Variation of P-wave reflectivity with offset and azimuth in anisotropic media

Transcription

1 GEOPHYSICS, VOL. 63, NO.3 (MAY-JUNE 1998); P ,7 FIGS., 4 TABLES. Variation of P-wave reflectivity with offset azimuth in anisotropic media Andreas Ruger* ABSTRACT P-wave amplitudes may be sensitive even to relatively weak anisotropy of rock mass. Recent results on symmetry-plane P-wave reflection coefficients in azimuthally anisotropic media are extended to observations at arbitrary azimuth, large incidence angles, lower symmetry systems. The approximate P-wave reflection coefficient in transversely isotropic media with a horizontal axis of symmetry (HTI) (typical for a system of parallel vertical cracks embedded in an isotropic matrix) shows that the amplitude versus offset (AVO) gradient varies as a function of the squared cosine of the azimuthal angle. This change can be inverted for the symmetry-plane directions a combination of the shear-wave splitting parameter y the anisotropy coefficient 8(v ). The reflection coefficient study is also extended to media of orthorhombic symmetry that are believed to be more realistic models of fractured reservoirs. The study shows the orthorhombic HTI reflection coefficients are very similar the azimuthal variation in the orthorhombic P-wave reflection response is a function of the shear-wave splitting parameter y two anisotropy parameters describing P-wave anisotropy for near-vertical propagation in the symmetry planes. The simple relationships between the reflection amplitudes anisotropic coefficients given here can be regarded as helpful rules of thumb in quickly evaluating the importance of anisotropy in a particular play, integrating results of NMO shear-wave-splitting analyses, planning data acquisition, guiding more advanced numerical amplitude-inversion procedures. INTRODUCTION Fracturing directional horizontal stress fields can cause azimuthal anisotropy in the subsurface. Knowledge of the direction of open fractures, the degree of fracturing, compartmentalization is critical in understing the flow of fluids or gas through the reservoir in making decisions on drilling locations optimization of reservoir productivity. Most upper crustal media many naturally fractured hydrocarbon reservoirs show azimuthal anisotropy of various types strength (Crampin, 1985). The most simple, firstorder model to describe azimuthal anisotropy is transverse isotropy with a horizontal axis of symmetry (HTI) that can, for example, describe a system of parallel penny-shaped vertical cracks embedded in an isotropic matrix (Thomsen, 1995). Any deviation from this model lowers the symmetry of the system. Media of orthorhombic symmetry that are believed to represent more realistic, azimuthally anisotropic models can be caused by a combination of horizontal layering vertically aligned cracks. Likewise, a system of two orthogonal but not necessarily identical crack systems or two identical crack systems at oblique angle lead to orthorhombic symmetry (Winterstein, 199). While seismic signatures of compressional waves in transversely isotropic media with a vertical axis of symmetry (VTI) have been an active study area in the last decade, research on HTI media has been predominantly focused on the propagation of shear waves. Shear waves are very sensitive to the direction amount of fracturing, several algorithms [mainly through analysis of shear-wave birefringence (Alford, 1986; Thomsen, 1988)] are used to extract this information from multicomponent data (Martin Davis, 1987; Michelena, 1995). While much attention has been devoted to the study of shear-wave splitting in HTI models, the dependence of compressional-wave data on the azimuthal anisotropy is much less understood, P-wave r data are rarely used to directly detect characterize fractured zones or "weak spots." ^I omit the qualifiers in quasi-p-wave quasi-s-wave Presented at the 66th Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor August 9, 1996; revised manuscript received June 19, *Formerly Department of Geophysics, Center for Wave Phenomena, Colorado School of Mines, Golden, CO; presently Lmark Graphics, 749 S. Alton Court, Englewood, CO reas@advance.com Society of Exploration Geophysicists. All rights reserved. 935

2 936 Huger Certainly, a method for the direct identification of fracturing caused by vertically aligned cracks, the isotropy plane coincides using P-wave data would be highly beneficial. The exploration with the fracture plane. community has become very sophisticated in the processing The equivalence between HTI VTI media is the key ob acquisition of P-wave data; additionally, compressional servation to derive the polarization vectors, the phase velocidata generally have better quality are cheaper than shear- ties, the approximate exact reflection coefficient in the wave data. symmetry-axis plane: the Christoffel equation that in general Recently, it became better known that P-wave amplitude differs for VTI HTI media is identical for waves propavariation with offset (AVO) signatures are sensitive to fractures gating in the symmetry-axis plane of HTI media vertical or cracks. Mallick Frazer (1991), for example, show on symmetry planes of VTI models. Consequently, all equations synthetic data that the P-wave AVO response from a fractured describing kinematic properties polarizations are identical layer is azimuthally dependent. Similar azimuthal variations in for transversely isotropic media with vertical symmetry axis the P-wave reflection response have been observed on field the symmetry-axis plane of HTI media (Ruger,1997). data by Lynn et al. (1995) Johnson (1995). The observed analogy also leads to the introduction of a Unlike traveltime methods, reflection amplitudes can pro- new set of dimensionless anisotropy parameters similar to vide a detailed measure of local anisotropy (highly resolved in Thomsen's (1986) coefficients. Specifically, HTI media are dedepth/time). To elucidate the relation between the P-wave re- scribed conveniently by the vertical P-wave velocity a, the verflection coefficients the medium parameters, Ruger (1997) tical velocity of the shear wave polarized parallel to the isotropy derived approximate reflection coefficients for the symmetry plane,b(=,8 11 ), three anisotropy parameters 6 (v), E ('' ), planes of HTI media. The reflection coefficient was linearized y (v). Anisotropy coefficients 8 (v), E (v>, y (v) are Thomsen with respect to the relative changes in isotropy parameters parameters defined with respect to vertical in the same way as in across the reflecting boundary new anisotropy parameters VTI media, i.e., they are different from the generic Thomsen similar to Thomsen's (1986) coefficients. Based on his results coefficients defined with respect to the horizontal symmetry a new exact description of NMO velocities in HTI media axis. The set of new parameters naturally evolves from the (Tsvankin, 1997), Ruger Tsvankin (1995) proposed a new analysis of the Christoffel system. Additionally, they represent AVO algorithm to improve the characterization of fractured the effective parameters describing the seismic signatures in reservoirs using P-wave data. HTI media. For example, a is the NMO velocity recovered This paper is a continuation of Ruger's work (1997). Specifi- from seismic surveys aligned with the isotropy plane; 8 (v), on cally, the fracture-detection algorithm based on the symmetry- the other h, determines the difference between the NMO plane P-wave reflection coefficients can be extended to ob- velocity in this plane in the symmetry-axis plane (Tsvankin, servations at arbitrary azimuth, to variations of reflection 1997). coefficients at large incidence angles, to lower symmetry In some contexts, it is also convenient to use fil, the slow systems. While the symmetry-plane coefficients describe the vertical shear-wave velocity, the symmetry-axis velocities magnitude of azimuthal change show how to relate it to V Vso. Table 1 summarizes the coefficients used in this the medium parameters, the newly derived expressions help to paper relates them to the generic Thomsen parameters. apply the fracture detection algorithm for any set of azimuths Also included in this table is Thomsen's parameter y that is characterize the functional type of the azimuthal variation directly related to the y ( v), the y-parameter of the equivalent caused by the different anisotropy parameters. VTI model. The former is chosen in the parameterization of the reflection coefficients below because of its importance in SYMMETRY PLANES OF HTI MEDIA: ANALOGIES the classical shear-wave splitting analysis. AND NOTATION HTI media have two vertical symmetry planes. The plane formed by the symmetry axis the vertical is called the symmetry-axis plane, while the plane perpendicular to the symmetry axis is called the isotropy plane. If the HTI symmetry is APPROXIMATE SCATTERING COEFFICIENTS FOR HTI MEDIA The continuity of displacement traction between elastic layers in welded contact leads to a system of equations that can Table 1. Anisotropy parameters used to study HTI media their relation to the generic Thomsen parameters. The c y representation corresponds to the symmetry axis pointing in the x1-direction. Parameterf in the equation for 61 is given byf = 1 Vsa/V. notation Generic Thomsen notation (exact) (Weak anisotropy) a C331P Vp 1 +2E Vp(1 +E) B c44/p Vso 1 +2y V5(1+Y) N 1 C55/P VSO Vs S(y) (C13 + C55) 2 (C33 C55)2 8 2E(1 + Elf) 2C33(c33 C55) (1 + 2E)(1 + 2E/f) E (V) C11 C33 E 2C33 1+2e E (V) C66 C44 y 2c ' Y Y C44 C66 n_ Y Y 8 2E

3 P-wave Reflectivity in Anisotropic Media 937 be inverted for the reflection transmission coefficients. The forward problem thus consists of solving a system of two, four, or six boundary conditions, depending on the symmetry the number of the wave modes above below a reflecting interface. The inverse problem of estimating medium parameters from the angular changes of the reflection response represents a much more difficult problem. Koefoed (1955) went through the laborious exercise of numerically investigating reflection coefficients for many different sets of elastic, isotropic parameters. Obviously, extending this approach to azimuthally anisotropic media is not feasible. Instead, it is more helpful to derive approximate scattering coefficients (Bortfeld, 1961; Richards Frasier, 1976; Aki Richards, 198; Banik, 1987; Thomsen, 1993; Ursin Haugen, 1996) learn about the influence of the anisotropy parameters on the reflection response. The derivation of P-wave reflection coefficients in azimuthally anisotropic media involves the study of six wave modes generated by the incident P-wave. Probably the first results on this topic were obtained (but not published) by Corrigan (199), who investigated the small-angle response at isotropic/orthorhombic boundaries. Using a Born-scattering approach, he derived the dependence of the initial slope of the reflection coefficient (also denoted as AVO gradient) B on a source-receiver azimuth of the form B = B 1 > + B (1 cos 2, (1) consistent with observations on synthetic data by Mallick Frazer (1991). To verify extend Corrigan's result to interfaces between two HTI media higher angular terms, I used a perturbation technique similar to that applied by Thomsen (1993) for (azimuthally isotropic) VTI media. This approach considers a perturbation from an isotropic background medium requires analytic expressions for P-wave polarization vectors, phase velocities, vertical slownesses as functions of the incidence phase angle i azimuthal phase angle (Figure 1) for weakly anisotropic HTI media. A derivation of the polarization vectors without investigating the eigenvector problem of the HTI Christoffel matrix is shown in Appendix A. It only requires applying the analogy of HTI VTI media some basic geometry to show that the approximate P-wave polarization vector d in HTI media can FIG. 1. The angle between the slowness vector of the incident wave vertical is denoted as i. The azimuthal angle 4' is defined with respect to the symmetry axis pointing in the xl -direction. be expressed through the incidence phase angle azimuthal phase angle as I(i, ) sin i cos d(i, 4') = m(i, 4') sin i sin 4, (2) m(i, 4') cos i where m 1 are given as m(i, 1p) = 1 f sing i cos2 QJ x [811'> + 2(Ewl 6l vl) sin2 i cos2 4] 1(i, ) = 1 + f (1 sin 2 i cos2 ) x [S') + 2(e ') 6 (v)) sine i cos2 ], (3) with f = a 2/(a 2,B 2). Phase velocities in HTI media as a function of the angle with the symmetry axis E(v) S(v ) are given in Tsvankin (1997). Expressed using incidence azimuthal angle, the P- wave phase velocity for weak anisotropy yields Vp (i, ) = a[1 + 8 (v) sin2 i cos2 +(e 'i Swl) sin4 i cos4 ]. (4) Knowledge of the polarizations phase velocities allows us to set up the perturbation equations for the reflection transmission coefficients of the waves scattered at HTI/HTI interfaces with the same orientation of the symmetry axis above below the interface. From the derivation shown in Appendix B, it follows that the compressional plane-wave reflection coefficient has the following dependence on the incidence (polar) azimuthal phase angles: 1 OZ 1 Da 2OG RP(i'Ø=2 Z + 2 a ( 2p a^ G 2 + OSl vi +2( 2^ Dy cost 4 sin2 i + 11 as + OEwi'cos + AS 1vi sine 4 cos 2.1 x sin2 i tan2 i, (5) where Z = pa is the vertical P-wave impedance G = pj 2 denotes the vertical shear modulus. The changes in the elastic parameters are expressed as relative differences. The average velocity a = 1/2(a2 + al) the difference Aa = a2 al can be written as functions of the vertical P-wave velocities al a2 in the upper lower layer, respectively. Corresponding expressions are defined for the shear modulus, the density, the P-wave impedance. This is similar for the anisotropy parameter: Ay,for example, denotes the difference in shear-wave splitting parameters of the reflecting medium the overburden. The simplicity of approximation (5) is striking, especially because the equation is linearized only in the small-contrast parameters anisotropy coefficients not in the incidence angle or its trigonometric functions (i.e, no angular terms have been neglected). Specifically, simple trigonometrical relations describe the anisotropic contribution as a function of azimuth.

4 938 For azimuth = 9, equation (5) reduces to the approximate reflection coefficient for the isotropy plane in HTI media: ( 1{ 2 G 1 Rp(i, n/2) 2 Z + 2 as ( 2a G sin i + 2 as sin 2 i tan2 i. (6) The velocities of waves propagating in the isotropy plane are angle independent, Rp(i, rr/2) has the same form as the isotropic approximation implied in Wright (1986) first published in Thomsen (199). As with the other approximations used in this study, equation (6) is linearized in small, relative differences (a useful assumption in many exploration contexts) is usually accurate enough for incidence angles not too close to the critical angle. Most importantly, it tells us what parameter combinations can be inverted for by investigating the intercept, gradient, higher-angle terms of the reflection coefficient hence forms the basis of conventional (isotropic) AVO analysis. For the second vertical symmetry plane (the symmetry-axis plane) at azimuth q5 =, the linearized reflection coefficient has the following form: = 1 OZ + 1 Au ( 2^ R i, Q 2 (AG 2O P( ) 2 Z 2 & G Y) I + ASw ) sine i + AEw) + 2 \ as x sine i tan2 i. (7) / Roger While P-waves incident in the symmetry-axis plane excite (slow) S-waves with the vertical velocity -, P-waves in the isotropy plane are coupled with the faster S 11 -wave. The difference in the vertical shear-wave velocities can be described by the shear-wave splitting parameter, i.e., ;8 1,B(1 y). That is why the parameter y influences the gradient (i.e., sin 2 i) AVO term in the symmetry-axis plane ( = ) elsewhere outside the isotropy plane. Because > describes the near-vertical P-wave velocity variations in the symmetry-axis plane, it is not surprising that it also enters the gradient term in equation (7). Likewise, the definition of e ( ' ) implies that it should contribute to the reflection coefficient at higher angles i [see the sine i tan2 i term in equation (7)]. Similar observations can be made by investigating the symmetry-plane shear-wave reflection response in azimuthally anisotropic media. Symmetry-plane shear-wave reflection transmission coefficients for HTI orthorhombic media have been used to study the viability of shear-wave AVO (Yardley et al., 1991; Kendall, 1995; Roger, 1998). Before studying equation (5) in more detail, it is instructive first to compare some numerically computed exact reflection coefficients with their linearized approximations. It is not important to achieve a high numerical accuracy [in that case, more complicated approximations such as those shown in Ursin Haugen (1996) should be used] but to see if the simple analytic approximation (5) of an otherwise incomprehensibly complex reflection coefficient helps to quickly analyze the influence of anisotropy on the reflection signature. In particular, it is of interest to study the accuracy of equation (5) for different values of the anisotropy parameters E (v), 8 (v), y. Figure 2 shows the reflection coefficients evaluated at boundaries between.8 C O U_.6 U C D U.4 b).4.2 C) c..4 O_ 4) 3 d).8...'...r. C (D.2 It e Incidence angle Incidence angle FIG. 2. Reflection coefficients for an isotropic layer overlying an HTI medium. Shown are the exact solution (solid lines) the weak elastic, weak anisotropic approximation (dashed) [equation (5)] for azimuths of, 3, 6, 9. Table 2 lists the model parameters.

5 P-wave Reflectivity in Anisotropic Media 939 isotropic HTI media for azimuths of, 3, 6, 9 incidence angles up to 4. The parameters of the models used in this test are listed in Table 2. The first test, shown in Figure 2a, corresponds to a reflecting HTI medium with a 1% shear-wave splitting parameter. The exact solutions (solid lines with decreasing thickness away from the symmetry-axis plane) the approximations (dashed) are in very good agreement for all azimuths. The approximations practically coincide with the exact curves for 3 azimuths with the symmetry axis, while a small deviation is visible for 6 9 azimuths large incidence angles. The second model has a nonzero coefficient 3 (v) in the reflecting medium. Equation (5) correctly predicts the split of the reflection coefficient curves. The slope of the exact curves for 3 azimuths is slightly overcorrected by the approximation leads to deviations for large angles of incidence. Model c Table 2. Models used to test the accuracy of equation (5). The upper medium is isotropic, the lower medium has the following isotropic parameters: compressional vertical velocity a2 = 2.5, faster shear-wave vertical velocity i32 = 1.5 density p2 = 2.7. &a/a AZ/z AG/G 6M E Y Model a Model b Model c Model d in Figure 2 has a nonzero value of parameter EM. According to equation (5), there should be no significant change in the AVO gradient (incidence angles < 2 ), indeed, the exact solution does not show any split in the reflection coefficient curves for low angles of incidence. Note that inaccuracies for 3 azimuths are smaller than for the 9 azimuth. The latter approximation coincides with the classical isotropic linearization of the reflection coefficient that is used successfully in conventional AVO analysis. Therefore, in this case, the accuracy of the anisotropic approximation is higher than that of the isotropic one. Model d in Figure 2 has three nonzero anisotropy parameters, including a shear-wave splitting coefficient of 15%. The accuracy of the approximation is acceptable for small values of incidence angles also shows the correct trend for higher incidence angles. Deviations for large angles of incidence 3 azimuths are caused by the approximation of shear-wave velocity f 1 through $ y hence are caused by the design of the approximation. For a better accuracy at small azimuths, approximation (5) should be rewritten in terms of Pl. FUNCTIONAL TYPE OF THE AZIMUTHAL VARIATION In addition to allowing the study of the reflection coefficient as a function of the incidence angle for several azimuths, equation (5) enables us to fix the incidence angle analyze the reflection coefficient as a function of azimuth. As seen in Figure 3, the approximations are highly accurate for all azimuths.8 C.6 C T_.4 b) C) d) C.4 N U.3 C c.2.1 Azimuth Azimuth FIG. 3. Reflection coefficients for an isotropic layer overlying an HTI medium as a function of the azimuthal phase angle. Shown are the exact solution (solid lines) the weak elastic, weak anisotropic approximation (dashed) [equation (5)] for incidence angles 1, 2, 3, 4. The reflection coefficient curves computed for model c 1 incident angle is invariant with azimuth. Table 2 lists the model parameters.

6 94 incidence angles of 1 2. For higher angles, the approximations increasingly deviate from the exact result but correctly predict the behavior of the azimuthal change. Clearly, the symmetry-plane directions at 9 azimuths coincide with the extrema of the reflection-coefficient curves plotted as a function of azimuth for fixed incidence angle. Study of the azimuthal variation in the reflection coefficients can help in finding the orientation of the natural coordinate system of the subsurface without any prior knowledge of the medium parameters. Moreover, because the locations of the extrema are the same for all incidence angles, analysis of stacked data for several azimuths can reveal the symmetry-plane directions, provided that the stacking is performed with the proper (azimuthally varying) NMO velocity. The curves generated for model c in Figure 3 show several features that are not observed for the other models. First, no azimuthal variation in the reflection coefficient is visible for the 1 incidence angle, the variation is small for the 2 incidence angle. Additionally, the dependence of the reflection coefficient on azimuth for large angles of incidence is different from the variations observed for the other models. As predicted by the cos4 term in equation (5), a stronger azimuthal change of the reflection coefficient away from the symmetry-axis plane is visible as compared with the azimuthal change close to the isotropy plane. The AVO gradient term The behavior of Rp(i, ) at small incidence angles is described by the AVO gradient B composed of the azimuthally invariant part B' 5 the anisotropic contribution Ba"` multiplied with the squared cosine of the azimuthal angle with the symmetry axis. If the symmetry-axis orientation is unknown, should be formally expressed by the difference between the azimuthal direction Ok of the kth observed azimuth the direction of the symmetry-axis plane t. The AVO gradient measured at azimuth Ok thus can be written as B(Ok) = Biso + Bani COS2 (Wk Osym)' (8) Roger symmetry-axis direction is enough to identify the symmetryaxis direction unambiguously. The dashed lines in Figure 4 show the approximate AVO gradient from equation (8). The "exact" gradient (solid line) is computed by averaging the slope of the exact reflection coefficient over to 2 incidence angles. The curves shown in Figure 4 are evaluated for the models shown in Table 2. The extrema denoting the symmetry-plane directions can be picked easily, even though more information is needed to distinguish between the symmetry-axis the isotropy planes. Table 3 compares the magnitude of the observed AVO-gradient change with the value of Bani that would be obtained using equation (5). If B(çbk) is known for several values of cb k the AVO gradient does not change sign, an alternative graphical interpretation is possible. Fitting an elliptical curve to the azimuthal variations in AVO gradients can help to determine the symmetry-plane directions the magnitude of the azimuthal change. The modulus of B(O k) can be assigned to the length of a radius vector at different azimuths k. The tip of this vector then delineates a curve that closely resembles an ellipse with the semiaxes aligned with the symmetry-plane directions. The semiaxes have the lengths (B`s ) (B's + B"') point toward the symmetry directions of the medium. An analogous elliptical dependence can be observed for azimuthal changes in NMO velocity. The azimuthal variation of NMO velocity always [except for certain complex areas such as those where common-midpoint (CMP) reflection time decreases with offset] represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium the direction of the reflector normal (Grechka Tsvankin, 1998). There are several alternative ways to represent the azimuthal changes in P-wave reflectivity. For example, the AVO gradient as predicted by equation (5) can be shown to be consistent with Corrigan's (199) approximation [equation (1)] with 4 a Bt > = 1/21 a ( 2^ AG O ag ) + 1 2AS (V) Y) with Bts = 1/2 as f 2a \ 2 (9) Banr = 1/2 Ab (v) +2() Dy. (1) Equation (8) is nonlinear with three unknowns: B's B n' Osy,,,. If the direction of the symmetry axis is known (for example, from S-wave dat, the equation becomes linear two independent measurements suffice to solve for B's B. In multiazimuth surveys, many more azimuths should be sampled, a least-squares approach can be used to find the optimum values of the unknowns in equation (8). Because of the nonlinearity of equation (8), the solution will not be unique will yield two possible directions of the symmetry axis orthogonal to each other. However, a simple rough estimate of the sign of B'' or an a priori knowledge of the approximate 2 Bltl = 1/2 1 -) Ay + 1/28 1 l (11) a In exploration situations, it may happen that fit, the fractureperpendicular shear velocity, is known instead of fi. To achieve a higher accuracy, the AVO gradient in HTI media should then be rewritten through,8 1. Table 3. Magnitude of the AVO-gradient variation obtained from the exact reflection coefficient the value predicted by equation (5) for the models given in Table 2. Banff Banff observed approx Model a Model b.57.5 Model c 5.4 x 1-3 Model d

7 P-wave Reflectivity in Anisotropic Media 941 AVO-gradient inversion in HTI media To eliminate the "isotropic" quantities in equation (9) in the special case of inverting AVO measurements in the isotropy symmetry-axis planes, it is best to use the difference between the symmetry-plane AVO gradients rather than inverting the gradients individually. Keeping just the two lowestorder terms assuming the incidence layer is isotropic, we find z B(4sym) B(q5sym+n/2) (2a Y (12) Equation (12) shows that the difference in the gradient depends on just two anisotropic parameters: the shear-wave splitting parameter the coefficient 5 (v). For,8/& 1/2, the difference between the gradient terms becomes y z + 1/26Z ). This means the weighting factor for the shear-wave splitting parameter is twice as large as that for SZ ). As shown by Tsvankin (1997), the parameter S (V) can be obtained from P-wave NMO velocities. If SZ ) has been determined, the difference in the gradients can be inverted for the shear-wave splitting parameter, which (for penny-shaped cracks) is close to the crack density, given an approximate value of the ratio 2,B/&. Also, the crack density can be obtained directly from the difference [B(øsy ) B(Osym + n/2)] in the important special case of the vanishing parameter E (E = ), typical for fractured, fluid-filled coal layers. If E =, then S(') y (Tsvankin, 1997), the crack density remains the only anisotropic parameter in equation (12). A modeling example to discuss the difference of reflection responses between rocks with fluid gas-filled cracks is shown in RUger Tsvankin (1997). One of the important lessons learned in these studies is that the shear-wave splitting parameter y coefficient 8 (" ) cannot be determined separately because both parameters influence the azimuthally dependent AVO gradient in the same way. On the other h, coefficient S (v) varies significantly as a function of crack filling, suggesting that azimuthal AVO has the potential to discriminate between fluid-filled dry cracks. Azimuthal variation of the higher-angle term For incidence angles i > 2, the AVO gradient term the higher-angle term will influence the reflection coefficient. For azimuths close to the symmetry axis large incidence angles, the coefficient c (v) will have a significant impact on the azimuthal variations. Here, we will discuss the interesting case of reflection coefficients with small or even nonexistent AVO gradient variations, i.e., the term [ASi' ) + 2(2$/ 2 Ay] being negligibly small. In this case (which I found typical for rocks of HTI symmetry with dry cracks), equation (5) still predicts an azimuthal change in the reflection coefficient at large angles of incidence for nonzero values of AE( ) A8 ( ' ). Note that both parameters have a different influence on the azimuthal change of the reflection coefficient. The value ASS ) has its major impact on.5 b) -.1 ctt ) Q C) d) 4- C N_ < Azimuth Azimuth FIG. 4. AVO gradients for an isotropic layer overlying an HTI medium as a function of azimuthal phase angle. Shown are the AVO gradients extracted from the exact reflection coefficients (solid lines) the weak elastic, weak anisotropic approximation (dashed) based on equation (8). Table 2 lists the model parameters.

8 942 Huger the large-angle reflection response at an azimuth of 45. The pa- words, all conclusions about the limited equivalence of VTI rameter that does not have any influence on the AVO-gradient HTI media remain valid for symmetry-plane propagavariation, (v), is primarily responsible for the azimuthal vari- tion in orthorhombic models. This leads to the introduction ation of the reflection coefficient at large angles of incidence of new dimensionless anisotropy parameters defined similarly azimuths close to the symmetry-axis direction. to the well-known Thomsen's (1986) coefficients e, 8, y (Tsvankin, REFLECTION COEFFICIENTS FOR ORTHORHOMBIC MEDIA 1) 2 the VTI parameter 6 in the symmetry plane [x 1, x3 ] normal to the x2-axis (close to the fractional difference between the P-wave velocities in the x 1- x3 - directions); Horizontal transverse isotropy is a useful model for studying the first-order influence of azimuthal anisotropy. More realistic models can be described by the orthorhombic symmetry system. Wave propagation in orthorhombic media is rather complex; analysis of wavefronts slowness surfaces in orthorhombic media can, for example, be found in Musgrave (197), Helbig (1994), Schoenberg Helbig (1997). Although the following investigation is valid for orthorhombic media of any origin, it is instructive to examine an orthorhombic model resulting from the combination of a VTI background medium with a system of aligned vertical cracks (Figure 5). This orthorhombic model (as in any other orthorhombic model with a horizontal symmetry plane) has two vertical symmetry planes: [Xi, x3] [x2, x3]). Following, I obtain approximations for reflection coefficients of P-waves incident in the two vertical symmetry planes of orthorhombic media to gain a better understing of the magnitude of azimuthal change of the AVO gradients relate this to the anisotropy in the subsurface. 6(2) C11 C33 (13) 2 c33 2) 8 2 the VTI parameter 8 in the [xl, x 3 ]-plane (responsible for near-vertical P-wave velocity variations; also influences SV-wave velocity anisotropy); Effective parameters for orthorhombic media A rigorous mathematical analysis of the symmetry-plane reflection responses is possible by studying the corresponding Christoffel equations for orthorhombic media in the same way as performed for HTI media. Study of the symmetryplane Christoffel equations helps to relate wave propagation in the [xi, x3] [x2, x3] planes of orthorhombic models to wave propagation in VTI media. This analysis shows that symmetry-plane propagation in orthorhombic media can be completely described by known VTI equations. In other 8 (2) = (C13 + C55)2 (C33 C55)2. (14) ZC33(C33 C55) 3) y (2) the VTI parameter y in the [xi, x3]-plane (close to the fractional difference between the SH-wave velocities in the x 1 - x3-directions); C66 C44 (15) 2 c44 4) E'> the VTI parameter E in the [X2, x3]-plane; (1) C22 C33 (16) ) 6' ) the VTI parameter 6 in the [X2, x3 ]-plane; 8 (1) _ (C23 + C44) 2 (C33 C44) 2 (17) y 2 C33(C33 C44) 6) y ( 1) the VTI parameter y in the [X2, x3]-plane; (l)c55 (18) 255 7) 5 (3) the VTI parameter 6 in the [xi, x21-plane (xi plays the role of the symmetry axis); 6 (3) _ (C12 + C66)2 (Cu1 C66)2 (19) 2 C11(cii C66) These seven dimensionless parameters, together with the following vertical velocities, can replace the nine independent stiffness components of the orthorhombic model. 8) a the vertical velocity of the P-wave; am!. (2) A 9),B the vertical velocity of the S-wave polarized in the x2-direction; f3 = p4, (21) As summarized in Table 4, both vertical horizontal transverse isotropy can be considered as degenerated special cases FIG. 5. Sketch of an orthorhombic model created by combining horizontal layering with a system of parallel vertical cracks. of orthorhombic media. An orthorhombic medium reduces Two vertical symmetry planes are determined by the crack oni- to the VTI model if the properties in all vertical planes are entation. identical the velocity of each mode in the [x 1, x2]-plane is

9 constant (although the velocities of the two S-waves generally differ from one another). Another special case is transverse isotropy with a horizontal axis of symmetry. If the symmetry axis is oriented along the x1-direction, the parameters E (2), 6 (2), y (2) coincide with the coefficients c (v), 6 (v), y (" ) introduced earlier. P-wave Reflectivity in Anisotropic Media 943 second vertical shear-wave velocity,b = c p the shearwave splitting parameter y = ( C44 C55)/2c55: [x1, x3] = 1 OZ 1 Da 2# 2 A G a / \ G 2 Y 1 P (^) 2 Z + 2 a ( P-wave reflections in the symmetry planes Seismic signatures in the [Xi, x3]-plane of media with orthorhombic symmetry can be evaluated by means of already known VTI equations. Published VTI phase-velocity equations in c;1 -notation, for example, exactly express phase velocity in symmetry planes of orthorhombic media (with appropriate substitutions in the [X2, x3]-plane; see details following). Additionally, reflection coefficients in the symmetry planes of orthorhombic media are the same as in VTI media if the upper lower media have the same orientations of the symmetryplanes. To compute exact symmetry-plane reflection coefficients to better underst their azimuthal variations, it is sufficient to use Graebner's (1992) VTI reflection-coefficient algorithm. Also, the approximate VTI reflection coefficients can be used in the symmetry planes: following the simple recipe of replacing S E with 6 (2) E(2) yields the approximate P-wave reflection coefficient in the [xi, x3]-symmetry-plane of orthorhombic media: R [x1 x3l (i) = 1 OZ P2 Z 2^' as AG + OSi2isine i ( l G + 1( as + DE(2)) sine i tan2 i, (22) with J3 1 = c551p. At the expense of reduced numerical accuracy, equation (22) can also be written as a function of the Table 4. Seven anisotropy parameters (plus two vertical velocities) completely describe orthorhombic anisotropy. Their relationship to the generic (VTI) Thomsen coefficients to HTI parameters S«e(V) are shown in the second third column. The expressions for 6(3) y (2) are approximate, the other relations are valid for any strength of anisotropy. Orthorhombic VTI HTI 5(2) = (c13 + C55) 2 (c33 c55)2 S S(r) 2 C33(C33 C55) 5(1) = (C23 + C 44)2 (C33 C44)2S 2 C33(C33 C44) 3) (C12 + C66) 2 (C11 C66)2 6(v) 2,W ) 2cii(cii c66) 11 C33^(V) 2 C33 22 C33 2 C33 2) = C66 C44Y Y 2C44 31(1) = C66 C55 y + OS (2) sing i DE (2^ \ as / x sin2 i tan2 i. (23) Basically the same observations can be made by comparing the Christoffel equations in the [X2, x 3]-plane of orthorhombic media with those for VTI media. Graebner's (1992) algorithm for VTI systems also can be used in this case, but one must make the substitutions C11 + C22, C55 -+ C44, C13 + C23 Thus, the [x2, xi]-plane of orthorhombic media can be replaced by a second equivalent VTI medium with E = c (l) S = 3 (1) [equations (16) (17)]. Substituting 8 (1) E^' ) into the approximate VTI reflection coefficient then yields the approximate reflection coefficient for the [x2, x3] symmetry plane in orthorhombic media: R [x2,x3](i ) _ 1 OZ P2 Z as (2^ G + [^8 (1) sing i + 1( Act + DE ( ) sine i tan2 i. (24) The results shown here apply to interfaces between isotropic, VTI, HTI, orthorhombic media, provided their vertical symmetry planes are aligned. The same approach can be used to derive the shear-wave reflection coefficients in the vertical symmetry planes of orthorhombic media; these results, together with all remaining pure- converted-mode scattering coefficients, are shown in Ruger (1998). Azimuthal variation of orthorhombic reflection coefficients The preceding discussion shows the approximate reflection coefficients in the two vertical symmetry planes of orthorhombic media are essentially identical to the reflection coefficients in the HTI symmetry-axis plane. This result makes it possible to use the difference in the P-wave reflection coefficients in the two vertical symmetry planes of orthorhombic media to invert for the shear-wave splitting parameter. To eliminate the "isotropic quantities" in equations (23) (24), I follow the approach suggested for HTI media find the difference between the two symmetry-plane coefficients: RP1X3I R1x2,x3] _ 2 2# Dy + 1 (Ob (2) AS (1) ( Ct 2 ) x sing i 2 {DE (2) AE 11) } sine i tang i. (25) For an isotropic overburden (yl = 6 = 512) = ), the difference in the AVO gradient depends on just three anisotropic parameters: the shear-wave splitting parameter y2 two new

10 944 parameters, SZl) S(Z), describing the anisotropy for nearvertical P-wave propagation in the symmetry planes. Tsvankin (1996 shows it is possible to obtain 8 (1) 5 (2) from shortspread P-wave moveout velocity in the vertical symmetry planes, provided an estimate of the vertical velocity is available (for example, from vertical velocities measured in a VSP experiment). Thus, we can get y by combining NMO AVO analysis. Note that for orthorhombic models with a single crack system, y gives an estimate of the crack density. The approximate (low-angle) reflection coefficient for an isotropic overburden a reflecting orthorhombic layer has been derived for arbitrary azimuth by Corrigan (199). Represented in the new parameterization, his solution can be generalized to the following form of the approximate low-angle reflection coefficient at arbitrary azimuth: 1 OZ 1 Da 2 z AG Rr(i'^)-2 Z + 2 a ( G 2 + A6 (2) +2( 2 ) Ay cos2 ^ + AS (1) sinesine i. (26) DISCUSSION AND CONCLUSIONS Analysis of reflection coefficients in azimuthally anisotropic media shows that the magnitude the functional type of the azimuthal variation of P-wave reflectivity can be inverted for the anisotropy parameters the symmetry-plane directions of the subsurface. The examples presented in this paper show that the derived approximate reflection coefficients describe the exact reflection response with acceptable accuracy. However, rather than achieving a high numerical accuracy, the new approximations most importantly create simple dependencies that interpreters can use to quickly evaluate the magnitude of anisotropy in a particular play. They describe the dependence of the AVO response on the individual anisotropy parameters show how independent data (such as those obtained by shear-wavesplitting or NMO analysis) can be integrated in the inversion procedure. Moreover, the approximations establish a physical foundation for (exact) numerical inversion algorithms. Evidently, there is little hope to carry out "blind" inversion of the AVO response for subsurface parameters. The linearized reflection coefficients given here indicate which parameters or parameter combinations can be inverted for at small or large angles of incidence. These approximations also can serve as a guide for a more efficient accurate inversion, e.g., in the iterative adjustment of the unknown parameters in the forward modeling process. To obtain a reasonably simple representation of the otherwise incomprehensibly complex reflection coefficients, a perturbation approach has been used in the derivation. The main assumption in this reflection-coefficient study small jumps in the elastic parameters across the reflecting interface weak anisotropy are geologically reasonable prove useful in many exploration contexts. Although the approximations may help to detect pronounced anomalies, they fail in any quantitative analysis at interfaces with a large contrast in the elastic Roger parameters strong anisotropy. The HTI approximations will also be inaccurate if the natural coordinate systems of the incidence reflected layer differ significantly in orientation, i.e., when the axis of symmetry changes abruptly across the boundary. Within the limits of these assumptions, the approximate P-wave reflection coefficient for HTI interfaces predicts a change of the AVO gradient as a function of the squared cosine of azimuth. This change can be inverted for the symmetryplane directions a combination of the shear-wave splitting parameter y the anisotropy coefficient SM. Coefficients y SM cause the same functional form of the azimuthal variation in the AVO gradient cannot be extracted individually. The two symmetry-plane directions can even be extracted in situations where no azimuthal AVO-gradient change can be observed. In this case, the azimuthal variation at large incidence angles is primarily dependent on the anisotropy coefficient E (V) (closely related to Thomsen's parameter E) provides enough information to detect the natural coordinate system of the subsurface. Certainly, realistic models of fractured reservoirs deviate from simple HTI media, it is interesting whether the reflection response in media of lower symmetry differs significantly from that in the HTI case, This study shows that although wave propagation is significantly more complex in orthorhombic media, the approximate reflection coefficient basically has the same form as in HTI models. The main observation of the study of reflection coefficients in orthorhombic media is that the difference between the P-wave reflection response in the vertical symmetry planes is a function of the shear-wave splitting parameter y two new parameters describing the anisotropy for near-vertical P-wave propagation in the symmetry planes. The practical implementation of the newly proposed AVO inversion faces challenges similar to or even greater than conventional AVO analysis. Prerequisite for any useful AVO-withazimuth investigation is a proper processing sequence that preserves azimuthally varying amplitude signatures. Moreover, a profound understing of lateral inhomogeneities is required. Wave propagation in an anisotropic overburden has a significant impact on the amplitude variations with offset azimuth (Tsvankin, 1995; Roger Tsvankin, 1995) need to be included in any meaningful AVO analysis. Other difficulties include thin layering, curved reflectors, or dipping symmetry axes. These issues are not within the scope of this investigation; however, they should be addressed in future research on AVO for fractured reservoirs. ACKNOWLEDGMENTS Thanks to Ilya Tsvankin, Ken Lamer, Vladimir Grechka, James Simmons, Jr., Bjorn Rommel for reviewing this manuscript. Dennis Corrigan was kind enough to share his programs, derivations, time. Jack Cohen's Mathematica packages were very helpful in the derivation of the linearized reflection coefficient. Thanks also to Leon Thomsen for pointing out the original author of approximation (6) to Geir Haugen for useful discussions. The support for this work was provided by the members of the Consortium Project on Seismic Inverse Methods for Complex Structures at CWP, Colorado School of Mines, by the U.S. Department of Energy (Velocity Analysis, Parameter Estimation, Constraints on Lithology for Transversely Isotropic Sediments project within the framework

11 P-wave Reflectivity in Anisotropic Media 945 of the Advanced Computational Technology Initiative). Partial funding for this project has also been provided by a scholarship from Phillips Petroleum Co. by a grant from Coleman Energy & Environmental Systems-Blackhawk Geoscience Division. REFERENCES Aki, K., Richards, P. G., 198, Quantitative seismology: Theory methods, 1: W. H. Freeman & Co. Alford, R. M., 1986, Shear data in the presence of azimuthal anisotropy: 56th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Abstracts, Banik, N. C., 1987, An effective anisotropy parameter in transversely isotropic media: Geophysics, 52, Bortfeld, R., 1961, Approximation to the reflection coefficients of plane longitudinal transverse waves: Geophys. Prosp., 9, Corrigan, D., 199, The effect of azimuthal anisotropy on the variation of reflectivity with offset: Workshop on Seismic Anisotropy: Soc. Expl. Geophys. 4IWSA, Crampin, S., 1985, Evidence for aligned cracks in the earth's crust: First Break, 3, Graebner, M., 1992, Plane-wave reflection transmission coefficients for a transversely isotropic solid (short note): Geophysics, 57, Grechka, V, Tsvankin, I., 1998, 3-D description of normal moveout in anisotropic media: Geophysics, 63, Helbig, K., 1994, Hbook of Geophysical Exploration, 22: Elsevier Science Publ. Co., Inc. Johnson, W. E., 1995, Direct detection of gas in pre-tertiary sediments?: The Leading Edge, 14, No. 2, Kendall, R. R., 1995, Modeling interpreting shear-waves fracture anisotropy in south-central Wyoming: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Abstracts, Koefoed,.,1955, On the effect of Poisson's ratio of rock strata on the reflection coefficients of plane waves: Geophys. Prosp., 3, Lynn, H. B., Bates, C. R., Simon, K. M., van Dok, R., 1995, The effects of azimuthal anisotropy in P-wave 3-D seismic: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Abstracts, Mallick, S., Frazer, L. N., 1991, Reflection/transmission coefficients azimuthal anisotropy in marine studies: Geophys. J. Int., 15, Martin, M. A., Davis, T. L., 1987, Shear-wave birefringence-a new tool for evaluating fractured reservoirs: The Leading Edge, 6, no. 1, Michelena, R. J., 1995, Quantifying errors in fracture orientation estimated from surface P-S converted waves: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Abstracts, Musgrave, M. J. P.,197, Crystal acoustics: Holden Day. Richards, P. G., Frasier, C. W., 1976, Scattering of elastic waves from depth-dependent inhomogeneities: Geophysics, 41, Rommel, B., 1994, Approximate polarization of plane waves in a medium having weak transverse isotropy: Geophysics, 59, Ruger, A., 1997, P-wave reflection coefficients for transversely isotropic models with vertical horizontal axis of symmetry: Geophysics, 62, , Analytic insight into shear-wave AVO for fractured reservoirs: Proceedings of 7IWSA, special SEG volume on seismic anisotropy, Internat. Workshop on Seismic Anisotropy. Ruger, A., Tsvankin, I., 1995, Azimuthal variation of AVO response for fractured reservoirs: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Exped Abstracts, , Using AVO for fracture detection: analytic basis practical solutions: The Leading Edge, Vol. 16, No. 1, Schoenberg, M., Helbig, K., 1997, Orthorhombic media: Modeling elastic wave behavior in a vertically fractured earth: Geophysics, 62, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, , Reflection seismology over azimuthally anisotropic media: Geophysics, 53, , Poisson was not a geophysicist: The Leading Edge, 9, no. 12, , Weak anisotropic reflections, in Castagna, J., Backus, M., Eds., Offset dependent reflectivity: Soc. Expl. Geophys. 1995, Elastic anisotropy due to aligned cracks in porous rock: Geophys. Prosp., 43, Tsvankin, I., 1995, Body-wave radiation patterns AVO in transversely isotropic media: Geophysics, 6, a, Effective parameters reflection seismic signatures for orthorhombic anisotropy: Center for Wave Phenomena report CWP b, P-wave signatures notation for transversely isotropic media: An overview: Geophysics, 61, , Reflection moveout parameter estimation for horizontal transverse isotropy: Geophysics, 62, Ursin, B., Haugen, G., 1996, Weak-contrast approximation of the elastic scattering matrix in anisotropic media: Reprinted from Pure Appl. Geophys., 148, No. 3/4, in Seismic waves in latterally inhomogeneous media, Part II: Birkhauser. Winterstein, D., 199, Velocity anisotropy terminology for geophysicists: Geophysics, 55, Wright, J., 1986, Reflection coefficients at pore-fluid contracts as a function of offset (short note): Geophysics, 51, Yardley, G. S., Graham, G., Crampin, S., 1991, Viability of shearwave amplitude versus offset studies in anisotropic media: Geophys. J. Int.,17, APPENDIX A POLARIZATION VECTORS IN HTI MEDIA One of the necessary elements in finding reflection transmission coefficients in anisotropic media is the derivation of polarization vectors. Equations describing the direction of compressional-wave polarizations in VTI media have been published by Rommel (1994) Tsvankin (1996b). Denoting the phase polarization angles with the symmetry axis as 9 y, respectively, one can show that cos y = (1 - f sin2 8[6 + 2(E - 8) sin2 6]) cos 9, sin y = (1 + f cos2 8[8 + 2(E - 3) sin2 8]) sin 6, (A-1) with f = a 2/(a2 - $2). Polarization vectors in HTI media can be derived using the analogy between VTI HTI media. First, replace Thomsen's (1986) parameters E l with Ew> S(v), the parameters of the equivalent VTI model. These substitutions yield the polarization vector for quasi-p-waves in the vertical plane containing the symmetry axis of HTI media, as seen, provides sufficient information to determine the polarization for P-waves with any incidence azimuthal angle. Equation (A-1), expressed in terms of angles y' ' with respect to the horizontal symmetry axis in HTI models, yields sin y' = (1 - f cos 2 o'[8 (11) +2(E (V) - 8 (V )) cost 8']) sin B' = m(8') sin 8' cos y' = (1 + f sin2 B'[Slvl + 2(E ( ^') - S (V)) cost 8']) cos B' = (8') cos 8', (A-2) where f = a 2/(a 2 - /3 2 ) as in equation (A-1) but a /3 denote the vertical compressional the vertical fractureparallel shear-wave velocity, respectively. Physical properties in transversely isotropic media are invariant for a fixed angle with the symmetry axis; hence, it is

12 946 Auger perfectly valid to generalize this equation allow the polarization vector to rotate about the horizontal axis. The next step in the derivation of the polarization vectors is to relate angles y' ' with the symmetry axis to their corresponding incidence angle azimuth. From simple geometry (Figure A-1), we can find a relation between the cosine of an arbitrary angle a with the horizontal axis its associated incidence polar angle i azimuth : cos a = sin i cos. (A-3) The Cartesian components of the unit polarization vector d = (d1, d2, d3)t the unit wave vector n = (n l, ny, n3)' normal to the wavefront can be written as i cos l^ ( sin id Cos d )( sin d = sin id sin 4d n = sin i sin (A-4) COSid COST ) Here, angles of the polarization vector the phase vector with the vertical axis are called i d i, respectively; similar notation applies to the azimuthal angles. Using equations (A-2) (A-3), one can relate the first components to obtain the simple expression d1 = n 1. Corresponding equations for the two remaining components can be found using geometrical observations. First note that in transversely isotropic media, the polarization vector n lies in the plane formed by the symmetry axis the phase vector. The two triangles shown in Figure A-2 thus have the same angle i, we find d2=a'b'= sin y'cos /r _ sin y' n2 sin B' = m n2 d3 = B'C' = sin y' sin J sin y' _ 3 sin 6' = m n3. (A-5) Summarizing, we see that the P-wave polarization vector in HTI media can be expressed as a function of the incidence azimuthal phase angle as 2(i, ) sin i cos d(i, ) = m(i, ) sin i sin (A-6) m(i, O) cos i where m [equation (A-2)] are given as m(i, ) = 1 f sin g i cos2 x [B (V) +2(e ('' S(v)) sinz i cosz OIl = 1 + f(1 sin2 i cos2 ) x [El ul + 2(c (V) 6 (v) ) sinz i cosz ]. Although the shear-wave polarizations are not necessary for this study, they can be derived using equation (A-6). Note that quasi-compressional shear-wave polarizations are mutually orthogonal, with the S 11 wave being a truly transverse mode (dsij I ii), with the polarization vector confined to the isotropy plane. FIG. A-1. Relation between an arbitrary angle a with the horizontal axis its associated incidence angle i azimuth. FIG. A-2. Phase vector fi polarization vector d lie in a common plane with the symmetry axis x 1. Angle,/rr is identical for triangles IABC DA'B'C'.

13 P-wave Reflectivity in Anisotropic Media 947 APPENDIX B DERIVATION OF AZIMUTHALLY VARYING APPROXIMATE REFLECTION COEFFICIENTS IN HTI MEDIA Here, the derivation of approximate reflection transmission coefficients for interfaces between VTI layers (Thomsen 1993) is extended to azimuthally anisotropic HTI media. The approximations are derived for a boundary between two weakly anisotropic media with the same symmetry-axis direction for small discontinuities in elastic properties across the boundary. Due to the considerable size of the matrices vectors considered in this derivation, it is impossible to state explicitly all quantities used in this appendix. However, I outline each step of the derivation provide all information necessary to reproduce the final result. The main idea of the derivation is to replace the exact boundary value problem AR = b, (B-1) where A is the boundary equation matrix formed by the scattered (reflected transmitted) wave types, R is the vector of reflection transmission coefficients, b is composed of the contribution of the incident wave to the boundary conditions, with a perturbation from the corresponding expression for the interface between two identical, isotropic, homogeneous media. Denoting the unperturbed quantities with the superscript u, Thomsen (1993) showed that or = (Au) -1 (ob - LARD). (B-2) Here, the operator A is defined as A = d; (3/ad1 ) with the vector d; of small deviations from the two identical homogeneous, isotropic media given by T di = y Da ^ a p 1 3i ' S2 Ei v> > E2, Y1 Y2) (B -3) The 6 x 6 matrix A" needs to be inverted analytically. This can be achieved by relating A" to the eigenvectors of the transformed wave equation (Ursin Haugen, 1996). To find the elements Ab AA, it is necessary to use the weak anisotropy approximations to polarization vectors phase velocities shown in equations (3) (4), while the approximate refracted angles can be derived using Snell's law. The element of AR of most interest is the P-wave reflection coefficient shown in equation (5). The sign notation is hereby chosen according to Aki Richards (198).