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Avalanche Crown-Depth Distributions

1 Donald Bren School of Environmental Science and Management University of California – Santa Barbara. 2 U.S.D.A. Forest Service National Avalanche Center. Avalanche Crown-Depth Distributions. Edward (Ned) Bair 1 , Jeff Dozier 1 , and Karl Birkeland 2.

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Avalanche Crown-Depth Distributions

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  1. 1 Donald Bren School of Environmental Science and Management University of California – Santa Barbara 2 U.S.D.A. Forest Service National Avalanche Center Avalanche Crown-Depth Distributions Edward (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

  2. We are surrounded by high variability data Forest Fires Earthquakes Stock Markets

  3. What is a power law? Normal distribution Power law distribution Linear scale Log scale Log scale Linear scale

  4. Power Laws: More Normal Than Normal • Power laws are the most parsimonious model for high variability data (Willinger et al 2004) • Why? • Strong invariance properties • specifically, for crown depths, maximization. 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

  5. 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.

  6. Debate in snow science on power laws • Do avalanches follow power law or lognormal distributions? • What is the generating mechanism? • Is a universal?

  7. Why is this important? • May answer why some avalanches are much deeper than others. • Paths with low a are stubborn! • A universal exponent would mean all paths have the same proportion of large to small avalanches.

  8. Data 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

  9. Methods d • Maximum likelihood • 3 tests of significance: • KS, c2,rank-sum • Ranked by probability of fit

  10. Distributions (excluding log normal) Truncated power law:

  11. Mammoth Patrol Crowns (N=3,106, a=3.3-3.5)

  12. 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 (a) varies significantly with path and area.

  13. 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.

  14. WAN Alyeska Crowns (N=4,562, a=3.5-3.7)

  15. WAN Snowbird Crowns (N=3,704, a=3.6-3.8)

  16. WAN Squaw Valley Crowns (N=3,926, a=3.9-4.1)

  17. WAN Alpine Meadows Crowns (N=3,435, a=4.2-4.4)

  18. Post Control Accident on a Stubborn Path Mammoth Mountain, CA Path: Climax 1:52pm 4/17/06.

  19. 4/17/2006 post-control video

  20. Fréchet exponent (a) for selected Mammoth paths Climax, a=2.5, is the most stubborn of the 34 paths plotted here

  21. Fréchet MDA requires the parent distribution to be scaling

  22. Acknowledgments Walter Rosenthal National Science Foundation Mammoth Mountain Ski Patrol USFS and Know Williams

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