Gephyscal Prspectng 010 58 599 617 d: 10.1111/j.1365-478.009.00857.x Mveut apprxmatn fr hrzntal transsely strpc and tcal transsely strpc layered medum. Part II: ectve mdel Zv Kren Igr Ravve and Rnt Levy Paradgm Gephyscal Research & Develpment Gav-Yam Center N. 9 Shenkar St. PO Bx 061 Herzlya B 4610 Israel Receved May 009 revsn accepted Nvember 009 ABSTRACT We use resdual mveuts measured alng cntnuus full azmuth reflectn angle gathers n rder t btan ectve hrzntal transsely strpc mdel parameters. The angle gathers are generated thrugh a specal angle dman magng system fr a wde range f reflectn angles and full range f phase velcty azmuths. The estmatn f the ectve mdel parameters s perfrmed n tw stages. Frst the backgrund hrzntal transsely strpc HTI/tcal transsely strpc VTI layered mdel s used alng wth the values f reflectn angles fr cntng the measured resdual mveuts r traveltme errrs nt azmuthally dependent nrmal mveut NMO velctes. Then we apply a dgtal Furer transfrm t cnt the NMO velctes nt azmuthal wavenumber dman n rder t btan the ectve HTI mdel parameters: tcal tme tcal cmpressn velcty Thmsen parameter delta and the azmuth f the medum s f symmetry. The methd als prvdes a relablty crtern f the HTI assumptn. The crtern shws whether the medum pssesses the HTI type f symmetry r whether the azmuthal dependence f the resdual traveltme ndcates t a mre cmplex azmuthal anstrpy. The ectve mdel used n ths apprach s defned fr a 1D structure wth a set f HTI VTI and strpc layers wth at least ne HTI layer. We descrbe and analyse the reductn f a mult-layer structure nt an equvalent ectve HTI mdel. The equvalent mdel yelds the same NMO velcty and the same ffset azmuth n the Earth s surface as the rgnal layered structure fr any azmuth f the phase velcty. The ectve mdel apprxmates the knematcs f an HTI/VTI layered structure usng nly a few parameters. Under the hyperblc apprxmatn the prpsed ectve mdel s exact. Key wrds: Anstrpy Mdellng Insn Parameter estmatn Velcty analyss. INTRODUCTION Acqured gephyscal data ften ndcate a clear azmuthal dependency f the prpagatn velcty. The smplest mdel Ths paper s based n extended abstracts P016 and P018 presented at the 71 st EAGE Cnference & Exhbtn Incrpratng SPE EUROPEC 009 8 11 June 009 n Amsterdam the Netherlands. E-mal: zv.kren@pdgm.cm that descrbes a medum wth azmuthally dependent prpertes s the hrzntal transse strpy. It can be descrbed by the tcal cmpressn velcty the azmuthal rentatn f the s f symmetry and Thmsen 1986 parameters. T establsh the parameters f the ectve hrzntal transsely strpc mdel we use resdual mveuts measured alng cntnuus full azmuth reflectn angle gathers. The angle gathers are generated thrugh a specal angle dman C 010 Eurpean Asscatn f Gescentsts & Engneers 599
600 Z. Kren I. Ravve and R. Levy magng system Kren et al. 007 008 fr a wde range f reflectn angles θ phs and full range f phase velcty azmuths ϕ phs. The estmatn f the ectve mdel parameters s perfrmed n tw stages. Frst the backgrund hrzntal transsely strpc HTI/tcal transsely strpc VTI layered mdel s used alng wth the values f reflectn angles fr cntng the measured resdual mveuts r traveltme errrs nt azmuthally dependent resdual nrmal mveut NMO velctes tθ phs ϕ phs V nm ϕ phs. 1 Then we update the NMO velcty by addng the resdual t the backgrund value V u nm ϕ phs = V b nm ϕ phs + V nm ϕ phs where superscrpt u stands fr updated and superscrpt b fr backgrund. The result s a number f seres f NMO velcty sus phase velcty azmuth. The seres dffer by ther reflectn angle θ phs whch s fxed fr each seres. In the secnd stage we apply a Furer transfrm t cnt each seres f the NMO velctes nt azmuthal wavenumber dman k ϕ V nm ϕ phs Ṽ nm k ϕ. 3 Actually we transfrm the updated NMO velctes squared. The NMO velcty n the Furer dman s further used t btan the ectve HTI mdel parameters: the tcal tme t the tcal cmpressn velcty V Thmsen parameter δ and the azmuth f the medum s f symmetry ϕ. The methd als prvdes a relablty crtern C f the HTI assumptn. Ths crtern shws hw clse the azmuthal dependence f the traveltme s t that f an HTI medum. The ectve mdel used n ths apprach s defned fr a 1D structure wth a set f HTI VTI and strpc layers wth gven prpertes and thckness and we assume that the package ncludes at least ne HTI layer. We shw that fr studyng the cmpressn waves wth near-tcal drectn f prpagatn ths layered structure s equvalent t a unque hmgeneus HTI layer whch can be cnsdered as the ectve mdel fr the rgnal layered structure. The equvalency means that the functns fr NMO velcty sus the azmuth f the phase velcty V nm ϕ phs are the same fr bth the ectve mdel and the rgnal mult-layer mdel. In addtn the functns fr the ffset azmuth n the Earth s surface sus the azmuth f the phase velcty ϕ ff ϕ phs are als the same fr the tw mdels under the accuracy f the hyperblc apprxmatn. The ectve mdel s vald fr small ffsets.e. near-tcal rays and therefre t ncludes fur parameters nly: the thckness f the ectve layer t the ectve tcal velcty V the ectve azmuth f symmetry ϕ and the ectve Thmsen parameter δ. The ect f parameter ε s gnred fr small-ffset rays. The derved ectve mdel apprxmates the knematcs f an HTI/VTI layered structure usng nly a few parameters. Under the hyperblc apprxmatn the prpsed ectve mdel s exact. Ths paper s structured as fllws. Frst we cnt the resdual mveut r traveltme errr t the NMO velcty and then we shw the methd t estmate the ectve mdel parameters frm the NMO velcty functn sus phase velcty azmuth. Next we defne the ectve mdel fr a cmplex structure that ncludes HTI VTI and strpc flat layers and prvde the relatnshps between the parameters f ths mdel and the prpertes f ndvdual layers. We prvde a real data numercal example where ectve mdels are establshed fr dfferent seres f reflectn angle. The detaled dervatns have been mved t appendces. In Appendx A we explan the transfrmatn f resdual traveltme nt resdual mveut velcty and update the backgrund NMO. In Appendx B we cnsder an mprtant partcular case when the backgrund mdel s strpc. Appendx C s devted t Furer analyss that enables the estmatn f the ectve mdel frm the NMO velcty functn. In Appendx D we further refne the parameters f the ectve mdel t best ft the measured NMO velcty. In Appendx E we explan the advantages f the ectve mdel and derve ts parameters gven the prpertes f the cmpnent layers. Fnally n Appendx F we derve the relatnshp f the NMO velcty sus the phase velcty azmuth startng frm a smlar relatnshp sus the ray grup velcty azmuth. NORMAL MOVEOUT VELOCITY FROM FULL AZIMUTH RESIDUAL MOVEOUTS The nrmal mveut NMO velcty sus the azmuth f the phase velcty V nm ϕ phs can be calculated frm resdual mveuts r traveltme errrs tθ phs ϕ phs specfed sus the reflectn angle and the azmuth f the phase velcty. We assume that the backgrund ectve mdel hrzntal transsely strpc HTI r strpc s gven and thus the backgrund NMO velcty V b nm can be cmputed fr each value f the phase velcty azmuth ϕ phs V b nm ϕ phs V b = 1 + 4δb 1 + δ b cs ϕ b ϕ phs 1 + δ b cs ϕ b ϕ. 4 phs The dervatn f ths frmula s based n the wrk by Tsvankn 1997 where a smlar relatnshp fr the NMO C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 601 velcty sus the azmuth f the ray grup velcty was btaned; see Appendx F fr detals. The updated NMO velcty V u nm needed fr the Furer analyss ncludes tw cunterparts: the backgrund value and the resdual; bth tems depend upn the phase velcty azmuth. The updated NMO velcty can be calculated by V u nm ϕ phs V b nm ϕ phs = 1 ξϕ phs tan θ phs tθ phsϕ phs t 5 see Appendx A fr detals. The range f the phase velcty azmuth s 0 ϕ phs <π. Functn ξϕ phs depends n the backgrund mdel parameter δ b and the backgrund azmuth f symmetry ϕ b ξ 1 ϕ phs = 1 + δ b cs ϕ b ϕ. 6 phs Nte that δ b s nrmally assumed negatve and thus ξϕ phs 1 where ξ = 1 crrespnds t the strpc backgrund mdel. In ther wrds the HTI anstrpy amplfes the nfluence f resdual traveltme n the resdual NMO velcty as cmpared t the strpc mdel. Next we cnsder a partcular practcal case when the backgrund mdel s strpc. In ths case the backgrund NMO velcty V b nm s cnstant.e. ndependent n the azmuth f the phase velcty. Assume that the updated mdel dffers nly slghtly frm the backgrund mdel.e. the anstrpy s weak. We ntrduce tw dmensnless small parameters the relatve majr and mnr NMO velctes α majr = Vmajr nm V b nm V b nm α mnr = Vmnr nm Vnm b. 7 V b nm Wth the assumptns n the strpc backgrund mdel and weak anstrpy n the updated mdel Thmsen parameter delta becmes δ α mnr α majr. 8 The detals are presented n Appendx B. The resdual traveltme depends n the majr and mnr NMO velctes tθ phs ϕ phs [ = tan θ phs αmajr sn ϕ ϕ phs t + α mnr cs ϕ ϕ phs ]. 9 A smlar apprach was used by Grechka and Tsvankn 1998 where the mveut vares sus the ray velcty azmuth. Nte that equatns 8 and 9 and Appendx B are the nly parts n ths study where the anstrpy s assumed weak; the degree f anstrpy s arbtrary elsewhere. EFFECTIVE MODEL PARAMETERS BY FOURIER ANALYSIS In the case when the nrmal mveut NMO velcty sus phase velcty azmuth V nm ϕ phs s a perdc functn wth perd π the medum may be a canddate fr a hrzntal transsely strpc HTI mdel. In ths sectn we descrbe the algrthm that makes t pssble t estmate the parameters f ectve mdel applyng the Furer analyss t the NMO velcty squared. In addtn the methd yelds a cnsstency crtern. Nte that nt all π-perdc NMO velcty functns crrespnd t an HTI mdel exactly. There may be dfferent azmuthally dependent NMO that crrespnd t mre cmplex types f azmuthal anstrpy. The cnsstency crtern ndcates hw clsely the nput data V nm ϕ phs matches the HTI mdel. In ths sectn we present the wrkflw f the Furer analyss and the detals f dervatn are gven n Appendx C. Assume that the NMO velcty s sampled n the unfrm grd n the range 0 ϕ phs <π. Assume als that the number f samples n s the Fast Furer Transfrm FFT number; therwse we resample the NMO values. The sampled values are V nmm = V nm ϕ phsm ϕ phsm = πm n m = {0 1...n 1}. 10 We apply a dgtal real-t-cmplex Furer transfrm t the NMO velcty squared and nrmalze the results by factr 1/n fr DC and /n fr AC F = 1 n 1 n V nm ϕ phsm ϕ phsm = πm n 11 m=0 F k = n 1 n πkm V nm ϕ phsm exp k= 1...N 1 n m=0 1 where N s the amunt f cmplex numbers n the Furer space N = n + 1 fr even n N = n + 1 fr dd n. 13 The ectve azmuth f the s f symmetry reads ϕ = arg F 1 π <ϕ π. 14 The ectve tcal velcty squared s deled by V N 1 = F + 1 k+1 F k. 15 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
60 Z. Kren I. Ravve and R. Levy Thmsen parameter δ δ V = N 1 35 can be btaned by F k. 16 T btan the cnsstency crtern we cmpute tw auxlary values ϕ m = arg F ± πm m = {0 1}. 17 4 We use ether plus r mnus fr m = 1 n equatn 17 t get the result wthn the range specfed n equatn 14. The cnsstency crtern C s the csne f the angular dfference between the azmuth f the symmetry s and ne f tw auxlary azmuths btaned n equatn 17. The auxlary azmuth clser t the s f symmetry shuld be taken C = m [ cs ϕ ϕ 1 cs ϕ ϕ ]. 18 As ne can see frm equatn 17 the tw angles ϕ 1 and ϕ dffer by π/ and therefre at least ne f the tw csnes n equatn 17 shuld be pstve s the range f the crtern s 0 < C 1. Fr the exact data that crrespnd t an HTI mdel r t a package f hrzntal transsely strpc tcal transsely strpc and strpc layers the cnsstency crtern C = 1. Otherwse ths crtern shws hw nsy the data are r hw dfferent the medum wth the gven azmuthal dependence f the NMO velcty s frm an HTI mdel. The methd descrbed abve s based n the Furer analyss and t yelds the NMO velcty n tw azmuthal drectns: alng the azmuth f the ectve symmetry s and alng the strpc azmuth nrmal t the s f symmetry. Alternatvely we can use the results lsted frm ths analyss as an ntal guess fr the true ectve parameters mnmzng the dscrepancy ntegral A = 1 [ ϕphs =π Vnm V δ ϕ ϕ phs V data ϕ phs =0 Vnm dataϕ phs nm ϕ phs ] dϕ phs A mn 19 where V nm V δ ϕ ϕ phs s the calculated value f the mveut velcty whle V data nm ϕ phs are the measured data. Althugh the calculated NMO velcty s always π-perdc the measured value may have perd π andnthscasethe upper lmt f the ntegral becmes π. As we mentned fr a true hrzntal transsely strpc/tcal transsely strpc layered structure the ntal guess leads t an exact slutn. Otherwse the ntal guess s nrmally a gd apprxmatn and the nn-lnear mnmzatn prblem can be slved see Appendx D fr detals. In many cases hwe the accuracy f the ntal guess des suffce and the teratve prcedure can be avded. MEDIUM WITH CRACKS AND LAYERING A medum wth bth cracks and layerng s rthrhmbc prvded the cracks and the layers are mutually rthgnal. In a mre general case wth an arbtrary angle between the nrmal t the fracture plane and the nrmal t the layerng plane the medum may have even lwer symmetry. A typcal example wuld be a sngle set f dppng cracks n a tcal transsely strpc backgrund that leads t a mnclnc anstrpy wth a sngle plane f symmetry shwn schematcally n Fg. 1. In the scheme α s the layerng plane and β s the fault plane and they are nt necessarly rthgnal. Lne A s an ntersectn f the layerng plane and the fault Fgure 1 Scheme f mnclnc anstrpy caused by dppng cracks n VTI backgrund. C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 603 plane. The mnclnc plane f symmetry γ s nrmal t lne A and therefre t s nrmal t bth the layerng plane α and the fault plane β. In ths wrk hwe we d nt study the anstrpc meda wth symmetry lwer than transse strpy. We assume a 1D layered structure such that n each layer.e. lcally there s ether nly a tcal transsely strpc VTI shale r nly a hrzntal transse strpy HTI caused by tcal cracks. We study the traveltme and the ffset cmpnents fr near-tcal rays shrt ffsets and we shw that fr these purpses the layered structure may be replaced by a sngle equvalent HTI layer wth ectve tcal cmpressn velcty ectve Thmsen parameter delta ectve azmuth f symmetry and thckness tcal tme. The equvalence s exact fr an arbtrary anstrpy: Thmsen parameters f the layers are nt necessarly small. Hwe the prpsed ectve mdel s vald fr shrt ffsets nly. The relatnshp between the ffset cmpnents and the traveltme makes t pssble t establsh the magntude and the azmuth f the NMO velcty f the ectve mdel. PARAMETERS OF EFFECTIVE MODEL VERSUS PROPERTIES OF LAYERS Gven a layered structure cnsstng f hrzntal transsely strpc HTI tcal transsely strpc VTI and strpc layers wth at least ne HTI layer ne can estmate the parameters f the equvalent ectve layer. Frst we assume that the tcal tme s preserved s that t = t. 0 Next we calculate three auxlary parameters: W x W y and U. Nte that nly HTI layers cntrbute t W x and W y and all layers cntrbute t parameter U W x = δ cs ϕ t V HTI W y = U = δ sn ϕ t V 1 VTI 1 + δ t V + ISO 1 + δ t V + t V. Assumng that the Thmsen parameter δ s negatve the ectve parameters are as fllws. The azmuthal rentatn Table 1 Parameters f ectve HTI mdel sus reflectn angle θ phs V mnr nm m/s V majr nm m/s δ ϕ C 10 10 30 0.040 41 0.996 15 135 35 0.036 37 0.996 0 151 37 0.031 33 0.998 5 169 39 0.09 7 0.999 30 180 140 0.05 3 0.999 Fr each data seres wth a fxed reflectn angle we btan a set f ectve parameters f the symmetry s s deled by cs ϕ = W x W sn ϕ = W y W 3 where W Wx + W y. 4 Nte that parameters U and W are bth pstve. The ectve tcal velcty s V = W + U. 5 t Fnally the ectve Thmsen parameter s δ = W W + U. 6 The full dervatn s gven n Appendx E. We cnsdered the data f Package 1 descrbed n Table 1 f Part I and we btaned the fllwng results: t = 1.76 s V =.948 km/s δ = 0.1403 ϕ = 0.9 rad = 5.84. FIELD EXAMPLE Full azmuth reflectn angle gathers were generated usng the magng methd descrbed n Kren et al. 008 fr an cean-bttm sesmmeter OBS data set. Fgure t the tp shws an example f such an angle gather where the data traces are rganzed n fve angle sectrs. Each sectr shares the same reflectn angle fxed half penng angle: 10 15 0 5 and 30 degrees respectvely and cntans 90 traces f phase velcty azmuths rangng frm 0 180 degrees. Cnsstent azmuthal mveut varatns can be clearly dentfed at a depth level f 3800 m. The event under nvestgatn s shwn n Fg. t the bttm wth the autmatcally pcked resdual mveuts lad. Gelgcally ths area can be classfed as a fractured carbnate layer. Assumng a hrzntal transsely strpc layered mdel we apply ur methd t analyse the ectve parameters frm each reflectn angle C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
604 Z. Kren I. Ravve and R. Levy Fgure T the tp: reflectn angle gather 5 angle sectrs. T the bttm: zm f the event under nvestgatn. sectr ndependently. The resultng parameters are shwn n Table 1 where V majr nm = V Vmnr nm = V 1 + δ 0.5 <δ < 0. 7 Althugh the btaned parameters at each sectr are slghtly dfferent the parameters can stll be classfed gruped t prvde valuable trend values especally fr the azmuth f the s f symmetry the rentatn f the fracture system. CONCLUSIONS We descrbed a new methd t cmpute ectve hrzntal transsely strpc mdel parameters frm full azmuth resdual mveuts. The methd explts the nput resdual mveuts btaned drectly as a functn f the reflectn angle and the azmuth f the phase velcty. Fr any package cnsstng f hrzntal transsely strpc HTI tcal transsely strpc VTI and strpc layers wth at least ne HTI layer an equvalent ectve HTI layer exsts that yelds the same magntude and drectn n the nrmal mveut velcty sus the phase velcty drectn. The ectve parameters are the thckness r tcal tme tcal velcty the azmuth f symmetry s and Thmsen parameter δ. We develped a cnsstency crtern that ndcates hw clsely the sampled data matches the HTI type f symmetry. Fr a true HTI/VTI 1D layered structure the slutn fr ectve parameters s exact under the hyperblc apprxmatn. ACKNOWLEDGEMENTS The authrs are grateful t Paradgm Gephyscal fr ther fnancal and techncal supprt f ths study and fr ther permssn t publsh ts results. Grattude s extended t the revewers frm Gephyscal Prspectng fr cnstructve remarks and suggestns that helped t mprve ths paper. REFERENCES Alkhalfah T. and Tsvankn I. 1995. Velcty analyss fr transsely strpc meda. Gephyscs 60 1550 1566. Grechka V. and Tsvankn I. 1998. 3-D descrptn f nrmal mveut n anstrpc nhmgeneus meda. Gephyscs 63 1079 109. Kren Z. Ravve I. Bartana A. and Kslff D. 007. Lcal angle dman n sesmc magng. 69 th EAGE meetng Lndn UK Expanded Abstracts. C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 605 Kren Z. Ravve I. Ragza E. Bartana A. and Kslff D. 008. Full-azmuth angle dman magng. 78 th SEG meetng Las Vegas Nevada USA Expanded Abstracts. Thmsen L. 1986. Weak elastc anstrpy. Gephyscs 51 1954 1966. Tsvankn I. 1997. Reflectn mveut and parameter estmatn fr hrzntal transse strpy. Gephyscs 6 614 69. Tsvankn I. and Thmsen L. 1995. Insn f reflectn traveltmes fr transse strpy. Gephyscs 60 1095 1107. APPENDIX A. RESIDUAL NORMAL MOVEOUT VELOCITY VERSUS RESIDUAL TRAVELTIME The algrthm that dels the ectve hrzntal transse strpy HTI parameters requres the nrmal mveut NMO velcty functn sampled sus azmuth f the phase velcty V nm ϕ phs. Hwe the avalable datum s the azmuthally dependent resdual traveltme tϕ phs. The updated NMO velcty may be cnsdered cnsstng f tw parts: that f the backgrund mdel and the resdual NMO V u nm ϕ phs = V b nm ϕ phs + V nm ϕ phs. A1 The backgrund NMO velcty can be btaned frm the backgrund HTI medum parameters Vnm b ϕ phs = 1 + 4 δb 1 + δ b cs ϕ b ϕ phs 1 + δ b cs ϕ b ϕ. A phs V b Equatn A fr the NMO velcty sus azmuth f the phase velcty was prved n Part I based n the analyss f the nfntesmal dfference π/ α phs where the phase angle α phs s the angle between the phase velcty and the HTI medum s f symmetry. An alternatve dervatn based n Tsvankn s 1997 equatn fr the NMO velcty sus azmuth f the ray velcty s gven n Appendx F. The backgrund medum parameters: the tcal velcty V b the rthrhmbc Thmsen parameter δb and the azmuth f the symmetry s ϕ b are assumed knwn n equatn A. In partcular ne can start frm the strpc backgrund mdel wth vanshng δ b n ths case the s azmuth ϕ b des nt matter and equatn A smplfes t V b nm ϕ phs = V b = cnst. A3 Therefre t btan the updated NMO velcty V u nm ϕ phs we need t estmate the resdual NMO V nm ϕ phs. In ths appendx we explan the methd that makes t pssble t establsh the azmuthally dependent resdual NMO gven the azmuthally dependent resdual traveltme tϕ phs V nm ϕ phs. We assume here that a fxed seres f resdual mveuts s beng prcessed.e. fr all azmuths ϕ phs the resdual traveltmes t crrespnd t the same fxed reference reflectn angle θ phs. Ths angle whch s als the zenth angle f the phase velcty s expected t be a small fnte knwn value. We start frm the defntn f the NMO velcty. Recall that the NMO velcty yelds the hyperblc apprxmatn fr the traveltme t = t + h Vnm A4 Equatn A4 apprxmates the resdual traveltme as t t = h V nm. A5 Vnm b 3 Fr the hyperblc apprxmatn the ffsets are assumed small s that t t and equatn A5 reduces t t = h t t V nm. A6 Vb nm V b nm The rat between the ffset and the layer thckness reads h z = h = tan θ t V b ray. A7 Accrdng t equatn 6 f Part I the zenth angle f the phase velcty θ phs s gven by cs α phs sn θ phs = csϕ ϕ phs. A8 Neglectng the hgh-rder nn-hyperblc term n equatn E7 f Part I and takng nt accunt that the devatn α phs f the phase angle frm the rght angle may be presented as α phs sn α phs = snπ/ α phs = cs α phs A9 we btan the hyperblc apprxmatn fr the zenth angle f the ray velcty sn θ ray = C h cs α phs csϕ ϕ phs. A10 Ths yelds the relatnshp between the zenth angles f the phase and ray velctes sn θ ray / sn θ phs = C h where fr the backgrund mdel 1 + δ b C h = 1 + 4 δb cs ϕ b ϕ phs A11. A1 Fr small zenth angles the rat f ther snes may be replaced by the rat f ther tangents tan θ ray sn θ ray = C h. A13 tan θ phs sn θ phs C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
606 Z. Kren I. Ravve and R. Levy Intrducng equatn A13 nt A7 gves h t = C h V b tan θ phs. Intrductn f equatn A14 nt A6 results n t t A14 = C h Vb V nm tan θ Vnm b Vnm b phs. A15 It fllws frm equatn A that V b V = 1 + δb cs ϕ b ϕ phs. A16 nm b C h Intrductn f equatn A16 nt A15 leads t t ] = [1 + δ b t cs ϕ b ϕ phs tan θ phs V nm. Vnm b A17 Intng equatn A17 we btan APPENDIX B. RESIDUAL TRAVELTIME VERSUS MAJOR AND MINOR NORMAL MOVEOUT VELOCITY Assume an strpc backgrund medum wth the cnstant nrmal mveut NMO velcty V b nm. Suppse that tw parameters α majr and α mnr are specfed related t the relatve resdual NMO velcty f the updated medum wth respect t the backgrund medum α majr = Vmajr nm V b nm V b nm α mnr = Vmnr nm Vnm b B1 V b nm where the majr and the mnr NMO depend n the tcal velcty V f the updated medum and the rthrhmbc Thmsen parameter δ V majr nm = V V mnr nm = V 1 + δ. B V nm V b nm 1 = 1 + δ b cs ϕ b ϕ phs t/t tan θ phs A18 Recall that Thmsen parameter δ s nrmally negatve. Equatn B1 leads t prvded the reference reflectn angle des nt vansh θ phs 0.e. the ray s near-tcal but nt strctly tcal. As we see the methd des nt requre keepng the reflectn angle cnstant and gemetry wth the data pnts f dfferent zenths and azmuths may wrk prvded the zenth values are small. Fr an strpc backgrund mdel equatn A18 smplfes t V nm V b nm = t/t. A19 tan θ phs Wth the ntatn 1 ξϕ phs 1 + δ b cs ϕ b ϕ A0 phs the resdual NMO velcty becmes V nm V b nm = ξϕ phs t tan θ t phs A1 and the updated NMO velcty fllws frm equatn A1 Vnm u Vnm b = 1 ξϕ phs t tan θ t phs where fr the strpc backgrund mdel ξ = 1. A V majr nm = V b nm 1 + αmajr V mnr nm = V b nm 1 + α mnr. B3 The tcal velcty f the updated medum becmes V = V b nm 1 + αmajr. B4 The Thmsen parameter becmes 1 + δ = Vmnr nm = 1 + α mnr B5 Vnm majr 1 + α majr and ths leads t δ = 1 + α mnr 1 + α majr 1 + α majr = α majr α mnr + α mnr + α majr 1 + α majr. B6 The NMO velcty sus phase velcty azmuth s gven by V nm ϕ phs V = 1 + 4 δ 1 + δ cs ϕ ϕ phs. B7 1 + δ cs ϕ ϕ phs Intrducng equatns B4 and B6 nt B7 gves V nm ϕ phs V b nm = 1 + α majr 4 sn ϕ ϕ phs + 1 + α mnr 4 cs ϕ ϕ phs 1 + α majr sn ϕ ϕ phs + 1 + α mnr cs ϕ ϕ phs. B8 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 607 The relatve resdual NMO s defned as V nm ϕphs V b nm = V nm ϕphs V b nm. B9 Vnm b Intrductn f equatn B8 nt B9 leads t V nm ϕphs 4 = 1 + αmajr sn ϕ ϕ phs + 1 + αmnr 4 cs ϕ ϕ phs 1 + αmajr sn ϕ ϕ phs + 1 + αmnr cs 1. B10 ϕ ϕ phs V b nm Frm equatn A16 fr the strpc backgrund mdel ξ = 1 the relatve resdual traveltme s t θ phs ϕ phs V = tan nm ϕphs θ phs. B15 t V b nm S far equatn B10 s vald fr any degree f anstrpy. Nw assume that the relatve resduals f the NMO velcty are small α majr 1 α mnr 1. B11 In ths case equatn B6 smplfes t δ α mnr α majr B1.e. the anstrpy s weak. Equatn B10 may be lnearzed fr small values f α mnr and α majr V nm ϕ phs V b nm α majr sn ϕ ϕ phs + α mnr cs ϕ ϕ phs. B13 Actually the accuracy f the apprxmatn depends nly n cndtn B1 and nt B11. We present three plts that cmpare exact and lnearzed slutns fr the nrmalzed resdual NMO velcty n the phase plane equatns B10 and B13 respectvely. In Fg. 3 the graph s pltted fr majr and mnr resdual NMO values f dfferent sgns: α majr = 0.08 α mnr = 0.06. Blue lnes crrespnd t the exact functn and red lnes t the lnearzed slutn. Thck lnes shw the pstve resdual velcty and thn lnes the negatve ne. In Fg. 4 the relatve resdual NMO velcty s pltted fr pstve and essentally dstnct majr and mnr values: α majr = 0.1 α mnr = 0.04.e. the anstrpy s large δ 0.08. As we see the resdual NMO cntur s cncave. In Fg. 5 the relatve resdual NMO velcty s pltted fr pstve and clse majr and mnr values: α majr = 0.4 α mnr = 0.18. The anstrpy s weaker than n the prevus case δ 0.06 and the cntur s cnvex. It fllws frm equatn B13 that the cntur n the phase plane s cnvex when the majr and mnr resdual NMO velctes are f the same sgn and when they d nt dffer t much α majr /α mnr < 3/. B14 We ntrduce equatn B13 nt B15 and btan the relatve resdual traveltme n the phase plane tθ phs ϕ phs [ = tan θ phs αmajr sn ϕ ϕ phs t + α mnr cs ϕ ϕ phs ]. B16 APPENDIX C. HORIZONTAL TRANSVERSELY ISOTROPIC EFFECTIVE PARAMETERS BY FOURIER ANALYSIS Assume that the nrmal mveut NMO velcty V nm ϕ phs s sampled n a unfrm azmuth grd. Our bjectve s t establsh the parameters f the ectve mdel. Fr a sngle hrzntal transsely strpc HTI layer e.g. fr the ectve mdel the NMO velcty sus the azmuth f the phase velcty reads V nm V = 1 + 4 δ 1 + δ cs ϕ ϕ phs 1 + δ cs ϕ ϕ phs Ths relatnshp may als be rewrtten as V nm ϕ phs V f ϕ phs. = 1 + δ 1 + δ cs ϕ ϕ phs 1 + δ cs ϕ ϕ phs C1 = f ϕ phs. C Ths s a perdc functn f the phase velcty and snce the csnes appear squared n the equatn then the perd s π and nt π. Assume that the data are sampled n the nterval [0 π] startng frm an arbtrary reference zer azmuth. T study the behavur f functn n the rght-hand sde f equatn C we expand f nt the Furer seres V nm ϕ phs V = A z + A k cskϕ phs + B k snkϕ phs where the ccents f the expansn are deled by A z = 1 π π 0 f ϕ phs dϕ phs C3 C4 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
608 Z. Kren I. Ravve and R. Levy Nrmalzed Resdual Mveut 0.06 0.04 Prjectn f Phase Velcty 0.0 0-0.0-0.04-0.06-0.08-0.06-0.04-0.0 0 0.0 0.04 0.06 0.08 Nrmal Prjectn Fgure 3 Relatve resdual NMO velcty n the phase plane wth majr and mnr values f dfferent sgns: α majr = 0.08α mnr = 0.06. Blue lnes exact functn red lnes lnearzatn; thck lnes pstve resdual velcty thn lnes negatve. 0.06 Nrmalzed Resdual Mveut 0.04 Prjectn f Phase Velcty 0.0 0-0.0-0.04-0.06-0.1-0.09-0.06-0.03 0 0.03 0.06 0.09 0.1 Nrmal Prjectn Fgure 4 Relatve resdual NMO velcty n the phase plane wth pstve majr and mnr values: α majr = 0.1α mnr = 0.04. Blue lnes exact functn red lnes lnearzatn. Cncave cntur α majr /α mnr > 3/. and Perfrmng the ntegratn we btan A k = π π 0 f ϕ phs cskϕ phs dϕ phs A z = 1 + δ 1 + δ 1 + = 1 1 + δ + 1 + δ C6 1 + δ and π B k = π 0 f ϕ phs snkϕ phs dϕ phs. C5 A k = M k cskϕ B k = M k snkϕ C7 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 609 Nrmalzed Resdual Mveut 0.18 0.1 Prjectn f Phase Velcty 0.06 0-0.06-0.1-0.18-0.4-0.18-0.1-0.06 0 0.06 0.1 0.18 0.4 Nrmal Prjectn Fgure 5 Relatve resdual NMO velcty n the phase plane wth pstve majr and mnr values: α majr = 0.4α mnr = 0.18. Blue lnes exact functn red lnes lnearzatn. Cnvex cntur α majr /α mnr < 3/. where M k = 1k+1 k+1 δ k 1 + δ k 1 + 1 + δ k = 1... C8 Alternatvely ccents A k and B k can be presented as real and magnary parts respectvely f a cmplex number A k = Re ˆM k B k = Im ˆM k C9 where ˆM k = M k expkϕ. C10 We nte that ccents ˆM k make an nfnte gemetrc seres wth decreasng abslute values ˆM k+1 ˆM k = δ expϕ 1 + 1 + δ. C11 Snce δ s nrmally negatve the rat M k+1 /M k s nrmally pstve. In Fg. 6 we plt the nrmalzed NMO velcty sus ray velcty azmuth blue lne and sus phase velcty azmuth red lne fr a sngle HTI layer r fr the equvalent layer f the package fr Thmsen parameter δ = 0.5. The nrmalzng factr s the tcal velcty V. The graph s pltted fr the fxed azmuthal rentatn f the HTI s f symmetry ϕ = 0. The green lne s the apprxmatn t the NMO velcty sus phase velcty azmuth wth the frst Furer harmnc. It yelds an essental errr. The grey lne s the apprxmatn wth tw Furer harmncs. It s already y clse t the exact value. The apprxmatn wth three harmncs thn black lne s s accurate that t can nt be dstngushed frm the exact graph. Nw assume that the Furer ccents are gven r can be calculated and the parameters f the ectve mdel are t be fund. Agan we need t make a chce whether we are lkng fr a pstve r a negatve δ. Assume δ s negatve. In ths case t fllws frm equatn C8 that all M k dd and even are negatve M k = A k + B k. C1 The azmuth f the s f symmetry can be btaned frm the frst par A 1 and B 1 ϕ = arctan B 1 A 1. C13 Nte that parameter M k n equatn C7 s negatve and ths leads t mnus sgns n equatn C13. Ths slutn s unque wthn the assumptn abut negatve δ. The secnd par gves tw slutns 4ϕ = arctan B A + πm m = {0 1} C14 and ne f these slutns the prper slutn shuld cncde wth the slutn C13. An arbtrary par A l B l gves l slutns lϕ = arctan B l A l + πm m = {0 1...l 1}. C15 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
610 Z. Kren I. Ravve and R. Levy Nrmalzed NMO Velcty vs. Ray and Phase Azmuth 0.8 Prjectn n Axs f Symmetry 0.4 0-0.4 Vs. Ray Azmuth Vs. Phs Azmuth Vs. Phs Furer 1 Vs. Phs Furer Vs. Phs Furer 3-0.8-1. -0.8-0.4 0 0.4 0.8 1. Nrmal Prjectn Fgure 6 NMO velcty sus azmuth f ray and phase velcty. Only ne f these slutns has a physcal sense and fr the nn-nsy data btaned frm the true hrzntal transsely strpc HTI/tcal transsely strpc VTI layered structure t cncdes wth the unque slutn C13. We explt ths prperty t check whether the medum under cnsderatn pssesses the HTI type f symmetry. After the azmuth f the symmetry s s establshed we can cmbne equatns C3 and C7 V nm V = A z + [ M k cs kϕphs cs kϕ r equvalently V nm V + sn kϕ phs sn kϕ ] = A z + M k cs k ϕ phs ϕ. C16 C17 Next we cnsder tw specfc values f the phase velcty azmuth ϕ phs = ϕ and V nm V = A z + M k ϕ phs = ϕ + π/ Vnm = A V z + M k cs kπ = A z + 1 k M k. C18 C19 The frst azmuth defnes the tcal plane f symmetry and the secnd ne defnes the strpc tcal plane. Bth cases C18 and C19 lead t an nfnte decreasng gemetrc seres: ne wth tems f the same sgn and anther wth tems f alternatng sgns. Intrducng M k frm equatn C8 we btan A z + M k = 1 + δ A z + 1 k M k = 1 C0 s that δ = 1 M k 1 1 k M k = 35 M k. C1 In practce we deal wth the NMO velcty squared whch s nt scaled by the tcal velcty squared. In ther wrds we deal wth V nm rather than wth V nm /V. Thus all the ccents dscussed btan factr V. T get the Furer ccents we apply the dscrete frward Furer transfrm t the NMO velcty squared n the nterval 0 ϕ phs <π real-t-cmplex. Next we nrmalze the DC value by factr 1/n and the ther AC values by factr /n F = 1 n 1 n F k = n 1 n V nm m=0 V nm m=0 ϕphsm ϕphsm = πm n C πkm ϕphsm exp n k = 1...N 1. C3 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 611 Nte that nrmally the standard frward Fast Furer Transfrm has a mnus sgn at the expnent n equatn C3. In ths case ne shuld nt the sgn f all magnary parts. Thus we btan the DC term and the AC terms F = A z V Re F k = A k V Im F k = B k V. The abslute AC values are F k = M k V. Next t fllws frm equatn B0 that = F N 1 1 k F k. V C4 C5 C6 N s the number f cmplex tems n the Furer space f realt-cmplex dgtal transfrm. Ths number ncludes the DC term and n case f even n als the Nyqust term. These tw terms are real whle the thers are cmplex N = n + 1 fr even n N = n + 1 fr dd n. C7 In practce the abslute values F k decay y quckly and 3 5 AC terms nrmally suffce. The rthrhmbc Thmsen parameter δ can be btaned frm equatn C1 N 1 δ V = 35 F k. C8 The values fr ectve parameters btaned n ths sectn may be cnsdered as an ntal guess and further refned t best ft the measured functn f the NMO velcty sus the phase velcty azmuth as shwn n Appendx D. APPENDIX D. REFINEMENT OF EFFECTIVE PARAMETERS We slve equatn 19 t fnd the ectve hrzntal transsely strpc parameters that best ft the gven data fr the nrmal mveut NMO velcty sus the azmuth f the phase velcty. Cmbnng equatns 19 and C1 we btan A V δ ϕ = 1 [ ϕphs =π V f δ ϕ ϕ phs V data nm ϕ ] phs ϕ phs =0 Vnm dataϕ dϕ phs phs A mn. D1 Accrdng t equatn C1 functn f s defned by f δ ϕ ϕ phs = 1 + 4 δ 1 + δ cs ϕ ϕ phs 1 + δ cs ϕ ϕ. phs D The NMO velcty depends n the ectve tcal velcty lnearly and ths makes t pssble t elmnate the ectve tcal velcty A V = 0 V δ ϕ = D δ ϕ B δ ϕ where the ccents B and D are deled by π f δ D δ ϕ ϕ ϕ phs dϕ 0 Vnm data phs ϕphs π f δ B δ ϕ ϕ ϕ phs dϕ phs. ϕphs 0 V data nm D3 D4 The ectve tcal velcty V s n mre an ndependent parameter and equatn D1 reduces t A δ ϕ = 1 ϕphs =π M δ ϕ δ ϕ dϕ phs mn ϕ phs =0 D5 where M δ ϕ ϕ phs V δ = δ ϕ ϕ phs =0 ϕphs =π f δ V data nm ϕphs. ϕphs ϕ ϕ phs Vnm data D6 T fnd the mnmum we slve a set f tw nn-lnear equatns wth tw unknwn values A ϕphs =π V Mdϕphs A ϕ = ϕ phs =0 δ f + V f δ V f + V ϕ f ϕ Vnm data Mdϕphs V data nm = 0 = 0. D7 The equatn set can be slved ectvely by the standard Newtn methd applyng the ntal guess fr the exact layered structure. On each teratn the crrectns δ and ϕ t the unknwn parameters can be fund frm the lnear set A A A δ δ ϕ A A δ δ = ϕ A. D8 δ ϕ ϕ ϕ C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
61 Z. Kren I. Ravve and R. Levy The secnd dervatves f the target functn are A δ = ϕphs =π ϕ phs =0 ϕphs =π + ϕ phs =0 V f + V δ V δ f f + V δ δ dϕphs V data nm f + V δ f δ Mdϕ phs Vnm data D9 A ϕphs =π ϕ = ϕ phs =0 + ϕphs =π ϕ phs =0 V ϕ V ϕ f + V f + V ϕ f ϕ dϕphs V data nm f + V ϕ f ϕ Mdϕ phs Vnm data D10 and the mxed dervatve s A ϕphs =π V = δ ϕ ϕ phs =0 δ V ϕ ϕphs =π + ϕ phs =0 + V ϕ f + V f + V V δ ϕ f + V δ f ϕ f δ dϕphs V data nm f + V δ f δ ϕ APPENDIX E. EFFECTIVE MODEL FOR LAYERED STRUCTURE f ϕ Mdϕphs V data nm. D11 We prved n Part I that the nrmal mveut velcty f a package f hrzntal transsely strpc HTI and tcal transsely strpc VTI layers s gven by V nm = where A ph t A pt A ph = A px + A py. E1 E Parameters A px and A py descrbe the lateral prpagatn n the drectn f the phase velcty azmuth and n the nrmal drectn respectvely whle parameter A pt descrbes the traveltme. Recall that parameters A px A py and A pt are btaned by summatn all layers: HTI VTI and strpc but nly HTI layers cntrbute nt A py A px = A pt = A HTI px + A HTI pt + A VTI px + A VTI pt + A ISO px A ISO pt. Fr an HTI layer the ffset ccents are A HTI px = C h cs ϕ t V A py = A HTI py E3 E4 AHTI py = C h sn ϕ t V E5 where ϕ s the azmuthal shft between the ray and the phase velctes fr an nfntesmal ffset wthn the hyperblc apprxmatn cs ϕ = 1 + δ cs ϕ ϕ phs C h sn ϕ = δ sn ϕ ϕ phs E6. C h Parameter C h s defned by C h = 1 + 4 δ 1 + δ cs ϕ ϕ phs E7 and the traveltme ccent s A HTI pt = C h cs ϕ t V. E8 Fr a VTI layer the ffset ccents are A VTI px = 1 + δ t V and the traveltme ccent s A VTI pt = 1 + δ t V. AVTI py = 0 E9 E10 An strpc layer can be cnsdered a partcular case f a VTI layer wth vanshng Thmsen parameter δ. Equatns E1 E10 lead t the fllwng resultng relatnshp fr the nrmal mveut NMO velcty f the entre package t V nm = A x + A y E11 Ax where fr an HTI layer A x = [ 1 + δ cs ] ϕ ϕ phs t V A y = δ sn ϕ ϕ phs t V. whle fr a VTI layer E1 A x = 1 + δ t V A y = 0. E13 The azmuth f the surface ffset s gven by tan ϕ ff = A y / A x. E14 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 613 Each HTI layer s characterzed by ts tcal velcty V rthrhmbc Thmsen parameter δ and the azmuth f the s f symmetry ϕ. Each VTI layer has the tcal velcty and TI Thmsen parameter δ and each strpc layer has the tcal velcty nly. In addtn each layer f any type has thckness t measured n unts f tme ne-way tme n ths study. Our gal s t fnd the parameters f the ectve mdel cnsstng f a unque HTI layer V δ and ϕ such that fr any azmuth f the phase velcty ϕ phs the magntude f the NMO velcty f the package wll match the magntude f the NMO velcty f the ectve mdel. The tcal tme f the ectve mdel s just the sum f tcal tmes f all layers s equatn E11 becmes A x + A y Ax Fr ths we requre Ax = A x = A x + A y A x Ay = A y.. E15 E16 Nte that f equatn E16 s satsfed then nt nly equatn E11 hlds but als equatn E14. In ther wrds ths means than the ectve mdel matches nt nly the magntude f the nrmal mveut velcty but als ts lateral drectn. That s t say the azmuthal devatn ϕ ff ϕ phs wll be the same fr the rgnal and the ectve mdels. We frst cnsder that the package cnssts f HTI layers nly wth arbtrary and dfferent azmuthal rentatns f ther es f symmetry and then we wll generalze ur fndngs fr a package that ncludes VTI and strpc layers as well. It fllws frm equatns E1 and E16 [ 1 + δ cs ] ϕ ϕ phs t V = [ 1 + δ cs ϕ ϕ ] phs t V δ sn ϕ ϕ phs t V = δ sn ϕ ϕ phs t V. E17 Cnsder the frst equatn f set E17 and expand the csnes squared [ 1 + δ + δ cs ] ϕ ϕ phs t V [ ] = 1 + δ + δ cs ϕ ϕ phs t V. E18 Cntnung the expansn we btan 1 + δ + δ cs ϕ cs ϕ phs + δ sn ϕ sn ϕ phs t V = 1 + δ + δ cs ϕ cs ϕ phs + δ sn ϕ sn ϕ phs t V. E19 We emphasze that equatn E19 shuld be satsfed fr any value f the phase velcty azmuth ϕ phs.termswth cs ϕ phs thse wth sn ϕ phs and the free terms whch d nt nclude ϕ phs are lnearly ndependent and they shuld be balanced apart. Therefre we balance the crrespndng ccents and btan three dstnct equatns δ cs ϕ t V = δ HTI δ sn ϕ t V = δ HTI 1 + δ t V = 1 + δ cs ϕ t sn ϕ t V V t V. E0 Fr the VTI and strpc layers ne can assume that the s azmuth cncdes wth the azmuth f the phase velcty ϕ = ϕ phs. These layers have n ect n the frst tw equatns f set E0 but cntrbute addtnal terms t the thrd equatn f ths set VTI 1 + δ t V + = 1 + δ t V ISO 1 + δ t V + t V. E1 Next we cnsder the secnd equatn f set E17 and expand the snes δ sn ϕ cs ϕ phs cs ϕ sn ϕ phs t V = δ sn ϕ cs ϕ phs cs ϕ sn ϕ phs t V. E Agan we balance the lnearly ndependent terms but nw ths peratn des nt lead t any new equatns. We btan the frst tw equatns f set E0.e. there s n new nf and n cntradctn wth the exstng nf. Thus we need t slve a system f three equatns that ncludes the frst tw C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
614 Z. Kren I. Ravve and R. Levy equatns f set E0 and equatn E1 δ cs ϕ δ sn ϕ HTI V V = W x t = W y t 1 + δ where W x = δ cs ϕ t V W y = and U = δ sn ϕ t V VTI 1 + δ t V + V = U E3 t ISO 1 + δ t V + E4 t V. E5 Nte that fr a gven layered mdel parameters values W x W y and U are cnsdered knwn as they can be cmputed usng explct frmulae. There may be tw slutns f equatn set E3: ne fr pstve ectve rthrhmbc Thmsen parameter δ and anther ne fr negatve. The tw slutns are abslutely equvalent. Althugh they lead t dfferent parameters f the ectve layer they netheless result n the same functn nrmal mveut velcty sus phase velcty azmuth V nm ϕ phs. We cnsder t s natural t assume that δ s negatve because nrmally ths parameter s ften negatve n HTI layers. Hwe negatve δ s nt a must even n the case when all δ f the layers are negatve. We can chse the sgn f δ arbtrarly but we need t make an explct chce. Fr a negatve ectve Thmsen delta the frst tw equatns f set E3 yeld the azmuth f the ectve s f symmetry ϕ = arctan W y W x π/ <ϕ π/ E6 where the nse tangent f tw arguments s used r equvalently cs ϕ = W x W where W Wx + W y. sn ϕ = W y W E7 E8 Nw the ectve azmuth can be elmnated frm equatn set E3 and the remanng equatns are δ V = W t V = W + U. E9 t Fnally the ectve delta s δ = W W + U. E30 Cmment: we assume that at least ne HTI layer s present n the package therwse the ectve azmuth f symmetry has n meanng. Mre fr a set cnsstng f VTI and strpc layers nly ne can calculate nly the resultng NMO velcty whch s als the rms velcty and ne can nt separate the ectve δ frm the rms velcty f such package. The NMO velcty f a VTI package s descrbed by a well-knwn frmula Thmsen 1986 t V nm = t V 1 + δ. E31 The trade-ff between the tcal velcty and parameter δ can nt be reslved even f the NMO velctes fr all three P SV and SH waves frm a hrzntal reflectr are knwn Tsvankn and Thmsen 1995. On the ther hand f the tcal velcty s knwn e.g. frm check shts r well lgs the mveut velcty can be used t btan δ Alkhalfah and Tsvankn 1995. In Fgs 7 and 8 we plt the NMO velcty fr a package f layers. The prpertes f the layers are gven n Table 1 f Part I Sectn Numercal Examples Package 1. The thck lne crrespnds t the NMO velcty f the package and the thn lne n Fg. 7 crrespnds t the NMO velcty f the equvalent layer wth the ectve prpertes. These graphs were pltted t test the algrthm that establshes the ectve mdel parameters. As expected the tw lnes cncde exactly thus the thn lne n legend f Fg. 7 can nt be seen n the graph. The functn s perdc wth the perd π radans. APPENDIX F. NORMAL MOVEOUT VELOCITY VERSUS RAY AND PHASE VELOCITY AZIMUTH In ths appendx we shw that the nrmal mveut NMO velcty functn sus phase velcty azmuth can be btaned frm the NMO velcty functn sus ray velcty azmuth fllwng smple gemetrc cnsderatns. The NMO velcty sus ray azmuth fr hrzntal transsely strpc medum s gven by Tsvankn 1997 V nm V = 1 + δ 1 + δ sn ϕ ray ϕ. F1 Equatn F1 can be rewrtten n a dfferent but stll equvalent frm cs ϕ ray ϕ V 1 + δ + sn ϕ ray ϕ = 1. F V Vnm C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 615 NMO Velcty fr Package f Layers 3 NMO Prjectn Y km/s 1 0-1 - Mult Layer Effectve -3-3 - -1 0 1 3 NMO Prjectn X km/s Fgure 7 NMO velcty dagram fr package f layers sus azmuth f phase velcty. NMO Velcty fr Package 3.9 NMO Velcty km/s.8.7.6.5.4 0 30 60 90 10 150 180 Phase Velcty Azmuth deg Fgure 8 NMO velcty fr package f layers sus phase velcty azmuth. Ths equatn descrbes an ellpse. Indeed cnsder an ellpse wth sem-es A and B as seen n Fg. 9. Its cannc equatn s n the ellptc cntur x = Rcs α y = Rsn α. F4 x A + y B = 1. F3 Intrducng equatn F4 nt F3 we btan cs α A + sn α B = 1 R. F5 Let α be the azmuth f the ellptc radus R measured frm the pstve sem-s x as shwn n Fg. 9. Then fr any pnt Cmparng equatns F and F5 we cnclude that F descrbes an ellptc lne where the sem-es and the radus C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
616 Z. Kren I. Ravve and R. Levy Fgure 9 Ellpse f NMO velcty sus ray velcty azmuth. are A = V 1 + δ B = V R = V nm. F6 Angle α shws the azmuth f the ray velcty wth respect t the azmuth f the symmetry s. Snce the NMO plt shws the wavefrnt cnfguratn then the nrmal t the ellptc lne shws the drectn f the phase velcty. Let β be the azmuth f the nrmal. Then α = ϕ ray ϕ β = ϕ phs ϕ. F7 We wll call α central angle and β nrmal angle. Equatn F5 can be reslved fr the radus R NMO velcty x prjectn f the NMO velcty n the medum s f symmetry and y prjectn f the NMO velcty n the drectn nrmal t the s f symmetry.e. n the strpc plane R = x = AB A sn α + B cs α. F8 AB cs α A sn α + B cs α y = AB sn α A sn α + B cs α. Accrdng t equatn F9 the dervatves f x and y cmpnents wth respect t the central angle are dx dα = A 3 B sn α A sn α + B cs α 3/ dy dα =+ AB 3 cs α A sn α + B cs α. F1 3/ Intrducng equatn F1 nt F11 we btan the arc length dervatve ds dα = AB A4 sn α + B 4 cs α A sn α + B cs α. F13 3/ Dervatves dx/ds and dy/ds defne the drectn f the lne tangent t the ellptc cntur. They als defne the drectn f the nrmal lne see Fg. 10 sn β = dx ds = dx/dα ds/dα cs β =+dy ds =+dy/dα ds/dα. F14 Intrduce equatns F1 and F13 nt F14 B cs α cs β = A4 sn α + B 4 cs α The arc length element s ds = dx + dy. The arc length dervatve reads ds/dα = dx/dα + dy/dα. F9 F10 F11 sn β = A sn α A4 sn α + B 4 cs α. Equatn F15 can be reslved fr the central angle α A cs β cs α = A4 cs β + B 4 sn β sn α = B sn β A4 cs β + B 4 sn β. F15 F16 C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617
Mveut apprxmatn n HTI/VTI layered medum 617 shft between the mveut drectn and the phase velcty azmuth. We can nt btan the exact relatnshp studyng the ellptc frnt but we can get the hyperblc apprxmatn fr ths shft. Recall that sn α β = sn α cs β cs α sn β cs α β = cs α cs β + sn α sn β. F0 Intrduce equatn F15 nt F0 δ sn β sn α β = 1 + 4δ 1 + δ cs β Fgure 10 Angle between the reference azmuth and the nrmal t wavefrnt. cs α β = 1 + δ cs β 1 + 4δ 1 + δ cs β. F1 Next we rewrte equatn F1 as V nm V = 1 + δ 1 + δ sn α F17 and substtute equatn F16 nt F17. Ths leads t V nm V = 1 + 4δ 1 + δ cs β F18 1 + δ cs β and we rewrte equatn F18 as V nm V = 1 + 4δ 1 + δ cs ϕ phs ϕ 1 + δ cs ϕ phs ϕ. F19 Ths relatnshp cncdes wth equatn 10 f Part I fr the NMO velcty. Eventually we can establsh the azmuthal Nte that ϕ ϕ phs = β. and therefre relatnshp F1 cmes t sn δ sn ϕ ϕ phs ϕ ray ϕ phs = 1 + 4δ 1 + δ cs ϕ ϕ phs F cs 1 + δ cs ϕ ϕ phs ϕ ray ϕ phs = 1 + 4δ 1 + δ cs. ϕ ϕ phs F3 Ths cncdes wth ur prevus results n equatn 10 f Part I. C 010 Eurpean Asscatn f Gescentsts & Engneers Gephyscal Prspectng 58 599 617