Avalanche Crown-Depth Distributions
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Avalanche Crown-Depth DistributionsEdward (Ned) Bair1, Jeff Dozier1, and Karl Birkeland2
Photo courtesy of Center for Snow and Avalanche Studies, Silverton, CO
Photo courtesy of Mammoth Mountain Ski Patrol
1 Donald Bren School of Environmental Science and ManagementUniversity of California – Santa Barbara2 U.S.D.A. Forest Service National Avalanche Center
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We are surrounded by high variability data
Earthquakes
Forest Fires
Stock Markets
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What is a power law?
Linear scale Log scale
Normal distribution Power law distribution
Log scale
Linear scale
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Power Laws: More Normal Than Normal
Why? • Strong invariance properties specifically, for crown depths, maximization.
•Power laws are the most parsimonious model for high variability data (Willinger et al 2004)
W. Willinger et al., “More "normal" than normal: scaling distributions and complex systems,” Proceedings of the 2004 Simulation Conference, p. 141. doi: 10.1109/WSC.2004.1371310
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Self Organized Criticality (SOC)
• Natural systems spontaneously organize into self-sustaining critical states.
Highly Optimized Tolerance (HOT)
• Systems are robust to common perturbations, but fragile to rare events.
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Debate in snow science on power laws
• Do avalanches follow power law or lognormal distributions?
• What is the generating mechanism?
• Is universal?
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Why is this important?•May answer why some avalanches are much deeper than others.
•Paths with low are stubborn!
•A universal exponent would mean all paths have the same proportion of large to small avalanches.
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Mammoth Mountain Ski Patrol (1968-2008)3,106 crowns > 1/3 meter)
Westwide Avalanche Network (1968-1995)61,261 crowns > 1/3 meter from 29 avalanche areas
Data
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dMethods
•Maximum likelihood
•3 tests of significance: KS, 2,rank-sum
•Ranked by probability of fit
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Distributions (excluding log normal)
Truncated power law:
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Mammoth Patrol Crowns (N=3,106, =3.3-3.5)
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Lex parsimoniae:All else being equal, the
simplest explanation is best.
• The generalized extreme value, and its special case, the Fréchet, provide the best fit.
• The simplest generating mechanism is a collection of maxima.
• The scaling exponent ( varies significantly with path and area.
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Smoothing
We had to smooth the WAN data:
1) We assume WAN data are rounded uniformly ± 0.5 feet (i.e. a 2ft crown is between 1.5 ft and 2.5 ft).
2) We add a uniform random number on the unit interval (0 to 1).
3) We then subtract 0.5 ft, and convert to meters.
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WAN Alyeska Crowns (N=4,562, =3.5-3.7)
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WAN Snowbird Crowns (N=3,704, =3.6-3.8)
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WAN Squaw Valley Crowns (N=3,926, =3.9-4.1)
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WAN Alpine Meadows Crowns (N=3,435, =4.2-4.4)
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Post Control Accident on a Stubborn Path
Mammoth Mountain, CA Path: Climax 1:52pm 4/17/06.
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4/17/2006 post-control video
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Fréchet exponent () for selected Mammoth paths
Climax, =2.5, is the most stubborn of the 34 paths plotted here
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Fréchet MDA requires the parent distribution to be scaling
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AcknowledgmentsWalter Rosenthal
National Science Foundation
Mammoth Mountain Ski Patrol
USFS and Know Williams