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1 Geophysical Journal International Geophys. J. Int. (04) GJI Seismology doi: 0.093/gji/ggu9 Effective orthorhombic anisotropic models for wavefield extrapolation Wilson Ibanez-Jacome Tariq Alkhalifah and Umair bin Waheed Earth Sciences and Engineering Program Physical Sciences and Engineering Division (PSE) King Abdullah University of Science and Technology Thuwal Saudi Arabia. Accepted 04 June 6. Received 04 June 4; in original form 03 June 30 INTRODUCTION SUMMARY Wavefield extrapolation in orthorhombic anisotropic media incorporates complicated but realistic models to reproduce wave propagation phenomena in the Earth s subsurface. Compared with the representations used for simpler symmetries such as transversely isotropic or isotropic orthorhombic models require an extended and more elaborated formulation that also involves more expensive computational processes. The acoustic assumption yields more efficient description of the orthorhombic wave equation that also provides a simplified representation for the orthorhombic dispersion relation. However such representation is hampered by the sixth-order nature of the acoustic wave equation as it also encompasses the contribution of shear waves. To reduce the computational cost of wavefield extrapolation in such media we generate effective isotropic inhomogeneous models that are capable of reproducing the firstarrival kinematic aspects of the orthorhombic wavefield. First in order to compute traveltimes in vertical orthorhombic media we develop a stable efficient and accurate algorithm based on the fast marching method. The derived orthorhombic acoustic dispersion relation unlike the isotropic or transversely isotropic ones is represented by a sixth order polynomial equation with the fastest solution corresponding to outgoing P waves in acoustic media. The effective velocity models are then computed by evaluating the traveltime gradients of the orthorhombic traveltime solution and using them to explicitly evaluate the corresponding inhomogeneous isotropic velocity field. The inverted effective velocity fields are source dependent and produce equivalent first-arrival kinematic descriptions of wave propagation in orthorhombic media. We extrapolate wavefields in these isotropic effective velocity models using the more efficient isotropic operator and the results compare well especially kinematically with those obtained from the more expensive anisotropic extrapolator. Key words: Numerical solutions; Non-linear differential equations; Seismic anisotropy; Wave propagation; Acoustic properties. One of the major challenges for any anisotropic pre-stack depth migration is to accurately and efficiently extrapolate wavefields in 3-D anisotropic media. Nowadays the most used and practical assumption to formulate and reproduce wave propagation in anisotropic media is based on transversely isotropic (TI) models. TI media are represented by a sequence of isotropic planes of mirror symmetry guided by a rotational symmetry axis. In addition a more specific case of these models can be defined in certain geological settings where horizontal finely layered sediments are present. In this particular case the anisotropy of the continuum is established in terms of vertical transversely isotropic media (VTI). TI models have gradually become a useful assumption of the Earth s subsurface representing in one form (tilted) the first order nature of the azimuthal anisotropy influence of the Earth. However a more realistic representation of the Earth s subsurface that includes the natural thin horizontal layering and a domain of oriented-parallel vertical cracks may be given by orthorhombic anisotropic models which assume three mutually orthogonal planes of mirror symmetry (Schoenberg & Helbig 997). Wavefield extrapolation is considered a very expensive step for any 3-D depth imaging method or inversion. The available methods are either based on eikonal or ray tracing equations (Beydoun & Keho 987; Vidale 990; Van & Symes 99). Along these lines Sethian & Popovici (999) proposed a fast-marching finite-difference eikonal solver in Cartesian coordinates which is very efficient and stable. Based on this approach the traveltime solution at each grid point in the model is estimated in a simple upwind fashion using the corresponding Downloaded from at King Abdullah University of Science and Technology on January 9 06 C The Authors 04. Published by Oxford University Press on behalf of The Royal Astronomical Society. 653

2 654 W. Ibanez-Jacome T. Alkhalifah and U. bin Waheed Table. Diagram comparing the general sequences used for wavefield extrapolation based on orthorhombic wave equation and the isotropic inhomogeneous wavefield operator. The highlighted blue series of steps represents the alternative approach presented in this study to reduce the cost of orthorhombic wavefield extrapolation. orthorhombic traveltime solution derived in this work. This reasonably accurate fast and unconditionally stable algorithm may also be implemented to compute traveltimes in more realistic anisotropic symmetries such as VTI and orthorhombic media. Wavefield extrapolation in orthorhombic media is well described by the numerical solution of the anisotropic elastic wave equation. For the acoustic case wave propagation is limited only to compressional waves such as P waves. Mixed-domain acoustic wave extrapolators for time marching may be applied using low-rank approximations (Fomel et al. 00; Song & Alkhalifah 0). The low-rank solution obtained for wavefield extrapolation used in this study is based on the acoustic approach introduced by Alkhalifah (003) where a sixth order pseudo-acoustic wave equation is used to describe wave propagation in orthorhombic media. This pseudo-acoustic approximation is defined by setting the shear wave velocities along the axes of symmetry to zero. In this paper a dispersion relation for acoustic vertical orthorhombic media is presented. We solve a sixth order polynomial equation to calculate the first arrival traveltime solution using the fast marching method (Sethian & Popovici 999). The process of computing traveltimes at each gridpoint follows an upwind concept where previously estimated values are required for continuous evolution of the traveltime field at the specific gridpoint. Because of the non-linearity aspect of the equation it does not define any initial marching direction. Thus after sorting from the smallest to the largest arrival time the corresponding solution can be found only by applying one pass in the region that contains the gridpoints. Then the upwind finite difference scheme is used to solve the eikonal equation. One of the major advantages of this approach is that the implied computational process is very efficient and easy to program. Subsequently the corresponding traveltime solution is used to explicitly calculate an effective isotropic velocity field that encompasses all the kinematic effects of the orthorhombic anisotropic model. Last we use the effective velocity field to extrapolate an approximate orthorhombic anisotropic wavefield using isotropic operators. A similar approach was applied by Alkhalifah et al. (03) for TI models using a finite difference scheme. We finally compare the new wavefields with those extracted from an actual orthorhombic wavefield extrapolator. The diagram presented in Table represents the general concept behind the effective velocity approach implemented in this work. The red-highlighted sequence symbolizes the standard method used to compute wavefields in orthorhombic media. On the other hand the blue-highlighted sequence describes the alternative method presented in this work for calculating wavefields in acoustic vertical orthorhombic media. The latter approach implies a significant decrease on computational cost. The approach presented in this study can be implemented in many depth migration methods that use depth extrapolation processes. These extrapolation methods are generally considered to be expensive so it is important to find the most efficient way of implementing them such as the effective velocity approach. Additionally inversion algorithms are usually bottle-necked by the computational cost of forward modeling tool it resorts to as several such modeling steps are needed for migration/inversion programs. The concept of effective model is useful in reducing the cost of the modeling step. Particularly for low to moderately complex media the effective model yields sufficient accuracy. As suggested by Alkhalifah et al. (03) the method can be used in a delayed shot reverse time migration algorithm for subsurface imaging. Downloaded from at King Abdullah University of Science and Technology on January 9 06

3 Effective orthorhombic anisotropic models 655 DISPERSION RELATION Kinematic signatures of P waves in pseudo-acoustic anisotropic orthorhombic media depend on only five anisotropic parameters and the vertical velocity as shown by Tsvankin (997). Instead of rigorously using the notation suggested by Tsvankin (997) we implement a different parametrization shown by Alkhalifah (003) where the anisotropic aspect of the medium is defined in terms of the anellipticity values η instead of the Thomsen s anisotropy parameters ε and δ (Thomsen 986). This convention is used to facilitate the interpretation process in terms of η. On the other hand the normal moveout (NMO) velocities are the natural extensions of their isotropic counterparts for small offsets (source receiver distance). Thus c33 v v ρ v c 3 (c 3 + c 55 ) + c 33 c 55 ρ(c 33 c 55 ) () c 3 (c 3 + c 44 ) + c 33 c 44 and v () ρ(c 33 c 44 ) define the P-wave vertical velocity and the NMO P-wave velocities for horizontal reflectors defined in the [x x 3 ]and[x x 3 ] planes of mirror symmetry respectively. The term ρ represents the density. The coefficients c ij correspond to the elastic modulus tensor components in Voigt notation (Thomsen 986) that characterize the elasticity of the medium. Now for the anisotropic parameters c (c 33 c 55 ) η c 3 (c 3 + c 55 ) + c 33 c 55 (3) η c (c 33 c 44 ) c 3 (c 3 + c 44 ) + c 33 c 44 δ (c + c 66 ) (c c 66 ) (5) c (c c 66 ) where the first two values η and η define the anellipticity in the [x x 3 ]and[x x 3 ] symmetry planes respectively and δ represents the anisotropic parameter in the [x x ] plane defined with respect to the x coordinate axis. In order to ease some of the derivations the parameter δ is used in the eikonal fast marching algorithm under the following definition γ + δ where the value of γ is also internally defined in the algorithm for convenience in notation. Now for a better understanding of the origin of the orthorhombic dispersion relation we consider with no loss of generality the equation of motion with no body forces using Einstein summation convention given by ρ(x) u i t = σ ij j where u i represents the displacement vector component and σ ij defines the stress tensor that accounts for the inhomogeneity as well as the anisotropy of the medium with the elasticity tensor being functions of position. Namely σ ij = ( c uk ijkl(x) + u ) l (7) l k with the elasticity coefficients c ijkl (x) fully described in tensor notation. Inserting eq. (7) into eq. (6) we obtain ρ(x) u i = ( c ijkl (x) uk + u ) l + ( ) t j l k c u k ijkl(x) + u l (8) j l j k representing the wave equation in anisotropic inhomogeneous continua. To represent a formulation of the solution for this equation let us consider a trial solution in terms of position x and time t represented by u(x t) = A(x) f (n) where A(x) is a vector function of position x andf(n) defines a scalar function whose argument is represented by n = v o [τ(x) t] with v o being a constant value defined in velocity units. The function τ(x) designates a domain that relates position x with the traveltime t. Hence after inserting the trial solution u(x t)into eq. (8) followed by a set of different operations rearranging the terms along with the application of the symmetry properties of the elastic tensor c ijkl (Slawinski 003) we obtain [ c ijkl (x) τ ] τ ρ(x)δ ik A k (x) = 0 (9) j l where δ ik is the Kronecker s delta and τ/ j = p j defines the phase-slowness vector which represents the slowness value of the wavefront propagation in a particular position x j. In terms of the gradient operator the term p = τ(x) describes a vector whose direction is normal to the wavefront and whose magnitude represents the wavefront slowness. Now we can rewrite eq. (9) in matrix notation as (4) (6) Downloaded from at King Abdullah University of Science and Technology on January 9 06 ρ(x)ɣ(p)a(x) = 0 (0)

4 656 W. Ibanez-Jacome T. Alkhalifah and U. bin Waheed where Ɣ(p) = c ijkl (x) τ τ /ρ(x) δ ik j l = c ijkl (x)p j p l /ρ(x) δ ik. () To gain insights into the physical meaning of eq. () we now focus our attention to the specific case of orthorhombic media. As shown in Slawinski (003) using Voigt notation we can write the elasticity matrix for an orthorhombic continuum as c c c c c c c 3 c 3 c c ortho =. () c c c 66 Using the pseudo-acoustic approximation shown in Alkhalifah (003) where the vertical velocity of the S wave v s = c 55 /ρ polarized in the x direction the S-wave velocity v s = c 44 /ρ polarized in the x direction and the S-wave velocity v s3 = c 66 /ρ polarized in the x direction but propagating in the x direction are all zero and using the definitions in eqs () (5) we can now rewrite Christoffel equation as px v ( + η ) γ p x p y v ( + η ) p x p z v v v Ɣ(p) = γ p x p y v ( + η ) p y v ( + η ) p y p z v v v p x p z v v v p y p z v v v p z v v where the values of p x p y and p z represent the Cartesian components of the phase vector p. Taking the determinant of Ɣ(p) setting the resultant linear equation to zero and solving for the squared vertical component of the phase vector pz yields the dispersion relation for orthorhombic media { [ + p y ( + η )γ v ( + η )v]} p z = ( + η )p y v ( + η )px v vv[ η p y v p x ( v η + γ p y v + 4η ( + η )γ p y v ( + η )γ p y v )]. (3) v + ( 4η η )p y v 3 TRAVELTIME EIKONAL SOLUTIONS Based on the phase-slowness vector definition expression (3) becomes a partial differential equation where ( ) τ ( ) τ ( + η ) v ( + η ) ( { ) ( ) τ v τ [( + + η )γ v ( + η ] } )v = ( ) (4) τ ζ τ with the term ζ in the denominator defined as ζ τ ) { ( ) τ ( ) [ τ ( ) τ ( ) τ = v v η v v η + γ v + 4η ( + η )γ v ( + η )γ ( ) τ v v + ( 4η η ) ) v ]}. (5) Setting v = v = v v η = η = 0andγ = increases the symmetry of eq. (3) in terms of the phase vector components and respectively provides the isotropic dispersion relation pz = /v px p y wherev represents the unique velocity field for isotropic media. Following the same approach for the VTI dispersion relation case v = v η = η = η and γ = or equivalently setting one of the phase vector components p y = 0orp x = 0 in eq. (3) yields p z = ( ) v p x (6) vv η v p x which represents the VTI acoustic dispersion relation. Rewriting eq. (3) in terms of the corresponding traveltime-spatial derivatives yields the orthorhombic eikonal equation shown in expression (4). To facilitate the implementation of a finite difference scheme in eq. (4) we factorize and rearrange all common coefficients in terms of the time derivative components. By doing this a simplified form of the eikonal eq. (4) may be found bringing together all Downloaded from at King Abdullah University of Science and Technology on January 9 06

5 medium properties in a new sequence of coefficients such that ( ) τ v ( + η ) + v ( + η ) ) + [ ( + 4η + 4η )γ v 4 ( + η + η + 4η η )v v + [ ( + η )γv 3 v v v + (4η η )v v v v ( + 4η + 4η )γ ] ( v 4 τ v v Effective orthorhombic anisotropic models 657 ) ) ( ) τ ] ) ( ) τ A η v v v ) ( ) τ η v v v ) ) + v v ) =. (7) Rewriting eq. (7) factorizing all velocities fields and anisotropic parameters into a new set of values we obtain ) ( ) τ ( ) τ ( ) τ ( ) τ ( ) τ ( ) τ ( ) τ ( ) τ ( ) τ + B + C + D + E + F + G ) ( ) τ = (8) where the sequence of terms that includes all medium properties is give by A = v ( + η ) B = v ( + η ) C = v v D = ( + 4η + 4η )γ v 4 ( + η + η + 4η η )v v E = η v v v F = η v v v G = ( + η )γv 3 v v v + (4η η )v v v v ( + 4η + 4η )γ v 4 v v. For the case of A = B E = F and D = G = 0 eq. (8) reduces to the VTI dispersion relation. Based on a first order finite difference approach and expanding all the resultant quadratic terms eq. (8) can be transformed into a sixth order polynomial equation as a function of the first arrival traveltime solution τ ijk defined in the algorithm at the gridpoint ijk. Therefore in order to approximate the derivatives of the first order nonlinear partial differential equation shown in expression (4) or equivalently in eq. (8) a 3-D backward finite difference method is implemented based on a first order scheme. Namely ( ) τi jk τ ( ) i jk τi jk τ ( ) i j k τi jk τ ( ) i jk τi jk τ ( ) i jk τi jk τ i j k A + B + C + D x y z x y ( ) τi jk τ ( ) i jk τi jk τ ( ) i jk τi jk τ ( ) i jk τi jk τ ( ) i j k τi jk τ i jk + E + G x z x y z ( ) τi jk τ ( ) i j k τi jk τ i jk + F =. (9) y z Expanding all quadratic terms in eq. (9) and collecting all common traveltime solutions τ i j k using exponential values that share the same power eq. (9) may be rewritten in a polynomial form as β 6 τ 6 ijk + β 5τ 5 ijk + β 4τ 4 ijk + β 3τ 3 ijk + β τ ijk + β τ ijk + β 0 = 0 where the set of coefficients β i represents real values of a combined contribution of physical properties related to the orthorhombic symmetry as well as the initial traveltime conditions and grid spacing values. To illustrate the form of the sequence of the coefficients β i the following equation represents the composition of the β 4 term D β 4 = x y + E x z + F y z + Gτ i jk x y z + 4Gτ i jk τ i j k + Gτ i j k x y z x y z + 4Gτ i jk τ i jk x y z + 4 Gτ i j kτ i jk + Gτ i jk x y z x y z. (0) Since the rest of coefficients share a similar structure but can be considerably larger and more complex eq. (0) will be the only coefficient shown here for practical informative purposes. The approach used in these equations facilitates the separation of medium properties and the finite difference contribution factors all used to find the required traveltime solution τ ijk. All these parameters are used in the finite difference algorithm based on the fast marching method. A sequential application of Bairstow s method (Press et al. 989) is implemented in order to solve for the roots of the polynomial equation P(τ ijk ) = 6 p=0 β pτ p ijk = 0 where the sequence of coefficients β p are considered to be real. This algorithm provides a numerical procedure to decompose a polynomial with real coefficients into a sequence of second order quadratic factors. Finding these second order quadratic factors from the original sixth order polynomial allows us to determine the corresponding polynomial roots (sometimes as complex conjugate pairs) by only solving quadratic formulas. Since complex conjugate roots may be found in the set of solutions only the real part of the roots are considered for this particular implementation. These complex solutions represent Downloaded from at King Abdullah University of Science and Technology on January 9 06

6 658 W. Ibanez-Jacome T. Alkhalifah and U. bin Waheed the consequence of not using the exact dispersion relation (acoustic approximation instead) to compute the traveltime solutions. Wavefield extrapolation in acoustic orthorhombic anisotropic media suffers from wave mode coupling from qsv (quasi-s-wave vertical component) and qp (quasi-p wave). Therefore the complex values of these solutions are related to the wavefield coupling modes implied in this process. 4 EFFECTIVE VELOCITY FOR WAVEFIELD EXTRAPOLATION Inserting the P-wave orthorhombic traveltime solutions τ ort (equivalent to τ ijk in the algorithm) into the isotropic dispersion relation leads to an equation describing the effective velocity field in the following form: v eff = ort ) ( + τort ) ( + τort ). The effective velocity in eq. () integrates all the first arrival kinematic effects of the anisotropic parameters and velocity fields from the orthorhombic model. As a result an isotropic wavefield extrapolation may be used in order to describe wave propagation in orthorhombic media. This alternative procedure only requires the corresponding v eff and an isotropic wavefield extrapolation which is computationally less expensive than regular orthorhombic wavefield extrapolations (Song & Alkhalifah 0). 5 NUMERICAL EXAMPLES In the following example we compare wavefields obtained using the more expensive orthorhombic wavefield extrapolators with the method proposed here. The model chosen here is suitable to show the benefits and limitations of the approach as it tries to match the first arrival kinematics. Fig. (i) shows an effective velocity model estimated from the traveltime field shown in Fig. (h) computed using the set of inhomogeneous v v v v η η and γ models shown in Fig. with a source located at x =.5 km y =.5 km and z =.5 km. These plots define -D slices of the 3-D cube representation of the data with different space or time domains over each side of the figures. For velocity time and anisotropic parameter models the vertical axis represents the corresponding model depths whereas the two horizontal axes correspond to inline and crossline directions of a potential 3-D survey. The blue lines in these figures represent the slice location that is projected to the respective parallel cube side. Figs (a) (c) show time snapshots of the orthorhombic wavefield modeled with the corresponding effective velocity shown in Fig. (i). In the depth slice figures a horizontal cut-section of the respective model is used whereas the inline and crossline slices represent vertical section on each of the domains respectively. An isotropic wavefield extrapolation based on the low-rank approximation approach (Fomel et al. 00) is applied to obtain the results shown in Figs (a) (c). In terms of the kinematic aspects of wavefield propagation equivalent results are found between the wavefield computed with the isotropic effective velocity approach and the orthorhombic wavefield extrapolation shown in Figs (d) (f). Note that the wavefield extrapolated with the effective velocity experiences a loss in amplitude around the region of highest velocity variation. Despite the difference in amplitude first arrival traveltimes as expected are found to be equivalent. Figs (g) (l) show the difference in amplitude and the corresponding match with respect to the first arrival traveltime solution represented in space domain. These traces represent cross-sections of the data over the different domains. The solid yellow curve superimposed on all the wavefield snapshots shown in Fig. represents the eikonal traveltime solution at the equivalent time. This traveltime solution is estimated using the orthorhombic eikonal solver proposed in this study. Despite only the first arrival matching with the isotropic wavefield extrapolator as Alkhalifah et al. (03) showed for the TI case this method can significantly reduce the computational cost of wavefield extrapolation in anisotropic media. Therefore as shown for the example presented in Fig. the implemented methodology provides accurate and stable results with a much simpler and less expensive technique used to generate wavefields in acoustic vertical orthorhombic media. Once the effective velocity model showninfig.(i) is constructed for a particular source located at x =.5 km y =.5 km and z =.5 km it is used to solve the isotropic wave equation. Solving the wave equation in this case involves wavefield extrapolation in inhomogeneous isotropic media which implies a much lower cost for the computational process. For this particular model the actual full orthorhombic wavefield extrapolation is found to be at least 3.5 times more expensive than the effective approach. With respect to the reflected or late events obtained in the effective wavefield extrapolation note that an accurate match is only given by the first arrival component of the corresponding two wavefields from effective and orthorhombic models. Since only a fitting process based on first arrival traveltime solution is applied it exclusively equals first arrival components. Thus represented by the comparison between Figs (a) (d) (b) (e) and (c) (f) reflected or late arrivals in the corresponding wavefields from effective and the actual orthorhombic approach do not lead to an accurate matching for most cases. Since only the first arrival traveltime fields are implemented for the inversion process we should not expect equivalent results for an accurate match of later arrivals. The sequence of traces shown in Fig. provides a much clearer representation of the delays (in space domain) found between the corresponding reflected or late events. The effective velocity model reproduces isotropic kinematic effects when anisotropy is zero. When anisotropy is considered this velocity model represents also an isotropic velocity field. However its variation or heterogeneity depends on the strength of anisotropy so the wavefield produced is a correction of the isotropic full wavefield which is expected to include the anisotropic correct traveltime for at least the first arrivals. Since imaging and inversion updates rely on transmissions no reflections this approach may serve to reduce the cost of these operations. () Downloaded from at King Abdullah University of Science and Technology on January 9 06

7 Effective orthorhombic anisotropic models 659 Downloaded from at King Abdullah University of Science and Technology on January 9 06 Figure. These plots define -D slices of the 3-D cube representation of the data with different space or time domains over each side of the figures. Velocity models v v (a) v (b) v (c) and the anisotropic parameters η (d) η (e) and γ (f) are shown respectively. Panel (g) represents a contour plot of the traveltime field computed with the orthorhombic eikonal solver presented in this study. For the calculation of this traveltime field the sequence of velocity and anisotropic-parameter models was required as shown in the finite difference scheme of eq. (9) where the final traveltime solution is computed using the fast marching algorithm within the corresponding sixth-order polynomial equation. Panel (h) shows the effective velocity model computed with the isotropic dispersion relation represented in eq. () based on the estimation of the solution traveltime-field τ ort. Notice the presence of head waves at approximately 3 km depth in the traveltime field and the expected effect in the inversion of the effective velocity model where a curved pattern of high velocity values is generated. The source is set in the middle location of the models on each side as indicated by the intersection points given by the blue lines highlighted by the.5 km values.

8 660 W. Ibanez-Jacome T. Alkhalifah and U. bin Waheed Downloaded from at King Abdullah University of Science and Technology on January 9 06 Figure. Sequence of slices of wavefield snapshots at t = 0.9 [s] from isotropic wavefield extrapolation using the effective velocity shown in Fig. (i); (a) depth slice (b) inline slice (c) crossline slice. Slices of wavefield snapshots at t = 0.9 [s] from orthorhombic wavefield extrapolation (Song & Alkhalifah 0) using the complete sequence of models v v v v η η and γ ; (d) depth slice (e) inline slice (f) crossline slice. The solid yellow curve on all the snapshots represents the corresponding traveltime solution estimated with the orthorhombic eikonal solver proposed in this study. Fig. (g) represents the overlapping between traces from the orthorhombic and effective wavefields at inline.5 km taken from the wavefields shown in Figs (a) and (d). Figs (h) (i) (j) (k) and (l) are equivalently generated from the different wavefield components. Dotted and solid curves represent the orthorhombic and effective wavefield solutions respectively.

9 Effective orthorhombic anisotropic models 66 6 CONCLUSIONS The numerical examples demonstrate that the presented algorithm based on the fast marching method is stable and accurate for calculating first arrival traveltimes. The high-frequency asymptotic solutions implicit in the orthorhombic eikonal solver fit adequately the wavefronts extrapolated from the orthorhombic low-rank solution. In addition the effective isotropic wavefield extrapolation approach is kinematically accurate when compared to results obtained from the orthorhombic wavefield extrapolation. The kinematic aspect of the corresponding wavefields calculated from the effective and the actual orthorhombic approach are found to be equivalent. This serves as a platform for evaluating approximate anisotropic wavefields using efficient isotropic extrapolators. However amplitude values mostly do not match especially in regions where large velocity gradients are located in the effective velocity model. Furthermore since the effective wavefields are computed from the first arrival traveltime solution they do not accurately reproduce reflected or late events in the data. Despite the presented dynamic variations between the respective solutions the method implemented in this study serves as a platform for evaluating approximate anisotropic wavefields using efficient isotropic extrapolators. This implies a significant decrease in computational cost without compromising on the accuracy of the kinematic aspect of wave propagation. Inversion algorithms are usually bottle-necked by the computational cost of forward modeling tool it resorts to as several such modeling steps are needed for migration/inversion programs. The concept of effective model is useful in reducing the cost of the modeling step. Particularly for low to moderately complex media the effective model yields sufficient accuracy. ACKNOWLEDGEMENTS We acknowledge KAUST for the financial support. We also thank the members of the Seismic Wave Analysis Group (SWAG) at KAUST for all their support. REFERENCES Alkhalifah T An acoustic wave equation for orthorhombic anisotropy Geophysics Alkhalifah T. Ma X. bin Waheed U. & Zuberi M. 03. Efficient anisotropic wavefield extrapolation using effective isotropic models in Proceedings of the 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 03 London UK 0 3 June 03 Extended Abstract EAGE doi:0.3997/ Beydoun W.B. & Keho T.H The paraxial ray method Geophysics Fomel S. Ying L. & Song X. 00. Seismic wave extrapolation using lowrank symbol approximation in Proceedings of the SEG Technical Program Expanded Abstracts SEG Denver 00 Annual Meeting pp Press W.H. Flannery B.P. Teukolsky S.A. & Vetterling W.T Numerical Recipes in FORTRAN: The Art of Scientific Computing nd edn Cambridge Univ. Press pp. 77 and Schoenberg M. & Helbig K Orthorhombic media: modeling elastic wave behavior in a vertically fractured earth Geophysics Sethian J.A. & Popovici A.M D traveltime computation using the fast marching method Geophysics 64() Slawinski M.A Seismic Waves and Rays in Elastic Media (Handbook of Geophysical Exploration: Seismic Exploration) Vol. 34 Elsevier Science. Song X. & Alkhalifah T. 0. Modeling of pseudo-acoustic p-waves in orthorhombic media with lowrank approximation in Proceedings of the SEG Technical Program Expanded Abstracts pp. 6. Thomsen L Weak elastic anisotropy Geophysics Tsvankin I Anisotropic parameters and p-wave velocity for orthorhombic media Geophysics Van Trier J. & Symes W.W. 99. Upwind finite-difference calculation of traveltimes Geophysics Vidale J.E Finite-difference calculation of traveltimes in three dimensions Geophysics Downloaded from at King Abdullah University of Science and Technology on January 9 06