California Earthquake Rupture Model Satisfying
Accepted Scaling Laws
Abstract
We are constructing an earthquake source model for the UCERF3
project. The basic philosophy is to start with the things we know best:
the magnitude distribution for the whole state, Omori’s law, total
moment rate, distribution of moment rate on major faults, the
instrumentally recorded earthquake catalog, long-term strain rate,
historic earthquake catalog, and paleo-event catalog, in that order. The
model is expressed as earthquake rate per unit area, time, magnitude,
and focal mechanism direction, with algorithms to simulate “realized”
earthquakes at random. The simulated earthquakes are described as
spatially tapered ruptures on hypothetical rectangular fault planes with
length, width, depth, strike, dip, and rake specified. Earthquake locations
and orientations will approximate those of mapped faults based on
empirical studies of actual earthquakes. The simulation scheme will
allow temporal and spatial clustering according to the Critical Branching
Model of Kagan et al. [2007] We would identify testable features of the
models and devise quantitative prospective and/or retrospective tests as
appropriate. We include a rule associating earthquakes with mapped
faults based on the proximity of their hypocenters to those faults. In that
way we can model fault slip rate and paleoseismic dates and
displacements.
Conclusions
1. By starting with magnitude distribution, we assure agreement with observed
statewide distribution.
2. Length, width, and slip scaling assumes self similarity: constant ratio of length,
width, and slip; all proportional to cube root of moment. Area scaling
consistent with 2008 Working Group model.
(SCEC 2010, 1-129)
David Jackson, Yan Kagan and Qi Wang
Department of Earth and Space Sciences, University of California Los Angeles
3. Weighting procedure puts largest earthquakes near major faults, with focal
mechanisms consistent with fault orientation.
4. Allows for rare surprises: large off-fault earthquakes, earthquakes over
magnitude 8.
5. Simple; no explicit stress calculation, no explicit fault-to-fault jumps. But
rupture length may exceed fault length.
Procedure
1. Choose random magnitude from Tapered Gutenberg-Richter
(Kagan) magnitude with corner magnitude 8.2.

2. Assign average length, width, and slip from assumed scaling
relationship.
3. Choose random hypocenter location from normalized probability
map
a. Weighted version two maps: one based on past earthquakes,
another on fault moment rates.
b. Weighting is linearly proportional magnitude: 100% earthquake
map at m=6.5; 100% fault map at m=8.5.
4. Choose random focal mechanism from normalized probability map
a. Weighted version two maps: one based on past earthquakes,
another on fault moment rates.
b. Weighting is linearly proportional magnitude: 100%
earthquake
map at m=6.5; 100% fault map at m=8.5.
Figure 7. Magnitude distribution of simulated earthquakes.
Figure 1. Extended sources representing large earthquakes in
California
Figure 2. Earthquake probability based on smoothed seismicity
Figure 3. Earthquake probability based on faults
Simple scaling model for length, width, and slip VS. magnitude
Kagan [2002] studied aftershock zone length vs moment for large shallow
earthquakes in the CMT catalog. This study used the most accurate,
comprehensive, and internally consistent data available. He concluded that length
is proportional to the cube root of moment, which implies that width and slip scale
the same way. Otherwise, one of them increases less strongly with moment and
the other more strongly. For either that would pose the problem of "inverse
saturation.“
5. Resolve fault plane – auxiliary plane ambiguity based on nearby
faults.
Assuming the self=similarity implied by the Kagan result, we adopted the
following forms for average length (L), average down-dip width (W) and average
slip (D) as a function of moment magnitude (m):
Data
Log10(L) = a + 0.5*m L in km; a = -1.65 from Kagan [2002]
Log10(D) = c + 0.5*m D in m; c = -3.50 from Wells and Coppersmith [1994]
Log10(W) = b + 0.5*m W in km; b = -2.55 from m*L*W*D = 10^(1.5*m+9)
The earthquake catalog used for seismicity calculation covers 1800 to
2009 with minimum magnitude 4.7 and maximum depth 30 kilometers.
This catalog was compiled by us from several previous catalogs. We
selected the most reliable location and magnitude from the catalogs, and
used regression relations to estimate the moment magnitude for each
earthquake. Compared to previous catalogs, ours is more complete and
includes more accurate magnitudes and hypocenter locations.
The fault information is derived from the SCEC Community Fault
Model, modified for the Uniform California Earthquake Rupture
Forecast, and provided to us by Kevin Milner. For distance from fault
calculations we simplified faults to line sources, at a depth of 7.5km in
most cases.
Figure 4. Smoothed focal mechanisms based on seismicity
Figure 5. Smoothed focal mechanisms based on faults
Figure 6. Simulated earthquakes and their ruptures. Rectangles
represent ruptures of earthquake larger than 7.5. Single lines show
the ruptures of other events.
We took m to be 5*1010 Nm. The sum a + b = -4.2 represents the area scaling, and
it coincidentally equals the value used in the “Ellsworth B” magnitude-area
relationship. The results:
Moment
Magnitude
Length
Width
Slip
Moment
Area
6.5
39.81
5.02
0.56
5.62E+18
200
7
70.79
8.93
1
3.16E+19
632
7.5
125.89
15.89
1.78
1.78E+20
2000
8
223.87
28.25
3.16
1.00E+21
6324
8.25
298.54
37.67
4.22
2.37E+21
11246
8.5
398.11
50.23
5.62
5.62E+21
19999
9
707.95
89.33
10
3.16E+22
63241
9.5
1258.93
158.85
17.78
1.78E+23
199986