Diffraction traveltime approximation for TI mediawith an inhomogeneous background

Umair bin Waheed1, Tariq Alkhalifah1, and Alexey Stovas2

ABSTRACT

Diffractions in seismic data contain valuable informationthat can help improve our modeling capability for better im-aging of the subsurface. They are especially useful for aniso-tropic media because they inherently possess a wide range ofdips necessary to resolve the angular dependence of velocity.We develop a scheme for diffraction traveltime computa-tions based on perturbation of the anellipticity anisotropyparameter for transversely isotropic media with tilted axisof symmetry (TTI). The expansion, therefore, uses anelliptically anisotropic medium with tilt as the backgroundmodel. This formulation has advantages on two fronts: first,it alleviates the computational complexity associated withsolving the TTI eikonal equation, and second, it providesa mechanism to scan for the best-fitting anellipticity param-eter η without the need for repetitive modeling of traveltimes,because the traveltime coefficients of the expansion are in-dependent of the perturbed parameter η. The accuracy of suchan expansion is further enhanced by the use of Shanks trans-form. We established the effectiveness of the proposed for-mulation with tests on a homogeneous TTI model andcomplex media such as the Marmousi and BP models.

INTRODUCTION

Diffracted waves carry valuable information regarding the sub-surface geometry and velocity. Such information can be used toimage geologic features beyond the classical Rayleigh limit of halfof seismic wavelength and for velocity model update (Khaidukovet al., 2004; Sava et al., 2005; Moser and Howard, 2008; Reshef and

Landa, 2009). In spite of that, diffractions have long been regardedas noise in seismic processing and migration. During the lastdecade, there has been a steady increase of interest in diffractedwaves. Attempts for diffraction imaging, however, have mainlyfocused on isotropic media (Landa et al., 1987; Kanasewich andPhadke, 1988; Landa and Keydar, 1998; Fomel et al., 2007). Atransversely isotropic (TI) model with tilted symmetry axis (TTI)is regarded as one of the most effective approximations to theearth’s subsurface, especially for imaging purposes (Zhou et al.,2006; Huang et al., 2008). Therefore, diffraction imaging basedon the TI approximation is expected to be more accurate, in additionto the potential that diffractions provide in estimating velocity varia-tion with angle.In anisotropic media, traveltime computations depend on

more than one model parameter. The P-wave traveltimes in threedimensions for anisotropic media, under the acoustic assumption,depend on the symmetry-axis velocity v0, the normal-moveout(NMO) equivalent velocity vnmo ¼ v0

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δ

p(where δ is a

Thomsen parameter; Thomsen, 1986) and the anellipticity param-eter η ¼ ϵ−δ

1þ2δ (where ϵ is also a Thomsen parameter [Thomsen,1986]). In addition, it also depends on the angle θ that the symmetryaxis makes with the vertical and the azimuthal angle ϕ of the planecontaining the symmetry axis with respect to the x-axis (Tsvan-kin, 1997).Traveltime computations for TI media using finite-difference

schemes are computationally tedious because they require solvinga quartic equation at each time evaluation step. Alkhalifah (2011a)used an elliptically anisotropic model as the starting point for atraveltime computation framework based on perturbation of η in aTaylor-series-type expansion. This simplification is useful becauseelliptically anisotropic media, although represent an uncommonmodel in practice, have the same order of complexity as isotropicmedia in terms of solving the eikonal equation. However, becauseelliptical anisotropy does not provide accurate focusing for media of

Manuscript received by the Editor 28 September 2012; revised manuscript received 15 April 2013; published online 2 August 2013; corrected versionpublished online 17 September 2013.

1King Abdullah University of Science and Technology (KAUST), Physical Sciences and Engineering (PSE) Division, Saudi Arabia. E-mail: umairbin.waheed@kaust.edu.sa; tariq.alkhalifah@kaust.edu.sa.

2Norwegian University of Science and Technology (NTNU), Department of Petroleum and Applied Geophysics, Trondheim, Norway. E-mail: alexey.stovas@ntnu.no.© 2013 Society of Exploration Geophysicists. All rights reserved.

WC103

GEOPHYSICS, VOL. 78, NO. 5 (SEPTEMBER-OCTOBER 2013); P. WC103–WC111, 10 FIGS.10.1190/GEO2012-0413.1

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

nonelliptical TI anisotropy (Alkhalifah and Larner, 1994), it can beused as the background medium for the perturbation expansion.As a result, Alkhalifah (2011b) used perturbation theory to de-

velop traveltime solutions for transversely isotropic media usingan elliptically anisotropic model with vertical symmetry axis asthe background. Thus, he had to perturb the tilt angle θ and the anel-lipticity parameter η. However, the accuracy of the expansion in sin θwould suffer from inaccuracies for large tilt values. Therefore,Stovas and Alkhalifah (2012) suggested the use of a tilted ellipticallyanisotropic (TEA) background model, thereby requiring perturba-tion in η only. The result is higher accuracy and less uncertainty com-pared with the earlier approach, especially if the tilt direction is fixedto be normal to the reflector dip, as it is commonly assumed.In this paper, we develop a scheme for diffraction traveltime com-

putation using perturbation theory. Specifically, we expand thesource and receiver traveltimes with regards to a fixed η. This resultsin a more accurate forward modeling scheme for diffraction datathan the simplified isotropic model of the earth. The accuracy ofsuch a formulation is further enhanced by using Shanks transform(Bender and Orszag, 1978). We demonstrate the applicability of ourformulation on a homogeneous TTI model and complex media likethe VTI Marmousi model (Alkhalifah, 1997) and the BP TTI model(Billette and Brandsberg-Dahl, 2005).

THE TI EIKONAL EQUATION

The 2D eikonal equation in vertically transverse isotopic (VTI)media, under the acoustic assumption, is given as (Alkhalifah,1998)

v2nmoð1þ 2ηÞ�∂τ∂x

�2

þ v02�∂τ∂z

�2�1 − 2ηv2nmo

�∂τ∂x

�2�

¼ 1; (1)

where τðx; zÞ is the traveltime measured from the source to a pointwith the coordinates ðx; zÞ, v0 and vnmo are the vertical and NMOvelocities, respectively, measured along the symmetry axis and ηdenotes the anellipticity parameter.For a TTI medium, the traveltime derivatives in equation 1 are

taken with respect to the tilt direction θ that the symmetry axismakes with the vertical. Thus, we use the following rotation oper-ator for the 2D case: �

cos θ sin θ− sin θ cos θ

�:

Consequently, the 2D eikonal equation for a TTI medium becomes

v2nmoð1þ 2ηÞ�cos θ

∂τ∂x

þ sin θ∂τ∂z

�2

þ v02�cos θ

∂τ∂z

− sin θ∂τ∂x

�2

×�1 − 2ηv2nmo

�cos θ

∂τ∂x

þ sin θ∂τ∂z

�2�

¼ 1: (2)

The numerical solution of equation 2 requires solving a quarticequation at each time step of the finite difference implementation.Alternatively, Alkhalifah (2011b) proposes the use of perturbationtheory (Bender and Orszag, 1978) by approximating equation 2

with a series of simpler linear equations. Here, we follow Stovasand Alkhalifah (2012) and use a TEA medium as a backgroundmodel and expand in terms of the parameter η.The 2D eikonal equation in TEA media resulting from setting

η ¼ 0 in equation 2 takes the form,

v2nmo

�cos θ

∂τ∂x

þ sin θ∂τ∂z

�2

þ v02�cos θ

∂τ∂z

− sin θ∂τ∂x

�2

¼ 1: (3)

The proposed trial solution is

τðx; zÞ ≈ τ0ðx; zÞ þ τ1ðx; zÞηþ τ2ðx; zÞη2; (4)

where τ0, τ1, and τ2 are coefficients of the expansion with dimen-sion of traveltime. For practical purposes, we consider only threeterms of the expansion.We substitute the trial solution in the TI eikonal equation 2,

expand the resulting equation as polynomial in η, and compare thecoefficients of powers of η, in succession, from the left-hand side tothose on the right-hand side. We note that τ0 satisfies the TEA ei-konal equation 3, and is obtained by solving equation 3 using a fastmarching-type eikonal solver (Sethian and Popovici, 1999). How-ever, τ1 and τ2 satisfy linear first-order PDEs having the followingform:

�ðv2nmo cos

2θ þ v20 sin2θÞ ∂τ0

∂xþ sin θ cos θ

�v2nmo − v20Þ

∂τ0∂z

�∂τi∂x

þ ðsin θ cos θðv2nmo − v20Þ∂τ0∂x

þ�v2nmo sin

2θ þ v20 cos2θ

�∂τ0∂z

�∂τi∂z

¼ fiðx; zÞ; (5)

where i ¼ 1; 2. The right-hand side functions fiðx; zÞ get morecomplicated for larger i and depend on terms that can be evaluatedsequentially starting with i ¼ 1. Appendix A presents full versionsof these equations needed to solve for τ1 and τ2.We can increase the accuracy of the traveltime expansion in equa-

tion 4 by using Shanks transform, which requires at least up to sec-ond-order terms of the expansion. Therefore, once τ0, τ1, and τ2have been evaluated, traveltimes can be calculated using the firstsequence of Shanks transform, given as

τðx; zÞ ≈ A0A2 − A21

A0 − 2A1 þ A2

; (6)

where

A0 ¼ τ0; A1 ¼ τ0 þ τ1η; A2 ¼ τ0 þ τ1ηþ τ2η2: (7)

This transform tends to predict the behavior of the higher-orderterms of the sequence, thereby improving the accuracy of the ex-pansion (Bender and Orszag, 1978).Substituting A0, A1, and A2 from equation 7 into equation 6, we

get the following traveltime expression:

τðx; zÞ ≈ τ0ðx; zÞ þητ21ðx; zÞ

τ1ðx; zÞ − ητ2ðx; zÞ: (8)

WC104 Waheed et al.

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Here, we present the 2D case for simplicity. A full 3D version ofequations 1, 2, 3, and 5 can be found in Appendix B.

DIFFRACTION TRAVELTIMES IN TTIHOMOGENEOUS MEDIA

Although the theoretical framework discussed in the previoussection can provide traveltimes corresponding to an inhomogeneousmedium allowing symmetry axis and velocities to vary, in this sec-tion we present equations for diffraction traveltimes in a TTI homo-geneous medium and assess their accuracy.The traveltime formulation discussed in equation 4 and the

Shanks representation of it in equation 8 describe a one-way wave.For a diffracted wave, we need to add two such components: onefrom the source to the diffractor and the other from the diffractor tothe receiver. For the diffraction geometry shown in Figure 1, ðxs; 0Þand ðxr; 0Þ denote the source and receiver locations, respectively,and the diffractor is placed at ðxd; zdÞ. The coefficients of the trav-eltime expansion in equation 4 are given as (Stovas and Alkhalifah,2012)

τ0 ¼ τ0sðxs; xd; zdÞ þ τ0rðxr; xd; zdÞ; (9)

where τ0s and τ0r are hyperbolic expressions with shifted minimaand are given as (Golikov and Stovas, 2012)

τ0sðxs; xd; zdÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis21v2nmo

þ s22v20

s; (10)

τ0rðxr; xd; zdÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21v2nmo

þ r22v20

s; (11)

where

s1 ¼ ðxs − xdÞ cos θ þ zd sin θ;

s2 ¼ −ðxs − xdÞ sin θ þ zd cos θ; (12)

r1 ¼ ðxd − xrÞ cos θ þ zd sin θ;

r2 ¼ −ðxd − xrÞ sin θ þ zd cos θ; (13)

where θ denotes the angle measured from the vertical.The coefficient τ1 is given as

τ1 ¼ −τ0sv40s

41

ðv20s21 þ v2nmos22Þ2−

τ0rv40r41

ðv20r21 þ v2nmor22Þ2; (14)

and τ2 is given as

τ2 ¼3τ0sv60s

61ðv20s21 þ 4v2nmos22Þ

2ðv20s21 þ v2nmos22Þ4

þ 3τ0rv60r61ðv20r21 þ 4v2nmor22Þ

2ðv20r21 þ v2nmor22Þ4; (15)

where τ0s; τ0r; s1; s2; r1, and r2 are defined by equations 10–13, re-spectively.

Substituting equations 9, 14, and 15 into the trial solutiongiven by equation 4 results in the following eikonal solutionapproximation:

τ ¼ τ0s

�1 −

v40s41η

ðv20s21 þ v2nmos22Þ2þ 3v60s

61ðv20s21 þ 4v2nmos22Þη22ðv20s21 þ v2nmos22Þ4

�

þ τ0r

�1 −

v40r41η

ðv20r21 þ v2nmor22Þ2þ 3v60r

61ðv20r21 þ 4v2nmor22Þη22ðv20r21 þ v2nmor22Þ4

�: (16)

The accuracy of this expansion is further enhanced by the use ofShanks transform. Substituting expressions for τ0; τ1, and τ2 intoequations 6 and 7, we get

τ ≈ ðτ0s þ τ0rÞ�1þ ðΦ1 þ Φ2Þη

1þ Φ2η

�; (17)

where

Φ1 ¼−τ0sv40s

41ðv20r21þ v2nmor22Þ2þ τ0rv40r

41ðv20s21þ v2nmos22Þ2

ðτ0sþ τ0rÞððv20s21þ v2nmos22Þ2þðv20r21þ v2nmor22Þ2Þ;

(18)

Φ2 ¼3v60

2ðv20s21þv2nmos22Þ2ðv20r21þv2nmor22Þ2

�

1

τ0sv40s41ðv20r21þv2nmor22Þ2þ τ0rv40r

41ðv20s21þv2nmos22Þ2

�× ðτ0ss61ðv20s21þ4v2nmos22Þðv20r21þv2nmor22Þ4þ τ0rr61ðv20r21þ4v2nmor22Þðv20s21þv2nmos22Þ4Þ: (19)

We could also consider the special case of δ ¼ 0 or vnmo ¼ v0, oftenconsidered in the absence of logging data or other a priori informa-tion to resolve the vertical velocity. In this case, the traveltime co-efficients τ0s and τ0r given by the equations 10 and 11, respectively,reduce to

Figure 1. Schematic plot showing a diffraction experiment for ahomogeneous TTI medium. The source and receiver are locatedat ðxs; 0Þ and ðxr; 0Þ, respectively, whereas the diffractor is posi-tioned at ðxd; zdÞ.

Diffraction traveltimes in TI media WC105

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

τ0s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffis21 þ s22

pv0

; τ0r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 þ r22

pv0

: (20)

Substituting the expressions for τ0s and τ0r from equation 20 inequation 9 yields

τ0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffis21 þ s22

pv0

þffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 þ r22

pv0

: (21)

Replacing s1; s2 from equation 12 and r1; r2 from equation 13, τ0simplifies to yield the isotropic homogeneous traveltime equationfor diffracted waves:

τ0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxs − xdÞ2 þ z2d

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxd − xrÞ2 þ z2d

pv0

; (22)

whereas the coefficients for first- and second-order terms are sim-plified to be

τ1 ¼ −ððxs − xdÞ cos θ þ zd sin θÞ4

v0ððxs − xdÞ2 þ z2dÞ3∕2

−ððxd − xrÞ cos θ þ zd sin θÞ4

v0ððxr − xdÞ2 þ z2dÞ3∕2; (23)

τ2 ¼�3ððxs − xdÞ cos θ þ zd sin θÞ6

2v0ððxs − xdÞ2 þ z2dÞ7∕2�

× ððxs − xdÞ2ð1þ 3sin2θÞ − 6ðxs − xdÞzd sin θ cos θ

þ z2dð4 − 3 sin2θÞÞ þ�3ððxd − xrÞ cos θ þ zd sin θÞ6

2v0ððxd − xrÞ2 þ z2dÞ7∕2�

× ððxd − xrÞ2ð1þ 3 sin2θÞ − 6ðxd − xrÞzd sin θ cos θ

þ z2dð4 − 3 sin2θÞÞ: (24)

To validate the accuracy of the traveltime approximation in equa-tion 17, we consider a TTI homogeneous model with v0 ¼ 2 km∕s,anisotropy parameters δ ¼ 0.1, η ¼ 0.2, and varying symmetrydirection. The diffractor is located beneath the source at a depthof 1 km whereas receivers extend up to an offset of 5 km.Figures 2a–2d present relative errors in traveltimes computed using

zeroth-, first-, and second-order approximations,and the Shanks transform representation for theconsidered model with varying tilt values. In allcases, the Shanks transform yields highly accu-rate traveltimes, even for an extremely largeη value.

DIFFRACTION TRAVELTIMES INTTI INHOMOGENEOUS MEDIUM

Let τs and τr represent the expansions in termsof η for traveltimes from the source to the diffrac-tor and from the diffractor to the receiver, respec-tively. We can then write, using equation 4,

τs ≈ τ0s þ τ1sηþ τ2sη2; (25)

τr ≈ τ0r þ τ1rηþ τ2rη2; (26)

where τ0s; τ1s, and τ2s are coefficients of theexpansion for the source-to-diffractor wave,whereas τ0r; τ1r, and τ2r denote the expansion co-efficients for the wave going from the diffractorto the receiver.Again, using Shanks transform can lead to

higher accuracy. By using the transform givenby equation 6, we get the following traveltimerepresentation for a diffracted wave:

τ ≈ ðτ0s þ τ0rÞ

þ ηðτ21s þ τ21rÞðτ1s þ τ1rÞ − ηðτ2s þ τ2rÞ

: (27)

Because the traveltime coefficients are com-puted using the background TEA inhomogeneous

Figure 2. Relative error in traveltimes versus offset for zeroth-, first-, and second-orderexpansions in η and the Shanks transform for a TTI homogeneous model havingv0 ¼ 2 km∕s, δ ¼ 0.1, η ¼ 0.2 with (a) θ ¼ 0°, (b) θ ¼ 30°, (c) θ ¼ 60°, and (d)θ ¼ 90°. The diffractor is located at a depth of 1 km beneath the source whereas thereceivers extend up to an offset of 5 km. The error curve for zeroth order in (c) and(d) is outside the plotted error range.

WC106 Waheed et al.

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

model, equation 27 allows us to search for the best η that could fitthe diffraction traveltime curve even in models as complex as theMarmousi model.

NUMERICAL TESTS

In this section, we test the accuracy of the Shanks transform ex-pansion for diffraction traveltimes (equation 27) in complex media.Specifically, we test the scheme on the VTI Marmousi model(Alkhalifah, 1997) and the BP TTI model (Billette and Brands-berg-Dahl, 2005). The VTI Marmousi model is interesting dueto severe faulting and folding present in velocity and η models thatinduces diffractions. However, the hallmark of the BP model is thehigh-velocity salt body present that causes complications in accu-rate forward modeling of traveltimes. In addition, it contains non-zero tilt θ. Both models represent a robust test for schemes modelingtraveltimes in anisotropic media.First, we consider a diffractor located at

(2400 m, 4800 m) in the VTI Marmousi model(see the background model in Figure 6 for thegeometry of the Marmousi model) and a sourcelocated at the top left of the model with coordi-nates (3000 m, 0 m). Receivers with a spacing of12 m are distributed along the profile. We solvethe TI eikonal equation 2, using a fast marching-type eikonal solver (Sethian and Popovici, 1999),for a diffraction traveltime curve at the surfaceusing interval η values given by the model(shown in the background of Figure 6b). The ob-tained moveout curve is shown in Figure 3a(solid black curve). We then assume completeignorance of the η model and scan for the effec-tive η value that best fits this moveout curve us-ing the formulation given by equation 27.Figure 3a also plots diffraction traveltime curvesassociated with η ranging from 0 to 0.1 in steps of0.02. The yellow dashed curve at the top corre-sponds to η ¼ 0, whereas the blue dashed curvein the bottom corresponds to η ¼ 0.1, moving se-quentially. In Figure 3b, we show relative errorassociated with these effective η values. Basedon this, we can choose the best effective η valuefor a particular source and diffractor position.Likewise, Figures 4 and 5 present η scan

curves for a source at the center (5000 m,0 m) and on the right (7000 m, 0 m) of the modeltop, respectively. The diffractor position is keptfixed at (2400 m, 4800 m). In each case, we canchoose the best-fitting η in a similar fashion.Using such analysis can lead to an extremely ac-curate fit for a diffraction traveltime curve. Notethat none of the η values exactly fits the modeledone as they are effective values. A coherencescan approach corresponding to the conventionalvelocity analysis is also possible here for deter-mining the best η curve.Based on Figures 3, 4, and 5, we pick η ¼ 0.02

as the best fit value for the diffractor located at(2400 m, 4800 m). We then plot traveltime con-tours in Figure 6 from this diffractor position,

comparing our perturbation formulation (dashed curves) with exacttraveltimes (solid curves) obtained by solving equation 2. We obtainhighly accurate traveltimes, even though we assumed completeignorance of the ηmodel and used an effective η obtained after scan-ning for it.Next, we perform tests on the BP TTI model (see the background

model in Figure 10 for the geometry of the BP model). We considera diffractor located at (30 km, 9 km) and a source positioned on thetop left of the model at (30 km, 0 km). Receivers with a spacing of10 m are distributed along the profile. Figure 7a depicts the exactdiffraction traveltime curve (solid black) obtained by solving equa-tion 2, taking into account the exact interval η values provided bythe model shown in the background of Figure 10b. Following ouranalysis for the Marmousi model, we assume complete ignorance ofthe η model and scan for it using equation 27. Figure 7a also plotsdiffraction traveltime curves associated with η ranging from 0 to

Figure 3. Scan for the effective η value in the VTI Marmousi model for a source locatedon the top left of the model at (3000 m, 0 m). (a) Diffraction traveltime curves observedat the surface and (b) relative error for these curves obtained for a range of η values from0 to 0.1 in steps of 0.02. The yellow dashed curve at the top corresponds to η ¼ 0,whereas the blue dashed curve at the bottom corresponds to η ¼ 0.1, moving sequen-tially. The black solid curve in (a) represents the exact diffraction traveltime curve. Thediffractor considered is located at (4800 m, 2400 m).

Figure 4. Scan for the effective η value in the VTI Marmousi model for a source locatedat (5000 m, 0 m). (a) Diffraction traveltime curves observed at the surface and (b) rel-ative error for these curves obtained for a range of η values from 0 to 0.1 in steps of 0.02.The yellow dashed curve at the top corresponds to η ¼ 0, whereas the blue dashed curveat the bottom corresponds to η ¼ 0.1, moving sequentially. The black solid curve in (a)represents the exact diffraction traveltime curve. The diffractor considered is located at(4800 m, 2400 m).

Diffraction traveltimes in TI media WC107

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

0.05 in steps of 0.01. The yellow dashed curve at the top corre-sponds to η ¼ 0 whereas the blue dashed curve in the bottomcorresponds to η ¼ 0.05, moving sequentially. Figure 7b plotsthe relative error for each of the considered η values. Thus, it allowsus a framework to choose the best effective η value for the consid-ered source and diffractor positions.

In a similar fashion, keeping the diffractor position fixed at(30 km, 9 km), we can choose effective η values for a source locatedat the center (32 km, 0 km) and to the right (34 km, 0 km) of themodel top from Figures 8 and 9, respectively.Next, based on Figures 7, 8, and 9, we pick η ¼ 0.03 as the best

effective value for the considered diffractor at (30 km, 9 km). As in

Figure 7. Scan for the effective η value in the BPmodel for a source located on the top left of themodel at (30 km, 0 km). (a) Diffraction traveltimecurves observed at the surface and (b) relative er-ror for these curves obtained for a range of η val-ues from 0 to 0.05 in steps of 0.01. The yellowdashed curve at the top corresponds to η ¼ 0,whereas the blue dashed curve at the bottom cor-responds to η ¼ 0.05, moving sequentially. Theblack solid curve in (a) represents the exact dif-fraction traveltime curve. The diffractor consid-ered is located at (30 km, 9 km).

Figure 6. Traveltime contours for the VTI Mar-mousi model using the Shanks transform expan-sion (dashed lines) and the exact TTI eikonalsolution (solid lines) mapped on (a) velocitymodel and (b) η model. The diffractor consideredis located at (4800 m, 2400 m). The exact solutionuses the interval η values given by the model in (b)whereas the Shanks transform expansion uses aneffective η value of 0.02.

Figure 5. Scan for the effective η value in the VTIMarmousi model for a source located on the topright of the model at (7000 m, 0 m). (a) Diffractiontraveltime curves observed at the surface and (b)relative error for these curves obtained for a rangeof η values from 0 to 0.1 in steps of 0.02. The yel-low dashed curve at the top corresponds to η ¼ 0,whereas the blue dashed curve at the bottom cor-responds to η ¼ 0.1, moving sequentially. Theblack solid curve in (a) represents the exact dif-fraction traveltime curve. The diffractor consid-ered is located at (4800 m, 2400 m).

Figure 8. Scan for the effective η value in the BPmodel for a source located at (32 km, 0 km). (a)Diffraction traveltime curves observed at the sur-face and (b) relative error for these curves obtainedfor a range of η values from 0 to 0.05 in steps of0.01. The yellow dashed curve at the top corre-sponds to η ¼ 0 whereas blue dashed curve atthe bottom corresponds to η ¼ 0.05, moving se-quentially. The black solid curve in (a) representsthe exact diffraction traveltime curve. The diffrac-tor considered is located at (30 km, 9 km).

WC108 Waheed et al.

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Figure 6, we compare the accuracy of our chosen effective η valuefor the BP TTI model in Figure 10. Again, we obtain remarkableaccuracy for most part of the model, keeping in view the fact that theperturbation solution (dashed curves) used an effective η value ob-tained after scanning for it.The above examples demonstrate the potential of equation 27 in

yielding highly accurate traveltimes, even for highly complexmedia, where the η model is unknown.

CONCLUSIONS

Accurate diffraction imaging requires efficient forward modelingschemes that can incorporate the complexities of real earth. A dif-fraction traveltime formulation based on perturbation theory allevi-ates the computational burden associated with solving the exact TTIeikonal equation, in addition to yielding high accuracy. This isachieved by expanding the solution of the TTI eikonal equationin terms of the independent parameter η. The perturbation expansionrequires solving a quadratic equation for a TEA background model

instead of a quartic equation needed for the original TTI eikonalsolution. The accuracy of the expansion is enhanced by usingShanks transform, which allows a better representation with fewerterms of the expansion, and consequently fewer equations to solve.An added advantage of this formulation lies in scanning for best ηthat fits the diffraction curve without the need to compute travel-times again. We demonstrated these assertions through tests on ahomogeneous TTI model and complex media such as the VTI Mar-mousi model and the BP TTI model. These tests also demonstratethe applicability of the proposed scheme for modeling diffractiontraveltimes even in highly complex media.

ACKNOWLEDGMENTS

We would like to thank KAUST and ROSE project for financialsupport. We also extend thanks to BP for releasing the benchmarksynthetic model. We acknowledge useful discussions with DavidKetcheson on implementing the TTI eikonal solver. We areextremely grateful to Claudia Vanelle, Stig-Kyrre Foss, Zvi Koren,

Figure 10. Traveltime contours for the BP modelusing the Shanks transform expansion (dashedlines) and the exact TTI eikonal solution (solidlines) mapped on (a) velocity model, (b) η model,(c) θ model, and (d) δ model. The diffractor con-sidered is located at (30 km, 9 km). The exact sol-ution uses the interval η values given by the modelin (b) whereas the Shanks transform expansionuses an effective η value of 0.03.

Figure 9. Scan for the effective η value in the BPmodel for a source located on top right of themodel at (34 km, 0 km). (a) Diffraction traveltimecurves observed at the surface and (b) relativeerror for these curves obtained for a range of ηvalues from 0 to 0.05 in steps of 0.01. The yellowdashed curve at the top corresponds to η ¼ 0,whereas the blue dashed curve at the bottom cor-responds to η ¼ 0.05, moving sequentially. Theblack solid curve in (a) represents the exact dif-fraction traveltime curve. The diffractor consid-ered is located at (30 km, 9 km).

Diffraction traveltimes in TI media WC109

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

and Sergius Dell for useful reviews that improved the quality ofthe paper.

APPENDIX A

CALCULATING COEFFICIENTS FORTRAVELTIME EXPANSION

The P-wave eikonal equation in 2D (for simplicity) TEA media isgiven as

v2nmoð1þ 2ηÞ�cos θ

∂τ∂x

þ sin θ∂τ∂z

�2

þ v02�cos θ

∂τ∂z

− sin θ∂τ∂x

�2

×�1 − 2ηv2nmo

�cos θ

∂τ∂x

þ sin θ∂τ∂z

�2�

¼ 1: (A-1)

To solve A-1 through perturbation theory, we assume that η issmall and therefore a trial solution expressed as expansion in ηabout η ¼ 0 can be formulated as

τðx; zÞ ≈ τ0ðx; zÞ þ τ1ðx; zÞηþ τ2ðx; zÞη2; (A-2)

where τ0; τ1, and τ2 are coefficients of expansion given in units oftraveltime and are terminated after three terms for practical consid-erations.Inserting the proposed solution A-2 in A-1 and setting η ¼ 0

yields

v2nmo

�cos θ

∂τ0∂x

þ sin θ∂τ0∂z

�2

þ v02�cos θ

∂τ0∂z

− sin θ∂τ0∂x

�2

¼ 1; (A-3)

which is the TEA eikonal. By equating the coefficients of thepowers of the independent parameter η, in succession, we obtainfirst the coefficients of the first power in η, which, after simplifica-tion using equation A-3, is given by (Stovas and Alkhalifah, 2012)

�ðv2nmo cos

2θþv20 sin2θÞ∂τ0

∂xþ sin θ cos θðv2nmo −v20Þ

∂τ0∂z

�∂τ1∂x

þ�sin θ cos θðv2nmo − v20Þ

∂τ0∂x

þðv2nmo sin2θþ v20 cos

2θÞ∂τ0∂z

�∂τ1∂z

¼−v2nmo

�cos θ

∂τ0∂x

þ sin θ∂τ0∂z

�2�v20

�cos θ

∂τ0∂z

− sin θ∂τ0∂x

�2

− 1

�:

(A-4)

Because τ0 has already been computed, this equation represents alinear partial differential equation in terms of τ1. The coefficients ofthe second power of η, after some algebra, result in the followingrelation:

�ðv2nmocos

2θþ v20sin2θÞ∂τ0

∂xþ sin θ cos θðv2nmo − v20Þ

∂τ0∂z

�∂τ2∂x

þ�sin θ cos θðv2nmo − v20Þ

∂τ0∂x

þðv2nmo sin2θþ v20 cos

2θÞ∂τ0∂z

�∂τ2∂z

¼−1

2

�v2nmo

�cos θ

∂τ1∂x

þ sin θ∂τ1∂z

�2

þ v20

�cos θ

∂τ1∂z

− sin θ∂τ1∂x

�2�

þ 2

�v20

�cos θ

∂τ0∂z

− sin θ∂τ0∂x

��cos θ

∂τ1∂z

− sin θ∂τ1∂x

�

−v2nmo

�cos θ

∂τ0∂x

þ sin θ∂τ0∂z

��cos θ

∂τ1∂x

þ sin θ∂τ1∂z

��

×�1− v20

�cos θ

∂τ0∂z

− sin θ∂τ0∂x

�2�: (A-5)

APPENDIX B

EXPANSION IN 3D

The eikonal equation for P-waves in VTI media, under the acous-tic assumption, is given as (Alkhalifah, 1998)

v2nmoð1þ 2ηÞ��

∂τ∂x

�2

þ�∂τ∂y

�2�

þ v20

�∂τ∂z

�2�1 − 2ηv2nmo

��∂τ∂x

�2

þ�∂τ∂y

�2��

¼ 1: (B-1)

For a TTI medium, the traveltime derivatives in equation B-1 aretaken with respect to the tilt direction, and thus, we use the follow-ing rotation operator in 3D:0

@ cos ϕ cos θ sin ϕ cos θ sin θ− sin ϕ cos ϕ 0

− cos ϕ sin θ − sin ϕ sin θ cos θ

1A;

which yields the TTI eikonal equation for 3D,

v2nmoð1þ 2ηÞ

��

cos θ cos ϕ∂τ∂x

þ cos θ sin ϕ∂τ∂y

þ sin θ∂τ∂z

�2

þ�− sin ϕ

∂τ∂x

þ cos ϕ∂τ∂y

�2�

þ v20

�− sin θ cos ϕ

∂τ∂x

− sin θ sin ϕ∂τ∂y

þ cos θ∂τ∂z

�2

× ð1− 2ηv2nmo

��cos θ cos ϕ

∂τ∂x

þ cos θ sin ϕ∂τ∂y

þ sin θ∂τ∂z

�2

þ�− sin ϕ

∂τ∂x

þ cos ϕ∂τ∂y

�2��

¼ 1: (B-2)

The proposed trial solution is

τðx; y; zÞ ≈ τ0ðx; y; zÞ þ τ1ðx; y; zÞηþ τ2ðx; y; zÞη2: (B-3)

Substituting the trial solution into equation B-2, expanding the re-sulting equation as polynomial in η, and comparing the coefficients

WC110 Waheed et al.

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

of the powers of η in sequence yields the traveltime coefficientsτ0; τ1, and τ2 needed by the trial solution.We obtain τ0 by solving the TEA eikonal equation in 3D,

v2nmoð1þ 2ηÞ��

cos θ cos ϕ∂τ∂x

þ cos θ sin ϕ∂τ∂y

þ sin θ∂τ∂z

�2

þ�− sin ϕ

∂τ∂x

þ cos ϕ∂τ∂y

�2�

þ v20

�−sin θ cos ϕ

∂τ∂x

− sin θ sin ϕ∂τ∂y

þ cos θ∂τ∂z

�2

¼ 1;

(B-4)

while τ1 and τ2 require solving a linear PDE of the form

�v2nmo

�ðcos2θcos2ϕþ sin2ϕÞ∂τ0

∂xþ sin ϕ cos ϕðcos2θ− 1Þ∂τ0

∂y

�

þ v02 cos ϕ sin2θ

�∂τ0∂x

cos ϕþ ∂τ0∂y

sin ϕ

�

þðv2nmo − v20Þ sin θ cos θ cos ϕ∂τ0∂z

�∂τi∂x

þ�v2nmo

�sin ϕ cos ϕðcos2θ− 1Þ∂τ0

∂xþðcos2θsin2ϕþ cos2ϕÞ∂τ0

∂y

�

þ v20sin2θ sin ϕ

�∂τ0∂x

cos ϕþ ∂τ0∂y

sin ϕ

�

þðv2nmo − v20Þ sin θ cos θ sin ϕ∂τ0∂z

�∂τi∂y

þ�ðv2nmo − v20Þ sin θ cos θ

�cos ϕ

∂τ0∂x

þ sin ϕ∂τ0∂y

�þv2nmo sin

2θ∂τ0∂z

þ v02cos2θ∂τ0∂z

�∂τi∂z

¼ fiðx;y;zÞ; (B-5)

where i ¼ 1; 2.The right-hand side function f1ð∂τ0∕∂x; ∂τ0∕∂y; ∂τ0∕∂zÞ is found

to be:

f1

�∂τ0∂x

;∂τ0∂y

;∂τ0∂z

�¼−v2nmo

�cos2ϕ

�cos2θ

�∂τ0∂x

�2

þ�∂τ0∂y

�2�

þ sin2ϕ

��∂τ0∂x

�2

þ cos2θ

�∂τ0∂y

�2�

þ 2 cos ϕ∂τ0∂x

�sin ϕðcos2θ− 1Þ∂τ0

∂yþ sin θ cos θ

∂τ0∂z

�

þ 2 sin ϕ sin θ cos θ∂τ0∂y

∂τ0∂z

þ sin2θ

�∂τ0∂z

�2���

−v0 sin θ�cos ϕ

∂τ0∂x

þ sin ϕ∂τ0∂y

�

þv0 cos θ∂τ0∂z

�2

− 1

�; (B-6)

whereas the right-hand side function f2ð∂τ0∕∂x; ∂τ0∕∂y; ∂τ0∕∂z;∂τ1∕∂x; ∂τ1∕∂y; ∂τ1∕∂zÞ results in a complicated expression and isnot presented here for brevity. However, it can be easily derived usingthe steps outlined above.

REFERENCES

Alkhalifah, T., 1997, Anisotropic Marmousi data set: Stanford ExplorationProject-95, 265–282.

Alkhalifah, T., 1998, Acoustic approximations for processing in transverselyisotropic media: Geophysics, 63, 623–631, doi: 10.1190/1.1444361.

Alkhalifah, T., 2011a, Scanning anisotropy parameters in complex media:Geophysics, 76, no. 2, U13–U22, doi: 10.1190/1.3553015.

Alkhalifah, T., 2011b, Traveltime approximations for transversely isotropicmedia with an inhomogeneous background: Geophysics, 76, no. 3,WA31–WA42, doi: 10.1190/1.3555040.

Alkhalifah, T., and K. Larner, 1994, Migration error in transversely isotropicmedia: Geophysics, 59, 1405–1418, doi: 10.1190/1.1443698.

Bender, C. M., and S. A. Orszag, 1978, Advanced mathematical methods forscientists and engineers: McGraw-Hill.

Billette, F. J., and S. Brandsberg-Dahl, 2005, The 2004 BP velocitybenchmark: 67th Annual Conference and Exposition, EAGE, ExtendedAbstracts, B305.

Fomel, S., E. Landa, and M. T. Taner, 2007, Poststack velocity analysis byseparation and imaging of seismic diffractions: Geophysics, 72, no. 6,U89–U94, doi: 10.1190/1.2781533.

Golikov, P., and A. Stovas, 2012, Traveltime parameters in a tilted ellipticalanisotropic medium: Geophysical Prospecting, 60, 433–443, doi: 10.1111/j.1365-2478.2011.01003.x.

Huang, T., S. Xu, J. Wang, G. Ionescu, and M. Richardson, 2008,The benefit of TTI tomography for dual azimuth data in Gulf ofMexico: 78th Annual International Meeting, SEG, Expanded Abstracts,222–226.

Kanasewich, E. R., and S. M. Phadke, 1988, Imaging discontinuitieson seismic sections: Geophysics, 53, 334–345, doi: 10.1190/1.1442467.

Khaidukov, V., E. Landa, and T. J. Moser, 2004, Diffraction imaging byfocusing-defocusing: An outlook on seismic superresolution: Geophysics,69, 1478–1490, doi: 10.1190/1.1836821.

Landa, E., and S. Keydar, 1998, Seismic monitoring of diffraction images fordetection of local heterogeneities: Geophysics, 63, 1093–1100, doi: 10.1190/1.1444387.

Landa, E., V. Shtivelman, and B. Gelchinsky, 1987, A method for detectionof diffracted waves on common-offset sections: Geophysical Prospecting,35, 359–373, doi: 10.1111/j.1365-2478.1987.tb00823.x.

Moser, T., and C. Howard, 2008, Diffraction imaging in depth: GeophysicalProspecting, 56, 627–641, doi: 10.1111/j.1365-2478.2007.00718.x.

Reshef, M., and E. Landa, 2009, Post-stack velocity analysis in the dip-angledomain using diffractions: Geophysical Prospecting, 57, 811–821, doi: 10.1111/j.1365-2478.2008.00773.x.

Sava, P. C., B. Biondi, and J. Etgen, 2005, Wave-equation migration velocityanalysis by focusing diffractions and reflections: Geophysics, 70, U19–U27, doi: 10.1190/1.1925749.

Sethian, J., and M. Popovici, 1999, 3-D traveltime computation using thefast marching method: Geophysics, 64, 516–523, doi: 10.1190/1.1444558.

Stovas, A., and T. Alkhalifah, 2012, A new traveltime approximationfor TI media: Geophysics, 77, no. 4, C37–C42, doi: 10.1190/geo2011-0158.1.

Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966,doi: 10.1190/1.1442051.

Tsvankin, I., 1997, Moveout analysis for tranversely isotropic media with atilted symmetry axis: Geophysical Prospecting, 45, 479–512, doi: 10.1046/j.1365-2478.1997.380278.x.

Zhou, H., G. Zhang, and R. Bloor, 2006, An anisotropic acoustic wave equa-tion for modeling and migration in 2D TTI media: 76th AnnualInternational Meeting, SEG, Expanded Abstracts, 194–198.

Diffraction traveltimes in TI media WC111

Dow

nloa

ded

01/0

9/16

to 9

8.11

0.16

.10.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/