Effective reflection coefficients for curved interfaces in TI ...€¦ · CWP-594 Effective reflection coefficients for curved interfaces in TI media Milana Ayzenberg 1, Ilya

CWP-594

Effective reflection coefficients for curved interfaces

in TI media

Milana Ayzenberg 1, Ilya Tsvankin 2, Arkady Aizenberg 3, & Bjørn Ursin 1

1 Norwegian University of Science and Technology, S.P. Andersens vei 15A, 7491 Trondheim, Norway2 Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA3 Institute of Petroleum Geology and Geophysics SB RAS, Pr.Ac.Koptyug 3, 630090 Novosibirsk, Russia

ABSTRACT

Plane-wave reflection coefficients (PWRC) are routinely used in amplitude-variation-with-offset (AVO) analysis and for generating boundary data inKirchhoff modeling. However, the geometrical-seismics approximation based onPWRC becomes inadequate in describing reflected wavefields at near- and post-critical incidence angles. Also, PWRC are derived for plane interfaces and breakdown in the presence of significant reflector curvature. Here, we discuss so-calledeffective reflection coefficients (ERC) designed to overcome the limitations ofPWRC for multicomponent data from heterogeneous anisotropic media.We show that the reflected wavefield in the immediate vicinity of a curved in-terface can be represented by a generalized plane-wave decomposition, whichapproximately reduces to the conventional Weyl-type integral computed foran “apparent” source location. The ERC is then obtained as the ratio of thereflected and incident wavefields at each point of the interface. To carry outdiffraction modeling, we combine ERC with the tip-wave superposition method(TWSM) extended to elastic media. This methodology is implemented forcurved interfaces separating an isotropic incidence halfspace and a transverselyisotropic (TI) medium with the symmetry axis orthogonal to the reflector.If the interface is plane, ERC represent the exact solution sensitive to theanisotropy parameters and source-receiver geometry. Numerical tests demon-strate that the difference between ERC and PWRC for typical TI models canbe significant, especially at low frequencies and in the post-critical domain. Forcurved interfaces, ERC provide a practical approximate tool to compute thereflected wavefield. We analyze the dependence of ERC on reflector shape anddemonstrate their advantages over PWRC in 3D diffraction modeling of PP andPS reflection data.

Key words:reflection coefficient, spherical wave, Kirchhoff modeling, AVO analysis,anisotropy, transverse isotropy, P-waves, converted waves, curved interfaces

INTRODUCTION

Plane-wave reflection and transmission coefficients pro-vide the basis for ray-theory treatment of seismic wave-fields in layered media. In the geometrical-seismicsapproximation, which represents the leading term

of the ray-series expansion, the amplitude of anywave mode is proportional to the product of thereflection/transmission coefficients along the raypath(Brekhovskikh, 1980; Cerveny, 2001). For example, thewell-known geometrical-seismics expression for a wavereflected from the bottom of a homogeneous layer in-cludes the plane-wave reflection coefficient (PWRC)

86 M. Ayzenberg et. al.

multiplied by the source-radiation function and dividedby the geometrical-spreading factor.

Geometrical seismics, however, becomes inaccu-rate for near- and post-critical incidence angles orwhen the source and/or receiver is located close (com-pared to the predominant wavelength) to the reflector(Brekhovskikh, 1980; Tsvankin, 1995). Deviations fromthe geometrical-seismics approximation become muchmore pronounced in the presence of even moderate seis-mic anisotropy (Tsvankin, 2005). Also, since PWRC arederived for plane interfaces, they cannot be used for ray-theory modeling in the presence of significant reflectorcurvature.

The limitations of the geometrical-seismics approx-imation pose serious problems for dynamic ray tracingand Kirchhoff integral modeling techniques (Frazer andSen, 1985; Hanyga and Helle, 1995; Ursin and Tygel,1997; Cerveny, 2001; Ursin, 2004). In particular, theboundary data used in conventional Kirchhoff modelingare obtained by simply multiplying the amplitude ofthe incident wave (which generally has a curved wave-front) with the PWRC. This approach produces artifi-cial diffractions on synthetic data due to the discontin-uous slope of the PWRC at the critical angle (Kampf-mann, 1988; Wenzel et al., 1990; Sen and Frazer, 1991).

Another practically important method based ongeometrical seismics is amplitude-variation-with-offset(AVO) analysis, which operates with PWRC estimatedfrom surface reflection data. Furthermore, because ofthe complexity of exact reflection coefficients, PWRCused in AVO processing are often linearized in the veloc-ity and density contrasts across the reflector. The weak-contrast approximation of PWRC is given by Shuey(1985) for isotropic media and extended by Thom-sen (1993) and Ruger (1997) to VTI (transverselyisotropic with a vertical symmetry axis) models. TheVTI expressions involve an additional linearization inthe anisotropy parameters on both sides of the inter-face, which helps to separate the reflection coefficientinto isotropic and anisotropic terms. Ruger (1997, 2002)generalizes the weak-contrast, weak-anisotropy PWRCequations for azimuthally anisotropic models and dis-cusses their application in fracture characterization us-ing wide-azimuth reflection data.

Whereas conventional PWRC are defined throughthe magnitude of the displacement vector, Schleicheret al. (2001) introduce linearized reflection coefficientsobtained from the ratio of the energy flux for the re-flected and incident waves. They show that applicationof the flux-normalized coefficients in Kirchhoff model-ing produces reciprocal reflected wavefields. The flux-normalized reflection coefficients are extended to vis-coelastic VTI media by Stovas and Ursin (2003).

The linearized approximations, however, lose accu-racy with increasing incidence angle and break downnear the critical ray. To overcome this problem, Down-ton and Ursenbach (2006) express the reflection coeffi-

cient as a function of the averaged incidence and trans-mission angles and develop an analytic continuation ofthe linearized PWRC in the post-critical domain. Forweak parameter contrasts across the interface, their ap-proximation remains close to the exact PWRC for post-critical angles.

Still, even exact PWRC employed in thegeometrical-seismics approximation cannot describethe post-critical reflected wavefield, which includes theinterfering head and reflected waves. To make PWRCsuitable for amplitude analysis in the post-criticaldomain, van der Baan and Smith (2006) propose toapply the τ − p transform to wide-angle reflection data.Although the transformed wavefield exhibits a betterfit to the corresponding PWRC, the τ − p techniquedoes not properly account for head waves and is limitedto laterally homogeneous models.

In an earlier publication (Ayzenberg et al., 2007),we introduce so-called effective reflection coefficients(ERC) for acoustic wave propagation and demonstratetheir advantages in Kirchhoff modeling. ERC are de-signed to generalize PWRC for wavefields from pointsources at curved interfaces, and are not limited tosmall incidence angles and weak parameter contrastsacross the reflector. In particular, Kirchhoff-type mod-eling with ERC removes the critical-angle artifacts men-tioned above and correctly reproduces the amplitudes ofthe head and reflected waves.

The goal of this paper is to extend ERC to curvedreflectors in heterogeneous anisotropic models and im-plement the new formalism for an interface betweenisotropic and TI media. We begin by defining ERCthrough a generalized plane-wave decomposition simi-lar to the one proposed by Klem-Musatov et al. (2004)for the acoustic problem. Although this solution in-volves integration over a curved reflecting surface, ERCcan be approximately obtained from Weyl-type integralscomputed for locally plane interface segments. By con-ducting numerical tests, we evaluate the difference be-tween ERC and PWRC for a plane interface and studythe dependence of ERC on the anisotropy parameters,frequency and local reflector shape. Finally, using thetip-wave superposition method (TWSM), we implementERC in 3D elastic diffraction modeling. Tests for curvedinterfaces of different shape confirm the ability of our al-gorithm to model reflection wavefields in the presenceof multipathing and caustics.

EFFECTIVE REFLECTION COEFFICIENTS

FOR ANISOTROPIC MEDIA

Wavefield representation using surface integrals

We consider the wavefield reflected from a smoothcurved interface S, which separates homogeneousisotropic and transversely isotropic (TI) halfspaces (Fig-ure 1). The symmetry axis of the TI medium is assumed

Effective reflection coefficients 87

Receivers

Bending TI layer

Symmetry axis orthogonal to reflector

VP0

, VS0

, , ,

Isotropic

VP, V

S,

Source

Figure 1. 2D sketch of the model. The isotropic incidencemedium is separated from the reflecting TI halfspace by acurved interface. The symmetry axis of the TI medium isorthogonal to the reflector.

to be orthogonal to the reflector at each point. Theisotropic medium is described by the P-wave velocityv(1)P , S-wave velocity v

(1)S and density ρ(1), and the TI

medium by the symmetry-direction velocities of P- andS-waves (v

(2)P0 and v

(2)S0 ), density ρ(2), and Thomsen pa-

rameters ε and δ (the parameter γ influences only SH-waves).

We consider only the primary P- and S-wave re-flections from the interface and neglect higher-orderscattering. Using the representation theorem (Pao andVaratharajulu, 1976; Aki and Richards, 2002), the re-flected wavefield can be described by the following sur-face integral:

u(x) =

∫ ∫

S

[u(x′) · T(x′,x) − t(x′) · G(x′,x)]dS(x′),

(1)where u(x′) and t(x′) are the displacement and tractionvectors at the interface, and G(x′,x) and T(x′,x) arethe Green’s displacement and traction tensors (Pao andVaratharajulu, 1976).

For a homogeneous isotropic medium, the reflectedwavefield 1 includes the primary PP- and PS-wave re-flections (uPP (x) and uPS(x), respectively). To evalu-ate integral 1, we split the reflector into small rhom-bic elements. As shown in Appendix F, the reflectedPP-wavefield can be represented as the sum of tip-wavebeams excited by each rhombic element in accordancewith the Huygens principle. Equations F-14 and F-15represent an extension of the tip-wave superpositionmethod to elastic media:

uPP (x) ≃∑

j

iω

v(1)P

lP [j](x) ∆BPP [j](x) , (2)

where the index j corresponds to a surface element,lP [j](x) = ∇gP (x′

[j],x)/|∇gP (x′[j],x)|, and gP (x′, x) is

the scalar P-wave Green’s function. ∆BPP [j](x) is thescalar contribution of the j-th element given by

∆BPP [j](x) =

∫ ∫

∆Π[j]

[∂gP (x′,x)

∂n′ d1,PP (x′)

− gP (x′,x)d2,PP (x′)

]dS′ , (3)

where d1,PP (x′) and d2,PP (x′) are the scalar boundaryvalues of the reflected PP-wave at the interface. Equa-tion F-13 expresses the boundary data d1,PP and d2,PP

through the incident wavefield and the PP-wave effec-tive reflection coefficient (ERC) introduced below.

We also show in Appendix F that the reflected PS-wavefield can be represented as the sum of the tip-wavebeams described by equations F-21 and F-22:

uPS(x) ≃∑

j

iω

v(1)S

lS[j](x) × ∆BPS[j](x), (4)

where lS[j](x) = ∇gS(x′[j],x)/|∇gS(x′

[j],x)|, gS(x′,x) isthe scalar S-wave Green’s function, and ∆BPS[j](x) isthe vector contribution of the j-th surface element:

∆BPS[j](x) =

∫ ∫

∆Π[j]

[∂gS(x′,x)

∂n′ d1,PS(x′)

− gS(x′,x)d2,PP (x′)

]dS′ ; (5)

d1,PS(x′) and d2,PS(x′) are the vector boundary valuesof the reflected PS-wavefield at the interface expressedthrough the corresponding ERC in equation F-20. Toevaluate integrals 3 and 5, we use the far-field approxi-mation 16 developed by Ayzenberg et al. (2007).

Wavefield at the interface in terms of ERC

In conventional Kirchhoff modeling, it is assumed thatthe reflected wavefield uPQ(x′) can be approximatelywritten as

uPQ(x′) ≃ RPQ(θ(x′))[u

incP (x′) · h−

P (x′)]h

+Q(x′), (6)

where RPQ(θ(x′)) is the plane-wave reflection coefficient(PWRC), θ(x′) is the incidence angle, and h−

P (x′) andh+Q(x′) are the unit polarization vectors of the incident

P-wave and reflected PQ-wave, respectively. This ap-proach, which is based on the geometrical-seismics ap-proximation, assumes that the wavefront curvature atthe reflector can be ignored, the reflector is plane, andthe medium near the reflector homogeneous. However,equation 6 is adequate only for sub-critical incidence an-gles and causes artificial diffractions due to the discon-tinuous slope of the PWRC at the critical angle (Kampf-mann, 1988; Wenzel et al., 1990; Sen and Frazer, 1991).


For a plane interface between homogeneous media,the assumption about the wavefront curvature can berelaxed by representing the incident wave in the form ofthe Weyl integral over plane waves (Aki and Richards,2002; Tsvankin, 1995). Each elementary plane wave inthe integrand is multiplied with the PWRC to obtain anexact integral expression for the reflected wavefield. Tohandle curved reflectors in heterogeneous media, Klem-Musatov et al. (2004) introduced a rigorous theory ofreflection and transmission for interfaces of arbitraryshape in acoustic models. They showed that the bound-ary data in the acoustic Kirchhoff integral can be repre-sented by a generalized plane-wave decomposition calledthe “reflection operator.” For curved interfaces, the de-composition is local and has to be evaluated separatelyfor each individual point at the interface. Ayzenberget al. (2007) proved that the exact action of the reflec-tion operator upon the incident wavefield may be ap-proximately described by multiplication of the incidentwavefield and the corresponding effective reflection co-efficient (ERC) for each point at the interface. This for-malism is not limited to small incidence angles and weakparameter contrasts across the interface. It also incor-porates the local interface curvature into the reflectionresponse.

Here we generalize the reflection operator forcurved interfaces between isotropic and TI media. InAppendix A we demonstrate that in the immediatevicinity of a curved interface there exist local exponen-tial solutions of the wave equation with variable coef-ficients in the form of generalized plane waves. Usingthese solutions as the basis, in Appendix B we intro-duce spectral integrals describing the decomposition ofthe displacement field into the generalized plane P-, S1-and S2-waves propagating to and from the interface. Inthe special case of a plane reflector separating two homo-geneous halfspaces, the generalized spectral integral re-duces to the known Weyl-type decomposition over con-ventional plane waves. For curved reflectors, the gener-alized spectral integrals satisfy the boundary conditions(i.e., the continuity of displacement and traction acrossthe interface) and are invariant with respect to the in-terface shape. In Appendix C, we rewrite the boundaryconditions in the form of reflection and transmission op-erators for anisotropic media. Here we concentrate onthe reflected displacement field, uPQ(x′). The tractiontPQ(x′) can be eliminated, which simplifies syntheticmodeling and reduces computing time.

As shown in Appendix C, the generalized plane-wave decomposition for the displacement component jof the PQ-mode reflected from a curved interface canbe represented as

uPQ,j(s1, s2, 0;x′) =

ω2

2π

∫ +∞

−∞

∫ +∞

−∞RPQ(p;x′)

h+Q,jx

′)

h−P,j(x

′)

×uincP,j(p1, p2, 0;x

′)eiω(p1s1+p2s2)dp1dp2 , (7)

where (s1, s2) are the curvilinear Chebyshev coor-dinates covering the interface S, (p1, p2) are the projec-tions of the slowness vector onto the plane tangentialto the interface at point x′, p =

√p21 + p2

2, RPQ(p;x′)is the PWRC at point x′, and h−

P,j(x′) and h+

Q,j(x′)

are the components of the unit polarization vectorsof the incident P-wave and reflected PQ-wave, respec-tively. For arbitrary interface geometry, the spectrumuincP (p1, p2, 0;x

′) of the incident wave has to be evalu-ated using the Fourier transform in the Chebyshev co-ordinates (s1, s2):

uincP (p1, p2, 0;x

′) =1

2π

∫ +∞

−∞

∫ +∞

−∞u

incP (s1, s2, 0;x

′)

× e−iω(p1s1+p2s2)ds1ds2. (8)

The generalized plane-wave decomposition 7 is local andhas to be computed at each point x′. It is valid withinan infinitely thin layer near the interface, and can beused only for calculation of the reflection response inthe immediate vicinity of the reflector.

The exact PWRC RPQ(p;x′) for a plane inter-face between two VTI media can be found in Graebner(1992) and Ruger (2002). In Appendix D we reproducethe derivation of PWRC in our notation and correcttypos in the published solutions.

In the special case of a plane interface, the de-compositions 7 reduce to the known Weyl-type inte-grals over conventional plane waves (Aki and Richards,2002; Tsvankin, 1995). For a horizontal reflector, thecurvilinear coordinates (s1, s2) coincide with the ordi-nary Cartesian coordinates (x1, x2). Also, the spectrumuincP (p1, p2, 0;x

′) in 8 does not depend on positon x′ andis represented by a known analytic function.

The integral formula 7 is much more complicatedthan the geometrical-seismics solution 6. In particular,evaluation of the spectrum uinc

P (p1, p2, 0;x′) from equa-

tion 8 involves extremely time-consuming integrationover the whole interface. However, the generalized de-composition 7 can be approximately reduced to the fol-lowing form similar to equation 6:

uPQ(x′) ≃ χPQ(θ(x′), L(x′))[u

incP (x′) · h−

P (x′)]h

+Q(x′) ,

(9)where χPQ(θ(x′), L(x′)) is the ERC and L(x′) =

ωR∗(x′)/v(1)P is a dimensionless frequency-dependent

parameter,

R∗(x′) = R(x′)2 − sin2 θ(x′)

2 − sin2 θ(x′) − 2R(x′)H(x′) cos θ(x′).

(10)Here, R(x′) is the distance between the source and pointx′ at the interface and H(x′) is the mean interface cur-vature (Ayzenberg et al., 2007). The parameter R∗(x′),which depends on the actual incidence angle θ(x′) andthe local interface curvature H(x′), has the meaning of


the distance between the source of an “apparent” inci-dent spherical P-wave and the interface.

In Appendix E we express the ERC in equation 9through the Fourier-Bessel integrals for the reflectedwavefield (Brekhovskikh, 1980; Aki and Richards, 2002):

χPP (θ(x′), L(x′))

=u∗

PP,norm(x′) cos θ(x′)+u∗PP,tan(x′) sin θ(x′)

(ikP − 1R∗ ) eikP R∗

R∗

,

χPS(θ(x′), L(x′))

=−u∗

PS,norm(x′) sin θS(x′)+u∗PS,tan(x′) cos θS(x′)


R∗

,

(11)where

u∗PQ,norm(x′)

= ω2∫ +∞0

RPQ(p)h+

Q,norm

h−P,norm

eiωlpP3J0(rωp)pdp ,

u∗PQ,tan(x′)

= −ω2∫ +∞0

RPQ(p)h+

Q,tan

h−P,tan

ieiωlpP3

pP3J1(rωp)p

2dp .

(12)The reflection S-wave angle θS(x′) is obtained fromSnell’s law as

θS(x′) = sin−1[(v

(1)S /v

(1)P ) sin θ(x′)

],

J0(rωp) and J1(rωp) are the zero-order and first-orderBessel functions,

pP3 =

√[v(1)P

]−2

− p2

is the vertical P-wave slowness,

l = R∗(x′) cos θ(x′) ,

and

r = R∗(x′) sin θ(x′) .

For the reflected PP-wave,

h+P,norm/h

−P,norm = −1

and

h+P,tan/h

−P,tan = 1 .

For the PS-wave,

h+S,norm/h

−P,norm = (v

(1)S p)/(v

(1)P p

(1)P3)

and

h+S,tan/h

−P,tan = (v

(1)S p

(1)S3 )/(v

(1)P p) ,

where

pS3 =

√[v(1)S

]−2

− p2

is the vertical S-wave slowness.

The ERC in equation 11 is defined as the ratio ofthe magnitude of the reflected PQ-wave and the in-cident P-wave. Therefore, they generalize the PWRCfrom equation 6 by taking into account the curvaturesof both the incident wavefront and the reflector. WhilePWRC depend on the stiffness and density contrastsacross the boundary and the incidence angle θ(x′), ERCare controlled by one more dimensionless parameter,L(x′), which incorporates the interface curvature. In thestationary-phase approximation, the ERC reduce to thecorresponding PWRC. In contrast to PWRC, ERC cor-rectly describe reflection phenomena at near-critical andpost-critical incidence angles.

Equation 10 shows how the local reflector curva-ture is incorporated into ERC. If the reflector is locallyplane, then H(x′) = 0, and the distance R∗(x′) reducesto R(x′). For particular parameter combinations, R∗(x′)may go to infinity, which means that the incident P-wave appears to be locally plane; in that case, the ERCreduces to the PWRC. For certain values of the prod-uct R(x′)H(x′), R∗(x′) may become negative. Then theapparent source represents the focus of an apparent con-verging spherical wave, and the ERC becomes complexconjugate.

PARAMETER SENSITIVITY STUDY AND

3D DIFFRACTION MODELING

Numerical study of effective reflection

coefficients

As follows from the formalism discussed above, ERCrepresent the exact reflection response for a plane reflec-tor and incident spherical P-wave. When the reflectinginterface is curved, ERC provide a practical approxi-mate tool to compute the reflected wavefield. Here, westudy the ERC for an interface between isotropic andTI media as a function of the parameter L, Thomsenanisotropy parameters of the reflecting halfspace, andthe local interface geometry incorporated into the dis-tance R∗. If the reflected wavefield is well-described bygeometrical seismics, ERC reduces to the correspondingPWRC. Therefore, the difference between the effectiveand plane-wave reflection coefficients helps to estimatethe error of the geometrical-seismics approximation.

Influence of the parameter L

First, we examine the dependence of ERC computed fora plane interface on the parameter L = ωR∗/v

(1)P (ω is

the angular frequency and R∗ is the distance from theapparent source to point x′ at the interface). Figure 2shows comparison of the ERC for PP- and PS-wavescomputed for a wide range of L with the correspond-ing PWRC. For both modes, the difference between theERC and PWRC decreases for larger values of L (i.e.,


0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

1.2a

Magnitude o

f P

P

Incidence angle (degrees)

PWRC

L = 10

L = 102

L = 103

0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

0.5b

Magnitude o

f P

S


PWRC

L = 10

L = 102

L = 103

Figure 2. Dependence of the magnitude of the (a) PP-waveand (b) PS-wave effective reflection coefficients (ERC) on theparameter L. The corresponding plane-wave reflection coeffi-

cients (PWRC) are shown for comparison. The reflector is ahorizontal plane located 1 km below the source. The param-

eters of the incidence isotropic medium are v(1)P = 2 km/s,

v(1)S = 1.2 km/s, and ρ(1) = 2.15 g/cm3; for the reflecting

TI medium, v(2)P0 = 2.4 km/s, v

(2)S0 = 1.4 km/s, ρ(2) = 2.35

g/cm3, ε = 0.2, and δ = 0.1.

for larger frequency or distance R∗). However, in con-trast to PWRC, ERC oscillate in the post-critical do-main even for L = 103 due to the interference of thereflected and head waves.

For the relatively small L = 10, the ERC (especiallythe one for PS-waves) substantially deviate from thePWRC even at sub-critical incidence angles. This meansthat for low values of L geometrical seismics can beused only for near-vertical incidence (i.e., small source-receiver offsets). Indeed, it is well known that the accu-racy of the geometrical-seismics approximation stronglydepends on the source-interface distance normalized bythe predominant wavelength (Tsvankin, 1995). If the

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

1.2a

Magnitude o

f P

P


= 0

= 0.1

= 0.2

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

1.2b

Magnitude o

f P

P


= -0.1

= 0.1

= 0.3

Figure 3. Dependence of the PP-wave ERC on theanisotropy parameters. (a) δ = 0.1 and ε = 0, 0.1, and 0.2;(b) ε = 0.2 and δ = −0.1, 0.1, and 0.3. The interface is a

horizontal plane located 1 km below the source. The medium

parameters are v(1)P = 2 km/s, v

(1)S = 1.2 km/s, ρ(1) = 2.15

g/cm3, v(2)P0 = 2.4 km/s, v

(2)S0 = 1.4 km/s, and ρ(2) = 2.35

g/cm3; the frequency f = 32 Hz

source (in our case, the apparent source) is located closeto the interface, the reflected wavefield is influenced bythe curvature of the incident wavefront and cannot beaccurately described by geometrical seismics.

Influence of the anisotropy parameters

The anisotropy parameters ε and δ contribute to theERC for the PP- and PS-waves mostly at near- andpost-critical incidence angles (Figures 3 and 4). Thecritical angle is controlled by the horizontal P-wave ve-locity in the TI medium that depends on ε (v

(2)P (90) =

v(2)P0

√1 + 2ε). Figures 3a and 4a confirm that the criti-

cal angle decreases for larger values of ε, which causesa horizontal shift of the ERC curves. Also, the PS-


0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7a

Magnitude o

f P

S


= 0

= 0.1

= 0.2

0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7b

Magnitude o

f P

S


= -0.1

= 0.1

= 0.3

Figure 4. Dependence of the PS-wave ERC on theanisotropy parameters for the model from Figure 3.

wave ERC in the post-critical domain substantially in-creases with ε . In general, the reflectivity of PS-waves ismore sensitive to the anisotropy parameters than thatof PP-waves, likely because shear-wave signatures arecontrolled primarily by the relatively large parameter σ

(σ =[v(2)P0/v

(2)S0

]2(ε− δ)). The magnitude of σ typically

is much larger than that of ε and δ; in our model, σvaries from -2.94 to 2.94.

Since ERC at post-critical incidence angles includethe contributions of both the head and reflected waves,Figures 3 and 4 do not provide enough information topredict the influence of ε and δ on the time-domainwavefield. The long-offset synthetic seismograms dis-cussed below help to separate the head and reflectedwaves and evaluate their dependence on the anisotropyof the reflecting medium.

-2 -1 0 1 2

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0a

Data window

Source

Reflecto

r depth

(km

)

Offset (km)

z = 0 km

z = 0.1 km

z = 0.2 km

-2 -1 0 1

-10

-5

0

5

10

15

20

25b

R* (

km

)

Offset (km)

z = 0 km

z = 0.1 km

z = 0.2 km

Figure 5. (a) Model with a curved reflector. (b) The corre-sponding apparent distance R∗. The source is placed at thesurface, and an array of 101 receivers is located at a depth

of 585 m with a step of 50 m. The reflector is described bythe equation x3 = −1.185 +△z tanh [2π(x1 − 0.75)]; the pa-rameter △z is marked on the plot.

Influence of the reflector shape

Here, we generate ERC for a curved interface that has aflexural shape governed by the parameter △z (Figure 5).When the reflector degenerates into a horizontal plane(△z = 0), the apparent distance R∗ reduces to the ac-tual source-reflector distance R, which has no singularpoints. The offset dependence of R∗ becomes more com-plicated with increasing reflector curvature (Figure 5b).

The ERC for both PP- and PS-waves are displayedin Figure 6 for three values of △z. We observe a rapidchange in both ERC near an offset of 0.75 km, where thedistance R∗ exhibits sharp spikes associated with theflexural segment of the reflector. The offset of the post-critical reflection on the left side of the model increaseswith reflector depth, which is controlled by △z.


-2 -1 0 1

0.0

0.2

0.4

0.6

0.8

1.0

1.2aM

agnitude o

f P

P

Offset (km)

z = 0 km

z = 0.1 km

z = 0.2 km

-2 -1 0 1

0.0

0.1

0.2

0.3

0.4

0.5b

Magnitude o

f P

S

Offset (km)

z = 0 km

z = 0.1 km

z = 0.2 km

Figure 6. Offset-dependent magnitude of the (a) PP-waveand (b) PS-wave effective reflection coefficients for the model

from Figure 5. The medium parameters are v(1)P = 2 km/s,

v(1)S = 1.2 km/s, ρ(1) = 2.15 g/cm3, v

(2)P0 = 2.4 km/s, v

(2)S0 =

1.4 km/s, ρ(2) = 2.35 g/cm3, ε = 0.2, and δ = 0.1.

Tip-wave superposition method for elastic

media

To model reflected wavefields for curved interfaces, weneed to evaluate the surface integral 1. We obtain thehigh-frequency (or far-field) approximation of the inte-gral using the tip-wave superposition method (TWSM)(Klem-Musatov and Aizenberg, 1985; Klem-Musatovet al., 1993, 1994). This published version of the methodis designed for modeling 3D wavefields in layered mod-els with complex interface geometries. The main as-sumption of the method is that the source-interface,receivers-interface and interface-interface distances obeythe Rayleigh principle (i.e., they are on the order of sev-eral wavelengths or larger).

Because the upper halfspace in our model isisotropic, in Appendix F we extend TWSM to isotropicelastic media. We show that TWSM generates the re-

flection response by the superposition of tip-diffractedwaves (which explains the name of the method) excitedat the reflector in accordance with the Huygens princi-ple.

Our implementation of TWSM involves splittingthe reflector into rhombic elements that conform to theChebyshev coordinates introduced earlier. Each elementacts as a secondary source emitting a tip-wave beam to-wards the receiver array, and the beams form what wecall the receiver matrix. We compute the boundary datausing either the ERC or PWRC, and form the sourcematrix for all rhombic elements at the interface. Thenthe two matrices are multiplied element-by-element togenerate the reflected wavefield and sum the reflectionresponses at each receiver. The superposition of the tip-wave beams in TWSM produces the correct reflectiontraveltimes at the receivers, but the amplitudes may besomewhat distorted by the high-frequency approxima-tions applied in the computation of the ERC and thesurface integral.

The TWSM with PWRC is computationally inex-pensive, but requires storing large matrices containinginformation about tip waves. Although the data stor-age may present a logistical problem, minor changesof the model can be incorporated without recalculatingall tip-wave beams. This advantage of TWSM becomesparticularly valuable for layered models and in surveydesign. Application of ERC in TWSM involves compu-tation of the Fourier-Bessel integrals for the entire fre-quency range of the initial wavelet instead of the simpleclosed-form PWRC expressions. Also, the disk-space re-quirements become even more demanding because thetip-wave matrices have to be stored separately for eachfrequency.

Modeling results

As illustrated by the numerical tests above, effectivereflection coefficients are sensitive to the elastic param-eters and the shape of the interface. Here, we combineERC with the tip-wave superposition method to gener-ate the time-domain wavefield and analyze its behaviorfor two different reflector shapes.

Influence of the anisotropy parameters

The seismograms in Figures 7–10 are computed for acurved reflector described by the function x3 = −1 +0.3 exp(−8x2

1 − 8x22). The reflection traveltimes of both

PP- and PS-waves exhibit a wide triplication (cusp) atthe far offsets, which corresponds to the caustic pro-duced at the anticlinal part of the Gaussian-cap reflec-tor.

In agreement with the ERC in Figure 3a, the PP-wave reflection amplitude at long offsets rapidly in-creases with ε (Figure 7). The amplitude at the largestoffset (2.5 km) is approximately four times higher for


1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

a

= 0.0

= 0.1

Offset (km)

Tim

e (

s)

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

b

= 0.1

= 0.1

Offset (km)

Tim

e (

s)

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

c

= 0.2

= 0.1

Offset (km)

Tim

e (

s)

Figure 7. Influence of ε on the vertical displacement of thePP-wave reflected from a curved interface. The source and anarray of 101 receivers are placed at the surface. The reflectoris described by x3 = −1 + 0.3 exp(−8x2

1 − 8x22), so that the

cap of the Gaussian anticline is located at a depth of 0.7

km below the source. The medium parameters are v(1)P = 2

km/s, v(1)S = 1.2 km/s, ρ(1) = 2.15 g/cm3, v

(2)P0 = 2.4 km/s,

v(2)S0 = 1.4 km/s, and ρ(2) = 2.35 g/cm3; the values of ε and

δ are marked on the plots.

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

a

= 0.2

= -0.1

Offset (km)

Tim

e (

s)

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

b

= 0.2

= 0.1

Offset (km)

Tim

e (

s)

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.0 0.4 0.8 1.2 1.6 2.0 2.4

c

= 0.2

= 0.3

Offset (km)

Tim

e (

s)

Figure 8. Influence of δ on the vertical PP-wave displace-ment for the model from Figure 7.


2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

a

= 0

= 0.1

Offset (km)

Tim

e (

s)

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

b

= 0.1

= 0.1

Offset (km)

Tim

e (

s)

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

c

= 0.2

= 0.1

Offset (km)

Tim

e (

s)

Figure 9. Influence of ε on the vertical PS-wave displace-ment for the model from Figure 7.

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

a

= 0.2

= -0.1

Offset (km)

Tim

e (

s)

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

b

= 0.2

= 0.1

Offset (km)

Tim

e (

s)

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.0 0.4 0.8 1.2 1.6 2.0 2.4

head

wave

PPS

c

= 0.2

= 0.3

Offset (km)

Tim

e (

s)

Figure 10. Influence of δ on the vertical PS-wave displace-ment for the model from Figure 7.


ε = 0.2 than for ε = 0. In contrast, the near-offset re-flections are weakly sensitive to ε. The influence of δ onPP-wave amplitudes is most visible at moderate offsetsbetween 1.5 km and 1.7 km (Figure 8). For the maxi-mum offset, the amplitude becomes about 20% higherwhen δ increases by 0.2.

The PS wavefield for a range of ε and δ valuesis shown in Figures 9 and 10. The influence of bothanisotropy parameters on the reflected wave can be gen-erally predicted from the corresponding ERC in Fig-ure 4. In particular, the moderate- and far-offset reflec-tion amplitudes noticeably increase with ε. The ampli-tude at the largest offset doubles when ε changes fromzero to 0.2. It is interesting that the amplitude of thePPS head wave (marked with an arrow for the right-most receiver) for the same change in ε decreases by only12%. Although the influence of δ is less pronounced, a0.2 increase in δ reduces the maximum-offset amplitudeof the reflected PS-wave by about 30%. The head-waveamplitude, however, is practically independent of δ.

Influence of the reflector shape

Synthetic PP-wave seismograms computed for a flexu-ral reflector with variable mean curvature (Figure 5) isdisplayed in Figure 11. The isotropic 2D version of thismodel has been used for testing finite-difference mod-eling and generalized ray tracing (Hanyga and Helle,1995). As the value of △z increases, the flexure pro-duces a strong caustic loop formed near zero offset. Thehead waves cannot be clearly identified due to the lim-ited length of the receiver array, which extends only upto the interference zone of the reflected and head waves.

For a plane reflector (△z = 0), we compared ourmodeling results with the exact wavefield computed bythe reflectivity method. As expected, the elastic ver-sion of TWSM based on the superposition of tip-wavebeams accurately reproduces traveltimes for the wholeoffset range. The amplitudes in Figure 11 are only a fewpercents higher than those produced by the reflectivityalgorithm.

To evaluate the errors of the conventional Kirchhoffmodeling technique, we also computed the wavefield us-ing the plane-wave reflection coefficient in TWSM (Fig-ure 12). The discontinuous slope of the PWRC at thecritical angles causes artificial diffractions for both plane(△z = 0) and curved reflectors. Additionally, the reflec-tion amplitudes for near-critical and post-critical offsetsare higher than those obtained with the ERC in Fig-ure 11.

Similar conclusions can be drawn from the PS-waveseismograms for the same model in Figures 13 and 14.The PS reflection also exhibits a caustic loop that be-comes more prominent for △z = 0.2 km. The critical off-set for the converted (PPS) head wave is smaller thanthat for the corresponding PPP-wave, which explainsthe separation of the head wave (marked with an ar-

1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

a

z = 0 km

Offset (km)

Tim

e (

s)

1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

b

z = 0.1 km

Offset (km)

Tim

e (

s)

1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

c

z = 0.2 km

Offset (km)

Tim

e (

s)

Figure 11. Vertical PP-wave displacement computed withthe ERC for the model from Figures 5 and 6. The sourceis placed at the surface, and an array of 101 receiversis located at a depth of 585 m with a step of 50 m.The reflector is described by the equation x3 = −1.185 +△z tanh [2π(x1 − 0.75)]; the parameter △z is marked on the

plots. The medium parameters are v(1)P = 2 km/s, v

(1)S = 1.2

km/s, ρ(1) = 2.15 g/cm3, v(2)P0 = 2.4 km/s, v

(2)S0 = 1.4 km/s,

ρ(2) = 2.35 g/cm3, ε = 0.2, and δ = 0.1.


1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

a

z = 0 km

Offset (km)

Tim

e (

s)

1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

b

z = 0.1 km

Offset (km)

Tim

e (

s)

1.8

1.6

1.4

1.2

1.0

-2 -1 0 1 2

c

z = 0.2 km

Offset (km)

Tim

e (

s)

Figure 12. Vertical PP-wave displacement computed withthe PWRC for the model from Figure 11.

2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

PPS

head

wave

a

z = 0 km

Offset (km)

Tim

e (

s)

2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

PPS

head

wave

b

z = 0.1 km

Offset (km)

Tim

e (

s)

2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

PPS

head

wave

c

z = 0.2 km

Offset (km)

Tim

e (

s)

Figure 13. Vertical PS-wave displacement computed withthe ERC for the model from Figure 11.


2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

a

z = 0 km

Offset (km)

Tim

e (

s)

2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

b

z = 0.1 km

Offset (km)

Tim

e (

s)

2.0

1.8

1.6

1.4

1.2

-2 -1 0 1 2

c

z = 0.2 km

Offset (km)

Tim

e (

s)

Figure 14. Vertical PS-wave displacement computed withthe PWRC for the model from Figure 11.

row for the left-most receiver) and reflected wave at thefar offsets in Figure 13. Although the artificial diffrac-tions caused by the PWRC in Figure 14 are not as pro-nounced as those for PP-waves, application of the ERC(Figure 13) yields a cleaner gather.

Our 3D modeling results obtained with TWSMagree well in the kinematic sense with the wavefieldscomputed by finite differences and generalized ray trac-ing for the corresponding isotropic 2D model (Hanygaand Helle, 1995). The amplitudes, however, are not thesame because of a different geometrical spreading in 2Dand 3D and the influence of anisotropy in our model.

DISCUSSION AND CONCLUSIONS

Effective reflection coefficients (ERC) provide a prac-tical tool for modeling near- and post-critical reflectedwavefields and for taking the interface curvature into ac-count. By extending a formalism suggested previouslyfor the acoustic problem, we gave a complete analyticdescription of ERC for curved reflectors in anisotropicmedia. The reflected wavefield can be expressed througha generalized plane-wave decomposition, which includesthe local spatial spectrum of the incident wave expressedthrough an integral over the whole interface.

Although this decomposition gives an accuratewavefield representation near a reflector of arbitraryshape, its computational cost for 3D anisotropic mod-els is prohibitive. Therefore, we suggested to approx-imately obtain the reflected wavefield from the con-ventional Weyl-type integral computed for an “appar-ent” source location, which depends on the incidenceangle and the mean reflector curvature. Then the ra-tio of the reflected and incident wavefields yields thespatially varying ERC along the reflector. To incor-porate ERC in 3D diffraction modeling, we employedthe tip-wave superposition method (TWSM) general-ized for elastic wave propagation. The superposition ofthe tip-wave beams corresponding to rhombic interfacesegments produces correct reflection traveltimes, whilethe accuracy of amplitudes depends on the validity ofthe high-frequency approximation used both in TWSMand in the computation of ERC. TWSM is also capa-ble of modeling multipathing and caustics produced bycurved segments of the reflector.

We implemented this formalism and studied theproperties of ERC for an interface separating isotropicand TI media. The symmetry axis in the reflecting TIhalfspace was assumed to be orthogonal to the reflec-tor, which is typical for anisotropic shale layers. For thespecial case of a plane interface, the ERC represents thefrequency-dependent exact wavefield governed by thevelocity and density contrasts, Thomsen anisotropy pa-rameters, and source-receiver geometry. Numerical testsshow that the ERC for PP-waves at post-critical inci-dence angles is particularly sensitive to the parameterε responsible for near-horizontal P-wave propagation inthe TI halfspace.

The ERC substantially deviates from the corre-sponding plane-wave reflection coefficient (PWRC) inthe post-critical domain, where the displacement fieldis influenced by the head wave. At low frequencies, thedifference between the ERC and PWRC may be signifi-cant even for sub-critical incidence angles. These resultsconfirm the limitations of the geometrical-seismics ap-proximation, which is based on PWRC, in describingpoint-source radiation in layered media.

We also presented synthetic examples illustratingthe importance of properly accounting for the reflector


curvature in the computation of ERC. When the reflec-tor is curved, the ERC may change rapidly along theinterface in accordance with variations of the local in-terface shape, thus influencing synthetic modeling.

The methodology developed here can be used togenerate accurate boundary data for 3D Kirchhoff-typemodeling in anisotropic media. In particular, our syn-thetic examples confirm that ERC eliminate the arti-facts produced by PWRC and provide more accurateamplitudes for large incidence angles and in the pres-ence of significant reflector curvature. Our results can bealso applied in anisotropic AVO analysis of long-offsetPP and PS reflection data.

ACKNOWLEDGMENTS

M.A. is grateful to the Center for Wave Phenomena forsupport during her five-month visit to Colorado Schoolof Mines. A.A. acknowledges the Russian Foundationfor Basic Research (grant 07-05-00671). This work waspartially funded by the Consortium Project on SeismicInverse Methods for Complex Structures at the Centerfor Wave Phenomena.

REFERENCES

Aki, K. and P. Richards, 2002, Quantitative Seismol-ogy (2nd edition): University Science Books.

Ayzenberg, M. A., A. M. Aizenberg, H. B. Helle, K. D.Klem-Musatov, J. Pajchel, and B. Ursin, 2007, Three-dimensional diffraction modeling of singly scatteredacoustic wavefields based on the combination of sur-face integral propagators and transmission operators:Geophysics, 72, SM19–SM34.

Ben-Menahem, A. and S. J. Singh, 1998, SeismicWaves and Sources: Dover Publications.

Brekhovskikh, L. M., 1980, Waves in Layered Media:Academic Press.

do Carmo, M. P., 1976, Differential Geometry ofCurves and Surfaces: Prentice Hall.

Downton, J. E. and C. Ursenbach, 2006, Linearizedamplitude variation with offset (AVO) inversion withsupercritical angles: Geophysics, 71, E49–E55.

Frazer, L. N. and M. K. Sen, 1985, Kirchhoff-Helmholtzreflection seismograms in a laterally inhomogeneousmulti-layered elastic medium - I. Theory: GeophysicalJournal of the Royal Astronomical Society, 80, 121–147.

Graebner, M., 1992, Plane-wave reflection and trans-mission coefficients for a transversely isotropic solid:Geophysics, 57, 1512–1519.

Hanyga, A. and H. H. Helle, 1995, Synthetic seis-mograms from generalized ray tracing: GeophysicalProspecting, 43, 51–75.

Kampfmann, W., 1988, A study of diffraction-likeevents on DECORP 2-S by Kirchhoff theory: Jour-nal of Geophysics, 62, 163–174.

Kennett, B. L. N., 1994, Representations of the seismicwavefield: Geophysical Journal International, 118,344–357.

Klem-Musatov, K. D. and A. M. Aizenberg, 1985,Seismic modelling by methods of the theory of edgewaves: Journal of Geophysics, 57, 90–105.

Klem-Musatov, K. D., A. M. Aizenberg, H. B. Helle,and J. Pajchel, 1993, Seismic simulation by the tipwave superposition method in complex 3D geologi-cal models: 55th Anual Meeting of EAGE, ExpandedAbstracts.

——–, 2004, Reflection and transmission at curvilinearinterface in terms of surface integrals: Wave Motion,39, 77–92.

Klem-Musatov, K. D., A. M. Aizenberg, J. Pajchel,and H. B. Helle, 1994, Edge and Tip Diffractions- Theory and Applications in Seismic Prospecting:Norsk Hydro.

Morse, F. M. and H. Feshbach, 1953, Methods of The-oretical Physics: McGraw-Hill Science.

Pao, Y.-H. and V. Varatharajulu, 1976, Hyugens’ prin-ciple, radiation conditions, and integral formulas forthe scattering of elastic waves: Journal of the Acous-tical Society of America, 59, 1361–1371.

Ruger, A., 1997, P-wave reflection coefficients fortransversely isotropic models with vertical and hor-izontal axis of symmetry: Geophysics, 62, 713–722.

——–, 2002, Reflection Coefficients and AzimuthalAVO Analysis in Anisotropic Media: SEG.

Schleicher, J., M. Tygel, B. Ursin, and N. Bleis-tein, 2001, The Kirchhoff-Helmholtz integral foranisotropic elastic media: Wave Motion, 34, 353–364.

Sen, M. K. and L. N. Frazer, 1991, Multifold phasespace path integral synthetic seismograms: Geophys-ical Journal International, 104, 479–487.

Shuey, R. T., 1985, A simplification of the Zoeppritzequations: Geophysics, 50, 609–614.

Stovas, A. and B. Ursin, 2003, Reflection and transmis-sion responses of layered transversely isotropic vis-coelastic media: Geophysical Prospecting, 51, 447–477.

Thomsen, L., 1993, Weak anisotropic reflections: Off-set Dependent Reflectivity (Castagna, J., and Backus,M., Eds.), 103–114.

Tsvankin, I., 1995, Seismic Wavefields in LayeredIsotropic Media: Samizdat Press.

——–, 2005, Seismic Signatures and Analysis of Re-flection Data in Anisotropic Media: Elsevier.

Ursin, B., 2004, Tutorial: Parameter inversion andangle migration in anisotropic elastic media: Geo-physics, 69, 1125–1142.

Ursin, B. and M. Tygel, 1997, Reciprocal volume andsurface scattering integrals for anisotropic elastic me-dia: Wave Motion, 26, 31–42.


van der Baan, M. and D. Smith, 2006, Amplitude anal-ysis of isotropic P-wave reflections: Geophysics, 72,C93–C103.

Cerveny, V., 2001, Seismic Ray Theory: CambridgeUniversity Press.

Weatherborn, C. E., 1930, Differential Geometry ofThree Dimensions. Vol. II: Cambridge UniversityPress.

Wenzel, F., K.-J. Stenzel, and U. Zimmermann, 1990,Wave propagation in laterally heterogeneous layeredmedia: Geophysical Journal International, 103, 675–684.


APPENDIX A: GENERALIZED PLANE WAVES

The conventional plane-wave decomposition of point-source radiation (i.e., the Weyl integral) can be used to obtain thereflected or transmitted wavefield for a plane interface between two homogeneous media. Here, we define generalizedplane waves, which help to extend the principle of plane-wave decomposition to interfaces of arbitrary shape and toaccount for local heterogeneity.

Let us consider wave propagation in a medium with a smooth curved interface S, which separates two hetero-geneous, arbitrarily anisotropic halfspaces D(1) and D(2). Each medium (superscript m) is described by the stiffness

tensor C(m)(x) =[c(m)ijkl(x)

]and density ρ(m); the unit vector n normal to the interface points toward D(1).

We define the curvilinear coordinates (s1, s2, s3) in the immediate vicinity of the interface S inside D(m), suchthat (s1, s2) form the Chebyshev coordinate mesh along the interface, and the axis s3 is normal to the interface andpoints inside D(m). Additionally, we define the local Cartesian coordinates (y1, y2, y3) with the origin at point x′.The axis y3 coincides with s3, while y1 and y2 are tangential to the curves s1 and s2 at x′.

In the vicinity of point x′, the Chebyshev and local Cartesian coordinates are related as (Weatherborn, 1930;do Carmo, 1976; Klem-Musatov et al., 2004; Ayzenberg et al., 2007):

s1(y1, y2, y3) = y1 +O(y3) ,s2(y1, y2, y3) = y2 +O(y3) ,s3(y1, y2, y3) = y3 − 1

2

[C1(x

′)y21 +C2(x

′)y22

]+O(y3) ,

(A-1)

where C1(x′) and C2(x

′) are the local curvatures of the interface along s1 and s2. The local and global Cartesiancoordinates are related by the linear transform:

yj(x1, x2, x3) = bij(x′)xj , (A-2)

where bij(x′) are the elements of the linear transform matrix, which is specified, for example, in Cerveny (2001).

We introduce a generalized plane wave in the vicinity of the interface as

u(m)(s1, s2, s3) = a(m)

[h

(m) + iv(m) s23

2

]eiω(p1s1+p2s2+p3s3) , (A-3)

where p1 and p2 can be treated as the components of the slowness vector tangential to the interface. The normalslowness p3, amplitude factor a(m) and polarization vector h(m) along with its perturbation v(m) have to be found.At the interface, where s3 = 0 and the term proportional s23 vanishes, equation A-3 describes a conventional planewave (Cerveny, 2001).

The unknown parameters of the generalized plane wave can be determined by substituting equation A-3 for apoint x′ into the wave equation in the frequency domain (the “stationary” wave equation). First, we rewrite the

stationary wave equation in the two-index notation C(m)jl (x′) =

[c(m)ijkl(x

′)]

(Kennett, 1994):

C(m)jl (x′)

∂2u(m)

∂xj∂xl(x′) +

∂C(m)jl

∂xj(x′)

∂u(m)

∂xl(x′) + ρω2

u(m)(x′) = 0. (A-4)

Substituting the generalized plane wave A-3 into equation A-4 and taking the coordinate transformations A-1 and A-2into account yields

−ω2[C

(m)ik (x′)pipk − ρ(m)I

]h(m)

−i[ωD(m)(x′)h(m) + C

(m)33 (x′)v(m)

]= 0 ,

(A-5)

where C(m)ik (x′) = bij(x

′)bkl(x′)C

(m)jl (x′) is the local stiffness tensor, and D(m)(x′) =

p3

[C1(x

′)C(m)11 (x′) + C2(x

′)C(m)22 (x′)

]− pl

∂C(m)jl

∂yj(x′) is the matrix that contains information about the local

interface curvature. Both the real and imaginary parts of the left-hand side of equation A-5 have to go to zero. Thereal part of equation A-5 reduces to the well-known Christoffel equation (Cerveny, 2001)

[C

(m)ik (x′)pipk − ρ(m)

I]h

(m) = 0 . (A-6)


The slowness components p(m)Q3 (p1, p2;x

′) of waves Q=P,S1 and S2 are obtained from the equation

det[C

(m)ik (x′)pipk − ρ(m)I

]= 0. By substituting p

(m)Q3 (p1, p2;x

′) into equation A-6, we find the mutually orthogo-

nal unit polarization vectors h(m)Q (x′). Note that the slownesses p

(m)Q3 (p1, p2;x

′) and polarization vectors h(m)Q (x′) are

functions of the medium parameters at point x′, but do not depend on the local interface curvature.The imaginary part of equation A-5 constrains the perturbation vectors:

v(m)Q (x′) = ω

[C

(m)33 (x′)

]−1

D(m)(x′)h

(m)Q (x′) . (A-7)

In the special case of a plane interface and homogeneous media, the derivatives∂C

(m)jl

(x′)

∂yjand curvatures C1(x

′) and

C2(x′) are equal to zero. Then the term D(m)(x′) and the perturbation v

(m)Q (x′) also vanish.

To solve the reflection/transmission problem, it is necessary to separate waves traveling towards the interface

(u(m)−Q (s1, s2, s3)) from those traveling away from it (u

(m)+Q (s1, s2, s3)) (Cerveny, 2001; Aki and Richards, 2002).

We assume that sorting is done according to the orientation of the group velocity vector. If the slownesses p(m)−Q3

and p(m)+Q3 correspond to waves traveling towards and away from the interface (respectively), the generalized plane

wave A-3 can be represented as

u(m)±Q (s1, s2, s3;x

′) = a(m)±Q

[h

(m)±Q (x′) + iv

(m)±Q (x′)

s232

]eiω(p1s1+p2s2+p

(m)±Q3 (x′)s3) . (A-8)

APPENDIX B: GENERALIZED PLANE-WAVE DECOMPOSITION AT THE INTERFACE

Here we introduce the generalized spectral integrals designed to decompose the displacement at the interface intothe generalized plane P-, S1- and S2-waves described in Appendix A. The total displacement inside D(m) can beexpressed as the sum of the waves traveling towards and away from the interface (equation A-8):

u(m)(s1, s2, s3) = u

(m)+(s1, s2, s3;x′) + u

(m)−(s1, s2, s3;x′) , (B-1)

with the displacements represented by the generalized plane-wave decomposition,

u(m)±(s1, s2, s3;x′) = ω2

2π

∫ +∞−∞

∫ +∞−∞

[H(m)± + iV(m)± s23

2

]

×E(m)±(s3)a(m)±eiω(p1s1+p2s2) dp1dp2 .

(B-2)

Equation B-2 is a generalization of the conventional Weyl-type integral for curved interfaces and locally heterogeneousmedia. Whereas the Weyl-type decomposition is valid everywhere in the halfspace D(m), the generalized expression B-2is restricted to an infinitely thin layer covering the interface. Therefore, our formalism can be used for calculation ofthe reflection response only in the immediate vicinity of the reflector.

The orthogonal polarization matrices H(m)± are similar to those introduced by Cerveny (2001) in his equa-tion 5.4.110,

H(m)±(x′) =

[h

(m)±P (x′); h

(m)±S1

(x′); h(m)±S2

(x′)].

V(m)±(x′) =

[v

(m)±P (x′); v

(m)±S1

(x′); v(m)±S2

(x′)]

are the perturbation matrices, and

E(m)±(s3;x

′) = diag

[eiωp

(m)±P3 (x′)s3 ; e

iωp(m)±S1,3 (x′)s3 ; e

iωp(m)±S2,3 (x′)s3

].

The vectors a(m)± = (a(m)±qP , a

(m)±qS1 , a

(m)±qS2 )T contain the unknown amplitudes of the generalized plane waves.

The generalized plane-wave decomposition B-2 is valid for interfaces of arbitrary shape in heterogeneousanisotropic media. If the interface is plane, the curvatures C1(x

′) and C2(x′) go to zero, and the curvilinear coordi-

nates (s1, s2, s3) coincide with the local Cartesian coordinate system. If, in addition, the medium near the interface ishomogeneous, the normal components of the slownesses and polarization vectors do not depend on the reference point


x′. Then integral B-2 reduces to the well-known Weyl-type decomposition over conventional plane waves (Cerveny,2001; Aki and Richards, 2002; Tsvankin, 1995, 2005).

At the interface (s3 → 0) equation B-2 reduces to the inverse Fourier integral,

u(m)±(s1, s2, 0;x

′) =ω2

2π

∫ +∞

−∞

∫ +∞

−∞H

(m)±a

(m)±eiω(p1s1+p2s2)dp1dp2 . (B-3)

APPENDIX C: REFLECTION AND TRANSMISSION OPERATORS IN ANISOTROPIC MEDIA

The results of Appendix B make it possible to introduce the generalized plane-wave representation of the reflectedwavefield at the interface. We assume that a point dislocation source is located in the upper halfspace D(1), and thereare no sources in the lower halfspace D(2). Then equations B-1 and B-3 can be written for D(1) as

u(1)(s1, s2, 0) = u

(1)+(s1, s2, 0;x′) + u

(1)−(s1, s2, 0;x′) , (C-1)

where u(1)−Q (s1, s2, 0;x

′) and u(1)+(s1, s2, 0;x′) may be considered as the incident and reflected wavefields (respec-

tively) at the interface. The reflected displacement u(1)+(s1, s2, 0;x′) is represented by the generalized spectral inte-

gral,

u(1)+(s1, s2, 0;x

′) =ω2

2π

∫ +∞

−∞

∫ +∞

−∞H

(1)+a

(1)+eiω(p1s1+p2s2)dp1dp2 . (C-2)

The amplitudes of the reflected (a(1)+) and incident (a(1)−) waves are related by the matrix R(p;x′) of the generalizedplane-wave reflection and transmission coefficients:

a(1)+ = R(p;x′)a(1)− , (C-3)

where p =√p21 + p2

2, and

R(p;x′) =

RPP RS1P RS2P

RPS1 RS1S1 RS2S1

RPS2 RS1S2 RS2S2

. (C-4)

The matrix C-4 coincides with the one introduced by Cerveny (2001), if the stiffness coefficients are fixed at locationx′, and the plane interface is tangential to the actual reflector at x′.

Because the matrix H(1)− is orthogonal, it satisfies the equality[H(1)−

]−1

=[H(1)−

]T. From equation B-3 it

follows that u(1)−(p1, p2, 0;x′) = H(1)−a(1)−, which allows us to obtain the amplitude vector of the incident wave in

the form

a(1)− =

[H

(1)−]T

u(1)−(p1, p2, 0;x

′) . (C-5)

Taking into account equations C-3 and C-5, the reflected wavefield C-2 can be represented as

u(1)+(s1, s2, 0;x′) = ω2

2π

∫ +∞−∞

∫ +∞−∞ H(1)+R(p;x′)

[H(1)−

]T

×u(1)−(p1, p2, 0;x′)eiω(p1s1+p2s2)dp1dp2 ,

(C-6)

where the spatial spectrum of the incident wavefield is expressed by the generalized Fourier integral over the curvedinterface:

u(1)−(p1, p2, 0;x

′) =1

2π

∫ +∞

−∞

∫ +∞

−∞u

(1)−(s1, s2, 0;x′)e−iω(p1s1+p2s2)ds1ds2 . (C-7)


For the incident spherical P-wave excited by a point source, u(1)−(s1, s2, 0;x′) = u

(1)−P (s1, s2, 0;x

′). The polar-ization matrix H(1)+ can be separated into the matrices for P- and S-waves:

H(1)+(x′) = H

(1)+P (x′) + H

(1)+S (x′) , (C-8)

H(1)+P (x′) =

[h

(m)±P (x′) 0 0

], H

(1)+S (x′) =

[0 h

(m)±S1

(x′) h(m)±S2

(x′)]. (C-9)

The reflected wavefield C-6 can be decomposed into the displacements of PP-waves and split PS-waves. The spectralrepresentation for PP-waves (Q=P) or converted PQ-waves (Q=S1 or S2) at the interface is given by

u(1)+PQ (s1, s2, 0;x

′) = ω2

2π

∫ +∞−∞

∫ +∞−∞ H

(1)+Q R(p;x′)

[H(1)−

]T

×u(1)−P (p1, p2, 0;x

′)eiω(p1s1+p2s2)dp1dp2.(C-10)

The displacement component orthogonal to the interface is

u(1)+PQ,norm(s1, s2, 0;x

′) = ω2

2π

∫ +∞−∞

∫+∞−∞ RPQ(p;x′)

h(1)+Q,norm(x′)

h(1)−P,norm(x′)

×u(1)−P,norm(p1, p2, 0;x

′)eiω(p1s1+p2s2)dp1dp2 .

(C-11)

For the two displacement components (j = 1, 2) tangential to the interface, we have

u(1)+PQ,j(s1, s2, 0;x

′) = ω2

2π

∫ +∞−∞

∫ +∞−∞ RPQ(p;x′)

h(1)+Q,j

(x′)

h(1)−P,j

(x′)

×u(1)−P,j (p1, p2, 0;x

′)eiω(p1s1+p2s2)dp1dp2.

(C-12)

APPENDIX D: PLANE-WAVE REFLECTION COEFFICIENTS FOR VTI MEDIA

The symmetry axis of the reflecting TI medium in our model is assumed to be orthogonal to the interface. Therefore,the plane-wave reflection coefficients in equations C-10–C-12 coincide with those for a horizontal interface betweenisotropic and VTI media. Also, for purposes of computing the reflection coefficient, the slowness vectors of the incident,reflected, and transmitted waves can be confined to the (x1, x3)-plane. The vertical slowness components q(m) areobtained from the eigenvalues of the Christoffel equation,

det

(c(m)11 p2 + c

(m)55 (q(m))2 − ρ(m) (c

(m)13 + c

(m)55 )pq(m)

(c(m)13 + c

(m)55 )pq(m) c

(m)33 (q(m))2 + c

(m)55 p2 − ρ(m)

)= 0 . (D-1)

The vertical slownesses of P- and SV-waves are given by

q(m)P = 1√

2

√K

(m)1 −

√K

(m)21 − 4K

(m)2 K

(m)3 ,

q(m)S = 1√

2

√K

(m)1 +

√K

(m)21 − 4K

(m)2 K

(m)3 ,

(D-2)

where

K(m)1 = ρ(m)

c(m)33

+ ρ(m)

c(m)55

− (c(m)11

c(m)55

+c(m)55

c(m)33

− (c(m)13 +c

(m)55 )2

c(m)33 c

(m)55

)p2 ,

K(m)2 =

c(m)11

c(m)33

p2 − ρ(m)

c(m)33

,

K(m)3 = p2 − ρ(m)

c(m)33

.

(D-3)

The eigenvectors of the Christoffel equation D-1 yield the directional cosines of the polarization vectors:


l(m)P =

√c(m)33 q

(m)2P

+c(m)55 p2−ρ(m)

(c(m)33 +c

(m)55 )q

(m)2P

+(c(m)11 +c

(m)55 )p2−2ρ(m)

,

m(m)P =

√c(m)55 q

(m)2P

+c(m)11 p2−ρ(m)

(c(m)33 +c

(m)55 )q

(m)2P

+(c(m)11 +c

(m)55 )p2−2ρ(m)

,

l(m)S =

√c(m)55 q

(m)2S

+c(m)11 p2−ρ(m)

(c(m)33 +c

(m)55 )q

(m)2S

+(c(m)11 +c

(m)55 )p2−2ρ(m)

,

m(m)S =

√c(m)33 q

(m)2S

+c(m)55 p2−ρ(m)

(c(m)33 +c

(m)55 )q

(m)2S

+(c(m)11 +c

(m)55 )p2−2ρ(m)

.

(D-4)

Next, we introduce a 4x4 matrix with the following elements:

m11 = l(1)P , m12 = m

(1)S , m13 = −l(2)P , m14 = −m(2)

S ,

m31 = m(1)P , m32 = −l(1)S , m33 = m

(2)P , m34 = −l(2)S ,

m21 = pl(1)P c

(1)13 + q

(1)P m

(1)P c

(1)33 , m22 = pm

(1)S c

(1)13 − q

(1)S l

(1)S c

(1)33 ,

m23 = −pl(2)P c(2)13 − q

(2)P m

(2)P c

(2)33 , m24 = pm

(2)S c

(2)13 − q

(2)S l

(2)S c

(2)33 ,

m41 = pm(1)P c

(1)55 + q

(1)P l

(1)P c

(1)55 , m42 = −pl(1)S c

(1)55 + q

(1)S m

(1)S c

(1)55 ,

m43 = pm(2)P c

(2)55 + q

(2)P l

(2)P c

(2)55 , m44 = −pl(2)S c

(2)55 + q

(2)S m

(2)S c

(2)55 .

(D-5)

(Note the misprint in the equivalent definition of the elements mij in Ruger (2002), pp. 51-52. In his notation, thenormalized stiffnesses aij should be replaced with cij .)

The cofactors of the matrix mij are

M11 = m22(m33m44 −m34m43) −m23(m32m44 −m34m42) +m24(m32m43 −m33m42),M21 = −m12(m33m44 −m34m43) +m13(m32m44 −m34m42) −m14(m32m43 −m33m42),M31 = m12(m23m44 −m24m43) −m13(m22m44 −m24m42) +m14(m22m43 −m23m42),M41 = −m12(m23m34 −m24m33) +m13(m22m34 −m24m32) −m14(m22m33 −m23m32),M12 = −m21(m33m44 −m34m43) +m23(m31m44 −m34m41) −m24(m31m43 −m33m41),M22 = m11(m33m44 −m34m43) −m13(m31m44 −m34m41) +m14(m31m43 −m33m41),M32 = −m11(m23m44 −m24m43) +m13(m21m44 −m24m41) −m14(m21m43 −m23m41),M42 = m11(m23m34 −m24m33) −m13(m21m34 −m24m31) +m14(m21m33 −m23m31),

(D-6)

Then the plane-wave reflection coefficients RPP (p) and RPS(p) can be found as

RPP (p) =−m11M11 −m21M11 +m31M11 +m41M11

m11M11 +m12M12 +m13M13 +m14M14, (D-7)

and

RPS(p) =−m11M12 −m21M22 +m31M32 +m41M42

m11M11 +m12M12 +m13M13 +m14M14. (D-8)

APPENDIX E: EFFECTIVE REFLECTION COEFFICIENTS FOR CURVED INTERFACES

For arbitrary interface geometry and heterogeneity, evaluation of integral 7 becomes complicated because it involvesgenerating the curvilinear mesh (s1, s2) and applying it in the computation of the spectrum u

(1)−P (p1, p2, 0;x

′) bymeans of the Fourier transform 8. However, the integration in equation 7 is performed over the tangential slownessplane (p1, p2) and is not explicitly related to the geometry of the mesh (s1, s2). This fact can be used to representthese integrals in the form similar to equation 6:

uPQ(x′) =[χPQ(x′)h+

Q(x′) + ǫPQ(x′)eQ(x′)] [

uincP (x′) · h−

P (x′)], (E-1)

where χPQ(x′) are the effective reflection coefficients (ERC), ǫPQ(x′) are the “spurious” reflection coefficients, andeQ(x′) are the unit vectors orthogonal to the polarization vectors h+

Q(x′).We define the effective and spurious reflection coefficients as


χPQ(x′) =uPQ(x′) · h+

Q(x′)


P (x′), (E-2)

ǫPQ(x′) =uPQ(x′) · eQ(x′)


P (x′). (E-3)

The ERC in equation E-2 is expressed through the projection of the displacement of the reflected PQ-mode ontothe polarization vector of the corresponding plane wave. Therefore, ERC generalize plane-wave reflection coefficients(PWRC) for point sources and curved interfaces. In the seismic frequency range, ERC describe the main componentof the reflected wavefield. Spurious reflection coefficients represent diffraction corrections, which are much smaller inmagnitude and can be neglected in equation E-1.

For acoustic wave propagation, integrals similar to those in equations C-11 and C-12 can be approximatelycomputed in the dominant-frequency approximation for an “apparent” source location and a plane interface tangentialto the actual reflector at point x′ (Ayzenberg et al., 2007). Then the problem reduces to the evaluation of Fourier-Bessel integrals similar to the ones for a plane interface. The same approach can be applied to elastic media becauseit is entirely based on the geometry of the incident P-wave. While the incidence angle θ(x′) stays the same, the actualsource moves along the ray to a new position located at the distance R∗(x′) from the plane interface:

R∗(x′) = R(x′)2 − sin2 θ(x′)

2 − sin2 θ(x′) − 2R(x′)H(x′) cos θ(x′), (E-4)

where H(x′) is the mean curvature of the interface. If the reflector is locally plane and H(x′) = 0, the distance R∗(x′)reduces to R(x′).

Adapting the results by Ayzenberg et al. (2007) for scalar integrals similar to 7, we replace the actual incidentP-wave uinc

P (s1, s2, 0;x′) in equation 8 by an apparent spherical wave u∗

P (s1, s2, 0;x′) and assume that the mesh

(s1, s2) belongs to the plane tangential to the actual reflector at point x′. Then the ERC in equation E-2 becomes

χPQ(x′) ≃ χPQ(θ(x′), L(x′)) =u∗PQ(x′) · h+

Q(x′)

u∗P (x′) · h−

P (x′), (E-5)

where L(x′) = ωR∗(x′)/v(1)P is a dimensionless frequency-dependent parameter. In contrast to integral 8, equation E-5

does not involve integration over the curvilinear mesh. For each point x′ at the curved reflector, the displacementu∗PQ(x′) is given by the conventional Weyl-type integral, while u∗

P (x′) describes the apparent incident P-wave in theplane tangential to the reflector at point x′.

Neglecting the term containing ǫPQ(x′), we rewrite equation E-1 as

uPQ(x′) ≃ χPQ(θ(x′), L(x′))[u

incP (x′) · h−

P (x′)]h

+Q(x′) . (E-6)

The apparent incident P-wave is described by

u∗P (s1, s2, s3;x

′) = gradeikPR

∗

R∗ =

(ikP − 1

R∗

)eikPR

∗

R∗

(xS∗1 − s1R∗ ,

xS∗2 − s2R∗ ,

xS∗3 − s3R∗

)T, (E-7)

where xS∗ = (xS∗1 , xS∗2 , xS∗3 ) are the apparent source coordinates in the global Cartesian system, R∗ =√l2 + r2,

l = |xS∗3 − s3|, and r =√

(xS∗1 − s1)2 + (xS∗2 − s2)2. Hereafter in this appendix, (s1, s2) are the local Cartesiancoordinates in the plane tangential to the actual reflector at point x′. Note that the product u∗

P (x′) · h−P (x′) from

E-5 is

u∗P (s1, s2, s3;x

′) · h−P (x′) =

(ikP − 1

R∗

)eikPR

∗

R∗ . (E-8)

The plane-wave decomposition of the displacement of the apparent incident P-wave has the form (Aki andRichards, 2002)


u∗P (s1, s2, s3;x

′) = grad

[ω

2π

∫ +∞

−∞

∫ +∞

−∞

ieiωlp(1)P3

p(1)P3

eiω(p1s1+p2s2)dp1dp2

]. (E-9)

Interchanging the order of differentiation and integration and setting s3 = 0, we obtain:

u∗P (p1, p2, 0;x

′) = −ω eiωlp

(1)P3

p(1)P3

(p1, p2,−p(1)

P3

)T. (E-10)

Thus, the unit polarization vectors of the incident P-wave (h(1)−P ) and reflected PP-wave (h

(1)+P ) are given by

h(1)−P = v

(1)P

(p1, p2,−p(1)

P3

)T= v

(1)P

(p cosψ, p sinψ,−p(1)

P3

)T,

h(1)+P = v

(1)P

(p cosψ, p sinψ, p

(1)P3

)T,

where ψ is the polar angle in the plane (p1, p2). It is straightforward to show that the polarization of the convertedPS-wave is

h(1)+S = v

(1)S

(p(1)S3 cosψ, p

(1)S3 sinψ,−p

)T.

Hence, for the PP-wave, h+P,norm/h

−P,norm = −1 and h+

P,tan/h−P,tan = 1. For the PS-wave, h+

S,norm/h−P,norm =

(v(1)S p)/(v

(1)P p

(1)P3) and h+

S,tan/h−P,tan = (v

(1)S p

(1)S3 )/(v

(1)P p).

Using equations E-7 and 7, we find the normal to the interface component of the displacement vector of thereflected PQ-mode:

u∗PQ,norm(s1, s2, 0;x

′) = ω2

2π

∫ +∞−∞

∫ +∞−∞ RPQ(p;x′)

h(1)+Q,norm(x′)

h(1)−P,norm(x′)

×eiωlp(1)P3eiω(p1s1+p2s2)dp1dp2 .

(E-11)

In the polar coordinates (p, ψ) and (r, ϕ), equation E-11 reduces to the Fourier-Bessel integral:

u∗PQ,norm(s1, s2, 0;x

′) = ω2

∫ +∞

0

RPQ(p;x′)h

(1)+Q,norm(x′)

h(1)−P,norm(x′)

eiωlp(1)P3J0(rωp)pdp , (E-12)

where J0 is the zero-order Bessel function,

J0(rωp) =1

2π

∫ 2π

0

eirωp cos(ψ−ϕ)dψ.

As follows from equation 7, the two tangential displacement components of the reflected PQ-wave are:

u∗PQ,j(s1, s2, 0;x

′) = −ω2

2π

∫+∞−∞

∫ +∞−∞ RPQ(p;x′)

h(1)+Q,j

(x′)

h(1)−P,j

(x′)

× eiωlp

(1)P3

p(1)P3

pjeiω(p1s1+p2s2)dp1dp2 .

(E-13)

In the polar coordinates (r, ϕ),

u∗PQ,tan(x′) = u∗

PQ,1(x′) cosϕ+ u∗

PQ,2(x′) sinϕ ,

and


u∗PQ,tan(s1, s2, 0;x

′) = −ω2

2π

∫ +∞−∞

∫ +∞−∞ RPQ(p;x′)

h(1)+Q,j

(x′)

h(1)−P,j

(x′)

× eiωlp

(1)P3

p(1)P3

p cos(ψ − ϕ)eiω(p1s1+p2s2)dp1dp2 .

(E-14)

Equation E-14 can also be reduced to the Fourier-Bessel integral:

u∗PQ,tan(s1, s2, 0;x

′) = −ω2

∫ +∞

0

RPQ(p;x′)h

(1)+Q,j (x′)

h(1)−P,j (x′)

ieiωlp(1)P3

p(1)P3

J1(rωp)p2dp , (E-15)

where J1 is the first-order Bessel function:

J1(rωp) = − i

2π

∫ 2π

0

cos(ψ − ϕ)eirωp cos(ψ−ϕ)dψ.

The normal and tangential to the reflector components of the polarization vectors can be written as h(1)+P,norm =

cos θ(x′), h(1)+P,tan = sin θ(x′), h

(1)+S,norm = − sin θS(x′) and h

(1)+S,tan = cos θS(x′), where θ(x′) is the P-wave incidence angle

and θS(x′) is the S-wave reflection angle determined from Snell’s law as θS(x′) = sin−1[v(1)S /v

(1)P sin θ(x′)

].

Finally, substitution of the Fourier-Bessel integrals E-12 and E-15 and the polarization components into thedefinition E-5 of the ERC yields

χPP (θ(x′), L(x′)) =u∗

PP,norm(x′) cos θ(x′)+u∗PP,tan(x′) sin θ(x′)


R∗

,

χPS(θ(x′), L(x′)) =−u∗

P S,norm(x′) sin θS(x′)+u∗PS,tan(x′) cos θS (x′)


R∗

.(E-16)

APPENDIX F: TIP-WAVE SUPERPOSITION METHOD FOR ISOTROPIC ELASTIC MEDIA

Here, we generalize the tip-wave superposition method (TWSM) for elastic media to obtain the PP- and PS-wavefieldsreflected from a curved interface. If the medium is homogeneous, it is possible to avoid evaluation of the tractionvector t(x′) and traction tensor T(x′,x) in the conventional wavefield representation 1 (Morse and Feshbach, 1953).We start by rewriting integral 1 in a form similar to equation 20 of Pao and Varatharajulu (1976):

u(x) = ρ(1)[v(1)P

]2 ∫ ∫S

(∇′ · G)(u · n′) − (G · n′)(∇′ · u)] dS(x′)

−ρ(1)[v(1)S

]2 ∫ ∫S

(u × n′) · (∇′ × G) − (n′ ×∇′ × u) · G] dS(x′).(F-1)

The reflected displacement field can be separated into the PP- and PS-modes (Ben-Menahem and Singh, 1998):

u(x) = uPP (x) + uPS(x) , (F-2)

which satisfy the equations

[v(1)P

]2∇ [∇ · uPP (x)] + ω2uPP (x) = 0 ,

−[v(1)S

]2∇× [∇× uPS(x)] + ω2uPS(x) = 0 .

(F-3)

Likewise, the Green’s displacement tensor can be split into the P- and S-wave components:

G(x′,x) = GP (x′,x) + GS(x′,x), (F-4)

where


GP (x′,x) = 1

ρ(1)ω2 ∇gP (x′,x)∇′ ,

GS(x′,x) = 1

ρ(1)ω2 ∇× [gS(x′,x) I] ×∇′

= 1

ρ(1)ω2

[ω2

[v(1)S

]2 gS(x′,x) I −∇gS(x′,x)∇′

],

(F-5)

and

gQ(x′,x) =eiωR/v

(1)Q

4πR, R = |x − x

′| , Q = P, S . (F-6)

Substituting equations F-2 and F-4 in F-1 and dropping the zero-value surface integrals, we obtain the reflectedPP-wavefield as

uPP (x) = ρ(1)[v(1)P

]2 ∫ ∫

S

[(∇′ · GP )(uPP · n′) − (GP · n′)(∇′ · uPP )

]dS′ . (F-7)

For the PS-wavefield,

uPS(x) = −ρ(1)[v(1)S

]2 ∫ ∫

S

(uPS × n

′) · (∇′ ×GS) − (n′ ×∇′ × uPS) · GS

]dS′ . (F-8)

Next, we rewrite the terms involving GP in equation F-7:

∇′ · GP = −∇ · GP = − 1

ρ(1)ω2 ∆gP ∇′

= 1

ρ(1)[v(1)P

]2 gP ∇′ = − 1

ρ(1)[v(1)P

]2 ∇gP ,(F-9)

and

GP · n′ =1

ρ(1)ω2∇gP ∇′ · n′ =

1

ρ(1)[v(1)P

]2 ∇[(n′ · ∇′)gP

]. (F-10)

Substituting equations F-9 and F-10 into equation F-7 yields

uPP (x) = ∇∫ ∫

S

[∂gP (x′,x)

∂n′ d1,PP (x′) − gP (x′,x) d2,PP (x′)

]dS(x′) , (F-11)

where

d1,PP (x′) = −

[v(1)P

]2

ω2

[∇′ · uPP (x′)

], d2,PP (x′) = uPP (x′) · n′ . (F-12)

The parameters d1,PP and d2,PP can be expressed through the incident wavefield and effective reflection coefficientχPP using approximation E-6:

d1,PP (x′) ≃ −[v(1)P

]2

ω2 χPP (x′)∇′ ·[[


P (x′)]h+P (x′)

]

≃ χPP (x′)gincP (x′,x) ,

d2,PP (x′) ≃ χPP (x′)[uincP (x′) · h−

P (x′)] [

h+P (x′) · n′] ,

(F-13)

where uincP (x′) = ∇′ginc

P (x′,x).


Because the surface integral in equation F-11 coincides with the acoustic surface integral 7 analyzed in Ayzenberget al. (2007), we can use their methodology (the tip-wave superposition method, or TWSM) to split the reflector intosmall rhombic elements. To extend TWSM to elastic media, we represent the PP-wavefield F-11 in a form similar toequations 11 and 12 from Ayzenberg et al. (2007):

uPP (x) ≃∑

j

iω

v(1)P

lP [j](x)∆BPP [j](x) , (F-14)

where the index j corresponds to a surface element, lP [j], (x) = ∇gP (x′[j],x)/|∇gP (x′

[j],x)|, and ∆BPP [j](x) is thescalar contribution of the j-th element:

∆BPP [j](x) =

∫ ∫

∆Π[j]

[∂gP (x′,x)

∂n′ d1,PP (x′) − gP (x′,x)d2,PP (x′)

]dS′ . (F-15)

To develop a similar expression for the PS-wavefield F-8, we rewrite the terms involving GS :

∇′ ×GS = −∇×GS

= −∇×[

1

ρ(1)ω2

[ω2

[v(1)S

]2 gSI−∇gS∇′

]]

= − 1

ρ(1)[v(1)S

]2∇× [gSI]

= − 1

ρ(1)[v(1)S

]2 [I ×∇gS] ,

−ρ(1)[v(1)S

]2(uPS × n′) · (∇′ ×GS) = (uPS × n′) · (I×∇gS)

= [(uPS × n′) · I] ×∇gS = (uPS × n′) ×∇gS

= −∇gS × (uPS × n′) = ∇× [−gS(uPS × n′)] ,

(F-16)

and

ρ(1)[v(1)S

]2(n′ ×∇′ × uPS) · GS = ρ(1)

[v(1)S

]2[GS × n′] · [∇′ × uPS]

= ∇×[∂gS(x′,x)

∂n′ d1,PS(x′)].

(F-17)

Substituting equations F-16 and F-17 into equation F-8, we find:

uPS(x) = ∇×∫ ∫

S

[∂gS(x′,x)

∂n′ d1,PS(x′) − gS(x′,x)d2,PS(x′)

]dS(x′) , (F-18)

where

d1,PS(x′) =

[v(1)S

]2

ω2

[∇′ × uPS(x′)

], d2,PS(x′) = uPS(x′) × n

′ . (F-19)

Approximation E-6 allows us to express the boundary data through the ERC χPS for PS-waves:

d1,PS(x′) ≃ −[v(1)S

]2

ω2 χPS(x′)∇′ ×[[


P (x′)]h+S (x′)

],

d2,PS(x′) ≃ χPS(x′)[uincP (x′) · h−

P (x′)] [

h+S (x′) × n′] .

(F-20)


The vector surface integral in equation F-18 is similar to the acoustic integral 7 in Ayzenberg et al. (2007), butthe boundary values d1,PS(x′) and d2,PS(x′) become vectors. Therefore we can adapt equations 11 and 12 fromAyzenberg et al. (2007) to obtain the following TWSM representation of the PS-wavefield F-18:

uPS(x) ≃∑

j

iω

v(1)S

lS[j](x) × ∆BPS[j](x), (F-21)

where lS[j](x) = ∇gS(x′[j],x)/|∇gS(x′

[j],x)|, and ∆BPS[j](x) is the vector contribution of the j-th surface element:

∆BPS[j](x) =

∫ ∫

∆Π[j]

[∂gS(x′,x)

∂n′ d1,PS(x′) − gS(x′,x)d2,PP (x′)

]dS′ . (F-22)

To evaluate integrals F-15 and F-22, we use the far-field approximation 16 of Ayzenberg et al. (2007).

Documents

Effective reflection coefficients for curved interfaces in TI ...€¦ · CWP-594 Effective reflection coefficients for curved interfaces in TI media Milana Ayzenberg 1, Ilya