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)
Graduate Student Jeff
Reissman
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
Location
Actual Locations
Synthetic Locations
CA Region, 1980-2000
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
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)

M t 
( M )     e
 M 

Mt  M
Mc
= the probability of an event with seismic moment larger than (or equal to) M
Mt = lower threshold seismic moment
Mc = ÒcornerÓmoment (parameter that controls the roll-off)
2
  b-value
3
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.

M t 
r     e
 M 
Mt  M
Mc
Magnitude

M t 
r     e
 M 
Mt  M
Mc
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)

M t 
r     e
 M 


Mt  M
Mc
Mt
Mc
Mt e
r   M
M e Mc
Magnitude

M e
Me


M
Mc
M
Mc

Mt  e

r

Mt  e
Mt
Mc
Mt
M c
1
r

Mt
M c
M
M
Mt  e
M c
e 
1
M c
M c r 
Magnitude
M
Now, with y 
, if x = yey, then y = W(x):
M c



Mt

Mc 
M
M

e
t

 W 
1
 M r  
M c
c


Mt


M c
M t  e 

M  M c W
 M r 1 
c


101.5( mt mc )

101.5(mt mc )  e 
2
2
m  log  mc  logW 
1

3
3

r






Magnitude
Halley’s Method was used (similar to
Newton’s Method)
wj e
wj 1  wj 
wj
e (wj  1) 

wj
x
(wj  2)(wj e
wj
 x)
2wj  2
For x ≥ 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
California, 1980-2000
Earthquakes occur randomly in time
Aftershocks occur after large EQs
Aftershocks decay over time
Time
Epidemic-Type Aftershock Sequence
(ETAS) model
 (mi m0 )
K 10
   
p
ti t (t  t i c)
 = (daily) rate of earthquakes at time t – ti after the i th event with magnitude mi
 = (daily) rate of background seismicity
m0 = lower magnitude threshold below which no aftershocks are generated
 = productivity constant
c = time offset constant
p = exponent of decay
K = GR constant

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, 
Total Eqs in CA
M≥3
Time
 (mi m0 )
K 10
   
p
ti t (t  t i c)
An estimate for  was calculated
Tried to fit the other parameters
K = [0.04, 0.09]
  = [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
Time vs Magnitude
For background EQs
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 
Create location probability map
Running the Program
Create background earthquakes
New  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
Synthetic Catalog
1984 - 2003
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.