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. 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