Reply to
“
Comment on
‘
Statistical Features of Short-Period and
Long-Period Near-Source Ground Motions
’
by Masumi
Yamada, Anna H. Olsen, and Thomas H. Heaton
”
by
Roberto Paolucci, Carlo Cauzzi, Ezio Faccioli,
Marco Stupazzini, and Manuela Villani
by Masumi Yamada, Anna H. Olsen, and Thomas H. Heaton
The comment by Paolucci and colleagues (
Paolucci
et al.
, 2011
) states that a probabilistic seismic hazard analysis
(
PSHA
) can provide
“
reliable prediction of long-period spec-
tral ordinates.
”
The result of such an analysis would be in
contrast to the more uncertain prediction suggested by our
empirical, and proposed theoretical, distribution of near-
source ground displacements in past, large magnitude earth-
quakes (
Yamada
et al.
, 2009
). After addressing two specific
concerns of Paolucci and colleagues, we use the balance of
this reply to discuss the apparent differences between a
PSHA
and our observations. These two approaches to understand-
ing the seismic hazard of long-period ground motions should
be consistent even though they view the problem from dif-
ferent perspectives.
Paolucci and colleagues prefer to use elastic spectral dis-
placement as the intensity measure of long-period ground
motions rather than peak ground displacement (
PGD
). Spec-
tral displacement and
PGD
, however, are highly correlated.
We calculate the elastic spectral displacements (
S
d
) of our
original set of recorded ground motions. We first find
S
d
for each horizontal component over a range of periods from
3 to 9 s at a 0.02 s interval, with damping at 5% of critical. At
each period, we take the square root of the sum of the
maximum squared
S
d
of each component. Then we find
the geometric mean over three ranges of periods: 3
–
5, 5
–
7,
and 7
–
9 s. We average the spectral displacements in each
range to find a more stable measure of the long-period
spectrum. We also calculate pseudo-
S
d
from the spectral
accelerations (
S
a
) reported in the current Next Generation
Attenuation (
NGA
) database (see the
Data and Resources
section).
1
Figure
1
shows that the logarithms of
PGD
and
S
d
are highly correlated, with correlation coefficients of 0.9405
(3
–
5 s), 0.9707 (5
–
7 s), and 0.9750 (7
–
9 s) for the records
collected in
Yamada
et al.
(2009)
. Also, we find similar
correlations of
PGD
and pseudo-
S
d
from the
NGA
database.
We prefer to use
PGD
because it is period independent, phys-
ically intuitive, and more concise than a family of spectral
curves. Although the value of
PGD
is certainly sensitive to
the processing of a recorded ground motion, our conclusions
do not depend on particular values of
PGD
.
In our original paper, we showed the distributions of
peak ground acceleration (
PGA
) and
PGD
from near-source
sites (that is, within 10 km of the surface projection of the
rupture, also known as a Joyner
–
Boore distance less than
10 km) of large magnitude (between 6.5 and 8) earthquakes
recorded in the years 1979 through 2004.
2
We now add near-
source records from similar sites since 2004 (Table
1
);
Figures
2
and
3
update the
PGA
and
PGD
distributions,
respectively, with these records. Our updated
PGA
and
PGD
distributions are consistent with the distributions presented in
Yamada
et al.
(2009)
. In the years 2005 through 2009, there
was no well-recorded large earthquake. Thus, we would not
expect our observed distribution of
PGD
to change signifi-
cantly. Figures
2
and
3
also overlay
PGA
s and
PGD
s from
the near source of past events with magnitudes
≥
6
:
5
as
reported in the current
NGA
database (see the
Data and
Resources
section). The
NGA
distributions are consistent
with our empirical distributions.
Furthermore, we perform statistical tests on the empiri-
cal distributions of
PGA
and
PGD
to determine whether they
are consistent with log-normal or log-uniform distributions.
We apply Lilliefors tests of the null hypothesis that each ob-
served
PGA
dataset is drawn from a log-normal population
distribution (
Lilliefors, 1967
). The null hypothesis cannot
be rejected for the three
PGA
datasets: (1)
YOH
2009,
p
-value
0
:
4617
; (2)
YOH
2009 updated,
p
-value
0
:
8070
; (3) Pacific Earthquake Engineering Research Center
(
PEER
)
NGA
,
p
-value
0
:
3236
; these observed PGAs could
have been drawn from a log-normal population distribution.
We apply Chi-square tests of the null hypothesis that each
1
The
NGA
project chose to employ the GMRotI50 algorithm (
Boore
et al.
,
2006
) to combine horizontal components of ground motion. The
NGA
S
a
are
reported at fewer periods than we calculate within the three ranges. Within
each range of periods, we find the geometric mean of the pseudo-
S
d
at the
periods given in the
NGA
database.
2
In our original paper we incorrectly stated the algorithm we use to cal-
culate
PGA
(or
PGD
) for the recorded ground motions. We calculate a peak
ground measure by first finding the maximum squared acceleration (dis-
placement) for each horizontal component time history. The
PGA
(
PGD
)
is the square root of the sum of the squared maxima.
919
Bulletin of the Seismological Society of America, Vol. 101, No. 2, pp. 919
–
924, April 2011, doi: 10.1785/0120100210
PGD
dataset is drawn from a truncated, uniform population
distribution on a logarithmic scale (see, for example,
Devore,
2000
, section 14.1). We limit the range of the log-uniform
distribution because we select records on a limited range
of magnitudes. These distributions,
p
uniform
, are defined as
p
uniform
1
;
1
:
65
≤
log
10
PGD
≤
0
:
05
0
;
otherwise
(1)
for our original and updated empirical distributions, and
p
uniform
1
;
1
:
35
≤
log
10
PGD
≤
0
:
15
0
;
otherwise
(2)
for the
NGA
distribution of
PGD
. The null hypothesis cannot
be rejected for the three
PGD
datasets (YOH2009:
p
-value
0
:
2630
; YOH2009 updated:
p
-value
0
:
2405
;
PEER NGA
:
p
-value
0
:
0799
); these observations could
have been drawn from a truncated, log-uniform population
distribution. To be clear, these statistical tests do not demon-
strate that the population distributions of
PGA
and
PGD
in
the near source of large earthquakes are log-normal and log-
uniform, respectively. Rather, the available observations are
consistent with these population distributions.
There is no question that seismic hazards exist through-
out the world. However, the question of how to conceptualize
and quantify these hazards remains. How do we as a seismo-
logical and earthquake engineering community understand
the likelihood and intensity of future ground motions? There
should be several, distinct approaches to quantifying seismic
10
−2
10
−1
10
0
10
1
10
−2
10
−1
10
0
10
1
10
−2
10
−1
10
0
10
1
10
−2
10
−1
10
0
10
1
PGD [m]
10
−2
10
−1
10
0
10
1
PGD [m]
10
−2
10
−1
10
0
10
1
PGD [m]
S
d
(T =3−5 s) [m]
S
d
(T =7−9 s) [m]
S
d
(T =5−7 s) [m]
R = 0.941
PEER NGA
YOH2009
R = 0.971
PEER NGA
YOH2009
R = 0.975
PEER NGA
YOH2009
(a)
(b)
(c)
Figure
1.
Peak ground displacement (
PGD
) versus spectral displacement (
S
d
) for three ranges of long-period (
T
)
S
d
: (a)
T
3
–
5
s;
(b)
T
5
–
7
s; and (c)
T
7
–
9
s. Circles represent data from the
NGA
database, and crosses represent data from
Yamada
et al.
(2009)
.For
periods of 3
–
5 s, the slope of
PGD
versus
S
d
is closest to one. For periods of 7
–
9 s, the correlation coefficient,
R
, between the logarithms of
PGD
and
S
d
is largest.
Table 1
Earthquakes since 2004 That Provided Near-Source Records
from Large Earthquakes*
Earthquake
M
w
†
N
Focal
Depth
†
Fault Model
2007 Noto-hanto
6.7
3
8.0
Kurahashi
et al.
,
2008
2007 Chuetsu-oki
6.6 10
12.0
Aoi
et al.
, 2008
2008 Wenchuan
7.9
6
19.0
Koketsu
et al.
,
2010
2008 Iwate
–
Miyagi
Nairiku
6.9
7
7.8
Suzuki
et al.
,
2010
2009 Suruga-wan
6.4
2
26.8
Aoi
et al.
, 2010
*Near-source records are within 10 km of the surface projection of the
rupture; large earthquakes are those with magnitudes
≥
6
:
4
.
†
The magnitude and focal depth for the Suruga-wan earthquake are from
the United States Geological Survey; the magnitudes and focal depths for all
other earthquakes are from the Harvard CMT model. This table augments
table 1 in
Yamada
et al.
(2009)
. (See the
Data and Resources
section for
descriptions of the data sources.)
1
10
100
1000
0
5
10
15
20
25
30
PGA [m/s
2
]
Count
YOH2009
YOH2009 updated
PEER NGA
Figure
2.
Distributions of recorded peak ground accelerations
(
PGA
). The YOH2009 distribution was published in
Yamada
et al.
(2009)
. The YOH2009 updated distribution adds records since 2004
to the original distribution. The
PEER NGA
distribution uses data
from the current
NGA
database. The three
PGA
distributions are
consistent with a log-normal distribution. There has been no
well-recorded large earthquake since 2004, which we expect would
change the
PGD
distribution in Figure
3
but not the
PGA
distribu-
tions here.
920
Reply
hazard, because, if the approaches provide consistent results,
the community can have greater confidence in our under-
standing of seismic hazard. If these various approaches pro-
vide inconsistent results, then we must identify the source of
the inconsistencies and work to improve our understanding.
Returning to the present discussion of future long-period
ground motions, we seek to determine whether different
techniques of quantifying this seismic hazard are consistent.
We begin with an example from recent work of the Tall
Buildings Initiative of the Pacific Earthquake Engineering
Research Center. Their study analyzes the performance, both
structural and economic, of 40- or 42-story buildings de-
signed according to three procedures (
PEER, 2010
). To this
end, the researchers use five sets of 15 ground motions to
represent five levels of seismic hazard for a site in downtown
Los Angeles (see the
Data and Resources
section). The ha-
zard levels are defined by return periods of 4975, 2475 (the
maximum considered earthquake), 475 (the design basis
earthquake), 43, and 25 years. The ground motions repre-
senting these hazard levels result from selecting recorded
or simulated ground motions and then scaling them accord-
ing to the technique of spectrum matching (
Jones and
Zareian, 2010
). Figure
3
shows the
PGD
s that represent the
4975- and 2475-year hazard levels compared with the em-
pirical distributions of near-source displacements. According
to the Tall Buildings Initiative, the 4975-year hazard level
represents
“
extremely rare shaking . . . which is well beyond
the ground motion level generally considered in the building
industry
”
(
PEER, 2010
). However, when we compare the
PGD
s representing the 4975- and 2475-year hazards to our
set of observed
PGD
s, these rare ground motions are smaller
than the largest recorded observations. In fact, the geometric
mean
PGD
within the 4975-year hazard level (1.00 m) is
much smaller than the most extreme observed
PGD
(2.99 m).
Given the considerable number of identified faults with the
capacity to generate large earthquakes, and an unknown
number of unidentified faults, in the Los Angeles basin and
proximate areas (
Dolan
et al.
, 1995
), why are 1-in-4975-year
ground motions for a site in downtown Los Angeles no larger
than worldwide observations from the last 30 years?
0.01
0.1
1
10
0
5
10
15
20
PGD [m]
Count
YOH2009
YOH2009 updated
PEER NGA
0.01
0.1
1
10
M 7.15
2475−yr
4975−yr
PGD [m]
(a)
(b)
Figure
3.
Distributions of (a) recorded peak ground displacement (
PGD
) and (b) box-and-whisker plots for sites in the Los Angeles area.
The YOH2009, YOH2009 updated, and
PEER NGA
distributions represent the same datasets described in the caption for Figure
2
. The three
PGD
distributions at the top of this figure are consistent with a truncated log-uniform distribution. In (b), we locate the quartiles of three
additional datasets. A probabilistic seismic hazard analysis for a site in downtown Los Angeles used 15 ground motions to represent the
4975-year hazard and an additional 15 motions to represent the 2475-year hazard.
Graves and Somerville (2005)
simulated five
M
7.15
ruptures on the Puente Hills fault system and generated ground motions at 648 sites in the Los Angeles region (for a total of 3240 motions
represented here).
Reply
921
We can also compare the
PGD
s of the ground motions
representing the seismic hazard at a site in downtown Los
Angeles to the
PGD
s of simulated ground motions resulting
from hypothetical ruptures of the Puente Hills fault system.
This fault system underlies downtown Los Angeles and the
areas immediately east; it has generated at least four large,
blind thrust earthquakes in the past 11,000 years (
Dolan
et al.
,
2003
). Converting this observation to a recurrence interval
suggests that the Puente Hills fault system generates a mag-
nitude 7.2 to 7.5 earthquake with an average recurrence of
2800 years.
Graves and Somerville (2005)
simulated broad-
band ground motions throughout the Los Angeles basin from
a hypothetical
M
7.15 rupture of this fault (see the
Data and
Resources
section). Figure
3
compares the
PGD
s at 648 sites
for five scenario ruptures with the
PGD
softhe
PSHA
de-
scribed previously. The Graves and Somerville simulations
of a roughly 2800-year event produce numerous
PGD
sin
excess of the
PGD
s representing the 4975-year hazard level.
Granted, the hazard analysis is site-specific, and the Graves
and Somerville ground motions are for sites throughout the
Los Angeles basin. However, is our knowledge of future
ground motions at sites in downtown Los Angeles so certain
that it precludes the possibility of displacements at least as
large as the largest displacements in these simulations?
Accurately characterizing the hazard of long-period
ground motions is important for assessing the performance
of long-period structures and, in particular, for designing tall
buildings in areas of high seismicity. In a recent study,
Jones
and Zareian (2010)
applied the 75, spectrum matched ground
motions just mentioned to 40-story building models of a
buckling-restrained braced frame, designed according to
three procedures: (1) the 2006 International Building Code
(abbreviated
CBD
, following Jones and Zareian); (2) the Los
Angeles Tall Buildings Structural Design Council Alterna-
tive Procedures for Tall Buildings (
PBD
); and (3) the
PEER
Tall Buildings Initiative Guidelines (
PBD
). Jones and
Zareian found that one of the 2,475-year hazard ground
motions caused an interstory drift ratio (
IDR
) greater than
3%, which they deem the collapse prevention limit, in the
PBD
building model. For the same building, three of the
4975-year hazard ground motions induced responses in
excess of this limit. For the
CBD
building model, one 4975-
year hazard ground motion induced a response greater than
this limit. None of the 2475- or 4975-year hazard ground
motions caused the
PBD
building model to exceed the
collapse prevention limit. We assume that the ground mo-
tions causing the largest building responses have the largest
PGD
s within each hazard level. Therefore, ground motions
with
PGD
s of roughly 1.3 m or greater can cause 40-story
building models of buckling-restrained braced frames,
designed with current techniques, to develop
IDR
s larger than
3%, the value associated with collapse prevention.
Jones and Zareian (2010)
also applied two sets of nine
near-fault ground motions from the
Graves and Somerville
(2005)
Puente Hills simulations. Of these 18 ground
motions: 28% caused the
CBD
model to exceed the collapse
prevention limit; 33% caused the
PBD
model to exceed this
limit; and 50% caused the
PBD
model to exceed this limit.
Certainly, nonlinear time history analyses of 40-story build-
ings with different lateral force-resisting systems, or analyses
of tall buildings with more or fewer stories, will find different
cutoffs for ground motions that cause buildings to exceed
their collapse prevention limits. Nonetheless,
PGD
sof1
–
2m
should be of concern when assessing tall buildings. A recent
PSHA
for a site in downtown Los Angeles at this level of
ground displacement seems to be inconsistent with the past
30 years of recorded ground motions and inconsistent with
realistic simulations of ruptures on a well-studied fault
system. If these hazard assessments are too low, then the
calculated probabilities of unsafe building responses are also
too low.
There are well-known concerns with each model of a
PSHA
(that is, the definition of the earthquake sources,
magnitude-frequency relationships, and ground-motion pre-
diction equations). Quantifying the uncertainties associated
with each model is robust for near-source short-period
ground motions because there is sufficient available data
from past events to establish that the distribution is approxi-
mately log-normal and independent of the fault slip. Data on
near-source long-period ground motions from large-slip
earthquakes are not so readily available, although the occur-
rence of just a small number of large urban earthquakes could
dramatically change that.
Estimating the probability of near-source shaking is
further complicated by the fact that there are numerous
examples of very damaging earthquakes caused by either
unknown or low slip-rate faults. For example, the 1994
Northridge (
M
w
6.7) and 2008 Iwate
–
Miyagi Nairiku
(
M
w
6.9;
Toda
et al.
, 2010
) earthquakes ruptured previously
unidentified faults. In a seismically active region, not iden-
tifying a fault has less severe consequences for short-period
ground-motion prediction as compared with long-period.
Since
PGA
saturates with magnitude, identifying a new fault
would not significantly change the expectation of a reason-
ably large
PGA
. If a major fault is not included in a
PSHA
,
there may be an unaccounted potential for a larger
PGD
than
otherwise expected.
Second, magnitude-frequency relationships are not well-
constrained at large magnitudes because of the paucity of
data. A
PSHA
might extrapolate the Gutenberg
–
Richter rela-
tionship from frequent events, or place an upper bound on
this relationship, or define a characteristic earthquake (
Kra-
mer, 1996
, section 4.4.1.2). These various approaches reflect
our incomplete knowledge of the recurrence of large earth-
quakes. Propagating this uncertainty to the prediction of
future ground motions should result in greater uncertainties
for long-period, but not necessarily for short-period, ground
motions. Further, there have been observations of large slips
(and thus, large magnitudes) in areas previously believed to
have only a moderate seismic hazard. The 1976 earthquake
in Tangshan, China (
Allen
et al.
, 1984
), and the 1995 Kobe
922
Reply
earthquake on Awaji Island (
Toda
et al.
, 1996
), are examples
of this.
Third, there is not a lot of data to define robust ground-
motion prediction equations in the near source of large earth-
quakes. We collect 174 near-source records from past large
earthquakes (146 in
Yamada
et al.
, 2009
and 28 additional
records here). In the current
NGA
database there are 129 such
records, accounting for only 3.6% of the total database.
Ground-motion prediction equations rely on the character of
empirical data from larger distances and smaller magnitudes
to inform the predictions of ground motions in the near source
of large magnitude earthquakes. These assumptions may not
be consistent with scaling relations for fault slip derived from
teleseismic observations of many large earthquakes.
Not only are the large uncertainties in future long-period
ground motions the result of a relatively short instrumental
record, these large uncertainties result from our limited the-
oretical knowledge of earthquake sources. As shown in our
original paper, magnitude and
PGD
are correlated in the near
source (also see
Aagaard and Heaton, 2004
). However, the
physical quantity measured by magnitude is the product of
crustal rigidity, the rupture area, and the average slip. If we
seek a predictive theory for
PGD
in the near source, we must
anticipate the distribution of slip on a fault (and thus average
slip) and anticipate the rupture area. Compilations of source
models (for example, the Finite-Source Rupture Model
Database,
Mai, 2010
) demonstrate that slip can have a con-
siderable spatial variability and that slip is difficult to reliably
estimate, as evidenced by many plausible slip distributions
inferred from the same past event. Theories of slip initiation
and termination are still open to debate, thus making any the-
oretical prediction of rupture area quite uncertain. We believe
the best current theory applicable to the prediction of
PGD
in
the near source are the scaling relations we employed in our
original paper.
At present, predictions of future long-period ground
motions from probabilistic seismic hazard analyses appear
inconsistent with and smaller than past observations and
plausible rupture simulations. Perhaps current
PSHA
s do not
properly account for the inherent uncertainties in predicting
near-source ground displacements. These uncertainties exist
because near-source ground displacements at the surface are
strongly correlated with the amplitude of slip on nearby
patches of fault rupture, and anticipating fault slip is highly
uncertain given our current knowledge. Until we have en-
ough data or a reliable theory, any prediction of long-period
ground motion in the near source of large earthquakes will
remain highly uncertain.
Data and Resources
The Pacific Earthquake Engineering Research Center
’
s
Next Generation Attenuation database flatfile can be down-
loaded at
http://peer.berkeley.edu/nga/
. We use public ver-
sion 7.3 in this work (accessed 12 June 2010). The strong
motion data for the 2007 Noto-hanto, 2007 Chuetsu-oki,
2008 Iwate
–
Miyagi Nairiku, and 2009 Suruga-wan earth-
quakes were recorded by the K-NET and KiK-net seismic
networks, operated by the National Research Institute
for Earth Science and Disaster Prevention in Japan (
www.
kyoshin.bosai.go.jp
; last accessed September 2010). The
dataset of the 2008 Wenchuan earthquake was recorded
by the National Strong Motion Observation Network System
of China, managed by the National Strong Motion Observa-
tion Center and local strong motion observation centers
(
www.csmnc.net/selnewxjx1.asp?id=749
; last accessed
September 2010).
Farzin Zareian, in personal communication of June 2010,
provided the 75 ground motions used in the
PEER
study of
downtown Los Angeles. In a personal communication of
March 2007, Robert Graves provided the ground-motion
time histories described in
Graves and Somerville (2005)
.
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Kyoto University
Gokasho, Uji, 611-0011 Japan
masumi@eqh.dpri.kyoto
‑
u.ac.jp
(M.Y.)
University of Colorado at Boulder
UCB 428
Boulder, Colorado 80309
anna.olsen@colorado.edu
(A.H.O.)
California Institute of Technology
MC104-44, 1200 East California Boulevard
Pasadena, California 91125
heaton_t@caltech.edu
(T.H.H.)
Manuscript received 3 August 2010
924
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