Acta Geophysica
vol. 62, no. 6, Dec. 2014, pp. 1214-1245
DOI: 10.2478/s11600-013-0199-9
Study of Combined Effects of Sediment Rheology
and Basement Focusing
in an Unbounded Viscoelastic Medium
Using P-SV-Wave Finite-Difference Modelling
Jay P. NARAYAN and Vinay KUMAR
Department of Earthquake Engineering,
Indian Institute of Technology Roorkee, India
e-mail: [email protected] (corresponding author)
Abstract
This paper consists of two parts. First, a fourth-order-accurate staggered-grid finite-difference (FD) program for simulation of P-SV-wave
in viscoelastic medium is presented. The incorporated realistic damping
is based on GMB-EK-model. The accuracy of program is validated by
comparing computed phase-velocity and quality-factors with same based
on GMB-EK-model and Futterman’s relations. The second part of paper
presents the combined effects of sediment damping and synclinal basement focusing (SBT) on ground motion. The results reveal SBT focusing, mode conversion and diffraction of incident waves. The response of
elastic SBT model reveals an increase of spectral amplification with increasing frequency. The viscoelastic response of SBT model reveals that
a particular frequency may get largest amplification for a particular set of
values for damping, focal-length and distance from tip of the SBT. This
frequency-dependent amplification may explain mysterious damage reported in some past earthquakes if predominantly amplified frequency
matches natural frequency of damaged structures.
Key words: viscoelastodynamic P-SV-wave equations, finite difference
program, fourth order spatial accuracy, basement topography effects,
local site effects.
© 2014 Institute of Geophysics, Polish Academy of Sciences
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1.
1215
INTRODUCTION
A frequency-dependent damping in time-domain simulations of basin effects
on the ground motion characteristics is essential to explain the very peculiar
damage patterns and seismic phase generations in basins (Gao et al. 1996,
Hartzell et al. 1997, Narayan 2005, 2012). The fourth-order-accurate staggered-grid finite-difference (FD) method, proposed by Madariaga (1976), is
one of the most useful numerical methods to simulate the ground motion
characteristics (Levander 1988, Moczo and Bard 2002, Narayan and Kumar
2008). The explicit staggered-grid FD has been used successfully to simulate
the scattering and diffraction phenomenon at the basin edges, underground
ridges and surface topography as well as the role of these phenomena in surface wave generation (Virieux 1986, Moczo and Bard 1993, Narayan and
Prasad Rao 2003, Narayan 2003, Narayan and Ram 2006, Narayan and
Richhariya 2008, Frehner et al. 2008, Narayan and Kumar 2010). In the past,
simple approaches have been used to incorporate the damping in timedomain FD simulations due to non-availability of required computational
memory and time (Narayan and Kumar 2008). Day and Minster (1984)
attempted to incorporate the realistic viscoelastic damping into a 2D timedomain FD program for the first time based on the Padé approximation.
Emmerich and Korn (1987) improved the incorporation of viscoelastic
damping in time-domain simulations using a rheological model widely
known as generalized Maxwell body – Emmerich and Korn (GMB-EK)
model, in which n-Maxwell bodies and one Hooke element are connected in
parallel (Moczo et al. 1997, Galis et al. 2008). A further improvement was
made by Kristek and Moczo (2003) by introducing a material independent
anelastic function, which is preferable in case of material discontinuities in
the FD grid. Saenger et al. (2005) studied the coupling mechanism of fluidsolid interaction using viscoelastic damping based on the GMB-EK model in
a rotated staggered FD grid. As a result, all the viscous parameters were
located at the centre of a grid cell, which allowed saturating the rock models
with realistic approximations of Newtonian fluids. On the other hand,
Carcione et al. (1988) used the rheology of a generalized Zener body (GZB
model), in which n-Zener bodies are connected in parallel.
Sometimes, very peculiar damage patterns observed in a basin during an
earthquake cannot be explained using simple soil amplification and resonance effects, as was for example reported in the Santa Monica area, Los
Angeles basin, during the Northridge earthquake of 1994 (Gao et al. 1996,
Hartzell et al. 1997, Alex and Olsen 1998, Davis et al. 2000). Based on the
analysis of aftershock records, Gao et al. (1996) suggested that the amplification in Santa Monica was due to basement focusing from a lens shaped
structure of the deep Los Angeles basin sediments. Alex and Olsen (1998)
estimated the contribution of the proposed “Lens Effects” on the amplificaUnauthenticated
Download Date | 6/18/17 11:08 PM
1216
J.P. NARAYAN and V. KUMAR
tion in Santa Monica. They simulated SH-wave and P-SV-wave responses in
the Santa Monica region and inferred that deep basement focusing can only
account for around 50% amplification of ground motion observed in Santa
Monica. Davis et al. (2000) carried out inversion of aftershock records using
SH-wave FD simulations and inferred that the damage in Santa Monica occurred due to focusing caused by the presence of several underground acoustic lenses at depths of around 3 km in the Los Angeles basin. They also
reported frequency-dependent amplification of ground motion due to basement focusing based on an analytical formulation.
Basement focusing effects on the ground motion characteristics were also observed in the form of consistent anomalous damage to only unreinforced brick chimneys, as shown in Fig. 1, in west Seattle, Washington,
during the 1949 Olympia earthquake of magnitude 7.1, the 1965 Tacoma
earthquake of magnitude 6.5, and the 2001 Nisqually earthquake of magnitude 6.8 (Ihnen and Hadley 1986, Booth et al. 2004, Stephenson et al. 2006).
Booth et al. (2004) carried out a detailed damage survey after the Nisqually
earthquake of 2001 and reported severe damage to collapses of chimneys only in west Seattle. They also inferred that the damage pattern does not correspond to epicentral distance, ground motion amplification caused by soft soil
deposit or surface topography. Based on SH-wave modelling, Stephenson et
al. (2006) inferred that basement focusing of direct waves played a major
role in the damage of chimneys in west Seattle during the Nisqually earthquake. Frankel et al. (2009) carried out 3D simulations for the Nisqually
earthquake and inferred the focusing of S-waves in west Seattle that experienced increased chimney damage. The question remains why only unreinforced chimneys were damaged selectively. Such a selective damage cannot
be explained by basement focusing alone, but an additional frequencydependent amplification is required.
In order to answer the above question, a parametric study is carried out
considering the seismic responses of simple synclinal basement topography
(SBT) models with a varying rheology of the overlying sediments. The first
aspect of this paper is to write an efficient second-order-accurate-in-time and
fourth-order-accurate-in-space (2, 4) explicit staggered-grid finite-difference
program based on the approximation of velocity-stress viscoelastic
P-SV-wave equations for heterogeneous media. The validation of an implementation of frequency-dependent damping in time-domain simulations is
done by comparing the numerically computed frequency-dependent quality
factor and phase velocity of both P- and SV-waves with the same computed
using the GMB-EK rheological model and the Futterman’s relation (Futterman 1962). The numerical grid dispersion and stability condition are also
analyzed in details. The second aspect of this paper is to study the effects of
basement focusing, focal length and sediment damping on ground motion
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1217
Fig. 1. Collapsed chimney during the Nisqually earthquake 2001, west Seattle,
Washington (after Booth et al. 2004).
characteristics. Seismic responses of an unbounded SBT model with different focal lengths on a vertical array along the focal length are simulated for
different sediment damping to infer whether there is a particular frequency
which gets maximum amplification for a particular set of values for sediment
damping, SBT focal length and SBT depth. Snapshots in a rectangular area
are also computed to identify the different seismic phases developed at the
base of the SBT.
2.
VISCOELASTIC P-SV-WAVE FINITE-DIFFERENCE PROGRAM
In this subsection, salient features of the developed viscoelastic second-order
accurate in time and fourth-order accurate is space P-SV-wave explicit staggered-grid finite-difference program based on the GMB-EK rheological
model is described in brief (Emmerich and Korn 1987, Kristek and Moczo
2003). The heterogeneous viscoelastodynamic P-SV-wave equations based
on the GMB-EK rheological model and using material-independent anelastic
functions (Kristek and Moczo 2003) are given below
∂σ xx
⎛ ∂U
= Ku ⎜
∂t
⎝ ∂x
ρ
∂U ∂σ xx ∂σ xz
=
+
,
∂t
∂x
∂z
(1)
ρ
∂W ∂σ xz ∂σ zz
=
+
,
∂t
∂x
∂z
(2)
⎞
⎛ ∂W
⎟ + λu ⎜ ∂Z
⎠
⎝
⎞
⎟−
⎠
m
∑ ⎡⎣Y α ( χ ) + Y λ ( χ )⎤⎦ ,
l
xx
l
l
zz
l
l = 1, 2,..., m
l =1
Unauthenticated
Download Date | 6/18/17 11:08 PM
(3)
1218
J.P. NARAYAN and V. KUMAR
∂σ zz
⎛ ∂W
= Ku ⎜
∂t
⎝ ∂z
⎞
⎛ ∂U
⎟ + λu ⎜ ∂x
⎠
⎝
∂σ xz
⎛ ∂U
= μu ⎜
∂t
⎝ ∂z
⎞
⎟−
⎠
⎞ ⎛ ∂W
⎟ + ⎜ ∂x
⎠ ⎝
m
∑ ⎡⎣Y α ( χ ) + Y λ ( χ )⎤⎦ ,
zz
l
l
l
xx
l
l = 1, 2,..., m
(4)
l =1
⎞
⎟−
⎠
m
∑ ⎡⎣Y μ ( χ )⎤⎦ ,
l
xz
l
l = 1, 2,..., m
(5)
l =1
where U and W are the particle velocity components in the x- and zdirections, respectively. σxx , σzz , and σxz are the stress components, and ρ is
the density. Ku , λu , and μu are the modified unrelaxed elastic parameters
and Yl α , Yl λ , and Yl μ are the modified anelastic coefficients. χlxx , χlzz , and
χlxz are the anelastic functions and ∂/∂x, ∂/∂z, and ∂/∂t are the differential
operators. Parameter m is the number of relaxation frequencies. Figure 2
shows the staggering technique, where normal stress components σxx and σzz ,
unrelaxed elastic parameters Ku and λu , anelastic coefficients Ylα and Ylλ and
Fig. 2. FD staggering technique for P-SV-wave modeling with fourth order spatial
accuracy, where dark circle denotes the defined position of normal stress components σxx and σzz, unrelaxed elasti parameters Ku and λu, anelastic coefficients Ylα
and Ylλ , and anelastic functions χlxx and χlzz at the nodes. Dark triangle and rectangle denotes the position of density and particle velocity components U and W in
the x- and z-directions, respectively. Open circle denotes the position of shear stress
σxz component, unrelaxed modulus of rigidity μu, anelastic coefficient Yl β , and
anelastic function χlxz .
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1219
anelastic functions χlxx and χlzz are defined at the nodes. The particle velocity components U and W in the x-direction and z-direction and density ρ are
defined midway between two adjacent nodes and the shear stress σxz , unrelaxed modulus of rigidity μu, anelastic coefficient Yl μ , and anelastic function
χlxz are defined at the centre of the grid cell.
In the present FD formulation, material-independent anelastic functions
(χl) have been computed at four relaxation frequencies (m = 4) using the following equation:
( χl )n +
1
2
=
1
2 − Δtωl
2Δtωl
ε,
( χl )n − 2 +
2 − Δtωl
2 + Δtωl
l = 1, 2,..., m ,
(6)
where superscript n denotes the time index, ε is the strain , and Δt is the time
increment.
The details of computation of the modified elastic parameters ( Ku , μu ,
(
)
and λu ) and the anelastic parameters Yl α , Yl μ , and Yl λ are given by Kristek
and Moczo (2003).
The unrelaxed elastic parameters Ku and μu for P- and S-waves have
been obtained using the phase velocity of P- (VP ,ωr ) and S-waves (VS ,ωr ), respectively, at a reference frequency (ω) and the following equations (Moczo
et al. 1997).
μu = ρVS2, wr
Ku = ρVP2, wr
ϑ1 = 1 −
m
⎡
1
∑ ⎢⎢Y 1 + (ω
l
l =1
⎣
r
R + ϑ1
2R
2
R + ϑ1
2R2
where R = ϑ12 + ϑ22 ,
(7)
where R = ϑ12 + ϑ22 ,
(8)
⎤
⎥ , ϑ2 =
2
ωl ) ⎥⎦
m
⎡
l
l =1
⎣
⎤
⎥ , l = 1, 2,..., m . (9)
2
ωl ) ⎥⎦
ωr ωl
∑ ⎢⎢Y 1 + (ω
r
The effective value of the unrelaxed modulus of rigidity, μu, and density
at the desired location are obtained using harmonic and arithmetic means, respectively (Moczo et al. 2000). The anelastic coefficients for P- and S-waves
have been computed using Futterman’s equation (Futterman 1962) and
a least square optimization technique.
In Equations 1-5, the time derivative was replaced by a second-order accurate central FD operator and space derivatives were replaced by a fourthorder staggered grid FD operator (Levander 1988, Narayan and Kumar
2008). Both the sponge boundary condition of Israeli and Orszag (1981) and
Unauthenticated
Download Date | 6/18/17 11:08 PM
1220
J.P. NARAYAN and V. KUMAR
absorbing boundary condition of Clayton and Engquist (1977) were implemented on all the four model edges to avoid edge reflections (Kumar and
Narayan 2008).
3.
VALIDATION OF PROGRAM
To validate the accuracy of the developed explicit FD program for P-SVwave as well as the implementation of viscoelastic damping in time-domain
simulations, seismic responses of unbounded homogeneous elastic and viscoelastic models were simulated using plane horizontal wave fronts of SVwave as well as P-wave separately. The simulated seismic responses were
compared with the analytical solutions.
3.1 Numerical grid dispersion and stability
To study the numerical grid dispersion, seismic responses of an unbounded
homogeneous elastic half-space model with VS = 3200 m/s, VP = 5542.5 m/s,
and density equals to 2800 kg/m3 were computed. A Ricker wavelet with
4.0 Hz dominant frequency with considerable spectral amplitude in a frequency range between 0.25 and 10.0 Hz was used as an excitation function
for generation of a horizontal plane wave front using five point sources per
dominant wavelength. The model was discretised using a square grid of
60 m side length. The computed seismic responses at two locations (60 m
apart) on a vertical array at a distances of 1200 and 1260 m from the linear
source were used for the numerical computation of phase velocity Vgrid(ω).
First, Fourier transforms of the seismic responses at the two locations were
used to determine the spectral phase difference Δϕ(ω), then the phase velocity was computed using the following equation:
V grid (ω ) =
ωΔH
,
Δϕ (ω )
(10)
where ∆H is the distance between the two considered receiver points.
The numerical grid dispersion curves for S- and P-waves are plotted as
the normalized phase velocity (Vgrid/V) and the number of grid points per
wavelength (λ/Δz). Parameter V is the S-/P-wave velocity assigned to each
grid point. Analytical grid dispersion curves for S- and P-waves were computed using the same model parameters and the methodology given by
Moczo et al. (2000) for a fourth-order spatial accuracy. Figure 3 shows
a comparison of the numerical grid dispersion curves with the analytical one
for S- and P-waves, respectively. A good agreement of numerical dispersion
curves with the analytical dispersion curves reveals that the error in the
numerically computed grid dispersion curves are within the permissible limit
when the number of grid points per shortest wavelength is more than six.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1221
Fig. 3. A comparison of numerical and analytical grid dispersion curves for S- and
P-waves with fourth order spatial accuracy.
Special attention may be required to make the explicit time stepping
scheme stable in case of simulating a viscoelastic model where velocity is
frequency dependent, particularly when the considered frequency bandwidth
is very large (Saenger et al. 2005). In case of the here developed code, the
computed unrelaxed moduli based on Eqs. 7-9 and anelastic coefficients are
used to find out the P-wave velocity for the purpose of finding the stability
condition. The P-wave velocity computed using unrelaxed moduli gives the
largest velocity for the considered model parameters and frequency bandwidth. This largest P-wave velocity can be conservatively used for computing the stability condition. The stability condition for the presented explicit
FD program was obtained based on various iterative numerical experiments
for a viscoelastic model i.e. using different time increments Δt for the simulation and then analyzing whether an error is arising (Δt was varied in a very
small steps near the upper limit of the stability condition). It was finally concluded that the scheme is stable for both the homogeneous and the heterogeneous viscoelastic models if the following stability condition is locally
satisfied (Moczo et al. 2000).
VP Δt
≤ 0.7 ,
min ( Δx, Δz )
(11)
where VP is the P-wave velocity computed using unrelaxed moduli, Δx and
Δz are the grid size in x- and z-directions, respectively at a particular node.
3.2 Phase velocity and quality factor computation
The accuracy of the presented FD program is also validated by comparing
the numerically computed phase velocities VP(ω) and VS(ω) and quality factors QP(ω) and QS(ω) with the same obtained using Futterman’s relations
and the GMB-EK model for an unbounded homogeneous viscoelastic medium. The responses of an unbounded model were computed at two locations
on a vertical array for two rheologies of the medium (HVM1 and HVM2).
The P- and S-wave velocities and quality factors at reference frequency
(FR = 1 Hz), density and the computed unrelaxed moduli for HVM1 and
Unauthenticated
Download Date | 6/18/17 11:08 PM
1222
J.P. NARAYAN and V. KUMAR
T ab l e 1
Rheological parameters for unbounded homogeneous viscoelastic models
with different damping (HVM1-HVM2)
Rheological
models
HVM1
HVM2
Velocity at FR
[m/s]
S-wave P-wave
3200
5542
3200
5542
Quality factor at FR
S-wave
32
64
P-wave
55
110
Density
[kg/m3]
2800
2800
Unrelaxed moduli
[GPa]
MU
KU
λU
31.455 90.732 27.821
30.021 88.323 28.280
HVM2 rheological models are given in Table 1. The reference frequency is
chosen randomly for this theoretical study. If simulations had to be done for
a specific site then the frequency at which quality factors and phase velocities are measured in the field has to be used as a reference frequency for the
computation of anelastic coefficients. Four relaxation frequencies were taken
as 0.02, 0.2, 2.0, and 20 Hz.
To compute the phase velocity and quality factor, horizontal plane
P- and SV-waves fronts were generated by applying only normal stress (σzz)
and shear stress (σxz), respectively at the position of different point sources
(Note: five point sources pre dominant wavelength was used to generate a
line source). The Fourier transform of the seismic responses at two locations
(∆H = 30 m apart) on a vertical array were used for the numerical computation of phase velocity using Eq. 10.
The frequency-dependent quality factors were numerically computed
using the phase velocity and the amplitude spectra of particle velocity
V(H, ω) of the two traces and the following relationships:
1
QS (ω )
1
QP (ω )
=−
=−
2VS (ω )
⎡ln V ( H + ΔH , ω ) − ln V ( H , ω ) ⎤ ,
⎦
ωΔH ⎣
2VP (ω )
⎡ln V ( H + ΔH , ω ) − ln V ( H , ω ) ⎤ ,
⎦
ωΔH ⎣
(12)
(13)
|V(H, ω)| is the mod of Fourier amplitude spectra of particle velocity at a distance H from the line source.
The phase velocities for P- and S-waves were also computed analytically
using the GMB-EK model (Moczo et al. 1997). First, relaxed moduli at the
desired frequencies were computed using unrelaxed moduli given in Table 1
with the help of following equations and then relaxed moduli were used to
compute phase velocity
⎧⎪
μ (ω ) = μu ⎨1 −
⎩⎪
m
∑
l =1
Yl μ
ωl2 ⎫⎪
⎬,
ωl2 + ω 2 ⎭⎪
Unauthenticated
Download Date | 6/18/17 11:08 PM
(14)
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
⎧⎪
K (ω ) = Ku ⎨1 −
⎩⎪
m
∑
l =1
Ylα
ωl2 ⎫⎪
⎬,
ωl2 + ω 2 ⎭⎪
1223
(15)
where μ(ω) and K(ω) are the relaxed moduli for S- and P-waves, respectively. The frequency dependent quality factors for P- and S-waves were
Fig. 4. Comparison of numerically computed phase velocity and quality factor for
S-wave in an unbounded homogeneous viscoelastic medium with the same computed analytically using Futterman’s relations and the GMB-EK rheological model.
Fig. 5. Comparison of numerically computed phase velocity and quality factor for
P-wave in an unbounded homogeneous viscoelastic medium with the same computed analytically using Futterman’s relations and the GMB-EK rheological model.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1224
J.P. NARAYAN and V. KUMAR
computed analytically using GMB-EK model with the help of following
equations:
1
QS (ω )
1
QP (ω )
m
=
l
l =1
m
=
ωωl
2
2
l +ω
⎡
⎢1 −
⎢⎣
ωωl
ωl2 + ω 2
⎡
⎢1 −
⎣⎢
∑Y β ω
∑
l =1
Ylα
m
ωl2 ⎤
⎥,
2
2
l +ω ⎥
⎦
(16)
ωl2 ⎤
⎥.
+ ω 2 ⎦⎥
(17)
∑Y β ω
l
l =1
m
∑Y α ω
l
l =1
2
l
The phase velocities and frequency dependent quality factors for P- and
S-waves were also computed using Futterman’s relationship (1962).
Figure 4 shows a comparison of numerically computed phase velocity
VS(ω) and quality factors QS(ω) with the same computed analytically using
GMB-EK model and the Futterman’s relationship. Analysis of Fig. 4 reveals
an excellent agreement between the numerically computed phase velocity
and quality factor for S-waves and the same computed using Futterman’s relation and the GMB-EK model. Similarly, Fig. 5 shows a comparison of numerically and analytically computed P-wave phase velocity and quality
factor. Analysis of Fig. 5 also reveals an excellent agreement between the
numerically computed P-wave phase velocity and quality factor and the
same computed using Futterman’s relation and the GMB-EK model. Figures
4 and 5 validate the accuracy of implementation of realistic viscoelastic
damping in the developed time-domain P-SV-wave FD program.
3.3 Response of bounded heterogeneous viscoelastic model
To further validate the accuracy of implementation of realistic viscoelastic
damping in a heterogeneous medium, seismic response of a bounded viscoelastic model containing a horizontal soil layer over the half-space was simulated at the free surface using S-wave horizontal line source. Rheology of
soil was changed by taking the different values of the quality factor at a reference frequency and at the same time maintaining the other model parameters constant. The velocities and quality factors for P- and SV-waves at
reference frequency (Fr), soil thickness, density, and unrelaxed moduli for
the soil layer corresponding to different considered models (SRM1-SRM3)
are given in Table 2. The velocities and quality factors for P- and SV-waves
at reference frequency (Fr), density and unrelaxed moduli for basement were
taken as that of the HVM1 model. Figure 6a depicts the simulated seismic
responses of half-space (without soil layer) and various heterogeneous viscoelastic models (SRM1-SRM3). The spectral amplifications were computed
by taking the ratio of spectra of seismic responses with and without soil
layer. Analysis of Fig. 6b reveals a decrease of spectral amplification with
an increase of sediment damping. This figure also depicts an increase of
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1225
T ab l e 2
Rheological parameters of soil considered in various inhomogeneous viscoelastic
models (SRM1-SRM3) and a comparison of obtained fundamental frequency (F0)
and spectral amplification at F0 (SAF) with the analytical one
Model parameters
Velocity at FR [m/s]
Densiy [kg/m3]
Quality factors at FR
Unrelaxed moduli
Fundamental frequency
of S-wave [Hz]
Amplification of S-wave
at fundamental frequency
S-wave
P-wave
S-wave
P-wave
[GPa]
Numerical
Analytical
Numerical
Analytical
SRM1
525.00
909.00
2000.00
10.00
17.00
7.476
19.725
4.762
1.65
1.66
5.20
5.10
SRM2
525.00
909.00
2000.00
20.00
34.00
6.401
18.030
5.227
1.64
1.65
6.40
6.38
SRM3
525.00
909.00
2000.00
50.00
85.00
5.847
17.106
5.411
1.64
1.64
7.50
7.53
Fig. 6. Unbounded (with no free surface) synclinal basement topography (SBT)
model with horizontal and vertical arrays. The distance of horizontal array from the
flat part of SBT is variable. The vertical array is passing along the focal length of
SBT. Note: on the vertical array, the distances of receiver’s points from the tip of
SBT are normalized with the focal length of SBT – mentioned as NDTSBT in the
manuscript.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1226
J.P. NARAYAN and V. KUMAR
fundamental and higher mode frequencies with an increase of sediment
damping. This finding corroborates with the GMB-EK rheology, where there
is an increase of phase velocity with an increase of sediment damping. The
spectral amplifications at fundamental frequency (SAF) were also computed
analytically using the phase velocity VS(ω) and quality factor QS(ω) corresponding to GBM-EK model and the following relationship (Bard and RieplThomas 2000)
⎛
πIC ⎞
SAF = IC ⎜1 +
,
⎜ 4Q (ω ) ⎟⎟
⎝
⎠
(18)
where IC is impedance contrast between soil layer and the half-space.
A comparison of the numerically obtained fundamental frequency (F0) and
the spectral amplification at F0 (SAF) with analytical solutions reveals an
excellent matching (Table 2).
4.
EFFECTS OF RHEOLOGY
AND SYNCLINAL BASEMENT TOPOGRAPHY
To study the combined effects of rheology of sedimentary deposits and synclinal basement topography (SBT) on the ground motion characteristics,
seismic responses of an unbounded 2D SBT model consisting of a single
sedimentary layer with different rheology overlying the SBT are simulated
using a horizontal plane SV-wave front. Figure 7 shows a 2D cross-section
(XZ plane) of a 3D SBT, which is extending infinitely perpendicular to the
considered cross-section. The shape of the SBT is semi-circular and numerically discretized in the form of a staircase since square FD grids are used. In
order to get the shape of the synform as close as possible to a semi-circular
SBT, finer grids (∆x = ∆z = 10 m) are used as compared to the required grids
per wavelength to avoid grid dispersion. In the considered XZ plane, positive
Z-coordinates are pointing upward from the tip of the SBT. All the distances
were measured with respect to the tip of the SBT. The radius of curvature of
the SBT is 3000 m. The P- and S-waves velocities and quality factors at reference frequency, density and unrelaxed moduli of the sedimentary deposit
and basement rock are given in Table 3 for different rheological models for
sediment deposit (BTM1-BTM4). The BTM1 model is elastic and the other
models are viscoelastic. A plane horizontal SV-wave front was generated at a
depth of 1000 m below the tip of the SBT (i.e., at a depth of 4000 m below
the top-flat part of the SBT; Fig. 7). To avoid edge reflections from the four
edges of the model, both the sponge boundary condition (Israeli and Orszag
1981) and absorbing boundary conditions of Clayton and Engquist (1977)
were implemented on all the model edges. The time increment was taken as
0.001 s, which fulfills the stability criterion (Eq. 18).
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1227
Fig. 7. Seismic responses of BTM1-BTM4 models, for an incident horizontal SVwave front, on a vertical array. Note: the amplitude scale used for normalizing the
vertical component is 104 times larger.
Basement rock
Sediment deposits
T ab l e 3
Rheological parameters for sediment with different damping and basement
for various basement topography models (BTM1-BTM4)
SBT
models
Velocity at FR
[m/s]
S-wave
P-wave
BTM1
1750
3031
BTM2
1750
BTM3
1750
BTM4
Density
[kg/m3]
Quality factor
at FR
Unrelaxed moduli
[GPa]
S-wave
P-wave
MU
KU
λU
2200
–
–
6.737
20.212
6.737
3031
2200
175
303
6.851
20.407
6.704
3031
2200
131
227
6.889
20.473
6.693
1750
3031
2200
85
151
6.967
20.605
6.670
BTM1
3200
5542
2800
–
–
28.672
86.016 28.672
BTM2
3200
5542
2800
320
554
28.935
86.453 28.582
BTM3
3200
5542
2800
320
554
28.935
86.453 28.582
BTM4
3200
5542
2800
320
554
28.935
86.453 28.582
Both the elastic (BTM1) and viscoelastic (BTM2-BTM4) seismic
responses of the SBT models were computed for the quantification of the
combined effects of sediment rheology and SBT on ground motion characUnauthenticated
Download Date | 6/18/17 11:08 PM
1228
J.P. NARAYAN and V. KUMAR
teristics. Seismic responses without SBT were also computed taking the
same model parameters and the source-receiver configuration but using a
horizontal interface between the sedimentary deposit and the basement rock
at the depth of the tip of the SBT.
4.1 Seismic response of SBT models
Seismic responses of SBT models were computed on both the horizontal and
vertical arrays, as shown in Fig. 7. The vertical array along the focal length
of the SBT extends from 450 m below the tip of the SBT to 6550 m above
the tip of the SBT with 15 equidistant (500 m apart) receiver points. On the
other hand, the horizontal array extends from 5000 m left to 5000 m right of
the SBT axis with 21 equidistant (500 m apart) receiver points. Further, the
position of the horizontal array is varied in vertical direction.
Response on vertical array
First, the seismic response of the elastic BTM1 model was simulated on
the vertical array along the focal length of the SBT. Figure 8a shows the
Fig. 8. Continued on next page.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1229
Fig. 8. Horizontal and vertical components of seismic responses of BTM1 model on
different horizontal arrays at a distance of 6750, 5750, and 4750 m from the tip of
SBT, respectively.
horizontal and vertical components of ground motion. The amplitude in the
vertical component is almost negligible as compared to the horizontal component (Note: amplitude scale used to normalise the traces in vertical component is 104 times larger). The normalised distance of receiver points from
the tip of the SBT with respect to the focal length (FL) of the SBT are given
in the brackets. The focal length (FL) of the SBT is 6620.7 m and was
obtained using the following equation:
FL =
r
,
1 −η
(19)
where r is the radius of the semi-circular SBT (3000 m) and η is the ratio of
the S-wave velocity in the sediment to that of the basement rock. Figure 8a
depicts that there is a tremendous increase of amplitude of transmitted SVwaves in the horizontal component towards the focus of the SBT, which may
be due to SBT-focusing effects. Diffracted body waves from the top corners
Unauthenticated
Download Date | 6/18/17 11:08 PM
1230
J.P. NARAYAN and V. KUMAR
of the SBT are clearly visible on receivers R2-R10 in the horizontal component. Their amplitude is highly variable due to divergence, damping and
interference effects. For example, maximum amplitude seems to be reached
at receiver R4 in the horizontal component. Diffracted waves are merged
with the transmitted SV-wave on receivers R11-R15. A decrease of amplitude of horizontal components of SV-waves at R15 very near the focus can
be inferred. This may be due to diffracted waves being out of phase with the
transmitted SV-waves. Maximum amplitude of SV-waves in horizontal component is obtained at receiver R14 at a distance of 6050 m from the tip of the
SBT. The normal incidence of SV-wave on the SBT and the recording on the
vertical array depicts that the mode converted P-wave should not be present
in any component and transmitted SV-wave should not be present in the vertical component. This means that the seismic phases visible in the vertical
component with negligible amplitude may be edge-reflections. However, the
maximum amplitude in the vertical component is of the order of 104-105
times smaller than that in the horizontal component and these edgereflections are therefore negligible. Further, the polarity of mode converted
P-waves, SV-waves, and the diffracted P- and SV-waves in the vertical component on the left side of the SBT axis is opposite to that on the right side
(see in Fig. 9) and cancel each other. On the other hand, these phases have
the same polarity in the horizontal component. Therefore, the diffracted Pand SV-waves cannot be present in the vertical component, even after the
SBT focusing. The presence of a direct P-wave in the horizontal component
first arrival (first arrival near the focus) may be attributed to the focusing of
the mode converted P-wave. Another reason may be the position of the horizontal component in the staggered grid, which is at an offset of half grid
spacing from the SBT axis. The receiver R1 located in the basement rock
shows both the up-going SV-wave and the reflected SV-wave from the SBT.
Similarly, Fig. 8b-d shows the horizontal and vertical components of the
seismic responses of the viscoelastic BTM2-BTM4 models, respectively.
It is surprising to notice the presence of the mode-converted P-wave, transmitted SV-wave, and diffracted P- and SV-waves in the vertical components,
as well as an increase of amplitude of the diffracted waves with the increase
of sediment damping. An increase of amplitude of the suspected modeconverted P-wave in the vertical component towards the focus point of the
SBT may be due to the interference with the suspected diffracted waves.
Further, an increase of amplitude of noise (edge-reflections) can also be
noticed with the increase of sediment damping. The only possible reason for
the presence of P- and SV-waves and an increase of amplitude of diffracted
waves in the vertical component is due to the somewhat poor performance
of the second-order accurate anelastic function at the sediment-basement
interface. A decrease of amplitude of transmitted SV-waves in horizontal
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1231
Fig. 9. Snapshots of horizontal component in a rectangular area (10 000 and 7480 m
in the horizontal and vertical directions, respectively) at different times for incident
SV-wave.
components with the increase of sediment damping can be inferred. Generally, the analysis of Fig. 8 reveals tremendous effects of SBT focusing and
rheology of sediments on the ground motion characteristics in the horizontal
components.
Response on horizontal array
To further study the effects of the SBT on the mode conversion of the incident SV-waves, seismic responses of the elastic BTM1 model were computUnauthenticated
Download Date | 6/18/17 11:08 PM
1232
J.P. NARAYAN and V. KUMAR
ed at a distance of 6750, 5750, and 4750 m from the tip of the SBT on a horizontal array (Fig. 7). Receivers HR5 and HR17 are located vertically above
the left and right top corners of the SBT. Figure 9 depicts the horizontal and
vertical components of the seismic responses. Analysis of Fig. 9 reveals a
strong mode conversion of SV-waves, particularly from the upper part of the
SBT, where the angle of incidence is larger. For example, the recorded mode
converted P-waves in the vertical component on receivers HR6-HR9 and
HR13-HR16 (Fig. 9a) have larger amplitudes than those of the transmitted
SV-wave in the horizontal component on the same receiver points. In other
words, at certain locations, maximum amplitude may be associated with Pwaves for an incident SV-wave due to mode conversion. Diffracted P-waves
are also visible in these seismograms (HR1-HR4 and HR18-HR21). The
analysis of Fig. 9 also shows that depending on the vertical distance of the
horizontal array to the focus point, there is a lot of spatial variation in ground
motion characteristics (particularly in the vertical component). So, depending on the depth of the basement, spatial variation in ground motion may occur due to intense mode conversion and diffraction phenomena. In the
vertical component, polarity of transmitted, mode converted and diffracted
waves on the right of the SBT-axis have opposite polarity to that on the left
of the SBT-axis. So, it can be concluded that basement focusing effects may
not occur in the vertical component for incident SV-wave. However, the spatial variability in the vertical component of ground motion may occur due to
the diffracted and the mode converted waves.
Snapshots
To further demonstrate the SBT-focusing effects, mode conversion and development of diffractions from the top corners of the SBT, snapshots for
both the horizontal and vertical components were computed in a rectangular
area at different times. Snapshots were computed in an area extending from
730 m below to 6750 m above the tip of the SBT and 5000 m left to 5000 m
right of the tip of the SBT. The snapshots at different times for the horizontal
and the vertical components are shown in Figs. 10 and 11, respectively. The
snapshots at time 0.1 s, shown in Figs. 10a and 11a, depicts that the incoming waves are below the considered area. Just entered SV-wave in the considered rectangular area can be seen only in the horizontal component of
snapshot at time 0.5 s (Fig. 10b). Figure 10c shows the SV-wave entered into
the synclinal part of the SBT and the reflected SV-wave from the bottom of
the SBT. On the other hand, the vertical component at time 1.0 s (Fig. 11c)
depicts the transmitted mode-converted P-wave and the transmitted SV-wave
in the basin as well as the reflected mode-converted P-wave in the basement
rock, moving downward. Further, the polarity of the mode converted P-wave
and the transmitted SV-wave towards the right of the SBT-axis is opposite to
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1233
Fig. 10. Snapshots of vertical component in a rectangular area (10 000 and 7480 m
in the horizontal and vertical directions, respectively) at different times for incident
SV-wave.
that towards the left of the SBT-axis, as was inferred from the records on the
horizontal arrays (Fig. 9). Similar effects can be seen in Figs. 10d and 11d
(snapshot time 1.5 s) except that the amplitude of the transmitted SV-wave in
Unauthenticated
Download Date | 6/18/17 11:08 PM
1234
J.P. NARAYAN and V. KUMAR
Fig. 11. Spectral amplification factors for horizontal component of SV-wave for
BTM1-BTM4 models.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1235
the horizontal component above the horizontal part of the SBT is larger than
that above the synclinal part of the SBT. This may be due to the intense
mode conversion of SV-wave to P-wave along the upper curved part of the
SBT.
The transmitted SV-wave and reflected SV-wave from the flat part of the
SBT along with SBT-focusing is more pronounced in the horizontal component at times 2.0 and 2.5 s. Diffracted waves are not well separated from the
transmitted SV-wave in Fig. 10e-f. On the other hand, the transmitted modeconverted P-wave and the transmitted SV-wave along with diffracted P- and
SV-waves are very clearly visible in Fig. 11e-f at snapshot times 2.0 and
2.5 s, respectively. The transmitted and diffracted SV-waves are only visible
in horizontal and vertical components of snapshots at times 3.0 and 3.5 s, as
shown in Figs. 10g-h and 11g-h, respectively. The focusing of the horizontal
component of the transmitted SV-wave towards the focus point can be
inferred in snapshots at times 2.0, 2.5, 3.0, and 3.5 s in Fig. 10e-h, respectively. The maximum focusing effects is visible in the horizontal component
of snapshot at time 3.5 s.
4.2 Amplification factors
To assess the combined effects of sediment rheology and SBT-focusing
quantitatively, spectral amplification of the SV-wave in each trace above the
SBT is computed with respect to a reference trace (response at the R2 receiver point of a model with horizontal interface between sediment and
basement rock passing through the tip of the SBT is used as reference trace).
The spectral amplifications at different normalized distances from the SBTtip (NDTSBT) are shown in Fig. 12. Diffracted P- and SV-waves were removed manually from the traces R2-R7 and R2-R10, respectively, before
computation of the amplitude spectra. Diffracted waves were removed to infer whether focusing is frequency dependent. Figure 12a shows the spectral
amplification for the elastic BTM1 model. This figure depicts an increase of
spectral amplification with increasing NDTSBT value. It can also be inferred
that spectral amplification is increasing with increasing frequency. Further,
the rate of increase of spectral amplification with frequency is increasing
with increasing NDTSBT value. However, spectral amplification is highly
affected by the presence of diffracted waves where manual removal of diffracted waves was not possible. The effect of diffraction can be seen in the
form of ups and downs in the spectral amplification instead of a steady increase of spectral amplification with frequency in the considered bandwidth.
Effect of diffraction is not visible in the traces with NDTSBT values less
than 0.53 since it has been removed from the traces before Fourier transform. Maximum effect of diffraction in the form of ups and downs of spectral amplification can be seen in the record at NDTSBT value equal to 0.76.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1236
J.P. NARAYAN and V. KUMAR
The spectral amplification at a NDTSBT value equal to 0.91 reveals a maximum value of 5.15 at 10.0 Hz. This finding suggests that higher frequencies
would be amplified even more if higher frequencies were used in the simulation. This corroborates with the finding of Davis et al. (2000), who predicted
elastic amplifications analytically, without considering diffractions.
The combined effects of SBT-focusing and the sediment rheology on the
spectral amplifications are shown in Fig. 12b-d. This figure depicts a decrease of spectral amplification with increasing sediment damping. Further,
higher frequencies are damped more compared to lower frequencies, even
though higher frequencies were amplified more due to the SBT-focusing as
was inferred from the elastic BTM1-model. The highest frequency, up to
which the spectral amplification increased with frequency, depends on the
Fig. 12. Continued on next page.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1237
Fig. 12. Effect of sediment damping on amplitude amplification of horizontal component of SV-wave in time domain and average spectral amplification factors, respectively.
NDTSBT value and sediment damping. For example, for a constant
NDTSBT value of 0.91 the spectral amplification is increasing with frequency up to 6.32, 5.51, and 4.29 Hz in BTM2-BTM4 models, respectively. Even
the amplification of certain higher frequencies and up to certain NDTSBT
values can be inferred in case of models having higher damping.
The computed amplification in time domain at different NDTSBT positions is shown in Fig. 13a. It was computed by taking the ratio of maximum
amplitude at different NDTSBT positions and the maximum amplitude in the
second trace of the model without SBT. There is an increase of amplification
Unauthenticated
Download Date | 6/18/17 11:08 PM
1238
J.P. NARAYAN and V. KUMAR
Fig. 13. Effects of focal length on amplitude amplification in time domain and average spectral amplitude amplification factors, respectively. Note: focal length is varied with varying sediment velocity.
in time domain with increasing NDTSBT value. This increase is to some extent linear up to an NDTSBT value of 0.76, thereafter the increase of amplification with NDTSBT value is tremendous (non-linear). The maximum
amplification is obtained at an NDTSBT value of around 0.91. The decrease
of amplification near the focus point may be attributed to the diffracted
waves. The average spectral amplifications in the considered frequency
range were also computed at different NDTSBT positions (Fig. 13b). There
is also an increase of average spectral amplification with increasing
NDTSBT value and decrease of sediment damping. It is also to some extent
linear up to a NDTSBT value of 0.76; thereafter, the rate of increase of the
average spectral amplification with NDTSBT value is very large. The maximum average spectral amplification is also obtained at NDTSBT value of
0.91. A comparison of panels (a) and (b) in Fig. 13 shows that the amplification in time domain is very similar to the average spectral amplification. This
may be due to the presence of diffracted waves and spectral shape of the
considered Ricker wavelet.
5.
EFFECT OF SBT FOCAL LENGTH
In this subsection, the effects of SBT focal length on ground motion characteristics are described. The SBT focal length is varied by varying both the
sediment velocity and the radius of curvature of the SBT for a fixed aperture
length (6000 m). Further, the damping of the sediment is kept constant for
the different considered models.
5.1 Variation of sediment velocity
First, the focal length of the SBT is varied by changing the sediment velocity. For this purpose, four SBT models (BTVM1-BTVM4) with different sediment velocities and densities but with a fixed SBT radius of 3000 m are
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1239
considered. The rheological parameters, namely velocity and quality factor
at reference frequency, density and unrelaxed moduli for the different types
of sediment and the basement are given in Table 4. Focal lengths for
BTVM1-BTVM4 models were obtained as 4363, 4923, 5647, and 6620 m,
respectively. The computed amplification in time domain and the average
spectral amplification at different NDTSBT positions are shown in
Fig. 14a-b, respectively. Both the amplification in time domain and the average spectral amplification at a particular NDTSBT value are decreasing with
decreasing velocity in the sediment (focal length). This may be due to an increase of mode conversion along the upper part of the SBT with increasing
impedance contrast. Further, amplification factors are highly variable near
the focus point. This may be due to different NDTSBT values available in
the different models at receiver points. A higher rate of decrease of ampliT ab l e 4
Rheological parameters of different sediment velocity
considered for various basement topography models (BTVM1-BTVM4)
SBT
models
Sediment
BTVM1
deposits
BTVM2
Basement
Velocity at FR
[m/s]
S-wave P-wave
Density
[kg/m3]
Q at FR
S-wave P-wave
Unrelaxed moduli
[GPa]
MU
KU
λU
1750
3031
2200
175
303
6.851
20.407
6.704
1500
2598
2050
175
303
4.690
13.970
4.590
BTVM3
1250
2165
1950
175
303
3.098
9.228
3.032
BTVM4
1000
1732
1800
175
303
1.830
5.452
1.791
3200
5542
2800
320
554
28.935 86.453
28.58
Fig. 14. Effects of focal length on amplitude amplification in time domain and average spectral amplification, respectively. Note: focal length is varied with varying the
sediment velocity.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1240
J.P. NARAYAN and V. KUMAR
fication with increasing distance on the far side of the focus point can be
observed compared to the near side of the focus point. This may be due to
the combined effects of sediment damping and SBT de-focusing.
5.2 Variation of radius of curvature
Second, the focal length was varied by taking different radii of curvature for
the SBT models. The radius of curvature was taken as 3000, 3250, 3500, and
3750 m for BTRM1- BTRM4 models but with a fixed aperture length of
6000 m (maximum width of SBT). The rheological parameters, namely velocity and quality factor at reference frequency, density and unrelaxed moduli for the sediment are given in Table 5. Focal lengths for BTRM1-BTRM4
models were obtained as 4363, 4727, 5090, and 5454 m, respectively. The
rheological parameters for the basement are the same as in the previous case.
The computed amplification in time domain and the average spectral amplifications are shown in Fig. 15a-b, respectively. Figure 15 depicts a decrease
of amplification factors with increasing focal length at a particular NDTSBT
value due to the larger distance travelled. Again, a much higher rate of
decrease of amplification on the far side of the focus point may be due to the
T ab l e 5
Rheological parameters of sediments for
BTRM1-BTRM4 basement topography models having different radii of curvature
BTRM1BTRM4
SBT
models
Velocity at FR
[m/s]
S-wave
P-wave
1000
1732
Density
[kg/m3]
2200
Q at FR
Unrelaxed moduli
[GPa]
S-wave
P-wave
MU
KU
MU
100
173
1.853
5.491
1.784
Fig. 15. Effects of focal length on amplitude amplification in time domain and average spectral amplification, respectively. Note: focal length is varied with varying the
radius of curvature of the SBT for a fixed value of aperture length.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1241
combined effects of sediment damping and SBT de-focusing. Finally, it can
be concluded that basement focusing effects not only depend on NDTSBT
value but also depends on focal length for a fixed value of sediment damping.
6.
DISCUSSION
This research work was motivated by the selective damage of chimneys in
west Seattle, Washington, consistently during the past three earthquakes
(Booth et al. 2004) as well as very peculiar damage pattern observed in the
Santa Monica area, Los Angeles basin, during the 1994 Northridge earthquake (Gao et al. 1996, Davis et al. 2000). Based on the simulated and
recorded seismograms, Stephenson et al. (2006) inferred that focusing of
direct waves played a major role in the damage of chimneys in west Seattle
during the 2001 Nisqually earthquake. Frankel et al. (2009) also carried out
3D simulations for the 2001 Nisqually earthquake and inferred focusing of
S-waves in west Seattle. However, the above works cannot explain the selective damage of chimneys only. The main conclusion of the presented
research work is that a particular frequency may be maximally amplified for
a particular value of NDTSBT value and sediment damping. This maximally
amplified frequency may cause selective damage if it matches with the natural frequency of a structure. This may be the reason behind the reported mysterious damage to only unreinforced brick chimneys in west Seattle during
the 2001 Nisqually earthquake.
Amplification factors caused by SBT focusing are highly affected by the
presence of diffracted waves from the very sharp edges at the transition from
the synform to the horizontal part of the SBT. In case of real geological
models there may not be such very sharp edges due to the erosion prior to
sedimentation. Further, SBT focusing may also be highly affected by variations of velocity in different sediment layers overlying the basement and
seismic anisotropy of the sediment. The considered 2D SBT models are applicable to simulate the response of elongated synclinal basement topography in a direction perpendicular to the plane of simulation, for example, an
elongated buried horst and graben structure below the sediment. However,
the considered 2D SBT models are not applicable to simulate the response of
a buried depression in the basin, which requires 3D models. The effect of
SBT focusing only on incident S-waves is studied and not on incident
P-waves since earthquake ground motion is dominated by S-waves and the
damage potential of P-wave is much lower than that of S-wave.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1242
7.
J.P. NARAYAN and V. KUMAR
CONCLUSIONS
A second-order-accurate-in-time and fourth-order-accurate-in-space P-SVwave staggered-grid viscoelastic finite-difference (FD) program is written.
Frequency-dependent damping in the time-domain FD simulations is incorporated based on the GMB-EK rheological model (Emmerich and Korn
1987, Kristek and Moczo 2003). An excellent match of numerically computed phase velocity and quality factors for both the SV- and the P-waves for an
unbounded viscoelastic homogeneous medium with the same computed analytically using Futterman’s relation (Futterman 1962) and the GMB-EK rheological model validates the accuracy of implementation of realistic damping
in the time-domain FD program. Grid dispersion and stability analysis reveals that the requirement for the number of grid point per wavelength and
stability condition are the same as reported by others (Moczo et al. 2000).
The snapshots along with the simulated responses revealed that SBT focusing effects depend on NDTSBT value and sediment damping. An increase of amplification with both increasing frequency and distance from the
tip of the SBT was obtained in case of an elastic model. A similar conclusion
was also drawn by Davis et al. (2000) using analytical response of an elastic
SBT model without considering diffractions. In addition, an increase of the
rate of amplification with frequency was inferred towards the focus point of
the SBT. However, the sediment damping masks the increase of amplification with frequency. For example, the largest amplification of 3.37, 3.05, and
2.59 was obtained at a NDTSBT value of 0.91 for 6.32, 5.51, and 4.29 Hz in
BTM2-BTM4 models, respectively. So, it is concluded that a particular frequency may be maximally amplified for a particular value of the SBT focal
length, distance of a site from the tip of SBT and sediment damping. It is also concluded that focusing of mode converted waves may not occur, but amplitudes of mode converted waves may be larger than the amplitudes of
transmitted waves at certain locations on the free surface.
A c k n o w l e d g e m e n t . Authors are grateful to Dr. Marcel Frehner, ETH
Zurich, and an anonymous reviewer for their insightful comments and suggestions which significantly improved the revised manuscript. The first
author is also thankful to the Ministry of Earth Sciences, New Delhi, and
Council of Scientific and Industrial Research, New Delhi, for financial assistance through Grant Nos. MES-484-EQD and CSR-569-EQD, respectively.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1243
References
Alex, C.M., and K.B. Olsen (1998), Lens-effect in Santa Monica?, Geophys. Res.
Lett. 25, 18, 3441-3444, DOI: 10.1029/98GL52668.
Bard, P.Y., and J. Riepl-Thomas (2000), Wave propagation in complex geological
structures and their effects on strong ground motion. In: E. Kausel and
G. Manolis (eds.), Wave Motion in Earthquake Engineering, International
Series on Advances in Earthquake Engineering, WIT Press, Southampton,
37-95.
Booth, D.B., R.E. Wells, and R.W. Givler (2004), Chimney damage in the greater
Seattle area from the Nisqually earthquake of 28 February 2001, Bull.
Seismol. Soc. Am. 94, 3, 1143-1158, DOI: 10.1785/0120030102.
Carcione, J.M., D. Kosloff, and R. Kosloff (1988), Wave propagation simulation in
a linear viscoacoustic medium, Geophys. J. Int. 93, 2, 393-407, DOI:
10.1111/j.1365-246X.1988.tb02010.x.
Clayton, R.W., and B. Engquist (1977), Absorbing boundary conditions for acoustic
and elastic wave equations, Bull. Seismol. Soc. Am. 67, 6, 1529-1540.
Day, S.M., and J. B. Minster (1984), Numerical simulation of attenuated wavefields
using a Padé approximant method, Geophys. J. Roy. Astr. Soc. 78, 1, 105118, DOI: 10.1111/j.1365-246X.1984.tb06474.x.
Davis, P.M., L.J. Rubinstein, K.H. Liu, S.S. Gao, and L. Knopoff (2000), Northridge
earthquake damage caused by geologic focusing of seismic waves, Science
289, 5485, 1746-1750, DOI: 10.1126/science. 289.5485.1746.
Emmerich, H., and M. Korn (1987), Incorporation of attenuation into time-domain
computations of seismic wave fields, Geophysics 52, 9, 1252-1264, DOI:
10.1190/1.1442386.
Futterman, W.I. (1962), Dispersive body waves, J. Geophys. Res. 67, 13, 52795291, DOI: 10.1029/JZ067i013p05279.
Frankel, A., W. Stephenson, and D. Carver (2009), Sedimentary basin effects in Seattle, Washington: Ground-motion observations and 3D simulations, Bull.
Seismol. Soc. Am. 99, 3, 1579-1611, DOI: 10.1785/0120080203.
Frehner, M., S.M. Schmalholz, E.H. Saenger, and H. Steeb (2008), Comparison of
finite difference and finite element methods for simulating two-dimensional
scattering of elastic waves, Phys. Earth Planet. Int. 171, 1-4, 112-121, DOI:
10.1016/j.pepi.2008.07.003.
Galis, M., P. Moczo, and J. Kristek (2008), A 3-D hybrid finite-difference-finiteelement viscoelastic modeling of seismic wave motion, Geophys. J. Int.
175, 1, 153-184, DOI: 10.1111/j.1365-246X.2008.03866.x.
Gao, S., H. Liu, P.M. Davis, and G.L. Knopoff (1996), Localized amplification of
seismic waves and correlation with damage due to the Northridge earthquake: Ewidence of focusing in Santa Monica, Bull. Seismol. Soc. Am. 86,
1B, S209-S230.
Unauthenticated
Download Date | 6/18/17 11:08 PM
1244
J.P. NARAYAN and V. KUMAR
Hartzell, S., E. Cranswick, A. Frankel, D. Carver, and M. Meremonte (1997), Variability of site response in the Los Angeles urban area, Bull. Seismol. Soc.
Am. 87, 6, 1377-1400.
Ihnen, S.M., and D.M. Hadley (1986), Prediction of strong ground motion in the
Puget Sound region: the 1965 Seattle earthquake, Bull. Seismol. Soc. Am.
76, 4, 905-922.
Israeli, M., and S.A. Orszag (1981), Approximation of radiation boundary conditions, J. Comp. Phys. 41, 1, 115-135, DOI: 10.1016/0021-9991(81)90082-6.
Kristek, J., and P. Moczo (2003), Seismic-wave propagation in viscoelastic media
with material discontinuities: A 3D fourth-order staggered-grid finite difference modeling, Bull. Seismol. Soc. Am. 93, 5, 2273-2280, DOI:
10.1785/0120030023.
Kumar, S., and J.P., Narayan (2008), Absorbing boundary conditions in a 4th order
accurate SH-wave staggered grid finite difference program with variable
grid size, Acta Geophys. 56, 4, 1090-1108, DOI: 10.2478/s11600-0080043-9.
Levander, A.R. (1988), Fourth-order finite-difference P-SV seismograms, Geophysics 53, 11, 1425-1436, DOI: 10.1190/1.1442422.
Madariaga, R. (1976), Dynamics of an expanding circular fault, Bull. Seismol. Soc.
Am. 66, 3, 639-666.
Moczo, P., and P.-Y Bard (1993), Wave diffraction, amplification and differential
motion near strong lateral discontinuities, Bull. Seismol. Soc. Am. 83, 1, 85106.
Moczo, P., E. Bystricky, J. Kristek, J.M. Carcione, and M. Bouchon (1997), Hybrid
modelling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures, Bull. Seismol. Soc. Am. 87, 5, 1305-1323.
Moczo, P., J. Kristek, and E. Bystricky (2000), Stability and grid dispersion of the
P-SV 4th-order staggered-grid finite-difference schemes, Stud. Geophys.
Geod. 44, 3, 381-402, DOI: 10.1023/A:1022112620994.
Moczo, P., J. Kristek, V. Vavryčuk, R.J. Archuleta, and L. Halada (2002), 3D heterogeneous staggered-grid finite-difference modelling of seismic motion
with volume harmonic and arithmetic averaging of elastic moduli and densities, Bull. Seismol. Soc. Am. 92, 8, 3042-3066, DOI:
10.1785/0120010167.
Narayan, J.P. (2003), Simulation of ridge-weathering effects on the ground
motion characteristics, J. Earthq. Eng. 7, 3, 447-461, DOI: 10.1080/
13632460309350458.
Narayan, J.P. (2005), Study of basin-edge effects on the ground motion characteristics using 2.5-D modeling, Pure Appl. Geophys. 162, 2, 273-289, DOI:
10.1007/s00024-004-2600-8.
Narayan, J.P. (2012), Effects of P-wave and S-wave impedance contrast on the characteristics of basin transduced Rayleigh waves, Pure Appl. Geophys. 169, 4,
693-709, DOI: 10.1007/s00024-011-0338-7.
Unauthenticated
Download Date | 6/18/17 11:08 PM
BASEMENT FOCUSING EFFECTS ON GROUND MOTION
1245
Narayan, J.P., and S. Kumar (2008), A fourth order accurate SH-wave staggered
grid finite-difference algorithm with variable grid size and VGR-stress
imaging technique, Pure Appl. Geophys. 165, 2, 271-294, DOI: 10.1007/
s00024-008-0298-8.
Narayan, J.P., and S. Kumar (2010), A 4th order accurate P-SV wave staggered grid
finite difference algorithm with variable grid size and VGR-stress imaging
technique, Geofizika 27, 1, 45-68.
Narayan, J.P., and P.V. Prasad Rao (2003), Two and half dimensional simulation of
ridge effects on the ground motion characteristics, Pure Appl. Geophys.
160, 8, 1557-1571, DOI: 10.1007/s00024-003-2360-x.
Narayan, J.P., and A. Ram (2006), Numerical modelling of the effects of an underground ridge on earthquake-induced 0.5-2.5 Hz ground motion, Geophys.
J. Int. 165, 1, 180-196, DOI: 10.1111/j.1365-246X.2006.02874.x.
Narayan, J.P., and A.A. Richharia (2008), Effects of strong lateral discontinuity on
ground motion characteristics and aggravation factor, J. Seismol. 12, 4,
557-573, DOI: 10.1007/s10950-008-9109-z.
Saenger, E.H., S.A Shapiro, and Y. Keehm (2005), Seismic effects of viscous Biotcoupling: Finite difference simulations on micro-scale, Geophys. Res. Lett.
32, 14, L14310, DOI: 10.1029/2005GL023222.
Stephenson, W.J., A.D. Frankel, J.K. Odum, R.A. Williams, and T.L. Pratt (2006),
Towards resolving an earthquake ground motion mystery in west Seattle,
Washington State: Shallow seismic focusing may cause anomalous
chimney damage, Geophys. Res. Lett. 33, 6, L06316, DOI: 10.1029/
2005GL025037.
Virieux, J. (1986), P-SV wave propagation in heterogeneous media; velocity-stress
finte-diffeence method, Geophysics 51, 4, 889-901, DOI: 10.1190/
1.1442147.
Received 14 November 2012
Received in revised form 9 May 2013
Accepted 14 May 2013
Unauthenticated
Download Date | 6/18/17 11:08 PM