Note: Descriptions are shown in the official language in which they were submitted.
CA 02730017 2011-01-06
WO 2010/014379 PCT/US2009/050220
METHOD FOR PROPAGATING PSEUDO ACOUSTIC
QUASI-P WAVES IN ANISOTROPIC MEDIA.
Field of the Invention
The present invention relates generally to geophysical prospecting using
seismic
signals, and in particular a. method for propagating pseudo-acoustic quasi-'
wave
propagation in variable tilted anisotropic media and using the propagated
wavelields
for subsurface property characterization.
I 0 Background of the invention
Anisotropy is Ubiquitously observed in many oil and gas exploration areas
(e.g., the
Gulf of Mexico, the North Sea, and offshore West Africa) because of preferred
ordering of minerals and defects related to stresses. In these regions, often
the .rock
properties can be characterized as transversely isotropic ("M) media with
either a
IS vertical or tilted axis of symmetry. Wave propagation in anisotropic
media exhibits
different kinematics and dynamics from that in isotropic media, thus, it
requires
=isotropic modeling and migration methods to image reservoirs properly for oil
and
gas exploration.
20 "Three-dimcnsional. ("3D") anisotropic seismic: modeling and migration,
however, are
computationally intensive tasks. Compared to prior art solutions of full
elasticity,
modeling and migration based on dispersion relations are computationally
efficient
alternatives. In one prior art method, Alkhalifah (2000), a pseudo-acoustic
approximation for vertical transversely isotropic (nun media was introduced,
In
25 the approximation of that prior art method, the phase velocity of shear
waves is set to
zero along the vertical axis of symmetry. This simplification doesn't
eliminate shear
waves in other directions as described by Grechka et al. (2004). Based on
Alkhalifah's approximation, several space- and time-domain pseudo-acoustic
partial
differential equations (PDEs) have been proposed (Alkhalifah, 2000; Zhou et
al.,
30 2006; and Du et al., 2008) for seismic modeling and migration in VII
media. These
systems of POEs are close approximations in kinematics to the solutions of
full
elasticity involving vector fields.
CA 02730017 2011-01-06
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As an extension from. Vii media, the axis of symmetry of a TI medium can be
tilted
("'M") as observed in regions associated with anticlinal -structures and/or
thrust
sheets: Zhou et al. (2006) extended their VTI pseudo-acoustic equations to a
system
for -2D TTI media by applying a rotation about the axis of symmetry.
Consequently
.5 the phase velocity of quasi-SV waves is zero in the direction parallel
or perpendicular
to the tilted axis. 'Usage et at. (2008) futther extended Zhou's rn system
from 2D to
3D based on the same phase velocity approximation. However, these prior art
pseudo-acoustic modeling and migration methods can become numerically unstable
due to rapid lateral variations in tilt and/or certain rock properties (when
the vertical
.10 velocity is greater than the horizontal velocity), and result in unstable
wave
propagation.
As one skilled in the art will appreciate, the plane-wave polarization vector
in.
isotropic, media is either parallel (for P-waves or orthogonal (for S-waves)
to the
15 slowness vector. Except for specific propagation directions, there are
no pure
longitudinal and shear waves in anisotropic Media. For that reason, in
anisotropic
wave theory the fast mode is awl referred to as the "quasi-P" wave and the
slow
modes "quasi-S." and "quasi-Se.
20 Summary of the Invention
The present invention provides both a pseudo-acoustic modeling method and a
pseudo-acoustic migration method for anisotropic media. Aspects of embodiments
of
the present invention include a computer-implemented method tbr pseudo
acoustic
quasi-P wave propagation which remain stable in variable-tilt anisotropic
media and
25 is not limited to weak anisotropic conditions. The method also includes
acquiring a
seismic exploration volume for a subsurface region of interest, and
determining a
modeling geometry for the seismic exploration volume. The method hardier
includes
propagating at least one wnvefteld through the seismic exploration volume
utilizing
the modeling geometry for initial conditions and preventing the accumulation
of
30 energy along the axis of symmetry as well as ensuring positive stiffness
coefficients
in the stress-strain relations through the use of a small finite quasi-S wave
velocity
thereby producing a stable wavefield. The method includes utilizing the stable
wavefield to generate subsurface images of the subsurface region of interest.
CA 02730017 2011-01-06
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Another embodiment of the present invention includes a geophysical seismic
migration method comprising the steps of establishing a seismic data set. and
a
velocity/anisotropy model corresponding to a seismic exploration volume,, and
for
each common shogreceiver record, setting boundary conditions to include
excitation
from source location(s). The embodiment thither includes propagating
wavefields
forward according to a pseudo-acoustic wave equation or its equivalents:
4
4, P=v10+20/2+,0-(f -1)./,OP vp0[2./f65-6.1ff2 +V -1X0+20/2 +)/k
{
12
0 .
where:
V. .
V
0,
2 a.' _;., , a" ,, a a 2 e .,, ,ea
4:::;sidq.,(c..os4¨ +sin %-----:: +snug ¨ ¨) +COs 610 ---27-d, 4-Sirwzm Co%
J
8e . -8 -4
C a a a 6 0
A =0-silf q.,coi ,00¨ +0 -ski' 4 sir? A.).¨ 4-sirl0----, -sitf 4, sin* - - --
sin2Nco4 -- sinC7-)
' ae & a <3, &
,. & e= az
Y 4.,
1;- ...f; + h . +. +
..i.' ez,
vso is the vertical velocity of quasi-SV waves, Tip is the vertical velocity
of qaasi-P
waves, Go is the tilt of the axis of symmetry with respect to the vertical in
a Ti
medium, 00 is the azimuth of the axis of symmetry, cõ, Sate the Thomsen
anisotropy
parameters, P is a scalar wavefield, and Q is an auxiliary function. The
embodiment
also includes for each common shotfreceiver record, setting boundary
conditions to
back propagate a recorded shot record, and propagating seismic data backward
according to the above pseudo-acoustic wave equations. The embodiment includes
applying imaging conditions such as but not limited to) cross correlation
between
the Computed forward wavefields and backward wavefields or their equivalent
Green's functions to derive subsurface images.
An additional embodiment of the present. invention also includes the step of
propagating wavefields or calculating Green's functions by reverse time
migration
3
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PCT/US2009/050220
(RTM), Gaussian beam migration, Kirchhoff migration or other wave equation
based
migrations.
An additional embodiment of the present invention also includes the step of
applying
imaging condition involving illumination normalization and/or reflection-angle
domain gather generation and/or phase-amplitude compensation in addition to
cross
correlation as options.
An additional embodiment of the present invention also includes the step of
processing common-shot/receiver signals and propagating wavefields in other
dependent domains, including but not limited to coramall offset, common
azimuth,
and common reflection-angle, and in other modeling and migration forms,
including
hut not limited to delayed shot, plane-way; and phase encoding.
An additional embodiment of the present invention also includes the step of
propagating wavefields or calculating Green's functions using other equivalent
terms
such as normal moveout velocity, horizontal velocity instead of Thomsen
parameters,
An additional embodiment of the present invention includes a geophysical
seismic
migration method comprising the steps of establishing a seismic data set and a
velocity/anisotropy model corresponding to a seismic exploration volume, and
for
each common shot/receiver record, setting boundary conditions to include
excitation
from source location(s). The embodiment also includes propagating wavefields
forward according to the fol lowing pseudo-acoustic wave equation or its
equivalents:
P=r 2Ra +24/2 +.4)¨Or¨D../AP+v 712i(15-4472s +Or ¨1)(0 +WI 4- ft MI?
ar Po =
[2]
- v 2P
po
The embodiment father includes for each common shot/receiver =cord, setting
boundary conditions to back propagate a recorded shot record, and propagating
seismic data backward according to the above pseudo-acoustic wave equations.
The
embodiment includes applying imaging conditions such as but not limited to)
cross
4
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correlation between the computed forward wavefields and backward wavefields or
their equivalent Green's functions to derive subsurface images..
Different embodiments of the present invention may utilize other pseudo-
acoustic
wave equations to propagate wavefields forward in geophysical seismic
migration.
For example, one embodiment of the present invention includes propagating
wavefields forward according to the pseudo-acoustic wave equation below or its
equivalents:
diP= v 'MI +2V2 E +.4)¨(f ¨1VDP+v 4[2f(6¨)ff t(f-1)04-.201-; 41)
.1s.i2
1
(02 Q.; 2 IA
1.0 where co is the angular frequency.
A further embodiment of the present invention that is utilized ibr geophysical
seismic
.migration 'includes propagating wavefields forward according to the pseudo-
acoustic
wave equation or its: equivalents:
1'32¨ P---1' v V4 4-244; +1;)---Or ---Df dP +1? ' 110 +26)-0 +24ifigi-
vp:Or¨INO+244 +.41g
a 2 iv , 0 w .
62
(2. fip
al +
[41
-----:-R
where Q and R are auxiliary functions.
Another embodiment of the present invention includes a geophysical seismic
migration method comprising the steps of establishing a seismic data set and a
velocity/anisotropy model corresponding to a seismic exploration volume, and
for
each common shot/receiver record, setting boundary conditions to include
excitation
from source location(s). The embodiment also includes propagating wavefields
forward according to a pseudo-acoustic wave equation and its equivalent
fommlations
for tilted media:
5
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WO 2010/014379
PCT/US2009/050220
.0 . 1. 1.a. = 0. ) . .vp.. : 'a . "
.............p 0. + 270,,,,,-- ----.4,1. +===-r-=.v. I+
:... . = = = = A.
.a.if = = ,,ax ay ). 1+.z.n. az
---,,u =-----.:r
:at eh:
= ----,- V = .----' r
Oi :a.)). 151
a liPt. a
'
arQ.... = ' = '= = .= R
=
.a .,.., 0 ,, ,...,
¨ .. tc .==.¨ .r ..i, :477¨ v .
.., = = ".
az (.',z7
v is the. vertical velocity of quasi-P waves 1,,,,,i0, #V,. .-0, i -=
:27.5 is the normal.
N. =.= = =
moveo.ut velocity of quasirP=waves., ri- ..(e =-=$:)./(1.1-2.0 is
the.:Alkhalif*Tsvarikin
satropy..parainetcr.:.(eNpressed in terms of the Thomsen anisotropy
.parnineters. e and.
0.õ .. P .t it Scalar =wavelteld, and ...Li; .P'',..0,..=pnd g are ani4liary
flinCtiOri* The.
embodiment further includes .for: each cOMITion= shot/receiver. record,
actting..boundary.-:
conditions to. hack 'propagate a recorded shot record, and
propagating;:stismie data
'Imekward.:.aceordn*tocthe..above 'pseudo-acoustic *tvt equations. The
embodiment
includes applying irnagingeonditions::such as. cross: correlation between the
computed
forward and 'backward .waveficida.ortheir equivalent ..Green's ftinetions to.
derive.
subsurface images
Different embodiments of the present: invention for geophysical seismic
may Otilitt 0thek..:pset40-aw0tio:.wave:.Npat1ons.t.O.ptop4gaito.Waveliekig
.foTwarti..tbt
15: tilted media. .For example, one embodiment of the present invention
incligica
monagating.waveticids forward Recording to a, pseudo-acoustic : wave equation
and its
.cquiN.,elentitbrmniations fottille$ media'
Tp..i.[0:+.20yõ..,,..4=01.;;Z:k.lip+f;..4.-Ø,,,:giP--400.4..2Øc1õ,g1Q-71
.0-..1.--EcOL.+=en.,Zik,9.-7,0141?
82
'tig*.v.lAP [61
..4 :
8
,---- R=v2itiP
, ?Ae.'2 13 =
k".
*lid*.
::6
CA 02730017 2011-01-06
WO 2010/014379
PCT/US2009/050220
a
a2
az -
a' a2
Cl/
Vro is the vertical velocity of quasi-P waves, v,õ +26 is the
normal-
moveout velocity tif quasi-P waves, a-is the square of the shear-wave to P-
wave
velocity ratio, rj (e 5)1(1 + 28) is the Alkhalifah-Tsvankin anisotropy
parameter
(expressed in terms of the Thomsen anisotropy parameters sand 6), and Q and
Rare
auxiliary functions.
A further embodimentof the present invention that is utilized for geophysical
seismic
migration includes propagatingwavelleids forward according to a pseudo-
acoustic
-10 wave equation and its equivalent fort-initiation for tilted media or
their derivative
.formulations/equivalents:
a4 a4 =a4
.=
F 2.77)v av, .14 + (1+ a)v --72=
a(127-1)1,= f
ax,2 ar " aZ
Vp-o 1 F F F
o4
ax = ay. ,
a 4'
2 2. a4 a4 a4
+ton+ a)v po _____________________ z F : õ F' 1+ av, ... F 0
' az ay- az-
where F is a. scalar wavefield.
Another embodiment of the present invention includes a geophysical seismic
migration method comprising the steps of establishing a seismic data set and a
velocity/anisotropy model corresponding to a seismic exploration volume, and
for
each. common shot/receiver record, setting boundary conditions to include
excitation
from source location(s). The embodiment also includes propagating wavefields
forward according to a pseudo-acoustic wave equation or its derivative
formulations/equivalents:
7
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WO 2010/014379 PCT/US2009/050220
641, a2P
+ 261:V3t .Pt2 + Old
Ot
+ v4 [2f 8).6/2 (f + 24; + fdfdP =
The embodiment further includes for each common shot/receiver record, setting
boundary conditions to back propagate a recorded shot record, and propagating
seismic data backward according to the above pseudo-acoustic wave equations.
The
embodiment includes applying imaging conditions such as (but not limited to)
cross
correlation between the computed forward and backward wavefields or their
equivalent Green's functiOns to deriVe subsurface images.
Another embodiment of the present invention- includes a geophysical seismic
modeling method comprising the steps of establishing a velocity/anisotropy
model
corresponding to a seismic exploration volume, and for each shot, setting
initial
conditions of wavefields. The embodiment also includes propagating wavefields.
forward according to apseudo-acoustic wave equation or its equivalents:
j--7:P=P4.1 WW2. 41,W1f3r+v132145-eW24-(f-4).(4+24./2µ 1610(0
0
P-47-3NO
where w(t) is a. source function, midi, is the vector of the source location.
The source
term and its form of insertion can be changed without affecting the governing
PDEs.
Another embodiment of the present invention that is utilized for geophysical
seismic
modeling includes propagating wavefields forward according to a pseudo-
acoustic
wave equation (equation 5) and its equivalent formulations for tilted media.
Another embodiment of the present invention that is utilized for geophysical
seismic
modeling includes propagating wavefields forward according to a pseudo-
acoustic
wave equation (equation 6) and its equivalent formulations for tilted media,
It. should also be appreciated that the present invention is intended to be
used with a
system which includes, in general, an electronic configuration including at
least one
8
CA 02730017 2016-07-05
processor, at least one memory device for storing program code or other data,
an
optional video monitor or other display device (i.e., a liquid crystal
display) and at least
one input device. The processor is preferably a microprocessor or
microcontroller-
based platform which is capable of displaying images and processing complex
mathematical algorithms. The memory device can include random access memory
(RAM) for storing event or other data generated or used during a particular
process
associated with the present invention. The memory device can also include read
only
memory (ROM) for storing the program code for the controls and processes of
the
present invention.
One such embodiment includes a system configured to perform pseudo acoustic
quasi -
P wave propagation which remain stable in variable tilt anisotropic media and
is not
limited to weak anisotropic conditions. The system includes a data storage
device
having computer readable data including a seismic exploration volume for a
subsurface region of interest, and a processor, configured and arranged to
execute
machine executable instructions stored in a processor accessible memory for
performing a method. The method for this particular embodiment includes
determining a modeling geometry for the seismic exploration volume, and
propagating at least one wavefield through the seismic exploration volume
utilizing
the modeling geometry for initial conditions and preventing the accumulation
of
energy along the axis of symmetry of anisotropic regions within the seismic
exploration volume and ensuring positive stiffness coefficients in the stress-
strain
relations thereby producing a stable wavefield. The method further includes
utilizing
the stable wavefield to generate subsurface images of the subsurface region of
interest.
These and other objects, features, and characteristics of the present
invention, as well
as the methods of operation and functions of the related elements of structure
and the
combination of parts and economies of manufacture, will become more apparent
upon
consideration of the following description with reference to the accompanying
drawings, all of which form a part of this specification, wherein like
reference
numerals designate corresponding parts in the various Figures. It is to be
expressly
understood, however, that the drawings are for the purpose of illustration and
9
CA 02730017 2016-07-05
description only and are not intended as a definition of the limits of the
invention. As
used in the specification and in the claims, the singular form of "a", "an",
and "the"
include plural referents unless the context clearly dictates otherwise.
In an aspect, there is provided a computer-implemented method of generating a
seismic data set corresponding to a computer-generated modeling geometry for a
subsurface region of interest comprising: establishing a seismic exploration
volume
for the subsurface region of interest; determining the modeling geometry for
the
seismic exploration volume; propagating at least one wavefield through the
seismic
exploration volume utilizing the modeling geometry for initial conditions;
preventing
the accumulation of quasi-shear wave energy along an axis of symmetry of
anisotropic regions located within the seismic exploration volume; ensuring
positive
stiffness coefficients in the stress-strain relations utilizing finite quasi-S
wave
velocities thereby producing a stable wavefield; and utilizing the stable
wavefield to
generate subsurface images of the subsurface region of interest; wherein each
of the
foregoing steps is performed by a processor operating in conjunction with a
data
storage device or memory, the processor being configured to execute
instructions to
perform each of the foregoing steps, and the resulting subsurface images
corresponding to the seismic data set are representative of pseudo acoustic
quasi-P
wave propagation configured to remain stable in variable tilt anisotropic
media that is
not limited by weak anisotropic conditions.
CA 02730017 2011-01-06
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Brief 'Description of the Drawings
Fig is a flow dud illustratitig a method it accorda.nee with one or
more:
embodiments atilt; present invention.
Fig. 2 is 4 flow 'chart illustrating a method in accordance with one or more
embodiments of the present livention.
Fig. 5 is: a flow chart illustrating a method in accordance : with one or more
omboditnentS: Of the present invention
Fig: 4 illustrates exemplary wave propagation :modeling According to the prior
=art
A Iklittlifatfs ,approximation wheret = 1,
Fig, 5: illustrates exemplary :wave propagation modeling according to one
:embodiment
of the preSentinvention.
Fig. 6 illustrates exemplary wave propagation: modeling:according:to one
embodiment
of the present: inventi ott where::1401?0=0,==; 0.01,
Fig. 7 illustrates exemplary wntin propagation modeling according to the Prior
art,
Mkhalifates: approximation wherers6,-",,. 0.
Fig, illustrates: an :c).c.crtiPlary phase Velocity distribution according to
the prior art,:
Alkhalifalfs approximation.
Fig 9 illustrates an exempary Omit) velocity :distribution according to the
prior :art,
AlkhalifaWs approximation..
mg. to illustrates an exemplary phase velocity distribution for one embodiment
of the
presernitvention whom Vsilff.po 0,01.
11
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Fig, I I illustrates an exemplary group velocity distribution for one
embodiment of the
present invention where rstirip) 0>01.
Fig. 12 illustrates an exemplary wave propagation modeling in. a medium with a
variable tilted axis of symmetry, according to one. embodiment of the present
invention utilizing a first-order 5x5 POE system.
.13 illustrates a schematic diagram of the geometry that is used in one
embodiment of the present invention.
Fig. 14 illustrates is a schematic. illustration of an embodiment of a system
for
pertbrining methods in accordance with embodiments of the present invention.
12
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Detailed Description
One embodiment. of the present invention is illustrated in Fig. I, wherein a
flow chart
describes a method for propagating quasi-P waves which remain stable in
=isotropic media with variable tilt. The present invention is not limited. to
Weak.
5. -- =isotropic conditions. This particular embodiment includes acquiring a
seismic
exploration volume of a subsurface region of interest 12, and determining a
modeling
geometry for the seismic exploration volume 14. The embodiment further
includes
propagating at least one wavefield through the seismic exploration volume
utilizing
the modeling. geometry for initial conditions and preventing the accumulation
of
10 -- energy along the axis of symmetry for the seismic exploration voltune
and ensuring
positive stiffness coefficients in. the stress-and-strain relations utilizing
finite quasi-S
wave velocities thereby producing a stable wavefield 16. The stable: wavefield
can.
then be utilized to generate subsurface images of the subsurface region of
interest 18.
-- As one in skilled in the art will appreciate, differing embodiments of the
present:
invention may provide, a pseudo-acoustic modeling method or a pseudo-acoustic
migration method for =isotropic media. For example, Fig. 2 illustrates a
flowchart
for one -embodiment of a pseudo-acoustic modeling method. for wave propagation
in =isotropic media with variable tilt, wherein the method. is not limited to
weak
20 -- anisotropic condhions. That enibodiment includes acquiring a seismic
exploration
volume for a subsurface region of interest 22 and determining a modeling
geometry
for the seismic exploration volume 24. The embodiment also includes
propagating at
least one wavefield through the seismic exploration volume utilizing the
modeling
geometry for initial conditions, wherein the artificial quasi-shear wave
velocity is
-- greater or equal to zero along the axis of symmetry for the seismic
exploration volume
thereby preventing the accumulation of energy along the axis a symmetry
thereby
producing a stable wavefield 26. The stable wavefield can then be utilized to
generate
subsurface images for the subsurface region of interest 28.
-- Fitt. 3 illustrates a flowchart 30 for another embodiment of the present
invention that
can be used for pseudo-acoustic migration. That embodiment includes acquiring
a
seismic exploration volume for a subsurface region of interest 32 and
determining a
model geometry for the seismic exploration volume 34. The embodiment also
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includes. propagating at least one=waVefield through the seismic :exploration
volume
utilizing the :modeling geometry :for initial conditionSi,.. wherOn
titiasi.7Shear. Wave:
energy dbeS: riot actunittlate along the axis of symmetry for the seismic
exploration.
volume .thereby producing a. stable WaVefield.:36... The gable .wavefield can
then be
utilitedlo generate.:sitbsurface.. images for the subsurface region.. Of
interest :38...
'The present invention provides :several :advantages relative 'to
coriventional..,acoustic,.
anisoiropic modeling and Migrationõ The. present :inventionproides a stable
:Way Of
wave propagation. in TI media with variable .tilt, thus simulated...wayefield
propagation:
1.0 .and
images of reflectivity can he Obtained. Prior .art pseudo4icoustic modeling
and
migration methods are based .on. ..AlkhatilaWs .approximatiOn in. which the
phase.
velocity .of shear.wavesis sotio zero along the. aXia, of symmetry.: Although
the .prior.
'art thethOds can work in a constanmilt Ti medium, the zero-speed: shear-
waves:tan..
make wave propagation unstable. (4. arnplitudes become unbounded) ii.. areas
where.
1$ tilt
various can locally concentrate:4*h energy. near the Oda. Of .symmetry.: Fig,
4
shows that prior AtI methods (fl) are unstable 40: in
medium (e.g, near
the treat of an :anticlinal l.nructure.),.. On the .contraryõ: Fig, $ shows
that wave.
propagation based on the present invention: it
remains Stable 42 in be. same.
Inediunt in addition, the present . invention can .provide.. the flexibility
of controlling:
20 Shear- to
P-Wave *doe ty ratios to:.:Optirnize. the results of modeling and migration: .
For
example, :shear- and P-wave .velocity ...ratios can be set. 'close to the
.actual values M.
approximate the kinematics in elastic =wave propagation,: Furthermore,. in
certain.
rocks, the vertical velocity maybe: greater than the horizontal velocity with:
respect to
the. =axis .of symmetry. In .giitt a case, wave 0414tion$. based on
Alktialifalfs
25
approximation will result in negative stiffness :Matrix thereby: producing
Unstable
wavectields.. :regardless numeircal implementation algorithm. The present
invention
can use. a tinite shear-wave Velocity 10 ensure :positive stiffness
coefficients in the
stress-strain. relations thereby: generating stable wave propagation
30 in prior
art methods based on Alkhalifah's approximation,. equating Shearwave phase
.velocity to zero :000 not. :eliminate !Shear WaVeS. Instead, high energy
concontrates
near the axis of symmetry,. The only eeptiori is elliptic anisotropyõ
4*51,: fo.17
Whig* Shear waves vanish. everywhere In the present invention,. vertical
Shear.
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velocity :is: relaxed from being :zero., hence, the 00-gy is. less
concentrated near the
axis .017.symmetry. Shear waves will not vanish 'bemuse of the presence of
additional
'cross derivatives eVen.if the conditions of elliptical anisotropy are
satisfied.
In terms of .computational cost, the PDES Utilized by embodiments Of the
present
invention :Involve additional spatial derivative terms to: be computed
compared. to
prior art methods In areas. VoTith.1,!atiabletilti.the additional workload
associated: with.
non,q,*ero V0 is :memory to achieve stability and reliability yelaixed. by
Seisinie
Modeling and migration. In. areas of nearly constant or very. smooth ilk the
additional
. .
workload mayMaybe Skipped,
It Will be Clear to one skilled in the art that the above embodiments be
altered. in.
many ways .without departing from the scope. of the invention. .For example,
as is
apparent to oneskilied in the art different initial Conditions. Or boundary
conditions or
a different linear combination of the the present ...invention . can be
used
modeling arid:migration convenient,
In one 'embodiment of the present invention, the ariiaotropic :modeling Method
includes establishing: a velocity and. .anisotropy model corresponding to a
seismic.
.eXploiatiOn Volnine;..serting initialebtiditiOns such 418 .source..etitatioh,
propagating.
wase in transversely isetwpic. media with Ø :0110 or = vertiCal .axis
01.:synurietryõ.
according to eq. [It or...its equivalent, .ifor forward modelina source
thaction of the
form.
needs10..be introduced in the right side of..eqttationain f4.. [1] or.
eq,121õ where .:Iiis.the.source. location, and w(r) isa source wavelet.
In the .aboVetdescribed embodiment of the present invention, the vertical
.shear-wave.
velocity in pq. [31 can be non.,zero:. (therefore [can be different: from:
io.Ontrasttti
the prior art method approximation witerefrounds off to 1. Accordingly, the
phase
velocity of shear waves in the :direction of both parallel and perpendieular
to.. the axis
. Of symmetry Can be nonzero in the present invention, In a medium with
variable tilt,.
the finite speed of ques$hear .waves can avoid: Ipeal :COncentratiOn.. of high
energy
15.
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which often occurs in the vicinity of the axis of .syinmetry. The present
invention
does not require weak anisotropy assumptions.
Utilizing the above PDEs, other enlOodiments of the present invention can be
derived
for anisotropie media. If tilt 04 =0, the above PDEs simplify to a 3D VTI
system, and
similarly. a 3D I-M system When 00 ==.0, A second-order 3x3 System for 3D VII
media. can take of the form in equation 6, This system of PDEs is extendable
to its
equivalent formulation for a tilted 11 medium by mplacing gj and .gz given in
eq. [61
byfi. and,f2 given in eq.. [I],
0
As an alternative to the PDEs eq, [1] or [2] or [4] when f=1, the first-order
5x5=
system of PD.F.,:s in eq. [51 is hyperbolic and stable in a Ti medium with
variable tilt.
This embodiment. of the present invention is symmetrizablly hyperbolic (well-
posed,
even With variable coefficients). This system is also extendable to variable-
tilt Tn.
The above complete first-order 5x5 system of PDEs in 31) reduces to 4x4 in
21).
As described above, additional embodiments of the present invention also
provide
pseudo-aconstie migration methodS. One embodiment includcs. the steps of:
establishing a seismic data set and a velocity/anisotropy model coxrcsponding
to a
seismic exploration volume; .setting boundary conditions of wave propagation;
propagating waves from source excitation and recorded seismic data separately
in
anisotropie media according to eq, [11, eq. Rh eq. [4], or eq, [6], or their
equivalents;
and applying imaging conditions such as, but not limited to, cross correlation
between
the two propagated wavefields to obtain subsurface images. Different initial
and/or
boundary conditions can be applied without affecting the scope of this
invention. An
exemplary boundary condition (e,.g., based on eq. [1]) for propagating a
source
wavoiet is as follows:
==0,0 = ¨IA w(e)cie
rio-j
y,z = 0;t) = 5(x x,)f 140 )di
= o
and the boundary condition for reverse time extrapolation of seismic data is
as
follows:
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IP (r, y,,z = 0;.1) ,---- .13(t, y,.xo y4.; t)
IQ (x, y,z -,z-- 0;1) = D(x, y, x, ys ;1) Ill]
wh&e w(t) is a source function, x, is the location of sourceõ
D(x.,..y.,;,y,;µ) is a shot
record to migrate.
The following example illustrates a further embodiment of the present
invention:
I, Establishing fourth-order dispersion relations for quasi-P wave in VII
media
Tsvartkires phase velocity relations for VII media for which Vso= is not set
to ?Inv lead
to the dispersion miation:
(04 -36692 c = Q [12]
Where:
1 B . R1 4, 21.7)v2 + avp:jk + (1. a)vplek,2
[131
C = 41+ 20v2 vp:k4 + [(217 + 012 Vpu 2 + al, 4]lek? + CiV 4k:I
estno pu ..
,
CO is angular.frequency, ic, is the vertical wavenumber, and k..! = k.2, +
ic'. is the [square
of the] magnitude of the horizontalverturnber Vector (Ico k) Eq. [121 admits
two.
pairs of solutions:
I (--- ¨
j114-VB'-4C
' [14]
4
(Ale,: correspond to quasi-P waves; 0,) correspond to quasi-SY waves.
2. Establishing a fourth-order PDE for quasi-P wave in VII media
A
Applying eq. [12] to the wave field F(kkõ.kx,w) in the Fourier domain and
taking
the inverse Fourier transform (F(x-,y,z,, 0) provide
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- av.1 F +24; F +(t+.0t. r
a.4 aea,4 cya,. poi &aa,z
( at
+41 4- 217K,J,1 ic +1074,4e.. [15]
= , . &iv 4,4
"s1 et1 \ 4fe
4-at,,,1-2=¨F+=---.----P +ay .. F)=0
--;ivazz iyae Po\.&/
3. Establishing a second-order 33 system of PDEs for VII media
Let:
P(x,y,z,t)=¨, F(x,y,z,t) p 6j
where F(x,y,zt) is a wavefield satisfying eq. [15]. Assuming the initial
conditions
'17]
leads to:
F4,y,z,t) = P(x,y,z,f)dt.dt [18]
Let:
... + fr " a' = az
rx, y, v F P +- P (x , y , ;1") di." de
4,2 i=))2
[19]
C.-
yb 3,49 = V, F = (x, y ,r)di"
az, ea.z =
Eq. [15] is then equivalent to the second-order 3x3 system of PDEs by eq. [4].
Fig. 6 shows a wavefront propagation in a Vii medium using the above PDEs for
that
particular embodiment of the present invention. Compared to the wavefronts
based
on the prior art methods (illustrated in Fig. 7), the outer qP-wavefront (44
in Fig. 6
and 48 in Fig. 7) remains almost identical, but the inner qSV-wavefront (46 in
Fig. 6
and 50 in Fig. 7) has a different form from a diamond shape. Fig. 8 and Fig. 9
show
the phase and group velocities, respectively, according to Alkh.alifah's
(prior art)
approximation. In contrast, Fig. 10 and Fig. 11 show the phase and group
velocities,
respectively, according to an embodiment of the present invention. Compared to
Alkhalifah's approximation, the phase velocities of q,SV waves are relaxed
from being
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zero along the axis of symmetry, Consequently, the maximum values of group
velocities or high energy are not so fbcused along the axis of symmetry as in
th. prior
=art methods, The same observations are applicable to a constant-tilt TT1
medium by
applying ..a rotation about the tilt.
4. Establishing a fint-order 545 system of FilEs for VT1 media
1)u et al. (2008) presents the following second-order 2x2 Systein of PDEs for
VII
media with v,4-:0:
-12- P ' 0 -4- 1 277)v 2P
g v '4 ""
a
¨q 2---
2 k
V,,g...1 p + VII 6 glq 1201
Where gi and g2 are given after eq. [61, p is a scalar wavefield, and q is an
auxiliary
function. A new wavefield P is defined and new auxiliary functions U, P, Q,
and K
by:
a a a 0 ' 0+24)131¨p) ,, =
P E.1
= ¨p, =--p, V.-----p, Q. J.
Pc= 0 +217)¨ q, [21]
at at, 0? a ',, 2/7 a=
Then equation. 5 is a complete first-order 5x5 system of PDEs. This system can
be.
shown to be hyperbolie by symmetrizingit Let;
.._,.... ........................................... - v
1
P = P, (.1 .---v\il + 217 ti, V =: vNti 4 24 r, Q= .vi 2,7 a R,.....- , R,
[22]
/l +2q
?hem
P P P P
U U U U
0 - - - 0 -
¨ v = M , ¨0 + .A.I ). 0 ¨ v [231
0 Q' 0 Q
¨ -
R R R R
where.
0 1+720
0 0 0 0
M , = 0 0 0 0 0 [24]
0 0 0 0 0
0 0 0 0 0
.... ...
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0 0 =võ417-1-2/- 0 0-
0 0 0 00
0 0 0 0 [25]
0 0 0 0
0 0 0 0 0
0 0 0 49.õ._. 0
0
0
m [
vp, 4-27i
____________________ 0 0 0 261
v1+2/
41+
v 417
0 0 0 .V.
µ11+
In 21), the variable V is eliminated and the third equation is deleted,
resulting in a
first-order 4x4 system.
Fig. 12 shows stable wavetiont propagation governed by such first-order PDEs M
a
variable-tilt medium.
5, Estoblishing fourth-order dispersion relations in TTI media
By relaxing Alkhalifalf a approximation that Vs0-0 (or fr!) along the axis of
symmetry, the following equation can he derived from Tsvankin's phase velocity
relations (2001);
2 1.1, 2 [ esin2C7H-1/2}.2
cos ) =
1,2 2./0(Ã ---,3).sites
V,
f =1
.A)
where phase velocity v has roots of two magnitudes: one for quasi-P waves, and
the
other for quasi-SV waves is the angle between the wavefront nonnal and the
axis of
symmetry, and other parameters are defined in eq. [FL
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According to the geometry Shown in Fig. 13, the wavefront normal CFO and the
axis
of symmetry (7 ) and the angle in between take the following form:
sint3icosit + sin Osini + cos fi
1 Oil cos Alt + sin 00 sill 43 + cos ook
Ft 4 I
EtMA = '7===- sin Ocos sin cos + sin sin Osin.00 sin 4 + cos cos ,
where fie is the tilt aft axis of symmetry with respect to the vertical in a
Ti medium,
and 00 isthe azimuth of the axis-of symmetry. Recognizing that:
.tsin Ocoscbkvfa
A sin 9.sin#
1cos 0 kv.1
the tbilowing fourth-order dispersion. relations can be derived:
42
v- [0 20,f2
Pe [271
+ v4 [2f -1X0 20i2+ AM) 0
Po
6. Establishing a fourth-order PDE hi Tit media
Multiplying both. sides. of the above fourth-order dispersion relation with a
scalar
wavefield P. and converting the frequency-wavenumber operators to the time-
space
domains the fourth-order PDE for TTINTI media takes oldie fonn of eq. [13).
7. Establishing a second-order 7,x2 system of PDEs for WI media
The above fourth-order pseudo-acoustic PDE for 111 media can be solved by the
2x2
time- and space-domain PDE system by eq. [3]:
fVk = vpolh + 2e
Where
the 2x2 system of PDEs can also take an equivalent form in terms of horizontal
velocity v and normal-moveout (ls1340) velocity vi,
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In a 2D medium as a special case, the above PDEs still remain valid with the
following simplified spatial derivative operators:
a a
= (sin20 2
_________________ + cos2 u 00 2 + s in 2,a0 ¨
ax
aZ 8X aZ
a 2
a ,
f2 = (cos2 2 e, + sin 2 00 2 sin 28 a0 ¨ ¨)
a 8Z 8X 08Z
5. Establishing a second-order 3x3 system of PDEs for TTI media
As an alternative, the fourth-order pseudo-acoustic PDE for TTI media can also
be
solved by a 3x3 time- and space-domain PDEs in eq. [4] or its equivalents
using a
different linear combination.
Embodiments of the present invention can be implemented on either co-processor
accelerated architectures, such as Field-Programmable-Gate-Arrays (FPGAs),
Graphics-Processing-Units (GPUs), Cells, or general-purpose computers. The
present
invention provides apparatus and general-purpose computers and/or co-
processors
programmed with instructions to perform a method for the present invention, as
well
as computer-readable media encoding instructions to perform a method of the
present
invention. A system for performing an embodiment of the present invention is
schematically illustrated in Fig. 14. A system 52 includes a data storage
device or
memory 54. The stored data may be made available to a processor 56, such as a
programmable general purpose computer. The processor 56 may include interface
components such as a display 58 and a graphical user interface 60. The
graphical user
interface (GUI) may be used both to display data and processed data products
and to
allow the user to select among options for implementing aspects of the method.
Data
may be transferred to the system 52 via a bus 62 either directly from a data
acquisition
device, or from an intermediate storage or processing facility (not shown).
Although the invention has been described in detail for the purpose of
illustration
based on what is currently considered to be the most practical and preferred
embodiments, it is to be understood that such detail is solely for that
purpose and that
the invention is not limited to the disclosed embodiments, but, on the
contrary, is
intended to cover modifications and equivalent arrangements that are within
the
22
CA 02730017 2016-07-05
scope of the appended claims. For example, though reference is made herein to
a
computer, this may include a general purpose computer, a purpose-built
computer, an
ASIC programmed to execute the methods, a computer array or network, or other
appropriate computing device. As a further example, it is to be understood
that the
present invention contemplates that, to the extent possible, one or more
features of
any embodiment can be combined with one or more features of any other
embodiment.
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