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

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

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

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 doi: /WSC

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.

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

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.

Mammoth Mountain Ski Patrol ( ) 3,106 crowns > 1/3 meter ) Westwide Avalanche Network ( ) 61,261 crowns > 1/3 meter from 29 avalanche areas Data

d Methods Maximum likelihood 3 tests of significance: KS,  2,  rank-sum Ranked by probability of fit

Distributions (excluding log normal) Truncated power law:

Mammoth Patrol Crowns (N=3,106,  = )

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.

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.

WAN Alyeska Crowns (N=4,562,  = )

WAN Snowbird Crowns (N=3,704,  = )

WAN Squaw Valley Crowns (N=3,926,  = )

WAN Alpine Meadows Crowns (N=3,435,  = )

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

4/17/2006 post-control video

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

Fréchet MDA requires the parent distribution to be scaling

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