Extreme Earthquakes: Thoughts on Statistics and Physics

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

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

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. Switzerland?

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