Normal moveout from dipping reflectors in anisotropic media

Transcription

1 GEOPHYSICS, VOL 6, NO1 (JANUARY-FEBRUARY 1995); P , 17 FIGS Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at Nrmal mveut frm dipping reflectrs in anistrpic media liya Tsvankin* ABSTRACT Descriptin f reflectin mveut frm dipping interfaces is imprtant in develping seismic prcessing methds fr anistrpic media, as well as in the inversin f reflectin data Here, I present a cncise analytic expressin fr nrmal-mveut (NMO) velcities valid fr a wide range f hmgeneus anistrpic mdels including transverse istrpy with a tilted in-plane symmetry axis and symmetry planes in rthrhmbic media In transversely istrpic media, NMO velcity fr quasi-p-waves may deviate substantially frm the istrpic csine-f-dip dependence used in cnventinal cnstant-velcity dip-mveut (DMO) algrithms Hwever, numerical studies f NMO velcities have revealed n apparent crrelatin between the cnventinal measures f anistrpy and errrs in the csine-f-dip DMO crrectin ("DMO errrs") The INTRODUCTION Cnventinal methds f seismic prcessing and interpretatin are designed fr istrpic velcity fields and, therefre, are subject t errr in anistrpic media It has been shwn that elastic anistrpy may seriusly distrt the results fvelcity analysis, nrmal-mveut (NMO) crrectin and stacking, migratin, etc (Banik, 1984; Thmsen, 1986; Tsvankin and Thmsen, 1994; Sams et ai, 1993; Lamer and Chen, 1993; amng thers) Clearly, dip-mveut prcessing cannt be an exceptin because mst existing algrithms rely n the behavir f mveut velcity with reflectr dip established fr istrpic mdels (Levin, 1971): V nm( <<\» = Vnm(O)/cs «\, (1) where «\ is the dip angle Fr hmgeneus istrpic media, reflectin mveut is purely hyperblic, and equatin (1) is exact fr any spread length analytic treatment develped here shws that fr transverse istrpy with a vertical symmetry axis, the magnitude f DMO errrs is dependent primarily n the difference between Thmsen parameters E and & Fr the mst cmmn case, E - &, the csine-fdip-crrected mveut velcity remains significantly larger than the mveut velcity fr a hrizntal reflectr DMO errrs at a dip f 45 degrees may exceed 2-25 percent, even fr weak anistrpy By cmparing analytically derived NMO velcities with mveut velcities calculated n finite spreads, I analyze anistrpy-induced deviatins frm hyperblic mveut fr dipping reflectrs Fr transversely istrpic media with a vertical velcity gradient and typical (psitive) values f the difference E - &, inhmgeneity tends t reduce (smetimes significantly) the influence f anistrpy n the dip dependence f mveut velcity It is well knwn that anistrpy may distrt the nrmal (shrt-spread) mveut velcity fr hrizntal reflectrs as well as enhance deviatins frm hyperblic mveut (Banik, 1984; Thmsen, 1986; Tsvankin and Thmsen, 1994) Therefre, it is natural t expect frmula (1) t becme inaccurate in the presence f anistrpy Levin (199) mdeled P-wave reflectin mveut fr dipping reflectrs beneath hmgeneus transversely istrpic media with tw different rientatins f the axis f symmetry (fr brevity, I will mit the qualifiers in "quasi P-wave" and "quasi-sv-wave") He shwed that if the axis is perpendicular t the reflectr, istrpic dip-mveut (DMO) frmula (1) hlds with gd accuracy Hwever, if the symmetry axis is kept vertical, the errr f equatin (1) fr ne f the mdels in Levin's study (the shale-limestne) reaches almst 4 percent at 6-degree dip Fr the ther three media used by Levin, the errrs were relatively small, Manuscript received by the Editr August 13, 1993; revised manuscript received June 1, 1994 "Center fr Wave Phenmena, Clrad Schl f Mines, Glden, CO Sciety f Explratin Gephysicists All rights reserved 268

2 Nrmal Mveut in Anistrpic Media 269 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at althugh ne f the mdels (Cttn Valley shale) may be ity with the mveut velcity calculated frm t 2 - x 2 cnsidered even mre "anistrpic" with respect t curves n cnventinal shrt spreads t verify analytic P-waves than the shale-limestne Indeed, anistrpic pa- slutins and t estimate the influence f nnhyperblic rameters E and 3 (Thmsen, 1986) fr the shale-limestne are mveut Fr values f E and 3 believed t be typical fr real E = 134, 3 =, while fr Cttn Valley shale E = 135, rcks, the csine-f-dip-crrected mveut velcity re- 3 = 25 Small errrs fr the ther tw mdels (Pierre mains significantly higher than the NMO velcity fr a Shale and Berea sandstne) are nt surprising since bth are hrizntal reflectr Finally, I use Lamer's (1993) raycharacterized by very weak P-wave anistrpy It is als tracing algrithm t analyze the cmbined influence f imprtant t mentin that Levin has nt fund nticeable transverse istrpy and vertical velcity gradient np-wave nnhyperblic mveut n cmmn-midpint (CMP) gath- NMO velcity frm dipping reflectrs ers with a spread length equal t the distance frm the CMP t the reflectr NORMAL-MOVEOUT VELOCITY IN ANISOTROPIC MEDIA Recently, it has been recgnized that depth-variable velcity may have a significant impact n dip-mveut prcessing Hwever, existing DMO algrithms built fr depth Let us cnsider a CMP gather ver a hmgeneus anistrpic medium; the CMP line is perpendicular t the variable velcity fields still ignre anistrpy (eg, Hale and strike f the reflectr (Figure 1) The nly assumptin made Artley, 1993) The cmbined influence f anistrpy and abut anistrpy at this stage is that the phase and grup inhmgeneity n DMO has been studied by Larner (1993), velcity vectrs d nt deviate frm the sagittal (incidence) wh has perfrmed calculatins similar t thse f Levin plane, ie, the sagittal plane is a plane f symmetry Fr (199) but fr "factrized" transversely istrpic mdels instance, the present treatment is valid fr any plane cntaining the symmetry axis in transversely istrpic media with a vertical velcity gradient ("factrized" means that the ratis f the elastic cnstants are independent f spatial (plus the istrpy plane), as well as fr symmetry planes in psitin) The main cnclusin f that wrk is that the DMO rthrhmbic media errrs remain clse t thse fund by Levin, prvided Out-f-plane phenmena cannt be neglected if the sagittal plane lies utside symmetry planes in azimuthally anis istrpic DMO crrectin takes the velcity gradient int accunt trpic media; still, the frmula derived belw remains a gd Thus, the existing numerical results shw n simple crrelatin between DMO errrs and the "degree f anistr apprximatin if azimuthal anistrpy is weak Azimuthally anistrpic mdels with rthrhmbic symmetry caused by py" Evidently, further insight int the character f DMO a cmbinatin f thin hrizntal layering and vertical fracture systems with a lw fracture density seem t be typical perfrmance requires analytic descriptin f reflectin mveut frm dipping planes in anistrpic media fr sedimentary basins (Leary et ai, 199) Such media are Anther imprtant aspect f this prblem is the pssibility characterized by relatively weak azimuthal anistrpy and f using the dip-dependence f nrmal-mveut (NMO) mre prnunced velcity variatins in vertical planes It is velcity in the inversin fr the anistrpic parameters likely that fr mdels f this type, the nrmal-mveut Tsvankin and Thmsen (1995) shwed that fr transversely equatin discussed here wuld be acceptable even utside istrpic media, P-wave reflectin mveut frm hrizntal symmetry planes interfaces is nt sufficient t reslve the vertical velcity and Our gal is t find an analytic expressin fr the nrmalmveut velcity in the CMP gemetry (Figure 1): anistrpic cefficients, even if lng spreads (twice as large as the reflectr depth) are used Mveut frm dipping reflectrs makes it pssible t extend the aperture f reflectin data withut recurse t large ffsets (ie, withut using nnhyperblic mveut) Byun (1982) and Uren et al (199b) derived analytic expressins fr nrmal-mveut velcity frm dipping reflectrs in elliptically anistrpic mdels Uren et ai (199b) als shwed that, fr elliptical anistrpy, reflectin mveut remains hyperblic irrespective f the rientatin f the z elliptical axes Hwever, elliptical anistrpy is n mre than a special case f transverse istrpy, hardly typical fr real rcks (Thmsen, 1986) Byun (1984) btained an analytic expressin fr nrmal-mveut velcity in general transversely istrpic media by applying a lcal elliptical fit t the wavefrnt The results discussed belw shw that Byun's frmula deviates frm the exact NMO velcity fr nnelliptical mdels Here, I derive a frmula fr nrmal-mveut velcity valid fr many anistrpic mdels f practical imprtance FIG} Cmmn-midpint gather ver a hmgeneus anistrpic medium Vgr and Vp,h are the grup and phase velcity Fr weak transverse istrpy with a vertical symmetry axis, vectrs, respectively Fr brevity, hencefrth In the text V this exact expressin fr NMO velcity is transfrmed int a p h!s rferre, t just as V Nte that the zer-ffset ("nrmalincidence ) ray IS nt necessanly perpendicular t the simple functin f the anistrpies E and 3 I then cmpare the exact and weak-anistrpy expressins fr NMO velc- reflectr

3 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at 27 Tsvankin Fr mdels with a hrizntally hmgeneus verburden, the ray parameter des nt change betweenthe reflectr and the surface In this case, it is cnvenient t represent the NMO velcity in the fllwing way (Hale et ai, 1992; Lamer, 1993), 2 2 dh V nm(4)) = - hm -, t h O dp where Ihl = x/2 is half the surce-receiver ffset (h in the dwn-dip directin), t is the tw-way traveltime alng the zer-ffset ("nrmal-incidence") ray, and p is the ray parameter Nte that the zer-ffset ray (h = ) is nt necessarily perpendicular t the reflectr in the presence f anistrpy; it is the phase-velcity vectr assciated with the zer-ffset ray that is nrmal t the reflectr Nrmal-mveut velcity (3) is derived under the assumptin that the reflectin pint dispersal can be ignred (Hale et ai, 1992) As shwn in Hubral and Krey (198, Appendix D), the difference between the true (specular) and zer-ffset reflectin pints changes nly the quartic and higher-rder mveut terms and des nt influence NMO velcity; their cnclusin hlds fr anistrpic media as well This implies that we can use z (Figure 1) as the depth f the zer-ffset reflectin pint Then h = z (tan I/J - tan I/Jn), where I/Jn and I/J are the grup-velcity (ray) angles fr the zer-ffset and nnzer-ffset rays, respectively Nw equatin (3) becmes 2 2z d tan I/J V nm(4)) = - hm--- t h O dp T evaluate equatin (4), we use the general relatin between grup and phase velcities in anistrpic media (Berryman, 1979) V a(kv) a(kv) a(kv) =--x+--y+--z gr akx ak ak' y z where k is the wave vectr with magnitude k, and V gr and V are the grup and phase velcities, respectively If the incidence plane [x, z] is bth a plane f symmetry and the dip plane f the reflectr, grup- and phase-velcity vectrs fr emp reflectins remain in the vertical plane and depend nly n the in-plane phase angle a (we measure a frm the z-axis, see Figure 1) Therefre, grup velcity may be represented as Vgr = (V sin a + : cs a)x + (V cs a - : sin a)z Nte that the vertical axis is nt necessarily an axis f symmetry Frm equatin (5), the grup angle I/J is given by (2) (3) (4) (5) The derivative d (tan I/J)/dp in equatin (4) may be written as then Using equatin (6), we find d tan I/J da Since p = sin a/v, da 1 dv tan a +- V da tan a dv tani/j=----- d tan I/J dp 1--- V da d tan I/J da da V dp -= dp d tan I/J dp tan a d2' cs? a ( V da tan a dv), cs a ( V da [ cs a( 1-V tan a dv)] 3 The vertical distance frm the zer-ffset reflectin pint t the surface is given by da z = &Vgr(I/Jn) t cs I/Jn' Using expressin (5) fr grup velcity yields 1 (tan a d z = - Vt cs a V da Since the phase angle a fr the zer-ffset ray is equal t the dip angle 4, z becmes 1 (tan 4 dv) z = 2 V(4)) t cs V(4)) da ' (8) where the derivative dv/da shuld be calculated at the angle 4 Substituting equatins (7) and (8) int frmula (4) fr NMO velcity, we finally btain / 1 d 2V V( 4) V 1 + V(;j;) d(j2 (6) (7) V nm (4)) = cs 4 tan 4 dv' (9) V(4)) da where bth derivatives f phase velcity shuld be evaluated at the dip angle 4 Equatin (9) is valid fr NMO velcity f P- and S -waves in symmetry planes f arbitrary anistrpic media Diflicul-

4 Nrmal Mveut in Anistrpic Media 271 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at ties in applicatin f frmula (9) can be expected nly in anmalus areas near shear-wave singularities and cusps, where the grup-velcity functin is multivalued This expressin is relatively simple t use because it des invlve just the phase-velcity functin, nt the cmpnents f the grup-velcity vectr f the zer-ffset ray (nte that the angle between the zer ffset ray and vertical is generally different frm <!» Fr example, it can be used in symmetry planes f rthrhmbic media just by substituting the apprpriate phase velcity functin and its derivatives Frm equatin (9), the result f the cnventinal istrpic dip-mveut crrectin may be far different frm the NMO velcity fr a hrizntal reflectr Vnm(O), if anistrpy is present Belw, I examine the behavir f the nrmal-mveut velcity and perfrmance f the istrpic DMO crrectin fr transversely istrpic media TRANSVERSE ISOTROPY WITH A TILTED AXIS OF SYMMETRY (1) Levin (199) shwed numerically that the csine-f-dip crrectin remains accurate in the case when the symmetry axis is perpendicular t the reflectr Equatin (9) gives a clear analytic explanatin fr this result If the reflectr's nrmal cincides with the symmetry directin, then dv/d6 at the dip angle is zer, and frmula (9) reduces t V(<!» / 1 d 2V V nm(<!» = cs <! V 1 + V(<!» d 2 (11) The values f V(<!» and d 2V/d6 2 at any dip crrespnd t the symmetry directin and, therefre, are independent f <! Hence, equatin (11) cincides with the istrpic equatin (1) This means that the istrpic DMO crrectin hlds if the symmetry axis is perpendicular t the reflectr Hwever, this result is derived fr nrmal (zer-spread) mveut velcities rather than fr mveut velcities measured n finite spreads If the medium abve the reflectr is hmgeneus and istrpic, reflectin mveut n emp gathers is purely hyperblic, and equatin (1) is exact irrespective f the maximum ffset In the presence f anistrpy, hwever, mveut is generally nnhyperblic (Tsvankin and Thmsen, 1994), and equatin (11) may becme inaccurate with increasing spread length Althugh the spreads used by Levin (199) are nt lng (equal t the nrmal distance t the reflectr), his results fr the mdels with the symmetry axis perpendicular t the reflectr shw small errrs in the csine-f-dip relatinship (1), indicative f the influence f nnhyperblic mveut n the mveut velcity While dip dependence f the NMO velcity is nt distrted by the anistrpy when the symmetry axis is perpendicular t the reflectr, the value f Vnm(O) is nt the same as the NMO velcity fr istrpic media The P-wave Vnm(O) depends n the anistrpic parameter 8, which is In this sectin, frmula (9) is cmpared with analytic expressins fr the nrmal-mveut velcity frm a hriznrespnsible fr P-wave velcity near the symmetry axis (Thmsen, 1986); this will be discussed in mre detail belw Applicatin f frmula (9) remains straightfrward in a mre general case when the symmetry axis is tilted at an arbitrary angle Phase velcity in transversely istrpic media is usually expressed thrugh the angle between the phase-velcity vectr and the symmetry axis The frmula fr the P-wave phase velcity in standard ntatin (using the elastic cefficients cij and density p) can be fund, fr instance, in White (1983): 2pV 2 ( 6 ) = (Cll + C44) sin? 6 + (C33 + C44) cs {[(Cll - C44) sin (C33 - C44) cs? 6]2 + 4(c13 + C44)2 sin 2 6 cs 2 6}1/2 (12) T get the SV-wave velcity, the plus sign in frnt f the radical shuld be replaced with a minus Thmsen (1986) gives analgus frmulas in his ntatin and transfrms them int much simpler expressins fr weakly anistrpic media T use any f these velcity equatins in the calculatin f the NMO velcity (9), they shuld be evaluated at the angle between the symmetry axis and the reflectr's nrmal TRANSVERSE ISOTROPY WITH A VERTICAL AXIS OF SYMMETRY Levin (199) pints ut that the rientatins f the symmetry axis mst likely t be encuntered in practice are clse t either the vertical r the nrmal t reflectrs In the previus sectin it was prved that in the latter case dipdependence f nrmal-mveut velcity remains the same as in istrpic media Therefre, althugh frmula (9) allws fr rather general anistrpy, in the fllwing I cncentrate n transversely istrpic media with a vertical symmetry axis, r vertical transverse istrpy (VTI) I will characterize VTI by the vertical P- and S-wave velcities (Vp = V C33/P and Vs = V C44/P), and the anistrpic parameters E, 8, and y, defined by Thmsen (1986): Cll - C33 E==---- 2C33 (C13 + C44)2 - (C33 - C44)2 8 == C33 (C33 - C44) C66 - C44 y== 2C44 (13) (14) (15) P- and SV-wave prpagatin is described fully by fur parameters: V p, V s, E, and 8 The SH-wave slwness surface and wavefrnt are elliptical with the phase velcity given (exactly) by Special cases: cmparisn with previus results

5 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at Tsvankln tal reflectr beneath VTI media (Hake et ai, 1984; Thmsen, 1986), and frm dipping reflectrs beneath elliptically anistrpic media (Byun, 1982; Uren et ai, 199b) Frmula (9) and the P-wave mveut velcity cmputed frm t 2 - X curves are als cmpared with analytically based calculatins fr general transverse istrpy presented by Byun (1984) In the case f a hrizntal reflectr (<! = ), equatin (9) reduces t V 1 d 2V Vnm(O) = V V d6 Using equatin (12) t evaluate the secnd derivative f phase velcity and substituting expressin (14) fr 8, we find fr the P-wave V 9 V nm(<!» =- cs <! (17) which cincides with Thmsen's (1986) result It shuld be emphasized that equatin (17), alng with the riginal equatin (9), is valid fr VTI media with an arbitrary degree f anistrpy, nt just fr weak transverse istrpy When the symmetry axis is perpendicular t a dipping reflectr, the P-wave NMO velcity is given just by equatin (17) and the csine-f-dip factr [equatin (1)] Similarly, we get Thmsen's expressin fr the zer-dip NMO velcity f the SV-wave Fr elliptical anistrpy, V(6) = VVJ cs? 6 + V sin 2 6, where V 9 is the hrizntal velcity The NMO velcity (9) then becmes (18) Equatin (18) agrees with the nrmal-mveut frmulas f Byun (1982) and Uren et ai (199b) In elliptically anistrpic media, reflectin mveut remains purely hyperblic irrespective f reflectr dip r the rientatin f the elliptical axes (Uren et ai, 199b) In general VTI media, equatins fr elliptical anistrpy are strictly valid nly fr the SH-wave If the SH-wave phase velcity is parametrized by "'( as in equatin (16), frmula (18) yields V sv 1 + 2"'( /---"- Vnm(<!»(SH) = vi + 2"'( sin 2 <! cs <! (19) Since fr the SH-wave Vnm(O) = V svi + 2"'(, equatin (19) may be represented as Vnm(O) Vs H(<!» Vnm(<!)(SH) =, (2) cs <! V s V S H(<!» is the SH-wave phase velcity at the dip angle Therefre, fr elliptical anistrpy the errr f the csinef-dip-dmo crrectin is determined directly by phase velcity variatins, ie, the errr is given just by the phase velcity at the dip angle divided by the vertical velcity I demnstrate belw that the frmula fr elliptical anistrpy may lead t significant errrs in the P-wave NMO velcity even fr "almst" elliptically anistrpic mdels Byun (1984) generalized his elliptical nrmal-mveut frmula fr arbitrary transverse istrpy by applying a lcal elliptical fit t the wavefrnt The resulting expressin fr the nrmal-mveut velcity invlves the grup velcity and grup angle f the nrmal-incidence ray The mveut velcity multiplied with the csine f the angle '"n between the zer-ffset ray and vertical ("emergence angle" in Byun's paper) Byun calls the "diffractrvelcity" Figure 2 shws the P-wave mveut velcity fr the limestnesandstne mdel used in Byun's study with his nrmalizatin The dtted curve is the analytic NMO velcity cmputed frm frmula (9); the slid curve is the mveut velcity recvered directly frm traveltimes (t 2 - x 2 curves) calculated by a ray-tracing cde ver a spread f 15 m The distance frm the emp t the reflectr in the traveltime calculatins n this and all subsequent plts (except fr Figure 16) is 3 m [fr a descriptin f the algrithm used t calculate traveltimes, see Lamer (1993)] Figure 2 was designed t reprduce the result in Figure 6a f Byun's (1984) paper Hwever, with increasing dip the mveut velcities in Figure 2 becme substantially higher than thse cmputed by Byun; a similar discrepancy was fund fr the secnd mdel used in Byun's wrk Since the analytic and numerical results in Figure 2 are clse t each ther (the small difference between the tw curves will be explained belw), it seems that Byun's frmula deviates (at least fr these tw mdels) frm the exact Vnm fr nnelliptical media We can nly speculate abut the reasn fr this inaccuracy One f the assumptins madebybyun in his derivatin is that V nm fr VTI media can be fund by fitting an ellipse x , ,----, -(J) 13 E " "" "" 1+---,---,---, FIG2 P-wave mveut velcity calculated frm frmula (9) (dtted curve) and frm traveltimes (slid curve) fr the limestne-sandstne mdel frm Byun (1984) Bth curves are cnverted int the "diffractr velcities" as suggested by Byun Mdel parameters are VPO = It/s, V s = 5753 ft/s, E = 183, 8 = -4

6 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at with vertical and hrizntal axes t the wavefrnt It is pssible that the crrect mveut velcity at nnzer dips is given by a tilted fitted ellipse, even thugh the symmetry axis is vertical Hwever, this is n mre than a tentative cnclusin; the nature f the abve discrepancy needs further investigatin Hw many parameters determine the P-wave DMO signature? This imprtant questin has t be answered befre starting a systematic study f the behavir f NMO velcities in transversely istrpic media In cnventinal ntatin, the P-wave phase velcity (12) is a functin f fur elastic cefficients: c 11, C 33, C 13, and c 44' This might lead ne t believe that dip dependence f the P-wave V nm is als determined by fur variables Hwever, it is pssible t cut dwn n the number f parameters by switching t Thmsen's (1986) ntatin First, nte that V PO is just a scaling cefficient fr the P-wave phase velcity, if VplVs, E, and 3 are kept cnstant Therefre, V PO des nt change the dependence f the P-wave nrmal-mveut velcity V nm n the dip angle <1 This cnclusin is illustrated in Figure 3, which shws that the nrmalizedp-wave NMO velcity Vnm(<I»IVnm(O) (Figure 3b) is independent f the vertical velcity V p, The velcity in Figure 3 is the mveut velcity in equatin (9) multiplied with cs <1, as it is cnventinally dne in the istrpic DMO crrectin Anther parameter that can be eliminated frm the P-wave dip-mveut prblem is the shear-wave vertical velcity V s (r the rati VpIVs) Althugh the P-wave phase velcity frmally depends n fur Thmsen parameters (Vp, V s, E, and 3), the cntributin f V s is practically negligible Indeed, in the weak-anistrpy apprximatin, the P-wave phase velcity is a functin just f V p and the anistrpic cefficients E and 3 (Thmsen, 1986) If anistrpy is nt weak, Thmsen's velcity equatin becmes inaccurate, but the P-wave phase velcity remains practically independent f V s (Tsvankin and Thmsen, 1994) Hence, even fr a wide range f V s' the crrespnding Nrmal Mveut in Anistrpic Media 273 variatins in the P-wave nrmal mveut are insignificant, as is supprted by the result in Figure 4 Thus, in VTI media the dip dependence f the P-wave NMO velcity is primarily a functin f just tw anistrpic cefficients E and 3 Mre than that, in the next tw sectins I shw that the angular behavir f the NMO velcity is mstly determined by a particular cmbinatin f these parameters; ie, by the difference E - 3 If the symmetry axis is inclined, the NMO velcity is als dependent n the tilt angle Weak-anistrpy apprximatin fr nrmal-mveut velcity A cnvenient way t understand the influence f anistrpy n nrmal-mveut velcity is t use the weak-anistrpy apprximatin (WAA) Althugh WAA is n substitute fr exact equatins [such as frmula (9)] in DMO crrectin, it can prvide us with simple analytic relatins elucidating - en E 5 - E c E - c: 5 " - 12 (ij 4 E 6, _r_-----r Dip(deg) FIG 4 Influence f Vs n the csine-f-dip--crrected P-wave nrmal-mveut velcity calculated frm frmula (9) The black curve crrespnds t VPOIVs = 15, the gray curve t VPOIVs = 25 Vp = 3 km/s, E = 3, and 3 = 1 are the same fr bth curves z 3 1 FIG 3 Influence f V PO n the csine-f-dip-crrectedp-wave nrmal-mveut velcity calculated frm frmula (9) The black curve crrespnds t the shale-limestne mdel with V p = 336 km/s, V ' = 1819 km/s, E = 134, 3 = O The gray curve is fr the mdel with V p = 42 km/s, and the same VplVs, E, and 3 (a) NMO velcity withut nrmalizatin; (b) NMO velcity nrmalized by the zer-dip value Vnm(O)

7 274 Tsvankln Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at the dependence f the NMO velcity n the parameters e and 8 In the case f weak anistrpy (e «1, 8 «1), phase velcities fp- and SV-waves can be significantly simplified by retaining nly the terms linear in e and 8 The P-wave phase velcity linearized in e and 8 is given by (Thmsen, 1986) V p(6) = V p (1 + 8 sin' 6 cs e sin" 6) (21) The derivatives f equatin (21) needed in the expressin fr NMO velcity (9) are then dvp(6 ) ---= V p sin 26(8 cs sin 2 6), d6 dvt(6) --2- = 2V p [8 cs sin 2 6 (1 + 2 cs 26 )] d6 After substitutin f the abve weak-anistrpy equatins int equatin (9) and further linearizatin in e and 8, we get Vp (<!» Vnm(<!» =--[ ( - 8) sm 2 <! (1 + 2 cs 2 <!»] cs <! (22) V p (<!» is the phase velcity given by equatin (21) In the istrpic DMO crrectin, multiplicatin f V nm (<!» with cs <! is suppsed t cnvert the mveut velcity at dip <! int the mveut velcity fr a hrizntal reflectr Hence, the anistrpy-induced DMO errr in the weak-anistrpy apprximatin is given by (again, nly the terms linear in e and 8 are retained) Vnm( <!» cs <! ---- Vnm(O) x [1 + 2( - 8) sin 2 <! (1 + 2 cs' <!»] (23) The structure f equatin (23) suggests that the P-wave dip-mveut errr in transversely istrpic media has tw majr cmpnents, which may be called the "elliptical errr" and "nnelliptical errr" Indeed, fr elliptical an- a) 6-r , Ẹ :: ;: I ,---r----I istrpy e = 8, and the errr is determined just by angular variatins in the P-wave phase velcity This result has already been discussed in the previus sectin [equatins (18) and (2)] The secnd, nnelliptical cmpnent f the DMO errr is the term cntaining the difference e - 8 T cmpare the tw cmpnents, I substitute the weakanistrpy apprximatin fr V p (<!» in equatin (21) int equatin (23) and drp the terms quadratic in the anistrpies e and 8 The dip-mveut errr then becmes Vnm(<!» cs <! = sin <! Vnm(O) + 3( - 8) sin 2 <! (2 - sin? <!» (24) The nnelliptical errr in the fully linearized expressin (24) is represented by the last term Analysis f the trignmetric cefficients in equatin (24) shws that, unless 1-8/ «181, the nnelliptical term usually makes the mst significant cntributin t the ttal errr Thus, fr typical values f the anistrpic cefficients, the difference - 8 determines, t a large degree, the angular behavir f the P-wave NMO velcity This cnclusin is supprted by exact numerical calculatins in the next sectin Nw we can explain the puzzling difference between the DMO signatures fr the mdels f Cttn Valley shale and the shale-limestne (Levin, 199; Lamer, 1993) Fr Cttn Valley shale (e = 135, 8 = 25), 8 is psitive, while e - 8 is small and negative As a result, the tw cmpnents f the DMO errr in equatins (23) and (24) almst cancel each ther, and the accuracy f the istrpic DMO crrectin is quite satisfactry Figure 5 shws the cmparisn between the mveut velcity calculated directly frm traveltimes (t 2 - x 2 curves) ver a spread f 3 m, the exact NMO velcity (9), and the weak-anistrpy nrmal-mveut apprximatin (22) fr the mdel f Cttn Valley shale All three curves display nly small variatins (-3-4 percent) in the crrected mveut velcity with angle, cnfirming the cnclusin abut the validity f the csine-f-dip crrectin fr this particular mdel Thugh Cttn Valley shale has a large value f 8, b) 5 ::t: ;:45 8 (j) ::E 3 -I-----,----r--,---"1 FIG5 Csine-f-dip--crrected P-wave mveut velcity fr (a) Cttn Valley shale and the (b) shale-limestne The slid curve is the mveut velcity calculated frm the traveltimes n a spread f 3m (the CMP-t-reflectr distance is als 3m); the dtted curve is the exact NMO velcity cmputed frm frmula (9); the dashed curve is the weak-anistrpy apprximatin (22) Parameters f Cttn Valley Shale are Vp = 4721 km/s, V s = 289 km/s, e = 135, 8 = 25; fr the shale-limestne, Vp = 336 km/s, V s = 1819 km/s, e = 134, 8 = O

8 Nrmal Mveut In Anistrpic Media 275 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at the weak-anistrpy result is clse t the exact NMO velcity (the difference is less than 2 percent) Fr the shale-limestne (E = 134, 8 = ), a psitive value f E - 8 leads t a prnunced increase in the csine-f-dip crrected mveut velcity with dip angle Nte that the accuracy f the weak-anistrpy apprximatin fr the shale-limestne is high A systematic cmparisn between the weak-anistrpy apprximatin and the exact NMO velcity is presented in the next sectin Frmula (9) can als be transfrmed int the weakanistrpy apprximatin fr the SV-wave nrmal-mveut velcity Using the weak-anistrpy expressin fr the SV-wave phase velcity (Thmsen, 1986) we btain V sv(6) = V s(l + U' sin 2 6 cs 2 6), Vsv(l\» Vnm(I\)(SV) = -- cs I\ x [1 + U' - 2U' sin 2 I\ (1 + 2 cs 2 1\»], (25) where U' is the effective parameter intrduced in Tsvankin and Thmsen (1994) t describe SV-wave prpagatin: ( VPO)2 U' == - (E - 8) V s Hence, the DMO signature fr the SV-wave is mstly determined by just ne anistrpic parameter-, Equatins (22) and (25) can be rewritten (nt dne here) fr the mre general case f transverse istrpy with a tilted axis f symmetry Dip-mveut signature fr P-waves Befre ding a systematic analysis fr vertical transverse istrpy, it is wrthwhile t explain the small difference between the mveut velcity calculated directly frm traveltimes (t 2 - x 2 curves), and NMO velcity frm frmula (9) in Figures 2 and 5 Since the mveut velcity was determined frm a least-squares fit t t 2 - x 2 curves n a finite spread length, it culd have been distrted by nnhyperblic mveut, while the analytic NMO velcity describes purely a) (i)6-r ;58 ' s, - \ -,, ",",, :, '",',",,'",,,,,',, ' I P"!,;:',, :::!: r-----r--,---j hyperblic mveut n very shrt spreads T check ut this pssibility, Figure 5 is reprduced in Figure 6, but with the mveut velcity calculated n a much shrter spread (1 m instead f 3 m), reduced t just 1/3 f the distance frm the CMP t the reflectr Nw the mveut velcity recvered frm the traveltimes (slid curve) practically cincides with the analytic slutin fr the NMO velcity (dtted curve) Therefre, the analytic and numerical results are in gd agreement with each ther Since the tw mdels studied abve exhibit such different behavir f the P-wave NMO velcity, it is imprtant t find ut what can be expected fr transversely istrpic media that are likely t be encuntered in the subsurface Existing labratry and field data indicate that in mst cases E 8 (Thmsen, 1986; Tsvankin and Thmsen, 1994) Fr instance, E 8 fr transversely istrpic media caused by thin bedding f istrpic layers (Berryman, 1979) This means that the csine-f-dip crrected mveut velcity in transversely istrpic media is usually higher than the mveut velcity fr a hrizntal reflectr [see frmulas (23) and (24)] Hence, the behavir f the crrected mveut velcity fr the shale-limestne mdel may be typical fr subsurface frmatins Rather than examine specific transversely istrpic mdels published in the literature, I present a systematic analysis f the P-wave DMO signatures fr transversely istrpic media parametrized by E and 8 Since the weak-anistrpy apprximatin suggests the difference E - 8 as the mst influential parameter in the DMO crrectin, I generate fur suites f plts fr E - 8 = -1,, 1, and 2 (Figures 7-1) The chice fthe values f E - 8 is explained abve; althugh E - 8 is believed t be predminantly psitive, the value f -1 is included fr cmpleteness Each plt cntains the same three types f curves shwn in Figures 5 and 6: the mveut velcity calculated frm t 2 - x 2 curves n the spread 3-m lng (slid), the exact analytic nrmal-mveut velcity cmputed frm equatin (9) (dtted), and the weak-anistrpy apprximatin fr V nm given by equatin (22) (dashed) Cmparisn between the first tw curves makes it pssible t estimate the influence f nnhyperblic mveut n the mveut velcity fr the typical spread length equal t the distance frm the CMP t the reflectr The difference between the secnd and third b) I 5 ;45 'g 4, 35 s :::!: 3+---r-----,---,----i FIG 6 Same as Figure 5 with (a) Cttn Valley shale and (b) shale-limestne, but the spread length used t calculate the mveut velcity frm t 2 - x 2 curves (slid curve) is 1 m instead f 3 m The analytic curves f the exact NMO velcity (dtted) and the weak-anistrpy apprximatin (dashed) have nt been changed

9 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at = b=1 =1 =2 12, '11 S 1 r-- 9 "" " -= -: : , , , 'g 11, "" S 1 '- ; -: :' -: 9 : ' ,----,r----,---1 =2 =3 =3 = , , - - ''511 ; : : '61 1 '' "" - : : : S 1 LILL, _ ;; ''';' :: -: ''- g ,---r--, S 1 r-----'-- g :, ;-----r------,--1 FIG 7 Csine-f-dip-crrected P-wave mveut velcity fr mdels with E - & = -1 The slid curve is the mveut velcity calculated frm (2 - x 2 curves n a spread length f 3 m (equal t the distance between the CMP and reflectr); the dtted curve is the exact NMO velcity frm frmula (9); and the dashed curve is the weak-anistrpy apprximatin frm frmula (22) On each plt in Figures 7-13, the vertical P-wave velcity V p is adjusted s that the exact analytic Vnm(O) (dtted curve) is 1 km/s The slid curve fr t = 3, & = 4 stps arund 52 degrees because the algrithm used t calculate traveltimes brke dwn at higher dips = 8= =1 b=1 13-r , " 'S 12 S i ,-,--,- t 'S 12 "ij) S 11 1 ::ie : :- -: " '"" "" " ,r-----,-----r------i =2 8=2 =3 b= , 13 -r ' - ;!:: 12 '" " '-g- 12 " ", ",,, ";'" : "ij) -' S11 : ':';";'-""'!' S11 --' g g -- 1-="':-:";-"!,",:1'"': : : 1 _-:-, : : ;r---,----r-----( ,r-----r----r-----i FIG 8 Csine-f-dip-crrected P-wave mveut velcity fr mdels with E - & = (elliptical anistrpy)

10 Nrmal Mveut in Anistrpic Media 277 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at curves shws the errr f the weak-anistrpy apprximatin On each plt, the vertical P-wave velcity V p is adjusted s that the exact analytic Vnm(O) (dtted curve) is 1 km/s First, I examine dip-dependence f the csine-f-dip crrected mveut velcity using the exact analytic expressin (9) (dtted curve) Later n, I discuss the accuracy f the weak-anistrpy apprximatin and the influence f nnhyperblic mveut The whle suite f plts in Figures 7-1 suggests that the P-wave DMO signature is cntrlled, t a significant degree (althugh nt entirely), by the difference 6-8 In spite f certain variatins frm ne pair f 6, 8 t anther, the general behavir and range f variatin f the mveut velcity are similar fr all curves with fixed 6-8, especially fr mderate anistrpies 161 < 2, 181 < 2 The dminant rle f 6-8 is particularly prnunced fr the mst typical case 6-8 (Figures 9 and 1) It is interesting that n mst f the plts, the exact NMO velcity, fr a fixed 6-8, shws even less dependence n a specific cmbinatin f 6 and 8 than des the weak-anistrpy result (fr instance, see Figure 1) When 6-8 = -1 (Figure 7), the csine-f-dip-crrected mveut velcity decreases with dip (fr mild dips), as predicted by the weak-anistrpy apprximatin This trend becmes less prnunced with increasing 6 and 8 because f a mre significant increase in the phase velcity with angle [frmula (23)] Nte that fr a fixed negative 6-8, the csine-f-dip crrectin becmes mre accurate (ie, the curves are clser t unity) with increasing anistrpies 6 and 8 On the whle, the DMO errr, determined by the amplitude f the angular variatins in the csine-f-dipcrrected mveut velcity, is relatively small (the "Cttn Valley shale" case) Fr elliptically anistrpic mdels (6-8 =, Figure 8), the anistrpy-induced distrtins f the csine-f-dip dependence are entirely determined by the amplitude f the phase-velcity variatins with angle The DMO errr fr elliptical anistrpy is mderate: the difference between the crrected mveut velcity and the zer-dip V nm fr <! < 6 degrees, 161 < 2, and 181 < 2 is less than 15 percent The csine-f-dip crrectin is, f curse, perfect fr the istrpic case, 6 = 8 = O If t - 8 is psitive (the mst cmmn case, figures 9 and 1), the anistrpy causes a prnunced increase in the csine-f-dip-crrected mveut velcity with dip angle Even fr relatively small 6-8 = 1, the dip-mveut errr reaches 25 percent at a 45-degree dip and 3-35 percent at a dip f 6 degrees ("the shale-limestne" case) Fr 6-8 = 2 (Figure 1), the crrected mveut velcity at a 6-degree dip is cnsistently abut 6 percent higher than the zer-dip mveut velcity! A remarkable feature f mdels e=o =-1 e=ol = , 16, , 'u 14 -:J 12 I 1 c:;:- ;-_l I e=o2 =1 e=3 = , e-, -'u 14 S 12 : ' 1 :,--_r_- r_- ; 16, , z- 'u 14 -:J 12,, :,, tit _r_-,r - FIG 9 Csine-f-dip-crrected P-wave mveut velcity fr mdels with 6-8 = 1

11 278 Tsvankln Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at with cnstant psitive 8 - B is the weakness f the dependence f the exact NMO velcity n 8 and B Thus, fr typical VTI media, with psitive 8 - B, the istrpic csine-f-dipcrrectin severelyunderstates mveut velcities at dips exceeding 2 t 3 degrees, even when the anistrpy is weak The range f dips n the plts abve was limited t 6 degrees Fr typical mdels with 8 - B, curves f the csine-f-dip--crrected mveut velcity flatten ut at dip angles exceeding 6 degrees (Figure 11) Therefre, the errr f the csine-f-dip dependence remains practically cnstant at steep dips in the 6-9 degree range The weak-anistrpy apprximatin fr the nrmalmveut velcity given by equatin (22) remains sufficiently accurate in the mst imprtant range f small and mderate values f 8 and B The errr f the weakanistrpy result, as cmpared with the exact NMO velcity frm equatin (9), des nt exceed 5 percent fr 181 :s 2, 181 :s 2 (the nly exceptin is the mdel with 8 =, B = -2) =-2-1 = 'g 16 ' :ie - 1-t-a;"--r-----,r-----r---i a 18, g 16 Q5 ' :ie 1 -t-"'o::::":'----;r-----r---r-- =2 = =3 = , , g 16 '5 14 g 12 :ie 1 -t--:---,r------r----r-----l a g16 ' :ie a F--"""'----,r------, FIG 1 Csine-f-dip-crrected P-wave mveut velcity fr mdels with 8-8 = 2 =1 5=-1 =2 = 18-r , 16 CD '5 14 g 12 1-t--L---r ,-----t a a ' J , '516 Q5 :; ,r---- r_----i FIG 11 Csine-f-dip-crrected P-wave mveut velcity fr steep reflectrs; 8-8 = 2

12 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at The abve suite f plts als presents a cmprehensive picture f the mveut-velcity distrtins at varius dips caused by nnhyperblic mveut The influence f nnhyperblic mveut manifests itself thrugh the difference between the mveut velcity, calculated frm (2 - x 2 curves (slid curves), and the exact NMO velcity (dtted curves) It is well knwn that deviatins frm hyperblic mveut rapidly increase with spread length; this trend may be enhanced by anistrpy (Tsvankin and Thmsen, 1994) Fr a maximum ffset-t-depth rati f 1, used in my calculatins, the cntributin f nnhyperblic mveut t the mveut velcity is nt significant, but the difference between the NMO and finite-spread velcities is clearly visible n sme f the plts Fr small dips, the distrtins f the mveut velcity caused by deviatins frm hyperblic mveut are in gd agreement with the analytic results in Tsvankin and Thmsen (1994) wh gave a descriptin f nnhyperblic mveut fr hrizntal reflectrs using the quartic Taylr series term fr (2 - x 2 curves Fr the P-wave, the influence f nnhyperblic mveut is largely prprtinal t the abslute value f e - &; if e - & is fixed, nnhyperblic mveut is mre prnunced fr smaller & The analytic analysis als shws that the P-wave mveut velcity measured n finite spreads is larger than the NMO velcity if 1 - &, and smaller than V nm if s - & < O The validity f these cnclusins is clearly seen in Figures 7-1 Fr elliptical anistrpy (1 = &)the mveut is purely hyperblic, and the dtted and slid curves fully cincide with each ther The bserved differences between the NMO velcity and the finite-spread mveut velcity seem t cntradict the results in Levin (199) and Lamer (1993), wh have nt nticed visible deviatins f their mveut curves frm hyperblas fr the same spread length Hwever, this is an apparent discrepancy Tsvankin and Thmsen (1994) shw that fr spread lengths clse t the depth f the reflectr, the best-fit hyperbla is clse t the actual mveut curve althugh the mveut velcity f this hyperbla may be different by several first percent frm the NMO velcity It is interesting that the difference between the mveut velcity n a finite spread and the NMO velcity changes sign with increasing dip (ie, the slid and dtted lines crss); mrever, the influence f nnhyperblic mveut fr steep reflectrs is typically smaller than fr zer dip (eg, Figure 11) I cnclude that if 1 - &1 < 15-42, nnhyperblic mveut des nt seriusly distrt the P-wave mveutvelcity n spreads cmmn fr emp acquisitin design, even if dips are large Apparent versus true dip In the discussin abve, we have nt made a distinctin between the true dip angle and an apparent dip used in cnstant-velcity DMO Since the true dip is usually unknwn, NMO velcity is cnventinally expressed thrugh ray parameter p as where p is determined frm the zer-ffset sectin ((y) : Nrmal Mveut In Anistrpic Media (26) 279 where V( <!» is the phase velcity at the dip angle This expressin frp is valid fr bth istrpic and anistrpic media Equatin (26) implies that the true dip angle <! is replaced in the DMO crrectin with the apparent dip </J (Lamer, 1993), Vnm(O) sin <! = pvnm(o) = sin <! (27) V(<!» In a hmgeneus istrpic medium Vnm(O) = V = V( <!), and the apparent and true dip angles cincide with each ther If the medium is vertically transversely istrpic, the zer-dip nrmal-mveut velcity given by equatin (17) (fr P-waves) is generally different frm the phase velcity at the dip angle V( cf» This means that substituting the apparent dip fr the true dip in the presence f anistrpy intrduces an additinal errr int a cnstant-velcity DMO prcess In principle, this new errr may either reinfrce r reduce the errr in mveut velcity discussed abve As shwn in Figures 12 and 13, the DMO crrectin using the apparent dip leads t higher DMO errrs fr bth 1 - & and E - & <, especially at steep dips (cf 45 degrees) Nte that after the crrectin fr the apparent dip, mveut velcities fr mdels with the same 1 - & remain clse if 1 - & and becme even clser if 1 - & < O The nly class f mdels fr which the intrductin f the apparent dip has a benign influence n the verall DMO perfrmance is elliptical anistrpy (e - & = ) It is interesting that fr elliptical mdels the crrectin f mveut velcity with cs </J instead f cs cf eliminates the DMO errr cmpletely (Figure 12); this result is easy t cnfirm analytically using equatin (27) and velcity equatins fr elliptical anistrpy Thus, the istrpic cnstant-velcity DMO crrectin is exact fr elliptically anistrpic mdels, f which istrpic mdels are a special case TRANSVERSELY ISOTROPIC MEDIA WITH VERTICAL VELOCITY GRADIENT The analysis in the previus sectins was carried ut fr hmgeneus transversely istrpic mdels Lamer (1993) has studied thep-wave dip-mveut errr fr factrized VTI media with a cnstant gradient in vertical velcity In terms f the ntatin used here, the velcity V PO in factrized transversely istrpic media varies with psitin, while the VplVs rati and the anistrpic cefficients e and & remain cnstant The fur mdels used in Lamer's wrk have the same anistrpic parameters and rt-mean-square (rms) vertical velcity dwn t the reflectr as the mdels in Levin's (199) study One f the interesting results reprted by Lamer is that fr the shale-limestne mdel with a typical value f the velcity gradient, the cnstant-velcity (csinef-dip) DMO crrectin gives a higher accuracy than des V(z) DMO (bth DMO crrectins ignre anistrpy) Cmparisn f the mveut velcities fr hmgeneus and inhmgeneus shale-limestne suggests that inhmgene-

13 28 Tsvankin Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at ity can cmpensate (t a certain degree) fr the distrtins f mveut velcity caused by the anistrpy The results btained in the previus sectin indicate that the behavir f NMO velcity fr the hmgeneus shalelimestne may be cnsidered typical fr a wide range f VTI mdels T verify whether the "cmpensatin effect," fund by Lamer, is typical fr inhmgeneus (factrized) VTI media, I carry ut the same calculatins as in the previus sectin, but fr factrized transversely istrpic media with a vertical velcity gradient f 6 s -1 (Figure 14) The mveut velcity is calculated frm t 2 - x 2 curves using Lamer's (1993) ray-tracing algrithm Cmparisn f Figure 14 with Figures 9 and 1 shws that fr typical psitive values f E - 8, angular variatins f the csine-f-dip-crrected mveut velcity are substantially suppressed by the velcity gradient It is ntewrthy that Lamer and Chen (1993) have fund a similar "cmpensatin effect" in their study f migratin errr in factrized transversely istrpic media When velcity increases with depth, small-ffset reflectins frm dipping interfaces travel mre clsely t vertical than in a hmgeneus medium This makes the "effective dip" f the reflectr smaller and reduces the increase in the mveut velcity with dip angle, bth in istrpic and anistrpic media Fr E - 8 = 1, the 'S 9 CD SO8 (I) f) 7 : ' r-----r---r-- influence f verticalvelcityvariatins even leads t "vercrrectin" in cnstant-velcity DMO, making the csinef-dip-crrected mveut velcity decrease with dip angle Figure 14 is reprduced in Figure 15, but with the DMO crrectin that hnrs inhmgeneity but ignres anistrpy [v(z) DMO, Lamer (1993)] Althugh the DMO errr caused by the anistrpy is smewhat smaller than in hmgeneus media with the same E and 8 (Figures 9 and 1), it is much larger than the errr f the simplest csinef-dip crrectin (Figure 14) Therefre, cnsistent with Lamer's results fr the shale-limestne mdel, fr typical factrized transversely istrpic mdels the DMO crrectin that ignres bth anistrpy and inhmgeneity is ften mre accurate than the crrectin that hnrs inhmgeneity but ignres anistrpy Anther imprtant cnclusin frm Figures 14 and 15 is that in factrized vertically inhmgeneus VTI media, the P-wave mveut velcity is still cntrlled primarily by the difference between E and 8, rather than by the individual values f these parameters Hwever, in V(z) media, dip dependence f the mveut velcity is als a functin f the velcity gradient, the rms vertical velcity, and the depth f the reflectr As illustrated by Figure 16, fr mre shallw reflectrs the influence f the velcity gradient is less prnunced, and the crrected -=-1 =1 =1 S=2 1--"""""': , "" '' 'g 9 CD SO8 (I) f) ,---, =1 S=Ol -8= =2 S=2 13-r , 'S 12 S 11 (I) "' '' : f) L:;;",; _l :!: r-----r---r-----i 13 -r _, z- 'g 12 S 11 (I) " '", : : : " " " f) 1 +--a:: l ,----r----r----; FIG 12 Cmparisn f DMO crrectins using the true and apparent dip angles fr mdels with E - 8 :5 O P-wave nrmal-il?-veut velcity is calculated frm frmula (9) and crrected wit the csine f the true dip angle 4 (dtted curves, same as 1 FIgures 7-1) and the apparent angle 4 frm equatin (27) (shd curves)

14 Nrmal Mveut in Anistrpic Media 281 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at = 8=-1-8= ' ' :; :; =2 =1 = z- z- ' ' :; :; =2 8=1 I, =2 = : FIG 13 Cmparisn f DMO crrectins using the true (dtted curve) and apparent (slid curve) dip angles fr E; - B O " Q!::! 1 (tj E9 z -8=1 -= , 12, , " ' ' ' ' - - '""-'"" ;: : : : ,----r " 1 J-_ (tj E 9 z, ,r-----,---r----l a FIG 14 P-wave mveut velcity crrected fr the csine f the true dip angle fr VTI mdels with a velcity gradient f 6 s -I The curves are nrmalized by the mveut velcity fr a hrizntal reflectr Each curve crrespnds t a different pair f E;, B On the left plt, E; =, B = -1 (black curve); E = 1, B = (gray curve); E = 2, B = 1 (dashed curve) On the right plt, E = 1, B = -1 (black curve); E; = 2, B = (gray curve); E; = 3, B = 1 (dashed curve) The distance frm the CMP t the reflectr and the spread length are 3 m; the rms vertical velcity dwn t 3 m is 35 mls

15 282 Tsvankln Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at mveut velcity is clser t the result fr a hmgeneus medium It is interesting that fr the rather typical mdel parameters used in the left prtin f Figure 16, the anistrpy and inhmgeneity cancel each ther's influence, and the csine-f-dip frmula yields almst an ideal crrectin S far in this sectin the DMO errr has been estimated using the true dip angle <1 The simplistic cnstant-velcity DMO apprach, described in the previus sectin, wuld result in the apparent dip <f given by equatin (27): Vnm(ZO, ) A sin <I = sin <I V(Zl, <1», (28) where the velcities V nm and V in FTI media depend n the depths f the zer-ffset reflectin pints Z (fr the hrizntal reflectr) and Z 1 (fr the dipping reflectr) Fr mdels with e - 8, the apparent dip angle turns ut t be smaller than the true ne and, cnsequently, the csine-f-dip-crrected mveut velcity becmes larger (cmpare Figure 17 with Figure 14) If e - 8 = 2, the intrductin f the apparent dip may lead t much higher errrs in cnstant-velcity DMO and a nticeable separatin f curves crrespnding t different pairs f e, 8 Fr e - 8 = 1, the difference between the apparent and true dip angles is smewhat smaller In analyzing these results, ne shuld keep in mind that the cmputatin f the apparent dip in FTI media using equatin (28) is strngly dependent n the relative spatial psitins f the hrizntal and dipping reflectrs The results in Figure 17 are btained fr the reflectrs lcated at the same distance frm the CMP pint If, instead, we cmpare the mveut velcities f reflectrs at the same zer-ffset time, the apparent dip usually becmes much clser t the true ne DISCUSSION In the presence f anistrpy, the dip dependence f mveut velcity deviates frm the istrpic csine-f-dip functin, thus leading t errrs in cnventinal istrpic DMO crrectin Here, I have given an analytic descriptin f NMO velcities that prvides a clear explanatin fr existing numerical results, such as Levin's (199) cnclusin that the csine-f-dip dependence f mveut velcity remains valid fr transverse istrpy if the symmetry axis is perpendicular t the reflectr Transversely istrpic mdels with a vertical symmetry axis (VTI media) were cnsidered in mst detail A simple weak-anistrpy apprximatin, derived frm the exact NMO expressin, relates the distrtins f NMO velcity t the anistrpic parameters The weak-anistrpy expressin fr the P-wave nrmal-mveut velcity is sufficiently accurate fr cmmn small and mderate values f e and 8 The errr f the weak-anistrpy result usually des nt exceed 5 percent fr lei s 2, 181 s 2 -=1-8= , , E 14 (ij E 12 Z :, ',, ;--;_- a E 14 (ij E 12 Z / ; '" :,$ / 1 -t-'-::::,-----r---r----l a FIG 15 P-wave mveut velcity after V(z) DMO crrectin All parameters are the same as in Figure 14 -=1 -= E 12 ; - (ij - (ij E 12 E 1 :: : E 1 Z 8 8 a a FIG 16 Csine-f-dip-crrected P-wave mveut velcity fr the same elastic parameters as in Figure 14, but fr a mre shallw reflectr: the distance frm the CMP t the reflectr and the spread length are 15 m z

16 Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at Dip dependence f the P-wave mveut velcity fr vn media is a functin f nly tw parameters-e and 8, with the influence f the S-wave vertical velcity V s being practically negligible Mre than that, the P-wave DMO signature is cntrlled, t a significant degree, by the difference E - 8 The systematic study f VTI med ia parameterized by E and 8 shws that fr E - 8 (the mst cmmn case), the csine-f-dip-crrected mveut velcity remains significantly larger than the mveut velcity fr a hrizntal reflectr Even fr relatively small E - 8 = 1 and E :5 2, the errr f the csine-f-dip frmula reaches 25 percent at a 45-degree dip and exceeds 3 percent at a dip f 6 degrees Fr E - 8 = 2 (als a feasible value), the csine-f-dipcrrected mveut velcity at 6-degree dip is almst 6 percent higher than the zer-dip mveut velcity The DMO errrs becme even higher if the true dip angle is replaced with the apparent dip calculated frm the cnventinal frmula used in cnstant-velcity DMO The analytic study f NMO velcities was supplemented by calculatins f the P-wave mveut velcity frm reflectin (2 - x 2 curves n relatively shrt-spread CMP gathers (ie, spread length = distance between CMP and reflectr), typical fr CMP acquisitin design Cmparisn between the analytic NMO velcity and the mveut velcity calculated n finite spreads makes it pssible t analyze the magnitude f nnhyperblic mveut (induced by the anistrpy) as a functin f reflectr dip The difference between the tw velcities changes sign with increasing dip, but is usually smaller fr large dips than fr hrizntal reflectrs If IE - 81 < 15-2, nnhyperblic mveut des nt seriusly distrt the P-wave mveut velcity n cnventinal length spreads, even fr steep reflectrs Significant errrs f cnventinal csine-f-dip DMO crrectin fr typical transversely istrpic mdels mean that it is imperative t develp dip-mveut algrithms fr anistrpic media Uren et a1 (199a) have generalized Gardner DMO fr elliptically anistrpic mdels; hwever, as shwn in the present paper, the elliptical P-wave DMO crrectin becmes inaccurate even fr " almst" elliptically anistrpic mdels The frmula fr NMO velcity, derived here, can prvide a basis fr building DMO algrithms fr general transversely istrpic and even rthrhmbic media 15"T""""" , E14 'l " " " -g 13 N 12 E =--,_- r_- -8=1 Nrmal Mveut In Anistrpic Media 283 One f the majr prblems in develping dip-mveut prcessing (as well as migratin, amplitude variatin with ffset algrithms, etc) in anistrpic media is recvery f the input anistrpic parameters with sufficient accuracy Fr VTI media, the parameter 8 can be determined using the P-wave NMO velcity frm a hrizntal reflectr and the true vertical velcity (such as frm check shts r VSP data) Hwever, the parameter E cannt be recvered frm shrt-spread P-wave data alne If the vertical P- and/r S-velcities (r reflectr depth) are knwn, bth Eand 8 can be determined frm the P- and SV-wave NMO velcities Tsvankin and Thmsen (1995) shw that it is pssible t find all fur anistrpic parameters gverning P-SV prpagatin (Vp, V s, E, 1» frm the cmbinatin flng-spreadp- and SV-traveltimes; hwever, this algrithm is nt easy t implement in practice Since E is directly related t the hrizntal velcity, it can als be determined frm headwave velcities r results f crsshle tmgraphy Anther way t vercme the ambiguity in the recvery f the anistrpic parameters is t include mveut frm dipping reflectrs in the inversin prcedure Appl icatin f the analytic NMO frmula develped here t inversin in anistrpic media will be discussed in a sequel paper CONCLUSIONS I have intrduced an analytic expressin fr nrmalmveut velcity frm dipping reflectrs valid in symmetry planes f hmgeneus arbitrary-anistrpic media The new frmula describes NMO velcities fp- and S-waves in many anistrpic mdels f practical imprtance, such as transverse istrpy with an in-plane symmetry axis, and symmetry planes in rthrhmbic media The dip dependence f P-wave NMO velcity in VTI media is determined mre by the difference between the parameters E and 8, than by the individual values f the anistrpic cefficients Fr the mst cmmn case, E - 8, the NMO velcity increases with dip much faster than in istrpic media, even fr mdels with mderate E - 8 = 1 t 2 This implies that cnventinal cnstant-velcity DMO algrithms, based n the istrpic csine-f-dip de- 15 E14 -g 13 N 12 Ẹ 11-8=2 1 FIG 17 P-wave mveut velcity crrected with the csine f the apparent dip angle <1 All parameters are the same as in Figure 14

17 284 Tsvankln Dwnladed 1/31/13 t Redistributin subject t SEG license r cpyright; see Terms f Use at pendence, are subject t significant errrs in transversely istrpic media Deviatins f P-wave NMO velcity frm the csine-fdip dependence are much less prnunced fr factrized VTI media with psitive e - 8 and an increase in vertical velcity with depth, than fr hmgeneus media Therefre, if a medium is nt nly anistrpic, but als has a vertical-velcity gradient, istrpic cnstant-velcity DMO can perfrm better than can be expected frm the results fr hmgeneus anistrpic media In principle, the expressin fr nrmal-mveut velcity derived here can be used fr nly the shrt-spread (hyperblic) prtin f the mveut curve Hwever, analysis f mveut velcity n cnventinal spreads clse t the distance between the CMP and the reflectr shws that the magnitude f anistrpy-induced nnhyperblic mveut fr P-waves is relatively small and tends t decrease at steep dips The frmula fr NMO velcity derived in the paper prvides a basis fr building dip-mveut algrithms in anistrpic media It can als be used t vercme the ambiguity in the inversin f reflectin mveuts fr anistrpic parameters by including dip dependence f mveut velcities in the inversin prcedure ACKNOWLEDGMENTS I am grateful t Ken Lamer fr many helpful discussins, fr use f his ray-tracing cde, and fr his thrugh review f the paper I wuld like t thank the reviewers fr useful cmments and Jhn Andersn (Mbil) fr his insight int the character f DMO perfrmance The supprt fr this wrk was prvided by the members f the Cnsrtium Prject n Seismic Inverse Methds fr Cmplex Structures at the Center fr Wave Phenmena (CWP), Clrad Schl f Mines, and by the United States Department f Energy, Grant Number DE-FG2-89ER1479 (this supprt des nt cnstitute an endrsement by DOE f the views expressed in this paper) REFERENCES Banik, N C, 1984, Velcity anistrpy f shales and depth estimatin in the Nrth Sea Basin: Gephysics, 49, Berryman, J G, 1979, Lng-wave elastic anistrpy in transversely istrpic media: Gephysics, 44, Byun, B, 1982, Seismic parameters fr media with elliptical velcity dependencies: Gephysics, 47, , Seismic parameters fr transversely istrpic media: Gephysics, 49, Hake, H, Helbig, K"t and Mesdag, C S, 1984, Three-term Taylr series fr t 2 - X curves ver layered transversely istrpic grund: Gephys Prsp, 32, Hale, D, and Artley, C, 1993, Squeezing dip mveut fr depthvariable velcity: Gephysics, 58, Hale, D, Hill, N R, and Stefani, J, 1992, Imaging salt with turning seismic waves: Gephysics, 57, Hubral, P, and Krey, T, 198, Interval velcities frm seismic reflectin time measurements: Sc Exp! Gephys Lamer, K, 1993, Dip-mveut errr in transversely istrpic media with linear velcity variatin in depth: Gephysics, 58, Lamer, K, and Chen, J, 1993, Migratin errr in factrized transversely istrpic media with linear velcity variatin with depth: Gephysics, 58, Leary, P C, Crampin, S, and McEvilly, T V, 199, Seismic fracture anistrpy in the Earth's crust: An verview: J Gephys Res, 95, B7, Levin, F K, 1971, Apparent velcity frm dipping interface reflectins: Gephysics, 36, , Reflectin frm a dipping plane-transversely istrpic slid: Gephysics, 55, Sams, M S, Wrthingtn, M H, and Khanshir, M S, 1993, A cmparisn f labratry and field measurements f P-wave anistrpy: Gephys Prsp, 41, 189,26 Thmsen, L, 1986, Weak elastic anistrpy: Gephysics, 51, Tsvankin, I, and Thmsen, L, 1994, Nnhyperblic reflectin mveut in anistrpic media: Gephysics, 59, , Inversin f reflectin traveltimes fr transverse istrpy: Gephysics, (Scheduled fr publicatin in September) Uren, N F, Gardner, G N F, and McDnald, J A, 199a, Dip mveut in anistrpic media: Gephysics, 55, b, Nrmal mveut in anistrpic media: Gephysics, 55, White, J E, 1983, Undergrund sund: Applicatin f sund waves: Elsevier Science Pub!