Chiou and Youngs PEER-NGA Empirical Ground Motion Model for ...

allows the interpretation of the coefficient C2 as the asymptotic value of magnitude scaling at
large distances from the source. This same argument should make us favor the functional
form of Equation (4) over that of Equation (3) in that with the function form of Equation (4)
the coefficient C4 can be interpreted as the geometric spreading at distances greater than
about 2×c(M). We experimented extensively with this form, but ultimately decided not to
use it because we judged that it resulted in too little distance scaling over the distance range
of 0 to 10 km for large-magnitude earthquakes.
Previous implementations of the functional form of Equation (3) (e.g. Sadigh et al., 1997)
have typically specified different parameters for large and small magnitude earthquakes.
This results in a step function for the magnitude scaling, as illustrated in the upper left-hand
plot of Figure 12. We prefer to have a smooth transition in the magnitude scaling over the
full magnitude range. This was accomplished by using the relationship:
{ max( − 3,
0)
}
c ( M) = C5
cosh C6
M
(7)
The use of Equation (7) has the property that c(M) varies smoothly from a constant at small
magnitudes to c( M) ∝ exp( C6M)
at large magnitudes.
Path Scaling at Large Distances: Many studies of the attenuation of ground motion Fourier
amplitudes with distance have indicated that there is a change in the rate of geometric
spreading from approximately proportional to 1/R at short distances to 1/ R at large
distances, with this transition occurring in the range of 40 to 70 km. This change has been
interpreted to be the combination of the effects of post-critical reflections from the lower
crust and transition from direct body wave to Lg wave spreading (e.g. Atkinson and Mereu,
1992). Models of the decay of Fourier spectra with distance in California have found or
assumed that the geometric spreading is proportional to 1/ R for distances greater than 40
km (Raoof et al., 1999; Erickson et al., 2004) and this form of geometric spreading was used
by Atkinson and Silva (2000) to model strong ground motions. Earlier, Atkinson and Silva
(1997) used a tri-linear form of attenuation similar to that defined by Atkinson and Mereu
(1992), but indicated that a bi-linear form would also work.
We explored this effect by modeling the broadband data assembled for three small (M 4.3 to
4.9) Southern California earthquakes (Anza, Yorba Linda, and Big Bear City, events 0163,
0167, and 0170, respectively). These data were fit with three functional forms, a 1-slope
model:
ln(
R
0
y) = C1
+ C2FBBC
+ C3FYL
+ C4
ln( R0
) + γR
+ φ1
ln( VS
30
=
R
2
+ 6
a 2-slope model with the second slope ½ of the first:
2
{ ln( R ) + ln( R ) / 2}
ln( y) = C1
+ C2FBBC
+ C3FYL
+ C4
1
2 + γR
+ φ1
ln( VS
R
0
=
R
2
2
+ 6 , R = min( R , C ), R = max( 1,
R
1
0
8
2
/ 400)
C&Y2006 Page 20
0
/ C
8
)
30
/ 400)
(8)
(9)