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

MODEL PARAMETER DEVELOPMENT
The general form of the ground motion model used to assess the model parameters is shown
in Equation (16):
f
Source
f
= c (
Path
f
HW
2
4
9
[ y ]
[ 1+
exp{
c ( c − M ) } ]
3
M n M
= c ln
b
2 2
[ RRUP
+ c5
cosh{
c6
max( M − cHM
, 0 ) } ] + ( c4a
− c4
) × ln[
RRUP
+ cRB
]
= φ ×
[ c + c / cosh{
max( M − c , 0}
]
tan
/ 2 ) ×
⎧ ⎡ VS
30 ⎤ ⎫
= φ1
min⎨ln
⎢
0⎬
+
⎩ ⎣1130⎥
, b
⎦ ⎭
2
γ 1
1130
⎛ c2
− c ⎞
− 6 ) + ⎜
⎟ × ln
⎝ cn
⎠
f
⎧ W cos( δ ) ⎫
⎨ ⎬
⎩2(
ZTOR
+ 1)
⎭ ⎧
× ⎨1−
π / 2 ⎩ R
{ ln[ y ] }
[ exp{
φ ( min( V , 1130 ) − 360)
} − expφ
( 1130 − 360)
]
3
] = c
γ 2
2
= c cos ( δ ) × tanh( R
Site
Site
ln
ln[ y
Surface
RUP
= ln[ y
1
+
f
S 30
1130
Source
] + f
−1
R ⎫
JB
⎬
+ 0.
001⎭
− 4 )
C&Y2006 Page 33
+
Site
f
Path
Site
+
γ 3
+ σ ⋅ z
f
HW
+ c
+ τ ⋅ z
1a
× R
ij
F
RUP
3
RV
+ c
1b
F
NM
RUP
⎡ exp 1130 + φ ⎤ 4
ln⎢
⎥
⎣ φ4
⎦
i
+ c ( Z
The parameter y1130 is the ground motion on the reference site condition (VS30 = 1130 m/sec).
Its level is based on the source scaling function fSource, the path scaling function, fPath, the
hanging wall function fHW, and a random effect τ zi that is modeled as a Gaussian random
variate with inter-event standard deviation τ. The log of the ground motion at a site is the
sum of the log of the reference motion and a nonlinear amplification, fSite, that is a function of
VS30 and the level of the reference motion. The ground motions at the site also include a
random Gaussian variate with intra-event standard deviation σ. The inter-event component
of randomness in included when computing the site amplification. Also note that the site
amplification function uses the reference motion for the same spectral period. The additional
parameters in Equation (16) are: RRUP , closest distance to the rupture plane (km); RJB ,
Joyner-Boore distance to the rupture plane (km); δ , rupture dip; W , rupture width (km); ZTOR
, depth to top of rupture (km); FRV , reverse faulting factor equal to 1 for 30º ≤ λ ≤ 150º, and 0
otherwise; FNM , normal faulting factor equal to 1 for -120º ≤ λ ≤ -60º, 0 otherwise; λ , slip
rake angle; VS30 , average shear wave velocity for top 30 m (m/s). Note, fHW applies to all
faulting styles.
The model parameters were obtained by fitting the model to the selected PEER-NGA data
using the nonlinear mixed effects method nlme implemented in the statistical packages S-
Plus and R. The process used to obtain these parameters is described below.
7
TOR
+
(16)