Extreme EQ

1. Extreme Earthquakes:
Thoughts on Statistics and Physics
Max Werner
29 April 2008
Extremes Meeting Lausanne

2. Magnitude Statistics
Relocated Hauksson Catalog of Southern California, 1984-2002
Gutenberg-Richter Law
b=1

3. Magnitude Physics
• Preferable to work with seismic moment, a measure of earthquake energy
(magnitude is a convention)
• Pdf of moment fit by power law with exponent 2/3
– If boxes are drawn around some “faults” (hard to define), other distributions may be relevant
(“characteristic earthquakes” as a bump in the tail)
• Average moment must be finite (only finite energy available for generating
earthquakes)  require change from pure power law!
– No obvious limit given by rupture physics, but there may be hints.
– [Are all earthquakes extreme (a continuous underlying stochastic process intermittently
escalates to produce observable quakes)?]
– But can a d.f. with infinite mean fit data in finite time window well?
• Where is the change from the power law?
– Do we (sometimes) observe it?
– Is the change point related to the thickness of the seismogenic zone?
– What is the relevant distribution beyond the change point?
– Is there a hard cut-off? Probably not.
– Evidence of differences in probability of large earthquakes between different tectonic zones

4. Magnitude Statistics
• Distributions
– Pure power law (ignore change-point)
– Truncated power law (ad-hoc)
– Exponential taper in density (gamma pdf)
– Exponential taper in cumulative df (“Kagan” df)
– Two-branch power law
– Others: Logarithmic taper, …
– EVT, GEV/GPD

5. 05/15/10

6. Parameter Estimation
• Methods:
– Maximum likelihood estimation
– Moment estimation
– Probability weighted
– Rank-ordering statistics
– Some simulation-based parameter uncertainty estimates (finite
sample)
– Last major AIC test (1999) suggests data does not warrant more than 2
parameter pdf
• But no uncertainties in data considered
– Only for traditional Gutenberg-Richter law (exponential magnitude df):
rounding and random error
• Some Bayesian approaches
• Usually requires “declustering” catalog to obtain independent
events

7. Tectonophysics & Geology?
• Estimate strain build-up from tectonic models
– Not all strain released seismically… (estimate of
proportion?)
– How accurate are the models?
• Some suggested scaling of magnitude with fault
length
– (“which fault can produce a M8 in Switzerland?”)
– Faults hard to define rigorously
– Rupture can jump faults, rupture many small ones
– Not all faults known and/or mapped

8. Some History
• Wadati (1932): power law d.f. of eq energies
• Ishimoto & Iida (1939): power law d.f. of amplitudes
• Gutenberg & Richter (1941, 1944): exponential d.f. of magnitudes
• First EVT paper Nordquist (1945)
• showed Gumbel approximates large magnitudes in California
• Aki (1965):
• MLE of pure exponential law (still used today)
• First major paper (Nature) Eppstein & Lomnitz (1966)
• derived Gumbel from Poisson process of exponential magnitude d.f.
• Knopoff & Kagan (1977):
• Require finite first moment
• Use full data sets for recurrence times (GR-law)
• Extreme value d.f.s give “unacceptable” uncertainties
• Problem with least squares fitting of Gumbel (bias in his plotting rule)
• Makjanic (1980, 1982):
• MLE of Gumbel and GEV and relation to GR law
• Dargahi-Noubary (1983, 1986, 1988):
• 1983: Confidence intervals based on de Haan (1981)
• 1986/1988:Excess modeling, GPD, POTs developed by Pickands (1975) (also see Davison, 1985, PhD!)
• Graphical estimation method based on Davison 1984
• Kijko (1983, 1988), Kijko & Dessokey (1987), Kijko & Sellevoll (1989, 1992), Kijko &
Graham (1998)

9. More History
• Pisarenko (1991), Pisarenko et al. (1996)
• Estimating hard cut-off, estimating bias
• Kagan (1991, 1993, 1997, 2002), Kagan & Schoenberg (2001) Bird & Kagan
(2004):
• Universality of the Gutenberg-Richter distribution, universality of exponent,
regional/tectonic variations of corner magnitude in exponential taper (“Kagan” d.f.)
• Pisarenko & Sornette (2003)
• MLE of GPD to tectonic zones
• Difference in power law exponents for mid-oceanic spreading ridges and subduction
zones (but see Bird & Kagan, 2004)
• Pisarenko & Sornette (2004)
• Hypothesis test for deviation from power law
• Simulation based significance levels
• GPD + tail (exponential or power law) (non-differentiable -> simulations)
• Need 1000 events to determine cross-over (only have a dozen)
• Estimated cross-over larger than seismogenic width…
• Pisarenko et a. (2007), Thompson et al. (2007) L-moments

10. Wish list
• Characterize tail of moment distribution
– Recovers power law in body
– Finite first moment
– “soft” cut-off
– Nb. parameters warranted by data (e.g. AIC)
– Keep all events (no declustering)
– Use a hierarchy of data sets (from quality to quantity)
– Full uncertainty characterization
• Data (random errors + rounding + missing events etc)
• Parameters (non-asymptotics, test MLE, ME, …)
• Bayesian Monte Carlo methods
– Compare or integrate results with
• Geological fault map & paleoseismic data
• Tectonic strain build-up
• Dynamical rupture physics