 1  mo )
o
U
o
 β (m m )
 e  β (m  m ) , for m o ≤m ≤mU
U
 b(m
N (m) = N (m ) e
&T
o
M o (1  10
4
)
Ne =
U
o
1.0  e  β (m  m )
c
c
b10b (1  10
 b(mU  1  mo ) b10 2
2)
(E3)
U
+
4
M o 10
c b
c
(E5)
whereβ = b ln(10) and b is the bvalue of the Gutenberg
Richter frequency law. Parameters β and N(mo) are
1
U
o
1 b ln(10) N e 10 b(m  4  m
 1)
estimated by fitting the recurrence relationship E3 to the
Nc = 2
observed recurrence rates obtained from a catalog of
 b(mU  1  mo )
1  10
4
historic seismicity. These parameters can be further
constrained by the geological slip rate, if it is available. An
&T
where M o is the rate of seismic moment release along a
example of such a truncated exponential recurrence
relationship is given on the upper left of Figure E1. The
U
fault and M o is the seismic moment for the upper limit
incremental recurrence rate λmi) is obtained by
(
event mU +3 . M T is estimated by Af S, where is the
&
discretizing the cumulative recurrence curves into narrow
o
shear modulus of fault zone rock (assumed to be 3⋅ 011
1
magnitude intervals as illustrated in the lower left of Figure
2
dyne/cm ), Af is the total fault surface area, S is the slip
E1.
rate, An example of such a characteristic recurrence
relationship is given on the upper right of Figure E1 and
(2) The characteristic earthquake recurrence
the incremental rate λmi) is given on the lower right.
(
model is based on the hypothesized fault behavior that
individual fault and fault segments tend to generate same
(3) In another implementation of the characteristic
size or characteristic earthquakes (Schwartz and
earthquake model (Wesnousky, 1986), no allowance is
Coppersmith, 1984; Youngs and Coppersmith, 1985a).
made for the occurrence of events of sizes other than the
"Samesize" usually means within about onehalf
characteristic size. The characteristic size (mc) is
magnitude unit. There are two implementations of the
proportional to fault length and can be determined using
characteristic earthquake model that are commonly used in
relations such as those in Wells and Coppersmith (1994).
PSHA. In the characteristic earthquake recurrence model
The recurrence rate for this characteristic size earthquake is
implemented by Youngs and Coppersmith (1985a), the
thus
maximum magnitude mU is taken to be the expected
magnitude for the characteristic event, with individual
events uniformly distributed in the range of mU 3
&T
Mo
c
λ(m ) =
(E6)
magnitude units, representing random variability in
c
Mo
individual "maximum" ruptures. The cumulative form of
the earthquake recurrence relationship thus becomes
c
where M o is the seismic moment of the characteristic size
earthquake mc. This version of the characteristic earthquake
U
mo )
o
 10  b(m

4
 b(m m )
1
e 10
+ N c,
N (m) = N
recurrence model (called the maximum magnitude model
U
o
1  10  b(m

4
m )
1
by Wesnousky, 1986) has been used by USGS (1996) and
for m o ≤m < mU 
others in PSHAs (e.g. Ferritto, 1994).
1
4
mU +
 m
1
c
for mU 
≤m ≤mU +
4
=N
c. Distance Probability Distribution. The distance
,
1
1
4
4
1
2
(E4)
geometry of earthquake sources and their distance from the
site; an assumption is usually made that earthquakes occur
where the terms Ne and Nc represent the rate of exponential
with equal likelihood on different parts of a source. The
and characteristic events, respectively. Ne and Nc are
function P(R=rj mi) also should incorporate the magnitude
specified by the slip rate of the individual fault using the
dependence of earthquake rupture size; largermagnitude
formulation of Youngs and Coppersmith (1985a).
earthquakes have larger rupture areas, and thus have higher
probability of releasing energy closer to a site than smaller
magnitude earthquakes on the same source. An example of
probability distributions for the closest distance to an
earthquake source is shown in Figure E2. In this
particular example, the source (fault) is characterized as a
line source and the probability distributions are based on
E2