CA2730017C - Method for propagating pseudo acoustic quasi-p waves in anisotropic media - Google Patents
Method for propagating pseudo acoustic quasi-p waves in anisotropic media Download PDFInfo
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- CA2730017C CA2730017C CA2730017A CA2730017A CA2730017C CA 2730017 C CA2730017 C CA 2730017C CA 2730017 A CA2730017 A CA 2730017A CA 2730017 A CA2730017 A CA 2730017A CA 2730017 C CA2730017 C CA 2730017C
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
- G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
- G01V1/28—Processing seismic data, e.g. analysis, for interpretation, for correction
- G01V1/30—Analysis
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
- G01V2210/00—Details of seismic processing or analysis
- G01V2210/60—Analysis
- G01V2210/67—Wave propagation modeling
- G01V2210/675—Wave equation; Green's functions
Abstract
A computer-implemented method for pseudo acoustic quasi-P wave propagation which remain stable in anisotropic media with variable tilt and is not limited to weak anisotropic conditions.
The method includes acquiring a seismic exploration volume for a subsurface region of interest, and determining a modeling geometry for the seismic exploration volume. The method further includes propagating at least one wavefield through the seismic exploration volume utilizing the modeling geometry and initial conditions and preventing the accumulation of energy along the axis of symmetry of the seismic exploration volume and ensuring positive stiffness coefficients in the stress-strain relations through the use of finite quasi-S
wave velocities thereby producing a stable wavefield. The method includes utilizing the stable wavefield to generate subsurface images of the subsurface region of interest.
The method includes acquiring a seismic exploration volume for a subsurface region of interest, and determining a modeling geometry for the seismic exploration volume. The method further includes propagating at least one wavefield through the seismic exploration volume utilizing the modeling geometry and initial conditions and preventing the accumulation of energy along the axis of symmetry of the seismic exploration volume and ensuring positive stiffness coefficients in the stress-strain relations through the use of finite quasi-S
wave velocities thereby producing a stable wavefield. The method includes utilizing the stable wavefield to generate subsurface images of the subsurface region of interest.
Description
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.
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.
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, P=v10+20/2+,0-(f -1)./,OP vp0[2./f65-6.1ff2 +V -1X0+20/2 +)/k {
0 .
where:
V. .
V
0,
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.
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.
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, P=v10+20/2+,0-(f -1)./,OP vp0[2./f65-6.1ff2 +V -1X0+20/2 +)/k {
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
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 (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
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 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 (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 .
(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:
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 (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 .
(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 .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?
'tig*.v.lAP [61 ..4 :
,---- R=v2itiP
, ?Ae.'2 13 =
k".
*lid*.
::6 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:
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 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 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 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.
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.
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.
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 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, 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 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.
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.
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], 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:
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]
(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 - 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 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 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
¨ v = M , ¨0 + .A.I ). 0 ¨ v [231 0 Q' 0 Q
¨ -R R R R
where.
0 1+720 M , = 0 0 0 0 0 [24]
.... ...
0 0 =võ417-1-2/- 0 0-0 0 0 0 [25]
0 0 0 49.õ._. 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
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:
v- [0 20,f2 Pe [271 + v4 [2f -1X0 20i2+ AM) 0 Po
.............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?
'tig*.v.lAP [61 ..4 :
,---- R=v2itiP
, ?Ae.'2 13 =
k".
*lid*.
::6 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:
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 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 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 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.
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.
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.
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 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, 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 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.
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.
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], 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:
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]
(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 - 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 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 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
¨ v = M , ¨0 + .A.I ). 0 ¨ v [231 0 Q' 0 Q
¨ -R R R R
where.
0 1+720 M , = 0 0 0 0 0 [24]
.... ...
0 0 =võ417-1-2/- 0 0-0 0 0 0 [25]
0 0 0 49.õ._. 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
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:
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, 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 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.
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, 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 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.
Claims (11)
1. 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.
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.
2. The method of claim 1, wherein propagating at least one wavefield through the seismic exploration volume further comprises employing an artificial quasi-shear wave velocity that is greater than or equal to zero along the axis of symmetry.
3. The method of claim 1, wherein propagating at least one wavefleld through the seismic exploration volume further comprises restricting accumulation of quasi-shear waves along the axis of symmetry.
4. The method of claim 1, further comprising propagating a plurality of wavefields through the seismic exploration volume.
5. The method of any one of claims 1 to 4, wherein propagating at least one wavefleld through the seismic exploration volume further comprises at least one of reverse time migration, a wave-equation based migration, Gaussian beam migration and Kirchhoff migration.
6. The method of any one of claims 1 to 4, further comprising propagating wavefields forwards and backwards through the seismic exploration volume and applying imaging conditions to the forward or backward waveflelds and equivalent Green's functions to derive the subsurface images.
7. The method of claim 6, wherein the step of applying imaging conditions further comprises cross-correlating at least one of the forward and backward waveflelds and equivalent Green's functions to derive the subsurface images.
8. The method of claim 7, wherein the step of applying imaging conditions further comprises at least one of illumination normalization, reflection-angle domain gather generation and phase-amplitude compensation.
9. The method of any one of claims 1 to 8, wherein the modeling geometry includes common offset, common azimuth and common reflection-angle domains.
10. The method of any one of claims 1 to 8, wherein propagating at least one wavefield further comprises utilizing at least one of delayed shot, plane-wave and phase encoding.
11. The method of any one of claims 1 to 10, wherein propagating at least one wavefield further comprises utilizing at least one of normal moveout velocity, horizontal velocity and Thomsen parameters.
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| PCT/US2009/050220 WO2010014379A2 (en) | 2008-07-30 | 2009-07-10 | Method for propagating pseudo acoustic quasi-p waves in anisotropic media |
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| US8385151B2 (en) * | 2010-06-24 | 2013-02-26 | Chevron U.S.A. Inc. | Reverse time migration with absorbing and random boundaries |
| EP2598914B1 (en) | 2010-07-28 | 2015-05-06 | Cggveritas Services SA | 3-d harmonic-source reverse time migration systems and methods for seismic data analysis |
| US9207342B2 (en) * | 2012-03-09 | 2015-12-08 | Chevron U.S.A. Inc. | Correction of shear log for elastic anisotropy |
| EP2885656B1 (en) | 2012-08-17 | 2018-09-26 | Landmark Graphics Corporation | Methods for imaging seismic data |
| US20150185347A1 (en) * | 2013-12-30 | 2015-07-02 | Chevron U.S.A. Inc. | System and method of mitigating instabilities in a pseudoacoustic wave propagator |
| GB2539592B (en) * | 2014-03-31 | 2020-12-30 | Logined Bv | Subsurface formation modeling with integrated stress profiles |
| US10215869B2 (en) | 2015-03-30 | 2019-02-26 | Chevron U.S.A. Inc. | System and method of estimating anisotropy properties of geological formations using a self-adjoint pseudoacoustic wave propagator |
| CN106597531B (en) * | 2015-10-16 | 2019-10-29 | 中国石油化工股份有限公司 | The Forward Modeling of the wave field propagation characteristic of shale containing vertical fracture |
| CN107102353B (en) * | 2017-05-08 | 2019-09-03 | 厦门大学 | Equations for elastic waves reverse-time migration imaging method based on High-order Difference Methods |
| CN111999770B (en) * | 2020-09-03 | 2024-01-16 | 中国地质科学院 | TTI medium conversion PS wave precise beam offset imaging method and system |
| CN113341455B (en) * | 2021-06-24 | 2024-02-09 | 中国石油大学(北京) | Viscous anisotropic medium seismic wave numerical simulation method, device and equipment |
| CN117233838B (en) * | 2023-09-20 | 2024-04-05 | 长江大学 | Elastic quasi-longitudinal and transverse wave field separation and reverse time migration imaging method in two-dimensional VTI medium |
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| GB2278920B (en) * | 1993-06-07 | 1996-10-30 | Geco As | Method of determining earth elastic parameters in anistropic media |
| GB9906995D0 (en) | 1998-09-16 | 1999-05-19 | Geco Prakla Uk Ltd | Seismic detection apparatus and related method |
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