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GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L02302, doi:10.1029/2007GL032214, 2008
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Discrete visco-elastic lattice methods for seismic wave propagation
G. S. O’Brien1
Received 1 October 2007; accepted 6 December 2007; published 19 January 2008.
[1] 2D and 3D discrete particle or lattice methods for the
simulation of seismic waves are presented for different
visco-elastic media. It is demonstrated that a numerical
implementation of the method is capable of modelling
visco-elastic seismic wave propagation. Lattice methods
represent the medium under investigation as particles or
nodes interacting through local force rules. Three schemes
are developed, a Maxwell body, a Kelvin-Voigt body and a
Zener Body (or standard linear solid). The force acting
between particles is chosen to represent each of these
different rheologies. Each of these schemes was tested
against analytical solutions for wave propagation in 2D and
3D unbounded visco-elastic media. The seismograms
generated for each different rheology fit well with the
expected theoretical seismograms with the maximum misfit
error energy being less than 3% for each different rheology.
As such, lattice methods offer an alternative approach to
seismic wave modelling in elastic and visco-elastic media.
Citation: O’Brien, G. S. (2008), Discrete visco-elastic lattice
methods for seismic wave propagation, Geophys. Res. Lett., 35,
L02302, doi:10.1029/2007GL032214.
1. Introduction
[2] The heterogeneous nature of geological materials
leads to many challenges when modelling a variety of
phenomena in natural structures. These include, for example, pore fluids, complex topography and fractures which
can be included as non-welded interfaces. The discontinuous nature of these natural flaws are often oversimplified in
order to define a set of differential equations which can then
be solved either analytically or numerically. An alternative
approach to continuum mechanics is to use discrete particle
methods (lattice methods, molecular dynamics methods).
Discrete particle methods have been successfully applied in
physics for the past 30 years. They originate from solid-state
physics models of crystalline materials [e.g., Hoover et al.,
1974]. These discrete methods do not solve continuum
equations directly, for example the wave equation. Instead,
they try to replicate the underlying physics at a microscopic
scale employing discrete micro-mechanical interaction rules
between discrete material particles. As these schemes solely
rely on particle-particle interactions, complex boundary
conditions can be readily implemented which makes them
ideally suited to model geological processes. Several
authors have applied discrete particle methods to a variety
of geological applications, e.g. passive and reactive fluid
transport in porous media [O’Brien et al., 2003], earthquake
1
School of Geological Sciences, University College Dublin, Belfield,
Ireland.
Copyright 2008 by the American Geophysical Union.
0094-8276/08/2007GL032214$05.00
dynamics [Mora and Place, 1998] and structural geology
[Schöpfer et al., 2006]. Discrete particle methods have been
successfully applied to elastic wave propagation in 2D.
Toomey and Bean [2000] used a 2D discrete particle scheme
where the particles are arranged on a triangular lattice and
interact through Hooke’s Law. Their method was restricted
to a fixed Poissons ratio of 0.25. Del Valle-Garcia and
Sanchez-Sesma [2003] used a similar method, but included
a bond-bending force term which removes the restriction on
the Poissons ratio. O’Brien and Bean [2004] used a cubic
lattice to model 3D wave propagation in the presence of
complex topography. All the above methods are restricted to
an elastic rheology and have ignored the effects of attenuation in the Earth’s crust. In this article a description of a
discrete approach to modelling seismic wave propagation in
three different visco-elastic media is presented. The numerical methods are then validated against different analytical
solutions to the visco-elastodynamic wave equation in
visco-elastic media.
2. Elastic Lattice Method
[3] An elastic solid can be represented by a series of
interconnected springs arranged on a regular lattice. Using
an irregular lattice requires calibration so the P-wave
velocity and Poissons ratio can be obtained. The P-wave
velocity may also be heterogeneous and anisotropic due to
the irregular lattice. By using a regular lattice these restrictions are easily overcome. The force Fij on an individual
node i from node j is given by
cuij
Fij ¼ Kij uij xij þ
jxij j2
ð1Þ
for both a 2D square lattice and 3D cubic lattice. Kij is the
elastic spring constant between particle i and j, c the bondbending constant. uij is the displacement vector (ui uj)
and xij is the vector connecting nodes xi and xj in the
undistorted lattice. The dependence of the spring constants
Kij on the lattice geometry remove the lattice anisotropy by
weighting each of the lattice directions [see Monette and
Anderson, 1998]. The Lamé constants are given by
c
c
m2D ¼ K þ 2
Dx2
Dx
K
2c
K
2c
¼
m3D ¼
þ
Dx Dx3
Dx Dx3
l2D ¼ K l3D
ð2Þ
where Dx is the lattice grid spacing. A detailed derivation of
the relationship between the discrete particle scheme and an
elastic continuum is given by Monette and Anderson [1994]
and O’Brien and Bean [2004]. The force acting on each
spring is calculated at each time step and the new position
of the lattice nodes and node velocities are updated using
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O’BRIEN: VISCO-ELASTIC LATTICE SEISMIC WAVE PROPAGATION
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3.1. Maxwell Body
[5] To model a Maxwell body, a dashpot with viscosity h
is included in series with a spring (Figure 1). The equation
for the time derivatives of the force acting between each
particle pair is then expressed as
1
F_ ðt Þ ¼ F_ elastic ðt Þ Fðt Þ
Q
Figure 1. Discrete particle methods represent rock as
distinct particles or nodes interacting through nearest
neighbour rules. (left) These schemes are capable of
modelling seismic wave propagation in 2D (square lattice)
and 3D (cubic lattice). By changing the local interaction
rules a variety of different rheologies can be modelled.
(right) Here we consider the three different visco-elastic
media.
the velocity-Verlet numerical integration scheme [Allen and
Tildesley, 1987]. This is a second-order in time and fourth
order in space finite difference approximation to the
equations of motion.
where F represents the total force, Felastic is the elastic force
given by equation (1) and Q is the quality factor for both Pand S-waves. Therefore, to model a Maxwell body the
forces acting at each node need to be updated following
equation (4). This can be done by numerically solving the
temporal derivatives using a simple Euler scheme. The
numerical method was tested against an analytical solution
and the comparison of the resultant seismograms gave a
maximum misfit error energy of 3%. The quality factor for a
Maxwell body is given by Q(w) = Q0w where w is the
angular frequency and hence is not readily applied to
seismic wave propagation but appears to be more appropriate for representing a visco-elastic fluid [Carcione,
2007].
3.2. Kelvin Voigt Body
[6] The Kelvin-Voigt body consists of a spring and
dashpot in parallel (Figure 1). To include a dashpot in
parallel the force term acting on node i is now written as:
cuij
Fij ¼ Kij uij xij þ
þ hu_ ij
jxij j2
z¼
2
Pt NUM
S
ðt Þ S AN ðt Þ
Pt AN 2
S ðt Þ
ð3Þ
where SAN(t) is the analytical seismogram and SNUM(t) is the
numerical seismogram.
ð5Þ
[7] Letting the displacement uij = uoeiwt+ikx and assuming a 1-D model the force equation (5) can be written as
c
kDx
w2 m ¼ 4 K 2 þ ihw Sin2
Dx
2
3. Viscoelastic Lattice Method
[4] To model a visco-elastic body the force acting between each particle pair has to be adjusted for the appropriate viscous body. A review of visco-elastic theory in
terms of rock mechanics is given by Jaeger et al. [2007] and
Carcione [2007]. In this article we will look at a Maxwell
body, a Kelvin-Voigt body and Zener or standard linear
solid (Figure 1). Each numerical scheme was tested against
either a 2D or 3D analytical solution. The 2D analytical
solutions are given by Carcione [2007] with the appropriate
complex velocities used in each case. The 3D analytical
solutions where derived from the Green’s functions for a 3D
elastic body [Aki and Richards, 2002] using the Fourier
transform and correspondence principle to transform to the
frequency domain in order to include the viscous attenuation. To quantify the comparison between the analytical and
numerical seismograms, we computed the misfit error
energy z given by
ð4Þ
ð6Þ
where m is the mass and in 1-D uij = (ui+1 + ui1 2ui).
2kDx
2
2
Assuming Dx 2p
k so 4Sin 2 = k Dx and using the
definition of Q as
QðwÞ ¼ Real½k 2 Imag½k 2 ð7Þ
we find
QðwÞ ¼
KDx2 þ c 1
1
¼
hDx2 w Q0 w
ð8Þ
where h = Q0(K + Dxc 2 ). As no shear forces are explicitly
included in the scheme this quality factor applies for both Pand S-waves. The assumption Dx 2p
k restricts the number
of grid points per minimum wavelength to greater than ten
[Toomey and Bean, 2000]. Separate P- and S-quality factors
can be included by resolving the force into tangential and
parallel components and damping the tangential component
with the S-quality factor and the parallel with the P-quality
factor. The scheme was tested in a 2D unbounded medium
with a P-wave velocity of 3500 m s1, S-wave velocity of
1800 m s1, density of 2000 kg m3 and Qp = 80 and Qs =
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Figure 2. (left) A comparison between the numerical (solid line) and analytical solution (dashed line) vertical components
for a Kelvin-Voigt body gives a visually excellent result. The seismograms, located at intervals of 500 m (in both the
vertical and horizontal) from the source, are produced by a Ricker wavelet 8 Hz vertical force in an unbounded 2D medium.
(right) The quantitative fit is shown with a maximum misfit error energy of 1.56% for the most distal seismogram. The
dashed asterisked line represents the misfit solution for an elastic medium, while the dashed triangle line represents the
misfit solution for a 6 Hz source. The vertical scale of the misfit energy is [0 1]% for each seismogram.
60. The source function was a 8 Hz Ricker wavelet vertical
force located at (0, 0). Figure 2 shows the comparison of the
numerical method with the analytical solutions. The left
panel shows 6 vertical numerical and analytical seismograms located at 500 m steps (in both directions) from the
source. Visually the fit is excellent. The quantitative fit is
shown in the right panel with a maximum misfit error
energy of 1.56% for the most distal seismogram. For
comparison the dashed asterisked line represents the misfit
solution for an elastic medium while the dashed triangle line
represents the misfit solution for a 6 Hz source in a Maxwell
body. The fit is better as, for a fixed recording position, the
numerical dispersion decreases with increasing seismic
wavelength. The general increase in error is a result of
numerical dispersion increasing with distance and the
increased effect of the finite numerical boundaries. Similar
results are found for the horizontal component.
3.3. Zener Body
[8] A Zener body or standard linear solid consists of a
Maxwell body in parallel with a spring (Figure 1). The force
acting between two particles can be written as
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Fij ðt Þ ¼ Ko uij þ K1 usij þ
cu_ dij
Dx2
ð9Þ
O’BRIEN: VISCO-ELASTIC LATTICE SEISMIC WAVE PROPAGATION
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Figure 3. (left) A comparison between the numerical (solid line) and analytical solution (dashed line) vertical components
for a 2D unbounded Zener body is shown. The seismograms are located at intervals of (500, 500) m from the source
(vertical Ricker wavelet 6 Hz). (right) The quantitative fit is shown with a maximum misfit error energy of 0.76% for the
most distal seismogram. The dashed asterisked line represents the misfit solution for an elastic medium using the same
method and parameters. The vertical scale of the misfit energy is [0 1]% for each seismogram.
where ud, us and u are the dashpot, K1 spring and total
displacement respectively. Substituting u = us + ud and
K1us = hu_ d into equation (9) gives
c c FðtÞ þ t s F_ ðt Þ ¼ Ko þ 2 uij þ Ko þ 2 t u_ ij
Dx
Dx
h
h
h
t ¼
þ
K1
K1 Ko þ Dxc 2
QðwÞ ¼
ð10Þ
where
ts ¼
[9] As in the Kelvin-Voigt body letting the displacement
uij = uoeiwt+ikx and assuming a 1-D model and Dx 2p
k the
force equation (10) yields
ð11Þ
1 þ w2 t s t wðt t s Þ
ð12Þ
[10] The scheme was tested in a 2D unbounded medium
with a P-wave velocity of 4000 m s1, S-wave velocity of
2500 m s1, density of 2500 kg m3 and Qp = Qs = 40. The
temporal derivatives in equation (10) were performed with a
simple Euler scheme. The dissipative forces can be resolved
into parallel and tangential (to the bond connecting the
nodes) forces allowing the inclusion of different P- and
S-Quality factors. The source function was a 6 Hz Ricker
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O’BRIEN: VISCO-ELASTIC LATTICE SEISMIC WAVE PROPAGATION
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across the fracture) and dynamic rupture (introducing a
bond failure criteria). We have focused only on dynamic
deformation. Static deformation can be modelled by applying external forces. Therefore, the method can be used in
examining changes in the seismic wavefield as a consequence of static deformation of different rheological models. The computational cost of these methods is similar to a
fourth-order finite-difference method for the same parameters. However, since a staggered grid fourth-order finitedifference method requires approximately 5 grid points per
minimum wavelength, the lattice method, which requires
about 10 grid points, requires more memory to avoid
numerical dispersion. However, the method only requires
nearest neighbour interaction, which reduces the communication overheads for parallel computing. There is no need to
include memory variables as is required for many wave
propagation methods. The only additional storage required
is the quality factors for each node and the force at the
previous time step for calculating the temporal derivatives.
The fit with the analytical solutions and ease of implementation means lattice methods offer an alternative approach to
modelling wave propagation where several added features
can be incorporated into numerical simulations.
Figure 4. A comparison between the numerical (solid line)
and analytical solution (dashed line) for a 3D unbounded
Zener body. The seismogram is located (1500, 1500, 1500)
m from the source (6 Hz Ricker wavelet applied in the
vertical direction). The maximum misfit is 1.8%.
wavelet vertical force. Numerical and analytical solutions
are compared in Figure 3. Six vertical numerical and
analytical seismograms located at intervals of (500,500) m
from the source give an excellent visual fit (Figure 3 (left)).
The total misfit error energy is less than 0.8% for all
the seismograms shown. The dashed vertical asterisked
line shows the total error energy for the elastic case. We
get similar results for the horizontal component. The
methodology can be readily applied to 3D Maxwell,
Kelvin-Voigt and Zener bodies. Figure 4 shows the results
for a 3D unbounded Zener body where the P-wave velocity
is 4000 m s1, the S-wave velocity is 2800 m s1, the
density is 2500 kg m3 and Qp = 50 Qs = 50. As in the
previous examples, a 6 Hz ricker wavelet vertical source
was used. As with the 2D results, we get an excellent visual
fit and a maximum misfit error of 2.0%.
4. Discussions and Conclusions
[11] The results show that the lattice method can be
applied to seismic wave propagation in 2D and 3D viscoelastic media. All the examples shown here are for an
unbounded homogeneous medium but as the scheme is
based on particle-particle interactions, arbitrary heterogeneity can be included by changing the spring and dashpot
constants on each bond. Topography can also be readily
introduced by simply removing any particles above the
required free surface. The discrete nature of the method
allows for several different features to be included such as
non-linear wave propagation (non-linear dashpots and
springs), fracture discontinuities (remove or weaken bonds
[12] Acknowledgments. This work was carried out in part by the
Cosmogrid Ireland project funded under the Irish Government Programme
for Research in Third Level Institutions and by the 6th framework EU
project VOLUME. The author wishes to acknowledge the SFI/HEA Irish
Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. The author would like to thank Prof. Chris
Bean for some helpful discussions and J. Carcione and an anonymous
reviewer for their comments.
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G. S. O’Brien, School of Geological Sciences, University College
Dublin, Belfield, Ireland. ([email protected])
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