Elastic wave-mode separation for VTI medianewton.mines.edu/paul/journals/2009_gpy00wb19.pdfElastic wave-mode separation for VTI media Jia Yan1 and Paul Sava1 ABSTRACT Elasticwavepropagationinanisotropicmediaiswellrep-resentedbyelasticwaveequations

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GEOPHYSICS, VOL. 74, NO. 5 �SEPTEMBER-OCTOBER 2009�; P. WB19–WB32, 19 FIGS.10.1190/1.3184014

lastic wave-mode separation for VTI media

ia Yan1 and Paul Sava1

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ABSTRACT

Elastic wave propagation in anisotropic media is well rep-resented by elastic wave equations. Modeling based on elas-tic wave equations characterizes both kinematics and dynam-ics correctly. However, because P- and S-modes are bothpropagated using elastic wave equations, there is a need toseparate P- and S-modes to efficiently apply single-modeprocessing tools. In isotropic media, wave modes are usuallyseparated using Helmholtz decomposition. However, Helm-holtz decomposition using conventional divergence and curloperators in anisotropic media does not give satisfactory re-sults and leaves the different wave modes only partially sepa-rated. The separation of anisotropic wavefields requires moresophisticated operators that depend on local material param-eters. Anisotropic wavefield-separation operators are con-structed using the polarization vectors evaluated at each pointof the medium by solving the Christoffel equation for localmedium parameters. These polarization vectors can be repre-sented in the space domain as localized filtering operators,which resemble conventional derivative operators. The spa-tially variable pseudo-derivative operators perform well inheterogeneous VTI media even at places of rapid velocity/density variation. Synthetic results indicate that the operatorscan be used to separate wavefields for VTI media with an ar-bitrary degree of anisotropy.

INTRODUCTION

Wave-equation migration for elastic data usually consists of twoteps. The first step is reconstructing the subsurface wavefield fromata recorded at the surface. The second step is applying an imagingondition that extracts reflectivity information from the reconstruct-d wavefields.

Elastic wave-equation migration for multicomponent data can bemplemented in two ways. The first approach is to separate recordedlastic data into compressional and transverse �P- and S-� modes and

Manuscript received by the Editor 19 November 2008; revised manuscript1Colorado School of Mines, Center for Wave Phenomena, Golden, Colorad2009 Society of Exploration Geophysicists.All rights reserved.

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o use these separated modes for acoustic wave-equation migration.coustic imaging of elastic waves is used frequently, but it is basedn the assumption that P and S data can be separated successfully onhe surface, which is not always true �Etgen, 1988; Zhe and Green-algh, 1997�. The second approach is not to separate P- and S-modesn the surface, extrapolate the entire elastic wavefield at once, theneparate wave modes before applying an imaging condition. Thelastic wavefields can be reconstructed using various techniques, in-luding reverse time migration �RTM� �Chang and McMechan,986, 1994� and Kirchhoff-integral techniques �Hokstad, 2000�.

The imaging condition applied to the reconstructed vector wave-elds directly determines image quality. The conventional crosscor-elation imaging condition does not separate the wave modes androsscorrelates the Cartesian components of the elastic wavefields.n general, P- and S-wave modes are mixed on all wavefield compo-ents and cause crosstalk and image artifacts. Yan and Sava �2008�uggest using imaging conditions based on elastic potentials, whichequire crosscorrelation of separated modes. Potential-based condi-ions create images that have clear physical meanings, in contrastith images obtained with Cartesian wavefield components, thus

ustifying the need for wave-mode separation.As the need for anisotropic imaging increases, more processing

nd migration are performed based on anisotropic acoustic one-wayave equations �Alkhalifah, 1998, 2000; Shan and Biondi, 2005;han, 2006; Fletcher et al., 2008�. However, much less research haseen done on anisotropic elastic migration based on two-way wavequations. Elastic Kirchhoff migration �Hokstad, 2000� obtainsure-mode and converted-mode images by downward continuationf elastic vector wavefields with a viscoelastic wave equation.avefield separation is done effectively with elastic Kirchhoff inte-

ration, which handles P- and S-waves. However, Kirchhoff migra-ion does not perform well in areas of complex geology where rayheory breaks down �Gray et al., 2001�, thus requiring migrationith more accurate methods, such as RTM.One of the complexities that impedes elastic wave-equation an-

sotropic migration is the difficulty of separating anisotropic wave-elds into different wave modes after reconstructing the elasticavefields. However, proper separation of anisotropic wave modes

s as important for anisotropic elastic migration as is the separation

d 15 February 2009; published online 28 September 2009.A. E-mail: [email protected]; [email protected].

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f isotropic wave modes for isotropic elastic migration. The mainifference between anisotropic and isotropic wavefield separation ishat Helmholtz decomposition is suitable only for separating isotro-ic wavefields and does not work well for anisotropic wavefields.

In this paper, we show how to construct wavefield separators forertically transversely isotropic �VTI� media applicable to modelsith spatially varying parameters. We apply these operators to an-

sotropic elastic wavefields and show that they successfully separatenisotropic wave modes, even for extremely anisotropic media.

The main application of this technique is in the development oflastic RTM. In this case, complete wavefields containing P- and-wave modes are reconstructed from recorded data. The recon-tructed wavefields are separated in pure wave modes before apply-ng a conventional crosscorrelation imaging condition. We limit thecope of our paper to the wave-mode separation procedure in highlyeterogeneous media, although the ultimate goal is to aid elasticTM.

SEPARATION METHOD

Scalar and vector potentials can be separated by Helmholtz de-omposition, which applies to any vector field W�x,y,z�. By defini-ion, the vector wavefield W can be decomposed into a curl-free sca-ar potential � and a divergence-free vector potential � accordingo the relation �Aki and Richards, 2002�

W� �� . �1�

quation 1 is not used directly in practice, but the scalar and vectoromponents are obtained indirectly by the applying the � · and ��perators to the extrapolated elastic wavefield:

P� � ·W, �2�

S� � �W . �3�

or isotropic elastic fields far from the source, quantities P and S de-cribe P- and S-wave modes, respectively �Aki and Richards, 2002�.

Equations 2 and 3 allow us to understand why � · and �� passompressional and transverse wave modes, respectively. In the dis-retized space domain, we can write

a)igure 1. The qP- and qS-mode polarization vec-ors as a function of normalized wavenumbers kxnd kz ranging from �1 to 1 cycle for �a� an isotro-ic model with VP�3 km /s and VS�1.5 km /s,nd �b� an anisotropic �VTI� model with VP0

3 km /s, VS0�1.5 km /s, ��0.25, and � �0.29. The red arrows are the qP-wave polariza-

ion vectors; the blue arrows are the qS-wave polar-zation vectors.


P� � ·W�Dx�Wx��Dy�Wy��Dz�Wz�, �4�

here Dx, Dy, and Dz represent spatial derivatives in the x-, y-, and-directions. Applying derivatives in the space domain is equivalento applying finite-difference filtering to the functions. Here, D� �epresents spatial filtering of the wavefield with finite-difference op-rators. In the Fourier domain, we can represent the operators Dx, Dy,nd Dz by ikx, iky, and ikz; therefore, we can write an expressionquivalent to expression 4 as

P̃� ik ·W̃� ikxW̃x� ikyW̃y � ikzW̃z, �5�

here k� �kx,ky,kz� represents the wave vector and W̃�kx,ky,kz� ishe 3D Fourier transform of the wavefield W�x,y,z�.

In this domain, the operator ik essentially projects the wavefield˜ onto the wave vector k, which represents the polarization direc-ion for P-waves. Similarly, the operator �� projects the wavefieldnto the direction orthogonal to k, which represents the polarizationirection for S-waves �Dellinger and Etgen, 1990�. For illustration,igure 1a shows the polarization vectors of the P-mode of a 2D iso-

ropic model as a function of normalized kx and kz ranging from1 to 1 cycle. The polarization vectors are radial because the

-waves in an isotropic medium are polarized in the same directionss the wave vectors.

Dellinger and Etgen �1990� suggest the idea that wave-mode sep-ration can be extended to anisotropic media by projecting the wave-elds onto the directions in which the P- and S-modes are polarized.his requires that we modify the wave separation expression 5 byrojecting the wavefields onto the true polarization directions U tobtain quasi-P �qP-� waves:

qP˜� iU�k� ·W̃� iUxW̃x� iUyW̃y � iUzW̃z. �6�

In anisotropic media, U�kx,ky,kz� is different from k, as illustratedn Figure 1b, which shows the polarization vectors of qP-wave modeor a 2D VTI anisotropic model with normalized kx and kz rangingrom �1 to 1 cycle. Polarization vectors are not radial because qP-aves in an anisotropic medium are not polarized in the same direc-

ions as wave vectors, except in the symmetry planes �kz�0� andlong the symmetry axis �kx�0�.

Dellinger and Etgen �1990� demonstrate wave-mode separationn the wavenumber domain using projection of the polarization vec-ors, as indicated in equation 6. However, for heterogeneous media,his equation does not work because the polarization vectors varypatially. We can write an equivalent expression to equation 6 in thepace domain for each grid point as

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qP��a ·W�Lx�Wx��Ly�Wy��Lz�Wz�, �7�

here Lx, Ly, and Lz represent the inverse Fourier transforms of iUx,Uy, and iUz and where L� � represents spatial filtering of the wave-eld with anisotropic separators. The values Lx, Ly, and Lz define theseudo-derivative operators in the x-, y-, and z-directions for an an-sotropic medium; they change from location to location accordingo the material parameters.

We obtain the polarization vectors U�k� by solving the Christoffelquation �Aki and Richards, 2002; Tsvankin, 2005�:

�G��V2I�U�0, �8�

here G is the Christoffel matrix Gij�cijklnjnl, in which cijkl is thetiffness tensor, and where nj and nl are the normalized wave-vectoromponents in the j and l directions, i,j,k,��1,2,3. The parametercorresponds to the eigenvalues of the matrix G. The eigenvalues V

epresent the phase velocities of different wave modes and are func-ions of k �corresponding to nj and nl in G�. For plane waves propa-ating in a VTI medium, we can set ky to zero and get

�c11kx2�c55kz

2��V2 0 �c13�c55�kxkz0 c66kx

2�c55kz2��V2 0

�c13�c55�kxkz 0 c55kx2�c33kz

2��V2��UxUy

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�0. �9�

The middle row of this matrix characterizes the SH-wave polar-zed in the y-direction; the qP-and qSV-modes are uncoupled fromhe SH-mode and are polarized in the vertical plane. The top and bot-om rows of this equation allow us to compute the polarization vec-or U� �Ux,Uz��the eigenvectors of the matrix� of the P- or SV-wave

ode, given the stiffness tensor at every location of the medium.We can extend the procedure to heterogeneous media by comput-

ng a different operator at every grid point. In the symmetry planes ofTI media, the 2D operators depend on the local values of the stiff-ess coefficients. For each point, we precompute the polarizationectors as a function of the local medium parameters and transformhem to the space domain to obtain the wave-mode separators. Wessume that the medium parameters vary smoothly �locally homoge-eous�; but even for complex media, the localized operators work inhe same way as the long finite-difference operators. If we representhe stiffness coefficients using Thomsen parameters �Thomsen,986�, then the pseudo-derivative operators Lx and Lz depend on �,, VP0, and VS0, which can be spatially varying parameters.We can compute and store the operators for each grid point in theedium and then use those operators to separate P- and S-modes

rom reconstructed elastic wavefields at different time steps. Thus,avefield separation in VTI media can be achieved by nonstationaryltering with operators Lx and Lz.

OPERATOR PROPERTIES

In this section, we discuss the properties of the anisotropic deriva-ive operators, including order of accuracy, size, and compactness.


perator orders

As we have shown, the isotropic separation operators �divergencend curl� in equations 4 and 5 are exact in the x- and k-domains. Thexact derivative operators are infinitely long series in the discretizedpace domain. In practice, when evaluating the derivatives numeri-ally, we must take some approximations to make the operators shortnd computationally efficient.

Usually, difference operators are evaluated at different orders ofccuracy. The higher-order the approximation is, the more accuratend longer the operator becomes. For example, Figure 2a shows thathe second-order operator has coefficients ��1 /2,�1 /2�, and the

ore accurate fourth-order operator has coefficients ��1 /12,2 /3,2 /3,�1 /12� �Fornberg and Ghrist, 1999�.In the wavenumber domain for isotropic media, as shown by the

lack line in Figure 2b, the exact difference operator is ik. Appendixshows the k-domain equivalents of the second-, fourth-, sixth-,

nd eighth-order finite-difference operators; they are plotted in Fig-re 2b. The higher-order operators have responses closer to the exactperator ik �black line�.

To obtain vertical and horizontal derivatives of different orders ofccuracy, we weight the polarization vector ik components ikx andkz by the weights shown in Figure 2c. For VTI media, similarly, weeight the anisotropic polarization vector iU�k� components iUx

nd iUz by these same weights. The weighted vectors are then trans-ormed back to the space domain to obtain the anisotropic stencils.

perator size and compactness

Figure 3 shows the derivative operators of second, fourth, sixth,nd eighth orders in the z- and x-directions for isotropic and VTI ��

0.25, � ��0.29� media. As we can see, isotropic operators

a)

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igure 2. Comparison of derivative operators of different orders ofccuracy �second, fourth, sixth, eighth order in space and the ap-roximation applied in Dellinger and Etgen �1990� — cosine taper�n �a� the x-domain and �b� the k-domain. �c� Weights to apply to theomponents of the polarization vectors.


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engthen when the order of accuracy is higher. Anisotropic opera-ors, however, change little in size. The central parts of the anisotrop-c operators look similar to their corresponding isotropic operatorsnd change with the order of accuracy, although the outer parts ofhese anisotropic operators look the same and do not change with therder of accuracy. This indicates that the central parts of the opera-ors are determined by the order of accuracy, and the outer parts rep-esent the degree of anisotropy.

Figure 4 shows anisotropic derivative operators with the same or-er of accuracy �eighth order in space� for three VTI media with dif-erent combinations of � and � . These operators have similar centralarts but different outer parts. This result is consistent with the obser-ation that the central part of an operator is determined by the orderf accuracy and the outer part is controlled by anisotropy parame-ers.

Figure 5a shows the influence of approximation to finite differ-nce �second and eighth orders, Figure 3h and e�. The anisotropicart �diagonal tails� is almost the same, and the difference comesrom the central part. Figure 5b shows the difference between opera-ors with different anisotropy �Figure 4a and b�. The difference

ainly lies in the tails of the operators.Acomparison between Figure 4a and b shows that when we have a

arge difference between � and � , the operator is big; when the dif-erence between � and � stays the same, � affects the operator size.

comparison between Figure 4b and c shows that when the differ-nce between � and � becomes smaller and � does not change, theperator gets smaller in size. This result is consistent with the polar-zation equation for VTI media with weak anisotropy �Tsvankin,005�:

�P�� B�� 2�� sin2 � �sin 2� , �10�

∂/∂z ∂/∂x

∂/∂z ∂/∂x

a)

b)

igure 5. �a� Difference between the eighth- and second-order oper-tors �Figure 3g and e� for a VTI medium with anisotropy ��0.25,��0.29 in the z- and x-directions. �b� Difference between the

ighth-order anisotropic operators for a VTI medium with anisotro-y ��0.25, � ��0.29 �Figure 4a� and a VTI medium with aniso-ropy ��0.54, � �0 �Figure 4b�.

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∂/∂z ∂/∂x

f)

∂/∂z ∂/∂x

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∂/∂z ∂/∂x

g)

∂/∂z ∂/∂x

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∂/∂z ∂/∂x

h)

∂/∂z ∂/∂x

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igure 3. Second-, fourth-, sixth-, and eighth-order derivative opera-ors for an isotropic medium �VP�3 km /s and VS�1.5 km /s� andVTI medium �VP0�3 km /s, VS0�1.5 km /s, ��0.25, and � �0.29�. �a-d� Isotropic operators; �e-h� anisotropic operators. From

∂/∂z ∂/∂x

∂/∂z ∂/∂x

∂/∂z ∂/∂x

a)

b)

c)

igure 4. Eighth-order anisotropic pseudo-derivative operators forhree VTI media: �a� ��0.25, � ��0.29, �b� ��0.54, � �0; andc� ��0.2, � �0.


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ngle, and �P is the P-wave polarization angle. Equation 10 demon-trates the deviation of anisotropic polarization vectors with the iso-ropic ones: the difference of � and � �which is approximately � foreak anisotropy� and � control the deviation of �P from � and there-

ore the size of the anisotropic derivative operators.

perator truncation

The derivative operators for isotropic and anisotropic media areery different in shape and size, and the operators vary with thetrength of anisotropy. In theory, analytic isotropic derivatives areoint operators in the continuous limit. If we can do a perfect Fourierransform to ikx and ikz �without doing the approximations to differ-nt orders of accuracy as in Figure 2�, we get point derivative opera-ors. This is because ikx is constant in the z-direction �see Figure 6a�,hose Fourier transform is a delta function; the exact expression of

kx in the k-domain also points the operator in the x-direction. This

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akes the isotropic derivative operators point operators in the x- and-directions. When we apply approximations to the operators, theyre compact in the space domain.

However, even if we do perfect Fourier transformation to iUx andUz �without the approximations for different orders of accuracy� forTI media, the operators will not be point operators because iUx and

Uz are not constants in the z- and x-directions, respectively �see Fig-re 6b�. The x-domain operators spread out in all directions �Figuree-h�.

This effect is illustrated in Figure 3. When the order of accuracyecreases, the isotropic operators become more compact �shorter inpace� but the anisotropic operators do not get more compact. Noatter how we improve the compactness of isotropic operators, we

o not get compact anisotropic operators in the space domain by theame means.

Because the size of the anisotropic derivative operators is usuallyarge, it is natural that one would truncate the operators to save com-utation. Figure 7 shows a snapshot of an elastic wavefield and cor-esponding derivative operators for a VTI medium with ��0.25nd � ��0.29. Figure 8 shows the attempt to separate using a trun-ated operator size of �a� 11�11, �b� 31�31, and �c� 51�51 out ofhe full operator size 65�65. Figure 8 shows that the truncationeaves wave modes incompletely separated. This is because the trun-ation changes the direction of the polarization vectors, projecting

Figure 6. �a� Isotropic and �b� VTI ��0.25, � ��0.29� polarization vectors �Figure 2� projectedonto the x-�left column� and z-directions �right col-umn�. The isotropic polarization vectors’ compo-nents in the z- and x-directions depend only on kzand kx, respectively. In contrast, the anisotropic po-larization vectors’ components are functions of kxand kz.


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he wavefield displacement onto the wrong direction.A comparison of the deviation of polarization vectors before and

fter truncation is plotted in Figure 9. For the operator size of 1111, the polarization vectors only deviate from the correct ones to aaximum of 10°; however, even this difference makes the separa-

ion incomplete.

EXAMPLES

We illustrate the anisotropic wave-mode separation with a simpleynthetic example and a more challenging elastic Sigsbee 2A modelPaffenholz et al., 2002�.

imple model

We consider a 2D isotropic model characterized by the VP, VS, andensity shown in Figure 10a-c. The model contains negative P- and-wave velocity anomalies that triplicate the wavefields. The source

s located at the center of the model. Figure 11a shows the verticalnd horizontal components of one snapshot of the simulated elasticavefield �generated using the eighth-order finite-difference solu-

a)

b)

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0.8

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40

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igure 7. �a� A snapshot of an elastic wavefieldhowing the vertical �left� and horizontal �right�omponents for a VTI medium ��0.25 and � �0.29�. �b� Eighth-order anisotropic pseudo-

erivative operators in the z-�left� and x-�right� di-ections for this VTI medium. The boxes show theruncation of the operator to sizes of 11�11, 31

31, and 51�51.


ion of the elastic wave equation�, Figure 11b shows the separation to- and S-modes using � · and �� operators, and Figure 11c shows

he mode separation obtained using the pseudo-derivative operatorsn the medium parameters. A comparison of Figure 11b and c indi-ates that the � · and �� operators and the pseudo-derivative opera-ors work identically well for this isotropic medium.

We consider a 2D anisotropic model similar to the model shown inigure 10a-c, characterized by � and � shown in Figure 10d and e.he parameters � and � vary gradually from top to bottom and left to

ight, respectively. The upper-left part of the medium is isotropic,nd the lower-right part is highly anisotropic. Because the differencef � and � is great at the bottom part of the model, the qS-waves inhat region are triplicated severely as a result of the strong anisotro-y.

Figure 12 illustrates the pseudo-derivative operators obtainedt different locations in the model defined by the intersectionsf x-coordinates 0.3, 0.6, 0.9 km and z-coordinates 0.3, 0.6,.9 km.Because the operators correspond to different combinationsf � and � , they have different forms. The isotropic operator at x

0.3 km and z�0.3 km �Figure 12a� is purely vertical and hori-ontal, whereas the anisotropic operators �Figure 12b-i� have tailsadiating from the center. The operators get larger at locations wherehe medium is more anisotropic, e.g., at x�0.9 km and z�0.9 km.

Figure 13a shows the vertical and horizontal components of onenapshot of the simulated elastic anisotropic wavefield, Figure 13b

/ z

0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

20 30 40 50 0 10 20 30 40 50

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hows the separation to qP- and qS-modes using conventional iso-ropic � · and �� operators, and Figure 13c shows the mode separa-ion obtained using the pseudooperators constructed with the local

edium parameters. A comparison of Figure 13b and c� indicateshat the spatially varying derivative operators successfully separatehe elastic wavefields into qP- and qS-modes, but the � · and �� op-rators only work in the isotropic region of the model.

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igsbee model

Our second model �Figure 14� uses an elastic anisotropic versionf the Sigsbee 2A model �Paffenholz et al., 2002�. In our modifica-ion, the P-wave velocity is taken from the original model, VP /VSanges from 1.5 to 2, � ranges from 0 to 0.48 �Figure 14d�, and �anges from 0 to 0.10 �Figure 14e�. The model is isotropic in the saltnd the top part of the model.Avertical point-force source is located

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Figure 8. Separation by eighth-order anisotropicpseudo-derivative operators of different sizes: �a�11�11, �b� 31�31, �c� 51�51, shown in Figure7b. The larger the operators, the better the separa-tion.

.6 0

0.6 0

0.6 0

(km)

qP-mode qS-mode


a

b

c

a

WB26 Yan and Sava

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Figure 9. Deviation of polarization vectors by trun-cating the operator size to �a� 11�11, �b� 31�31,and �c� 51�51 out of 65�65. The left columnshows polarization vectors from �1 to 1 cycle inthe x- and z-directions; the right column zoomsfrom 0.3 to 0.7 cycles. The green vectors are theexact polarization vectors; the red ones are the ef-fective polarization vectors after truncation of theoperator in the x-domain.

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0.2

0.4

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m)

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) b) c)

d) e)

Figure 10. A 1.2�1.2-km model with parameters�a� VP0�3 km /s except for a low-velocity Gauss-ian anomaly at x�0.65 km and z�0.65 km, �b�VS0�1.5 km /s except for a low-velocity Gaussiananomaly at x�0.65 km and z�0.65 km, �c� ��1 g /cm3 in the top layer and 2 g /cm3 in the bot-tom layer, �d� � smoothly varying from 0 to 0.25from top to bottom, �e� � smoothly varying from 0to �0.29 from left to right. A vertical point forcesource is located at x�0.6 km and z�0.6km,shown by the dot in �b–e�. The dots in �a� corre-spond to the locations of the anisotropic operatorsshown in Figure 12.

Downloaded 07 Oct 2009 to 138.67.12.184. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

a

b

c

a

b

c


)

)

)

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2qP-mode qS-mode

qP-mode qS-mode

Displacement of z Displacement of x

z(k

m)

z(k

m)

z(k

m)

x (km) x (km) Figure 11. �a� A snapshot of the isotropic wavefieldmodeled with a vertical point-force source at x�0.6 km and z�0.6 km for the model shown inFigure 10a-c�. �b� Isotropic P- and S-wave modesseparated using � · and ��. �c� Isotropic P- andS-wave modes separated using pseudo-derivativeoperators. Both �b� and �c� show good separationresults.

∂/∂z ∂/∂x

g)

∂/∂z ∂/∂x

h)

i)

∂/∂z ∂/∂x

)

∂/∂z ∂/∂x

)

)

∂/∂z ∂/∂x

d)

∂/∂z ∂/∂x

e)

f)

Figure 12. The eighth-order anisotropic pseudo-de-rivative operators in the z- and x-directions at theintersections of x�0.3, 0.6, 0.9 km, and z�0.3,0.6, 0.9 km, for the model in Figure 10.

∂/∂z ∂/∂x∂/∂z ∂/∂x ∂/∂z ∂/∂x


aw

wsoram

sem

tniwm

tahmcl�

p

FfixFrmtmpttu

WB28 Yan and Sava

t x�14.5 km and z�5.3 km to simulate the elastic anisotropicavefield.

Figure 15 shows one snapshot of the modeled elastic anisotropicavefields using the model in Figure 14. Figure 16 illustrates the

eparation of the anisotropic elastic wavefields using the � · and ��perators. Figure 17 illustrates the separation using our pseudo-de-ivative operators. Figure 16 shows the residual of unseparated P-nd S-wave modes, such as at x�13 km and z�7 km in the qP-ode panel and at x�11 km and z�7 km in the qS-mode panel.

The residual of S-waves in the qP-mode panel of Figure 16 is veryignificant because of strong reflections from the salt bottom. Thisxtensive residual can be harmful to subsalt elastic or even acousticigration if not removed completely. In contrast, Figure 17 shows

a)

b)

c)

00.0

0.2

0.4

0.6

0.8

1.0

1.2

0 00.0

0.2

0.4

0.6

0.8

1.0

1.2

00.0

0.2

0.4

0.6

0.8

1.0

1.2

z(km)

z(km)

z(km)

igure 13. �a� A snapshot of the anisotropic wave-eld modeled with a vertical point-force source at�0.6 km and z�0.6 km for the model shown inigure 10. �b� Anisotropic qP- and qS-modes sepa-ated using � · and ��. �c�Anisotropic qP- and qS-odes separated using pseudo-derivative opera-

ors. The separation of wavefields into qP- and qS-odes in �b� is incomplete, which is obvious at

laces such as at x�0.4 km, z�0.9 km. In con-rast, the separation in �c� is much better becausehe correct anisotropic derivative operators aresed.


he qP- and qS-modes better separated, demonstrating the effective-ess of the anisotropic pseudo-derivative operators constructed us-ng the local medium parameters. These wavefields composed ofell-separated qP- and qS-modes are essential to produce clean seis-ic images.To test the separation with a homogeneous assumption of aniso-

ropy in the model, we show in Figure 18 the separation with ��0.3nd � �0.1 in the k-domain. This separation assumes a model withomogeneous anisotropy. The separation shows that residual re-ains in the separated panels. Although the residual is much weaker

ompared to separating using a isotropic model, it is still visible atocations such as at x�13 km and z�7 km, and x�13 km and z

4 km, in the qP-panel and at x�16 km and z�2.5 km in the qS-anel.

.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

x (km) x (km)

isplacement of z Displacement of x

qP-mode qS-mode

qP-mode qS-mode

0.2 0

.2 0

0.2 0

D


FSr


a) b) c)

d) e)

x x x

xx

z z z

z z

igure 14. ASigsbee 2Amodel in which �a� is the P-wave velocity �taken from the original Sigsbee 2Amodel �Paffenholz et al., 2002��, �b� is the-wave velocity, where VP /VS ranges from 1.5 to 2, �c� is the density ranging from 2.2 to 2.7 g /cm3, �d� is � ranging from 0.2 to 0.48, and �e� is �anging from 0 to 0.10 in the rest of the model.

x (km)10 11 12 13 14 15 16 17 18 19

2

3

4

5

6

7

8

9

z(xm)

Displacement of z Displacement of x

x (km)10 11 12 13 14 15 16 17 18 19

Figure 15. Anisotropic wavefield modeled with avertical point force source at x�14.3 km and z�5.3 km for the model shown in Figure 14.

x (km)10 11 12 13 14 15 16 17 18 19

x (km)10 11 12 13 14 15 16 17 18 19

2

3

4

5

6

7

8

9

z(x

m)

qP-mode qS-mode

Figure 16. Anisotropic qP- and qS-modes separat-ed using � · and �� for the vertical and horizontalcomponents of the elastic wavefields shown in Fig-ure 15. Residuals are obvious at places such as x�13 km and z�7 km in the qP-mode panel andat x�11 km and z�7 km in the qS-mode panel.


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WB30 Yan and Sava

DISCUSSION

The separation of P- and S-wave modes is based on the projectionf elastic wavefields onto their respective polarization vectors. ForTI media, P- and S-mode polarization vectors can be obtained con-eniently by solving the Christoffel equation. The Christoffel equa-ion is a plane-wave solution to the elastic wave equation. Becausehe displacements, velocity, and anisotropy field have the same formf elastic wave equation, the separation algorithm applies to all ofhese wavefields. The P- and SV-mode separation can be extended toransverse isotropy with a tilted symmetry axis �TTI� media by solv-ng a TTI Christoffel matrix and obtaining TTI separators. Physical-y, the TTI medium is a rotation of VTI media.

In TI media, SV- and SH-waves are uncoupled most of the time,here the SH-wave is polarized out of plane. We only need to de-

ompose P- and SV-modes in the vertical plane. The plane-wave so-ution is sufficient for most TI media, except for a special case wherehere exists a singularity point at an oblique propagation angle in theertical plane �a 3D line singularity�, at which angle SV- and SH-ave velocities coincide.

10 11

2

3

4

5

6

7

8

9

z(x

m)

igure 17. Anisotropic qP- and qS-modes separat-d using pseudo-derivative operators for the verti-al and horizontal components of the elastic wave-elds shown in Figure 15. They show better separa-

ion of qP- and qS-modes.

10 11

2

3

4

5

6

7

8

9

z(x

m)

igure 18. Anisotropic qP- and qS-modes separat-d in the k-domain for the vertical and horizontalomponents of the elastic wavefields shown in Fig-re 15. The separation assumes ��0.3 and �

0.1 throughout the model. The separation is in-omplete. Residuals are still visible at places sucht x�13 km and z�7 km, and at x�13 km and,�4 km in the qP-mode panel and at x�16 kmnd z�2.5 km in the qS-mode panel.


At this point, the SV-wave polarization is not defined uniquely byhe Christoffel equation. S-waves at the singularity are polarized in alane orthogonal to the P-wave polarization vector. However, this isot a problem because we define SV-waves polarized in verticallanes only, removing the singularity by using the cylindrical coor-inates. This situation is similar to S-wave-mode coupling in ortho-hombic media, where there is at least one singularity in a quadrant.owever, as pointed out by Dellinger and Etgen �1990�, the singu-

arity in orthorhombic media is a global property of the media andannot be removed; therefore, the separation using polarization vec-ors to 3D orthorhombic is not straightforward.

The anisotropic derivative operators depend on the anisotropicedium parameters. In Figure 19, we show how sensitive the sepa-

ation is to the medium parameters. The elastic-wavefield snapshote use is shown in Figure 7a for a VTI medium with VP /VS�2 and�0.25, � ��0.29. We try to separate the P- and SV-modes with

a� ��0.4, � ��0.1, �b� ��0, � ��0.3, and �c� ��0, � �0.he separation shows that parameters �a� have good separation,howing the difference in � and � is important. The worst-case sce-ario is shown by parameters �c�, where isotropy is assumed for thisTI medium.

x (km)14 15 16 17 18 19

x (km)10 11 12 13 14 15 16 17 18 19

qP-mode qS-mode

x (km)14 15 16 17 18 19

x (km)10 11 12 13 14 15 16 17 18 19

qP-mode qS-mode

12 13

12 13


dacrTthm

fs

Fmtwt


CONCLUSIONS

We have presented a method of obtaining spatially varying pseu-o-derivative operators with application to wave-mode separation innisotropic media. The main idea is to utilize polarization vectorsonstructed in the wavenumber domain using the local medium pa-ameters and then transform these vectors back to the space domain.he main advantage of applying the pseudo-derivative operators in

he space domain constructed in this way is that they are suitable foreterogeneous media. The wave-mode separators obtained with thisethod are spatially variable filtering operators that can be used to

a)

b)

c)

0 0.2 0.4 0.6 0.8 1 1.2x (km)

0 0.2 0.4 0.x (km

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z(km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z(km)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z(km)

qP-mode qS-m

igure 19. P- and SV-wave-mode separation for a snapshot shown inedium parameters are ��0.25, � ��0.29. The separation assum

ers of �a� ��0.4, � ��0.1, �b� ��0, � ��0.3, and �c� ��0, �as applied to show the weak events. The plot shows that the differe

ropy parameters influences wave-mode separation.


separate wavefields in VTI media with an arbi-trary degree of anisotropy. This methodology isapplicable for elastic RTM in heterogeneous an-isotropic media.

ACKNOWLEDGMENTS

We acknowledge the support of the sponsors ofthe Center for Wave Phenomena at ColoradoSchool of Mines.

APPENDIX A

FINITE-DIFFERENCEAPPROXIMATIONS TO DIFFERENT

ORDERS OF ACCURACY

This appendix summarizes the wavenumber-domain weighting of the ideal derivative functionikj, j�1,2,3 for operators of various degrees ofaccuracy. Starting from the coefficients of a de-rivative stencil in the space domain, we can useconventional Z-transforms to construct a wave-number-domain weighting function representingthe same order of accuracy. We apply the weight-ing functions to the isotropic and anisotropic pro-jections of the polarization vectors to obtainspace-domain derivative operators of a similarorder of accuracy.

For the case of second-order centered deriva-tives, the stencil has the following Z-transformrepresentation:

D2�Z��1

2�Z1�Z�1� . �A-1�

Transforming from Z to the wavenumber domain,we obtain

D2�k��1

2�eik�e�ik�, �A-2�

which enables us to define the second-orderweight

W2�k��sin�k�

k. �A-3�

Here, the wavenumber k is normalized and henceis dimensionless.

Similarly, we can construct weights forourth-, sixth-, and eighth-order operators from Z-transform repre-entations:

D4�Z��2

3�Z1�Z�1��

1

12�Z2�Z�2�, �A-4�

D6�Z��3

4�Z1�Z�1��

3

20�Z2�Z�2��

1

60�Z3�Z�3�,

�A-5�

1 1.2

e 7a. The trueium parame-ard clippingate of aniso-

6 0.8)

ode

Figures med�0. H

nt estim


w

T

w

thhEf

A

A

—

C

—

D

E

F

F

G

H

P

S

S

TT

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Z

WB32 Yan and Sava

D8�Z��4

5�Z1�Z�1��

1

5�Z2�Z�2��

4

105�Z3�Z�3�

�1

140�Z4�Z�4�, �A-6�

hich transform to the wavenumber-domain weights:

D4�k��2

3�eik�e�ik��

1

12�e2ik�e�2ik�, �A-7�

D6�k��3


3

20�e2ik�e�2ik��

1

60�e3ik

�e�3ik�, �A-8�

D8�k��4


1

5�e2ik�e�2ik��

4

105�e3ik

�e�3ik��1

280�e4ik�e�4ik� . �A-9�

hese lead to the following weighting functions:

W4�k��4 sin�k�

3k�

sin�2k�6k

, �A-10�


2k�

3 sin�2k�10k

�sin�3k�

30k,

�A-11�


5k�

2 sin�2k�5k

�8 sin�3k�

105k�

sin�4k�140k

.

�A-12�

The derivatives are shown in Figure 2a and b in the space and

avenumber domains, respectively. Comparing these weights and


heir corresponding frequency responses, we see that we need to useigher-order difference operators to have better approximations toigh frequencies. For comparison, the weight used in Dellinger andtgen �1990� is 1 /2�1�cos�k��. The cosine taper attenuates middle

requencies compared to high-order finite-difference operators.

REFERENCES

ki, K., and P. Richards, 2002, Quantitative seismology, 2nd ed.: UniversityScience Books.

lkhalifah, T., 1998, Acoustic approximations for processing in transverselyisotropic media: Geophysics, 63, 623–631.—–, 2000, An acoustic wave equation for anisotropic media: Geophysics,65, 1239–1250.

hang, W. F., and G. A. McMechan, 1986, Reverse-time migration of offsetvertical seismic profiling data using the excitation-time imaging condi-tion: Geophysics, 51, 67–84.—–, 1994, 3-D elastic prestack, reverse-time depth migration: Geophys-ics, 59, 597–609.

ellinger, J., and J. Etgen, 1990, Wave-field separation in two-dimensionalanisotropic media: Geophysics, 55, 914–919.

tgen, J. T., 1988, Prestacked migration of P- and SV-waves: 58thAnnual In-ternational Meeting, SEG, ExpandedAbstracts, 972–975.

letcher, R., X. Du, and P. J. Fowler, 2008, A new pseudo-acoustic waveequation for TI media: 78th Annual International Meeting, SEG, Expand-edAbstracts, 2082–2086.

ornberg, B., and M. Ghrist, 1999, Spatial finite difference approximationsfor wave-type equations: SIAM Journal on Numerical Analysis, 37,105–130.

ray, S. H., J. Etgen, J. Dellinger, and D. Whitmore, 2001, Seismic migrationproblems and solutions: Geophysics, 66, 1622–1640.

okstad, K., 2000, Multicomponent Kirchhoff migration: Geophysics, 65,861–873.

affenholz, J., B. McLain, J. Zaske, and P. Keliher, 2002, Subsalt multiple at-tenuation and imaging: Observations from the Sigsbee2B synthetic dataset: 72ndAnnual International Meeting, SEG, ExpandedAbstracts, 2122–2125.

han, G., 2006, Optimized implicit finite-difference migration for VTI me-dia: 76th Annual Interntional Meeting, SEG, Expanded Abstracts,2367–2371.

han, G., and B. Biondi, 2005, 3D wavefield extrapolation in laterally vary-ing tilted TI media: 75th SEGAnnual Interntional Meeting, SEG, Expand-edAbstracts, 104–107.

homsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.svankin, I., 2005, Seismic signatures and analysis of reflection data in an-isotropic media, 2nd ed.: Elsevier Science Publ. Co., Inc.

an, J., and P. Sava, 2008, Isotropic angle domain elastic reverse time migra-tion: Geophysics, 73, no. 6, S229–S239.

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Geophysics, 62, 598–613.


Documents

Elastic wave-mode separation for VTI medianewton.mines.edu/paul/journals/2009_gpy00wb19.pdfElastic wave-mode separation for VTI media Jia Yan1 and Paul Sava1 ABSTRACT Elasticwavepropagationinanisotropicmediaiswellrep-resentedbyelasticwaveequations