DATA NEEDS FOR IMPROVED SEISMIC
HAZARD ANALYSIS
John G. Anderson, James N. Brune, Abdolrasool
Anooshehpoor and Matthew D. Purvance
Seismological Laboratory and Department of Geological Sciences
University of Nevada
Reno, Nevada 89557
Figure 1. Map of part of southern California. The heavy line shows the approximate extent of
rupture of the 1857 Fort Tejon, California, earthquake, Mw=7.9. The large 4-pointed star
shows the location of Lovejoy Buttes (~34.60N, 117.85W). The Lovejoy Buttes are significant
for the presence of 10,000 year old, semi-precarious rocks that have not been toppled by
earthquakes. The inset photo (upper right) shows some of these rocks, with the San Andreas
fault in the background. The inset perspective view (lower left) views the region along the
direction of rupture propagation in the 1857 earthquake.
A
B
Figure 6. A) Illustration of a geometry where several stations all record the ground motion at
the same distance from a vertical strike-slip fault.
B) Peak acceleration results of a numerical experiment by Yuehua Zeng (personal
communication) in which the experiment in part A is implemented using a numerical scheme to
calculate synthetic seismograms (Zeng et al, 1994). The experiment was performed at several
distances, as given by the horizontal axis.
C) Estimated standard deviation at each distance.
Peak Acceleration (g)
Figure 7. Illustration of the difference between epistemic and aleatory uncertainty, based on
the numerical experiment in Figure 6. The green line shows the distribution of mean values of
peak acceleration over the set of equidistant stations from the fault. The blue line shows the
probability distribution for repeated recordings of the ground motion at a single site.
Precarious Rock at Lovejoy Buttes,
a
4
σ T =0.38 , mu=0.38g
Purvance probability of topplin g
GM prob density, d F /d x
2
0
0
1.5
0.5
1
1.5
b
2
2.5
3
Probability density of toppling, per event
Cumulative probability of toppling, per event
1
0.5
0
0
1
0.5
1
1.5
Peak Acceleration (g )
c
2
2.5
Probability of surviving (250 year repeat time)
0.5
0
3
Age of this rock
0
1000
2000
3000
4000
5000
6000
Age of Rock
7000
8000
9000
10000
Figure 8. A) Fragility curve for toppling a balanced rock with parameters typical of Lovejoy
Butte. The monotonic increasing curve gives the probability that the rock will topple as a
function of increasing peak acceleration in a suite of records with frequency content
appropriate to a magnitude 8 earthquake. This is a highly nonlinear problem, where the
outcome for any individual record is either toppling or not, and switches between the two
domains as peak acceleration increases. This curve is smoothed and approximately monotonic
due to summing the curves for a large suite of records. The peaked curve on this plot is the
probability density distribution for the peak acceleration based on the Sadigh et al (1997)
ground motion prediction equation. conditional on the earthquake happening.
B) The peaked line gives the probability density for the rock to topple, conditional on the
earthquake happening, obtained as the product of the two curves in part A. The monotonic
increasing line is the cumulative probability of toppling per event. Under the assumptions used
here, the overall probability of toppling is about 0.3 in each characteristic earthquake.
C) Probability of the rock surviving as a function of it’s age. With a probability of toppling of
about 50% per earthquake, the half life of an individual rock of this geometry is about 400
years, or two earthquake cycles. The bold vertical line shows the age of the rock.
Lovejoy Buttes Constraint
0.5
Boore et al., 1997
0.45
Abrahamson and Silva, 1997
0.4
Sadigh et al., 1997
0.35
0.3
Not Allowed
0.25
0.2
Allowed
0.15
t
0.1
0.25
1/2
t
=5000 year s
0.3
0.35
1/2
=2000 years
0.4
0.45
0.5
Mean Peak Acceleration (g)
Figure 9. One way of evaluating ground motion prediction equations for the specific case of
Lovejoy Buttes. The contours show two half-live lines in the peak acceleration – aleatory
uncertainty plane. Since the rocks at Lovejoy Buttes are over 10,000 years old, ground motion
prediction equations substantially to the right of the 2000 year half-life line are inconsistent.
Lovejoy Buttes Constraint
0.5
Boore et al., 1997
0.45
Abrahamson and Silva, 1997
0.4
Sadigh et al., 1997
0.35
0.3
Not Allowed
0.25
0.2
0.15
90%
0.1
0.25
50%
0.3
10%
0.35
Prob. to survive 10
0.4
4
years
0.45
0.5
Mean Peak Acceleration (g)
Figure 10. An alternative way of evaluating ground motion prediction equations. The
contours on the peak acceleration – aleatory uncertainty plane give the probability for a rock
with a certain geometry to survive 104 years.