dataca1culatedforthebackground.Aforwardoperatorforsuchalinearinversion
eachrayconnectingthesourcefreceiverandthescatterers.Thebackgroundvelocitymodelmugibeinhomogeneousforgeneralapplications.However,inmanycasesitcanbeassumedsmoothatleastinafirststep(VERSTEEG,1991).Modeldiscontinu-
density criterion
Overthrust
Linear
repeated en sures uniform
inversion
evaluation
based
ities can be reconstructed 1G.LAMBARÉ1andA.HANYGA2Abstract-Analgorithmforcomputingmultivaluedmapsfortraveltime,amplitudeandanyotherrayrelatedvariablein3Dsmoothvelocitymodelsispresented.Itisbasedontheconstructionofsuccessiveisochronsbytracingauniformlydensediscretesetofraysbyfixedtravel-timesteps.RaytracingisbasedonHamiltonianformulationandinc1udescomputationofparaxialmatrices.Aray
P. S. LuCIo,
on the ray
of amplitudes,
+Born(LAMBARÉetal.,1992;FORGUESetal.,1994;ETTRICHandGAJEWSKI,1995)orray+Rytovapproximationrequires
Thecurrentinterestin3Dseismicimagingbasconsiderablyincreasedtheimportanceofraytracingmethodsinwavefieldcomputations.Amongseismicmodellingmethods,raytracingmethodsprovideareasonablecompromisebe-tweenaccuracyandcomputationalefficiency(HANYGAandHELLE,1994)whilethealternativemethodssuchasfinitedifferences(AMINZADEHetal.,1994a,b;AMINZADEHetal.,1995)andspectralmethodsrequiresubstantialcomputing
Applicationstocomplexmodelsareshawn.Keywords:3DRaytracing,traveltime,amplitude,Hamiltonian,Lagrangianmanifold,3D
3DMultivaluedTravelTimeandAmplitudeMaps
PAGEOPH, Vol. 148, Nos. 3/4 (1996)
0033-4553/96/040449-31$1.50 + 0.20/0 cg 1996 Birkhauser Verlag, Basel
ray density along isochrons
travel
timesand
by linear inversion over the entire ray field inc1uding caustics.
mode!.
1. Introduction
power.
other
J École des Mines de Paris, Centre de Recherche en Géophysique, Fontainebleau, France. 2 lnstitute for Solid Earth Physics, University of Bergen, Allegaten
ray re1ated variables
35 rue Saint
Honoré,
41, 5007 Bergen,
for
of data residuals with respect to the
77 305
Norway.
theLagrangianmanifoldAcalibedeterminedbyparaxialraytracing.TheraydensitycriterionofLAMBARÉetal.(1996)isbasedonthexandpcurvaturesof
slownessvector.ThesetofbicharacteristicsassociatedwithasourcespansaregularLagrangianmanifoldAinthephasespace
e1993b)andSUN(1992)resultedinadrasticundersamplinginthesezones,(LAMBARÉetal.,1996).lnLAMBARÉetal.(1996)rayequationsarerecastintheHamiltonianfaIm(CHAPMAN,1985).lntheHamiltonianformulation,raysintheconfigurationspace
Themethodwasimprovedin2DsuccessivelybySUN(1992)andLAMBARÉetal.(1996)byintroducingmoreefficientraydensitycriteria.TheraydensitycriterionofLAMBARÉetal.(1996)ensuresauniformsamplingoftherayfield,in
when necessary. ln
Wepresentinthispaperanextensiontothe3Dcaseofthe2Dalgorithm(LAMBARÉetal.,1996).Wefirstrecallraytheory,Hamiltonianequationsforraytracingandparaxialraytracing.Wethendescribethenumericalschemeforthewavefrontconstructionmethodin3D.FinallywepresentSailleexamples.
in the phase et
isochrons distance VINJE
between
space. al.
particular Forthe2Dcase,variousmethodshavebeenproposed(LAMBARÉetal.,1992;FORGUESetal.,1994;VINJEetal.,1992,1993a;SUN,1992).ThewavefrontconstructionmethodproposedbyVINJEetal.appearsreasonablyefficient,eveninthe3Dcase(VINJEetal.,1993b).Itisbasedonsubdivisionoftherayfieldintoelementarycellsdefinedbyadjacentraysandsuccessiveisochrons.Acriterioncontraistheraydensityovertheisochronsandnewraysareaddedbyinterpolation
andamplitudesispossiblebydynamicraytracing(FARRAandMADARIAGA,1987;FARRA,1993)ofadenselysampledrayfieldandevaluationoftraveltimes
amplitude
(1993a)
the metric 1988;PODVINandLECOMTE,1991).However,FDcomputationoftraveltimesleadstopoorimagingincomplexmedia(GEOLTRAINandBRAC,1991),due
difference
information.
adjacent
(x) are replaced by bicharacteristics such
unreliable
finite
a
the
Asignificantbreakthroughinthetravel-timecomputationwasachieved
ray related variables
criterion
through
or any other
computationofGreen'sfunctionsinthebackgroundmodelisavailable.Weaddressheretheproblemofefficientdeterminationofmultivalued3Dmapsfortraveltime,
provided
(FD) calculation
Simultaneous
in caustic regions, while the criteria of was
amplitude approximation, an efficient
throughout
VINJE
(WEINSTEIN, formulated
DALE,
by the Born
of the first-arrival
computation of
et al. (1993a), in
is given
travel
terms
to
PAGEOPH,
P. S. Lucio et al.
time
travel
VINJE
of
450
algorithm for numerical
the target zone froID any
shot and receiver position at the surface and, indirectly, for numerical evaluation of asymptotic Green's functions by ray tracing. (VI-
times
and amplitudes at given points by interpolation. The ray density must be controlled in aIder to ensure accuracy as well as computational efficiency of the algorithm.
rays.
et al.
in the phase space (x, p), where p denotes the
'1979). Tangent planes to
.
Vol. 148, 1996
2.AsymptoticRayTheoryWeshaH
3DMultivaluedTravelTimeandAmplitudeMaps
451
discuss the asymptotic of the scalar wave equation:
computation of the Green's
function
G(x, t, s)
a2 t, s)
+ U2(X)
at2 G(x,
(1)
t, s) = b(x, s)b(t)
whereudenotestheslownessofthemedium.Thezero-orderraytheoryyieldsthe
~G(x,
high
frequency
elementary
asymptotic
Green's
functions
in the
form
of a superposition
of
arrivaIs
G(x, t)
"-'
l
Ak(X)
-
YeCtk[b](t
(2)
Tk(X))
whereAkdenotestheamplitude,Tkisthetraveltime,CXkistheKMAHindexofthe
k
arrivaI
and
satisfy partial
;It denotes
differential
and the transport
equations,
equation
transformation.
respectively
A \l2T + 2\1 A . \lT
time and amplitudes
are calculated
Travel
time
and
the eikonal equation
amplitudes
(\lT)2
=
u2
by integration
of ordinary
differential
equations(O.D.E.)alongtherays.Arayisdefinedbythesecond-orderdifferential
Travel
the Hilbert
=O.
kth
equation
ax
a a{J
(
U2(X)
1
)
a{J
(3)
=2:u-_(X)\lu2(X)
=Xoanddxjd{J({Jo)=u-1(XO)twheretisthe
and by initial conditions, x({Jo) unitary initial direction of the ray, and
?
is a sampling
parameter
along
rays, with
thedimensionofthelime.AswedealwithGreen'sfunctionswechoose{Jo=0at
{J
source
and
in this
way
(J cali be associated
with
the
d{J u(x)
ax _a
travel
lime.
Thetravellimeiscalculatedbyintegratingtheslownessalongrays
the
T({Jo)
(J
+ 1
T({Jo)
+{J
-
{Jo.
( 4)
Theamplitudeisgivenfronlthegeometricalrayspreading/by
=
(J
(JO
T({J) =
1
= 4n
u(x({J))lf({J,
(5) (Jo)!.
Thegeometricalrayspreadingisdefinedherebytheexpressionf({J,(Jo)=
A({J, (Jo)
U(x({Jo))
where
sectionat{JanddQistheelementarysolidangleassociatedwiththeraytubeatthe
dS jdQ
TheKMAHindexisanintegernumberwithazerovalueaithesource.Itincreasesbyoneeachlimetheraytouchesacuspoidcaustic(CHAPMAN,1985).
source
ln
{Jo.
dS is the elementary
oriented
surface
of an orthogonal
ray
tube
ln the Hami for ray app as pr of th b H(x p, cr) =2: U2 -1 , and the bich sat the ass Ha sy o e We also note here tha fro eq. (8) it is po to de eq (3 b t i = 0 s alo an th bi at On a o e The surf in the con spa (x) or in th ph sp (x p db the
in turn
space
phase
eikonal
are
defined
as
the
equation
cali be
written
in the
H(x,
The
integral
curves
of
the
Hamilton
equations
in
(x,p).TheHamiltonianH(x,p,cr)isdefinedinsuchawaythatthe
tics, which
PAGEOPH,3.HamiltonianFormulationofRayTracing
P. S. Lucio et al.
452
Hamiltonian
form
VT(x),
(6)
cr)= o.
for an isotropie medium
cali be written in the form:
p2
1
(7)
(
2
-=VH=u-(x)p
ax
)
P
acr
(8)
ap
1
The
eikonal
equation
=-VxH=2:p2U-4(X)VU2(X).
acr
is satisfied if the
Hamiltonian
vanishes
aIl along
the
bicharacteristics.
account
the
identity
H
(8) the Hamiltonian that it vanishes
is constant
at the source
an
every
bicharacteristic;
it is sufficient to ensure
point.
isochron.
4.ParaxialRayTraeing
cr = est represents
along
interpolation
of ray field variables
as weIl as amplitude
computation
basedonparaxialraytracing(CHAPMAN,1985;FARRAandMADARIAGA,1987).Aperturbationofthebicharacteristic(bX,bp)aroundacentralone(Figure1)satis-
Linear
fies in the first-order
approximation
a linear
system
-a/5p
2
equations:
1
p2
2
=-VxVxH'bx-VpVxH'/5p=-4()VVu(x)/5xcr2uX
a
1
p2
-
U6(X)
1
+
U4(X)VU(x)(p'/5p)(9)
a(J
p
differential
=VxVpH'bx+VpVpH'/5p=-U4(X)(Vu(x)./5x)+U2(X)/5p
a/5x
of ordinary
VU2(X)(VU2(X)'
2
/5x)
is
A3Drayfieldisnaturallyparameterizedbyapairofcoordinates,«Pl'-3l\j
::rD (bg
++
Zo (b
++
.....
+++
++++
+
++
+++
~
+
+
0 '"1 ~
+
0
+
....
0
°
+
00
+
::rD (b°
+
>-3 0 "C 0
+
+
+ 00 00
++++++++++*
0 OQ '
