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Model for Basin Effects on Long-Period

Response Spectra in Southern

California

StevenM.Day,

a)

Robert Graves,

b)

Jacobo Bielak,

c)

Douglas Dreger,

d)

Shawn Larsen,

e)

Kim B. Olsen,

a)

Arben Pitarka,

b)

and

Leonardo Ramirez-Guzman

c)

We propose a model for the effect of sedimentary basin depth on long-

period response spectra. The model is based on the analysis of 3-D numerical

simulations (ﬁnite element and ﬁnite difference) of long-period

共2–10 s兲

ground motions for a suite of sixty scenario earthquakes (Mw 6.3 to Mw 7.1)

within the Los Angeles basin region. We ﬁnd depth to the

1.5 km/s S-wave

velocity isosurface to be a suitable predictor variable, and also present

alternative versions of the model based on depths to the 1.0 and

2.5 km/s

isosurfaces. The resulting mean basin-depth effect is period dependent, and

both smoother (as a function of period and depth) and higher in amplitude than

predictions from local 1-D models. The main requirement for the use of the

results in construction of attenuation relationships is determining the extent to

which the basin effect, as deﬁned and quantiﬁed in this study, is already

accounted for implicitly in existing attenuation relationships, through (1)

departures of the average “rock” site from our idealized reference model, and

(2) correlation of basin depth with other predictor variables (such as

Vs

30

). 关DOI: 10.1193/1.2857545兴

INTRODUCTION

The entrapment and ampliﬁcation of seismic waves by deep sedimentary basins pro-

duces important effects on seismic waveﬁelds (e.g., King and Tucker 1984; Field 1996;

Joyner 2000) and response spectra (e.g., Trifunac and Lee 1978; Campbell 1997; Field

2000; Choi et al. 2005). These ampliﬁcation effects are three-dimensional, and they are

difﬁcult to quantify empirically with currently available strong motion data, especially

for periods longer than

1 second and sedimentary thicknesses exceeding 3kmor so

(Campbell and Bozorgnia 2008).

Numerical simulations of earthquake ground motion have the potential to comple-

ment empirical methods for the study of basin effects. A number of studies have simu-

lated 3-D seismic wave propagation in regional geological models that include basin

a)

Department of Geological Sciences, San Diego State University, San Diego CA 92182

b)

URS Corporation, 566 El Dorado Street, Pasadena CA 91101

c)

Department of Civil and Environmental Engineering, Carnegie-Mellon University, Pittsburgh PA 15213

d)

Berkeley Seismological Laboratory, University of California, Berkeley, Berkeley CA 94720

e)

Lawrence Livermore National Laboratory, Livermore CA 94550

257

Earthquake Spectra, Volume 24, No. 1, pages 257–277, February 2008; © 2008, Earthquake Engineering Research Institute

structure (e.g., Frankel and Vidale 1992; Olsen 1994; Olsen et al. 1995; Pitarka et al.

1998; Graves et al. 1998). Moreover, the recent availability of comprehensive 3-D earth

models for southern California (e.g., Magistrale et al. 2000; Kohler et al. 2003; Suss and

Shaw 2003) has substantially advanced our capability for simulating ground motion in

that region (e.g., Olsen 2000; Komatitsch et al. 2004; Olsen et al. 2006). In the current

study, we simulate ground motions for a set of earthquake scenarios for southern Cali-

fornia, in an effort to quantify the effects of sedimentar y basins on long-period

共ⱖ ⬃ 2 seconds兲 response spectra. The study employs both ﬁnite element (FE) and ﬁnite

difference (FD) methods to compute ground motion from propagating earthquake

sources, using the Southern California Earthquake Center (SCEC) Community Velocity

Model (CVM), a 3-D seismic velocity model for southern California (Magistrale et al.

2000).

For the current investigation, we compute long-period ground motion in the SCEC

CVM for a suite of 60 earthquake scenarios. The 3-component ground motion time his-

tories from these scenarios are saved on a grid of 1600 sites covering the Los Angeles

region, including sites in the Los Angeles, San Fernando, and San Gabriel basins, as well

as rock sites in the intervening areas. The results from the study take 2 forms: (1) We

have saved and archived a library of time histories from the 60 scenarios. In cooperation

with the SCEC Community Modeling Environment project, these time histories are

available online, through a web interface specialized to engineering applications (http://

sceclib.sdsc.edu/LAWeb). These long-period time histories capture basin ampliﬁcations,

rupture-propagation-induced directivity, and 3-D seismic focusing phenomena. They are

suitable for the engineering analyses of large, long-period structures, and smaller struc-

tures undergoing large, nonlinear deformations. (2) The results of the simulation suite

have been analyzed to estimate response spectral ampliﬁcation effects as a function of

basin depth and period. The resulting mean response has been characterized parametri-

cally and provided to the Next Generation Attenuation (NGA) project (Power et al.

2008) to guide development of attenuation relations in the empirical (NGA-E) phase of

the project.

COMPUTATIONAL METHODS

The computational program was a multi-institutional collaboration requiring major

computing resources. In order to make effective use of supercomputing resources avail-

able to the respective collaborators, as well as to have cross-checks on the methodology,

we employed ﬁve independently developed 3D wave propagation codes to do the nu-

merical simulations. Four of these are FD codes (Olsen 1994; Larsen and Schultz 1995;

Graves 1996; Pitarka 1999). These four are very similar in their mathematical formula-

tion. Each uses a uniform, structured, cubic mesh, with staggered locations of the ve-

locity and stress components, and a differencing scheme that is fourth-order accurate in

space and second-order accurate in time. The codes differ in their computational ap-

proaches, their implementation of absorbing boundary conditions, and their implemen-

tation of anelastic attenuation. We used a FE code (Bao et al. 1998) for some of the

simulations. The FE code uses unstructured meshing, and is second-order accurate in

space and time.

258 DAYETAL.

Two of the FD schemes (Olsen, Graves) approximate a frequency-independent seis-

mic quality factor

共Q兲 by implementing anelastic losses using the coarse memory vari-

ables representation (Day 1998; Day and Bradley 2001; Graves and Day 2003). Another

(Larsen) uses a standard linear solid formulation that represents the absorption spectrum

with a single Debye peak. The fourth FD scheme (Pitarka) represents attenuation by the

method of Graves (1996), which is equivalent to mass-proportional Rayleigh damping

and results in

Q proportional to frequency (i.e., a red absorption spectrum). The FE

scheme also uses the mass-proportional Rayleigh damping approximation to represent

anelastic loss.

Comparison of results among these codes is useful for verifying the mathematical

soundness of the ﬁve simulation codes. Such comparisons also permit assessment of any

artifacts attributable to the absorbing boundary and attenuation implementations. Day

et al. (2001, 2003) carried out a set of test simulations using all ﬁve codes. The com-

parisons between FD and FE solutions are particularly informative, as they permit an

assessment of solution variability introduced by the model discretization. We show an

example of such a comparison in a later section. The comparisons verify that all ﬁve

codes are accurate for the class of problems relevant to this study.

These tests also enabled us to improve computational efﬁciency by modifying the

SCEC CVM to eliminate very low seismic wavespeeds. Computing time is sensitive to

the ratio of highest to lowest wavespeed present in the model, with low-wavespeed vol-

umes requiring ﬁner meshing than high-wavespeed volumes to ensure a given accuracy

over a given bandwidth. The unstructured meshing possible with the FE method permit-

ted us to perform a few simulations that include the very low-velocity, near-surface sedi-

ments present in the CVM (

S velocity as low as 180 m/ s). We compared these with cal-

culations in which we put a lower threshold on the velocity model to exclude

S wave

velocity values in the CVM that fall below

500 m/ s (replacing the lower values with the

500 m/ s threshold value). The tests conﬁrm that imposing this threshold (for the sake of

computational efﬁciency) has negligible effect within the target bandwidth of

0–0.5 Hz.

Olsen et al. (2003) carried out simulations of the 1994 Northridge, California, earth-

quake, using the SCEC CVM, with the same FD method used in the current study, and

their comparisons of synthetic and recorded ground velocities demonstrate the ability of

the numerical modeling procedures to capture basin ampliﬁcation effects. Further vali-

dation is provided by comparison of synthetic (FD and FE) and recorded seismic wave-

forms of small southern Califor nia earthquakes. For example, Chen et al. (2007) ﬁnd

that FD synthetics computed with the SCEC CVM model give a variance reduction of

greater than 60% in both phase-delay times and log of amplitude, relative to a standard

1-D model. Additional validation for the velocity structure used in this study is provided

by sonic log data, on the basis of which Stewart et al. (2005) estimate that uncertainties

in basin depth in the SCEC CVM introduce uncertainties in ground motion (up to 0.1

natural log units) that they judge to be small compared with typical error terms in

attenuation relations.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 259

EARTHQUAKE SCENARIOS

We model sources on ten different faults, or fault conﬁgurations (for example, the

Puente Hills fault is modeled in 3 different segmentation conﬁgurations). For each fault,

we simulate 6 sources, using combinations of 3 different static slip distributions and 2

hypocenter locations. These are kinematic simulations: Rupture velocity, static slip, and

the form of the slip velocity function are all speciﬁed a priori.

The areal coverage for the 3-D models is the

100 km⫻ 100 km region outlined by

the large gray box in Figure 1. In all simulations, the boundaries of the computational

domain (i.e., absorbing boundaries) lie at or outside of this area and extend to a depth of

at least

30 km. For the FD calculations, a uniform grid spacing of 200 m was used. The

FE grid uses a variable element size, with near-surface elements as small as

30 m in

dimension.

Figure 1. Map of scenario events and model region. See Table 1 for fault names and event

magnitudes. Each rectangle is the surface projection of one of the faults, with the upper edge

shown as a solid line and the other three edges shown as dashed lines. The large gray rectangle

indicates the computational domain of the simulations.

260 DAYETAL.

We use the 10 faults listed in Table 1 for the scenario calculations. The fault surfaces

are simpliﬁed representations of the f ault geometry given by the SCEC Community

Fault Model, and our choice of rupture scenarios was guided in part by the geologic con-

siderations surveyed by Dolan et al. (1995). The surface projections of these faults are

also shown in Figure 1. The longitude and latitude coordinates in this table refer to the

geographic location of the top center of the fault, that is, the point on the surface that is

directly above the midpoint of the top edge of the fault. Strike, dip and rake follow Aki

and Richards’ (1980) convention. Length, width and depth are all given in km. The

depth refers to the depth below the surface of the top edge of the fault (0 corresponding

to a surface-rupturing event).

For each of the fault geometries, we generate 3 random slip distributions, as realiza-

tions of a stochastic model, for use in the simulations. The slip distributions are gener-

ated following some empirical rules for the size and distribution of asperities as given by

Somerville et al. (1999). The slip values on the fault are drawn from a uniformly dis-

tributed random variable, then spatially ﬁltered to give a spectral decay inversely pro-

portional to wavenumber squared, with a corner wavenumber at approximately

1/L,

where

L is f ault length. Finally, the slip values are scaled to the target moment of the

scenario. As an example, Figure 2 shows the slip distribution functions generated by this

procedure for one of the faults (Compton, fault number 8 in Figure 1). The two hypo-

center locations (shown as stars in Figure 2) are deﬁned as follows for each fault: Hy-

pocenter 1 is located at an along-strike distance of 0.25 of the fault length and at a

down-dip distance of 0.7 of the fault width (measured within the fault plane from the top

edge of the fault, not the ground surface). Hypocenter 2 is located at an along strike

distance of 0.75 of the fault length and at a down-dip distance of 0.7 of the fault width.

Table 1. List of fault rupture scenarios

Fault Lon Lat M

w

Length Width Strike Dip Rake Depth

r

1)smad −118.178 34.242 7.0 61 18 288 53 90 0 1.4

2)smon1

−118.479 34.039 6.3 14 14 261 36 45 1 0.63

3)hwood

−118.343 34.099 6.4 14 19 256 69 70 0 0.71

4)raym2

−118.128 34.139 6.6 26 17 258 69 70 0 0.89

5)ph2e

−118.004 33.904 6.8 25 27 268 27 90 3 1.1

6)phla

−118.228 34.003 6.7 21 26 293 28 90 3 1.0

7)phall

−118.102 33.967 7.1 46 27 289 27 90 2 1.6

8)comp

−118.344 33.843 6.9 63 14 306 22 90 5 1.3

9)nin

−118.202 33.868 6.9 51 16 319 90 180 0 1.3

10)whitn

−117.876 33.933 6.7 35 15 297 73 180 0 1.0

(1) Lengths in kilometers, times in seconds, angles in degrees

(2)

r

is slip duration (from Equation 1)

(3) Geographical and depth coordinates refer to center of upper fault edge

(4) Faults are Sierra Madre (smad), Santa Monica (smon1), Hollywood (hwood), Puente Hills (northern segment

is ph2e, southern segment is phla, combined scenario is phall), Compton (comp), Newport-Inglewood (nin), and

Whittier (whitn).

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 261

The slip velocity function for each simulation is an isosceles triangle with a base of

duration

r

. The value of

r

is magnitude dependent and given by the empirically derived

expression (Somerville et al. 1999):

log

10

r

= 0.5M

w

− 3.35, 共1兲

where M

w

is moment magnitude and

r

is in seconds. Values used for

r

are given in

Table 1. Rupture velocity is constant for all faults and all slip models. This value is set

at

2.8 km/s. The rupture starts at the hypocenter and spreads radially outward from this

point at the speciﬁed velocity. The simulated duration for each scenario is

80 seconds.

All simulations use the SCEC CVM, Version 2, except for modiﬁcations described

below to impose a lower limit on the velocities and add anelastic attenuation. The un-

modiﬁed model is described in Magistrale et al. (2000). The SCEC model is modiﬁed as

follows: We replace the SCEC model

S velocity with the value 500 m/ s whenever the

SCEC model value falls below

500 m/ s. Whenever this minimum S velocity is imposed,

the

P wave velocity is set equal to 3 times the S velocity (1500 m / s in this case). Den-

sity values follow the SCEC model without modiﬁcation. The quality factors for

P and

S waves, respectively, Q

p

and Q

s

, are set to the preferred Q model of Olsen et al. (2003).

The 3-component time histories are saved on a

2km⫻ 2kmgrid covering the inner

80 km⫻ 80 km portion of the model area (Figure 3). No ﬁltering is applied to the out-

put. The resulting synthetic data set contains 3-component records for 1600 sites (4800

time histories) for each scenario simulation. For all 60 scenarios, and all sites, we com-

pute response spectral acceleration (Sa), for 5% damping, as a function of period, for

Figure 2. The three slip distributions and two hypocenter locations (stars) used in combination

to provide sources for the six Compton Fault simulations. The distributions were generated by

the stochastic approach described in the text.

262 DAYETAL.

each component of motion. This is done for 26 periods in the range 2 – 10 seconds:

Spectral acceleration is computed at

0.2 second intervals between 2 and 5 seconds, and

at

0.5 second intervals between 5 and 10 seconds.

The earthquake scenarios in this study are limited to events on faults near or within

the main sedimentary basins of the Los Angeles region. We have excluded earthquake

scenarios for the more distant San Andreas fault, for which several earthquake scenarios

of Mw greater than 7.5 have been proposed (e.g., Working Group on California Earth-

quake Probabilities 1995). Olsen et al. (2006; 2008) simulate large San Andreas fault

earthquakes, ﬁnding unusually strong basin effects for some events. In particular, in their

simulations the sequence of basins south of the Transverse Ranges, between the San An-

dreas and Los Angeles, acts as a waveguide. The resultant channeling of seismic energy

into the Los Angeles region produces anomalously high amplitudes at some relatively

distant basin sites.

Figure 3. Map showing the grid of time-history output sites (dots). Sites shown as triangles are

those at which we cross-checked results from difference simulation codes.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 263

COMPARISON OF FE AND FD SOLUTIONS

As noted above, the FD simulations used a uniform grid spacing of

200 m. Since

seismic velocities as low as

500 m/ s are present in the model, and the target frequency

band of the simulations is

0–0.5 Hz, the FD simulations resolve the minimum wave-

length with about 5 grid intervals. At this resolution, phase-velocity errors over homo-

geneous paths are typically very small (for the fourth-order staggered-grid FD method

used here), of the order of 1% or less (Levander 1988). However, it is difﬁcult to trans-

late this measure of accuracy into accuracy of ground motion time histories calculated

over complex paths. To conﬁr m that the FD simulations indeed predict ground motion

time histories accurately within our target frequency band, we repeated some of them

using the FE method, with a highly oversampled mesh (feasible because of the unstruc-

tured meshing capability of the FE method). The FE simulations employed node spacing

as small as

30 m in the low-velocity parts of the model. Thus, even accounting for the

approximate factor of two difference in points-per-wavelength requirement for a given

phase-velocity accuracy between the (fourth-order accurate) FD method and the

(second-order accurate) FE method, the FE simulations can be expected to have about

three times better resolution of the minimum wavelength in the basins. Figure 4 com-

pares three-component velocity time-histories for one of the scenarios (Newport-

Inglewood fault), as computed by FE and FD methods, respectively. The time histories

are for the 16 locations denoted by triangles in Figure 3. In nearly all cases the differ-

ences between the solutions are negligible, in that relative phase-delay times for the

dominant arrivals never exceed a few tenths of a second, and their amplitude differences

rarely reach 10%.

Figure 5 provides a quantitative comparison in the frequency domain. The ﬁgure was

constructed from smoothed (

0.1 Hz band averages) Fourier spectra of the 16 FD and 16

FE EW-component velocity time histories in Figure 4. The ﬁgure shows means and stan-

dard deviations (over the 16 recordings) of the natural logarithm of the FD to FE spectral

ratio, for each independent frequency band. The FD/FE bias is less than 10% (and scat-

ter less than 25%) for all frequencies in our target band of

0–0.5 Hz. Given the heavy

oversampling achieved for the FE solution, this ag reement provides strong evidence that

both methods have adequate resolution to be accurate throughout the target frequency

band. It also shows that there are no signiﬁcant biases introduced by the different ways

in which the two methods model anelastic attenuation.

We have made similar comparisons for 10 such simulation pairs (i.e., comparing re-

sults from either a pair of FD codes or from an FD FE pair). Because the FD and FE

computational meshes are very different, sampling the SCEC CVM at different points

(and with much higher resolution in the near surface in the case of the FE grid), the

FE/FD comparison represents the worst case for achieving agreement between codes.

Simulations of the same scenario computed with different FD codes produce time his-

tories and spectra that are almost indistinguishable.

264 DAYETAL.

REFERENCE SIMULATIONS

To aid us in quantifying the effect of sedimentary basins on the computed ground

motions, we perform several auxiliary, or “reference,” simulations (using the same FE

and FD methods used for all the other simulations). For each of the 10 faults, we select

one rupture scenario, and repeat that simulation using the same source model, but re-

placing the SCEC CVM with a horizontally stratiﬁed (1-D) model. The stratiﬁed refer-

ence model corresponds to an artiﬁcially high-velocity, unweathered hard-rock site. This

reference velocity model was constructed by laterally extending a vertical proﬁle of the

Figure 4. Comparson of ﬁnite element and ﬁnite difference solutions for one of the Newport-

Inglewood rupture-scenario simulations. Velocity time histories are shown for the 16 sites

shown as triangles in Figure 3. North-south (NS), east-west (EW) and up-down (UD) compo-

nent traces are given. The upper number of each pair on the left designates the station number,

as given in Figure 3. The lower number of each pair gives the peak velocity (for the component

having the largest peak), in cm/s.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 265

SCEC CVM located at (−118.08333, 34.29167), in the San Gabriel Mountains. As

noted, surface

S velocities are ar tiﬁcially high 共3.2 km/ s兲 in the resulting model, since

this part of the SCEC model does not account for a weathered layer. The principal pur-

pose of the reference simulations is to provide a spectral normalization for the results

from the simulations done in the full SCEC CMV, as an approximate means of isolating

basin effects from source effects.

Figure 6 shows velocity time histories for a Newport-Inglewood scenario (source and

recording sites are the same as used for the FD/FE comparison shown in Figure 4). The

ﬁgure compares three-component velocities for the simulation done with the SCEC

CVM with the corresponding velocities for the reference simulation. The comparison

gives an idea of the importance of 3-D structure, which introduces effects that are espe-

cially pronounced at long period and late in the time series.

RESPONSE SPECTRAL AMPLIFICATIONS

Basin ampliﬁcation effects result from interaction of the waveﬁeld with basin mar-

gins, and depend in a complex, poorly understood manner on period, source location,

source distance, basin geometry, sediment velocity distribution, and site location within

the basin. The 60 scenarios provide synthetic data that can be used to improve our un-

derstanding of these effects. We take an initial step in this direction by attempting to

isolate the effects of period and local basin depth. To isolate these two effects, we aver-

age over sources. As response spectral values vary much more between ruptures on dif-

ferent faults than between ruptures on a given fault, we have computed averages using

only 1 of the 6 scenarios from each fault, giving us a 10-event subset of the simulations.

This subset misses a small amount of the variability in basin response present in the full

60-event suite, but allows us to work with spectral values normalized to the reference

Figure 5. Ratio of smoothed (over 0.1 Hz bands) Fourier spectra of the EW-component FD and

FE solutions shown in Figure 4. Solid circles are averages (over the 16 sites) of the natural

logarithm of the FD to FE spectral ratio, and error bars give the corresponding standard

deviations.

266 DAYETAL.

structure, without requiring 60 reference-structure simulations. Tests using a small num-

ber of additional events conﬁrm that source effects have been adequately removed by

this procedure.

The synthetic time histories are band limited, and, although there is no abrupt spec-

tral cutoff at the

0.5 Hz limit, the synthetics rapidly become spectrally deﬁcient above

this frequency. This limitation will have the effect of biasing response spectral estimates

downward at frequencies near, yet still below, the

0.5 Hz cutoff (since each response

spectral ordinate is a ﬁnite-bandwidth measure of ground motion), compared with values

that would be calculated from full-bandwidth time histories. We have made a quantita-

tive estimate of this bias by calculating response spectra from 25 recordings of the 1992

Figure 6. Comparison of time histories for one of the 3D (SCEC CVM) simulations (Newport-

Inglewood rupture scenario) with the time histories for the corresponding reference (1-D rock

model) simulation.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 267

Landers, California, earthquake. We calculated the response spectra both before and af-

ter applying a low-pass ﬁlter to remove Fourier spectral components above

0.5 Hz.Ata

period of

2s(i.e., right at the upper frequency cutoff), the low-passed case has its re-

sponse spectrum biased downward by 45%. However, at

3speriod the bias is only 15%

and falls rapidly as the period lengthens further (e.g., to 4% at

4s). The actual bias in

our case will be even lower, as the synthetics do not have as sharp a spectral cutoff as we

created by ﬁltering the Landers earthquake data. Further more, any bias will be further

reduced because our presentation is in terms of spectral ratios. That is, in all cases we

normalize the response spectra by dividing them by response spectra for reference so-

lutions that have the same Fourier spectral limits (and as a result have similar response-

spectral bias). The effectiveness of the normalization in removing the short-period bias

is difﬁcult to quantify, but is qualitatively supported by the consistency and smoothness

with which the

2sspectra extrapolate trends deﬁned at longer period (as seen in the

results to be presented later). Therefore, we present the normalized response spectra for

periods as low as

2s, but the ordinates at periods below 3sshould be interpreted with

caution; only at periods of

3sand longer do we have quantitative corroboration that the

response spectra are nearly unbiased (i.e., within

⬃15% tolerance).

METHOD

We ﬁrst bin the sites according to the local basin depth D at a site, with D

j

denoting

the depth at site

j. For this purpose, we deﬁne the depth D to be the depth to a speciﬁed

S-wave velocity isosurface. We present results for the case

D=Z

1.5

, where Z

1.5

is depth

to the

1.5 km/s isosurface. Note, however, that in the SCEC CVM, the depths of dif-

ferent

S velocity isosurfaces are strongly correlated, and therefore very similar results

(apart from a scaling of the depth variable) are obtained using the 1.0 or

2.5 km/s iso-

surface (

Z

1.0

or Z

2.5

) instead of the 1.5 km/ s isosurface. The binning is represented

through a matrix

W. We deﬁne N

bin

bins by specifying depths D

q

bin

, q=1, ...N

bin

,atthe

bin centers, spaced at equal intervals

⌬D (i.e., D

q

bin

=共q −1/2兲⌬D, and then form W,

W

qj

=

再

1if共D

q

bin

− ⌬D/2兲 艋 D

j

⬍ 共D

q

bin

+ ⌬D/2兲

0 otherwise

冎

. 共2兲

For consistency with most empirical attenuation relations, we work with response

spectral values averaged over the two horizontal components. For the

ith event and jth

site, we form the ratio

Sa

ij

共T

k

兲/Sa

ij

ref

共T

k

兲, where Sa

ij

共T

k

兲 is the absolute spectral accel-

eration (geometrical mean of the two horizontal components) from SCEC-CVM event

i

at site j and period T

k

, and Sa

ij

ref

共T

k

兲 is the corresponding quantity for the corresponding

reference-model event. Then we form the source-averaged basin response factor

B共D

q

,T

k

兲 by taking the natural logarithm and averaging over all N

site

sites 共N

site

=1600兲, and over all N

ev

events, where in this case N

ev

is 10:

B共D

q

,T

k

兲 =

冉

N

ev

兺

j=1

N

site

W

qj

冊

−1

兺

i=1

N

ev

兺

j=1

N

site

W

qj

ln关Sa

ij

共T

k

兲/Sa

ij

ref

共T

k

兲兴. 共3兲

268 DAYETAL.

RESULTS

Figure 7 summarizes the results of this procedure (for 200 m bins). The upper frame

shows

B as a function of depth and period. The lower frame shows basin ampliﬁcation

calculated by the same procedure, but replacing the spectral acceleration ratio

Sa

ij

共T

k

兲/Sa

ij

ref

共T

k

兲 at each site by the vertically incident plane-wave ampliﬁcation factor

Figure 7. Top: Natural logarithm of basin ampliﬁcation versus depth (to 1.5 km/ s S-velocity

isosurface) and period, calculated from 3D simulations. Bottom: Natural logarithm of basin am-

pliﬁcation calculated by same procedure, but replacing the 3D results with 1-D plane-wave am-

pliﬁcation f actors calculated using the local 1-D wavespeed and density proﬁles (from the

SCEC CVM) at each of the 1600 sites.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 269

for that site. The latter factors were computed using a plane-layered structure speciﬁc to

each site, and corresponding to the SCEC-CVM shear wavespeed and density depth-

proﬁles directly beneath that site. The main results from Figure 7 are the following: (1)

Source-averaged basin ampliﬁcation is period dependent, with the highest ampliﬁcations

occurring for the longest periods and greatest basin depths. (2) Relative to the very-hard

rock reference structure, the maximum ampliﬁcation is about a factor of 8. (3) Com-

pared with 1-D theoretical predictions, the 3-D response is in most cases substantially

higher. (4) The 3-D response is also smoother, as a function of depth and period, than is

the 1-D prediction. We attribute the smoother depth and period dependence to the pres-

ence of laterally propagating waves in the 3-D case that smooth out the resonances

present in the 1-D case.

Figure 8 shows the standard deviations

s of the logarithm of ampliﬁcation, as a func-

tion of depth and period, that is,

s

2

共D

q

,T

k

兲 =

冉

N

ev

兺

j=1

N

site

W

qj

冊

−1

兺

i=1

N

ev

兺

j=1

N

site

W

qj

兵ln关Sa

ij

共T

k

兲/Sa

ij

ref

共T

k

兲兴 − B共D

q

,T

k

兲其

2

. 共4兲

Most values fall between 0.5 and 0.6, and there is a mild tendency for s to increase

at the short-period end of our range. The differences are small, but some period-

dependence of this sort is what one might expect on the basis of simple physical argu-

ments. Short-period waves are subject to short-wavelength variations due to local focus-

Figure 8. Standard deviation of the natural logarithm of basin ampliﬁcation, as function of

depth and period, from the 3D simulations.

270 DAYETAL.

ing and interference effects. Very long-period waves, in contrast, represent oscillations

that are coherent over large scale lengths and are inﬂuenced principally by large-scale

averages of the seismic velocity structure.

Figure 9 presents the basin-depth dependence of

B in the form of mean ampliﬁcation

values (open circles) and their standard deviations (vertical bars) for each of 3 periods

(2, 4, and

8s). For depths 共Z

1.5

兲 in the range of roughly 500–1000 m, mean ampliﬁca-

tion tends to decrease slightly with period, though the effect is small compared with the

scatter. This result is, at least qualitatively, in agreement with expectations from 1-D

theory: Shallow sediments will have diminished effect as the wavelength becomes long

relative to sediment depth. For depths exceeding about

2000 m, mean ampliﬁcation in-

creases systematically with period. This is a 3-D effect: Higher-mode resonances present

in the 1-D case are smoothed out by lateral scattering, so that the longer-period reso-

nances dominate. Numerical values of

B are given in Table 2 for a range of representa-

tive periods and depths.

PARAMETRIC MODEL

It is useful to have a simple functional form that captures the main elements of the

period- and depth-dependent basin ampliﬁcation behavior observed in the simulations.

One purpose of such a representation is to provide a functional form for representing

basin effects in regression modeling of empirical ground motion data. We constructed a

Figure 9. Natural logarithm of the basin ampliﬁcation factor, as a function of depth to the

1.5 km/ s S velocity isosurface. Solid circles of a given color represent the mean ampliﬁcation

factor for one response-spectral period, and the error bars give the standard deviations. For clar-

ity, only three periods (2, 4, and 8 s) are shown, out of the 26 periods calculated. Further nu-

merical results are given in Table 2. The dashed curves are the corresponding basin ampliﬁca-

tion factors calculated from the parametric model (Equation 5a and 5b and Table 3) ﬁt to the 3D

simulation results.

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 271

preliminary representation of this sort to provide guidance to the NGA development

teams. Our approximate representation,

B

˜

共D , T兲 takes the following form:

B

˜

共D,T兲 = a

0

共T兲 + a

1

共T兲关1 − exp共D/300兲兴 + a

2

共T兲关1 − exp共D/4000兲兴, 共5a兲

where

a

i

共T兲 = b

i

+ c

i

T, i = 0,1,2, 共5b兲

with T given in seconds and D in meters. This functional form is not itself based directly

upon physical considerations, but rather serves to summarize the practical results of the

simulations (which of course are themselves based on the physics of seismic wave

propagation). The particular function in Equation 5a and 5b was chosen because (i) it

allows for a basin-depth dependence with decreasing slope at increasing values of the

depth parameter, as required to capture the behavior shown in Figure 9, and (ii) it per-

mits the depth dependence to vary with period, as also required by Figure 9. The 6 pa-

rameters

b

i

, c

i

were calculated in a two-step procedure. Separate least squares ﬁts (at

each period

T

k

)ofB

˜

共D , T

k

兲 to B共D

q

,T

k

兲) gave individual estimates of the a

i

共T

k

兲 values

for each period

T

k

. Then parameters b

i

and c

i

, for each i =0,1,2, were obtained by least-

squares ﬁtting of these 26 individual

a

i

共T

k

兲 estimates. The resulting values are shown in

Table 3. We repeated the full analysis (normalizing, binning, and parameter ﬁtting) using

the 1.0 and

2.5 km/s isosurfaces, respectively, as depth parameters (i.e., setting

D=Z

1.0

and D=Z

2.5

, respectively), and those results are also shown in Table 3.

Table 2. Mean (and standard deviation) of natural log of ampliﬁcation, versus basin depth and

period

Z

1.5

(km)

Period (s)

23456810

0.3 0.54(0.66) 0.38(0.63) 0.44(0.57) 0.52(0.55) 0.59(0.54) 0.63(0.48) 0.63(0.48)

0.5 1.00(0.65) 0.89(0.55) 0.90(0.51) 0.94(0.53) 0.97(0.54) 0.94(0.51) 0.89(0.55)

0.7 1.16(0.72) 1.07(0.57) 1.04(0.54) 1.05(0.58) 1.08(0.59) 1.03(0.60) 0.98(0.63)

0.9 1.27(0.65) 1.23(0.54) 1.22(0.54) 1.25(0.58) 1.28(0.57) 1.21(0.57) 1.13(0.64)

1.1 1.34(0.66) 1.32(0.58) 1.35(0.53) 1.37(0.53) 1.36(0.51) 1.29(0.50) 1.21(0.56)

1.3 1.37(0.65) 1.37(0.57) 1.49(0.56) 1.56(0.57) 1.55(0.53) 1.47(0.49) 1.36(0.51)

1.5 1.45(0.66) 1.44(0.56) 1.57(0.57) 1.69(0.56) 1.71(0.51) 1.64(0.48) 1.51(0.48)

1.7 1.57(0.65) 1.57(0.56) 1.64(0.54) 1.76(0.54) 1.81(0.53) 1.77(0.52) 1.65(0.53)

1.9 1.64(0.62) 1.64(0.53) 1.73(0.51) 1.83(0.54) 1.92(0.53) 1.89(0.56) 1.80(0.58)

2.1 1.64(0.65) 1.63(0.58) 1.73(0.52) 1.84(0.53) 1.92(0.53) 1.91(0.53) 1.85(0.55)

2.3 1.62(0.59) 1.65(0.51) 1.75(0.54) 1.87(0.51) 1.97(0.52) 1.98(0.56) 1.96(0.59)

2.5 1.70(0.60) 1.70(0.52) 1.79(0.55) 1.94(0.50) 1.99(0.51) 2.07(0.55) 2.06(0.56)

2.7 1.90(0.55) 1.90(0.50) 2.07(0.53) 2.13(0.56) 2.15(0.54) 2.21(0.53) 2.21(0.51)

272 DAYETAL.

Three of the resulting ampliﬁcation curves (obtained by evaluating Equations 5a and

5b), for periods 2, 4, and

8s, are shown as dashed curves in Figure 9. These expressions,

despite their simplicity, represent the mean predictions of the numerical simulations

quite well, and can serve as a starting point for modeling basin effects in empirical stud-

ies. Because they give a compact representation of complex wave propagation effects

captured by the numerical simulations, they provide a physical basis for extrapolation of

empirical models to periods greater than 2 or

3 seconds, where reliable data on basin

effects are especially scarce. The standard deviations

s of the simulation results, given in

Figure 9 and Table 2, provide appropriate estimates of the standard errors of prediction

for use with Equations 5a and 5b (the misﬁt of Equations 5a and 5b to

B is very small

compared with

s, and has negligible effect on the prediction error).

DISCUSSION AND CONCLUSIONS

We have characterized the source-averaged effect of basin depth on spectral accel-

eration using depth to the

1.5 km/s S velocity isosurface 共 Z

1.5

兲 as the predictor variable.

The resulting mean basin-depth effect is period dependent, and both smoother (as a

function of period and depth) and higher in amplitude than predictions from local 1-D

models. For example, relative to a reference hard-rock site, sites with

Z

1.5

equal to

2.5 km (corresponding to some of the deeper L.A. basin locations) have a predicted

mean ampliﬁcation factor of approximately 5.5 at

3speriod, and approximately 7.8 at

10 s period.

The basin ampliﬁcation estimates described here are intended to guide the design of

functional forms for use in attenuation relationships for elastic response spectra. In par-

ticular, they should be useful guides for extrapolating the period-dependence of basin

terms to periods longer than a few seconds, where empirical data provide little con-

straint. More direct, quantitative use of the results may become possible in the future,

however. The main requirement is that we ﬁrst carefully assess the extent to which the

basin effect, as deﬁned and quantiﬁed in this study, is already accounted for implicitly in

existing attenuation relationships, through (1) departures of the average “rock” site from

our idealized reference model, and (2) correlation of basin depth with other predictor

variables (such as

Vs

30

, i.e., the average S velocity in the upper 30 m). A preliminary

assessment of the reference model bias is presented in Day et al. (2005). They ﬁnd that

the reference-model simulations under-predict the rock regression model of Abraham-

son and Silva (1997) by a factor of 2 at long period (

5 seconds, which is much too long

a period to be affected by any response spectral biases associated with bandwidth limi-

Table 3. Coefﬁcients for basin ampliﬁcation factor (Equations 5a and 5b)

Isosurface

Depth (km)

b

0

b

1

b

2

c

0

c

1

c

2

1.0 −0.609 2.26 0.421 0.083 −0.189 0.560

1.5

−1.06 2.26 1.04 0.124 −0.198 0.261

2.5

−0.95 1.35 1.84 0.132 −0.167 0.091

MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 273

tations of the simulations, as discussed earlier). They argue that at the long periods con-

sidered, both source details and

Vs

30

will have minimal effects, and that this factor of 2

is likely representative of a seismic velocity shift (between the average engineering rock

site and the reference model) extending to depths of the order of half a kilometer or

more.

The NGA relationships all use

Vs

30

as a predictor variable. The correlation between

Vs

30

and basin depth is sufﬁciently strong to complicate the identiﬁcation of the basin

effect in the residuals after having ﬁt a regression model to

Vs

30

. Chiou and Youngs

(2006) tested the basin effect model proposed here (Equation 5) for three small (Mw

4-5) earthquakes in southern California, and found that the model compared very well

with spectral ampliﬁcations observed at over two hundred broadband stations of the

Southern California Seismic Network. However, they concluded that, because of the ba-

sin depth-

Vs

30

correlation, they would have had to remove the Vs

30

site term from their

NGA relationship in order to use the basin term. Doing so would make sense from a

physical standpoint, for the long periods

共3–10 s兲 considered here, because simple

wavelength arguments make it clear that

Vs

30

is unlikely to have signiﬁcant physical ef-

fect at long period, and it is predictive of long-period response only to the extent that it

is statistically correlated with overall sediment thickness. However,

Vs

30

information is

widely available for strong motion recording sites, whereas

Z

1.5

(as well as Z

1.0

and Z

2.5

)

data are not available for all sites. Therefore, from a practical standpoint, Chiou and

Youngs (2008) found it expedient to develop their NGA model with

Vs

30

retained as a

predictor variable even at long period, to which they added a

Z

1.0

term to capture that

part of the basin effect not fully accounted for by the correlation between

Vs

30

and Z

1.0

.

The correlation of basin effects with

Vs

30

is discussed further by Choi et al. (2005), who

propose data analysis procedures for separating these effects.

For their NGA model, Campbell and Bozorgnia (2008) were able to empirically

identify a residual basin-depth effect after applying the

Vs

30

term in their model, but

only for sites for which

Z

2.5

⬍ 3km (corresponding to Z

1.5

⬍ ⬃1.5 km). For

Z

2.5

⬎ 3km, they found existing data too sparse to extend the empirical model and its

period dependence to greater sediment depth. Campbell and Bozorgnia’s NGA model

uses the parametric basin-effect model from the current study (Equations 5a and 5b) to

extrapolate the basin term into the

Z

2.5

⬎ 3kmregime.

Our parametric model is based on simulations for the southern California region.

That region is characterized by deep sedimentary basins with relatively low

S wa ve ve-

locity. Sedimentary basins in other regions can have signiﬁcantly different characteris-

tics. For example, the San Francisco Bay region of California is characterized by later-

ally juxtaposed geologic blocks having relatively high

S velocity and relatively shallow

basins (e.g., Santa Clara basin). There is thus a need for additional region-speciﬁc stud-

ies of basin ampliﬁcation effects, including empirical analysis as well as further

simulation-based analysis. In addition, simulations, including the ones done for this

study, should be used to assess the utility of other predictor variables besides basin

depth. As an example, Choi et al. (2005) have taken a step in this direction by examining

the effect on spectral ampliﬁcation of source location relative to basin boundaries.

274 DAYETAL.

ACKNOWLEDGMENTS

We beneﬁted from helpful reviews by Roberto Paolucci, Charles Langston, and an

anonymous reviewer. This work was supported by Paciﬁc Earthquake Engineering Re-

search (PEER) Center Lifelines Program (Tasks 1A01, 1A02, and 1A03), the National

Science Foundation under the Southern California Earthquake Center (SCEC) Commu-

nity Modeling Environment Project (grant EAR-0122464), and by SCEC. SCEC is

funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agree-

ment 02HQAG0008. The SCEC contribution number for this paper is 1101.

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MODEL FOR BASIN EFFECTS ON LONG-PERIOD RESPONSE SPECTRA IN SOUTHERN CALIFORNIA 277