Earthquake Input Motion - Fundamentals of Earthquake Engineering - page 127

3.3.4 Attenuation Relationships for North America
Several attenuation relationships have been derived for North America since the early 1970s (e.g. Esteva
and Villaverde, 1973; McGuire, 1978; Joyner and Boore, 1981 , 1988 ; Boore et al. , 1997 ; Chapman,
1999, among others). Most of these are calibrated for western USA conditions, which exhibit high
recurrence of earthquakes. Two relationships for PGA are presented hereafter. They can be employed
for the central and eastern areas of the USA and in western North America, respectively.

(i) Central and Eastern United States
The Mid-America Earthquake Center developed a ground-motion model to predict horizontal PGA in
the central and eastern United States (CEUS) region (Fernandez and Rix, 2006). The attenuation rela-
tionship is based on a stochastic method and employs three source models, i.e. Atkinson and Boore
( 1995 ), Frankel et al. ( 1996 ) and Silva et al. (2003). It was developed for soil sites in the Upper Mis-
sissippi Embayment in the New Madrid seismic zone, which is a low probability- high impact source
of earthquakes. The region has been hit by three great earthquakes in 1811 and 1812 (e.g. Reiter,
1990 ).
The ground -motion model is defi ned by the following equations:

R

⎡
⎢

⎛
⎝

⎞
⎠

⎤
⎥ +

2

( ) =+ ( ) + ( ) + ln PGA ccM cM + − 6 c ln R c max ln , 0 c 6M (3.10.1)

1 2w 3 w 4 M 5

70

where the equivalent distance term R M is given by:

( ) RRc =+ exp cM (3.10.2)

M 7 8 w

and the logarithmic standard deviation of PGA, termed σ ln(PGA) , is considered to be magnitude-
dependent. It is obtained from the following equation:

σ ln PGA

) =

cM + c

10

(3.10.3)

(

9 w

In the above equations, R is the epicentral distance (in km), M w is the moment magnitude and c 1
through c 10 are the regression coeffi cients. The value of the peak ground acceleration PGA is expressed
in g. In equations (3.10.1) and (3.10.2), the epicentral distance R is the distance from the observation
point to the surface projection of the hypocentre. The c i coeffi cients, which depend on the source model,
the stress drop, the soil profi le, dynamic soil properties and depth, can be found in Fernandez (2007 ).
These coeffi cients are computed for epicentral distances uniformly distributed between 1 and 750 km
for eight values of magnitudes M w varying between 4 and 7.5. In equations (3.10.1) and (3.10.2) ,
the equivalent distance term R M accounts for the increase in travelling distance by the seismic waves
due to the increase in fault rupture size. The exponential term in equation (3.10.2) accounts for the
magnitude-dependence of the energy release. The effects of inelastic soil behaviour are incorporated
in the above attenuation relationship. A number of ground-motion models to predict the horizontal
components of PGAs in the CEUS for rock sites can be found in the literature (e.g. Atkinson and Boore,
1995 , 2006 ; Frankel et al. , 1996 ; Silva et al. , 2003 ).

(ii) Western North America
Boore et al. ( 1997 ) and Boore ( 2005) formulated the following equation to predict peak ground accel-
erations in western North America:

2

( ) =+ ( ) + ( ) + () + ( ) log PGA bbM − 6 bM − 6 bRb + log R b log v − log v (3.11.1)

1 2 w 3 w 4 5 V S A