Traveltime approximations for inhomogeneous transversely isotropic media with a horizontal symmetry axis

Transcription

1 Geophysical Prospecting, 2013, 61, doi: /j x Traveltime approximations for inhomogeneous transversely isotropic media with a horizontal symmetry axis Tariq Alkhalifah Physical Sciences and Engineering King Abdullah University of Science and Technology Mail box # 1280 Thuwal Saudi Arabia Received November 2011, revision accepted January 2012 ABSTRACT Traveltime information is crucial for parameter estimation, especially if the medium is described by a set of anisotropy parameters. We can efficiently estimate these parameters if we are able to relate them analytically to traveltimes, which is generally hard to do in inhomogeneous media. I develop traveltime approximations for transversely isotropic media with a horizontal symmetry axis (HTI as simplified and even linear functions of the anisotropy parameters. This is accomplished by perturbing the solution of the HTI eikonal equation with respect to the anellipticity parameter, η and the azimuth of the symmetry axis (typically associated with the fracture direction from a generally inhomogeneous, elliptically anisotropic background medium. Such a perturbation is convenient since the elliptically anisotropic information might be obtained from well velocities in HTI media. Thus, we scan for only η and the symmetry-axis azimuth. The resulting approximations can provide a reasonably accurate analytical description of the traveltime in a homogenous background compared to other published moveout equations. They also help extend the inhomogenous background isotropic or elliptically anisotropic models to an HTI one with a smoothly variable η and symmetry-axis azimuth. Key words: Azimuth, HTI media, Isotropic. INTRODUCTION In addition to transversely isotropic media with a vertical symmetry axis (VTI or tilted one (TTI, wave propagation in the Earth s subsurface encounters media with horizontal symmetry axis TI (HTI usually attributable to parallel vertical fractures. Even if a VTI component is present in such a medium, which makes the symmetry orthorhombic, HTI can predict many of the azimuth variation features, including the strength of the azimuthal anisotropy (Grechka and Tsvankin Developing simple traveltime formulations for such a model can help in many applications, including traveltime tomography and integral-based Kirchhoff imaging. For homogeneous or smoothly varying HTI media, traveltime approximations, especially those for moveout descrip- tariq.alkhalifah@kaust.edu.sa tion, are based on Taylor s series type expansions around the zero-offset and thus, despite the enhancements applied to handle far offsets, the accuracy tends to degrade with offset (Al-Dajani and Tsvankin Nevertheless, these traveltime approximations have been instrumental in estimating anisotropy parameters of HTI models and the corresponding fracture parameters (Contreras, Grechka and Tsvankin 1999; Bakulin, Grechka and Tsvankin For inhomogeneous media, traveltimes are conventionally evaluated by solving a nonlinear partial differential equation (PDE, known as the eikonal equation. Among the most popular methods for solving this equation are finite-difference approximations. Finite-difference solutions of the eikonal equation have been recognized as one of the most efficient means of traveltime calculations (Vidale 1990; van Trier and Symes 1991; Popovici 1991; Alkhalifah and Fomel In anisotropic media, traveltime computation is dependent on more than one parameter field. However, through specific C 2012 European Association of Geoscientists & Engineers 495

2 496 T. Alkhalifah parametrization of HTI media, P-wave traveltimes in 3D, under the acoustic assumption, become dependent on only three parameters and the symmetry-axis azimuth. These parameters include the vertical velocity, v v, the normal-moveout velocity, v = v v 1 2δ (V (where δ (V corresponds to the vertical direction (Tsvankin 1997; hereafter V is dropped for brevity, and the anellipticity parameter η = ɛ(v δ 12δ (ɛ (V is also defined with respect to the vertical direction; hereafter V is also dropped for brevity. This is evident in the eikonal equation for TI media developed by Alkhalifah (1998, Numerically solving the HTI (or even TI in general eikonal equation using finite-difference schemes usually requires finding the root of a quartic equation at each computational step (Wang Nemeth and Langan 2000, which is hard to do analytically. However, traveltime computation for a simpler (albeit not practical elliptically anisotropic model is far more efficient. Thus, it is used here as the background medium for perturbing traveltime for the more practical HTI model. (Alkhalifah 2011a developed an eikonal-based scanning scheme to search for the anisotropy parameter η, which provides the best traveltime fit to the data in a general inhomogeneous background medium. He also used this concept to approximate traveltimes in transversely isotropic media with a tilted axis of symmetry (Alkhalifah 2011b. In that paper the focus was on perturbations with respect to the symmetry-axis tilt from the vertical. In this paper, I derive multi-parameter expansions of traveltime as a function of η and the symmetry-axis azimuth, φ, with coefficients estimated using linearized forms of the eikonal equation. The accuracy of these expansions is enhanced using Shanks transform (Bender and Orszag 1978 to obtain higher-order representations. The expansion with respect to η and φ makes practical sense as the inhomogeneous elliptical anisotropic background can be built by combining seismic data with well velocities. THE HTI EIKONAL FOR ARBITRARY SYMMETRY-AXIS AZIMUTH In HTI media, the eikonal equation in the acoustic approximation can be extracted from the VTI version (Alkhalifah 1998 by a simple rotation of the axis and thus, has the form: ( τ 2 vv 2(1 2δ(1 2η v v ( ( ( τ 2 1 2ηv 2 v (1 2δ ( τ ( τ 2 = 1, (1 where τ(x, y, z is the traveltime (eikonal measured from the source to a point with the coordinates (x, y, z. To formulate a well-posed initial-value problem for equation (1, it is sufficient to specify τ at some closed surface and to choose one of the two solutions: the wave going from or toward the source. The level of nonl-inearity in this quartic (in terms of τ equation is higher than that for the isotropic or elliptically anisotropic eikonal equations. This results in much more complicated finite-difference approximations of the HTI eikonal equation. For HTI media, with an arbitrary symmetry axis directon in the x y plane, the traveltime derivatives in equation (1 are taken with respect to the symmetry axis azimuth. Therefore, we have to rotate the derivatives in equation (1 using the following Jacobian in 3D: cos φ sinφ 0 sin φ cos φ 0, ( to obtain an eikonal equation corresponding to the reference coordinates governed typically by the acquisition. In equation (2, φ corresponds to the azimuth of the horizontal symmetry axis measured from the x-axis (typically the in-line direction of the acquisition. Equation (1 can be solved using the perturbation theory (Bender and Orszag 1978 by approximating equation (1 with a series of simpler linear equations. Considering φ and η to be constant and small, we can represent the traveltime solution as a series expansion in φ and η. This will result in a solution that is globally representative in the space domain and, despite the assumption of small φ and η, the accuracy for even large values of these parameters, as we will see later, is high as a result of using series prediction methods. The constant φ and η assumptions assume a factorized medium (Alkhalifah 1995 in these parameters (valid for smooth φ or η estimation applications under an effective medium assumption. However, the other two parameters, v v and δ, are allowed to vary freely. Figure 1 illustrates the concept of the global expansion as we predict the traveltime for any φ from its behavior at φ = 0 for the full traveltime field using, in this case, a quadratic approximation. Specifically, we substitute the following trial solution, τ(x, y, z τ 0 (x, y, z τ η (x, y, zη τ φ (x, y, zsinφ τ η2 (x, y, zη 2 τ ηφ (x, y, zη sin φ τ φ2 (x, y, zsin 2 φ (3

3 Traveltimes for HTI media 497 The φ expansion does not adapt well to the Shanks transform requirements for predicting the behavior of the higher-order terms in φ. In equation (5, we use only the first-order approximation in φ. Including the second-order term yields τ(x, y, z τ 0 (x, y, z τ φ (x, y, zsinφ τ φ2 (x, y, zsin 2 φ η ( τ η (x, y, z τ ηφ (x, y, zsinφ (6 τ η (x, y, z τ ηφ (x, y, zsinφ ητ η2 (x, y, z. Later, I will test the accuracy of both expansions. A detailed description of solutions for such coefficients can be found in Alkhalifah (2011c. For η and φ scan applications, the coefficients (τ 0, τ η, τ φ, τ η2, τ ηφ and τ φ2 need to be evaluated only once and can be used with equation (5 to search for the best traveltime fit to those traveltimes extracted from the data. Of course, as we will see later, the accuracy of these equations depends on the azimuth of the symmetry axis as the expansion is with respect to φ = 0. However, a similar expansion around φ = 90 o (Appendix B can extend the accuracy limit to those angles near φ = 90 o. A hybrid approximation can combine the two through a simple linear (in sin 2 φinterpolation given by the following: Figure 1 Schematic plot showing the relation between a background traveltime field for φ = 0andφ>0. The round dot at the top of the φ = 0 plane represents a source. into equation (1, resulting in linear first-order partial differential equations having the following general form: v 2 τ ( 0 τ i τ0 τ i v2 v τ 0 τ i = f i (x, y, z, (4 with i = η, φ, η 2, ηφ, φ 2 and τ 0 satisfies the eikonal equation for an elliptical anisotropic background model, where f 0 = 1. The function f i (x, y, z becomes more complicated for i corresponding to the second-order term and it depends on terms for the first-order and background medium solutions (Appendix A. Therefore, these linear partial differential equations must also be solved in succession starting with i = η and i = φ. As soon as all τ i coefficients are evaluated, they can be used, as Alkhalifah (2011a showed, to estimate the traveltime using the first-sequence of Shanks transform (Bender and Orszag 1978, which has the form (Appendix A: τ(x, y, z τ 0 (x, y, z τ φ (x, y, zsinφ ητη 2 (x, y, z (5 τ η (x, y, z ητ η2 (x, y, z. τ(x, y, z = τ 0 o(x, y, zcos 2 φ τ 90 o(x, y, zsin 2 φ, (7 where τ 0 o(x, y, z corresponds to the traveltime approximation extracted using the perturbation from φ = 0(orsinφ = 0, Appendix A and τ 90 o(x, y, z corresponds to the traveltime approximation extracted using the perturbation from φ = 90 o (or cos φ = 0, Appendix B. A first-order hybrid corresponds to τ obtained from equation (5, and the second order corresponds to τ evaluated using equation (6. TEST FOR HOMOGENEOUS MEDIA Though the equations above are developed for a general inhomogeneous background medium, I examine their accuracy in representing reflection moveout in the homogeneous case. This is convenient since most parameter scan-type applications (i.e., semblance velocity analysis are performed considering an effective homogeneous medium. I use the simple traveltime relation for an elliptically anisotropic homogeneous background to recursively solve for the coefficients of the traveltime expansion in φ and η and thus obtain analytic representations for the coefficients τ 0, τ φ, τ η, τ φ2, τ η2 and τ ηφ. Setting δ = 0, to allow for a simplified presentation, I obtain the following analytic expression for traveltime in HTI media: r ( Dx 3 η 2r 4 (x 4y sin(φ τ(x, y, z = ( v v 2r 4 (x 4y sin(φ 3x 3 η ( x 2 4 ( y 2 z 2, (8

4 498 T. Alkhalifah where D = x 2 16xy sin(φ 4y 2 ( 3 8sin 2 (φ 12z 2, (9 and r is the distance between the source, located at x = y = z = 0 and a point given by the x, y and z coordinates; so r 2 = x 2 y 2 z 2.Ifwesetx = 0 in equation (8, we obtain y2 z τ(y, z = 2, (10 v v and if we set y = 0 x 2 z 2 ( x 4 (η 2 4x 2 z 2 (3η 1 2z 4 vv 2 τ(x, z = 3x 2 η ( x 2 4z 2 2 ( (11 x 2 z 2, 2 both independent of φ for δ = 0, a consequence of using this type of dual expansion, the traveltimes will become inaccurate for large φ. Thus, the hybrid approach (equation (7 can introduce some φ dependency. To test the accuracy of the expansion in φ for the general equation (δ 0, I set η=0 and φ = 90 o, its maximum value considering the periodic symmetry of the HTI model and thus obtain the following relations for the traveltime in each of the x and y directions, respectively, x 2 τ(x, z = 1 2δ z2 τ(y, z = v 2 v y 2 z 2 v 2 v x 2 δ v v x2 z 2 (1 2δ, y 2 δ v v (1 2δ y 2 z 2. (12 Clearly, in both equations the errors in traveltime are given by the second term, which depends on δ and increase with offset; however, this is for the extreme case of φ = 90 o.sucherrors will be reduced using the suggested hybrid form. Equation (8 basically represents a moveout equation for traveltime in HTI media as a function of offset (or x and y and can be compared with equations developed specifically to represent reflection moveout in HTI media. Al-Dajani and Tsvankin (1998 derived the exact quartic moveout coefficient (i.e., the fourth-order term of the Taylor series expansion for squared traveltime for pure (nonconverted reflections in an HTI layer and substituted it into the moveout equation of Tsvankin and Thomsen (1994 to obtain: t 2 (X,φ = t0 2 X2 Vnmo 2 (φ A 4(φX 4 A(φX 2 1, (13 where V nmo is described by the NMO ellipse, Vnmo 2 (φ = v2 v (1 2δ 2δ sin 2 φ 1, (14 A 4 (φ = 2η cos4 φ, (15 t0 2v4 v (1 2δ2 and A(φ = A 4 (φ. 1 v 2 (1 2η 1 vv 2 (1 2δ (16 Here t 0 is the two-way zero-offset time and X = 2 x 2 y 2 is the offset. For zero azimuth, equation (13 reduces to t 2 (X,φ = 0 = t v 2 X2 2ηX 4 v 2 [t 2 0 v2 (1 2ηX 2 ], (17 which is the familiar nonhyperbolic equation for VTI media, considering that v 2 = vv 2 (1 2δ. Using an elliptically anisotropic background model with the vertical velocity equal to 2 km/s, δ =0.1 and symmetry-axis azimuth φ = 0 o, I compare the traveltime errors of the moveout equations extracted from our eikonal-based formulations with those of a pure moveout approximation (equation (13. For a reflector at depth z = 2 km, ɛ =0.25 and zero azimuth of symmetry axis, Fig. 2 shows the percentage traveltime errors as a function of in-line and cross-line offsets for the equations given above. Clearly, equation (13 (the right-hand plot is less accurate in describing overall traveltime behavior than the new formula (equation (5 represented on the left-hand plot. Of course, the comparison here is purely for the η perturbation as the background medium has the same symmetry direction as the desired model. In this case, we benefit from the Shanks higher-order transform to improve the expansion approximation, which is reflected in the results. However, for a more practical problem, in which the azimuth is unknown, we will have to perturb it. Thus, using the same model parameters as above but with an azimuth of 20 o, Fig. 3 shows the percentage error in the traveltime prediction as a function of the in-line and cross-line offsets. On the left (a we have the results for equation (13 and on the right (b the result corresponding to new approximation (6 withtwo termsof the expansionin φ. Obviously, we generally have slightly higher errors with the new expansion. These errors will increase for larger φ, as the expansion assumes small φ. To improve the accuracy, we use the hybrid approach of equation (7, which should reduce the bias as the expansion is performed from both φ = 0 o and φ = 90 o. Figure 4(a shows the traveltime errors for the hybrid approach for the one-term expansion in φ, while Fig. 4(b shows it for the twoterms expansion. Now the errors are comparable to those in Fig. 3(a.

5 Traveltimes for HTI media 499 Figure 2 Percentage traveltime error as a function of X and Y offsets for a model with v = 2km/s,δ =0.1, ɛ = -0.25, φ = 0, and a reflector depth z = 2 km for the nonhyperbolic moveout equation (13 (a, and the new expansion (5 in φ (b. Figure 3 Percentage traveltime error as a function of X and Y offsets for a model with v = 2km/s,δ =0.1, ɛ = -0.25, φ = 20 0, and a reflector depth z = 2 km for the nonhyperbolic moveout equation (13 (a and the new expansion (6 in φ with two terms of the expansion (b. The arrow marks the symmetry axis. Figure 4 Percentage traveltime error as a function of X and Y offsets for a model with v = 2km/s,δ =0.1, ɛ = -0.25, φ = 20 0 and a reflector depth z = 2 km for the hybrid expansion (7 in φ with one (a and two (b terms of the expansion. The arrow marks the symmetry axis. To examine the accuracy of the new formulas for a large deviation in the symmetry axis from the background axis (the reference in-line direction, I repeat the experiment in Fig. 4 but for 40 o and 60 o azimuths. Note that these are large azimuths for the approximation because as we pass 45 o,the traveltime can be approximated by the expansion from the y-axis (90 o, through the hybrid approach. Yet, the accuracy is generally acceptable (Fig. 5. The similarity between errors for the 40 o (a and 60 o (b azimuths is due to the nature of the interpolation in the hybrid implementation (equation (7, which is symmetric with respect to the 45 o azimuth.

6 500 T. Alkhalifah Figure 5 Percentage traveltime error as a function of X and Y offsets for a model with v = 2km/s,δ =0.1, ɛ = and a reflector depth z = 2 km for the hybrid expansion (7 for φ = 40 0 (a and φ = 60 0 (b. The arrow marks the symmetry axis. PARAMETER SEARCH The perturbation PDEs developed here are with respect to a generally inhomogeneous and possibly anisotropic, background medium. If a generally inhomogeneous isotropic velocity field is available (for example from conventional migration velocity analysis, in addition to a map of the well-to-seismic misties, which can be used to develop a vertical velocity field, then it is possible to construct an initial background elliptically anisotropic model with a horizontal symmetry axis in the in-line direction of the survey (the x-axis. We can utilize this model to compute traveltimes in the elliptically anisotropic medium as a background model and then use the computed background traveltime to solve for expansion coefficients in equations (4. These coefficients can be used with, for example, equation (5 to search explicitly for η and φ in 3D, which provides the best traveltime fit to the observed data. This process can be implemented in a semblance-type search or incorporated as part of a tomographic inversion. Though the scans are based on the factorized representation of the perturbation parameters, η and φ, we can allow them to vary smoothly with location and thus, produce effective values. In 3D, the search for η and the azimuth direction, φ, can be applied either sequentially or to both parameters at once. A sequential search, though faster and easier, may propagate some of the errors of an initial (wrong symmetry direction into the estimation of the parameter η; similarly, an initial wrong value of η can influence the estimate of the azimuth, φ. However, this influence of η is expected to be weak as the symmetry axis direction tends to be insensitive to the strength of anisotropy. The azimuth resolution ability, on the other hand, depends on the strength of anisotropy. Thus, if a sequential search is applied, it would be more practical to search first for the optimal symmetry direction and then for η. Nevertheless, a multi-parameter search for both η and φ would be optimal. CONCLUSIONS Expanding 3D traveltime solutions of the HTI eikonal equation in a power series in terms of independent parameters, the anelipticity parameter η and the symmetry-axis azimuth, φ, provides a direct tool to estimate these parameters in a generally inhomogeneous background medium. For a homogeneous background, I obtain analytic nonhyperbolic moveout equations for anisotropic media that are generally simple and yet accurate. Furthermore, the formulations can help estimate η and the symmetry-direction azimuth for a general inhomogeneous background medium. This is practical since conventional seismic experiments combined with well information may provide us with a background elliptically anisotropic inhomogeneous model. ACKNOWLEDGEMENTS I am grateful to KAUST for its financial support. I thank Ilya Tsvankin and Alexey Stovas for their critical review of the paper and helpful suggestions. REFERENCES Al Dajani A. and Tsvankin I Nonhyperbolic reflection moveout for horizontal transverse isotropy. Geophysics 63, Alkhalifah T Efficient synthetic seismogram generation in transversely isotropic, inhomogeneous media. Geophysics 60, Alkhalifah T Acoustic approximations for processing in transversely isotropic media. Geophysics 63, Alkhalifah T An acoustic wave equation for anisotropic media. Geophysics 65, Alkhalifah T. 2011a. Scanning anisotropy parameters in complex media. Geophysics 76, U13 U22. Alkhalifah T. 2011b. Traveltime approximations for transversely isotropic media with an inhomogeneous background. Geophysics 76, WA31 WA42.

7 Traveltimes for HTI media 501 Alkhalifah T. 2011c. Traveltime approximations for transversely isotropic media with an inhomogeneous background. Geophysics 76, WA31 WA42. Alkhalifah T. and Fomel S Implementing the fast marching eikonal solver: spherical versus Cartesian coordinates. Geophysical Prospecting 49, Bakulin A., Grechka V. and Tsvankin I., Estimation of fracture parameters from reflection seismic data Part I: HTI model due to a single fracture set. Geophysics 65, Bender C. M., and Orszag S. A Advanced mathematical methods for scientists and engineers. McGraw-Hill. Contreras P., Grechka V. and Tsvankin I Moveout inversion of P-wave data for horizontal transverse isotropy. Geophysics 64, Grechka V. and Tsvankin I D moveout velocity analysis and parameter estimation for orthorhombic media. Geophysics 64, Popovici M Finite difference travel time maps, in SEP-70. Stanford Exploration Project Tsvankin I., Reflection moveout and parameter estimation for horizontal transverse isotropy. Geophysics 62, Tsvankin I. and Thomsen L Nonhyperbolic reflection moveout in anisotropic media. Geophysics 59, van Trier J. and Symes W. W Upwind finite-difference calculation of traveltimes. Geophysics 56, Vidale J. E Finite-difference calculation of traveltimes in three dimensions. Geophysics 55, Wang Y., Nemeth T. and Langan R. T An expandingwavefront method for solving the eikonal equations in general anisotropic media. Geophysics 71, T129 T135. which is simply the eikonal formula for elliptical anisotropy. By equating the coefficients of the powers of the independent parameter sin φ and η, in succession starting with the first powers of the two parameters, we end up first with the coefficients of first-power in sin φ and zeroth power in η, simplified by using equation (A2 and given by (1 2δ τ 0 τ φ τ 0 τ φ τ 0 τ φ =2δ τ 0 τ 0, (A3 which is a first-order linear partial differential equation in τ φ. The coefficients of zero-power in sin φ and the first-power in η is given by (1 2δ τ 0 τ η τ 0 τ η τ ( 0 τ 4 η τ0 =v2 v (1 2δ2, (A4 The coefficients of the square terms in sin φ, with some manipulation, result in the following relation 2(1 2δ τ 0 τ φ2 2 τ 0 =4δ τ 0 ( (1 2δ τ φ 2δ ( τ φ2 2 τ 0 τ φ2 2 τ 0 ( τφ τ φ ( τφ ( τ0 ( τφ. ( τ0 (A5 APPENDIX A: EXPANSION IN SIN φ AND η For an expansion in φ and η, I use the following trial solution: τ(x, y, z τ 0 (x, y, z τ η (x, y, zη τ φ (x, y, zsinφ τ η2 (x, y, zη 2 τ ηφ (x, y, zη sin φ τ φ2 (x, y, zsin 2 φ, (A1 in terms of the coefficients τ i,wherei corresponds to η, φ, η 2, ηφ and φ 2. Inserting the trial solution, equation (A1, into equation (1 yields again a long formula but by setting both sin φ = 0andη = 0, I obtain the zeroth-order term given by ( vv 2(x, y, z(1 2δ(x, y, z τ0 ( ( vv 2(x, y, z τ0 ( τ0 = 1, (A2 This is again a first-order linear partial differential equation in τ φ2 with an obviously more complicated source function given by the right-hand side. The coefficients of the square terms in η, with also some manipulation, result in the following relation 2(1 2δ τ 0 τ η2 2 τ 0 τ η2 2 τ 0 ( ( ( = 2vv 2(1 2δ τ 0 2 τ η τ0 2 τ ( 0 τ0 τ η τ 0 4(1 2δ τ 0 τ η τ η ( τη τ η2 ( τ0 (1 2δ ( τη, ( τη (A6 which is again a first-order linear partial differential equation in τ η2 with again a complicated source function. Finally, the coefficients of the first-power terms in both sin φ and η result also in a first-order linear partial differential

8 502 T. Alkhalifah equation in τ ηφ given by (1 2δ τ 0 τ φη τ 0 = vv 2(1 2δ τ 0 τ ( ( 0 τ0 τφ (1 2δ τ η τ φη τ 0 τ φη (( ( τ0 τ 0 ( τφ τ 0 ( ( τ0 τφ τ 0 τ φ τ 0 (1 2δ τ 0 τ η 2(1 2δ τ 0 τ 0 τ 0 τ η τ 0 τ η 2(1 2δ τ 0 τ φ τ η τ φ τ η τ φ. (A7 Though the equation seems complicated, many of the variables of the source function (right-hand side can be evaluated during the evaluation of equations (A3 and (A4 in a fashion that will not add much to the cost. Using Shanks transforms (Bender and Orszag 1978 we can isolate and remove the most transient behavior of expansion (A1 in η (the φ expansion did not improve with such a treatment by first defining the following parameters: A 0 = τ 0 τ φ sin φ τ φ2 sin 2 φ A 1 = A 0 ( τ η τ ηφ sin φ η A 2 = A 1 τ η2 η 2 (A8 The first sequence of Shanks transforms uses A 0, A 1 and A 2, and thus, is given by τ(x, y, z A 0 A 2 A 2 1 A 0 2A 1 A 2 = τ 0 (x, y, z τ φ (x, y, zsinφ τ φ2 (x, y, zsin 2 φ η ( τ η (x, y, z τ ηφ (x, y, zsinφ τ η (x, y, z τ ηφ (x, y, zsinφ ητ η2 (x, y, z. (A9 APPENDIX B: EXPANSION IN COS φ AND η To allow for higher accuracy for a large azimuth (90 o, we expand once again, but with respect to cos φ instead of sin φ and thus, I use the following trial solution: τ(x, y, z τ 0 (x, y, z τ η (x, y, zη τ φ (x, y, zcosφ (B1 τ η2 (x, y, zη 2 τ ηφ (x, y, zη cos φ τ φ2 (x, y, zcos 2 φ, in terms of the coefficients τ i,wherei corresponds to η, φ, η 2, ηφ and φ 2. Inserting the trial solution, equation (B1, into equation (1 yields again a long formula, but by setting both cos φ = 0andη = 0, I obtain the zeroth-order term given by ( ( vv 2(x, y, z τ0 vv 2 (x, y, z(1 2δ(x, y, z τ0 ( vv 2(x, y, z τ0 = 1, (B2 which is simply the eikonal formula for elliptical anisotropy but now the symmetry axis of the background medium is along the y-axis. By equating the coefficients of the powers of the independent parameter cos φ and η, in succession starting with the first powers of the two parameters, we end up first with the coefficients of first-power in cos φ and zeroth power in η, simplified by using equation (B2 and given by τ 0 τ φ (1 2δ τ 0 τ φ τ 0 τ φ =2δ τ 0 τ 0, (B3 which is a first-order linear partial differential equation in τ φ. The coefficients of zero-power in cos φ and the first-power in η is given by τ 0 τ η (1 2δ τ 0 τ η τ ( 0 τ 4 η τ0 =v2 v (1 2δ2, (B4 The coefficients of the square terms in cos φ, with some manipulation, results in the following relation 2 τ 0 τ φ2 2(1 2δ τ 0 τ φ2 2 τ 0 =4δ τ ( 0 τ φ 2δ τ0 τ φ ( τφ 2δ ( τ0 ( τφ τ φ2 ( τφ, (B5 which is again a first-order linear partial differential equation in τ φ2 with an obviously more complicated source function given by the right-hand side. The coefficients of the square terms in η, with also some manipulation, results in the

9 Traveltimes for HTI media 503 following relation 2 τ 0 τ η2 2(1 2δ τ 0 τ η2 2 τ 0 ( ( ( = 2vv 2(1 2δ τ 0 2 τ η τ0 2 τ ( 0 τ0 τ η τ 0 τ η 4(1 2δ τ ( 0 τ η τη τ η2 ( τ0 (1 2δ ( τη ( τη, (B6 which is again a first-order linear partial differential equation in τ η2 with again a complicated source function. Finally, the coefficients of the first-power terms in both cos φ and η results also in a first-order linear partial differential equation in τ ηφ given by τ 0 τ φη (1 2δ τ 0 = vv 2(1 2δ τ 0 τ ( ( 0 τ0 τφ (1 2δ τ η τ φη τ 0 τ φη (( ( τ0 τ 0 ( τφ τ 0 ( ( τ0 τφ τ 0 τ φ τ 0 (1 2δ τ 0 τ η 2(1 2δ τ 0 τ 0 τ 0 τ η τ 0 τ η 2(1 2δ τ 0 τ φ τ η τ φ τ η τ φ. (B7