# Elastic Wave Propagation for VSP Geometry

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```					3-D Elastic Wave Propagation for a VSP Geometry                                                                         SB2.5
D. Kessler* and Dan KoslofJ El Aviv Univ., Israel

Summary                                                  medium undergoing infinitesimal deformation.

We presenta method for a solutionof the three            The momentum conservation equations are

dimensional elastic wave equation for VSP                given by (Fung, 1965):

geometry. This solution is operatedon a three

dimensional cylindrical grid using the multi-

domain approach. Discretization of the wave-
pu, = la      +a           1 a%ll          a%
field is carried out on a grid of r 0 and z ,                                    +--+-+ffa
r2 Jr (      I       r       ae      aZ
where r is the distancefrom the center, 0 is the
1 a%,            a%
+--+-+ff,
angular angle and z is depth. Spatial deriva-                                   r ae                aZ
where U, , iJe and U, are respectivelythe dis-
tives are performedusing the Chebychevexpan-
placements r , 9 and z directions;err , 0,
in
sion along the radial direction, and using the
, CJ,, , o,a , o,, and 0,           are the stresscom-
Fourier expansion along the angular and the
ponents; , fe andf, are body forcesper unit
f,
vertical direction.

We combine the equations of conservationof               volume, and p denotesthe density.

momentum with the stress-strainrelation to                               relationsare:
The stress-strain

yield a system of nine equationsfor the dis-

This system is
placementsand for the stresses.

resorted to a first order system which includes

the variablesthat are needed for boundarycon-

ditions constructionand for domain decomposi-

tion.

Boundary conditions and patching of grids are                 _rau, all,_&
by
constructed the use of the characteristic
vari-
u
rem ae
r  fP
ar           r   ue
ables of the wave equation. The numericalal-

gorithm is tested on the Cray YMP supercom-
OaZ= px
au, 7~ w
+
p                . h is the rigidity
puter.
modulusand p is the shearmodulus.
Equations of Motion
Solution Scheme
The equationsof motion we use are formed in
For the treatment of the boundary conditions
cylindricaI coordinates.We combine the aqua-
concurrentvalues of the variables er, ire, c, ,
cons of momentum conservation with the

stress-strainrelation for an isotropic elastic

859
2                                                         3-D elastic wave propagation
T-

of
Thus we recast[l] and [2] to a system nine                                                                                             PI

which are given
coupledfirst order equations                                                Y\$       = iK#                      where K, = j-1
by:
and j = 1, 2,        , N,
[31           Along the verticaldirection,we again use the
\$V =A \$V+B \$V+C                           \$V+U+W                                         wherenow:
Fouriermethod,

whereV is thevariables
vector                                                                                                        El
.   .    .                                            T
u,,    ue9 u,,   o,r,   bee, OZZ’ o,,,   o,,   00,           >
\$j       = i   K,F          with K, = - 2A         (j-l)
(                                                     I                                                               Nz .Dz

A, B, and C are the 9x9 termsmatrices                                       and j = 1, 2,        , N,

which includesmaterialparameters, is the
u                                           Integrationin time is carried out using the

(Dablain,1986):
fourthorderTaylorseries

[71

U(r+At) = U(t)+         &      tiU(t)+f       1
and W is the vectorwhichcontains source
the
and M is the right sizeof [3]; f is the source
terms.
term,and& is thetimestepincrement.
algorithm based solution
The numerical        is   on       of

system                          [3]
[3]. Solutionof the system requires

of                 in
operation spatialderivatives r , 6 and z                                   Boundary Conditions andDomain Composition

and a scheme time integration.
directions,          for                                                          conditions applied thecenter
Boundary        are      at       of

we
Along r direction usethe Chebychev
expan-                                                           of
the grid and at theedges the grid at the radi-

sionfor thederivative:                                                                 and             of
al direction, at thetwo edges the grid in

141          the verticaldirection.At the angulardirection,

is
theoperation periodic.
k = 0, 1, ... , N,
Boundary         are
conditions appliedby correcting

variables
the valuesof the nine component       vec-
Chebychev
whereTk are the discrete       polynomi-
tor (equation[3]). The values of each com-
for           and
als,bk arethe coefficients thederivative
by
ponentaredefined theuseof the characteris-
*
-1 < rj < 1 is the j’         samplingpoint (Gottlieb
tic variables of the elastic wave equation
and Orzag 1977, Kosloff et. al. 1989, Kessler
(Gottlieb et. al., 1982, Baylis et. al., 1990,
direction,
and Kosloff 1990);Along the angular
Kosloff et. al., 1990, Kessler and Kosloff,
for         of
we use the Fourier method construction
variables
1990). The nine characteristic      are:
(Hamming,1978):
thederivative
3-D elastic wave propagation                                            3

PI
45 X 45 X 225 in     the interior grid and

45 X 125 X 225 in the exteriorgrid. The total

number of variablesrequire for solving this

problem      the    59,000,OOO.very
reaches number        A

is                  of
powerfulcomputer neededfor execution

job.         we
thiscomputer The computer foundmost

is
appropriate the Gray YMP. The Cray YMP
and   %z                                                                  possessesvery largephysical
supercomputer      a
the
whereVP and V, are respectively, pressure                     memory and multiple CPUs which sharethe
and shearwavesvelocity. The above charac-                     same memory. Figures 2 and 3 shows
variables
teristic               one    and
describes sided onedi-                                 of
snapshots the numericalsolution.Figure 2
wave propagation. orderto create
mensional              In                                             an
describes horizontal(r,e) cut of the radial

the boundaryconditionneeded,we keep con-                      velocity field at times 0.135 and 0.27 sec.,
stantor zero the termsof the variableswhich                   respectively.

motionof energy
describes                     or
inwards outwards
The white spot at the middle is the borehole,
and
(Kessler Kosloff,1990).
by
and it is surrounded the interior grid, and

grid. Figure3 shows vertical
thentheexterior                 a
The numberof domainsthat we use are two.
(r,z ) cut of the radial velocity field at times
an            of
Figure 1 shows (r,O) section the 3D nu-
At
0.28 and 0.42 sec.,respectively. thesesnap
merical grid. The interior grid has less grid
wave front
shotswe can identify the pressure
direc-
pointsin boththe radial and the angular
followedby the shearwave front and alsothe
of
tionsthan the exteriorone. Composition the
wavespartitionalongtheborehole.
is        usingthe sametech-
two domains performed

of
nique we use for the performance the boun-                    References
meansby correcting vari-
dary conditions,                the                                                 B.J. LeMesurier E.
A. Baylis, K.E. Jordan,             &
[3])
ablesof the nine termsvector(equation us-                                         accurate
Turkel, A forth-order      finite-difference
variablesof the wave
ing the characteristic                                              for            of
scheme the computation elasticwaves,
(equation
equation       181).                                                                    society america,
Bulletinof the seismological    of

Vol. 76, 1986.
ExampleandComputerImplementation
of
M. A. Dablain, The application high-order
wave propagation
In this examplewe consider
to                       Geo-
differencing the scalarwave equation.
medium. The pressure
in a homogeneous                 wave
Vol
physics, 51, 1986.
wave velocity
velocity2ooOm/s and the shear
of
Y.C. Fung, Foundations solid mechanics,
is 1300 m/s. The depthof the cylindrical
strut-
Inc.,        1965.
Prentice-hall New Jersey,
ture is 1000 m. The numberof grid pointsis

861
4                                          3-D elastic wave propagation

and
D. Gottlieb,M. D. Gunzberger E. Turkel,

boundary
On numerical              for
treatment hyperbolic

SIAM J. Numer.Anal. Vol. 19, 1982.
systems.

D. Gottlieband S. Orszag,Numericalanalysis

theory and applications.
of spectralmethods,

SocietyFor IndustrialAnd AppliedMathemat-

ics, 1977.

methods
R. Hamming,Numerical              and
scientists

McGrawHill bookcompany,
engineers,                    1978.

and D. Kosloff,Acoustic
D. Kessler                     wave pro-

Geo-
pagationin 2-D cylindricalcoordinates.

J.
phys. Int., Vol. 103, 1990.

A.
D. Kosloff,D. Kessler, Q. Filho, E. Tessmer,

Solutions the
A Behlle and R. Strahilevits,       of

of                by
equations dynamicelasticity a Chebychev

spectral     Geophysics, 55, 1990.
method.        Vol.                                                     at
Fig. 2. (r,e)snapshots times0.135 set and0.27 sec.

Acknowledgements                                               DISTANCE                   DISTANCE

our          to
We wish to express appreciation George

and                       Inc.
Stevenson Ted Clee of Cray Research

of
for makingavailableto us the resources the

computer.
Cray YMP super

at
Fig. 3. (r,z) snapshots times0.28 set and0.42 sec.
Fig. 1. Horizontal cut of the 3-D grid.

862

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