(*P(R=r*j |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

probabilities or weights of the alternative models and

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|m*i,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|m*i,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

illustrated in Figure E-3.

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

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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