  Kevin E. Trenberth,et al.  The Definition of El Niño. , 1997 .

[15]  A. Grimm.  Interannual climate variability in South America: impacts on seasonal precipitation, extreme events, and possible effects of climate change , 2011 .

[16]  R. Abers.  Organizing for Governance: Building Collaboration in Brazilian River Basins , 2007 .

[17]  R. Bras.  Hydrology : an introduction to hydrologic science , 1990 .

[18]  Michel Lang,et al.  Bayesian Methods for Non-stationary Extreme Value Analysis , 2013 .

[19]  Bruno Merz,et al.  Flood trends and variability in the Mekong river , 2009 .

[20]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[21]  Mauro Naghettini,et al.  A Bayesian approach for estimating extreme flood probabilities with upper-bounded distribution functions , 2010 .

[22]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Jonathan A. Tawn,et al.  Statistical models for overdispersion in the frequency of peaks over threshold data for a flow series , 2010 .

[24]  George Casella,et al.  Statistical Inference Vol. 70 , 1990 .

[25]  R. Stouffer,et al.  Stationarity Is Dead: Whither Water Management? , 2008, Science.

[26]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[27]  R. T. Clarke,et al.  Hydrological prediction in a non-stationary world , 2007 .

[28]  T. Ouarda,et al.  Bayesian Nonstationary Frequency Analysis of Hydrological Variables 1 , 2011 .

[29]  A. Grimm,et al.  ENSO and Extreme Rainfall Events in South America , 2009 .

[30]  M. Barreiro.  Influence of ENSO and the South Atlantic Ocean on climate predictability over Southeastern South America , 2010 .

[31]  J. Shukla,et al.  Current status of ENSO prediction skill in coupled ocean–atmosphere models , 2008 .

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

[33]  L. Haan,et al.  Residual Life Time at Great Age , 1974 .

[34]  V. Barros,et al.  Climate Variability in Southern South America Associated with El Niño and La Niña Events. , 2000 .

[35]  Elizabeth C. Kent,et al.  Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century , 2003 .

[36]  T. Ouarda,et al.  A Nonstationary Extreme Value Analysis for the Assessment of Changes in Extreme Annual Wind Speed over the Gulf of St. Lawrence, Canada , 2008 .

[37]  Taha B. M. J. Ouarda,et al.  Long‐term prediction of precipitation and hydrologic extremes with nonstationary oscillation processes , 2010 .

[38]  James H. Lambert,et al.  Risk of Extreme Events Under Nonstationary Conditions , 1998 .

[39]  Richard L. Smith,et al.  Models for exceedances over high thresholds , 1990 .

[40]  Thomas Yee,et al.  VGAM: Vector Generalized Linear and Additive Models 1.0-4 , 2017 .

[41]  M. Bayazit,et al.  Effect of the occurrence process of the peaks over threshold on the flood estimates , 2001 .

[42]  Daniel Cooley,et al.  Extreme value analysis and the study of climate change , 2009 .

[43]  Tom M. L. Wigley,et al.  The effect of changing climate on the frequency of absolute extreme events , 2009 .

[44]  Comparisons of Weibull- and exponential-based partial duration stochastic flood models , 1993 .

[45]  P. K. Bhunya,et al.  Flood analysis using negative binomial and Generalized Pareto models in partial duration series (PDS) , 2013 .

[46]  Sylvie Parey,et al.  Trends and climate evolution: Statistical approach for very high temperatures in France , 2007 .

[47]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[48]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[49]  M. Parlange,et al.  Statistics of extremes in hydrology , 2002 .

[50]  J. Salas,et al.  Revisiting the Concepts of Return Period and Risk for Nonstationary Hydrologic Extreme Events , 2014 .

[51]  Bernard Bobée,et al.  Towards operational guidelines for over-threshold modeling , 1999 .

[52]  P. K. Bhunya,et al.  Flood analysis using generalized logistic models in partial duration series , 2012 .

[53]  Francesco Serinaldi.  On the relationship between the index of dispersion and Allan factor and their power for testing the Poisson assumption , 2013, Stochastic Environmental Research and Risk Assessment.

[54]  H. Madsen,et al.  Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events: 1. At‐site modeling , 1997 .

[55]  E. Zelenhasić.  Theoretical probability distributions for flood peaks , 2007 .

[56]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[57]  H. Akaike.  A new look at the statistical model identification , 1974 .

[58]  Graham A. Tobin,et al.  Applied hydrology, McGraw-Hill series in water resources and environmental engineering; book review , 1989 .

[59]  A. Gupta,et al.  A Bayesian Approach to , 1997 .

[60]  Xavier Kergadallan,et al.  A two-step framework for over-threshold modelling of environmental extremes , 2014 .

[61]  R. Katz,et al.  Non-stationary extreme value analysis in a changing climate , 2014, Climatic Change.

[62]  Philip J. Ward,et al.  Annual flood sensitivities to El Niño–Southern Oscillation at the global scale , 2013 .

[63]  Jose D. Salas,et al.  Quantifying the Uncertainty of Design Floods under Nonstationary Conditions , 2014 .

[64]  D. Dacunha-Castelle,et al.  Different ways to compute temperature return levels in the climate change context , 2010 .

[65]  Félix Francés,et al.  Non-stationary flood frequency analysis in continental Spanish rivers, using climate and reservoir indices as external covariates , 2013 .

[66]  Daniel Cooley,et al.  Return Periods and Return Levels Under Climate Change , 2013 .

[67]  S. Wood.  Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models , 2011 .

[68]  Anthony C. Davison,et al.  Local likelihood smoothing of sample extremes , 2000 .

[69]  Upmanu Lall,et al.  Statistical Prediction of ENSO from Subsurface Sea Temperature Using a Nonlinear Dimensionality Reduction , 2009, Journal of Climate.

[70]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[71]  C. Cunnane,et al.  A note on the Poisson assumption in partial duration series models , 1979 .

[72]  P. McCullagh,et al.  Monograph on Statistics and Applied Probability , 1989 .

[73]  R. Clarke,et al.  Likelihood-based confidence intervals for estimating floods with given return periods , 1993 .

[74]  A. Davison,et al.  Generalized additive modelling of sample extremes , 2005 .

[75]  Henrik Madsen,et al.  Prediction in partial duration series with generalized pareto‐distributed exceedances , 1992 .

[76]  William Kirby.  On the Random Occurrence of Major Floods , 1969 .