CA2730017A1 - Method for propagating pseudo acoustic quasi-p waves in anisotropic media - Google Patents
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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. for interpretation or for event detection
- 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
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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 ACOU'STIC
QUASI- WAVES ,IN A1`QI SO1F.RO1F IC MEDIA.
Field of the Invention The, p:. cnf. inv i, .-Lion relates gene rally to geophysical prospecting using seismic Signals, and in particular a m :thod for propagating psetudo-acoustic quasi-P
wav propagation in variable tilted an-isotropic media and using the propagated wave-fields, fir subsurface. property char..ct :rÃzation.
Background of the Invention Anisotropy is ubiquitously observed in many oil and gas exploration areas (e-g, : he Gulf of Mexico, the North Sea, and of shore esÃ. Africa) bmause of preferred ordering of mineral : and de.lbct s related to stresses. In these regions, often tile rock pro perti.::s can be characterized is trans ersely isotropic ("'N" media with either a vertical or tilted axis of symmetry. Wave propagation in anÃsotrooplc à edÃa exhibits different, kinematics and dynamics from that in isotropic media, thus, it requires arà Isotropic m adeling and migration Tncethods to image reservoirs properly for oil and ,rags exploration.
Three -dimensional ('31)") arnisotrÃ3p#c ss isà - ic. r iodeling and migration, however, are computationally intensive tasks. Compared to prior art. solutions of full 0-1atsticity, n odelÃng and migration based on dispersion relations are, computation 0y efficient alternatives. In one, prior art method, Alkhalif rh (2000), a pseudo-acoustic approximation for vertical transversely isotropic ("VIT) Media was introduced, In the approximation of that, prior art me:tho d, the phase velocity of shear waves is set to zero along the vertical axis of symmetry, This Simplification doesn't el'.
urinate shear w Laves in other directions as described by Grechka .t- al. (2004). Based on Alkhaiitah's approxiin tion, several space- and time domain pseudo acoustic partial Jiff :rentiarl equations (PI f s) have been proposed (Aikhalit'a , 2000; Zhou et aal.,
QUASI- WAVES ,IN A1`QI SO1F.RO1F IC MEDIA.
Field of the Invention The, p:. cnf. inv i, .-Lion relates gene rally to geophysical prospecting using seismic Signals, and in particular a m :thod for propagating psetudo-acoustic quasi-P
wav propagation in variable tilted an-isotropic media and using the propagated wave-fields, fir subsurface. property char..ct :rÃzation.
Background of the Invention Anisotropy is ubiquitously observed in many oil and gas exploration areas (e-g, : he Gulf of Mexico, the North Sea, and of shore esÃ. Africa) bmause of preferred ordering of mineral : and de.lbct s related to stresses. In these regions, often tile rock pro perti.::s can be characterized is trans ersely isotropic ("'N" media with either a vertical or tilted axis of symmetry. Wave propagation in anÃsotrooplc à edÃa exhibits different, kinematics and dynamics from that in isotropic media, thus, it requires arà Isotropic m adeling and migration Tncethods to image reservoirs properly for oil and ,rags exploration.
Three -dimensional ('31)") arnisotrÃ3p#c ss isà - ic. r iodeling and migration, however, are computationally intensive tasks. Compared to prior art. solutions of full 0-1atsticity, n odelÃng and migration based on dispersion relations are, computation 0y efficient alternatives. In one, prior art method, Alkhalif rh (2000), a pseudo-acoustic approximation for vertical transversely isotropic ("VIT) Media was introduced, In the approximation of that, prior art me:tho d, the phase velocity of shear waves is set to zero along the vertical axis of symmetry, This Simplification doesn't el'.
urinate shear w Laves in other directions as described by Grechka .t- al. (2004). Based on Alkhaiitah's approxiin tion, several space- and time domain pseudo acoustic partial Jiff :rentiarl equations (PI f s) have been proposed (Aikhalit'a , 2000; Zhou et aal.,
2 06 and Du c i. of, 2008) or seism iic .modeling and migration in VTI i edia.
These systems of PDE.,s are close approximations in kinematics to the solutions of full elasticity involving, vector fields.
an extension from V"11 .meth t, tlt axis of s; aametry of a,171 medium c. ln' e tilted ... .j as {~I,: t~~tts..C. with a~~ti~.lrlaa3~ structures Qaladli~ thuu ae1 F Ir; ms`s` ,Fi%GS i in rLt, Sti,aa~ %tss.f~ ~' , ~'r, sheets. Zhou ;::'t Al. (2006) extonextended their VTI psea o acoustic equations to a system l tr 2) I'll media by auply>ing, a rotation about the axis of syi aaris tr Consequerrtly e phase Z elocity of gaaasi V waves is zero in the -irGc'=tion pmaflol or perpendicular to the LLtilted axissage :~.til. (3.008 t ._. er extende hou {s, 't~:`T' systom from'-'D to ~~N:ii4'yytt 2 on the :iFsame phase 4~~..~ \` L}' approximation. 2o 4'ter, these prior art p Feu o-aicoustic modeling and migration [a3 , ;i~~ can become n.m7,ericallyunstable.
due to rapid lateral variations in tilt and'.or certain rock properties (when the vert.ic~:.11 velocity is greater than the horizontal v'eloeiq) and result in, unstable wave propagation, As one ,killed in the art will appreciate, the plan ,- ave polarization vector in isotropic t ediit is either parallel (for P-waves ) or onhogonal (for ~* t a i s to tit.
slowness v`etoa. Except for specific l 13.i?,z>w, ai?.a directions, there are no pure anisot 1 21.
longitudinal and shear waves in miaisotropic= For that rason, in 'wave theory the fast mode is often ref :rred to as the quasi- , wave and the slow modes `quasi-S~" and `quasi-S,", r.# mum yof r Inve ton 110 Present Ãn' ention prof ides both a pseudo=.aeousti modeling method and a pseudo-acoustic migration method for ar isotropic: m diaa. Aspects of embodiments of ttt.~> t i;
the present invention include a c> aÃ~ purer-im (~Qa ente method f r pseudo quasi- P wmi propagation hiich remain stable: in v'arÃiable-tilt anisotropic media and '25 is not limited to weak aniso$ opic conditions. The method also includes acquiring a ei mic e: l rbation v }Ila. Ã' o a subsurfiice region of interest aid determining a modeling geometry or the s64mic exploration' The method furtla~:r includes propagating at least one wavefeld through the seismic exploration volume utilizing f he model nrg geometry for initial conditions and preventing the accumulation of 30 energy along the axis of' symmetr . as well as ensuring Positive stiffs es coefficients in the stress-strain relations thi.-m ,,h A e use of a small finite quasi-S Z
ave velocity' method includes utilizing t e stable ;:m uc'iat : a stable ve#aeld, The way c : teld to generate subsura Images of the. :subsurface region of interest.
Another embodiment of the present invention in dudes a pophysical seismic migration meth .od coialprising the steps of establishing a seismic data set and a velocity/anisotropy model Corresponding to a seismic exploration volume- and for each record, sett ng boundary conditions to include ex itatioÃ, from source location.(:). The embodiment tuner includes propagating wavefields forward according to a pseuu do-acouslic wave equation or its equivalents, +1, f2 ` 1 tjPr {., ( 1 t,t à {
t~A
where'.
s~:; t t~ - t - u t f:
# iõ t = ~ FÃ fr, C 4# i 0, r ,ti#C#7~ -Cos }2 # 7 s t`, C;ti {t # n#i _ qqt
These systems of PDE.,s are close approximations in kinematics to the solutions of full elasticity involving, vector fields.
an extension from V"11 .meth t, tlt axis of s; aametry of a,171 medium c. ln' e tilted ... .j as {~I,: t~~tts..C. with a~~ti~.lrlaa3~ structures Qaladli~ thuu ae1 F Ir; ms`s` ,Fi%GS i in rLt, Sti,aa~ %tss.f~ ~' , ~'r, sheets. Zhou ;::'t Al. (2006) extonextended their VTI psea o acoustic equations to a system l tr 2) I'll media by auply>ing, a rotation about the axis of syi aaris tr Consequerrtly e phase Z elocity of gaaasi V waves is zero in the -irGc'=tion pmaflol or perpendicular to the LLtilted axissage :~.til. (3.008 t ._. er extende hou {s, 't~:`T' systom from'-'D to ~~N:ii4'yytt 2 on the :iFsame phase 4~~..~ \` L}' approximation. 2o 4'ter, these prior art p Feu o-aicoustic modeling and migration [a3 , ;i~~ can become n.m7,ericallyunstable.
due to rapid lateral variations in tilt and'.or certain rock properties (when the vert.ic~:.11 velocity is greater than the horizontal v'eloeiq) and result in, unstable wave propagation, As one ,killed in the art will appreciate, the plan ,- ave polarization vector in isotropic t ediit is either parallel (for P-waves ) or onhogonal (for ~* t a i s to tit.
slowness v`etoa. Except for specific l 13.i?,z>w, ai?.a directions, there are no pure anisot 1 21.
longitudinal and shear waves in miaisotropic= For that rason, in 'wave theory the fast mode is often ref :rred to as the quasi- , wave and the slow modes `quasi-S~" and `quasi-S,", r.# mum yof r Inve ton 110 Present Ãn' ention prof ides both a pseudo=.aeousti modeling method and a pseudo-acoustic migration method for ar isotropic: m diaa. Aspects of embodiments of ttt.~> t i;
the present invention include a c> aÃ~ purer-im (~Qa ente method f r pseudo quasi- P wmi propagation hiich remain stable: in v'arÃiable-tilt anisotropic media and '25 is not limited to weak aniso$ opic conditions. The method also includes acquiring a ei mic e: l rbation v }Ila. Ã' o a subsurfiice region of interest aid determining a modeling geometry or the s64mic exploration' The method furtla~:r includes propagating at least one wavefeld through the seismic exploration volume utilizing f he model nrg geometry for initial conditions and preventing the accumulation of 30 energy along the axis of' symmetr . as well as ensuring Positive stiffs es coefficients in the stress-strain relations thi.-m ,,h A e use of a small finite quasi-S Z
ave velocity' method includes utilizing t e stable ;:m uc'iat : a stable ve#aeld, The way c : teld to generate subsura Images of the. :subsurface region of interest.
Another embodiment of the present invention in dudes a pophysical seismic migration meth .od coialprising the steps of establishing a seismic data set and a velocity/anisotropy model Corresponding to a seismic exploration volume- and for each record, sett ng boundary conditions to include ex itatioÃ, from source location.(:). The embodiment tuner includes propagating wavefields forward according to a pseuu do-acouslic wave equation or its equivalents, +1, f2 ` 1 tjPr {., ( 1 t,t à {
t~A
where'.
s~:; t t~ - t - u t f:
# iõ t = ~ FÃ fr, C 4# i 0, r ,ti#C#7~ -Cos }2 # 7 s t`, C;ti {t # n#i _ qqt
3.
t} t`
_ #33 ? wf 5 ' :n#i -h-.'-0 --s:#CÃ co, ft i Ã> 1# i l# s ;# #~# t3 :: ' tfF t S F ~
---- ----- ----`Y --k, 1' '; is the 3 ertic ai VelocÃty, of uasi-SV waves, L p 0 is t e vertical ve oc tz- .3i quasi-P
waves. rY is the tilt of the axis of symmetry with respect to the vertical in aTI.
il' me r\', x c, f 'are the homs'13 ~aniso r[~p medium. ~hfJ is the a irnuth of the axis of parameters, Pis a scalar , avefeld, and 'i is an auxiliary function. The enibodirnent also includes for each common? C tf s .i i c record, sating boundary conditions to back propagate a recorded shot record, and pro agaiinf seismic data backward according to the above pseudo-acoustic wave equations. The embodiment include.-s' applying imaging conditions such as (btit, not limited to) cross correlation between .0 the computed for aad wavef olds and backward w avefields or their equivalent Greens functions to drive subsur c .ima e .
An addition zl embodiment of the presenà nvention als include"', the step of propagating w4r i fields or calculating Green's fu n.et_i ns by reverse time migration (RT'. I), t; {tEissian beam migration, Kirchhoff migration or other wave.
equation based rnigratÃons,.
An additional embodiment of the present invention also includes the s ep of applying imaging condmon mvol'4 #n .lilu'.` inau on normal,zation and/0'r r S ction^ 3 domain zither generation and/or phase-amplitude compensation in addition to cross correlation as option.s.
An additional ernbodiment of the present invention also includes tine step of processing common-si?(?t/receiver s_i à ais and propagating wave-fields in other dependent domains, including but not limited to common oà swt, con.,=)n azimuth, and common reflection--angle, and in other modeling and migration forms, including hut not limited to delayed shot, plane-wave, and phase: encoding.
An additional embodiment of the present invention ;l:nc includes the step of gating wavef elcls or calculating GreeÃn' r .tI ctions using, other equivalent germs pro pa such as normal nnnovreout velociÃy, horizontal velocity instead offhomsen parameters, An addition :,M embodiment (if the present invention includes a geophysical ei srr rc, migration method comprising the steps of establishing a seismic data set. and a eki aà '{ <rariso'tÃot?z model corresponding to a seismic exploration volume, and. for ea :h cwnvnorn shoti ro e.iver record, setting boundary conditions to include excitation from sourzce lc?ca it?;nt;:s The embodiment also includes propagating waetiel s for s'iard according to the kollowinig pseudo-acous.ic wave equation or its e taivuic its.
y _ y 21 [ .r The ,3 3 embodiment further includes for each common shot/receiver record, setting boundary conditions to back propagate a recorded shot :cord, and propagating seismie data backward according to the above pseu~io_aeocist.lc Wave equations. The embodiment includes anal lying imaging conditions, such as (hut not limited to) cross
t} t`
_ #33 ? wf 5 ' :n#i -h-.'-0 --s:#CÃ co, ft i Ã> 1# i l# s ;# #~# t3 :: ' tfF t S F ~
---- ----- ----`Y --k, 1' '; is the 3 ertic ai VelocÃty, of uasi-SV waves, L p 0 is t e vertical ve oc tz- .3i quasi-P
waves. rY is the tilt of the axis of symmetry with respect to the vertical in aTI.
il' me r\', x c, f 'are the homs'13 ~aniso r[~p medium. ~hfJ is the a irnuth of the axis of parameters, Pis a scalar , avefeld, and 'i is an auxiliary function. The enibodirnent also includes for each common? C tf s .i i c record, sating boundary conditions to back propagate a recorded shot record, and pro agaiinf seismic data backward according to the above pseudo-acoustic wave equations. The embodiment include.-s' applying imaging conditions such as (btit, not limited to) cross correlation between .0 the computed for aad wavef olds and backward w avefields or their equivalent Greens functions to drive subsur c .ima e .
An addition zl embodiment of the presenà nvention als include"', the step of propagating w4r i fields or calculating Green's fu n.et_i ns by reverse time migration (RT'. I), t; {tEissian beam migration, Kirchhoff migration or other wave.
equation based rnigratÃons,.
An additional embodiment of the present invention also includes the s ep of applying imaging condmon mvol'4 #n .lilu'.` inau on normal,zation and/0'r r S ction^ 3 domain zither generation and/or phase-amplitude compensation in addition to cross correlation as option.s.
An additional ernbodiment of the present invention also includes tine step of processing common-si?(?t/receiver s_i à ais and propagating wave-fields in other dependent domains, including but not limited to common oà swt, con.,=)n azimuth, and common reflection--angle, and in other modeling and migration forms, including hut not limited to delayed shot, plane-wave, and phase: encoding.
An additional embodiment of the present invention ;l:nc includes the step of gating wavef elcls or calculating GreeÃn' r .tI ctions using, other equivalent germs pro pa such as normal nnnovreout velociÃy, horizontal velocity instead offhomsen parameters, An addition :,M embodiment (if the present invention includes a geophysical ei srr rc, migration method comprising the steps of establishing a seismic data set. and a eki aà '{ <rariso'tÃot?z model corresponding to a seismic exploration volume, and. for ea :h cwnvnorn shoti ro e.iver record, setting boundary conditions to include excitation from sourzce lc?ca it?;nt;:s The embodiment also includes propagating waetiel s for s'iard according to the kollowinig pseudo-acous.ic wave equation or its e taivuic its.
y _ y 21 [ .r The ,3 3 embodiment further includes for each common shot/receiver record, setting boundary conditions to back propagate a recorded shot :cord, and propagating seismie data backward according to the above pseu~io_aeocist.lc Wave equations. The embodiment includes anal lying imaging conditions, such as (hut not limited to) cross
4 corrF;`anon het eei? the ~c~ a ?sac orw rc -, xavefekis and backward wav .tie:. s or their equivalent r ai3`s functions to derive subsur ace imaà es.
l:)ÃtTerent embodiments of the present invention may utilizo othu pseudo-:ALL?
us ie wave equations to p o ita o wave fields for and in geop ys#.s_ al seisms is migration.
For exampbm . one L~i~ilt?L i ent of th pre sent Ãn xention. inclu&F
Propagating `avehelds fbr ward ac :omdi-ng to the pseudo-aeoustie wave e wltÃon below or its, equi'' aknts:
.~: > t 3' t\ v i i 3 ]P+-v,. i f # ~srj: ~.x , i ;F. \
y, L.~-i rJ', 0 ==P
0 where (v is the angular frequency.
A further embodiment of t >e present invention that is utilized for geophy iii al seismic migration includes propagating w `avtfields tons "ard ac-cording to the, pscmdo-auwstic wave equation. or its equiva .%.nts:
fa )P 1"
[41 R= U
3~w where O and R are au i[i Ary> functions.
Another embodiment of the 3 s ratio `~ includes a ge oph lien. seismic migration i aethod comprising the steps of establishing a seise ic~ data set and a 20 velo it /anisotr ? ?L model corresponding to a seismic explara ion v'olurne. and for common shot recei er record, setting boundary conditions to include excitatiort a]"oin ;~C`urce iocatio.u s). f l' e L 2-ibodii'21ent ids ? ncludes propagatii g '4' :fields forward c cording to a pseudo ac 3ust.,c. wave equation ail.d its izt sent:
f+3'iT?ii: cttion , for tilted media `a t r t f, OV I+ 2y a~~
Ã. r }
i R P + 2q, r 1 ; ' N ) is the vertical voh > t j of gmisi-P waves, ?';at v i"\ v i 2t is the n 3r:'rnaly '37.o c[fi:i: v l'I6i%:]t of i i.... 3 waves, r ^- F ii j!E 1 {) is the <' it h S4' tl' ?.i'>
c .isotropy parr meter (expos'wd in terms of f hi: ' l<?.m iE ii anisotropy parameters ` and ), P is a scalar wavyfie1 . and a . F', 0, and R am auxiliary %inctio .s. ' x embodiment further includes or each common shot/ coiver record., sett boundary conditions to back pro agate a recorded shot record., and propagating seismic data backwar(Li cormr1? to t' cquatio.n<+ ''r embodiment includes applying m,ag r g Ãy' i ditt-ms such as cros eon'eiation between the cornputed Ihrward and wavef :lis or their equivalent Gre n's functions to derive ace Bi?*ag 5:
I tkicuà embodiment=' of thti ?': s nt li??. e- do n for g oph-ysica. seismic afigration may w li c. other ,:is. & o <u ou'ti~, ... e Iua iom to pi'E,pa c i awf--Szelds lonvard for tilted ini dla, lYo c\ample, one: ert l?odir?: ena of z1be pi . nt yr rention indu :s prop i;Fat?'ng vr=`asiekield i fonvand. accoi'ldin if) a }`i t=. ~&E}..
hÃ=C?tai>ti~ L ':z~ '4 = Ii iit,:,tz : t: its equivalent torrnud ttiorr> for tilled mwdia', ..r .t =~Ei.,~=~ r~~t..ttn:. t~~;.4~:~~..r ~ ~. ~~ ~ ;ri: ~ ti: g..Q..~1~"^r~
t s -6 f ~r.
i PO
`.7 z t % C='`
3 is the vatica' voIcscity of iqu fsi P , i '"es vi:asi> v ./= .`Li is the i f.=, nalx ,nnoveout: velocity of quasi -P waves, .., is the square of the s a - ave to V-wave .' 5 r :loe~iIy rafto, 1i (k' ....,y) /(i "" 2 8) is i he AIkhaIifi -Tsi 3.nkin ani-,o ropy a p ra (express d in terms of t1 w Thomsen is ni t3' i i3 parameters r and dt, and Q
Fand- tare auxiliary functions.
A .i# it er embodiment of the pits #i invert ion th that is utilized fo ophysicai s ~ismic-` Aga \ a R t r: t C
it?i~. iÃi%:li:~i`=::: i'Ã3i~,,;i.$Ã`iE~, ~'vC ~,r..3t~;~ .ty;`~i?c~.~.tE3 according w p\s~i~C# Iacoti:6L`=
n-fi,?Ã";.i~.
m . , a e equation a d its -.`r tiled. media on their deri,vwive ----- --------Lr, p "" ik i~ y..._ L.~.~' J` ;F. [r ~~1 }= '^" r .f Y
r {)# 4 ti r i~
+ =~ + + i'A' # 3 E cut? 3~ 3F f a' }
\:L CH a L l fu, where F is a scalar w'av of .,ld.
Another embodiment of the pies :Ãnt invention includes a geoph sisal seism cc ÃF igrat.io: i .i::th ?d c mprI'an J w st4qof .tiithh.i sh*.Ã"iu a sei nie data, set and veto '; tiÃ... t? A ? ?I z iodel wz?.i i , kia 1M; to zi seismic exploration volui7 e, and l or each co mon Aot/r cel ei' reco y settin ', boundary conditions to include from source Location(s), The embodiment also in ~. a+..:k;. 1p,op gatinv . `0 forward ac or'dÃ.)ng to p a eudo- courtÃC= b?iavo, equation or its, derivative ft?f.,fi uii~tio s/~,..qØ Sv Yf:243:its.
rl `f t_r y ~` r J
Pu 6t The embodiment fi.Fu ha in nudes for each e o t s~tozn sho Ã/ receiver Ã
:. ord, Setting boundary conditions to back p )pag: to a recorded shot m cord, and propagating seismic: data backward according to the 3 rc~z' pseur.o-<..oi.Fstic wave equation s. The embodiment. includes applying imaging, condition, such as (but not limited to) Gros, orrdatioF ~~t een the con puted forward andbackward o.r `' .ir r, equivalent Green's fiina.tions to derÃv ubsurt w-- iti a s .
s tiff `.3? :' t.Ã3 both en of the Present invention Ãà Z lushes a g eoph'c sical seismic modeling method c o mp isimg,, the steps of 4st bl shffi a eloci zr',.Ãri ~c?trt? ?~ model corresponding to a sà ismio exploration v'ox?.iÃ?'?e, and for each shot, sotting initial conditions of zi eieltfs, The embodiment also includes propagating A-vave field . ? and according to a pseudo acousti . waw equation or es equi alert .
a., -Y2 i.$ "r ltt;~{tà f;,u'~f r l;#t ''si"rili~p 1:5 where w(/) is a iource. function, and ', is the Vector of the sou ce loi Lion. ,, he source team and its form of insertion can In changed without affo-e ng the governing PD Es, Another embodiment of the present invention that. is for geophysical seismic modeling includes pr'opag..tf=." fie: s forward according to a p. seud'o acou atic wave equation (equation and it equivalent tormu suÃons for tilted media.
Another embodiment (if the :l" ~' invention that is utilized. for geophysical sàs is modeling includes propa flri wav'efields fo ,ard according to a l- seudo -cousti ,.
wavy. equation equation 6) and its egtdvaalc. ,: for tiled media.
It .sshoul :l also be #pl` reci lted that the, inZ ention Is iF?Ctn... ? t =
be used with a syst i?3 Which include", in goner 6l, an z:* ctronic configuration i-t luding at least one t?a aet~sc r. at j east one i e:moiy device for storing. program code or Other data, an optional Video à n for or other display device ;.c , a liquid crystal isplay :ind at least one Fi3tli it', The processor is preferably a microprocessor or =~~.icr<?controller-put based platform which is capable of displaying images and processing complex _1T e Ã3 4' à iathe n ;xt.}kcal algoritt`}1ÃF..s. : he memory vice can include random access (RAM) for storin4 event i? other data gene'Ãated or used during a particular.
Pro..'--v's associated with the r : 'FF :3 t #3t#CY11. The Ã12à z33o iC e can ,11s o in :l zd read only memory (ROM) for string the rog=rw code tear à 3e .~ ..trots and processes of the ilrt s.e at ixiventi n, One such embodiment includes a systefxi configured to perform pseudo acoustic quasi 13 t 'aSe propagation v hich Femain st.able in variable salt ?.isotropic media and is not limited to veak snisotropicc conditions, The `f s'.em includes a data torage de ite having computer readable data inc.ludin a exploration volume for a.
subsurface region of ..TFtere at, and a processor, t is}`i' tt C and arranged to execute machine executable instructions _re in a p #:ocessor accessible n-i mor'y for per'fOrini i a method. The method for this particular eTnbodi-rrient includes determining a modeling geometry f, t' e seismic exploration -"oh.ii'f'ie, and f ?tip a satin at least one b!' avei..3d , as?z.1 ', tia4 seismic exploF ation volume utilizing the modeling 3 nu ti" for initial. t tsl à itiou and preventing the accumulation o energy along the axis of symmetry of a õt>C? ?iC re-glom" Within the seismic exploration volume, and ensuring 1?,2s1 `.. coefficients in the stress-strain relaÃtio.ns thereby pr-o uc,1-ng a stable w avefi ld. The method 'forth u includes utilizing the stable wsa-v: a field to generate subsurface Images of the subsur1:a . F
ion of These and other obiects, features, and ci arac=teristit s of the Present intention, as 'ell as the methods of operation and id functions of the related aeelements of structure and the combination of parts and economies of .111 i1ÃFt3.tbctu e, will become more apparent upoii 3 consideration of the following desc ".ipdun ..:k ? the appended claims with T,, . ienc to the acco p nying di' 1''<3 ings, all of which firtm a. put of fl-us wherein file reference FFf:F orals desi'.F? it corresponding parts in the various, l i 11 s. it is,, to be =e ,pFessly understood, }?~?:4 , that the. dra vingi are for flac'purpose of illustration a et description only and am not intended as a definition of the lin _its of the invention.
As used in the specÃ?` caitio and inthe i ,ainns' the Singular .form of "a"' c and 'the" include: plural >e_ ,.ren s unk ss the :.Ã..kar y dictates, o erw sW' Brief P 4,s4 rxi t n the Drawi s I aA . o'w chart i li} .ai<A F: a .t :a tli in %iii c~atla u 'wiali one or i 3i`i'r embodiments of the present inveui.;<on.
Fig, 2 is a flo w chars illiistratmg as method in. accordance with one or more ernbodirnenu of the pr:Ã sent lrw ru o .
F:in. ? is,,, a flow chart Ãl t .,.r.~v#; a. r iethod In accordaaice: z ritl urn or more 1 S embodiments oft he present invention>
Fig, 4 illustrates ~a. =:i i l it > wave propagation modeling- <a cord nig to the, prior iin, Aixhaali :ah's approximation whc,,re,,/':::: 1, 15 1 i . 5 illustrates exempla 5' wave propagation .modt. i ng according to one embodiment of the present Fig. Ã> illustrates, exemplary wave propagation modeling according to once.
embodiment of he present invention w '':here F1 2tr7;7 :__: O' I .
`%0 `ig. "i ill istrat:`,s exemplary wave propagation modeling according to the prior a t, A khaaalit h's approximation where H& . l illustrates is exemplary phase v>>ekc=.it istribu io according to the prior art, 25 khauifa-i's appioximation.
Fig,. 9 illustrates an exernplaar group velocity distribu ioiF. to tht art, z l li. lit' ih s a pproximtiti23n.
30 Fig: 10 illustrates an xemphi:y phase velocity distribution tbbr one embodiment of he present invention ~Z}icere 4pi 0.01.
Fig. 11 illllstrat an exemplary gm up Velocity distribution for one embo a5men of the present invention where y k # `,} 001 Fig. 12 illustrates u l exemplary wave propaga don Ãnod l.:n4 in a Tedium vial: a 'Lai`Ãabk,Z ti led axis of sy'mi >et. y ac.cording to t} .le embodiment of the present invention u i i a 3 ; a à rsÃ-or er 5x P1)11 s stem .
Fig, 13 illustrates a ch ::i t tia diagram of the geometry that is used in one nibodimz Ãi. of the pi sent invention, Fig. 14 illustrates is a schematic illustration of all embodiment of a s st;.]lms for, p :rfonnmi g met cods in accordance with [mhodiinents of "he present invention, Detailed Description One i:=n nodÃm .:nt of the prosent invention is illustrated in Fig. 1, wherein a ffl(nk` Ãõ' art.
describes a m t_hod for propagating quasi-lx waves which remain à is in an-isotropic media with variab ~ tilt, ''be pr sent .n enÃ.ion is not limited to weak.
l:)ÃtTerent embodiments of the present invention may utilizo othu pseudo-:ALL?
us ie wave equations to p o ita o wave fields for and in geop ys#.s_ al seisms is migration.
For exampbm . one L~i~ilt?L i ent of th pre sent Ãn xention. inclu&F
Propagating `avehelds fbr ward ac :omdi-ng to the pseudo-aeoustie wave e wltÃon below or its, equi'' aknts:
.~: > t 3' t\ v i i 3 ]P+-v,. i f # ~srj: ~.x , i ;F. \
y, L.~-i rJ', 0 ==P
0 where (v is the angular frequency.
A further embodiment of t >e present invention that is utilized for geophy iii al seismic migration includes propagating w `avtfields tons "ard ac-cording to the, pscmdo-auwstic wave equation. or its equiva .%.nts:
fa )P 1"
[41 R= U
3~w where O and R are au i[i Ary> functions.
Another embodiment of the 3 s ratio `~ includes a ge oph lien. seismic migration i aethod comprising the steps of establishing a seise ic~ data set and a 20 velo it /anisotr ? ?L model corresponding to a seismic explara ion v'olurne. and for common shot recei er record, setting boundary conditions to include excitatiort a]"oin ;~C`urce iocatio.u s). f l' e L 2-ibodii'21ent ids ? ncludes propagatii g '4' :fields forward c cording to a pseudo ac 3ust.,c. wave equation ail.d its izt sent:
f+3'iT?ii: cttion , for tilted media `a t r t f, OV I+ 2y a~~
Ã. r }
i R P + 2q, r 1 ; ' N ) is the vertical voh > t j of gmisi-P waves, ?';at v i"\ v i 2t is the n 3r:'rnaly '37.o c[fi:i: v l'I6i%:]t of i i.... 3 waves, r ^- F ii j!E 1 {) is the <' it h S4' tl' ?.i'>
c .isotropy parr meter (expos'wd in terms of f hi: ' l<?.m iE ii anisotropy parameters ` and ), P is a scalar wavyfie1 . and a . F', 0, and R am auxiliary %inctio .s. ' x embodiment further includes or each common shot/ coiver record., sett boundary conditions to back pro agate a recorded shot record., and propagating seismic data backwar(Li cormr1? to t' cquatio.n<+ ''r embodiment includes applying m,ag r g Ãy' i ditt-ms such as cros eon'eiation between the cornputed Ihrward and wavef :lis or their equivalent Gre n's functions to derive ace Bi?*ag 5:
I tkicuà embodiment=' of thti ?': s nt li??. e- do n for g oph-ysica. seismic afigration may w li c. other ,:is. & o <u ou'ti~, ... e Iua iom to pi'E,pa c i awf--Szelds lonvard for tilted ini dla, lYo c\ample, one: ert l?odir?: ena of z1be pi . nt yr rention indu :s prop i;Fat?'ng vr=`asiekield i fonvand. accoi'ldin if) a }`i t=. ~&E}..
hÃ=C?tai>ti~ L ':z~ '4 = Ii iit,:,tz : t: its equivalent torrnud ttiorr> for tilled mwdia', ..r .t =~Ei.,~=~ r~~t..ttn:. t~~;.4~:~~..r ~ ~. ~~ ~ ;ri: ~ ti: g..Q..~1~"^r~
t s -6 f ~r.
i PO
`.7 z t % C='`
3 is the vatica' voIcscity of iqu fsi P , i '"es vi:asi> v ./= .`Li is the i f.=, nalx ,nnoveout: velocity of quasi -P waves, .., is the square of the s a - ave to V-wave .' 5 r :loe~iIy rafto, 1i (k' ....,y) /(i "" 2 8) is i he AIkhaIifi -Tsi 3.nkin ani-,o ropy a p ra (express d in terms of t1 w Thomsen is ni t3' i i3 parameters r and dt, and Q
Fand- tare auxiliary functions.
A .i# it er embodiment of the pits #i invert ion th that is utilized fo ophysicai s ~ismic-` Aga \ a R t r: t C
it?i~. iÃi%:li:~i`=::: i'Ã3i~,,;i.$Ã`iE~, ~'vC ~,r..3t~;~ .ty;`~i?c~.~.tE3 according w p\s~i~C# Iacoti:6L`=
n-fi,?Ã";.i~.
m . , a e equation a d its -.`r tiled. media on their deri,vwive ----- --------Lr, p "" ik i~ y..._ L.~.~' J` ;F. [r ~~1 }= '^" r .f Y
r {)# 4 ti r i~
+ =~ + + i'A' # 3 E cut? 3~ 3F f a' }
\:L CH a L l fu, where F is a scalar w'av of .,ld.
Another embodiment of the pies :Ãnt invention includes a geoph sisal seism cc ÃF igrat.io: i .i::th ?d c mprI'an J w st4qof .tiithh.i sh*.Ã"iu a sei nie data, set and veto '; tiÃ... t? A ? ?I z iodel wz?.i i , kia 1M; to zi seismic exploration volui7 e, and l or each co mon Aot/r cel ei' reco y settin ', boundary conditions to include from source Location(s), The embodiment also in ~. a+..:k;. 1p,op gatinv . `0 forward ac or'dÃ.)ng to p a eudo- courtÃC= b?iavo, equation or its, derivative ft?f.,fi uii~tio s/~,..qØ Sv Yf:243:its.
rl `f t_r y ~` r J
Pu 6t The embodiment fi.Fu ha in nudes for each e o t s~tozn sho Ã/ receiver Ã
:. ord, Setting boundary conditions to back p )pag: to a recorded shot m cord, and propagating seismic: data backward according to the 3 rc~z' pseur.o-<..oi.Fstic wave equation s. The embodiment. includes applying imaging, condition, such as (but not limited to) Gros, orrdatioF ~~t een the con puted forward andbackward o.r `' .ir r, equivalent Green's fiina.tions to derÃv ubsurt w-- iti a s .
s tiff `.3? :' t.Ã3 both en of the Present invention Ãà Z lushes a g eoph'c sical seismic modeling method c o mp isimg,, the steps of 4st bl shffi a eloci zr',.Ãri ~c?trt? ?~ model corresponding to a sà ismio exploration v'ox?.iÃ?'?e, and for each shot, sotting initial conditions of zi eieltfs, The embodiment also includes propagating A-vave field . ? and according to a pseudo acousti . waw equation or es equi alert .
a., -Y2 i.$ "r ltt;~{tà f;,u'~f r l;#t ''si"rili~p 1:5 where w(/) is a iource. function, and ', is the Vector of the sou ce loi Lion. ,, he source team and its form of insertion can In changed without affo-e ng the governing PD Es, Another embodiment of the present invention that. is for geophysical seismic modeling includes pr'opag..tf=." fie: s forward according to a p. seud'o acou atic wave equation (equation and it equivalent tormu suÃons for tilted media.
Another embodiment (if the :l" ~' invention that is utilized. for geophysical sàs is modeling includes propa flri wav'efields fo ,ard according to a l- seudo -cousti ,.
wavy. equation equation 6) and its egtdvaalc. ,: for tiled media.
It .sshoul :l also be #pl` reci lted that the, inZ ention Is iF?Ctn... ? t =
be used with a syst i?3 Which include", in goner 6l, an z:* ctronic configuration i-t luding at least one t?a aet~sc r. at j east one i e:moiy device for storing. program code or Other data, an optional Video à n for or other display device ;.c , a liquid crystal isplay :ind at least one Fi3tli it', The processor is preferably a microprocessor or =~~.icr<?controller-put based platform which is capable of displaying images and processing complex _1T e Ã3 4' à iathe n ;xt.}kcal algoritt`}1ÃF..s. : he memory vice can include random access (RAM) for storin4 event i? other data gene'Ãated or used during a particular.
Pro..'--v's associated with the r : 'FF :3 t #3t#CY11. The Ã12à z33o iC e can ,11s o in :l zd read only memory (ROM) for string the rog=rw code tear à 3e .~ ..trots and processes of the ilrt s.e at ixiventi n, One such embodiment includes a systefxi configured to perform pseudo acoustic quasi 13 t 'aSe propagation v hich Femain st.able in variable salt ?.isotropic media and is not limited to veak snisotropicc conditions, The `f s'.em includes a data torage de ite having computer readable data inc.ludin a exploration volume for a.
subsurface region of ..TFtere at, and a processor, t is}`i' tt C and arranged to execute machine executable instructions _re in a p #:ocessor accessible n-i mor'y for per'fOrini i a method. The method for this particular eTnbodi-rrient includes determining a modeling geometry f, t' e seismic exploration -"oh.ii'f'ie, and f ?tip a satin at least one b!' avei..3d , as?z.1 ', tia4 seismic exploF ation volume utilizing the modeling 3 nu ti" for initial. t tsl à itiou and preventing the accumulation o energy along the axis of symmetry of a õt>C? ?iC re-glom" Within the seismic exploration volume, and ensuring 1?,2s1 `.. coefficients in the stress-strain relaÃtio.ns thereby pr-o uc,1-ng a stable w avefi ld. The method 'forth u includes utilizing the stable wsa-v: a field to generate subsurface Images of the subsur1:a . F
ion of These and other obiects, features, and ci arac=teristit s of the Present intention, as 'ell as the methods of operation and id functions of the related aeelements of structure and the combination of parts and economies of .111 i1ÃFt3.tbctu e, will become more apparent upoii 3 consideration of the following desc ".ipdun ..:k ? the appended claims with T,, . ienc to the acco p nying di' 1''<3 ings, all of which firtm a. put of fl-us wherein file reference FFf:F orals desi'.F? it corresponding parts in the various, l i 11 s. it is,, to be =e ,pFessly understood, }?~?:4 , that the. dra vingi are for flac'purpose of illustration a et description only and am not intended as a definition of the lin _its of the invention.
As used in the specÃ?` caitio and inthe i ,ainns' the Singular .form of "a"' c and 'the" include: plural >e_ ,.ren s unk ss the :.Ã..kar y dictates, o erw sW' Brief P 4,s4 rxi t n the Drawi s I aA . o'w chart i li} .ai<A F: a .t :a tli in %iii c~atla u 'wiali one or i 3i`i'r embodiments of the present inveui.;<on.
Fig, 2 is a flo w chars illiistratmg as method in. accordance with one or more ernbodirnenu of the pr:Ã sent lrw ru o .
F:in. ? is,,, a flow chart Ãl t .,.r.~v#; a. r iethod In accordaaice: z ritl urn or more 1 S embodiments oft he present invention>
Fig, 4 illustrates ~a. =:i i l it > wave propagation modeling- <a cord nig to the, prior iin, Aixhaali :ah's approximation whc,,re,,/':::: 1, 15 1 i . 5 illustrates exempla 5' wave propagation .modt. i ng according to one embodiment of the present Fig. Ã> illustrates, exemplary wave propagation modeling according to once.
embodiment of he present invention w '':here F1 2tr7;7 :__: O' I .
`%0 `ig. "i ill istrat:`,s exemplary wave propagation modeling according to the prior a t, A khaaalit h's approximation where H& . l illustrates is exemplary phase v>>ekc=.it istribu io according to the prior art, 25 khauifa-i's appioximation.
Fig,. 9 illustrates an exernplaar group velocity distribu ioiF. to tht art, z l li. lit' ih s a pproximtiti23n.
30 Fig: 10 illustrates an xemphi:y phase velocity distribution tbbr one embodiment of he present invention ~Z}icere 4pi 0.01.
Fig. 11 illllstrat an exemplary gm up Velocity distribution for one embo a5men of the present invention where y k # `,} 001 Fig. 12 illustrates u l exemplary wave propaga don Ãnod l.:n4 in a Tedium vial: a 'Lai`Ãabk,Z ti led axis of sy'mi >et. y ac.cording to t} .le embodiment of the present invention u i i a 3 ; a à rsÃ-or er 5x P1)11 s stem .
Fig, 13 illustrates a ch ::i t tia diagram of the geometry that is used in one nibodimz Ãi. of the pi sent invention, Fig. 14 illustrates is a schematic illustration of all embodiment of a s st;.]lms for, p :rfonnmi g met cods in accordance with [mhodiinents of "he present invention, Detailed Description One i:=n nodÃm .:nt of the prosent invention is illustrated in Fig. 1, wherein a ffl(nk` Ãõ' art.
describes a m t_hod for propagating quasi-lx waves which remain à is in an-isotropic media with variab ~ tilt, ''be pr sent .n enÃ.ion is not limited to weak.
5. ani:sorrop.ii ià à ditio This particular embodiment includes acquiring aF
seismic exploration volume of a subsurface region of interest 1 2 < and determining a modeling geometry for the ..e isimÃG= exploration volume 14. The embodiment further includes Propagating at one wavefiell l t aririar. Ã thee seismic .. `ration volume utilizing the, k" odeling g.: for initial coud:i'^. n and orev -ting the accumulation of tt) energy along the a_ is of symmetry for t`L exploration vo i.ii ne and ensuring positive stiffness oe:t .14i nts in the relations utilizing -;mitre quasi-S
velocities hers by producing a stable. wa , :iel d 16> The stabl wai efin d c.a then be utilized to generate subsurface images of the subsurface region of interest ` c8 As one in ,killed in the at` will anpreciate.ditYenng ee.mbodÃments of the present invention may provide a pseeudo-actin 'ic= modeling method or as pseudo"-acoustÃc migration mothod for ~.inisotropic ruAi a., For example, Fig. ' illustrates a 'fi ch an for oiie embodiment of a pseud acouit i aodeling method for wave propagation in sni , 2 pls.:: Y edia with variablc tilt, wherein the method is w x limited to weak 20 an'isotropic condition?. That embodiment includes acquiring a seismic exploration volume for a subsui Pce re ion of interest 212 and t :t iminiril a modeling geometry for the I oluim. 24, The embodiment also includes propagating at least one wavefleka through the seismic exploration volume utilizing the.
modeling >g" -ometry for initial ccn :;::IS.. wherein the artificial quasi ';.:hear w air ~ velocity is ' f <#.requal to Zero adoÃ:~ the axis of sy mR' metry for the seismic exploration volume thereby pros tinting the accumulation s ^", energy long the axis of symmetry thereby producing as; ?1 e wavefie l ' Ã?. Thy t :.. le wa vctie.l~ can then be utilized to .gene ate subsurface images for the subsurface region of into. i St 218..
Fig. 3 ill ;~tia ; a flowchart 30 for another embodiment oftl the pre tint invention that can be used #`i i. l+ieudo aFeo stir mi rattion. That. embodiment i a.clude:s acquiring a seismic ex 1lo: Lion volume for a subsurface region of interest 32 and determining a model geometry for the seismic, exploration volume 34. The em bodim en.t: also 3ropagatarg at iOast Ã3 w Wave-field through the seismic. exploration volume izi tli rrtd}d lang geomerry for initial conditions, h min quasi-.hear, wwave energy does not accumulate along the xis of symmetry for the seist_n?i =
exploration volume thereb }' producing a stanbh~ wa'v efield ?6. The stable wa efie.ki can t ?en be utilized i~ <a= to t T f.~~.?~4ry~t0 st~Et. } ig s for the ` ~=Y '. utilized ~'ti.4~ :~'~C23Ca E?~ft..MS... of ?.Ii L.ro:fit .'i?
The present inven?tion provi;des several c3< t-<Ei?. e relative to eo Ev ul-ti >'ial acou itic, i blew v of s aEn>=at#ieF. ?? modeling and n igration.. The present inwntnon provides a t e wave propagation in Ti media with variabl0 tilt:, Onus simulated wav e.t.:el pw agatic?r and images of ' lle c6 dtyr can be obtained, Prior ar ps.eudo-acousdc modeling and migration m hods are based on Alkhahual?'s approximation in which the Phase velzocity~ of shear waves is sc to zero along the axis of symmetry. Although t 1w prior an methods can work in a constant-tilt Ti medium', the zero speed e ~ a es can make wave unstable (i.e. amplitudes become unbounded ) in area- where 1.5 tilt Variations can locally concentrate high energy die axis of sprinietry. Fig. 4 sh w that prior art meth ?<7s i t ' i. zi? ur.stait}le 4() in a '. "Iab1e- 1 +
.i3'dium ear the crest of an anti .final stru tur'e) Oi? the Contrary, Fig, 5 shows that wave Propagation bas e-%,! On the present invention (f remains stable 4-2 in the same medium, In andd :_r ? the present invention O an Provide the flex bilitys of controlling shear- td 1> ave \.,eioclty ratios to à itimi re the results of mod ling, and mi gnat on?, For cmi-11pi . sl#ear and P Wa ve VOOCIty rc .. Man be set dose to ti'le actual values to #i??It &' the kinematics in elastic w,.i".e prropagation. t 1nrther fore, in ceeraffi rc ks, the rtican.l vviocity ntrar be greater than the horizontal velocity wit n respect to axis of s 'nr metro'. In sucI-i a Oasea wave equations based, on Alkhalita ':s a>p.oxi ation will : _sn It in negative stiffness matrix thereby producing unstable wat'e.tiel.ds regardless nu ri `.:ircal implementation algorithms. The present invention can i. se a finite shear -wave Velocity I< positive stift'bess coetfici :nets in the stress strain relations thereby generating stable w avi.e, propagation..
3 In prior art metho bused on Alkhalitah's appmxinrm Lion, equating shear-wave phase ie#~.z.i.y` to lg t - t 3 Zen') not ~...ln#nfitile'~' ty shear a~ ,. 4~<..=..
' insten.d, 1 .y,h energy concentrates , dC ~ni not rve~. ~.
near the 'an, is of symmetry. The only exception is elliptic a-JI-Sotropy, (i e., c.. for ltr, : :i invention?. vertical shear' ., ea waf'v.. i lE 11 the is, re l"AXQ' Er am being m o,, hem e, Ã e energy is less concentrated near the axis of symmei.q. Shear wave", will not vanish ecause of he pr'esencce.. of additional cross th:.rivativ'es Q von ii the conditions of elliptical anisotropy are satisfied.
In terrn_zs of computational asst, the P1DEs utilized by eaibodi 'ieerns of the present t invention . tnvol e additional spatial d'rivative terms to be computed comp `red to prior an methods. In. are w,,, with variable tilt, the additional workload associated with 1."S.. is necessary to achie' e stability and reliability nequired by seismic to areas ;~i .z.:ad constant or very smooth tilt, the additional t= `orkload t ayr be skipped, It will be clear to one skilled in the a.:t. that à to bove embodiments may be altered in miry .`t without departing from the scope of the inver tion. For example, as is t ons o `
apparent to one skilled in the art, cif t rent initial ~?t:tÃ:li i~?i or boundary Condi a different linear combination. of the PI)Es it the present invention can b)e used in modeling and migration as convenienKt.;, in. one.:nibod n-mnt of the present invention5 the anisotmo is modeling method includes establishing a velocity x.w' anisotropy model corrLspondin to ii seismic exploration volume setting initial ~.Lndit ons such a source excitations prop gati:i`ig waves in transversely isotropic media 1 1"; a tiled or vertical axis of syuunetry, to e: , H] or A'S =quiv'alent. For tbrward modeling, a source :t.:mnciion of the accor~h.n g 'I or ? E c t t ' . a` t) needs to be introduced in à e right side of equations in, s [1' 4' l 1' v here t- is the source I ~, ion and iw'(t ) IS a source wavelet, In tt above-described embodiment of the present invention, the vertical sh ar-#~'~r ave velocity in eq. [3 can be non-zero ~therei r ,f `can be different from 1.) in contrast to the prior art m t'lod approximation where ; `rounds off to. 1. Acc<o dingly, Ã
e phase velocity of shear waves in the direction of both parallel and perpendicular to t!1 e. axis of symmetr can be non zero in the present in.Vention. in a rm..dÃÃam with variable tilt, th finite speed of quasi-shear wavos can avoid local of high energy is which o#'[,ia occurs in the vicinity of the axis 3.t s`ii metry. Il.:e:
present invention does not require weak anisotropy t Ãion .
Utilizing the above PDEs, other embodiments of t be present invention can be derived fb anisotr'op :. media. If tilt O 0, the above PPDEs s mpl.fy to a 31.) VTI
system, and similarly as 3D I T11 system when ~v'f) _0. A second--order R33, system tar ID
VTl media can n 1x of the f mmi in equ it.ion 6, This sy+st Ãn of PDiis is extendable to iv s equivalent 1:6-MI ulataor: for a tilted TI r ediu~ . by replacing gi and ,,, given in eq. $:
b =:i i and,tx givi~m in eq. f1 j.
As an i ltea`a?ati ;ve to the III- I~s in eq= [11 or [21 or `l.] when t-A, the first-order 5x s ti iri of PI)l- s in e q., 151 is by erbolic w id stable in a TI medium with variable tilt.
This embodiment of the present i >enÃioÃ^ is +raiai tii : is . hyperbolic, (Well posed, even vi' th variable t? alt ci :nt ".this system is al w extendable to v;
.able-til; VF 1.
Tli i . complete ...,-Order 5:x5 system of in 31) reduces 1o 4x4 in 21).
As described above, additional embodiments of the present invention also provide i? e.udo~< C?.i 1. t' migration methods. One embodiment includes the steps o establishing a sty ismic data set and a velocity anisot opy .model corresponding to a seismic `.xploraÃion vole e; sei g boundary conditions of wave, propagation;;
?r 3p'3w ating w i' from source '''?.:.. at on and recordw, seismic data se par tciiy in wiisoi.ropic media ac-cording to .q, [1 e [2]. e [.4], or eq. [61, or their .
ak ms;
and applying i na ing conditions such as, 'but not limited to, cross co3reliation between the. two prop #g 3{.'d waav etieli s to obtain sia1. surhice images, Diffi.r`ent initial widjor boundary, conditions can be applied without . aflect.ing the scope of this invention. An.
t= ro ) Z 2 g Gg sA
e,~x, t;`: kr), boundary .ondit.ii on s +'z; I- asi;.:.l on -q. [1)) for ra.
wati.ra4, ir source w;i.l. i:t is, s follows:
I P! `Y ; t) (i., W( ) tF
x") woOdf ?
and the boundary condition for reverse tin fi l i l; , .,itl ra of seismic data is as tollowws:
f ` Y , It) D?( x A yY. ~=,, v ; t) ?~i+' !} p (x, ,z 0; -fY-r t 4".. ''r\ .rt.t \ v1\ w4'h4" \ is a source tur ctiojn, is the location of sourceis a slot record to i ni ;rate.
The following example lliustr..3t'es a fue'l-wr cI,; bo'd i tllent of th present #n'S? .ntion ., Establishing fourth-order dlsp rs n relations for quaii-P wave in VTl media tiank.ie ' phase veloi:ity relytiom for VTl m dia or which is not set to zero le ,d 1 3 ti,11 ~ t .As< ion relation:
[(+ 2 ) v , .: ai'+,` J#";< (I. 4 iv ,+z:' g .` as( .~:+y V i, f `i2q , a ,v A +i i + NiS f~ }i F
r f cr' 3 .i3 t '. fe ue tcy, k, is the vertical G~avenumber, and .+i: r+S; f is the [square I S of I hC p of the i 3 I T F#BF: ? t<i ' .iii e her vector k) Eq, [121 admits two pair of solutions -------------- --------t1 4= t r 4( - ---- I l 4] rff -{rB 4C
60, = = s - ------fzf corre`, p w i d to quasivP waves-, e correspond to quua ii- SV waves.
20 2. Establishing .a fourth-order P DE for quasi-P wave in VTl marl`
Applying eq 11211 to war .. L (, krj , >) the Fourier domlaai and %mkin the inverse Fourier ti'il`?. Srrtl #1 fX,I Th provide:
l d t .' (t33> 1 F
{ F J 3 nm; it 4r t2 ,; o C$i f ffx#xe t tft ,.^ f ={;.?t EEEE :,ci f: Lry f '' EEE
eta z l 3 ' i -- --- -------------CZ 'a ax, az 3r Establishing ascot . -order @yst . of DEa for VII media Let her . z t } 3 is a wave f ice:?: eq. [15 1, Assuming. the a hi al conditions, leads tot \ }+'jp tX t 1.0 Le., ?
Cry t c tft __: `r ) P } r a .', V, z , i<)#fn f. ft s ;.S' p 91 t w .t j 1~ t! fC: T
Eq. [15' is then eq al.valen to tsar s-cored-order 3>3) s stem of PDEs by eq, E4L, 1 shows a \Z,a e 2ont propagation in a Vi,i it.tedi n u ing the above POE
ho~t~. that 1S paa.,cui';ar embodiment: of the present invention Compared to the %xavc Fronts ased on the prior art methods (illustrated in Fag 7), the o- r q wave root (-41-11-in Fig. 6 and 48 in Fig, 7) remains almost idantical, but the (461n, Fig , 6 and 50 in Fig. 7) has a different .form from a dia-mcs.ad shape. Fi . $ and Fig, 9 show t ho phrase end group velocities, rspectiv ly>, according to Alkhah to "s (prior , 20 4a zrox ;: ;.iu . In con rest, Fig.. 10 aà d Fig. 11 show the Phase and ;group scat i ataa:..
res ec=tively according to in. embodiment- of the pies, n t .3nv ea,taon, Compared to Al. hali ah xa ?rza .s:a t at~c?:ah, the phase v& locit`ies of qS V waves are relaxed from being ,Y<:l? awn ; the ,uxas of 4 r'`13Fmetry, 3miS i `F FI fl'; the ? ii31 iu i`s v iL .F group 4'z. tt..ix > or high energy are not so wised along the a.? is of sy-n-,mi tr'y as in the prior art methods . The same <k ~T~'ations are applicable to a consta`-E.th TJ
medium by applying a rotation aboul the tilt, Establishing a fl or (:r 5x5 system of PDF"A for VIA media D .F e i al, (2008) presents the following ?s cort or 3er 2x2 system of II E) S for VT
r, yx p (1 + 4? )1rN{fJfO'.r.P ,+ A?gjq her L t ; and 4't are g i v e n I CF' eq, à ? , pis e3 velar" we vefeld, and q à F all a .diary uncÃion. A new P is defined and iE X10 auxiliary ` fi ctions 4.='. wd.R
by:
Pi, t j 2 } Q. 1.21.]
Then equation 5 . sF a complete, first-arder 5x5 system of P DEs. 'This system can be 15 shown to he hyperbolic by symmetrizifagià La ' 3 _.F \. U
rvq: +
g_ R. I J: y} 3 2q # r {r {t } r TI, [23.1 R I sC R
wherC
r t# l', rr I 2?/ 0 0 0 ---------------z t$`
20 # >x 0 0 i t 0 {c F
----------- - -0 0 0 "J l I q 0 0 t 0 0 0 0 0 t}' ._-___---tI # Ã (Ã
it $) (1 + 2q g +~p~
In 21), the variable V is <l r .: ~<3 d and t.1 e third equation is d : etcd, resel-Ong in a fir; t-order 4x4 system.
Fig. I2 sho stable wavefront. propagatioa governed by such first-order PDEs in a vari abk-tilt à tedium, ., ;staI fishing fourth-order dispersion r lafions T TI m itt By relaxing Alkhalifah's approximation ft at (of _:. i) along the axis 'Y try, the l owing equation can In derived from Tvank 's phase v 1ocity relations 00I
seismic exploration volume of a subsurface region of interest 1 2 < and determining a modeling geometry for the ..e isimÃG= exploration volume 14. The embodiment further includes Propagating at one wavefiell l t aririar. Ã thee seismic .. `ration volume utilizing the, k" odeling g.: for initial coud:i'^. n and orev -ting the accumulation of tt) energy along the a_ is of symmetry for t`L exploration vo i.ii ne and ensuring positive stiffness oe:t .14i nts in the relations utilizing -;mitre quasi-S
velocities hers by producing a stable. wa , :iel d 16> The stabl wai efin d c.a then be utilized to generate subsurface images of the subsurface region of interest ` c8 As one in ,killed in the at` will anpreciate.ditYenng ee.mbodÃments of the present invention may provide a pseeudo-actin 'ic= modeling method or as pseudo"-acoustÃc migration mothod for ~.inisotropic ruAi a., For example, Fig. ' illustrates a 'fi ch an for oiie embodiment of a pseud acouit i aodeling method for wave propagation in sni , 2 pls.:: Y edia with variablc tilt, wherein the method is w x limited to weak 20 an'isotropic condition?. That embodiment includes acquiring a seismic exploration volume for a subsui Pce re ion of interest 212 and t :t iminiril a modeling geometry for the I oluim. 24, The embodiment also includes propagating at least one wavefleka through the seismic exploration volume utilizing the.
modeling >g" -ometry for initial ccn :;::IS.. wherein the artificial quasi ';.:hear w air ~ velocity is ' f <#.requal to Zero adoÃ:~ the axis of sy mR' metry for the seismic exploration volume thereby pros tinting the accumulation s ^", energy long the axis of symmetry thereby producing as; ?1 e wavefie l ' Ã?. Thy t :.. le wa vctie.l~ can then be utilized to .gene ate subsurface images for the subsurface region of into. i St 218..
Fig. 3 ill ;~tia ; a flowchart 30 for another embodiment oftl the pre tint invention that can be used #`i i. l+ieudo aFeo stir mi rattion. That. embodiment i a.clude:s acquiring a seismic ex 1lo: Lion volume for a subsurface region of interest 32 and determining a model geometry for the seismic, exploration volume 34. The em bodim en.t: also 3ropagatarg at iOast Ã3 w Wave-field through the seismic. exploration volume izi tli rrtd}d lang geomerry for initial conditions, h min quasi-.hear, wwave energy does not accumulate along the xis of symmetry for the seist_n?i =
exploration volume thereb }' producing a stanbh~ wa'v efield ?6. The stable wa efie.ki can t ?en be utilized i~ <a= to t T f.~~.?~4ry~t0 st~Et. } ig s for the ` ~=Y '. utilized ~'ti.4~ :~'~C23Ca E?~ft..MS... of ?.Ii L.ro:fit .'i?
The present inven?tion provi;des several c3< t-<Ei?. e relative to eo Ev ul-ti >'ial acou itic, i blew v of s aEn>=at#ieF. ?? modeling and n igration.. The present inwntnon provides a t e wave propagation in Ti media with variabl0 tilt:, Onus simulated wav e.t.:el pw agatic?r and images of ' lle c6 dtyr can be obtained, Prior ar ps.eudo-acousdc modeling and migration m hods are based on Alkhahual?'s approximation in which the Phase velzocity~ of shear waves is sc to zero along the axis of symmetry. Although t 1w prior an methods can work in a constant-tilt Ti medium', the zero speed e ~ a es can make wave unstable (i.e. amplitudes become unbounded ) in area- where 1.5 tilt Variations can locally concentrate high energy die axis of sprinietry. Fig. 4 sh w that prior art meth ?<7s i t ' i. zi? ur.stait}le 4() in a '. "Iab1e- 1 +
.i3'dium ear the crest of an anti .final stru tur'e) Oi? the Contrary, Fig, 5 shows that wave Propagation bas e-%,! On the present invention (f remains stable 4-2 in the same medium, In andd :_r ? the present invention O an Provide the flex bilitys of controlling shear- td 1> ave \.,eioclty ratios to à itimi re the results of mod ling, and mi gnat on?, For cmi-11pi . sl#ear and P Wa ve VOOCIty rc .. Man be set dose to ti'le actual values to #i??It &' the kinematics in elastic w,.i".e prropagation. t 1nrther fore, in ceeraffi rc ks, the rtican.l vviocity ntrar be greater than the horizontal velocity wit n respect to axis of s 'nr metro'. In sucI-i a Oasea wave equations based, on Alkhalita ':s a>p.oxi ation will : _sn It in negative stiffness matrix thereby producing unstable wat'e.tiel.ds regardless nu ri `.:ircal implementation algorithms. The present invention can i. se a finite shear -wave Velocity I< positive stift'bess coetfici :nets in the stress strain relations thereby generating stable w avi.e, propagation..
3 In prior art metho bused on Alkhalitah's appmxinrm Lion, equating shear-wave phase ie#~.z.i.y` to lg t - t 3 Zen') not ~...ln#nfitile'~' ty shear a~ ,. 4~<..=..
' insten.d, 1 .y,h energy concentrates , dC ~ni not rve~. ~.
near the 'an, is of symmetry. The only exception is elliptic a-JI-Sotropy, (i e., c.. for ltr, : :i invention?. vertical shear' ., ea waf'v.. i lE 11 the is, re l"AXQ' Er am being m o,, hem e, Ã e energy is less concentrated near the axis of symmei.q. Shear wave", will not vanish ecause of he pr'esencce.. of additional cross th:.rivativ'es Q von ii the conditions of elliptical anisotropy are satisfied.
In terrn_zs of computational asst, the P1DEs utilized by eaibodi 'ieerns of the present t invention . tnvol e additional spatial d'rivative terms to be computed comp `red to prior an methods. In. are w,,, with variable tilt, the additional workload associated with 1."S.. is necessary to achie' e stability and reliability nequired by seismic to areas ;~i .z.:ad constant or very smooth tilt, the additional t= `orkload t ayr be skipped, It will be clear to one skilled in the a.:t. that à to bove embodiments may be altered in miry .`t without departing from the scope of the inver tion. For example, as is t ons o `
apparent to one skilled in the art, cif t rent initial ~?t:tÃ:li i~?i or boundary Condi a different linear combination. of the PI)Es it the present invention can b)e used in modeling and migration as convenienKt.;, in. one.:nibod n-mnt of the present invention5 the anisotmo is modeling method includes establishing a velocity x.w' anisotropy model corrLspondin to ii seismic exploration volume setting initial ~.Lndit ons such a source excitations prop gati:i`ig waves in transversely isotropic media 1 1"; a tiled or vertical axis of syuunetry, to e: , H] or A'S =quiv'alent. For tbrward modeling, a source :t.:mnciion of the accor~h.n g 'I or ? E c t t ' . a` t) needs to be introduced in à e right side of equations in, s [1' 4' l 1' v here t- is the source I ~, ion and iw'(t ) IS a source wavelet, In tt above-described embodiment of the present invention, the vertical sh ar-#~'~r ave velocity in eq. [3 can be non-zero ~therei r ,f `can be different from 1.) in contrast to the prior art m t'lod approximation where ; `rounds off to. 1. Acc<o dingly, Ã
e phase velocity of shear waves in the direction of both parallel and perpendicular to t!1 e. axis of symmetr can be non zero in the present in.Vention. in a rm..dÃÃam with variable tilt, th finite speed of quasi-shear wavos can avoid local of high energy is which o#'[,ia occurs in the vicinity of the axis 3.t s`ii metry. Il.:e:
present invention does not require weak anisotropy t Ãion .
Utilizing the above PDEs, other embodiments of t be present invention can be derived fb anisotr'op :. media. If tilt O 0, the above PPDEs s mpl.fy to a 31.) VTI
system, and similarly as 3D I T11 system when ~v'f) _0. A second--order R33, system tar ID
VTl media can n 1x of the f mmi in equ it.ion 6, This sy+st Ãn of PDiis is extendable to iv s equivalent 1:6-MI ulataor: for a tilted TI r ediu~ . by replacing gi and ,,, given in eq. $:
b =:i i and,tx givi~m in eq. f1 j.
As an i ltea`a?ati ;ve to the III- I~s in eq= [11 or [21 or `l.] when t-A, the first-order 5x s ti iri of PI)l- s in e q., 151 is by erbolic w id stable in a TI medium with variable tilt.
This embodiment of the present i >enÃioÃ^ is +raiai tii : is . hyperbolic, (Well posed, even vi' th variable t? alt ci :nt ".this system is al w extendable to v;
.able-til; VF 1.
Tli i . complete ...,-Order 5:x5 system of in 31) reduces 1o 4x4 in 21).
As described above, additional embodiments of the present invention also provide i? e.udo~< C?.i 1. t' migration methods. One embodiment includes the steps o establishing a sty ismic data set and a velocity anisot opy .model corresponding to a seismic `.xploraÃion vole e; sei g boundary conditions of wave, propagation;;
?r 3p'3w ating w i' from source '''?.:.. at on and recordw, seismic data se par tciiy in wiisoi.ropic media ac-cording to .q, [1 e [2]. e [.4], or eq. [61, or their .
ak ms;
and applying i na ing conditions such as, 'but not limited to, cross co3reliation between the. two prop #g 3{.'d waav etieli s to obtain sia1. surhice images, Diffi.r`ent initial widjor boundary, conditions can be applied without . aflect.ing the scope of this invention. An.
t= ro ) Z 2 g Gg sA
e,~x, t;`: kr), boundary .ondit.ii on s +'z; I- asi;.:.l on -q. [1)) for ra.
wati.ra4, ir source w;i.l. i:t is, s follows:
I P! `Y ; t) (i., W( ) tF
x") woOdf ?
and the boundary condition for reverse tin fi l i l; , .,itl ra of seismic data is as tollowws:
f ` Y , It) D?( x A yY. ~=,, v ; t) ?~i+' !} p (x, ,z 0; -fY-r t 4".. ''r\ .rt.t \ v1\ w4'h4" \ is a source tur ctiojn, is the location of sourceis a slot record to i ni ;rate.
The following example lliustr..3t'es a fue'l-wr cI,; bo'd i tllent of th present #n'S? .ntion ., Establishing fourth-order dlsp rs n relations for quaii-P wave in VTl media tiank.ie ' phase veloi:ity relytiom for VTl m dia or which is not set to zero le ,d 1 3 ti,11 ~ t .As< ion relation:
[(+ 2 ) v , .: ai'+,` J#";< (I. 4 iv ,+z:' g .` as( .~:+y V i, f `i2q , a ,v A +i i + NiS f~ }i F
r f cr' 3 .i3 t '. fe ue tcy, k, is the vertical G~avenumber, and .+i: r+S; f is the [square I S of I hC p of the i 3 I T F#BF: ? t<i ' .iii e her vector k) Eq, [121 admits two pair of solutions -------------- --------t1 4= t r 4( - ---- I l 4] rff -{rB 4C
60, = = s - ------fzf corre`, p w i d to quasivP waves-, e correspond to quua ii- SV waves.
20 2. Establishing .a fourth-order P DE for quasi-P wave in VTl marl`
Applying eq 11211 to war .. L (, krj , >) the Fourier domlaai and %mkin the inverse Fourier ti'il`?. Srrtl #1 fX,I Th provide:
l d t .' (t33> 1 F
{ F J 3 nm; it 4r t2 ,; o C$i f ffx#xe t tft ,.^ f ={;.?t EEEE :,ci f: Lry f '' EEE
eta z l 3 ' i -- --- -------------CZ 'a ax, az 3r Establishing ascot . -order @yst . of DEa for VII media Let her . z t } 3 is a wave f ice:?: eq. [15 1, Assuming. the a hi al conditions, leads tot \ }+'jp tX t 1.0 Le., ?
Cry t c tft __: `r ) P } r a .', V, z , i<)#fn f. ft s ;.S' p 91 t w .t j 1~ t! fC: T
Eq. [15' is then eq al.valen to tsar s-cored-order 3>3) s stem of PDEs by eq, E4L, 1 shows a \Z,a e 2ont propagation in a Vi,i it.tedi n u ing the above POE
ho~t~. that 1S paa.,cui';ar embodiment: of the present invention Compared to the %xavc Fronts ased on the prior art methods (illustrated in Fag 7), the o- r q wave root (-41-11-in Fig. 6 and 48 in Fig, 7) remains almost idantical, but the (461n, Fig , 6 and 50 in Fig. 7) has a different .form from a dia-mcs.ad shape. Fi . $ and Fig, 9 show t ho phrase end group velocities, rspectiv ly>, according to Alkhah to "s (prior , 20 4a zrox ;: ;.iu . In con rest, Fig.. 10 aà d Fig. 11 show the Phase and ;group scat i ataa:..
res ec=tively according to in. embodiment- of the pies, n t .3nv ea,taon, Compared to Al. hali ah xa ?rza .s:a t at~c?:ah, the phase v& locit`ies of qS V waves are relaxed from being ,Y<:l? awn ; the ,uxas of 4 r'`13Fmetry, 3miS i `F FI fl'; the ? ii31 iu i`s v iL .F group 4'z. tt..ix > or high energy are not so wised along the a.? is of sy-n-,mi tr'y as in the prior art methods . The same <k ~T~'ations are applicable to a consta`-E.th TJ
medium by applying a rotation aboul the tilt, Establishing a fl or (:r 5x5 system of PDF"A for VIA media D .F e i al, (2008) presents the following ?s cort or 3er 2x2 system of II E) S for VT
r, yx p (1 + 4? )1rN{fJfO'.r.P ,+ A?gjq her L t ; and 4't are g i v e n I CF' eq, à ? , pis e3 velar" we vefeld, and q à F all a .diary uncÃion. A new P is defined and iE X10 auxiliary ` fi ctions 4.='. wd.R
by:
Pi, t j 2 } Q. 1.21.]
Then equation 5 . sF a complete, first-arder 5x5 system of P DEs. 'This system can be 15 shown to he hyperbolic by symmetrizifagià La ' 3 _.F \. U
rvq: +
g_ R. I J: y} 3 2q # r {r {t } r TI, [23.1 R I sC R
wherC
r t# l', rr I 2?/ 0 0 0 ---------------z t$`
20 # >x 0 0 i t 0 {c F
----------- - -0 0 0 "J l I q 0 0 t 0 0 0 0 0 t}' ._-___---tI # Ã (Ã
it $) (1 + 2q g +~p~
In 21), the variable V is <l r .: ~<3 d and t.1 e third equation is d : etcd, resel-Ong in a fir; t-order 4x4 system.
Fig. I2 sho stable wavefront. propagatioa governed by such first-order PDEs in a vari abk-tilt à tedium, ., ;staI fishing fourth-order dispersion r lafions T TI m itt By relaxing Alkhalifah's approximation ft at (of _:. i) along the axis 'Y try, the l owing equation can In derived from Tvank 's phase v 1ocity relations 00I
6' i#3 +,-- rl,12 C
----------------------xt 'who're phase velocity v has roots of two rna"nitudes: one for gwis. P navm and the ) }Cl TJv2{}we, .,n the 22tt=`?~ ~:>Tow no m al and ti ifs. ..'.!i~irli S o other C f'~Tr quay 5e }ter ~av-t-R(-0 is the angle f v' -lmc tr-' a a d other parameters are de.i.izne \f in 4q. [ i'.3.
A :<cordi.iig to he geometry shown in Fig, 0, the wavefront nom al f r) and the axis of symmetry `_i and the angle in between tak the tffl[owsing form.
r?i sill. f'r cos f~ + SUI O siri w, `i^ COS
sin ( co 's. i,,I + sin imi {2 rr' + cos- 0,,k ;~ilua fr{--- _ sin 0 Cos ~i3~in fir;o"?iF1`FF3h-ii t~ Si Ãi${~; co- sfi:O.y "=here 0;' Is, the tilt oftli axis ~t ' liil i t r` with respect to d- w ' eriical in a Ti mediwn, nd e isthe. ~V'imlii:i 3~ hl . axis of sy mi'.1 try, Recognizing that s> iil 0 Q'os- k, '' {I.`F
sin 6stn k 1 6 c OS rr U i (l.F
the i't}Ilowing f >urth-order dispersion relatis ns can be derived:
{{ PO
6. Est bis k g a fourth-order FIDE hi 1'T media Multiplying both sides of à e abo 4 fourth-order dispersion re at.t n with a ses: t:rw'a efieitd P. iiiet3 coiwei`tinc the taequency-wavi:,numbei o'per'ators to . e time-space domain the fourth-order I 'DE fo i'l, ' l 1 i ledic takes of the tbnn of e q, E8,
----------------------xt 'who're phase velocity v has roots of two rna"nitudes: one for gwis. P navm and the ) }Cl TJv2{}we, .,n the 22tt=`?~ ~:>Tow no m al and ti ifs. ..'.!i~irli S o other C f'~Tr quay 5e }ter ~av-t-R(-0 is the angle f v' -lmc tr-' a a d other parameters are de.i.izne \f in 4q. [ i'.3.
A :<cordi.iig to he geometry shown in Fig, 0, the wavefront nom al f r) and the axis of symmetry `_i and the angle in between tak the tffl[owsing form.
r?i sill. f'r cos f~ + SUI O siri w, `i^ COS
sin ( co 's. i,,I + sin imi {2 rr' + cos- 0,,k ;~ilua fr{--- _ sin 0 Cos ~i3~in fir;o"?iF1`FF3h-ii t~ Si Ãi${~; co- sfi:O.y "=here 0;' Is, the tilt oftli axis ~t ' liil i t r` with respect to d- w ' eriical in a Ti mediwn, nd e isthe. ~V'imlii:i 3~ hl . axis of sy mi'.1 try, Recognizing that s> iil 0 Q'os- k, '' {I.`F
sin 6stn k 1 6 c OS rr U i (l.F
the i't}Ilowing f >urth-order dispersion relatis ns can be derived:
{{ PO
6. Est bis k g a fourth-order FIDE hi 1'T media Multiplying both sides of à e abo 4 fourth-order dispersion re at.t n with a ses: t:rw'a efieitd P. iiiet3 coiwei`tinc the taequency-wavi:,numbei o'per'ators to . e time-space domain the fourth-order I 'DE fo i'l, ' l 1 i ledic takes of the tbnn of e q, E8,
7. Establishing a second-order 2%2 system of PDEs forTTI Media The above fourth-ordeer pse dl.o-acoustic- PD[: or'lii media can be solved by the 'N2 6.m e- and spac -doma n ME syste by eq= {3.1 Where 1 +-t t2:
+, the 2..c2 system of can also take an equivalent form in t .rimms of horizontal velocity v,:. and. nor al-m i ~eotit (NMlO) velocity _;
~Ã~ aF ..,D I3 iei.mri as ~ special the iava7tFvc I'DE'S still remain valid wÃ-fl-i t~~~~`:
following. simplified spatà derivatÃve o .r kk )F-S, {
r F ~? 3 .{ }- Scab ;`i Kira `r , :stahli, h i s econd Nord aMia st n of PD Es for TTI media As an ai(ernafve, the (du r:. i -or :.i pseud s-zacoustic PDE for l ri a e i<a. can also be 4 i h.f by a >: time- and spa i oSF #WÃIà PDEs in j or its equivalentt.using of the present invention can he implemented on either co processor ,:i t:.4ziF fà Y'CiaitEE=tt^ <:C, Suih, as .Fi if1- rogrf~f zaa nbie ( t ÃÃ%L' t` ~~z c , Grapbi :sõPro= essÃn; -U:Units (GPU s Cei`. or eneral-gar ose cot puters. The present invention provides apparatus and general-purpose computers and/or co-proce'ssor.
programmed with instructions to pertE nn a mete od for the presenat invention, as W01, a com ute -re Ãadahk media. encoding :nstructions to perform aI method of the present invention, A system for pedonuinÃg an embodiment of the present invent .on is schematically 111ustraled in Fig, 14. A system `< includ .s a data storage device or memory 54, The stored data may be rnad to a processor 5$. such as >i r: +
~laZa~^ interface t'?~~,r~ata`ir:'iti~ili: #;~,rai:.zai ~3f:r?tb~4f=~~%3F~?fa~:.r. The ~
i'f?coSa'.a?t` 56 may include t.
comp iients as la display ' 'and a graphical user interface 60, The graphical user interface ((-'sUI) may be used both to display data and processed data products a d to allow the user to ,anions options for implementing aspects of dee method, Data r nay be transferred fo the system 52 via a bus $2 either directly from a data acquisition device. fir from an intermediate storage or processing facility (not..s""u'fvii:).
i-\ }lough the. invention has been. described for the purpose of illustution biassed on what is currently considered to be the most practical and prof tued embodimenis, it is to be understood that,uch detail is solely for t. fat purpose an . that he invention is not limited to the. disclosed but, on the contra a is '0 intended: to cover F '#odi`Fcsation and equivalent ua ,gem nzs that zi wit in the spirit and 3 `C of the appended claims> For example, though reference is i'. de ?
rei to a computer, i s may include a genera` purpose compute', a Ã. r os -huÃ.t co puter, a AS IC , ~ 3 m d to execute the methods, a y or zz or other xap ropyÃat computing device. As a firth r A is to he understood. that the present invention contemplates, that, to à e extent possible, one or ml-,ore features of any embodiment can he combined with one or more finA ,es of any other em bedl Trent .
+, the 2..c2 system of can also take an equivalent form in t .rimms of horizontal velocity v,:. and. nor al-m i ~eotit (NMlO) velocity _;
~Ã~ aF ..,D I3 iei.mri as ~ special the iava7tFvc I'DE'S still remain valid wÃ-fl-i t~~~~`:
following. simplified spatà derivatÃve o .r kk )F-S, {
r F ~? 3 .{ }- Scab ;`i Kira `r , :stahli, h i s econd Nord aMia st n of PD Es for TTI media As an ai(ernafve, the (du r:. i -or :.i pseud s-zacoustic PDE for l ri a e i<a. can also be 4 i h.f by a >: time- and spa i oSF #WÃIà PDEs in j or its equivalentt.using of the present invention can he implemented on either co processor ,:i t:.4ziF fà Y'CiaitEE=tt^ <:C, Suih, as .Fi if1- rogrf~f zaa nbie ( t ÃÃ%L' t` ~~z c , Grapbi :sõPro= essÃn; -U:Units (GPU s Cei`. or eneral-gar ose cot puters. The present invention provides apparatus and general-purpose computers and/or co-proce'ssor.
programmed with instructions to pertE nn a mete od for the presenat invention, as W01, a com ute -re Ãadahk media. encoding :nstructions to perform aI method of the present invention, A system for pedonuinÃg an embodiment of the present invent .on is schematically 111ustraled in Fig, 14. A system `< includ .s a data storage device or memory 54, The stored data may be rnad to a processor 5$. such as >i r: +
~laZa~^ interface t'?~~,r~ata`ir:'iti~ili: #;~,rai:.zai ~3f:r?tb~4f=~~%3F~?fa~:.r. The ~
i'f?coSa'.a?t` 56 may include t.
comp iients as la display ' 'and a graphical user interface 60, The graphical user interface ((-'sUI) may be used both to display data and processed data products a d to allow the user to ,anions options for implementing aspects of dee method, Data r nay be transferred fo the system 52 via a bus $2 either directly from a data acquisition device. fir from an intermediate storage or processing facility (not..s""u'fvii:).
i-\ }lough the. invention has been. described for the purpose of illustution biassed on what is currently considered to be the most practical and prof tued embodimenis, it is to be understood that,uch detail is solely for t. fat purpose an . that he invention is not limited to the. disclosed but, on the contra a is '0 intended: to cover F '#odi`Fcsation and equivalent ua ,gem nzs that zi wit in the spirit and 3 `C of the appended claims> For example, though reference is i'. de ?
rei to a computer, i s may include a genera` purpose compute', a Ã. r os -huÃ.t co puter, a AS IC , ~ 3 m d to execute the methods, a y or zz or other xap ropyÃat computing device. As a firth r A is to he understood. that the present invention contemplates, that, to à e extent possible, one or ml-,ore features of any embodiment can he combined with one or more finA ,es of any other em bedl Trent .
Claims (15)
1. A computer-implemented method for pseudo acoustic quasi - P wave propagation which remain stable in variable tilt anisotropic media and is not limited to weak anisotropic conditions, the method includes;
acquiring a seismic exploration volume for a subsurface region of interest;
determining a modeling geometry for the seismic exploration volume;
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 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.
acquiring a seismic exploration volume for a subsurface region of interest;
determining a modeling geometry for the seismic exploration volume;
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 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.
2. The method of claim 1 wherein propagating at least one wavefield through the seismic exploration volume includes an artificial quasi-shear wave velocity being greater or equal to zero along the axis of symmetry of anisotropic regions within the seismic exploration volume.
3. The method of claim 1 wherein propagating at least one wavefield through the seismic exploration volume includes restricting accumulation of quasi-shear waves along the axis of symmetry for the seismic exploration volume.
4. The method of claim 1 wherein a plurality of wavefields are propagated through the seismic exploration volume.
5. The method of claim 1 wherein propagating at least one wavefield through the seismic exploration volume includes utilizing one of reverse time migration, a wave-equation based migration, Gaussian beam migration or Kirchhoff migration.
6. The method of claim 1 which includes propagating wavefields forwards and backwards through the seismic exploration volume and applying imaging conditions to the forward and backward wavefields or equivalent Green's functions to derive the subsurface images.
7. The method of claim 6 wherein the step of applying imaging conditions includes cross correlation between the forward and backward wavefields or equivalent Green's functions to derive the subsurface images.
8. The method of claim 7 wherein the step of applying imaging conditions includes at least one of illumination normalization, reflection-angle domain gather generation and phase-amplitude compensation.
9. The method of claim 1 wherein the imaging output geometry includes common offset, common azimuth and common reflection-angle domains.
10. The method of claim 1 wherein propagating at least one wavefield includes utilizing at least one of delayed shot, plane-wave and phase encoding.
11. The method of claim 1 wherein propagating at least one wavefield includes utilizing at least one of normal moveout velocity, horizontal velocity and Thomsen parameters.
12. 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 comprising:
data storage device having computer readable data including a seismic volume for a subsurface region of interest, a processor, configured and arranged to execute machine executable instructions stored in a processor accessible memory for performing a method comprising:
determining a modeling geometry for the seismic exploration volume;
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 utilizing finite quasi-S wave velocities thereby producing a stable wavefield;
and utilizing the stable wave-field to generate subsurface images of the subsurface region of interest.
data storage device having computer readable data including a seismic volume for a subsurface region of interest, a processor, configured and arranged to execute machine executable instructions stored in a processor accessible memory for performing a method comprising:
determining a modeling geometry for the seismic exploration volume;
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 utilizing finite quasi-S wave velocities thereby producing a stable wavefield;
and utilizing the stable wave-field to generate subsurface images of the subsurface region of interest.
13. The system of claim 12 wherein propagating at least one wavefield through the seismic exploration volume includes an artificial quasi-shear wave velocity being greater or equal to zero along the axis of symmetry of anisotropic regions within the seismic exploration volume.
14. The system of claim 12 wherein propagating at least one wavefield through the seismic exploration volume includes restricting accumulation of qua-shear waves along the axis of symmetry for the seismic exploration volume.
15. The system of claim 1 a plurality of wavefields are propagated through the seismic exploration volume.
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| US8467208P | 2008-07-30 | 2008-07-30 | |
| US61/084,672 | 2008-07-30 | ||
| 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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| AU2011252076B2 (en) | 2010-05-12 | 2014-04-10 | Shell Internationale Research Maatschappij B.V. | Seismic P-wave modelling in an inhomogeneous transversely isotropic medium with a tilted symmetry axis |
| 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 |
| CA2881807C (en) | 2012-08-17 | 2017-10-17 | Landmark Graphics Corporation | Systems and 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 |
| CA2944375C (en) | 2014-03-31 | 2023-01-24 | John Adam Donald | 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 |
| US6819111B2 (en) * | 2002-11-22 | 2004-11-16 | Baker Hughes Incorporated | Method of determining vertical and horizontal resistivity, and relative dip in anisotropic earth formations having an arbitrary electro-magnetic antenna combination and orientation with additional rotation and position measurements |
| JP4105139B2 (en) * | 2004-09-30 | 2008-06-25 | 石油資源開発株式会社 | Three-dimensional image generation method and program |
| KR100660562B1 (en) * | 2005-09-30 | 2006-12-22 | (주)지 텍크 | Ultra-shallow 3d super resolution s-wave seismic survey device and method |
| US9519072B2 (en) * | 2006-05-11 | 2016-12-13 | Schlumberger Technology Corporation | Method and apparatus for locating gas hydrate |
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| CA2730017C (en) | 2017-03-14 |
| AU2009276976A1 (en) | 2010-02-04 |
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| EP2308025A4 (en) | 2016-11-16 |
| WO2010014379A3 (en) | 2010-03-25 |
| EA201170261A1 (en) | 2011-08-30 |
| EP2308025A2 (en) | 2011-04-13 |
| CN102105900B (en) | 2013-03-20 |
| BRPI0916637A2 (en) | 2019-03-26 |
| AU2009276976B2 (en) | 2015-07-16 |
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