Avalanche risk evaluation and protective dam optimal design using extreme value statistics

In snow avalanche long-term forecasting, existing risk-based methods remain difficult to use in a real engineering context. In this work, we expand a quasi analytical decisional model to obtain simple formulae to quantify risk and to perform the optimal design of an avalanche dam in a quick and efficient way. Specifically, the exponential runout model is replaced by the Generalized Pareto distribution (GPD), which has theoretical justifications that promote its use for modelling the different possible runout tail behaviours. Regarding the defence structure/flow interaction, a simple law based on kinetic energy dissipation is compared with a law based on the volume stored upstream of the dam, whose flexibility allows us to cope with various types of snow. We show how a detailed sensitivity study can be conducted, leading to intervals and bounds for risk estimates and optimal design values. Application to a typical case study from the French Alps, highlights potential operational difficulties and how they can be tackled. For instance, the highest sensitivity to the runout tail type and interaction law is found at abscissas of legal importance for hazard zoning (return periods of 10-1000 a), a crucial result for practical purposes.

[1]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[2]  D. Dantzig.  Economic decision problems for flood prevention , 1956 .

[3]  Howard Raiffa,et al.  Decision analysis: introductory lectures on choices under uncertainty. 1968. , 1969, M.D.Computing.

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

[5]  K. Lied,et al.  Empirical Calculations of Snow–Avalanche Run–out Distance Based on Topographic Parameters , 1980 .

[6]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[7]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[8]  David M. McClung,et al.  Statistical and geometrical definition of snow avalanche runout , 1987 .

[9]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[10]  Christian Wilhelm.  Wirtschaftlichkeit im Lawinenschutz , 1996 .

[11]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[12]  Christopher J. Keylock,et al.  Avalanche risk mapping by simulation , 1999 .

[13]  Kristján Jónasson.  Estimation of avalanche risk , 1999 .

[14]  B. Salm,et al.  Calculating dense-snow avalanche runout using a Voellmy-fluid model with active/passive longitudinal straining , 1999, Journal of Glaciology.

[15]  David M. McClung.  Extreme avalanche runout in space and time , 2000 .

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

[17]  Christopher J. Keylock,et al.  Snow avalanche impact pressure: vulnerability relations for use in risk assessment , 2001 .

[18]  D. Mcclung,et al.  Extreme avalanche runout: a comparison of empirical models , 2001 .

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

[20]  F. Savi,et al.  Effects of Release Conditions Uncertainty on Avalanche Hazard Mapping , 2002 .

[21]  S. Margreth,et al.  Winter opening of high alpine pass roads—analysis and case studies from the Swiss Alps , 2003 .

[22]  Eric Parent,et al.  Encoding prior experts judgments to improve risk analysis of extreme hydrological events via POT modeling , 2003 .

[23]  J. Bernier,et al.  Décisions et comportement des décideurs face au risque hydrologique , 2003 .

[24]  Jacques Bernier,et al.  Bayesian POT modeling for historical data , 2003 .

[25]  T. Faug,et al.  Varying Dam Height to Shorten the Run-Out of Dense Avalanche Flows: Developing a Scaling Law from Laboratory Experiments , 2003 .

[26]  T. Faug,et al.  An equation for spreading length, center of mass and maximum run-out shortenings of dense avalanche flows by vertical obstacles , 2004 .

[27]  Christophe Abraham,et al.  Asymptotic global robustness in bayesian decision theory , 2004, math/0410074.

[28]  Mohamed Naaim,et al.  Dense snow avalanche modeling: flow, erosion, deposition and obstacle effects , 2004 .

[29]  F. Savi,et al.  Risk assessment in avalanche-prone areas , 2004, Annals of Glaciology.

[30]  Kristján Jónasson,et al.  Avalanche hazard zoning in Iceland based on individual risk , 2004, Annals of Glaciology.

[31]  Simulation sur modèle réduit de l'influence d'un obstacle sur un écoulement à surface libre : application aux ouvrages de protection contre les avalanches de neige , 2004 .

[32]  Rudolf Sailer,et al.  Empirical Estimate Of Vulnerability RelationsFor Use In Snow Avalanche Risk Assessment , 2004 .

[33]  C. Ancey,et al.  Towards a conceptual approach to predetermining long-return-period avalanche run-out distance , 2004 .

[34]  Ian Jordaan.  Decisions under Uncertainty: Probabilistic Analysis for Engineering Decisions , 2005 .

[35]  Christopher J. Keylock,et al.  An alternative form for the statistical distribution of extreme avalanche runout distances , 2005 .

[36]  Sven Fuchs,et al.  The net benefit of public expenditures on avalanche defence structures in the municipality of Davos , Switzerland , 2005 .

[37]  Andreas Paul Zischg,et al.  The long-term development of avalanche risk in settlements considering the temporal variability of damage potential , 2005 .

[38]  Nicolas Eckert,et al.  Revisiting statistical–topographical methods for avalanche predetermination: Bayesian modelling for runout distance predictive distribution , 2007 .

[39]  Michael Bründl,et al.  Avalanche Hazard Mitigation Strategies Assessed by Cost Effectiveness Analyses and Cost Benefit Analyses—evidence from Davos, Switzerland , 2007 .

[40]  Ian Owens,et al.  Modified avalanche risk equations to account for waiting traffic on avalanche prone roads , 2008 .

[41]  Peter Gauer,et al.  Overrun length of avalanches overtopping catching dams: Cross‐comparison of small‐scale laboratory experiments and observations from full‐scale avalanches , 2008 .

[42]  Nicolas Eckert,et al.  Optimal design under uncertainty of a passive defense structure against snow avalanches: from a general Bayesian framework to a simple analytical model , 2008 .

[43]  M. Barbolini,et al.  Snow avalanche risk assessment and mapping : A new method based on a combination of statistical analysis, avalanche dynamics simulation and empirically-based vulnerability relations integrated in a GIS platform , 2008 .

[44]  Michael Bründl,et al.  The risk concept and its application in natural hazard risk management in Switzerland , 2009 .

[45]  Nicolas Eckert,et al.  Bayesian optimal design of an avalanche dam using a multivariate numerical avalanche model , 2009 .

[46]  Jakob Rhyner,et al.  Dealing with the White Death: Avalanche Risk Management for Traffic Routes , 2009, Risk analysis : an official publication of the Society for Risk Analysis.

[47]  David Bertrand,et al.  Physical vulnerability of reinforced concrete buildings impacted by snow avalanches , 2010 .

[48]  Bruno Merz,et al.  Review article "Assessment of economic flood damage" , 2010 .

[49]  N. Eckert,et al.  Long-term avalanche hazard assessment with a Bayesian depth-averaged propagation model , 2010, Journal of Glaciology.

[50]  N. Eckert,et al.  Cross-comparison of meteorological and avalanche data for characterising avalanche cycles: The example of December 2008 in the eastern part of the French Alps , 2010 .

[51]  P. Gauer,et al.  Can we learn more from the data underlying the statistical α–β model with respect to the dynamical behavior of avalanches? , 2010 .

[52]  Jordi Corominas,et al.  Vulnerability of simple reinforced concrete buildings to damage by rockfalls , 2010 .

[53]  Stefan Margreth,et al.  Effectiveness of mitigation measures against natural hazards , 2010 .

[54]  Nicolas Eckert,et al.  Return period calculation and passive structure design at the Taconnaz avalanche path, France , 2010, Annals of Glaciology.

[55]  Anthony C. Davison,et al.  Spatial modeling of extreme snow depth , 2011, 1111.7091.

[56]  N. Eckert,et al.  Using spatial and spatial-extreme statistics to characterize snow avalanche cycles , 2011 .

[57]  T. Glade,et al.  Physical vulnerability assessment for alpine hazards: state of the art and future needs , 2011 .

[59]  N. Eckert,et al.  Relative influence of mechanical and meteorological factors on avalanche release depth distributions: An application to French Alps , 2012 .

[60]  Nicolas Eckert,et al.  Quantitative risk and optimal design approaches in the snow avalanche field: Review and extensions , 2012 .

[61]  N. Gilardi,et al.  Network design for heavy rainfall analysis , 2013 .

[62]  Nicolas Eckert,et al.  A reliability assessment of physical vulnerability of reinforced concrete walls loaded by snow avalanches , 2013 .

[63]  N. Eckert,et al.  Dense avalanche friction coefficients: influence of physical properties of snow , 2013, Journal of Glaciology.

[64]  N. Eckert,et al.  Mapping extreme snowfalls in the French Alps using max‐stable processes , 2013 .

[65]  H. Joe,et al.  A Bayesian extreme value analysis of debris flows , 2013 .

[66]  G. McMillan.  On reliability. , 2014, Ear and hearing.

[67]  Nicolas Eckert,et al.  Sensitivity of avalanche risk to vulnerability relations , 2014 .

[68]  N. Eckert,et al.  Validation of extreme snow avalanches and related return periods derived from a statistical-dynamical model using tree-ring techniques , 2014 .

[69]  E. Xoplaki,et al.  A fast nonparametric spatio‐temporal regression scheme for generalized Pareto distributed heavy precipitation , 2014 .

[70]  P. Naveau,et al.  A limiting distribution for maxima of discrete stationary triangular arrays with an application to risk due to avalanches , 2016 .

[71]  P. O. Boks,et al.  Empirical Calculations of Snow–Avalanche Run–out Distance Based on Topographic Parameters , 1980, Journal of Glaciology.