geophysical journal international - princeton · geophysical journal international geophys. j. int....

Geophysical Journal InternationalGeophys. J. Int. (2014) 198, 1653–1661 doi: 10.1093/gji/ggu229

GJI Seismology

Effective orthorhombic anisotropic models for wavefieldextrapolation

Wilson Ibanez-Jacome, Tariq Alkhalifah and Umair bin WaheedEarth Sciences and Engineering Program, Physical Sciences and Engineering Division (PSE), King Abdullah University of Science and Technology, Thuwal,Saudi Arabia. E-mail: [email protected]

Accepted 2014 June 16. Received 2014 June 14; in original form 2013 June 30

S U M M A R YWavefield extrapolation in orthorhombic anisotropic media incorporates complicated but re-alistic models to reproduce wave propagation phenomena in the Earth’s subsurface. Com-pared with the representations used for simpler symmetries, such as transversely isotropicor isotropic, orthorhombic models require an extended and more elaborated formulation thatalso involves more expensive computational processes. The acoustic assumption yields moreefficient description of the orthorhombic wave equation that also provides a simplified repre-sentation for the orthorhombic dispersion relation. However, such representation is hamperedby the sixth-order nature of the acoustic wave equation, as it also encompasses the contributionof shear waves. To reduce the computational cost of wavefield extrapolation in such media, wegenerate effective isotropic inhomogeneous models that are capable of reproducing the first-arrival kinematic aspects of the orthorhombic wavefield. First, in order to compute traveltimesin vertical orthorhombic media, we develop a stable, efficient and accurate algorithm based onthe fast marching method. The derived orthorhombic acoustic dispersion relation, unlike theisotropic or transversely isotropic ones, is represented by a sixth order polynomial equationwith the fastest solution corresponding to outgoing P waves in acoustic media. The effectivevelocity models are then computed by evaluating the traveltime gradients of the orthorhombictraveltime solution, and using them to explicitly evaluate the corresponding inhomogeneousisotropic velocity field. The inverted effective velocity fields are source dependent and produceequivalent first-arrival kinematic descriptions of wave propagation in orthorhombic media.We extrapolate wavefields in these isotropic effective velocity models using the more efficientisotropic operator, and the results compare well, especially kinematically, with those obtainedfrom the more expensive anisotropic extrapolator.

Key words: Numerical solutions; Non-linear differential equations; Seismic anisotropy;Wave propagation; Acoustic properties.

1 I N T RO D U C T I O N

One of the major challenges for any anisotropic pre-stack depth migration is to accurately and efficiently extrapolate wavefields in 3-Danisotropic media. Nowadays, the most used and practical assumption to formulate and reproduce wave propagation in anisotropic mediais based on transversely isotropic (TI) models. TI media are represented by a sequence of isotropic planes of mirror symmetry guided by arotational symmetry axis. In addition, a more specific case of these models can be defined in certain geological settings where horizontalfinely layered sediments are present. In this particular case, the anisotropy of the continuum is established in terms of vertical transverselyisotropic 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 subsurfacethat includes the natural thin horizontal layering and a domain of oriented-parallel vertical cracks, may be given by orthorhombic anisotropicmodels, which assume three mutually orthogonal planes of mirror symmetry (Schoenberg & Helbig 1997).

Wavefield extrapolation is considered a very expensive step for any 3-D depth imaging method or inversion. The available methods areeither based on eikonal or ray tracing equations (Beydoun & Keho 1987; Vidale 1990; Van & Symes 1991). Along these lines, Sethian &Popovici (1999) proposed a fast-marching finite-difference eikonal solver in Cartesian coordinates, which is very efficient and stable. Basedon this approach, the traveltime solution at each grid point in the model is estimated in a simple upwind fashion using the corresponding

C© The Authors 2014. Published by Oxford University Press on behalf of The Royal Astronomical Society. 1653

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1654 W. Ibanez-Jacome, T. Alkhalifah and U. bin Waheed

Table 1. Diagram comparing the general sequences used for wavefield extrap-olation based on orthorhombic wave equation and the isotropic inhomogeneouswavefield operator. The highlighted blue series of steps represents the alternativeapproach presented in this study to reduce the cost of orthorhombic wavefieldextrapolation.

orthorhombic traveltime solution derived in this work. This reasonably accurate, fast and unconditionally stable algorithm, may also beimplemented 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. Forthe acoustic case, wave propagation is limited only to compressional waves, such as P waves. Mixed-domain acoustic wave extrapolatorsfor time marching may be applied using low-rank approximations (Fomel et al. 2010; Song & Alkhalifah 2012). The low-rank solutionobtained for wavefield extrapolation used in this study is based on the acoustic approach introduced by Alkhalifah (2003), where a sixth orderpseudo-acoustic wave equation is used to describe wave propagation in orthorhombic media. This pseudo-acoustic approximation is definedby 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 tocalculate the first arrival traveltime solution using the fast marching method (Sethian & Popovici 1999). The process of computing traveltimesat each gridpoint follows an upwind concept, where previously estimated values are required for continuous evolution of the traveltime fieldat the specific gridpoint. Because of the non-linearity aspect of the equation, it does not define any initial marching direction. Thus, aftersorting from the smallest to the largest arrival time, the corresponding solution can be found only by applying one pass in the region thatcontains the gridpoints. Then, the upwind finite difference scheme is used to solve the eikonal equation. One of the major advantages of thisapproach is that the implied computational process is very efficient and easy to program. Subsequently, the corresponding traveltime solutionis used to explicitly calculate an effective isotropic velocity field that encompasses all the kinematic effects of the orthorhombic anisotropicmodel. 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. (2013) for TI models using a finite difference scheme. We finally compare thenew wavefields with those extracted from an actual orthorhombic wavefield extrapolator. The diagram presented in Table 1 represents thegeneral concept behind the effective velocity approach implemented in this work. The red-highlighted sequence symbolizes the standardmethod used to compute wavefields in orthorhombic media. On the other hand, the blue-highlighted sequence describes the alternative methodpresented in this work for calculating wavefields in acoustic vertical orthorhombic media. The latter approach implies a significant decreaseon 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 forwardmodeling tool it resorts to, as several such modeling steps are needed for migration/inversion programs. The concept of effective model isuseful in reducing the cost of the modeling step. Particularly for low to moderately complex media, the effective model yields sufficientaccuracy. As suggested by Alkhalifah et al. (2013), the method can be used in a delayed shot reverse time migration algorithm for subsurfaceimaging.

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Effective orthorhombic anisotropic models 1655

2 D I S P E R S I O N R E L AT I O N

Kinematic signatures of P waves in pseudo-acoustic anisotropic orthorhombic media depend on only five anisotropic parameters and thevertical velocity as shown by Tsvankin (1997). Instead of rigorously using the notation suggested by Tsvankin (1997), we implement adifferent parametrization shown by Alkhalifah (2003), where the anisotropic aspect of the medium is defined in terms of the anellipticityvalues η, instead of the Thomsen’s anisotropy parameters ε and δ (Thomsen 1986). This convention is used to facilitate the interpretationprocess in terms of η. On the other hand, the normal moveout (NMO) velocities are the natural extensions of their isotropic counterparts forsmall offsets (source–receiver distance). Thus,

vv ≡√

c33

ρ, v1 ≡

√c13(c13 + 2c55) + c33c55

ρ(c33 − c55), (1)

and v2 ≡√

c23(c23 + 2c44) + c33c44

ρ(c33 − c44), (2)

define the P-wave vertical velocity, and the NMO P-wave velocities for horizontal reflectors defined in the [x1, x3] and [x2, x3] planes ofmirror symmetry, respectively. The term ρ represents the density. The coefficients cij correspond to the elastic modulus tensor components inVoigt notation (Thomsen 1986) that characterize the elasticity of the medium. Now, for the anisotropic parameters,

η1 ≡ c11(c33 − c55)

2c13(c13 + 2c55) + 2c33c55− 1

2, (3)

η2 ≡ c22(c33 − c44)

2c23(c23 + 2c44) + 2c33c44− 1

2, (4)

δ ≡ (c12 + c66)2 − (c11 − c66)2

2c11(c11 − c66), (5)

where the first two values, η1 and η2 define the anellipticity in the [x1, x3] and [x2, x3] symmetry planes, respectively, and δ represents theanisotropic parameter in the [x1, x2] plane, defined with respect to the x1 coordinate axis. In order to ease some of the derivations, the parameterδ is used in the eikonal fast marching algorithm under the following definition γ ≡ √

1 + 2δ, where the value of γ is also internally definedin 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 equationof motion with no body forces using Einstein summation convention, given by

ρ(x)∂2ui

∂t2= ∂σi j

∂x j, (6)

where ui represents the displacement vector component and σ ij defines the stress tensor that accounts for the inhomogeneity as well as theanisotropy of the medium, with the elasticity tensor being functions of position. Namely,

σi j = 1

2ci jkl (x)

(∂uk

∂xl+ ∂ul

∂xk

), (7)

with the elasticity coefficients cijkl(x) fully described in tensor notation. Inserting eq. (7) into eq. (6), we obtain

ρ(x)∂2ui

∂t2= 1

2

∂ci jkl (x)

∂x j

(∂uk

∂xl+ ∂ul

∂xk

)+ 1

2ci jkl (x)

(∂2uk

∂x j∂xl+ ∂2ul

∂x j∂xk

), (8)

representing the wave equation in anisotropic inhomogeneous continua. To represent a formulation of the solution for this equation, let usconsider 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 positionx, and f(n) defines a scalar function whose argument is represented by n = vo[τ (x) − t], with vo being a constant value defined in velocityunits. The function τ (x) designates a domain that relates position, x, with the traveltime t. Hence, after inserting the trial solution u(x, t) intoeq. (8), followed by a set of different operations, rearranging the terms along with the application of the symmetry properties of the elastictensor cijkl (Slawinski 2003), we obtain,[

ci jkl (x)∂τ

∂x j

∂τ

∂xl− ρ(x)δik

]Ak(x) = 0, (9)

where δik is the Kronecker’s delta and ∂τ/∂xj = pj defines the phase-slowness vector, which represents the slowness value of the wavefrontpropagation in a particular position xj. In terms of the gradient operator, the term p = ∇τ (x) describes a vector whose direction is normal tothe wavefront and whose magnitude represents the wavefront slowness. Now, we can rewrite eq. (9) in matrix notation as

ρ(x)(p)A(x) = 0, (10)

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where

(p) = ci jkl (x)∂τ

∂x j

∂τ

∂xl/ρ(x) − δik,

= ci jkl (x)p j pl/ρ(x) − δik . (11)

To gain insights into the physical meaning of eq. (11), we now focus our attention to the specific case of orthorhombic media. As shownin Slawinski (2003), using Voigt notation, we can write the elasticity matrix for an orthorhombic continuum as

cortho =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (12)

Using the pseudo-acoustic approximation shown in Alkhalifah (2003), where the vertical velocity of the S wave vs1 = √c55/ρ polarized in

the x1 direction, the S-wave velocity vs2 = √c44/ρ polarized in the x2 direction and the S-wave velocity vs3 = √

c66/ρ polarized in the x2

direction but propagating in the x1 direction, are all zero, and using the definitions in eqs (1)–(5), we can now rewrite Christoffel equation as

(p) =

⎛⎜⎜⎜⎝

p2xv

21(1 + 2η1) − 1 γ px pyv

21(1 + 2η1) px pzv1vv

γ px pyv21(1 + 2η1) p2

yv22(1 + 2η2) − 1 py pzv2vv

px pzv1vv py pzv2vv p2z v

2v − 1

⎞⎟⎟⎟⎠,

where the values of px, py and pz represent the Cartesian components of the phase vector p. Taking the determinant of (p), setting theresultant linear equation to zero and solving for the squared vertical component of the phase vector p2

z , yields the dispersion relation fororthorhombic media,

p2z = 1 − (1 + 2η2)p2

yv22 − (1 + 2η1)p2

xv21

{1 + p2

y

[(1 + 2η1)γ 2v2

1 − (1 + 2η2)v22

]}v2

v

[1 − 2η2 p2

yv22 − p2

xv21

(2η1 + γ 2 p2

yv21 + 4η1(1 + η1)γ 2 p2

yv21 − 2(1 + 2η1)γ p2

yv1v2 + (1 − 4η1η2)p2yv

22

)] . (13)

3 T R AV E LT I M E E I KO NA L S O LU T I O N S

Based on the phase-slowness vector definition, expression (13) becomes a partial differential equation, where,

(∂τ

∂z

)2

=1 − (1 + 2η2)

(∂τ

∂y

)2v2

2 − (1 + 2η1)(

∂τ

∂x

)2v2

1

{1 +

(∂τ

∂y

)2 [(1 + 2η1)γ 2v2

1 − (1 + 2η2)v22

]}

ζ(

∂τ

∂x , ∂τ

∂y

) , (14)

with the term ζ in the denominator defined as

ζ

(∂τ

∂x,∂τ

∂y

)= v2

v

{1 − 2η2

(∂τ

∂y

)2

v22 −

(∂τ

∂x

)2

v21

[2η1 + γ 2

(∂τ

∂y

)2

v21 + 4η1(1 + η1)γ 2

(∂τ

∂y

)2

v21

−2(1 + 2η1)γ

(∂τ

∂y

)2

v1v2 + (1 − 4η1η2)

(∂τ

∂y

)2

v22

]}. (15)

Setting v1 = v2 = vv , η1 = η2 = 0 and γ = 1 increases the symmetry of eq. (13) in terms of the phase vector components and respectivelyprovides the isotropic dispersion relation p2

z = 1/v2 − p2x − p2

y , where v represents the unique velocity field for isotropic media. Followingthe same approach for the VTI dispersion relation case, v1 = v2, η1 = η2 = η and γ = 1 or equivalently setting one of the phase vectorcomponents py = 0 or px = 0 in eq. (13), yields

p2z = 1

v2v

(1 − v2

1 p2x

1 − 2η1v21 p2

x

), (16)

which represents the VTI acoustic dispersion relation.Rewriting eq. (13) in terms of the corresponding traveltime-spatial derivatives yields the orthorhombic eikonal equation shown in

expression (14). To facilitate the implementation of a finite difference scheme in eq. (14), we factorize and rearrange all common coefficientsin terms of the time derivative components. By doing this, a simplified form of the eikonal eq. (14) may be found, bringing together all

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medium properties in a new sequence of coefficients, such that

v21(1 + 2η1)

(∂τ

∂x

)2

+ v22(1 + 2η2)

(∂τ

∂y

)2

+ [(1 + 4η1 + 4η2

1)γ 2v41 − (1 + 2η1 + 2η2 + 4η1η2)v2

1v22

] (∂τ

∂x

)2 (∂τ

∂y

)2

+ [2(1 + 2η1)γ v3

1v2v2v + (4η1η2 − 1)v2

1v22v

2v − (1 + 4η1 + 4η2

1)γ 2v41v

2v

] (∂τ

∂x

)2 (∂τ

∂y

)2 (∂τ

∂z

)2

−2η1v21v

2v

(∂τ

∂x

)2 (∂τ

∂z

)2

− 2η2v22v

2v

(∂τ

∂y

)2 (∂τ

∂z

)2

+ v2v

(∂τ

∂z

)2

= 1. (17)

Rewriting eq. (17), factorizing all velocities fields and anisotropic parameters into a new set of values, we obtain

A

(∂τ

∂x

)2

+ B

(∂τ

∂y

)2

+ C

(∂τ

∂z

)2

+ D

(∂τ

∂x

)2 (∂τ

∂y

)2

+ E

(∂τ

∂x

)2 (∂τ

∂z

)2

+ F

(∂τ

∂y

)2 (∂τ

∂z

)2

+ G

(∂τ

∂x

)2 (∂τ

∂y

)2 (∂τ

∂z

)2

= 1,

(18)

where the sequence of terms that includes all medium properties is give by

A = v21(1 + 2η1), B = v2

2(1 + 2η2), C = v2v ,

D = (1 + 4η1 + 4η21)γ 2v4

1 − (1 + 2η1 + 2η2 + 4η1η2)v21v

22,

E = −2η1v21v

2v ,

F = −2η2v22v

2v ,

G = 2(1 + 2η1)γ v31v2v

2v + (4η1η2 − 1)v2

1v22v

2v − (1 + 4η1 + 4η2

1)γ 2v41v

2v .

For the case of A = B, E = F and D = G = 0, eq. (18) reduces to the VTI dispersion relation. Based on a first order finite difference approachand expanding all the resultant quadratic terms, eq. (18) can be transformed into a sixth order polynomial equation as a function of the firstarrival traveltime solution τ ijk, defined in the algorithm at the gridpoint ijk. Therefore, in order to approximate the derivatives of the firstorder nonlinear partial differential equation shown in expression (14) or equivalently in eq. (18), a 3-D backward finite difference method isimplemented, based on a first order scheme. Namely,

A

(τi, j,k − τi−1, j,k

�x

)2

+ B

(τi, j,k − τi, j−1,k

�y

)2

+ C

(τi, j,k − τi, j,k−1

�z

)2

+ D


�x

)2 (τi, j,k − τi, j−1,k

�y

)2

+ E


�x

)2 (τi, j,k − τi, j,k−1

�z

)2

+ G


�x

)2 (τi, j,k − τi, j−1,k

�y

)2 (τi, j,k − τi, j,k−1

�z

)2

+ F

(τi, j,k − τi, j−1,k

�y

)2 (τi, j,k − τi, j,k−1

�z

)2

= 1. (19)

Expanding all quadratic terms in eq. (19) and collecting all common traveltime solutions τ i, j, k using exponential values that share thesame power, eq. (19) may be rewritten in a polynomial form as

β6τ6i jk + β5τ

5i jk + β4τ

4i jk + β3τ

3i jk + β2τ

2i jk + β1τi jk + β0 = 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 followingequation represents the composition of the β4 term,

β4 = D

�x2�y2+ E

�x2�z2+ F

�y2�z2+ Gτ 2

i, j,k−1

�x2�y2�z2+ 4Gτi, j,k−1τi, j−1,k

�x2�y2�z2+ Gτ 2

i, j−1,k

�x2�y2�z2+ 4Gτi, j,k−1τi−1, j,k

�x2�y2�z2

+4 Gτi, j−1,kτi−1, j,k

�x2�y2�z2+ Gτ 2

i−1, j,k

�x2�y2�z2. (20)

Since the rest of coefficients share a similar structure but can be considerably larger and more complex, eq. (20) will be the only coefficientshown here for practical informative purposes. The approach used in these equations facilitates the separation of medium properties and thefinite difference contribution factors, all used to find the required traveltime solution τ ijk. All these parameters are used in the finite differencealgorithm based on the fast marching method. A sequential application of Bairstow’s method (Press et al. 1989) is implemented in order tosolve for the roots of the polynomial equation P(τi jk) = ∑6

p=0 βpτp

i jk = 0, where the sequence of coefficients βp are considered to be real.This algorithm provides a numerical procedure to decompose a polynomial with real coefficients, into a sequence of second order quadraticfactors. Finding these second order quadratic factors from the original sixth order polynomial, allows us to determine the correspondingpolynomial roots (sometimes as complex conjugate pairs), by only solving quadratic formulas. Since complex conjugate roots may be foundin the set of solutions, only the real part of the roots are considered for this particular implementation. These complex solutions represent

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the consequence of not using the exact dispersion relation (acoustic approximation instead) to compute the traveltime solutions. Wavefieldextrapolation 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 E F F E C T I V E V E L O C I T Y F O R WAV E F I E L D E X T R A P O L AT I O N

Inserting the P-wave orthorhombic traveltime solutions τort (equivalent to τ ijk in the algorithm) into the isotropic dispersion relation leads toan equation describing the effective velocity field in the following form:

veff = 1√(∂τort∂x

)2 + (∂τort∂y

)2 + (∂τort∂z

)2. (21)

The effective velocity in eq. (21) integrates all the first arrival kinematic effects of the anisotropic parameters and velocity fields from theorthorhombic model. As a result, an isotropic wavefield extrapolation may be used in order to describe wave propagation in orthorhombicmedia. This alternative procedure only requires the corresponding veff and an isotropic wavefield extrapolation which is computationally lessexpensive than regular orthorhombic wavefield extrapolations (Song & Alkhalifah 2012).

5 N U M E R I C A L E X A M P L E S

In the following example we compare wavefields obtained using the more expensive orthorhombic wavefield extrapolators with the methodproposed here. The model chosen here is suitable to show the benefits and limitations of the approach as it tries to match the first arrivalkinematics. Fig. 1(i) shows an effective velocity model estimated from the traveltime field shown in Fig. 1(h), computed using the set ofinhomogeneous vv , v1, v2, η1, η2 and γ models shown in Fig. 1, with a source located at x = 2.5 km, y = 2.5 km and z = 2.5 km. Theseplots define 2-D slices of the 3-D cube representation of the data with different space or time domains over each side of the figures. Forvelocity, time and anisotropic parameter models, the vertical axis represents the corresponding model depths, whereas the two horizontalaxes correspond to inline and crossline directions of a potential 3-D survey. The blue lines in these figures represent the slice location that isprojected to the respective parallel cube side.

Figs 2(a)–(c) show time snapshots of the orthorhombic wavefield modeled with the corresponding effective velocity shown in Fig. 1(i).In the depth slice figures, a horizontal cut-section of the respective model is used, whereas the inline and crossline slices represent verticalsection on each of the domains, respectively. An isotropic wavefield extrapolation based on the low-rank approximation approach (Fomel et al.2010) is applied to obtain the results shown in Figs 2(a)–(c). In terms of the kinematic aspects of wavefield propagation, equivalent results arefound between the wavefield computed with the isotropic effective velocity approach and the orthorhombic wavefield extrapolation shown inFigs 2(d)–(f). Note that the wavefield extrapolated with the effective velocity experiences a loss in amplitude around the region of highestvelocity variation. Despite the difference in amplitude, first arrival traveltimes, as expected, are found to be equivalent. Figs 2(g)–(l) show thedifference in amplitude and the corresponding match with respect to the first arrival traveltime solution, represented in space domain. Thesetraces represent cross-sections of the data over the different domains. The solid yellow curve superimposed on all the wavefield snapshotsshown in Fig. 2 represents the eikonal traveltime solution at the equivalent time. This traveltime solution is estimated using the orthorhombiceikonal solver proposed in this study. Despite only the first arrival matching with the isotropic wavefield extrapolator, as Alkhalifah et al.(2013) 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. 1, the implemented methodology provides accurate and stable results with a muchsimpler and less expensive technique, used to generate wavefields in acoustic vertical orthorhombic media. Once the effective velocity modelshown in Fig. 1(i) is constructed for a particular source located at x = 2.5 km, y = 2.5 km and z = 2.5 km, it is used to solve the isotropicwave equation. Solving the wave equation in this case involves wavefield extrapolation in inhomogeneous isotropic media, which implies amuch lower cost for the computational process. For this particular model, the actual full orthorhombic wavefield extrapolation is found to beat 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 bythe first arrival component of the corresponding two wavefields, from effective and orthorhombic models. Since only a fitting process basedon first arrival traveltime solution is applied, it exclusively equals first arrival components. Thus, represented by the comparison betweenFigs 2(a)–(d), (b)–(e) and (c)–(f), reflected or late arrivals in the corresponding wavefields, from effective and the actual orthorhombicapproach, do not lead to an accurate matching for most cases. Since only the first arrival traveltime fields are implemented for the inversionprocess, we should not expect equivalent results for an accurate match of later arrivals. The sequence of traces shown in Fig. 2 provides amuch 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 velocitymodel represents also an isotropic velocity field. However, its variation or heterogeneity depends on the strength of anisotropy, so the wavefieldproduced is a correction of the isotropic full wavefield which is expected to include the anisotropic correct traveltime for at least the firstarrivals. Since imaging and inversion updates rely on transmissions, no reflections, this approach may serve to reduce the cost of theseoperations.

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Figure 1. These plots define 2-D slices of the 3-D cube representation of the data with different space or time domains over each side of the figures. Velocitymodels vv (a), v1 (b), v2 (c) and the anisotropic parameters η1 (d), η2 (e) and γ (f) are shown, respectively. Panel (g) represents a contour plot of thetraveltime field computed with the orthorhombic eikonal solver presented in this study. For the calculation of this traveltime field, the sequence of velocityand anisotropic-parameter models was required, as shown in the finite difference scheme of eq. (19), where the final traveltime solution is computed using thefast marching algorithm within the corresponding sixth-order polynomial equation. Panel (h) shows the effective velocity model computed with the isotropicdispersion relation represented in eq. (21), based on the estimation of the solution traveltime-field τort. Notice the presence of head waves at approximately3 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 isgenerated. 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 the2.5 km values.

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Figure 2. Sequence of slices of wavefield snapshots at t = 0.9 [s] from isotropic wavefield extrapolation using the effective velocity shown in Fig. 1(i);(a) depth slice, (b) inline slice, (c) crossline slice. Slices of wavefield snapshots at t = 0.9 [s] from orthorhombic wavefield extrapolation (Song & Alkhalifah2012) using the complete sequence of models vv , v1, v2, η1, η2 and γ ; (d) depth slice, (e) inline slice, (f) crossline slice. The solid yellow curve on all thesnapshots represents the corresponding traveltime solution estimated with the orthorhombic eikonal solver proposed in this study. Fig. 2(g) represents theoverlapping between traces from the orthorhombic and effective wavefields at inline 2.5 km taken from the wavefields shown in Figs 2(a) and (d). Figs 2(h), (i),(j), (k) and (l) are equivalently generated from the different wavefield components. Dotted and solid curves represent the orthorhombic and effective wavefieldsolutions, respectively.

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6 C O N C LU S I O N S

The numerical examples demonstrate that the presented algorithm based on the fast marching method is stable and accurate for calculatingfirst arrival traveltimes. The high-frequency asymptotic solutions implicit in the orthorhombic eikonal solver fit adequately the wavefrontsextrapolated from the orthorhombic low-rank solution. In addition, the effective isotropic wavefield extrapolation approach is kinematicallyaccurate when compared to results obtained from the orthorhombic wavefield extrapolation. The kinematic aspect of the correspondingwavefields, calculated from the effective and the actual orthorhombic approach, are found to be equivalent. This serves as a platformfor 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 arecomputed from the first arrival traveltime solution, they do not accurately reproduce reflected or late events in the data. Despite the presenteddynamic variations between the respective solutions, the method implemented in this study serves as a platform for evaluating approximateanisotropic wavefields using efficient isotropic extrapolators. This implies a significant decrease in computational cost, without compromisingon the accuracy of the kinematic aspect of wave propagation. Inversion algorithms are usually bottle-necked by the computational cost offorward modeling tool it resorts to, as several such modeling steps are needed for migration/inversion programs. The concept of effectivemodel is useful in reducing the cost of the modeling step. Particularly for low to moderately complex media, the effective model yieldssufficient accuracy.

A C K N OW L E D G E M E N T S

We acknowledge KAUST for the financial support. We also thank the members of the Seismic Wave Analysis Group (SWAG) at KAUST forall their support.

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