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SeismoMath!

SeismoMath!. Math Colloquium #7 Nancy Ikeda April 13, 2010. Problem. Q: How can earthquake forecasting models be tested? Most often, researchers have to just wait to see if their predicted earthquake occurs. Solution. A: Use a Monte Carlo simulation

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SeismoMath!

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  1. SeismoMath! Math Colloquium #7 Nancy Ikeda April 13, 2010

  2. Problem • Q: How can earthquake forecasting models be tested? • Most often, researchers have to just wait to see if their predicted earthquake occurs

  3. Solution • A: Use a Monte Carlo simulation • Create a realistic synthetic earthquake catalog to test the forecasting method • The synthetic catalog represents the null hypothesis: there is nothing “causing” the earthquakes and they occur “randomly”

  4. Synthetic EQ Project • Collaborated with Dr. David Bowman, Chair, Dept. of Geological Sciences at CSUF (my advisor) • Graduate Student Jeff Reissman • Worked in “The Field” at CSUF • Used Macintosh computers • Programmed in IDL

  5. Earthquake Catalog • Location • Magnitude • Time • Depth

  6. Location • Must be in the form of latitude and longitude • Should be located near a tectonic plate boundary • Aftershocks should be located near the mainshock

  7. Location http://mineralsciences.si.edu/tdpmap/

  8. Location

  9. Location Actual Locations CA Region, 1980-2000 Synthetic Locations CA Region, 1980-2010

  10. Location • Probability map • 5 km x 5 km cells • Find the number of EQ in each cell with some aftershocks removed (declustered) • Use random number generator to select a cell • Randomly offset the earthquake from the center of the cell

  11. Location • Aftershocks are located near the mainshock • They are placed a random distance and direction from the mainshock • Distance is based on mainshock magnitude

  12. Location Background only BG + Aftershocks

  13. Magnitude • Gutenberg-Richter (GR) Law N = 10a - bm or log N = a - bm Number of events (cumulative) Global Catalog 1984 - 2003 Magnitude

  14. Magnitude • tapered GR distribution [Kagan and Jackson, 2000; Kagan, 2002] • has an exponential taper applied to the cumulative number of events (for higher magnitudes)

  15. Magnitude Used Felzer et al’s [2002] inverse transform technique to generate a random magnitude: Since Kagan’s formula is in the form of a cumulative distribution, it follows that it will take on values between 0 and 1.

  16. Magnitude To generate a magnitude from a random number r, we must solve this equation for m. But how?!?!

  17. Magnitude • Use the Lambert W function, W(x) • It is the inverse of the function f(x) = x·ex Thus, for x = yey, then y = W(x)

  18. Magnitude

  19. Magnitude Now, with , if x = yey, then y = W(x):

  20. Magnitude • Halley’s Method was used (similar to Newton’s Method) • Forx ≥ e, W(x) can be approximated by ln x – ln(ln x) • For x < e, an approximation of the function for argument values near 0 had to be found

  21. Magnitude • fit a quartic curve to the Lambert W function • y = -0.0285x4 + 0.1892x3 – 0.508x2 + 0.9138x • R2 = 0.99995 • 5 iterations • Then plug into magnitude formula

  22. Time • Earthquakes occur randomly in time • Aftershocks occur after large EQs • Aftershocks decay over time California, 1980-2000

  23. Time • Epidemic-Type Aftershock Sequence (ETAS) model

  24. Total Eqs in CA M ≥ 3 Time • To use the formula, time and magnitude have to be plugged in • All of the parameters had to be approximated also: K, , c, p, 

  25. Time • An estimate formwas calculated • Tried to fit the other parameters • K = [0.04, 0.09] • a = [0.4, 0.8] • C = 0.02 (about 30 minutes) • P = [1.5, 1.75] • Picked parameter values for a region • Each aftershock sequence has a new set of parameters based on selected regional parameters

  26. Time Time vs Magnitude For background EQs Time vs Magnitude For All Synthetic EQs

  27. Depth • found the average depth of events for a region • And the average depth of events in the 5 km x 5 km cell • Assigned events a depth based on the cell average, following a normal distribution • If a cell had no previous events, it was assigned the average depth for the region

  28. Running the Program • Load in file for real data (ANSS) • 1984 - 2003 • Minimum magnitude = 3.5 • Depth = 40 km • Load in region boundary data (including ETAS parameters) • Select earthquakes from a region • Estimate m • Create location probability map

  29. Running the Program • Create background earthquakes • New m is generated for each year • Use poissonian distribution for day of event • Assign random time on day • Assign location based on a-value map • Assign magnitude • Run ETAS on each background event • New ETAS parameters are generated for each background event • ETAS parameters are fixed for each aftershock sequence • Run daily to determine number of aftershocks per day • Assign aftershocks a time, location and magnitude

  30. Running the Program • Run ETAS on all aftershocks individually • New set of parameters are used again • This continues until the end of the catalog • Index the events • Create final catalog • Originally 40 years • Cut out the first 10 years • Cut out any events that happened after 40 years • Write events to a file

  31. Global Synthetic Catalog Magnitude Distributions Real Catalog 1984 - 2003 Synthetic Catalog 1980 - 2010

  32. Global Synthetic Catalog Time vs Magnitude Real Catalog Synthetic Catalog

  33. What’s left/next? • Use synthetic catalog to test the accelerating moment release (AMR) method • Write a paper on the use of the Lambert W function for generating magnitudes • Find even more realistic formulas and start over using Matlab (instead of IDL)

  34. References Corless, R.M., Gonnet, G. H., Hare, D.E.G., Jeffrey, D. J., and D.E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, vol. 5, p. 329-359, 1996. Felzer, K.R., Becker, T. W., Abercrombie, R. E., Ekstrom, G., and J. R. Rice, Triggering of the 1999 Mw 7.1 Hector Mine earthquake by aftershocks of the 1992 Mw 7.3 Landers earthquake, JGR, v. 107, B9, 2190, 2002. Helmstetter, A., and D. Sornette, Sub-critical and Super-critical Regimes in Epidemic Models of Earthquake Aftershocks, JGR, 107, B10, 2237, 2002. http://mathworld.wolfram.com/LambertW-Function.html http://mineralsciences.si.edu/tdpmap/ Kagan, Y. Y., Universality of the Seismic Moment-frequency Relation, Pure and Applied Geophysics, 155, p. 537-573, 1999. Kagan, Y. Y., and D. D. Jackson, Probabilistic earthquake forecasting, GJI, v. 143, p. 438-453, 2000. Kagan, Y. Y., Seismic moment distribution revisited: I. Statistical results, GJI, v. 148, p. 520-541, 2002. Ogata, Y., Seismicity Analysis through Point-process Modeling: A Review, Pure and Applied Geophysics, 155, p. 471-507, 1999. Ogata, Y., and J. Zhuang, Space-time ETAS models and an improved extension, Tectonophysics, 413, p. 13-23, 2006.

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