SeismoMath!

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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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### 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
• 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”
Synthetic EQ Project
• Collaborated with Dr. David Bowman, Chair, Dept. of Geological Sciences at CSUF (my advisor)
• Worked in “The Field” at CSUF
• Used Macintosh computers
• Programmed in IDL
Earthquake Catalog
• Location
• Magnitude
• Time
• Depth
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
Location

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

Location

Actual Locations

CA Region, 1980-2000

Synthetic Locations

CA Region, 1980-2010

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

Background only

BG + Aftershocks

Magnitude
• Gutenberg-Richter (GR) Law

N = 10a - bm or log N = a - bm

Number of events (cumulative)

Global Catalog 1984 - 2003

Magnitude

Magnitude
• tapered GR distribution

[Kagan and Jackson, 2000; Kagan, 2002]

• has an exponential taper applied to the cumulative number of events (for higher magnitudes)
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.

Magnitude

To generate a magnitude from a random number r, we must solve this equation for m.

But how?!?!

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)

Magnitude

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

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
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
Time
• Earthquakes occur randomly in time
• Aftershocks occur after large EQs
• Aftershocks decay over time

California, 1980-2000

Time
• Epidemic-Type Aftershock Sequence (ETAS) model

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, 
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
Time

Time vs Magnitude

For background EQs

Time vs Magnitude

For All Synthetic EQs

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
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
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
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
Global Synthetic Catalog

Magnitude Distributions

Real Catalog

1984 - 2003

Synthetic Catalog

1980 - 2010

Global Synthetic Catalog

Time vs Magnitude

Real Catalog

Synthetic Catalog

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)

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.