[CEUS-earthquake-hazards] alternative hazard maps

[Chris Harold Cramer (ccramer)]

Seth,

As you know it is difficult to make statistical inferences from small sample
populations. Estimates of mean recurrence interval may have some value from
small (less than 10) samples with some reliability, but estimates of
variance hold little or no meaning from such small populations.  The
paleoseismic data of Tuttle et al., 2002 provide only two recurrence
interval estimates (~900 AD to ~1450 AD and ~1450 AD to 1811-1812) and a
mean recurrence interval of about 500 years.  Adding another recurrence
interval (~300 AD to ~ 900 AD) as suggested by Tuttle et al., 2005 only
changes the estimate of the mean to about 550 years (only a 10% difference).

However, estimates of variance (standard deviation) are essentially
meaningless from 2 or 3 samples of recurrence interval.  This can be seen
from some longer sequences of characteristic earthquakes. A sequence of
tsunami deposits at Eureka, CA, provide nine recurrence intervals with a
natural lognormal standard deviation of 0.43 for the rupture of the
southernmost Cascadia subduction zone (Cramer et al., 2000, p 6).  A
sequence of similar deposits at Willapa Bay, WA, provide seven recurrence
intervals with a natural lognormal standard deviation of 0.58 for the
rupture of the northern Cascadia subduction zone (Petersen et al., 2002, p
2154). In Southern California at Pallet Creek on the San Andreas fault, a
sequence of 24 recurrence intervals indicates a natural lognormal standard
deviation of 0.77 (Cramer et al., 2000, p 6).  Subsets of 2 or 3 samples
from these sequences could easily indicate much more regular recurrence
intervals with smaller lognormal standard deviations on the order of half of
the values from the longer sequences.  Thus longer sequences of recurrence
intervals show more variability than very short sequences.  This is why a
world-wide average lognormal standard deviation of 0.5 should be used for a
few recurrence interval estimates such as are available from New Madrid
paleoseismic results.

Unfortunately, your lognormal conditional probabilities as posted are
incorrectly calculated, judging from your posted spreadsheet.  The
recurrence interval mean and standard deviation cannot be used to estimate
the lognormal conditional probability.  The coefficient of variation
(standard deviation divided by the mean) of the recurrence intervals is not
the same as the lognormal standard deviation for a lognormal distribution of
those recurrence intervals.  Further, the mean of a lognormal distribution
of recurrence intervals is equal to the median of those recurrence intervals
and not the mean.  Please refer to Benjamin and Cornell, 1970, p 265,
equation 3.3.25 for the proper way of calculating a conditional probability
for a lognormal distribution (or see Cramer et al., 2000, p 3, bottom of
first column).

I think we both can agree that a negative recurrence interval (the next
earthquake in a sequence occurring before its preceding earthquake) is a non
sequitur. Yet, as an alternative, you have used a Gaussian distribution of
recurrence times, which allows for the possibility of negative recurrence
intervals, to calculate conditional probabilities.   Because negative
recurrence intervals are possible, a Gaussian distribution is not an
acceptable model for recurrence intervals.  That is why lognormal
distributions are used instead.

Using a lognormal model of recurrence times, a Monte Carlo sampling of the
uncertainty in the New Madrid paleoseismic dates (Cramer, 2001, p 255-256),
and a world-wide average of 0.5 for the lognormal standard deviation in
recurrence intervals, a 7% conditional probability for the next fifty years
is obtained for the New Madrid seismic zone.  This is not significantly
different in seismic hazard from the Poisson probability for New Madrid of
10% in 50 years.  Using the same approach for the Charleston, SC, seismic
zone yields a 2% in the next 50 years condition probability, which is
significantly different from the 10% in 50 year Poisson probability for
Charleston.  But the current background seismicity in the Charleston area
still yields some seismic hazard in that region.  So time dependent hazard
estimates yield much lower seismic hazard than time independent (Poisson)
estimates in the Charleston region, but not in the New Madrid region.

References:

Benjamin, J.R., and C.A. Cornell (1970). Probability, Statistics, and
Decision for Civil Engineers, McGraw-Hill Publishing Company, New York, 686
pp.

Cramer, C.H. (2001). A seismic hazard uncertainty analysis for the New
Madrid seismic zone, Engineering Geology 62, 251-266.

Cramer, C.H., M.D. Petersen, T. Cao, T.R. Toppozada, and M. Reichle (2000).
A time-dependent probabilistic seismic-hazard model for California, Bull.
Seism. Soc. Am. 90, 1-21.

Petersen, M.D., C.H. Cramer, and A.D. Frankel (2002).  Simulations of
seismic hazard for the Pacific Northwest of the United States from
earthquakes associated with the Cascadia subduction zone, Pure and Applied
Geophysics 159, 2147-2168.

Tuttle, M. P., E. S., Schweig, J. D. Sims, R. H. Lafferty, L. W. Wolf, and
M. L. Haynes, (2002). The earthquake potential of the New Madrid seismic
zone, Bulletin of the Seismological Society of America, 92, 2080-2089.

Tuttle, M. P., E. S., Schweig,  J. Campbell, P.M. Thomas, J. D. Sims, and R.
H. Lafferty (2005).  Evidence for New Madrid earthquakes in A.D. 300 and
2350 B.C., Seismological Research Letters 76, 489-501.


Chris Cramer
CERI, Univ. of Memphis