(2) Formulation for frequency of exceedance. The

annual frequency of ground motion exceedance, *v(z), *is

evaluated using the following expression:

∑ ∑ λ (m ) ∑ P (R = *r*

) (

)

(E-2)

describe details of the methodology used in probabilistic

seismic hazard analysis (PSHA) to develop site-specific

in which

response spectra. More general aspects of the site-specific

approach are presented in Chapter 3. In paragraph E-2, the

=

the annual frequency of occurrence of

formulation of the basic probabilistic model is described.

earthquakes on seismic source *n *in a

Paragraph E-3 discusses the incorporation of uncertainty in

magnitude interval centered at *m*i. *m*i is

PSHA. Paragraph E-4 describes the results of a PSHA and

above a minimum size of engineering

how they can be analyzed to determine the dominant

significance, *m*o , and below the

contributors to the seismic hazard and sources of

maximum event size, *m*U .

uncertainty. In paragraph E-5, two examples of

applications of PSHA to develop site-specific response

the probability of an earthquake of

spectra are presented.

certain distance *r*j from the site

will be exceeded, given an earthquake of

(1) Formulation for probability of exceedance.

Thus, for a given source, the annual frequency or rate of

The methodology used to conduct PSHA was initially

exceeding a certain ground motion level at the site is

developed by Cornell (1968). The formulation of the basic

obtained by summing over all magnitudes (the second

seismic hazard model is summarized herein. Additional

summation of Equation E-2) and source-to-site distances

discussion and guidance for conducting a PSHA is

(the last summation of Equation E-2) for that source.

described in several publications, including National

Then, the total rate of ground motion exceedance at the

Research Council (1988), Earthquake Engineering

site, *v(z)*, is obtained by adding the rates for all the sources

Research Institute (1989), and Ferritto (1994, 1997). Using

(the first summation of Equation E-2). The components of

a Poisson probability model, the probability of exceedance,

equation E-2 are discussed in paragraphs b, c, and d below.

pz, (z), of a ground motion level, z, in an exposure time or

design time period, t, at a site is related to the annual

frequency (or rate) of ground motion exceedance at the site,

incremental rate of earthquakes occurrence *λ (m*i) is

obtained from earthquake recurrence relationships. Two

recurrence models are typically used in PSHA, the

(E-1)

truncated exponential model and the characteristic

earthquake recurrence model. These two recurrence models

A PSHA is carried out to obtain *v(z) *and *p*z (z) can then be

are also discussed in paragraph 3-4e(3)(b) of Chapter 3. For

obtained using Equation E-1. The return period (RP) for

convenience, the subscript *n *for the source region is

ground motion exceedance at a site is equal to the

eliminated in the following paragraphs.

reciprocal of *v(z)*. The results of a PSHA are, in practice,

expressed in terms of one or more of the parameters, *p*z(z),

(1) The truncated exponential model of Cornell

and Vanmarcke (1969) represents the truncation of the

(approximately ≤0.1) *p*z(z) is approximately equal to

Gutenberg-Richter (1954) earthquake frequency law at a

(*v(z)⋅t)*. For larger values of (*v(z)⋅t)*, *p*z (z) is less than

(*v(z)⋅t)*.

which expresses the rate of occurrence of earthquakes equal

to or greater than a certain magnitude *m*, is specified by

E-1

Integrated Publishing, Inc. |