Geophys. J. Int. (1996) 125, F5-F9
F A S T - T R A C K PAPER
A 2.5-D time-domain elastodynamic equation for
plane-wave incidence
Hiroshi Takenaka'." and Brian L.N. Kennett2
Department of Earth and Planetary Sciences, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-81, Japan
Research School of Earth Sciences, Australian National University, Canberra ACT 0200, Australia
Accepted 1996 March 11. Received 1996 March 7; in original form 1996 February 15
SUMMARY
Full 3-D modelling of seismic wave propagation is still computationally intensive.
Recently, as a compromise between realism and computational efficiency, two-and-ahalf-dimensional (2.5-D)methods for calculating 3-D elastic wavefields in media varying
in two dimensions have been developed. Such 2.5-D methods are an economical
approach for calculating 3-D wavefields, and require a storage capacity only slightly
larger than those of the corresponding 2-D calculations.
In this paper, a 2.5-D elastodynamic equation in the time domain is constructed for
seismic wavefields in models with a 2-D variation in structure but obliquely incident
plane waves. The approach does not require wavenumber summation, and as a result
requires much less computation time than in previous techniques. The modelling of
such seismic wavefields for a 2.5-D situation with an incident plane wave is of
considerable practical importance: for example, this approach can be applied to the
modelling of the local response of an irregular basin structure, or to teleseismic body
waveforms from shallow earthquakes occurring in subduction zones, where the laterally
heterogeneous medium can have a large effect on the waveform.
All the variables in the 2.5-D time-domain elastodynamic equation are real-valued,
so the propagation characteristics can be efficiently calculated using 2-D time-domain
numerical techniques such as the finite-difference method or the pseudospectral method
with less computation time and memory than for other implementations.
Key words: elastic-wave theory, seismic modelling, seismic waves, synthetic seismograms,
wave equation.
1
INTRODUCTION
Recent advances in high-performance computers have brought
full 3-D elastic modelling for seismic wave propagation just
within reach. Finite-difference calculations have been performed for large-scale 3-D media by Frankel & Leith (1992),
Frankel & Vidale (1992) and Yomogida & Etgen (1993). Full
3-D elastic modelling by the pseudospectral method has also
been performed using a supercomputer (Reshef et al. 1988), a
parallel computer and a workstation cluster (Furumura,
Kennett & Takenaka 1996).Such full 3-D calculations are still
very expensive to perform, however, because of their large
memory requirements. Nevertheless, in order to provide a
*On leave at: Research School of Earth Sciences, Australian National
University, Canberra ACT 0200, Australia.
0 1996 RAS
quantitative analysis of real seismic records from complex
regions we need to be able to calculate the 3-D wavefields.
An economical approach to the modelling of seismic wave
propagation that includes many important aspects of the
propagation process is to examine the 3-D response of a model
where the material parameters vary in two dimensions. Such
a configuration, in which a 3-D wavefield is calculated for a
medium varying in two dimensions, is sometimes called a
'2.5-D problem' (e.g. Eskola & Hongisto 1981).2.5-D modelling
can provide useful results for many problems of practical
interest. Bleistein ( 1986) developed the ray-theoretical implications of 2.5-D modelling for acoustic problems. Luco, Wong
& De Barros (1990) proposed a formulation for a 2.5-D
indirect boundary method using Green's functions for a
harmonic moving point force in order to obtain the 3-D
response of an infinitely long canyon, in a layered half-space,
for plane elastic waves impinging at an arbitrary angle with
F5
F6
H . Takenaka and B.L.N. Kennett
respect t o the axis of the canyon. Pedersen, Sanchez-Sesma &
Campillo ( 1994) also presented a 2.5-D indirect boundary
element method based on moving Green’s functions to study
3-D scattering of plane elastic waves by 2-D topographies.
Takenaka, Kennett & Fujiwara (1996) have developed the
2.5-D discrete wavenumber-boundary integral equation method,
coupled with a Green’s function decomposition into P- and
S-wave contributions, to consider the problem of the interaction of the seismic wavefield excited by a point source with
2-D irregular topography. Randall (1991) developed a 2.5-D
velocity-stress finite-difference technique to calculate waveforms for multipole excitation of azimuthally non-symmetric
boreholes and formations. Okamoto (1994) also presented a
2.5-D finite-difference method, coupled with the reciprocal
principle, to simulate the teleseismic records of a subduction
earthquake. Furumura & Takenaka ( 1996) have developed an
efficient 2.5-D formulation for the pseudospectral method for
point-source excitation and have applied this approach successfully to modelling the waveforms recorded in a refraction
survey. Such 2.5-D methods can calculate 3-D wavefields
without huge computer memory requirements, since they
require storage of the same size as for the corresponding 2-D
calculations.
In this paper we consider a 2.5-D elastodynamic equation
in the time domain as a means of modelling seismic wavefields
in media with a 2-D variation in structure for obliquely
incident plane waves. For a 2-D medium, when we apply a
spatial Fourier transform to the 3-D time-domain elastodynamic equation in the direction along which the material
parameters are constant, we obtain equations in the mixed
coordinate-wavenumber domain. These can be solved as
independent sets of 2-D equations for a set of wavenumbers.
Okamoto (1994) solved the equations for each wavenumber
by a staggered-grid finite-difference method and then applied
an inverse Fourier transform over wavenumber (i.e. wavenumber summation) in order to obtain theoretical seismograms
in the spatial domain. His time-domain approach solves
the source-free elastodynamic equation in the time domain
and needs to perform a number of 2-D calculations. O n the
other hand, frequency-domain methods, such as the indirect
boundary methods mentioned above, require only one 2-D
calculation for solving plane-wave incidence problems; they do
not require wavenumber summation because in 2.5-D planewave incidence problems the wave slowness is invariant, and
so at each frequency the wavenumber is invariant and equal
to that of the incident wave.
The aim of the present paper is to derive a 2.5-D timedomain elastodynamic equation for plane-wave incidence that
does not require wavenumber summation. In the following
section, we present a derivation based on the physical character
of 2.5-D wavefields. We confirm the derivation by a n alternative
route using the 2.5-D equations in the frequency-wavenumber
domain. Finally we discuss the relation between the new 2.5-D
time-domain elastodynamic equation and the corresponding
2-D equation, and the efficiency of the proposed 2.5-D
equation.
2 DERIVATION F R O M PHYSICAL
INSPECTION
In this section we use the physical properties of the wavefield
to derive a 2.5-D elastodynamic equation in the time domain
for the situation of an incident plane wave. Throughout this
paper we employ a Cartesian coordinate system [ x, y , z],
where x and y are the horizontal coordinates and z is the
vertical one.
For an isotropic linear elastic medium, the source-free 3-D
elastodynamic equation in the time domain is given by
pa,,u = a,z,,
+ ayz,, + a,Z,,,
pd,,w = axz,,
+ aytrz+ a,~,,,
where [u, u, w]= u = [u, 0, w ] ( x ,y, z) are the displacements
at a point ( x , y , z ) at time t, the stress components are
T,, = z,,(x, y, z, r), (r, s = x, y, z). The density p = p(x, y, z), and
we have used a contracted notation for derivatives: a,, = a2/at2,
and 5 ,d/ar,
~ (r = x, y, 2). The stress and displacement components are related by the 3-D Hooke’s law through the Lamb
constants 1 = i(x, y, z) and p = p ( x , y, z ) as follows:
t,,=(1+21()a*u+n(a,u+a,W),
7,,=(n+2p)a,w+1(a,u+a,u),
t,, = p(a,w
+ a,u),
t
,
7Syp=p~a,U+ayw),
= p(a,u
+ a,u)
Numerical modelling schemes such as the finite-difference and
pseudospectral methods can directly compute discretized
versions of eqs (1) and (2), where the bounded computational
domains are usually represented by grids.
Now we derive a 2.5-D equation of motion for a 3-D
wavefield in a medium varying in two dimensions which is
invariant with respect to one coordinate and varies with the
other two coordinates. We will assume that the medium is
invariant in the y-direction throughout the rest of this paper,
so the material properties take the form
2 = 4x,4, p = Ax,4, p = p ( x , 4 .
(3)
Furthermore, we assume that the medium includes a homogeneous half-space underlying the 2-D heterogeneous region
whose top may be bounded by a free surface.
Consider an upgoing plane wave with horizontal slowness
[p,, p,], which passes a point in the homogeneous half-space
[x,, yo, z,] at a time t = to.This wavefield has the characteristic
of repeating itself with a certain time delay for different
observers along the medium-invariant axis. For instance, the
wavefield in the vertical plane y = yo at the time t = to must
be identical to that in the vertical plane y = O at the time
t = to - p,y,. This means that
0,Y,x, r ) = u(x, 0, z, t - P,Y),
(4)
M x ,Y>x, t ) = z,,(x, 0, z,t - P,Y), (1; s = x, Y,4 .
On using these relations, the derivatives of the displacement
and the stress with respect to y can be expressed as
a,u(x, y, x, t ) = -Pyalu(X, y, z,t ) ,
(5)
ayz,,(x, y, x, t ) = -pya,z,,(x, y , z, t ) ,
where a, s a/&. The equivalent relations for the stress in (4)
and (5) can be derived directly from those for the displacement
in (4) and (5), and eqs (2) and (3).
O n substituting (5) into (1) and (2) we obtain
Panu = a s x x- P
+a z L
y ~ l L y
3
(6)
0 1996 RAS, G J l 125, FS-F9
A 2.5-0 equation for plane-wave incidence
After changing the order of the integration, and inserting the
following relation between the wavenumber k, and the slowness
and the stress representations
z,,=(i+2p)a,u+~.(-p,atv+a,W),
Py :
- ( / ~ + 2 p ) p , a , U + q a , ~ +a,u),
tyy=
F7
(7)
t,,=(a+2p~)a,W+/~(a,u-p,a,u),
~yz=~(azu-Py~,W),
t,,=p(a,w + a z U ) , t,y=p(-pPyatU+a,U).
This set of equations represents the elastodynamic response of
a 2.5-D medium in the absence of a source and incorporates
Hooke's law. We note that all the variables in this set of
equations are real-valued. When we solve eqs (6) and (7), we
can set y = y o , so that these equations are reduced to 2-D
ones. Once eqs ( 6 ) and ( 7 ) have been solved for y = yo, we can
deduce the wavefield at any y from that at y = y o by shifting
the time origin by py(y - y o ) (see eq. 4).
In the next section, we will give an alternative derivation
of these time-domain 2.5-D eqs ( 6 ) and (7), from the 2.5-D
equations in the frequency-wavenumber domain, and recover
the characteristic of the 2.5-D wavefield, eq. (4), in the process
of deriving these equations.
(12)
k, = u p , ,
eq. ( 11 ) leads to
m
(13)
where 2 in the time-slowness domain has been defined as
dw eiorlolg(x,wp,, z , w ) .
(14)
Then, we find
k J-,
f m
I
dw eiore-'kyYlwlii(x,k,,
Z,
w ) = ii(x, p,,,
Z,
t -p y y ) ,
(15)
3 DERIVATION VIA THE EQUATION I N
THE FREQUENCY-WAVENUMBER
DOMAIN
and
O n performing a Fourier transform of the 3-D eqs ( 1 ) and (2)
with respect to t and y , we obtain the following source-free
2.5-D elastodynamic equation in the frequency-wavenumber
domain:
= a,f,,
-w 2 p f i
-w2p5=
+ a,fz,, ,
i k y f Y ,+ azfyz,
-ikyfxy
a,?,-
(8)
- ~ ~ p ~ = a , f ~ ~ - i k , ~ , ~ + a ~ f ~ ~ ,
and the stress-displacement relations
f,,
+ 2p)a,E + A(-ik,v" + a,$),
f,,,,=(A + 2 p ) ( - i k Y ) 5 + / i ( a , $ + + , f i ) ,
(A + 2p)a,$
+ A(8,fi
fz,,=p(a,~+a,a),
- iky6),
fYye
= p(a,v"- ik,G),
(9)
~,,y=p(-iikya+a,q
Here we have used the y-invariance of the medium, i.e. eq. (3),
and have used the notation
s3
sI_
,e+ikyy
g(x, k,, z, w ) = 2.n -dy
dt e-'O'g(x, y, z , t )
y'
Y
lk,lwlWx, k,,
2,
w ) = pyatQ(x,P,, Z, t - p , ~ ) .
(16)
For an incident plane wave in a 2.5-D situation, the horizontal wavenumber k, is invariant for each w, and equal to
that of the incident plane wave. Furthermore, from (12) we
require p , to be invariant and equal to the y-component of the
slowness of the incident plane wave, which represents 'Snell's
law' for 2.5-D problems. Thus, when the slowness of the
incident wave is p y o , a(x, p,, z , t - p,y) can be represented as
=
w,
P y o , 29
t -P y O Y P ( P , -Pyo)
3
(17)
where 6(x) is the Dirac delta function. Applying the inverse
transform ( 11) to the displacement in the frequencywavenumber domain a(x, k,, z , w), and using eqs (13), (15)
and (17), we obtain
U(X, Y , z, t ) =
Pyo,
z, t - P y o Y ) .
(18)
Then
(10)
for the transform to the frequency-wavenumber domain. For
a fixed value of the wavenumber k,, eqs (8) and (9) depend
on only two space coordinates, i.e. x and z . For each value of
k,, these equations can therefore be solved as independent 2-D
equations. The invariance of the medium in the y-direction
means that there is no coupling between different k, components, whereas for full 3-D problems there would be coupling
between different k, wavenumber components. For 2.5-D
problems with an incident plane wave, we need to consider
only one k, for each w , which is that of the incident plane wave.
The inverse transform of the double Fourier transform (10)
is
0 1996 RAS, CJI 125, F5-F9
eiwt e - i k
-m
a(x, Pya z>t - P Y A
= (1
frz =
lrn
dw
271
We can recover (4) from (18) by appropriate substitutions:
setting y to 0 we obtain an equation in time t and then making
the particular choice t - pyoy gives
u(x, 0, z, t - P y o Y ) = w , P y a , z, t - P,OY)
= u(x, Y,z,t ) .
(20)
In a similar way, we can obtain the corresponding equation
for the stress. Applying the partial Fourier inversions (15) and
(16) to ( 8 ) and (9), and using eqs (18) and (19), we recover
the earlier forms (6) and (7).
4
DISCUSSION
Now we discuss the relation between the new 2.5-D timedomain elastodynamic equation and the corresponding 2-D
equation. When we substitute (7) into ( 6 ) and rearrange the
resultant equations we obtain the following form, which allows
H. Takenaka and B.L.N. Kennett
F8
a n easy comparison between the 2.5-D and 2 - 0 equations:
where tl =E(x,z) and j=P(x,z) are the P- and S-wave
velocities, respectively. T,?;~ = z?iD(x,z, t ) (r, s = x, y, z) are the
2-D stress components related to the displacement by the 2-D
Hooke's law that can be derived by setting p , = 0 in the 2.5-D
Hooke's law (7), and we have defined a? ( r = x, y, z ) as
0, =
aU a
+ -(Lu),
ax ax
p-
au a
+ -((nu).
aZ aZ
a,=/,-
Compared with the 2-D elastodynamic equation, the 2.5-D
elastodynamic equation has the following characteristics:
(1) the inertia coefficient (coefficients multiplied by the
acceleration) is anisotropic, i.e. its components are different
between the y-direction and the other directions;
(2) the components of the inertia coefficient can change
their sign, i.e. all are positive if 0 < p , < l/cc (supersonic region);
the x- and z-components are positive, and the y-component is
negative if l/a < p y < 1/p (trans-sonic region); and all are
negative if p y > lip (subsonic region);
(3) the in-plane motion [u, w ] and the anti-plane motion u
a r e coupled by the terms &or (r = x, y, z ) which act as a type
of damping.
The modelling of seismic wavefields for models with a 2-D
variation in structure but obliquely incident waves (2.5-D
model) is of considerable practical importance. For example,
such an approach can be applied to the modelling of the local
response of an irregular basin structure (Luco et al. 1990;
Pedersen et al. 1994), or teleseismic body waveforms from
shallow earthquakes occurring in subduction zones (Okamoto
1994), where the laterally heterogeneous medium has a large
effect on the waveform. The displacement at teleseismic distances can be calculated by using seismic reciprocity (see e.g.
Bouchon 1976). Instead of solving a forward problem in which
the earthquake source is located in the subduction zone, we
can solve the reciprocal problem in which a virtual force is
applied at infinity (i.e. at the station location). When we are
only concerned with the far-field body wave, the wavefield
due to this virtual force can be approximated by a plane
wave. Thus the calculation of teleseismic waveforms is
mathematically a plane-wave incidence problem.
Okamoto (1994) used this reciprocal approach in combination with a direct Fourier transform of the source-free elastic
equations with respect to y, so that, for example,
pattii = a,.t;,
T x x=(A
- ik,?,,
+ a,?,,
,
+ 2/*)d,G+ i.(-ik,fi + a,*),
where we have used the notation
dy eikyyg(x,y, z, t ) .
(24)
The transformed variables are here complex-valued, so that
memory requirements are twice those of the space domain.
Furthermore, in the time domain the wavenumber k , is not
invariant even for the plane-wave incidence problem, although
it is invariant at each frequency. In order to recover the
theoretical seismograms in the space domain from the mixed
time-wavenumber domain, wavenumber summation needs to
be carried out. This requires the solution of (23) and its
analogues to be performed many times.
The alternative 2.5-D time-domain equations derived in this
paper, (6) and (7), are well suited to the use of numerical
techniques for 2-D time-domain problems such as the finitedifference method (e.g. Levander 1988) or the pseudospectral
method (e.g. Kosloff, Reshef 62 Loewenthal 1984). In contrast
to the usual equations for plane-wave incidence (23), no
wavenumber summation is required since the invariance of the
horizontal slowness p y is exploited, and only real variables are
employed. This gives significant advantages, of at least a factor
of two, in both computation time and memory.
The new 2.5-D equations will be useful even for non-plane
waves if we. use the Radon transform over slownesses p ,
(Chapman 1978) (cJ: eq. 13) instead of the Fourier transform
over wavenumber k,. This will be the subject of further study.
ACKNOWLEDGMENTS
This work was undertaken when the first author (HT) was a
Visiting Fellow at the Research School of Earth Sciences,
Australian National University, a position supported by an
overseas research program of the Japan Society for the
Promotion of Science (JSPS fellowship for Research at Centers
of Excellence Abroad). The comments of two anonymous
referees were very useful for improving the manuscript.
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