C
ALIFORNIA
I
NSTITUTE OF
T
ECHNOLOGY
E
ARTHQUAKE
E
NGINEERING
R
ESEARCH
L
ABORATORY
C
HARACTERIZING AVERAGE PROPERTIES OF SOUTHERN
C
ALIFORNIA GROUND MOTION AMPLITUDES AND
ENVELOPES
BY
G
EORGIA
C
UA AND
T
HOMAS
H.
H
EATON
R
EPORT
N
O
.
EERL
2009-05
P
ASADENA
,
C
ALIFORNIA
F
EBRUARY
2008
A
R
EPORT ON
R
ESEARCH
S
UPPORTED BY THE
C
ALIFORNIA
I
NSTITUTE OF
T
ECHNOLOGY
U
NDER THE
S
UPERVISION OF
T
HOMAS
H
EATON
.
Characterizing ave
rage
prop
erties of sou
thern California grou
nd motion
amplitudes
a
nd envelop
es
G
eorgia
Cua
1
and
Thom
as
H. Heaton
2
1
Swiss Seismologi
cal Service, ETH Zurich, Switzerland
2
California Institute of Technol
ogy
2
Abstract
1
2
W
e exam
i
ne ground
motion
enve
lope
s of hor
izontal and
vertical acceleration,
velocity,
3
and
filtered displacement recorded within 200
km
from
s
out
hern California earthqua
kes
4
in the magni
tude
range
2 < M
!
7.3
. We introduc
e a parameterization
that decom
pos
es
5
the obs
erved ground
motion
enve
lope
into P
-
wave
train
, S
-
wave
train
, and
ambient noi
se
6
enve
lope
s. The
shape of the
body
wave enve
lope
s
as a func
tion
of time
is
further
7
parameterized
by
a rise time, a dur
ation, a constant amplitude
, and
2
coda
decay
8
par
ameters. Each obs
erved ground
motion
enve
lope can thus
be described by
11
enve
lope
9
parameters. We fit this parameterization
to
30,000
obs
erved
ground
motion
time
10
histories, and
develop
attenua
tion
relations
hips describing
the magni
tude
, distance,
and
11
site depende
nce of these 11
enve
lope
parameters. We use these
relations
hips to study
1)
12
magni
tude
-
depende
nt saturation
of peak
amplitude
s on
rock
and
soil sites
for peak
13
ground
acceleration,
peak ground
velo
city, and
peak filtered displacement
, 2) magni
tud
e
14
and
distance scaling
of P
-
and
S
-
waves, and
3) the reduc
tion
of unc
ertainty in predicted
15
ground
motions
due
to the appl
ication
of site
-
specific station
corrections
.
We
develop
16
extende
d magni
tude
range
atte
nua
tion
relations
hips for PGA and
PGV
valid ove
r the
17
magni
tude
range
2 < M < 8
by
suppl
ement
ing
our
dataset of S
-
wave enve
lope
amplitude
s
18
with
the Next Generation
Attenua
tion
(NGA) strong
motion
dataset.
We com
pare
19
extende
d magni
tude
range
attenua
tion
relations
hips with
the Campbe
ll and
Bozorgni
a
20
(2008)
and
Boor
e and
Atkinson
(2008
) NGA relations
hips
.
Our
extende
d magni
tude
21
range
attenua
tion
relat
ions
hips exhi
bit
a
stronge
r inter
-
depende
nce between distance and
22
magni
tude
scaling
. This character of ground
motion
scaling
becom
es
evident when
23
3
examining
ground
motion
amplitude
s ove
r an extende
d magni
tude
range
, but
is not
24
appa
rent
when cons
idering
data
within a more limited magni
tude
range
, for instance, the
25
M>5
range
typi
cally cons
idered for strong
motion attenua
tion
relations
hips.
26
27
28
4
Introd
uction
28
29
The widespread deploym
ent of seismic stations
in
s
out
hern California unde
r the TriNet
30
project resulted i
n an unpr
ecedented dataset of recorded ground
motions
(Mori et al.,
31
1998)
. We analyzed a large por
tion
of this dataset
as part of a study
on
seismic early
32
warning
(Cua, 2005)
. We studi
ed enve
lope
s of ground
motion, as oppos
ed to the fully
33
sampled
time histories, due
to
our
interests in
developi
ng
a seismic early
warning
34
methodol
ogy
for deploym
ent on
the Sout
hern California Seism
ic Network (SCNS); peak
35
ground
motion
information
(acceleration,
velocity, and
displacement)
ove
r 1
-
second
36
window
lengt
hs are among
the data packets that arrive in closest to real
-
time at the
37
central processing
facility of the SCSN. In this study,
we defin
e ground
motion
enve
lope
s
38
as the peak
ground
motion
value ove
r non
-
overlappi
ng
one
-
second
window
s; this
39
definition
is cons
istent with the type
of data
streams that can be realistically produc
ed by
40
seismic networks in real
-
time.
41
42
We develope
d a parameter
ization
that decom
pos
ed the obs
erved ground
motion enve
lope
43
time history into P
-
wave
train
, S
-
wave
train
, and
ambient noi
se enve
lope
s. Each
wavetrain
44
enve
lope
is described by
a rise time, a peak amplitude
, a dur
ation,
and
two
coda
decay
45
parameters.
We analyz
ed 9 com
pone
nts of ground
motion:
2
hor
izont
al and
1
vertical
46
com
pone
nt
of
acceleration,
velocity,
and
filtered
displacement.
With
this
47
parameterization,
the evol
ution
of
each com
pon
ent of
ground
motion
amplitude
as a
48
func
tion
of
time is described by
1
1 enve
lope
parameters (5 P
-
wave parameters, 5 S
-
wave
49
parameters, and
1 cons
tant to describe ambient noise levels). We use the neighbor
hood
50
5
algor
ithm
, a nonl
inear direct search algor
ithm
(Sambridge
, 1999a
, 1999b)
to find
the set
51
of 11
maximum
likelihood
enve
lope
parameters for each enve
lope
wavetrain
in the
52
database.
53
54
We develope
d
attenua
tion
relations
hips
that
describ
e
each
of these 11
env
el
ope
55
parameters as a func
tion
of magni
tude
, distance,
site condi
tion,
com
pone
nt, and
type
of
56
ground
motion parameter (acceleration,
velocity, displacement).
In this paper, we focus
57
the discussion
on
the attenua
tion
relations
hips for peak P
-
and
S
-
wave ampl
itude
s of
58
hor
izont
al and
vertical ground
motion
acceleration,
velocity, and
filtered displacement on
59
rock
and
soil sites. We use these attenua
tion
relations
hips to
study
1) magni
tude
-
60
depende
nt saturation
of peak amplitude
s on
rock and
soil sites, 2) magni
t
ude
and
distance
61
scaling
of P
-
and
S
-
waves, and
3) the reduc
tion of unc
ertainty in predicted ground
62
motions
due
to the appl
ication
of site
-
specific station
corrections
.
63
64
The fact that the TriNet project provi
des well calibrated broad
-
band
motions
ove
r a
very
65
large amplitude
range
allow
s us the oppor
tun
ity
to
study
the interdepende
nce of
66
magni
tude
scaling
and
distance scaling
for acceleration,
velocity, and
displacement.
In
67
previous
studi
es that cons
ider onl
y strong
motions
from
large earthqua
kes,
the mag
nitude
68
range
is small enough
that
empirical
prediction
equa
tions
that
cons
ist of inde
pende
nt
69
distance decay terms and
magni
tude
scaling
terms
can appr
oxi
mately capture trend
in the
70
data
(Boor
e and
Atkinson,
2008;
Campbe
ll and
Bozorgni
a, 2008)
.
How
ever, using
a
71
data set with a much larger range
of magni
tude
s, w
e
find
com
pelling
evidence
that
72
amplitude
decay with distance and
magni
tude
scaling
cannot
be separated.
Fo
r example
,
73
6
we find
that
near
-
sour
ce
peak accelerations
change
their magni
tude
scaling
from
74
10
3
2
M
for small magni
tude
s to
com
ple
te sa
turat
ion
a
t large
magni
tude
s
.
In cont
rast, p
eak
75
near
-
sour
ce
displacement
s change
their magni
tude
scalin
g
from
10
3
2
M
for small
76
magni
tude
s to
10
1
2
M
for large magni
tude
s
.
77
78
Since
the data set in our
study
is large, we can
derive
separate prediction
equa
tions
for
79
rock and
soil sites
. We attribut
e differences in the
pre
diction
equa
tions
(except for an
80
amplification
factor) to nonl
inear behavior of soil sites.
In particular, we find
that near
-
81
sour
ce peak accelerations
from
small earthqua
kes
are about
twice as large at soil sites
82
than at
rock sites, where
as
near
-
sour
ce pe
ak accelerations
from
large earthqua
kes are
83
appr
oxi
mately the same for
soil sites
and
ro
c
k sites.
This behavior is cons
istent with
84
yielding
of
soil sites
at large amplitude
s
that serves to
nonl
inearly increase effective
85
damping
for soil sites.
We also f
i
nd
that P
-
wave amplitude
s appe
ar to exhi
bit stronge
r
86
saturation
characteristics than S
-
wave amplitude
s, particularly in the hor
izont
al direction.
87
88
In this study
we
also use the Next Generation
Attenua
tion
(NGA) strong
motion
dataset
89
(http://peer.berkeley.edu/
nga
)
to suppl
ement our
s
out
hern California data set to d
erive
90
extende
d magni
tude
range
attenua
tion
relations
hips for peak ground
acceleration
(PGA)
91
and
peak
ground
velocity
(PGV)
valid
up
to 200
km
epicentral distance ove
r the
92
magni
tude
range
2
<
M
<
8
. We com
pare the median groun
d motion
leve
ls predicted by
93
our
extende
d
magni
tude
range
relations
hips with
thos
e predicted
by
the Boor
e and
94
7
Atkinson
(2008)
and
Campbe
ll and
Bozorgni
a
(2008)
relations
hips develope
d as part of
95
the NGA project.
96
97
The NGA relations
hips, and
the majority of attenu
ation
relations
hips in the literature, are
98
used in seismic hazard analyses to provi
de estimates of eithe
r the median, geom
etric
99
mean, or random
com
pone
nt of the hor
izont
al ground
motions
.
How
ever, n
one
of these
100
are representative of the maximum
ground
motion
level expe
rienced by
a given bui
lding
101
dur
ing
an earthqua
ke
, which is the vector amplitude
of the hor
i
zont
al ground
motions
.
102
W
e also develop
conve
rsion
factors between the vector amplitude
of hor
izont
al ground
103
motion
with other com
monl
y used measures of
hor
izont
al
ground
motion.
104
105
Method
106
Wave
form dataset
107
Wa
veforms for this study
were obt
ained from
1) the
S
outhern California Earthqua
ke
108
Center (SCEC) database
(http://www.data.scec.org)
which
archives
waveform
data
109
recorded by
the Sout
hern California Seismic Network (SCSN), and
2) the Con
sortium
of
110
Organizations
for
Strong
Motion
Observation
Systems
(COSMOS)
database
111
(http://db.
cosmos
-
eq.org)
, whic
h archives strong
motion
data from
the U.S. Geologi
cal
112
Survey, California Geologi
cal Survey, and
other strong
motion
arrays worldw
ide.
Many
113
SCSN stations
have co
-
located broadba
nd
and
strong
motion
instrum
ents, and
con
tribut
e
114
3 com
pone
nts of broad
-
b
and
seismom
eter records (for small to mode
rate motions
) and
3
115
com
pone
nts of accelerom
eter records
(for mode
rate to large motions
).
W
e typi
cally
used
116
8
the
broadba
nd
velocity waveforms.
How
ever, i
f we found
evidence of clippi
ng
(visual
117
examination,
or peak v
elocities exceeding
13
cm/s, the typi
cal clip level of an STS
-
2
118
seismom
eter),
then
we dow
nloaded the
strong
-
motion
accelerom
eter data
instead
.
119
120
We performed
gain
and
baseline corrections
on
the dow
nloaded
waveforms and
121
integrated and/
or differentiated to
obt
ain acceleration,
velocity, and
displacement time
122
histories. The displacement waveforms were filtered using
a 3
-
second,
4
-
pol
e high
-
pass
123
Butterworth
filter to
reduc
e the influence of microseisms
on
small amplitude
124
displacements
.
This filter also remove
s
long
-
period noi
se introduc
ed in the processing
of
125
strong
motion
records.
126
127
We examined ground
motions
recorded at 150
Sout
hern California Seismic Network
128
(SCSN) stations
located within 200
km
epicentral distance of 70
Sout
hern California
129
events in the ma
gni
tude
range
2 < M
!
7.3
.
In addition
to SCSN data, we also include
d
130
strong
motion
records from
the COSMOS database from
the 1989
M=7.0
Lom
a Prieta,
131
1991
M=5.8
Sierra Madre, 1992
M=7.3
Lande
rs, 1992
M=6.4
Big Bear, and
1994
132
M=6.7
Northridge
(and
a
M=5.1
a
ftershoc
k) earthqua
kes. Ground
motion
enve
lope
s time
133
series were obt
ained from
the 100
-
or 80
-
sample per second
time series by
taking
the
134
maximum
amplitude
s ove
r one
-
second
non
-
ove
rlappi
ng
window
s.
135
136
Site classificatio
n
137
9
We adopt
ed a binary (rock
-
soil) si
te classification
based on
the
s
out
hern California site
138
classification
map of Wills et al
(2000)
, which was based on
correlating
the average shear
139
wave velocity in the uppe
r 30 m (Vs30)
with geologi
c uni
ts. Wills et al
(2000)
created
140
intermediate categor
ies B
C and
CD to accom
moda
te geologi
c uni
ts that had Vs30
values
141
near the bounda
ries of the existing
NEHRP
-
UBC site classes. In
our
binary
site
142
classification,
“rock” sites are thos
e assign
ed to classes BC and
above
(Vs30
> 464
m/s)
,
143
and
“soil” sites are tho
se
with classification
C and
below
(Vs30
!
464
m/s)
.
Of the SCSN
144
stations
we used, 35
stations
were classified as rock, and
129
stations
were classified as
145
soil stations
. Separate attenua
tion
relations
hips for the various
enve
lope
parameters were
146
develope
d fo
r rock and
soil sites, allow
ing
us to inve
stigate differences in the average
147
prope
rties of ground
motions
on
rock and
soil sites ove
r the magni
tude
and
distance
148
range
s cove
red by
our
dataset. Since SCSN stations
, which are almost all located on
rock
149
or sti
ff soil sites,
cont
ribut
e the majority of the ground
motions
in our
dataset, this study
150
doe
s not
include
records from
very soft soils
(
E class, or Bay mud
-
type
sites
)
.
151
152
Next Generation
Attenuation
(NGA) stron
g motion
dataset
153
The S
-
wave enve
lope
amplitude
for hor
izont
al acceleration
or velocity for a given record
154
is equi
valent to the maximum
acceleration
or velocity obs
erved on
a given channe
l. We
155
can relate these enve
lope
amplitude
s to peak grou
nd
acceleration
(PGA) and
peak ground
156
velocity (PGV), which a
re fundamental qua
ntities of interest in seismic hazard analyses.
157
When deriving
attenua
tion
relations
hips for these particular enve
lope
parameters, we
158
suppl
ement the sout
hern California S
-
wave enve
lope
amplitude
s with amplitude
s from
a
159
subs
et of the NGA st
rong
motion
database used by Boor
e and
Atkinson
(2008)
. We will
160
10
refer t
o this subs
et of the NGA database as the NGA dataset for brevity. The NGA
161
dataset cont
ribut
es 50
addi
tiona
l records to
the rock
categor
y, and
1557
addi
tiona
l
162
records to the soil categor
y. The largest event from the NGA dataset is the 2000
M
=
7.9
163
Denali, Alaska earthqua
ke. It shoul
d
be empha
sized
that general analysis of the
164
waveform enve
lope
s and
the associated enve
lope
parameters uses the sout
hern California
165
dataset. The NGA dataset is used as a suppl
ement onl
y for the attenua
tion
of
the S
-
wave
166
enve
lope
amplitude
s for horizont
al acceleration
(PGA)
and
velocity
(PGV)
.
167
168
Figur
e
1
show
s the distribut
ion
in magni
tude
and distance
space
of the data (sout
hern
169
California enve
lope
dataset and
NGA strong
mot
ion
dataset
) used in this study.
E
ach
170
poi
nt on
these plots for the sout
hern California dataset cont
ribut
es waveforms for 9
171
channe
ls of ground
motion
(vertical, North
-
Sout
h, and
East
-
West com
pone
nts for each of
172
acceleration,
velocity, and
filtered displace
ment). For each channe
l of ground
motion,
173
there are 958
records from
rock sites, and
2,630
records from
soil sites.
174
175
Param
eterization
of grou
nd motion
envelop
es
176
We mode
led the obs
erved ground
motion
enve
lope
s as a com
bination
of P
-
wave
train
, S
-
177
wave
train
, and
ambient noi
se enve
lop
es. The P
-
wave
train
, S
-
w
ave
train
, and
ambient
178
noi
se enve
lope
s of a given record com
bine according
to the rule:
179
E
obs
(
t
)
=
E
P
2
(
t
)
+
E
S
2
(
t
)
+
E
am
bient
2
+
!
(
t
)
(
1
)
180
11
where
E
obs
(t)
is the
obs
erved
ground
motion
enve
lope
,
E
P
(t)
,
E
S
(t)
,
E
ambient
are the
181
mode
led
P
-
wave
train
, S
-
wave
train
, and
ambient noi
se enve
lope
s, and
"
(t)
is the
182
difference between the obs
erved and mode
led envelope
.
183
The ambient noi
se enve
lope
for a given time history,
E
am
bient
,
, is mode
led as a cons
tant.
184
The time depende
nce of the P
-
and
S
-
wave
train
enve
lope
s,
E
P
(t)
and
E
S
(t)
, is piece
-
wise
185
linear with Omori
-
type
decay. Each of
E
P
(t)
and
E
S
(t)
is described by
a rise time (
tr
),
186
cons
tant amplitude
(
A
) with an associated dur
ation
(
#
t
), and
two decay parameters (
$
,
%
).
187
We found
that using
a singl
e decay parameter woul
d typi
cally fit the ove
rall coda
, but
188
with large misfits immediately after the peak P
-
or S
-
wave amplitude
s. Jenni
ngs
et al
189
(1968)
also
requi
re two
parameters to
describe the decay
of enve
lope
amplitude
s
190
follow
ing
the peak ground
motion.
Using
two decay parameters improve
s the fit between
191
the mode
le
d
and
obs
erved
enve
lope
s at the cost of introduc
ing
trade
-
offs in
the
192
parameterization.
193
12
E
ij
(
t
)
=
0
for
t
<
T
i
A
ij
tr
ij
(
t
!
T
i
)
for
T
i
"
t
<
T
i
+
tr
ij
A
ij
for
T
i
+
tr
ij
"
t
<
T
i
+
tr
ij
+
#
t
ij
A
ij
(
t
!
T
i
!
tr
ij
!
#
t
ij
+
$
ij
)
%
ij
for
t
&
T
i
+
tr
ij
+
#
t
ij
'
(
)
)
)
)
*
)
)
)
)
whe
re
i
=
P-, S
-wave
T
i
=
P-, S
-wave arrival times
j
=
!
!
u
Z
,
!
!
u
N
!
S
,
!
!
u
E
!
W
!
u
Z
,
!
u
N
!
S
,
!
u
E
!
W
u
Z
,
u
N
!
S
,
u
E
!
W
with
!
!
u
de
not
ing a
cceleration
!
u
de
not
ing ve
locity
u
de
not
ing di
splacement
(
2
)
194
A total of 11
enve
lope
parameters (5 each for the P
-
and
S
-
wave enve
lope
s, and
a
195
cons
tant for the ambient noi
se) are used to describe a singl
e obs
erved ground
motion
196
enve
lope
.
197
198
Th
e
parameterization
described by Eqns
.(1
)
and
(
2
)
allow
s for a separate characterization
199
for P
-
and
S
-
wave
train
s. It makes intuitive sense that each of the body
wave enve
lop
es
200
has a rise time, an amplitude
with a finite dur
ation,
and
parameters describing
its cod
a
201
decay. Unfortuna
tely, this intuitive parameterization
is qui
te non
-
linear, due
to tr
ade
-
offs
202
between the various
parameters. For instance, we identified strong
trade
-
offs between rise
203
time and
dur
ation,
and
between the coda
decay parameters
!
and
!
for bot
h P
-
and S
-
204
wave enve
lope
s. Addi
tio
nal difficulties arose in unique
ly characterizing
the P
-
wave coda
205
decay at close distances (less than 20
km
), when there is less than 3 second
s of P
-
wave
206
13
data before the on
set of the S
-
wave arrival.
Our aim was to qua
ntify the time
-
depende
nce
207
of the shape
of ground
motions
enve
lope
s on
magni
tude
, distance, freque
ncy band,
and
208
site condi
tion.
209
210
In
principle, we coul
d
pos
tulate how
the various
enve
lope
parameters depend
on
211
magni
tude
, distance, and
site, and
along
with Eqn.
(
2
)
, find
the mode
l parameters that best
212
fit all enve
lope
time histories in our
database in a singl
e very large and
highl
y nonl
inear
213
inve
rsion
(Figur
e
2
a). Instead, we use an iterative appr
oach where the singl
e large and
214
nonl
inear inve
rse probl
em is replaced by
num
erous
small nonl
inear inve
rse probl
ems
215
(Figur
e
2
b). In
this iterative appr
oach, we use the neighbor
hood
algorithm
(NA)
216
(Sambridge
, 1999a
, 1999b
)
to find
the set of 11
enve
lope
parameters that minimize
"
in
217
Eqn.
(
1
)
in a
least squa
res sense
for each ob
served enve
lope
time history in our
dataset
.
218
Figur
e
3
a show
s the ground
motion acceleration
recorded at SCSN station
Dom
enegoni
219
Reservoi
r (DGR) dur
ing
the 1994
M=6.7 Northridge
earthqua
ke. Figur
e
3
b show
s its
220
ground
motion
en
velope
and
the 11
least squa
res enve
lop
e parameters from
the NA
221
inve
rsion.
The set of enve
lope
parameters carried to the next stage of the analysis for
222
each given ob
served enve
lope
time series was not
necessarily the onl
y good
solution
for
223
that particular
time series. There were families of “good”
solutions
in the neighbo
ring
224
regions
of the parameters space, due
to the trade
-
offs between the rise time and
dur
ation
225
parameters, as well as between the two coda
decay parameters. Fortuna
tely, the P
-
and
S
-
226
wave e
nve
lope
amplitude
parameters from the NA inve
rsions
were robus
t relative to
227
these trade
-
offs.
228
229
14
Typi
cally, each station
has 1 vertical and
2 hor
izont
al (from 2 orthogona
lly oriented
230
hor
izont
al sensors) time series available. These were differentiated and/
or
integrated to
231
yield 9 waveforms for each station
(1 vertical and
2 hor
izont
al channe
ls for each of
232
acceleration,
velocity, and
filtered displacement). For each station,
the NA was appl
ied to
233
all 9 waveforms. For each ground
motion
com
pon
ent (acceleration,
velocity, and
filtered
234
displacement) at each station,
the 2 sets of ho
rizont
al enve
lope
parameters (from
2
235
orthogona
lly oriented sensors) were com
bined in a root
mean squa
re sense to define a
236
singl
e set of hor
izont
al enve
lope
parameters. Separate regress
ions
were develope
d for 6
237
channe
ls (1
each
of vertical and
hor
izont
al acceleration,
velocity, and
filtered
238
displacement) channe
ls of enve
lope
parameters.
239
240
Envelop
e attenuation
relation
ships
for magn
itude an
d distan
ce
241
242
Rise time, duration
, and decay
param
eters
243
We mode
led the
loga
rithm of
rise time (
tr
),
loga
rithm
of
durations
(
#
t
), and
coda
decay
244
parameters (
$
,
%
) as linear func
tions
of magni
tude
, distance, and
log
distance.
245
log(
env
_
par
am
ij
)
=
!
ij
M
+
"
ij
R
+
#
ij
log
R
+
μ
ij
whe
re
e nv
_
par
am
ij
=
{
tr
ij
,
$
t
ij
,
%
ij
,
&
ij
}
(
3
)
246
where subs
cripts
i
,
j
are as in Eqn.
(
2
)
. The least squa
res
mode
l coefficients for these
247
paramet
ers are listed
Tables
2
.1
-
2
.4. The
se Tables
can also
be dow
nloaded
from
248
Appe
ndi
x
C of Cua
(2005)
,
http://resolver.
caltech.edu/
CaltechE
TD:etd
-
02092005
-
249
125601
.
250
251
15
P
-
and S
-
wave
envelop
e param
eters
252
Of the 11 enve
lope
parameters, the P
-
and
S
-
wave amplitude
s were expe
cted to have the
253
stronge
st magni
tude
and
distance depende
nce. We used Eqn.
(
4
)
to mode
l the magni
tude
,
254
distance, and
site depende
nce of P
-
and
S
-
wave amplitude
s for the 6 channe
ls of ground
255
motion.
256
257
log
Y
ijk
=
a
i
M
k
+
b
i
(
R
1
k
+
C
ik
(
M
k
))
+
d
i
log(
R
1
k
+
C
ik
(
M
k
))
+
e
ij
+
!
ijk
whe
re
i
=
1,
...
, 24
(P-, S
-wave amplitude
s on r
ock a
nd s
oil sites for 6 c
hanne
ls)
j
=
1,
...
, num
ber of
stations
k
=
1,
...
, num
ber of
records
Y
ijk
=
body w
ave amplitude
from
NA algor
ithm
inve
rsion on gi
ven r
ecord
M
k
=
repor
ted m
agni
tude
(mom
ent m
agni
tude
for
M
>
5)
R
k
=
epicentral di
stance for
M
<
5, fault di
stance for
M
"
5
R
1
k
=
R
k
2
+
9
(assum
ing a
n a
verage de
pth of
3 km
for sout
hern Ca
lifornia events)
C
ik
(
M
)
=
c
1
i
exp(
c
2
i
(
M
k
#
5))
$
arctan(
M
#
5)
+
%
2
(
)
e
ij
=
e
1
i
+
e
2
ij
(cons
tant term pl
us station-
specific corrections
)
!
i
=
statistical or
pr
ediction e
rror,
!
NID
(0,
&
i
2
)
(
4
)
258
259
For the ground
motions
at a given station,
the ho
rizont
al body
wave amplitude
s are the
260
root
mean
squa
res of the respective body
wave enve
lope
amplitude
s from
the 2
261
(orthogona
l) hor
izont
al records. Base
-
10
logs
are used throughout
this paper. In a later
262
section
of this paper, we derive factors that can be used to conve
rt different measures of
263
hor
izont
al ground
motion
(for instance, geom
etric mean, larger random
com
pone
nt, root
264
mean squa
re
) to the maximum
vector amplitude
of the hor
izont
al ground
motions
, which
265
16
corresponds
to the maximum
ground
motions
amplitude
expe
rienced at a given site for a
266
given earthqua
ke.
267
268
Eqn.
(
4
)
has strong
influences from
traditiona
l strong
motion
attenua
tion
relations
hips, in
269
particular, from
the work of Boor
e and
Joyne
r
(19
82)
, Boor
e, Joyne
r, and
Fum
al
(1997)
,
270
and
Campbe
ll
(1981;
2004)
.
In the subs
eque
nt discussion,
the subs
cripts
i,j,k
are droppe
d
271
for brevity.
The phys
ical motivations
for the various terms are as enum
erated in the early
272
literature
on groun
d motion
attenua
tion
:
273
274
•
logY
&
a
M
is cons
istent with the definition
of magni
tude
as the loga
rithm
of
275
ground
motion amplitude
(Richter, 1935)
276
•
logY
&
log
R
-
d
is cons
istent with
the geom
etric attenua
tion
of the seismic
277
wavefront
away from
the sour
ce
278
•
logY
&
b
R
is cons
istent with anaelastic attenua
tion due
to material damping and
279
scattering
280
•
logY
&
e
, where e is partitione
d into a cons
tant and station
-
specific site correction
281
terms, is cons
istent with the multiplicative nature of site effects
282
•
C
(
M
)
=
c
1
exp(
c
2
(
M
!
5))
"
arctan(
M
!
5)
+
#
2
(
)
is
a
magn
i
t
ude
-
depende
nt
283
saturation
term that allow
s ground
motion
amplitude
s at close distances to large
284
earthqua
kes (
M>5
) to be relatively inde
pende
nt of magni
tude
. Ground
motion
285
simulations
sugge
st that the shape of attenua
tion
curves is mag
nitude
-
depend
ent,
286
with ground
motion amplitude
s in the near
-
sour
ce region
of large earthqua
kes
287
appr
oaching
a limiting
value
(Hadley and
Helmberger, 1980)
. Campbe
ll
(1981)
288