(P(R=rj |mi)), which is obtained by discretizing the curves
been widely used to incorporate scientific uncertainty in a
PSHA (Kulkarni et al., 1984; Youngs et al., 1985;
occur at closer distances for longer rupture lengths (larger
Coppersmith and Youngs, 1986; National Research
magnitudes) can be noted by comparing Figure E-2
Council, 1988; SSHAC, 1997). Figure E-4 shows an
(diagram b) with E-2 (diagram a). Note that the distance to
example of a logic tree used in a PSHA. Although only a
the earthquake rupture must be expressed in terms of the
few branches of the logic tree are shown, there may be
same definition of distance as used in the ground motion
many thousands of branches in the tree. Each path through
attenuation relationships. Typically, some form of closest
the tree to an end branch (on the right-hand side of the
distance to rupture definition is used for attenuation
Figure E-4) defines a set of parameters that are used to
relationships (variations in this definition include: closest
conduct a basic seismic hazard analysis for that path and
distance to rupture, closest distance to rupture of the
end branch using Equation E-2. Basic hazard analyses are
seismogenic zone (at some depth below ground surface),
carried out for each path. Each path also has an associated
closest horizontal distance to surface projection of rupture,
probability or weight that is determined by the product of
etc.).
the relative probabilities or weights assigned to the various
models and parameters along the path. (The relative
d. Ground Motion Exceedance Probability
probabilities or weights of the alternative models and
Distribution. The conditional probability of exceeding a
parameters are illustrated by the numbers in parentheses in
ground motion level for a certain earthquake magnitude
Figure E-4.) The basic hazard analysis results for all the
and distance, P(Z>z|mi,rj), is determined from the ground
paths are combined using the associated weights to arrive at
motion attenuation relationships selected for the site. As
best estimates (mean or median values) for the frequencies
noted in paragraph 3-4f of Chapter 3 and illustrated in
of exceedance of ground motions as well as uncertainty
Figure 3-11, attenuation relationships are available for
bands for the estimates. Through the approach of
response spectral values as well as for peak ground
incorporating scientific uncertainty, PSHA incorporates the
acceleration. Uncertainty in the median attenuation curves
alternative hypotheses and data interpretations that may
is incorporated, as illustrated in Figures 3-3, 3-4 and 3-11.
significantly affect the computed results. The display and
The function P(Z>z|mi,rj) is usually evaluated assuming
analysis of uncertainty in the seismic hazard is discussed in
that ground motion values are log-normally distributed
the following section.
about the median value; the calculation of this function is
E-4. Analysis Results.
illustrated in Figure E-3.
a. Basic Results. The basic results of a PSHA are
E-3. Treatment of Modeling and Parameter
Uncertainties in PSHA.
seismic hazard curves (curves of the amplitude of a ground
motion parameter at a site vs. frequency of exceedance).
The basic probability formulations in Equations E-1 and E-
An example of the typical form of results is illustrated in
2 incorporate the randomness of the physical process of
Figure E-5 for the parameter of peak ground acceleration.
A distribution of seismic hazard curves ranging from the 5th
earthquake generation and seismic wave propagation.
to the 95th percentile is shown. This distribution results
Although these formulations incorporate the inherent
uncertainty due to randomness, they do not incorporate
from the incorporation of scientific uncertainty in the
additional sources of uncertainty that may be associated
PSHA through the use of logic trees as discussed above.
Typically, the mean curve or median (50th percentile) curve
with the choice of particular models or model parameters.
For example, there could be uncertainty as to which ground
is used to obtain design parameters, while the various
motion attenuation relationship is most applicable to a site,
percentiles of the distribution are a measure of the
uncertainty as to whether an exponential or characteristic
uncertainty in the result. Note in Figure E-5 that the mean
earthquake recurrence model is most applicable,
curve lies above the median curve. This result is typical of
uncertainty in the geometry of earthquake sources,
seismic hazard analysis. In general, the mean curve rather
uncertainty in the values of maximum earthquake
than the median curve is the preferred measure of the
magnitude, uncertainty in earthquake recurrence
hazard results. The use of hazard curve results to develop
parameters, etc. In a deterministic analysis, these
response spectra is described in paragraph 3-4h of Chapter
uncertainties, which are termed epistemic uncertainties, are
3.
usually treated by applying conservatism in selecting design
b. Analysis of Contribution to the Seismic Hazard. A
earthquakes and estimating ground motions. In PSHA,
these uncertainties can be directly modeled within the
hazard curve incorporates contributions from different
analysis framework to provide an assessment of the
earthquake sources, magnitudes, and source-to-site
uncertainty in the result. The technique of "logic trees" has
distances. The results can be analyzed to determine the
E-5
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