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

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 scale

Linear 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. 141. doi: 10.1109/WSC.2004.1371310

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 (1968-2008)3,106 crowns > 1/3 meter)

Westwide Avalanche Network (1968-1995)61,261 crowns > 1/3 meter from 29 avalanche areas

Data

dMethods

•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, =3.3-3.5)

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, =3.5-3.7)

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

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

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

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

AcknowledgmentsWalter Rosenthal

National Science Foundation

Mammoth Mountain Ski Patrol

USFS and Know Williams