Session: Probabilistic methods in avalanche hazard assessment Session moderator: Marco Uzielli Norwegian Geotechnical Institute
Session plan Title Presenter Affiliation A probabilistic framework for avalanche hazard assessment Climate statistics and triggering probability estimation using a fuzzy logic approach How do we take into account the release probability? Probabilistic characterization of the alpha-beta avalanche runout model Discussion Marco Uzielli Galina Ragulina Peter Gauer Marco Uzielli NGI NGI NGI NGI
A probabilistic framework for avalanche hazard assessment Marco Uzielli Norwegian Geotechnical Institute
Purpose of presentation Illustrates a preliminary proposal for the GISbased probabilistic spatial estimation and mapping of avalanche hazard
Scale of investigation Hazard is estimated individually for discretized areas of a site under investigation, e.g., for individual cells of a spatial grid into which the area may be subdivided.
Conceptual sequence Occurrence of triggering conditions (meteorological, climatic, etc.) Occurrence of triggering (not certain, even if conditions are present) Spatial propagation following triggering
From concept to method The quantitative procedure is based on conditional probabilities and on the total probability theorem, and relies on the availability of: a set of reference return periods, for which it may be of interest to estimate hazard; data series for potential avalanche triggering factors (meteorological, climatological, geographic, topographic, etc.); frequency-intensity relationships, quantifying the probability of attainment of a given level of intensity (and combinations thereof) at a given spatial location in the set of reference return periods; a triggering model, relating the magnitude of avalanche intensity at one spatial location to the occurrence of avalanche triggering at the same spatial location; a runout model, relating the occurrence of avalanche triggering at one spatial location and the impact (exposure) to the avalanche at all other target spatial locations. The triggering and runout models must be expressed in a probabilistic format to allow Monte Carlo simulation of triggering and runout, respectively.
General probabilistic framework n-th intensity level in the k-th source cell q-th return period P M j W k, I k n, T r q = P M j W k P W k I k n P I k n T r q P T r q impact on the j-th target cell triggering in the k-th source cell
Probabilistic framework: operational modules time-series analysis for triggering factors time-series analysis for triggering factors P M j W k, I k n, T r q = P M j W k P W k I k n P I k n T r q P T r q runout analysis scenarios definition
Cell-level to site-level approach Conditional probability formulation operates on a cell-to-cell basis Site-level estimates are obtained through convolution of cell-level outputs (total probability theorem) j-th target cell k-th source cell
Other possible output formats (examples) From the total probability theorem, the probability of impact on the j-th cell for the q- th return period T rq is : P M j T r q = N K n=1 k=1 P M j W k P W k I k n P I k n T r q P T r q Considering the set of Q reference return periods T r1,,t rq, the probability of exposure of the j-th target cell is given by the convolution of P(M j T rq ) over the set of return periods: P M j = Q N K q=1 n=1 k=1 P M j W k P W k I k n P I k n T r q P T r q
Main strengths Direct dialogue with GIS for enhanced mapping Quantitative estimation of hazard Quantitative inclusion of uncertainty in hazard factors and models Explicit quantification of spatial variability of hazard Flexibility to accommodate different methods for climatological and meteorological data analysis, triggering analysis and runout analysis
Possible future developments Refinement of triggering models Volume-specific runout models Definition of triggering areas through clustering methods 2-D and 3-D runout analysis (e.g., considering avalanche width and depth)
Climate statistics and triggering probability estimation using a fuzzy logic system Galina Ragulina Norwegian Geotechnical Institute
Meteorological and climatological elements of importance for avalanche triggering Forecasting Snow height (SA) Precipitation (RR) Wind velocity and direction (FF, DD) Temperature and its tendency (t, dt) Insolation Hazard mapping Expected maximum snow height per given time period (return period) Expected maximum precipitation per given time period (return period)
Fuzzy inference system Based on fuzzy logic Algorithm for transposing qualitative knowledge (including vagueness) into quantitative parameters Able to model complex systems Able to replicate human expert information and reasoning Able to accommodate quantitative and qualitative inputs Relies on user-input membership functions and rules Here, used to quantify avalanche triggering probability from climatic, meteorological, physical and geographical factors
Qualitative judgement and the parameter-range definitions Qualitative categories Very low Low Medium High Very high Negative/Positive Increased/Decreased Windward/Parallel/Leeward Range table (example) t, C dt, C SA, cm RR24, mm RR72, mm FF, m/s DD, ATP "top" -15 0.08 v.low lowest -40 0 highest -8 0.12 "top" -5 4 80 8 25 4 180 0.2 low lowest -12 0 0 0 0 0 100 0.08 highest -2 6 120 15 30 6 180 0.25 "top" 0 10 200 20 60 13 90 0.35 medium lowest -3 5 100 8 25 4 45 0.15 highest 3 15 250 35 100 17 135 0.55 "top" 2 15 500 50 90 20 0 0.5 high lowest 0 10 200 25 75 15 0 0.35 highest 20 30 1000 300 600 35 80 0.7 "top" 0.65 v.high lowest 0.5 highest 1
Qualitative judgement and the parameter-range definitions Membership functions (example)
Qualitative judgement: Rules for Fuzzy-system All possible scenaria based on categories of chosen parameters Assigning Avalanche Triggering Probability (ATP) category to each scenario Example: t dt Ddt SA RR24 RR72 FF DD ATP if medium medium positive high low low medium parallel high v.low low negative low medium medium medium windward low
Error management of expert -judgement in Fuzzysystem Many dependent on each other elements Large amount of rules Degraded boundaries between categories Inter-cancellation of eventual errors
Challenges and ideas for overcoming them Too heavy system t, dt, ddt, RR24, RR72, SA, FF, DD + Slope and forecasted values = > 10 parameters Each parameter is divided into 2-5 categories Over 60 000 rules Must be subdivided in several separate systems! How to divide the meteorological elements into groups which would still make sense?
Status now Fuzzy-system1: t, (dt+ddt), FF, DD, FF_f, DD_f, (dt+ddt)_f Fuzzy-system2: RR72, RR24, RR_f Fuzzysystem_Major: Impact1, Impact2, Impact3 Fuzzy-system3: SA, dsa_f, Slope Avalanche Triggering Probability
Statistical methods in avalanche hazard mapping: How to take into account the release probability? Peter Gauer Norwegian Geotechnical Institute, Oslo, Norway
Avalanche hazard Avalanche hazard is a combination of: precipitation (snow or rain) and wind snow pack conditions (probability of avalanche release) runout of the avalanche
Probability for avalanche release? Bakkehøi (1987) presented the Gaussian normal cumulative probability of measuring HNW3d at the day of an avalanche; that is the conditional probability P(HNW3d A) However, in the most cases, we are interested in probability of having an avalanche under the condition we observe HNW3d; That is the conditional probability P(A HNW3d) Bakkehøi 1987
Cumulative probability distribution of the conditional probabilities P(HNW 3d A) versus P(A HNW 3d )
Cumulative probability distribution of the conditional probabilities P(A HNW 3d ) for different regions and paths Avalanche probability/frequency P A = 0 P A HNW id P HNW id d(hnw) Expected annual avalanche frequency F(A) 2 4 OK with observations
Initial volume (??) An important input for avalanche models
Fracture height D rel HWL HN WL D rel Performance function G Rest Load < 0 => release HS (Photo J. Schweizer) Load = ρhs + HNW g cos φ sin φ Rest = c + μ ρ s HS + HNW g cos 2 φ + F i A
Monte-Carlo approach Performance function G Rest Load 0 Required are distributions for: HNW HS wl etc. Example of distributions
Comparison between observed and simulated P(A HNW 3d ) for the Ryggfonn path
Fracture height vs mean slope angle T r
Thank you Acknowledgment: The work was partially funded by: a NGI research grant from Oil and Energy Department (OED) through the Norwegian Water Resources and Energy Directorate (NVE)
Probabilistic characterization of the alpha-beta avalanche runout model Marco Uzielli and Sylfest Glimsdal Norwegian Geotechnical Institute
Runout modeling The goal of runout analysis in the context of the proposed framework is the quantification of the probability of an avalanche impacting a given cell assuming that it has been triggered in another cell Runout analysis is conducted by coupling probabilistic methods and GIS GIS provides, for every potential triggering cell, the most plausible runout cross-section based on geographic attributes (e.g., slope) Runout analysis is applied to each potential triggering (source) cell The runout analysis entails probabilistic simulation of the alpha-beta model
Alpha-beta model Predicts the extreme runout distance for an avalanche solely on the basis of topography b: average inclination of the avalanche path between the starting point and the 10 inclination along the path profile (found to be the most important significant topographic parameter) a: average inclination of the total average path (represents runout distance)
Source database Alpha beta model was developed empirically using correlation analysis based on the topographic parameterization (a vs. b) of a large set of Norwegian avalanche paths
Probabilistic vs. deterministic formulation The alpha-beta model is expressed probabilistically as α prob = α det ε in which: a det is a deterministic analytical model and e is a multiplicative transformation uncertainty factor e accounts for the "true" scatter in data points around the deterministic models, as well as for epistemic uncertainties in the sets of source parameters a and b. A power function is selected as deterministic model: α det = p 1 + p 2 β
Deterministic expression α det = p 1 + p 2 β Estimators for the deterministic coefficients (by generalized least squares regression): p 1 =2.014, p 2 =0.580, p 3 =1.105.
Uncertainty characterization: approach The characterization of the multiplicative transformation uncertainty factor e is pursued through a Markov Chain Monte Carlo computational Bayesian approach. It is assumed that e is a beta-distributed multiplicative transformation uncertainty factor. Gaussian priors were assumed for the parameters of the beta distribution Posterior distributions were obtained for the parameters of the beta distribution
MCMC Bayesian characterization Markov chains and PDFs of posterior distributions of the parameters of the beta-distributed transformation uncertainty factor
Transformation uncertainty factor: characterization PDF and CDF of the trasformation uncertainty factor (betadistributed)
Uncertainty characterization: outputs Functions corresponding to specific probabilities of non-exceedance and database values degree of conservatism
Probabilistic estimation of impact The probability of impact on a given cell is calculated through Monte Carlo simulation. To count the number of impact instances on each cell, runout distance is retrieved for each simulation instance. Subsequently, the results are sent to the GIS which relates distance to cells, and determines which cells are affected in each simulation instance. Impact probability is calculated as the frequentist ratio of the number of occupations to the total number of simulations flow direction k-th source cell
Software implementation The model computes the probability to be impacted by using: Digital terrain model (raster) Flow direction (raster) Slope angles (raster) Probability of triggering (fuzzy system) Based on Open Source/freeware: Written in Python Rasters prepared by GRASS GIS Flow direction 1-128
Computations For each potential source cell, we determine: Slide path (following the flow direction) a 0 Calculating a distribution of the run-out Holding track of influenced cells (number of occupations for each cell downslope) Calculating the frequentist impact probability Compiling the spatial distribution of probability of being impacted
Status and future developments Status Ready for testing the first version Based on freeware/open Source Can easily be included into ArcGIS Future developments Obtain volume-specific runout models (requires database) Fill in the gaps between the «fingers» (extend to 2-D and 3-D)