GEOPHYSICS, VOL. 64, NO. 3 MAY-JUNE 1999); P. 815 819, 9 FIGS., 1 TABLE. Effect of vector attenuation on AVO of offhore reflection J. M. Carcione ABSTRACT Wave tranmitted at the ocean bottom have the characteritic that, for any incidence angle, the attenuation vector i perpendicular to the ocean-bottom interface auming water a lole medium). Such wave are called inhomogeneou; in thi cae, the inhomogeneity angle coincide with the propagation angle. The vector character of thi tranmitted pule affect the amplitude variation with offet AVO) repone of deeper reflector. The analyi of the reflection coefficient i performed for a hale the ocean-bottom ediment) overlying a chalk, auming no lo in the ea floor and lo with an incident homogeneou wave and an incident inhomogeneou wave. Beyond the elatic critical angle the difference are important, mainly for the incident homogeneou wave. Thee difference depend not only on the propertie of the media but alo on the inhomogeneity of the wave. INTRODUCTION In offhore eimic exploration, the wave tranmitted at the ocean bottom have a particular characteritic. Auming that water i lole, their attenuation vector are perpendicular to the ocean-bottom interface. Thi fact affect the amplitude variation with offet AVO) repone of reflection event generated at the lower layer. Wintertein 1987) invetigate the general problem from a kinematic point of view. He analyze how the angle between propagation and maximum attenuation varie in an anelatic layered medium and how that departure from elatic-wave raypath can be large. On the other hand, compreional wave reflection coefficient for different incidence inhomogeneity angle are compared by Krebe 1984). He how that the deviation from the elatic cae can be important at upercritical angle. In thi paper I invetigate the AVO repone for an inhomogeneou wave generated at the ocean bottom and incident at a lower interface eparating two vicoelatic tranverely iotropic TI) media. Unlike the analyi performed by Krebe 1984), the inhomogeneity angle i not contant with offet but i equal to the incidence angle, ince the interface i aumed to be parallel to the ocean bottom ee Figure 1). The interface may eparate two finely layered formation whoe contact plane i parallel to the tratification or two media with intrinic aniotropic propertie, uch a hale and limetone. CONSTITUTIVE EQUATIONS A conitent tre-train model for aniotropic vicoelaticity i given by Carcione 1995). The convention i to denote with ν = 1) and ν = 2) the quai-dilatational and quai-hear deformation, repectively. The complex tiffnee relating tre and train for a 2-D TI medium can be expreed a and p 11 = c 11 1 2 c 11 + c 33 ) + [ 1 2 c 11 + c 33 ) c 55 + c 55 M 2, 1) p 33 = c 33 1 2 c 11 + c 33 ) + [ 1 2 c 11 + c 33 ) c 55 + c 55 M 2, 2) p 13 = c 13 1 2 c 11 + c 33 ) + [ 1 2 c 11 + c 33 ) c 55 + c 55 2 M 2 ), 3) p 55 = c 55 M 2. 4) The elatic contant c IJ, I, J = 1,...,6 are the unrelaxed or high-frequency limit tiffnee; M ν ω) are dimenionle complex moduli decribing the amount of attenuation. In the purely elatic cae ω ) M ν 1. Manucript received by the Editor April 14, 1997; revied manucript received Augut 7, 1998. Oervatorio Geofiico Sperimentale, P.O. Box 2011 Opicina, 34016 Triete, Italy. E-mail: jcarcione@og.triete.it. c 1999 Society of Exploration Geophyicit. All right reerved. 815
816 Carcione GENERATION OF INHOMOGENEOUS WAVES Let u aume that the poitive z-axi point downward. A general olution for the particle velocity field v = v x,v z )i v = iωu exp[iωt x x z z), 5) where x and z are the component of the complex lowne vector, t i the time variable, and U i a complex vector. The real lowne vector and the attenuation vector R [Re x ), Re z ) 6) α [Im x ), Im z ) 7) in general will not point in the ame direction; when they do not, wave are called inhomogeneou. Otherwie, the wave are homogeneou, a in 1-D pace. Figure 1 depict a tranmitted inhomogeneou wave generated at the ocean bottom. Since the attenuation vector of wave propagating in the water layer i zero, vicoelatic Snell law Wennerberg, 1985) implie that the tranmitted attenuation vector i perpendicular to the ocean bottom. Note that the inhomogeneity angle i equal to the propagation angle θ. The complex lowne relation of a vicoelatic TI medium ha the form e.g., Auld, 1990) p11 x 2 + p 55z 2 ρ) p 33 z 2 + p 55x 2 ρ) p 13 + p 55 ) 2 x 2 2 z = 0 8) and ha two olution, correponding to the quai-compreional qp) and quai-hear qs) wave. The complex lowne component below the ocean bottom are x = R in θ, z = R co θ i α ω, 9) where R and α are the magnitude of R and α, repectively. For a given angle θ, R and α can be computed from equation 8); ubtitution of thee quantitie into equation 9) yield the lowne component of the incident inhomogeneou wave. However, thi method require the numerical olution of two fourth-degree polynomial. A impler approach i the following. Firt, aume a given propagation angle θ h for a hypothetical tranmitted homogeneou wave. Then, the complex lowne i = 1 2ρ p55 + p 11 in 2 θ h + p 33 co 2 θ h ± E ) 1/2, 10) where ρ i the denity and E = {[ p 33 p 55 ) co 2 θ h p 11 p 55 ) in 2 θ h 2 + p 13 + p 55 ) 2 in 2 2θ h } 1/2, 11) with the plu ign correponding to the qp-wave and the minu ign to the qs-wave e.g., Carcione, 1997). Next, chooe x for the inhomogeneou wave equal to Re) in θ h a real quantity according to Snell law), ince the projection of α on the interface i zero. Then compute z from equation 8). Finally, compute the incidence propagation angle θ for the inhomogeneou wave a x θ = arcin ). 12) 2 x + [Re z ) 2 In thi way, a vector x, z ), atifying equation 8) and input to the reflection-tranmiion problem, can be obtained for each incidence angle θ. The price we pay for thi implicity i that the ray angle doe not reach 90, but thi i not important ince the offet of interet in exploration geophyic are ufficiently covered. REFLECTION-TRANSMISSION PROBLEM The problem of reflection and refraction at an interface between two TI media whoe repective ymmetry axe are perpendicular to the interface ha been invetigated by Graebner 1992) and Carcione 1997) in the elatic and anelatic cae, repectively. He conidered a homogeneou incident wave and obtained the attribute of the reflected and tranmitted wave uch a, for intance, the energy reflection coefficient, the phae and energy velocitie, the quality factor, and the interference coefficient. To ditinguih between downward- and upward-propagating wave, the lowne relation 8) i olved for z, given the horizontal lowne x. It yield z =± 1 ) 1/2 K 1 pv K1 2 4K 2K 3, 13) 2 FIG. 1. Snell law for a plane wave incident on the oceanbottom interface. The diagram how the continuity of the horizontal component of the complex lowne vector. In the ocean thi vector i real, ince water i aumed to be lole. In the hale layer the attenuation vector i perpendicular to the ocean bottom. where K 1 = ρ 1 + 1 ) + 1 [ p13 p 13 + 2p 55 ) p 11 x 2 p 55 p 33 p 55 p, 33 K 2 = 1 p 33 p11 2 x ρ), K 3 = 2 x ρ p 55
Vector Attenuation and AVO Effect 817 and pvz) 1/2 denote the principal value of the quare root of the complex number z. The ign correpond to +, ) = downward qp-wave, +, +) = downward qs-wave,, ) = upward qp-wave, and, +) = upward qs-wave. Application of welded boundary condition generate the following matrix equation for the reflection and tranmiion coefficient R and T : β P1 β S1 β P2 β S2 R PP β P1 γ P1 γ S1 γ P2 γ S2 R PS Z P1 Z S1 Z P2 Z S2 T PP = γ P1 Z P1. W P1 W S1 W P2 W S2 T PS W P1 14) The upper layer i denoted by the ubcript 1 and the lower layer by the ubcript 2. The ymbol P and S indicate the qp- and qs-wave, repectively. The quantitie β and γ are the horizontal and vertical complex polarization, repectively, given by [ p 55 x 2 β = pv + p 33z 2 ρ 1/2 p 11 x 2 + p 33z 2 + p ) 15) 55 2 x + z 2 2ρ angle, i hown in Figure 2, where E correpond to the elatic cae i.e., elatic hale), H to an incident vicoelatic homogeneou wave, and I to an incident inhomogeneou wave with the characteritic indicated in Figure 1 the chalk i aumed anelatic in the three cae). In the purely elatic cae, i.e., hale and chalk both elatic Wright, 1987), there i a critical angle between 40 and 50. It can be hown that the energy vector of the refracted qp-wave point downward for all incident angle. Thu, there i no critical angle in the trict ene. However, the hape of the E and I curve indicate that a quai-evanecent wave propagate through the interface. Thi character i lot in the H curve. In the near-offet up to 20 ), the three coefficient follow the ame trend and are very imilar each other. The difference with the elatic cae E) i becaue of the anelatic propertie of the hale. Beyond 30 the difference are important, mainly for the incident homogeneou wave. Thi can alo be oberved in the phae, where the H curve ha the oppoite ign with repect to the other curve. A imilar effect i reported by Krebe 1984). Figure 3 repreent the energy velocitie of the reflected qp-wave for the three cae. The variation with offet are mainly due to hale aniotropy and, a before, the difference with the elatic cae E) are from the anelatic propertie of the and [ 1/2, p 11 x 2 γ =±pv + p 55z 2 ρ p 11 x 2 + p 33z 2 + p ) 16) 55 2 x + z 2 2ρ where the plu and minu ign correpond to the qp- and qswave, repectively. Moreover, W = p 55 γ x + β z ) and Z = βp 13 x + γ p 33 z, 17) and the ray angle i where Carcione, 1997). tan ψ = Reβ X + γ W ) Reβ W + γ Z), 18) X = βp 11 x + γ p 13 z 19) RESULTS AND DISCUSSION The material propertie of the incidence and tranmiion media the hale and the chalk, repectively) are given in Table 1, where V IJ = c IJ /ρ. The unrelaxed velocitie are indicated in the table, and attenuation i quantified by the parameter Q ν = ReM ν )/ImM ν ). Wright 1987) calculated the reflection coefficient for the elatic cae, which i obtained in the unrelaxed limit. The comparion between the abolute value of the qp-wave reflection coefficient, together with the correponding phae Table 1. Material propertie. V 11 V 33 V 55 V 13 ρ Rock m/) m/) m/) m/) Q 1 Q 2 g/cm 3 ) Shale 3810 3048 1402 1828 10 5 2.3 Chalk 5029 5029 2621 3414 100 70 2.7 FIG. 2. Comparion between the abolute value of the R PP reflection coefficient together with the correponding phae angle, where E correpond to the elatic cae i.e., elatic hale), H to an incident vicoelatic homogeneou wave, and I to an incident inhomogeneou wave with the characteritic indicated in Figure 1.
818 Carcione hale. The elatic velocitie are higher than the vicoelatic velocitie ince the elatic limit correpond to the high-frequency limit. The attenuation of the reflected qp-wave i hown in Figure 4. For an incident homogeneou wave H), the variation with angle are olely becaue of the aniotropic effect. On the contrary, the variation for the I ) curve are attributable to the inhomogeneou character of the wave. It can be hown that a imilar trend i obtained for an iotropic hale. The interference coefficient, diplayed in Figure 5, are the reult of the interaction of the tre and particle velocity field of the incident and reflected qp-wave. Much of the energy flow i becaue of interference beyond the elatic critical angle. For ome incident angle, the interference coefficient can have the ame magnitude a the reflection coefficient. To complete the reflection problem, Figure 6 8 how the correponding curve for the reflected qs-wave reulting from an incident qp-wave. A can be appreciated, mode converion and anelatic effect are ignificant beyond the elatic critical angle. The value of the hale quality factor in Table 1 correpond to a very unconolidated ea-floor ediment. Typical value for marine ediment can be found in Hamilton 1972), with compreional quality factor of approximately 30 S-wave quality factor are not reported). Let u conider Q 1 = 30 and Q 2 = 10, which are cloe to the value meaured by McDonal et al. 1958) in Pierre Shale. Figure 9 compare the abolute value of the qp-wave reflection coefficient and the correponding phae angle for the three cae illutrated in Figure 2. A can be appreciated, the difference are till important, mainly at upercritical angle. FIG. 5. Comparion between the interference coefficient of the incident homogeneou H) and inhomogeneou I ) wave. The coefficient correpond to the interaction of the tre and particle velocity field of the incident and reflected qp-wave. FIG. 3. Comparion between the energy velocitie of the reflected qp-wave for the three cae indicated in Figure 2. FIG. 4. Comparion between the attenuation magnitude of the reflected qp-wave for the three cae indicated in Figure 2. FIG. 6. Comparion between the abolute value of the R PS reflection coefficient together with the correponding phae angle. The different cae are indicated in Figure 2.
Vector Attenuation and AVO Effect 819 FIG. 7. Comparion between the attenuation magnitude of the reflected qs-wave a the reult of an incident qp-wave. The different cae are indicated in Figure 2. FIG. 9. Comparion of the abolute value of the R PP reflection coefficient and the correponding phae angle for the three cae indicated in Figure 2, with a ea-floor attenuation defined by Q 1 = 30 and Q 2 = 10. FIG. 8. Comparion between the interference coefficient for incident homogeneou H) and inhomogeneou I ) wave. The coefficient correpond to the interaction of the tre and particle velocity field of the incident qp-wave and reflected qs-wave. CONCLUSIONS AVO tudie in the preence of a highly attenuating ocean bottom e.g., unconolidated ediment) hould not be baed on forward model and proceing technique that aume implified rheologie or neglect the vector attenuation character of the eimic pule. Thee propertie affect not only the analyi of the hallow layer but alo the inverion of the deeper reflector. Amplitude and phae difference are ignificant at upercritical angle. Variation of the attenuation depend on both the aniotropic propertie and the inhomogeneity of the wave. Moreover, for certain offet, energy flow produced by interference of tre and particle velocity can be comparable to the energy flux of the reflected wave, an effect that doe not occur in perfect elaticity. The analyi doe not take into account the amplitude variation with angle of the incident inhomogeneou wave generated at the ocean bottom. Thi i an additional effect to conider in the AVO inverion proce. ACKNOWLEDGMENTS I thank the Source Rock project Nork Hydro) for financing the reearch. REFERENCES Auld, B. A., 1990, Acoutic field and wave in olid, 1: Robert E. Krieger, Publ. Co. Carcione, J. M., 1995, Contitutive model and wave equation for linear, vicoelatic, aniotropic media: Geophyic, 60, 537 548. 1997, Reflection and refraction of qp-qs plane wave at a plane boundary between vicoelatic tranverely iotropic media: Geophy. J. Internat., 129, 669 680. Graebner, M., 1992, Plane-wave reflection and tranmiion coefficient for a tranverely iotropic olid: Geophyic, 57, 1512 1519. Hamilton, E., 1972, Compreional wave attenuation in marine ediment: Geophyic, 37, 620 646. Krebe, E. S., 1984, On the reflection and tranmiion of vicoelatic wave Some numerical reult: Geophyic, 49, 1374 1380. McDonal, F. J., Angona, F. A., Mill, R. L., Sengbuh, R. L., Van Notrand, R. G., and White, J. E., 1958, Attenuation of hear and compreional wave in Pierre hale: Geophyic, 23, 421 439. Wennerberg, L., 1985, Snell law for vicoelatic material, Geophy. J. Roy. Atr. Soc., 81, 13 18. Wintertein, D. F., 1987, Vector attenuation: Some implication for plane wave in anelatic layered media: Geophyic, 52, 810 814. Wright, J., 1987, The effect of tranvere iotropy on reflection amplitude veru offet: Geophyic, 52, 564 567.