elastic-wave inverse scattering with active ...scalar waves [39, 24, 1] to elastic waves. we make...

Proceedings of the Project Review, Geo-Mathematical Imaging Group (Purdue University, West Lafayette IN),Vol. 1 (2010) pp. 145-168.

ELASTIC-WAVE INVERSE SCATTERING WITH ACTIVE AND PASSIVE

SOURCE REFLECTION DATA

VALERIY BRYTIK∗, MAARTEN V. DE HOOP† , AND ROBERT D. VAN DER HILST ‡

Abstract. We develop a comprehensive theory and microlocal analysis of reverse-time imaging – also referred toas reverse-time migration or RTM – for the anisotropic elastic wave equation based on the single scattering approx-imation. We consider a configuration reminiscent of the seismic inverse scattering problem. In this configuration,we have an interior (point) body-force source which generates elastic waves, which scatter off discontinuities in theproperties of earth’s materials (anisotropic stiffness, density), and are observed at receivers on the earth’s surface.The receivers detect all the components of displacement. We introduce (i) an anisotropic elastic-wave RTM inversescattering transform, and for the case of mode conversions (ii) a microlocally equivalent formulation avoiding knowl-edge of the source via the introduction of so-called array receiver functions. These allow a seamless integration ofpassive source and active source approaches to inverse scattering.

Key words. elastic wave equation, inverse scattering, receiver functions, microlocal analysis

AMS subject classifications. 86A15, 35R30

1. Introduction. We develop a program and analysis for elastic wave-equation inverse scat-tering, based on the single scattering approximation, from two interrelated points of view, known inthe seismic imaging literature as “receiver functions” (passive source) and “reverse-time migration”(active source).

We consider an interior (point) body-force source which generates elastic waves, which scatter offdiscontinuities in the properties of earth’s materials (anisotropic stiffness, density), and are observedat receivers on the earth’s surface. The receivers detect all the components of displacement. Wedecompose the medium into a smooth background model and a singular contrast and assume thesingle scattering or Born approximation. The inverse scattering problem concerns the reconstructionof the constrast given a background model.

In this paper, we extend the original reverse-time imaging or migration (RTM) procedure forscalar waves [39, 24, 1] to elastic waves. We make use of the integral formulation of Schneider[31] and the inverse scattering integral equation of Bojarski [2]. Elastic-wave RTM has recentlybecome a subject of considerable interest. The current developments, however, have been limited toapproaches based on ad hoc scalar-wave approximations [34, 46, 20, 22, 9, 10, 12]. In this framework,the RTM imaging condition is tied to a decomposition into polarizations (for such a decompositionin quasi-homogeneous media, see [42, 43]).

We develop a comprehensive theory and microlocal analysis of reverse-time imaging for theanisotropic elastic wave equation. We construct the corresponding normal operator and an inverse-scattering transform which naturally removes the “smooth artifacts” discussed in [44, 27, 11, 41, 19].Our work is based on results presented in [4, 33] while assuming a common-source data acquisition.The three main results are: (i) the development of an anisotropic polarized-wave equation formu-lation, (ii) the introduction of a new (anisotropic) elastic-wave RTM inverse scattering transform,and (iii) the reformulation of (ii) using mode-converted wave constituents removing the knowledgeof the source by introducing the notion of array receiver functions which generalize the notion ofreceiver functions.

Under the assumption of absence of source caustics (the generation of caustics between thesource and scattering points), the RTM inverse scattering transform (cf. (i)) defines a Fourier in-tegral operator the propagation of singularities of which is decscribed by a canonical graph. Thusthis transform is amenable to expansions into wave packets or curvelets and partial reconstruction

∗Center for Computational and Applied Mathematics, Purdue University, West Lafayette, IN 47907, USA.†Center for Computational and Applied Mathematics, Purdue University, West Lafayette, IN 47907, USA

([email protected]).‡Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge

MA 02139, USA

145

146 V. BRYTIK, M. V. DE HOOP, AND R. D. VAN DER HILST

as developed in [5]. Moreover, if a set of sources is available, the RTM imaging condition lendsitself to a (linearized) reflection tomography formulation in terms of (linearized) transmission to-mography. The polarized-wave equation formulation (cf. (ii)) is well-suited for a frequency-domainimplementation of the type presented in [38]. The array receiver functions (cf. (iii)) provide aseamless integration of passive source and active source approaches to inverse scattering.

Over the past decades, converted seismic waves have been extensively used in global seismol-ogy to identify discontinuities in earth’s crust, lithosphere-asthenosphere boundary, and mantletransition zone. The method commonly used has been the one of receiver functions, which were in-troduced and developed by Vinnik [37] and Langston [21]. In this method, essentially, the converted(scattered) S -wave observation is deconvolved (in time) with the corresponding incident P -wave ob-servation at each available receiver, and assumes a planarly layered earth model. Various refinementshave been developed for arrays of receivers. We mention binning according to common-conversionpoints (Dueker and Sheehan [7]) and diffraction stacking (Revenaugh [29]); an analysis of (imagingwith) receiver functions starting from plane-wave single scattering has been given by Rydberg andWeber [30]. (Plane-wave) Kirchhoff migration for mode-converted waves was considered by Bostock[3] and Poppeliers and Pavlis [28], while its extension to wave-form inversion was developed byFrederiksen and Revenaugh [13]. Receiver functions, however, being bilinear in the data, do notfit a description directly in terms of Kirchhoff migration, being linear in the data. We resolve thisissue by making precise under which limiting assumptions receiver function imaging is equivalentwith (Kirchhoff-style) RTM via the synthesis of source plane waves.

The outline of the paper is as follows. In the next section, we summarize the construction of thepolarized-wave equation, the construction of the elastic-wave parameterix, the WKBJ approxima-tion, and introduce the single scattering approximation. In Section 3, we develop the (anisotropic)elastic-wave RTM inverse scattering transform. In Section 4, we introduce array receiver functionsand reformulate this inverse scattering transform by back extrapolating the observed incident field.In Section 5, we discuss how the receiver functions used in global seismology can be recovered fromarray receiver functions in flat, planarly layered earth models using the WKBJ approximation. InSection 6 we conclude with some final remarks.

2. Single scattering: Common source acquisition geometry. The propagation and scat-tering of seismic waves is governed by the elastic wave equation, which is written in the form

(2.1) Pilul = fi,

where

(2.2) ul =√ρ(x)(displacement)l, fi =

1√ρ(x)

(volume force density)i,

and

(2.3) Pil = δil∂2

∂t2−

∂

∂xj

cijkl(x)

ρ(x)

∂

∂xk+ l.o.t. ,

where l.o.t. stands for lower order terms. Here, x ∈ Rn and the subscripts i, j, k, l ∈ {1, . . . , n}. Thesystem of partial differential equations is assumed to be of principal type. It supports different wavetypes (modes). System (2.1) is real, time reversal invariant, and its solutions satisfy reciprocity.

2.1. Polarizations. First, we consider a smoothly varying medium. Decoupling of the modesis then accomplished by diagonalizing the system. We describe how the system (2.1) can be decou-pled by transforming it with appropriate matrix-valued pseudodifferential operators, Q(x,Dx)iM ,Dx = −i

∂∂x , see Taylor [35], Ivrii [45] and Dencker [6]. Since the time derivative in Pil is already in

diagonal form, it remains only to diagonalize its spatial part,

Ail = −∂

∂xj

cijkl(x)

ρ(x)

∂

∂xk+ l.o.t. .

ELASTIC-WAVE INVERSE SCATTERING 147

The goal becomes finding QiM and AM such that

(2.4) Q(x,Dx)−1Mi Ail(x,Dx)Q(x,Dx)lN = diag(AM (x,Dx) ; M = 1, . . . , n)MN .

The indices M,N denote the mode of propagation. Then

(2.5) uM = Q(x,Dx)−1Miui, fM = Q(x,Dx)

−1Mifi

satisfy the uncoupled equations

(2.6) PM (x,Dx, Dt)uM = fM ,

where Dt = −i∂∂t , and in which

PM (x,Dx, Dt) =∂2

∂t2+AM (x,Dx).

(Pseudodifferential equation (2.6) can be used in implementing elastic-wave inverse scattering in thereverse-time migration (RTM) approach.)

Because of the properties of stiffness related to (i) the conservation of angular momentum, (ii)the properties of the strain-energy function, and (iii) the positivity of strain energy, subject to the

adiabatic and isothermal conditions, the principal symbol Aprinil (x, ξ) of Ail(x,Dx) is a positive,symmetric matrix. Hence, it can be diagonalized by an orthogonal matrix. On the level of principalsymbols, composition of pseudodifferential operators reduces to multiplication. Therefore, we letQ

priniM (x, ξ) be this orthogonal matrix, and we let A

prinM (x, ξ) be the eigenvalues of A

prinil (x, ξ), so that

(2.7) QprinMi (x, ξ)−1A

prinil (x, ξ)Q

prinlN (x, ξ) = diag(A

prinM (x, ξ))MN .

The principal symbol QpriniM (x, ξ) is the matrix that has as its columns the orthonormalized polar-

ization vectors associated with the modes of propagation. If the AprinM (x, ξ) are all different, Ail canbe diagonalized with a unitary operator, that is, Q(x,Dx)

−1 = Q(x,Dx)∗; see Appendix A.

Remark 2.1. In the isotropic case, for n = 3, the symbol matrix Aprinil (x, ξ) attains the form,

ρAprinil (x, ξ) =

(λ+ µ) ξ21 + µ |ξ|2 (λ+ µ) ξ1ξ2 (λ+ µ) ξ1ξ3

(λ+ µ) ξ1ξ2 (λ+ µ) ξ22 + µ |ξ|

2 (λ+ µ) ξ2ξ3(λ+ µ) ξ1ξ3 (λ+ µ) ξ2ξ3 (λ+ µ) ξ

23 + µ |ξ|

2

,

where λ = λ(x) and µ = µ(x) denote the Lamé parameters. We find that

Q̃prin = Q̃prin(ξ) =

| | |

Q̃P Q̃SV Q̃SH| | |

,

which is independent of x and where

Q̃P =

ξ1ξ2ξ3

, Q̃SH = n× Q̃P =

−ξ2ξ10

, Q̃SV = Q̃P × Q̃SH =

−ξ1ξ3−ξ2ξ3ξ21 + ξ

22

,

with n = (0, 0, 1)t, diagonalizes Aprinil (x, ξ):

diag(ρAprinM (x, ξ) ; M = 1, . . . , n) =

(λ+ 2µ) |ξ|2 0 00 µ |ξ|2 00 0 µ |ξ|2

.


Upon normalizing the columns of Q̃prin, we obtain the unitary symbol matrix, Qprin, with

(Qprin)−1 = (Qprin)∗ =

ξ1|ξ|

ξ2|ξ|

ξ3|ξ|

−ξ1ξ3(ξ21+ξ

22)

1/2|ξ|−ξ2ξ3

(ξ21+ξ22)

1/2|ξ|

(ξ21+ξ22)

1/2

|ξ|−ξ2

(ξ21+ξ22)

1/2

ξ1(ξ21+ξ

22)

1/2 0

.

We note that Q̃SV and Q̃SH are zero if ξ ‖ n. This reflects the fact that it is not possible to constructa non-vanishing continuous tangent vector field on S2, which is the consequence of the non-vanishingof the Euler characteristic of S2 (which is 2).

With the projections onto P and S, it follows that

Qprini1 (Q

prin)∗1juj =

−∇(−∆−1(∇ ·

u1u2u3

))

i

and

[Qprini2 (Qprin)∗2j +Q

prini3 (Q

prin)∗3j ]uj =

∇× (−∆−1(∇×

u1u2u3

))

i

in accordance with the Helmholtz decomposition of u.

Having assumed that Wil is of principal type, the multiplicities of the eigenvalues AprinM (x, ξ)

are constant, whence the principal symbol QpriniM (x, ξ) depends smoothly on (x, ξ) and microlocallyequation (2.7) carries over to an operator equation. Taylor [35] has shown that if this condition issatisfied, then decoupling can be accomplished to all orders.

The second-order equations (2.6) inherit the symmetries of the original system, such as time-reversal invariance and reciprocity. Time-reversal invariance follows because the operatorsQiM (x,Dx), AM (x,Dx) can be chosen in such a way that QiM (x, ξ) = −QiM (x,−ξ), AM (x, ξ) =AM (x, ξ). Then QiM , AM are real-valued. Reciprocity for the causal Green’s function Gij(x, x0, t−t0) means thatGij(x, x0, t−t0) = Gji(x0, x, t−t0). Such a relationship also holds (modulo smoothingoperators) for the Green’s function GM (x, x0, t− t0) associated with (2.6).

2.2. The Green’s function. To evaluate the Green’s function, we use the first-order systemfor uM that is equivalent to (2.6),

(2.8)∂

∂t

(uM∂uM∂t

)=

(0 1

−AM (x,Dx) 0

)(uM∂uM∂t

)+

(0fM

).

This system can be decoupled also, namely by the matrix-valued pseudodifferential operator(

1 1−iBM (x,Dx) iBM (x,Dx)

),

where BM (x,Dx) =√AM (x,Dx), which is a pseudodifferential operator of order 1 that exists

because AM (x,Dx) is positive definite. The principal symbol of BM (x,Dx) is given by BprinM (x, ξ) =√

AprinM (x, ξ). Then

(2.9) uM,± =12uM ±

12 iBM (x,Dx)

−1 ∂uM

∂t, fM,± = ±

12 iBM (x,Dx)

−1fM

satisfy the two first-order (“half wave”) equations

(2.10) PM,±(x,Dx, Dt)uM,± = fM,±,


where

(2.11) PM,±(x,Dx, Dt) =∂

∂t± iBM (x,Dx), PM,+PM,− = PM .

We construct operators GM,± with Lagrangian distribution kernels GM,±(x, x0, t) that solve theinitial value problem for (2.10). The operators GM,± are Fourier integral operators. Their con-struction is well known, see for example Duistermaat [8], Chapter 5. Singularities are propagatedalong the bicharacteristics, that are determined by Hamilton’s equations generated by the principalsymbol ω ±BprinM (x, ξ) of (2.10),

(2.12)

∂x

∂λ= ±

∂

∂ξB

prinM (x, ξ) ,

∂t

∂λ= 1,

∂ξ

∂λ= ∓

∂

∂xB

prinM (x, ξ) ,

∂ω

∂λ= 0.

Naturally, the solution may be parameterized by t. We denote the solution of (2.12) with the + signand initial values (x0, ξ0) at t = 0 by (xM (x0, ξ0, t), ξM (x0, ξ0, t)). The solution with the − sign isfound upon reversing the time direction and is given by (xM (x0, ξ0,−t), ξM (x0, ξ0,−t)).

A complete view of the propagation of singularities is provided by the canonical relation of theoperator GM,±, given by

(2.13) CM,± = {(xM (x0, ξ0,±t), t, ξM (x0, ξ0,±t),ω = ∓BM,±(x0, ξ0);x0, ξ0)}.

A convenient choice of phase function for the oscillatory integral representation for the kernel ofGM,± is described in Maslov and Fedoriuk [23]. They state that one can always use a subset of thecotangent vector components as phase variables. Let us assume coordinates for CM,+ of the form

(2.14) (xI , x0, ξJ ,ω),

where I ∪ J is a partition of {1, . . . , n}. It follows from Theorem 4.21 in Maslov and Fedoriuk [23]that there is a function SM (xI , x0, ξJ ,ω), such that locally CM,+ is given by

(2.15)

xJ = −∂SM

∂ξJ, t = −

∂SM

∂ω,

ξI =∂SM

∂xI, ξ0 = −

∂SM

∂x0.

Here we take into account the fact that CM,+ is a canonical relation, which introduces a minus signfor ξ0. A nondegenerate phase function for CM,+ is then found to be

(2.16) φM,+(x, x0, t, ξJ ,ω) = SM (xI , x0, ξJ ,ω) + 〈ξJ , xJ〉+ ωt.

The canonical relation CM,− can be obtained from CM,+, viz.

CM,− = {(x, t,−ξ,−ω;x0,−ξ0) | (x, t, ξ,ω;x0, ξ0) ∈ CM,+}.

Thus a phase function for CM,− is φM,−(x, x0, t, ξJ ,ω) = −φM,+(x, x0, t,−ξJ ,−ω).A theorem of Hörmander (see also Theorem 5.1.2 of Duistermaat [8]) implies that the operator

GM,± is microlocally a Fourier integral operator. Hence, we have an expression for its kernel,GM,±(x, x0, t), in the form of an oscillatory integral,

(2.17) GM,±(x, x0, t)

= (2π)−|J|+1

2 −2n+1

4

∫aM,±(xI , x0, ξJ ,ω) exp[iφM,±(x, x0, t, ξJ ,ω)] dξJ dω.


The amplitude aM,±(xI , x0, ξJ ,ω) satisfies a transport equation along the bicharacteristics(xM (x0, ξ0,±t), ξM (x0, ξ0,±t)).

We may define the canonical relation for GM as CM = CM,+ ∪ CM,− and a phase function

φM =

{φM,− if ω > 0,φM,+ if ω < 0.

It follows from (2.9) that the Green’s function for the second-order decoupled equation is given by

(2.18)

∫GM (x, x0, t− t0)fM (x0, t0) dx0dt0

= 12 i

∫[GM,+(x, x0, t− t0)−GM,−(x, x0, t− t0)]BM (x0, Dx0)

−1fM (x0, t0) dx0dt0,

where BM (x0, Dx0)−1 is of order −1. We have

(2.19) GM (x, x0, t) = (2π)−

|J|+12 −

2n+14

∫aM (xI , x0, ξJ ,ω) exp[iφM (x, x0, t, ξJ ,ω)] dξJ dω,

in which, to highest order,

(2.20) |aM (xI , x0, ξJ ,ω)| = (2π)1/4

∣∣∣∣det∂(x0, ξ0, t)

∂(xI , x0, ξJ ,ω)

∣∣∣∣1/2

1

2|ω|.

The determinant can be obtained by solving the perturbed Hamilton system following from (2.12).

The amplitude is an element of MCM ⊗ Ω1/2(CM ), the tensor product of the Keller-Maslov

bundle MCM and the half-densities on the canonical relation CM . The Keller-Maslov bundle givesa factor ik, where k is an index, which we will absorb in the amplitude.

In case J = ∅, the generating function SM reduces to frequency, ω, times the negative of traveltime, which we denote by TM = TM (x, x0).

Flat, smoothly layered media. Here, we make use of results in [40, 16, 17, 14, 15, 32]. Weintroduce coordinates x = (x′, z) if xn = z is the (depth) coordinate normal to the surface,and write cjk;il = (cjk)il = cijkl. We consider the displacement, ρ

−1/2ui, and the traction,∑nk,l=1 cnk;il

∂(ρ−1/2ul)∂xk

, and form

(2.21) W =

(ρ−1/2ui∑n

k,l=1 cnk;il∂(ρ−1/2ul)

∂xk

), F =

(0fi

), i = 1, . . . , n.

The elastic wave equation (cf. (2.1)-(2.3)) can then be rewritten as the system of equations,

(2.22)∂Wa

∂z= i

2n∑

b=1

Cab(x′, z,Dx′ , Dt)Wb + Fa,

with

(2.23) Cab(x′, z,Dx′ , Dt)

= −i

(−∑n−1

q=1

∑nj=1(cnn)

−1ij cnq;jl

∂∂xq

(cnn)−1il

−∑n−1

p,q=1∂

∂xpbpq;il

∂∂xq

+ ρδil∂2

∂t2 −∑n−1

p=1∂

∂xpcpn;ij(cnn)

−1jl

)

ab

, i, l = 1, . . . , n,


where bpq;il = cpq;il −∑n

j,k=1 cpn;ij(cnn)−1jk cnq;kl. Diagonalizing the system, microlocally, involves

(2.24) Cab(x′, z,Dx′ , Dt) =

2n∑

µ,ν=1

L(x′, z,Dx′ , Dt)aµ

diag(Cµ(x′, z,Dx′ , Dt);µ = 1, . . . , 2n)µνL(x

′, z,Dx′ , Dt)−1νb .

The principal parts of the symbols Cµ(x′, z, ξ′,ω) are the solutions for ζ of

AprinM (x

′, z, ξ′, ζ) = ω2.

In smoothly layered media one can Fourier transform (2.22) with respect to x′ and t and obtain a

system of ordinary differential equations for W̃ (z) = W̃ (ξ′, z,ω) =∫W (x′, z, t) exp[−i(

∑n−1j=1 ξjxj+

ωt)] dx′dt,

(2.25)∂W̃a

∂z= i

2n∑

b=1

Cab(z, ξ′,ω)W̃b + F̃a.

We choose the Cµ such that the homogeneity property, Cµ(z, ξ′,ω) = ωCµ(z,ω

−1ξ′, 1), extends toω < 0. We have

(2.26) Laµ(z, ξ′,ω) =

(QiM(µ)(z, (ξ

′, Cµ(z, ξ′,ω)))∑n

k,l=1 cnk;il(−i)(ξ′, Cµ(z, ξ

′,ω))kQlM(µ)(z, (ξ′, Cµ(z, ξ

′,ω)))

)

aµ

,

with inverse

(2.27) L−1(z, ξ′,ω) = N(z, ξ′,ω)Lt(z, ξ′,ω) J, where J =

(0 InIn 0

).

Here, N(z, ξ′,ω) is a diagonal normalization matrix, diag(Nµ(z, ξ′,ω))µν . It follows that

(2.28) Nµ(z, ξ′,ω)−1 =

n∑

i=1

QiM(µ)(z, (ξ′, Cµ(z, ξ

′,ω)))

n∑

k,l=1

[cnk;il + cnk;li] (−i)(ξ′, Cµ(z, ξ

′,ω))kQlM(µ)(z, (ξ′, Cµ(z, ξ

′,ω))).

The index mapping µ → M(µ) assigns the appropriate mode to the depth component of the wavevector.

We cast (2.25) into an equivalent initial value problem. Let W̃ab(z, z0) be the solution to

∂W̃a

∂z= i

2n∑

b=1

Cab(z, ξ′,ω)W̃b, W̃ (z0) = I2n.

Then W̃a(z) =∫ zz0

∑2nb=1 W̃ab(z, z0)F̃b(z0) dz0 solves (2.25). We introduce

(2.29) Ẇ =

(ρ−1/2ui∑n

k,l=1 cnk;ilD−1t

∂(ρ−1/2ul)∂xk

), Ḟ =

(0

D−1t fi

);

with ξ′ = ωp′, we identify ˜̇W (p′, z,ω) = W̃ (ωp′, z,ω) whereas

∂˜̇W a

∂z= iω

2n∑

b=1

Cab(z, p′, 1)˜̇W b + ˜̇F b.


In the WKBJ approximation, in the absence of turning rays (the characteristics are nowhere hori-zontal), we have

(2.30) ˜̇W ab(z, z0) ≈2n∑

µ=1

Laµ(z, p′, 1)Yµ(z, p

′, 1)

exp

[iω

∫ z

z0

Cµ(z̄, p′, 1) dz̄

]Yµ(z0, p

′, 1)−1L−1µb (z0, p′, 1)

=2n∑

µ=1

Laµ(z, p′, 1)Yµ(z, p

′, 1) exp

[iω

∫ z

z0

Cµ(z̄, p′, 1) dz̄

]Yµ(z0, p

′, 1)(Lt(z0, p′, 1)J)µb.

Here, Yµ(z, p′, 1) = [Nµ(z, p

′, 1)]1/2. We identify the “vertical” travel time

τµ(z, z0, p′) = −

∫ z

z0

Cµ(z̄, p′, 1) dz̄.

To obtain the tensor Gij , we substitute a δ source for fi, yielding JF̃ =

(In0

)δ(.− z0):

(2.31) Gij(x′, z, x′0, z0, t− t0) ≈

2n∑

µ=1

′

1

(2π)n

∫∫QiM(µ)(z, (p

′, Cµ(z, p′, 1)))Yµ(z, p

′, 1)

exp

[iω

(−τµ(z, z0, p

′) +

n−1∑

l=1

p′l(x′ − x′0)l + t− t0

)]

Yµ(z0, p′, 1)QtM(µ)j(z0, (p

′, Cµ(z, p′, 1))) dp′|ω|n−1dω;

which values of µ contribute depends on whether z > z0 (“downgoing”) or z < z0 (“upgoing”). The(negative) values of the components of p′ associated with the ray connecting (z0, x

′

0) with (z, x′) is

the solution of the equation

∂p′τµ(z, z0, p′) = x′ − x′0.

2.3. The single scattering approximation. In the contrast formulation the total valueof the medium parameters ρ, cijkl is written as the sum of a smooth background constituentρ(x), cijkl(x) and a singular perturbation δρ(x), δcijkl(x), viz. ρ + δρ, cijkl + δcijkl; we assumethat δρ, δcijkl ∈ E

′(X) with X a compact subset of Rn. This decomposition induces a perturbationof Pil (cf. (2.3)),

δPil = δilδρ(x)

ρ(x)

∂2

∂t2−

∂

∂xj

δcijkl(x)

ρ(x)

∂

∂xk.

We denote the causal solution operator associated with (2.1) by Gil and its distribution kernel byGil(x, x0, t− t0). The first-order perturbation, δGil, of Gil admits the representation

(2.32) δGjk(x̂, x̃, t) = −

∫ t

0

∫

X

Gji(x̂, x0, t− t0) δPil(x0, Dx0 , Dt0)Glk(x0, x̃, t0) dx0 dt0,

which is the Born approximation. Here, x̃ denotes a source location, x̂ a receiver location, andx0 a scattering point location. We restrict our time window (of observation) to (0, T ) for some0 < T < ∞. Because the background model is smooth the operator δGil contains only the singlescattered field.


We introduce the MN contribution, δGMN , to δGjk as follows,

(2.33)

∫δGjk(x̂, x̃, t− t̃)fk(x̃, t̃) dx̃dt̃

= Q(x̂, Dx̂)jM

∫δGMN (x̂, x̃, t− t̃) (Q(x̃, Dx̃)

−1Nkfk)(x̃, t̃) dx̃dt̃.

We apply reciprocity in (x̂, x0) to the integrand of the right-hand side and obtain

(2.34) δGMN (x̂, x̃, t) = −

∫ t

0

∫

X

(Q(x0, Dx0)−1)∗iMGM (x0, x̂, t− t0)

δPil(x0, Dx0 , Dt0)Q(x0, Dx0)lNGN (x0, x̃, t0) dx0 dt0

= −

∫ t

0

∫

X

(Q(x0, Dx0)−1)∗iMGM (x0, x̂, t− t0)

∂

∂(t0, x0,j)

(δil

δρ(x0)

ρ(x0),−

δcijkl(x0)

ρ(x0)

)

∂

∂(t0, x0,k)Q(x0, Dx0)lNGN (x0, x̃, t0) dx0 dt0

=

∫ t

0

∫

X

∂

∂(t0, x0,j)(Q(x0, Dx0)

−1)∗iMGM (x0, x̂, t− t0)

(δil

δρ(x0)

ρ(x0),−

δcijkl(x0)

ρ(x0)

)

∂

∂(t0, x0,k)Q(x0, Dx0)lNGN (x0, x̃, t0) dx0 dt0

=

∫

X

{ ∫ t

0

∂

∂(t0, x0,j)Q(x0, Dx0)iMGM (x0, x̂, t− t0)

∂

∂(t0, x0,k)Q(x0, Dx0)lNGN (x0, x̃, t0) dt0

}

(δil

δρ(x0)

ρ(x0),−

δcijkl(x0)

ρ(x0)

)dx0

upon integration by parts; see Appendix A. Reciprocity implies thatGM (x0, x̂, t−t0) = GM (x̂, x0, t−t0) while GN (x0, x̃, t0) = GN (x̃, x0, t0).

Microlocally, we can substitute (2.19) for GM , with appropriate substitutions for its arguments:

(x̂, ξ̂Ĵ) for (x, ξJ) while Î ∪ Ĵ = {1, . . . , n}; for GN we have the substitutions (x̃, ξ̃J̃) for (x, ξJ) while

Ĩ ∪ J̃ = {1, . . . , n}. It follows that the composition

∂

∂(t0, x0,j)(Q(x0, Dx0)

−1)∗iMGM

is a Fourier integral operator with the same phase as GM , and amplitude that to highest orderequals the product

aM (x̂Î , x0, ξ̂Ĵ ,ω)Q(x0, ξ̂0)iM i (ω, ξ̂0,j),

subject to the substitution ξ̂0 = ξ0(x̂Î , x0, ξ̂Ĵ ,ω) (cf. (2.15)).We write (2.34) in the form of an oscillatory integral. To this end, we introduce the radiation

patterns (wMN ;0, wMN ;ijkl) as

wMN ;0(x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω) = −QiM (x0, ξ̂0)QiN (x0, ξ̃0)ω2,(2.35)

wMN ;ijkl(x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω) = QiM (x0, ξ̂0)QlN (x0, ξ̃0) ξ̂0,j ξ̃0,k,(2.36)

subject to the substitutions ξ̂0 = ξ0(x̂Î , x0, ξ̂Ĵ ,ω), ξ̃0 = ξ0(x̃Ĩ , x0, ξ̃J̃ ,ω) (cf. (2.15)). We introducethe amplitude,

(2.37) bMN (x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω) = (2π)−

n−14 aM (x̂Î , x0, ξ̂Ĵ ,ω) aN (x̃Ĩ , x0, ξ̃J̃ ,ω)


and the phase function

(2.38) ΦMN (x̂, x̃, t, x0, ξ̂Ĵ , ξ̃J̃ ,ω) = φM (x̂, x0, t, ξ̂Ĵ ,ω) + φN (x̃, x0, t, ξ̃J̃ ,ω)− ωt.

Then

(2.39) δGMN (x̂, x̃, t) = (2π)−

|Ĵ|+|J̃|+12 −

3n+14

∫ ∫

X

bMN (x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω)

(wMN ;0(x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω)

δρ(x0)

ρ(x0)+ wMN ;ijkl(x̂Î , x̃Ĩ , x0, ξ̂Ĵ , ξ̃J̃ ,ω)

δcijkl(x0)

ρ(x0)

)

exp[iΦMN (x̂, x̃, t, x0, ξ̂Ĵ , ξ̃J̃ ,ω)] dx0 dξ̂Ĵdξ̃J̃dω

modulo lower-order terms in amplitude.Data are measurements of the scattered wave field which we relate here to the Green’s function

perturbation in (2.34): They are assumed to be representable byδGMN (x̂, x̃, t) for (x̂, x̃, t) in some acquisition manifold, which contains the receiver points and time.Let

ŷ /→ x̂(ŷ), ỹ /→ x̃(ỹ)

be coordinate transformations, such that y = (ŷ, ỹ, t), y′ = (ŷ′, t), y′′ = (ŷ′′, ỹ) with ŷ = (ŷ′, ŷ′′),and the acquisition manifold is given by Y = Σs × {0}× (0, T ), that is, ỹ = 0 while Σs ⊂ {ŷ

′′ = 0}.This is the case in which the source is fixed; the dimension of ŷ′′ is 1, whence the codimension ofthis acquisition manifold is c = n+ 1. We sometimes write x̃(0) = s and ŷ′ = r.

Remark 2.2. The operator Q(x̂, Dx̂)jM in (2.33) can be removed from the data. In the furtheranalysis, we will consider the case of a (locally) flat acquisition hypersurface, and write x̂ = (x̂′, ẑ)or ŷ′ = r = x̂′ and ŷ′′ = ẑ. Assuming upgoing waves at the receivers, we can replace Q(x̂, Dx̂)jMby an operator, Q̄(x̂′, ẑ, Dx̂′ , Dt)jM(µ), with principal symbol QjM(µ)(x̂

′, ẑ, (ξ̂′, Cµ(x̂′, ẑ, ξ̂′,ω))). The

action of this operator can be restricted naturally to ẑ = 0. It also has a parametrix, whence,microlocally, the polarizations in the data can be separated.

In this framework, with fk = ekδx̃δ0, the scattered-field data, dMN (r, t; s), are modeled by theoperator,

(2.40) FMN :

(δil

δρ(x)

ρ(x),δcijkl(x)

ρ(x)

)→

∫δGMN (x̂, x̃

′, t)Q−1Nk′(x̃′, x̃)ek′ dx̃

′

∣∣∣∣x̂=x̂(r,0),x̃=s

(no sum over N);

here Q−1 denotes the kernel of Q−1. Incident field data, dN (r, t; s), are modeled by∫

GN (x̂, x̃′, t)Q−1Nk′(x̃

′, x̃)ek′ dx̃′

∣∣∣∣x̂=x̂(r,0),x̃=s

(no sum over N).

To simplify the analysis, we will consider a polarized source, fk = QkN (., s)δ0, leading to the

introduction of F̄MN : (δilδρ(x)ρ(x) ,

δcijkl(x)ρ(x) ) → δGMN (x̂, x̃, t)|x̂=x̂(r,0),x̃=s, modelling data d̄MN (r, t; s),

and the incident field data d̄N (r, t; s).

We investigate the propagation of singularities by the mapping in (2.40). Let ω = ∓BM (x0, ξ̂0),and

x̂ = xM (x0, ξ̂0,±t̂), x̃ = xN (x0, ξ̃0,±t̃), t = t̂+ t̃,

ξ̂ = ξM (x0, ξ̂0,±t̂), ξ̃ = ξN (x0, ξ̃0,±t̃)


denote the bicharacteristics in modes N and M , originating at x0. We then obtain(y(x0, ξ̂0, ξ̃0, t̂, t̃), η(x0, ξ̂0, ξ̃0, t̂, t̃)) by transforming (x̂, x̃, t̂+ t̃, ξ̂, ξ̃,ω) to (y, η) coordinates. We invokethe following assumptions

Assumption 1. There are no elements (y′, 0, η′, η′′) with (y′, η′) ∈ T ∗Y \0 such that, for some

ξ̂0

y′ = y′(x0, ξ̂0,−ξ̂0, t̂, t̃), η′ = η′(x0, ξ̂0,−ξ̂0, t̂, t̃), and ŷ

′′(x0, ξ̂0, t̂) = 0, ỹ(x0,−ξ̂0, t̃) = 0.

This assumption is related to the condition under which the pull back of a distribution is againa distribution (Gel’fand and Shilov, 1958). If N 1= M this assumption is generically satisfied; ifN = M , this excludes scattering over π. In the further analysis we will focus on the conversionwhere N corresponds with qP and M corresponds with qSV, in particular with a view to developingarray receiver functions.

Assumption 2. The matrix

(2.41)∂ŷ′′

∂(x0, ξ̂0, t̂)has rank 1.

This assumption essentially states the absence of grazing receiver rays. With Assumptions 1 and2 it follows that FMN in (2.40) defines a Fourier integral operator. The propagation of singularitiesby (2.40) is governed by the canonical relation

(2.42) ΛFMN = {(ŷ′(x0, ξ̂0, t̂), t̂+ t̃, η̂

′(x0, ξ̂0, t̂),ω;x0, ξ̂0 + ξ̃0) |

BM (x0, ξ̂0) = BN (x0, ξ̃0) = ∓ω, ỹ(x0, ξ̃0, t̃) = 0, ŷ′′(x0, ξ̂0, t̂) = 0}

⊂ T ∗Y \0× T ∗X\0.

3. Imaging converted waves: Common source. An image is obtained by appling theadjoint, F ∗MN , to the data, dMN :

(3.1) IMNijkl (x0; s) = (F∗

MNdMN )ijkl(x0)

=

∫ ∫ ∫

Σs

∂

∂x0,jQ(x0, Dx0)iMGM (x0, x̂(r, 0), t− t0) dMN (r, t; s)drdt

∂

∂x0,kQ(x0, Dx0)lN

∫GN (x0, x̃

′, t0)Q−1Nk′(x̃

′, s)ek′ dx̃′ dt0.

and similarly for the density contrast upon replacing ∂∂x0,j

and ∂∂x0,k

by ∂∂t0

.

Assumption 3. (Bolker condition) No caustics form between the source and scattering pointsin mode N .

Let SM (ŷ′

I , x0, η̂′

J ,ω) denote the generating function associated with the canonical relation ofthe parametrix in mode M (cf. (2.19)) subjected to the restriction to the surface ŷ′′ = 0. We write

x̂0 = x̂0(x0, ξ̂0) = xM (x0, ξ̂0, t̂0) where t0 is determined by ŷ′′(x0, ξ̂0, t̂0) = 0.


Theorem 3.1. The common-source normal operator, F ∗MNFMN , acting on density and stiffnesstensors, is pseudodifferential of order n− 1. Its principal symbol is given by

(2π)−n−12

∣∣∣∣det∂(x̂)

∂(ŷ)

∣∣∣∣−1

ŷ′′=0

1

4ω2aN (s, x0,ω)

2

∣∣∣∣dŷ′′

dt̂

∣∣∣∣−1

[1− 〈

∂TN (x0, s)

∂x0,∂B

prinM

∂ξ

∣∣∣∣x0,ξ=ξ0+ω

∂TN (x0,s)

∂x0

〉

]−1

,

ω = ω(x0, ξ0, s), BprinM

(x0,ω(x0, ξ0, s)

−1ξ0 +∂TN (x0, s)

∂x0

)= 1,

up to radiation pattern factors.Proof. We evaluate the symbol of the normal operator associated with F̄MN . We use (2.39); in

the absence of “source-ray” caustics, we have J̃ = ∅. We find that (cf. (2.20))

(3.2) |bMN (ŷ′

I , s, x0, η̂′

J ,ω)| = (2π)−

n+12

∣∣∣∣det∂(x̂)

∂(ŷ)

∣∣∣∣−1/2

ŷ′′=0∣∣∣∣∣det

∂(ŷ′I , ŷ′′, η̂′J ,ω)

∂(x̂Î , ξ̂Ĵ ,ω)

∣∣∣∣∣

−1/2

ŷ′′=0

(2π)1/4

∣∣∣∣∣det∂(x0, ξ̂0, t̂)

∂(x̂Î , x0, ξ̂Ĵ ,ω)

∣∣∣∣∣

1/21

2|ω|aN (s, x0,ω)

(aN (s, x0,ω) contains a factor (2π)1/4), while ΦMN = φM + φN − ωt with

(3.3) φN (s, x0, t,ω) = −ω(TN (x0, s)− t),

φM (ŷ′, x0, t, η̂

′

J ,ω) = SM (ŷ′

I , x0, η̂′

J ,ω) + 〈η̂′

J , ŷ′

J〉+ ωt

(I ∪ J = {1, . . . , n− 1}). We can express (ξ̂0, ξ̃0) as functions of (ŷ′

I , s, x0, η̂′

J ,ω):

ξ̃0 = ω∂TN (x0, s)

∂x0, ξ̂0 = −

∂SM (ŷ′

I , x0, η̂′

J ,ω)

∂x0.

Subject to such a substitution, we transform the radiation patterns using (2.35)-(2.36),

wMN ;0(ŷ′

I , s, x0, η̂′

J ,ω) = −QiM (x0, ξ̂0)QiN (x0, ξ̃0)ω2,

wMN ;ijkl(ŷ′

I , s, x0, η̂′

J ,ω) = QiM (x0, ξ̂0)QlN (x0, ξ̃0) ξ̂0,j ξ̃0,k.

We expand the phase of the oscillatory integral representation of F̄ ∗MN F̄MN , and carry out theintegration over ŷJ . We obtain an oscillatory integral with phase function,

〈ω

[1

ω

∂SM (ŷ′

I , x0, η̂′

J ,ω)

∂x0−

∂TN (x0, s)

∂x0

], (x̄0 − x0)〉+ l.o.t.,

and integration variables (ŷ′I , η̂′

J ,ω); here, we exploit the fact that this operator is pseudodifferential.The oscillatory integral representation of the normal operator (up to the principal symbol)

contains the factor and measure,

(3.4)

∣∣∣∣∣det∂(x0, ξ̂0, t̂)

∂(x̂Î , x0, ξ̂Ĵ ,ω)

∣∣∣∣∣

∣∣∣∣∣det∂(ŷ′I , ŷ

′′, η̂′J ,ω)

∂(x̂Î , ξ̂Ĵ ,ω)

∣∣∣∣∣

−1

ŷ′′=0

dŷ′Idη̂′

Jdω

=

∣∣∣∣dŷ′′

dt̂

∣∣∣∣−1 ∣∣∣∣

∂BprinM

∂ξ

∣∣∣∣−1

x0,ξ=ξ̂0

dΣM,x0(ξ̂0)dω,


where the integration in ξ̂0 is over (part of) the surface ΣM,x0 : BprinM (x0, ξ̂0) = ω. We have

dΣM,x0(ξ̂0)dω = |ω|n−1dΣM,x0(p̂0)dω if ξ̂0 = ωp̂0; p̂0 =

1ω

∂SM (ŷ′

I ,x0,η̂′

J ,ω)∂x0

. Furthermore,

∣∣∣∣dŷ′′

dt̂

∣∣∣∣ =∣∣∣∣〈

∂ŷ′′

∂x̂,∂B

prinM

∂ξ

∣∣∣∣x̂(ŷ),(

∂(ŷ)∂(x̂)

)tη̂

〉 ∣∣∣∣ .

We then consider a change of variables of integration, (ω, p̂0) → ξ0, with

ξ0 = ω

[p̂0 −

∂TN (x0, s)

∂x0

]

for given x0; see Fig. 1. Let ν(x0, .) denote the normal to ΣM,x0 , that is,

ν(x0, p̂0) =

∣∣∣∣∂B

prinM

∂ξ

∣∣∣∣−1

∂BprinM

∂ξ

∣∣∣∣x0,ξ=p̂0

, p̂0 ∈ ΣM,x0 .

Then

dξ0 = 〈p̂0 −∂TN (x0, s)

∂x0, ν〉 |ω|n−1dωdΣM,x0(p̂0).

We note that

〈p̂0, ν〉 =

∣∣∣∣∂B

prinM

∂ξ

∣∣∣∣−1

x0,ξ=p̂0

, p̂0 ∈ ΣM,x0 .

Hence,

(3.5) dξ0 =

∣∣∣∣∂B

prinM

∂ξ

∣∣∣∣−1

x0,ξ=p̂0

[1− 〈

∂TN (x0, s)

∂x0,∂B

prinM

∂ξ

∣∣∣∣x0,ξ=p̂0

〉

]|ω|n−1dωdΣM,x0(p̂0).

Combining (3.4) with (3.5) yields the final Jacobian. The principal symbol of the normal operatorfollows from multiplying this Jacobian with the square of the amplitude in (3.2) and the dyadicproduct of (3.4) with itself.

We can compensate for the factor

∣∣∣∣dŷ′′

dt̂

∣∣∣∣−1

[1− 〈

∂TN (x0, s)

∂x0,∂B

prinM

∂ξ

∣∣∣∣x0,ξ=p̂0

〉

]−1

,

upon replacing F̄ ∗MN by an imaging operator, H̄MN , say, making use of ωp̂0 =∂SM∂x0

, the homogeneity

of BprinM , and Egorov’s theorem. We define pseudodifferential operators Ξ and Θ with principalsymbols

Ξ0(x0, ξ0,ω) = ω, Ξj(x0, ξ0,ω) = ξ0j

and

Θ0(x0, ξ0,ω) = ω, Θj(x0, ξ0,ω) =∂B

prinM

∂ξj(x0, ξ0)ω.

Let S(ω) = 1ω2

and

Ψ(ŷ′, 0, η̂′,ω) =

∣∣∣∣〈

∂ŷ′′

∂x̂,∂B

prinM

∂ξ

∣∣∣∣x̂(ŷ′,0),(

∂(ŷ)∂(x̂)

)t(η̂′,Cµ(ŷ′,0,η̂′,ω))

〉 ∣∣∣∣ ,


then

(H̄MNdMN )ijkl(x0) =

∫S(Dt0)

n∑

p=0

[Θp(x0, Dx0 , Dt)

∂

∂x0,jQ(x0, Dx0)iM

∫ ∫

Σs

(Ψ(r, 0, Dr, Dt)GM )(x0, x̂(r, 0), t− t0) dMN (r, t; s) drdt

]

Ξp(x0, Dx0 , Dt)∂

∂x0,kQ(x0, Dx0)lNGN (x0, s, t0) dt0

and similarly for the density contrast upon replacing ∂∂x0,j

and ∂∂x0,k

by ∂∂t0

.

Σ x̃

x̃

Fig. 1. Integration over receivers, the construction of ξ0 (isotropic case). The ray with single arrow correspondswith the incident field, which is also observed at the boundary; the ray with double arrows corresponds with thescattered field. Σs indicates the array.

Remark 3.2. Adjoint state formulation: (i) In the decoupled formulation, using (2.6), weobtain (F ∗MNdMN )ijkl upon successively solving

PN (x,Dx, Dt)wN ;s = δ(t)Q−1Nk′(x, s)ek′ , wN ;s|t=0 = 0, ∂twN ;s|t=0 = 0,

and

PM (x,Dx, Dt)u∗

M =

∫dr dMN (r, s, T − t)δ(x− x̂(r, 0)), u

∗

M |t=0 = 0, ∂tu∗

M |t=0 = 0,

and evaluating the cross correlation

∫ T

0

∂

∂x0,jQ(x0, Dx0)iMu

∗

M (x0, T − t; s)

∂

∂x0,kQ(x0, Dx0)lNwN ;s(x0, t; s)dt = I

MNijkl (x0; s).


(ii) The corresponding adjoint state formulation associated with the (full) elastic wave equation (2.1)is given by the forward equation

Piq(x,Dx, Dt)wq;s = δ(t)δ(x− s)ek, wq;s|t=0 = 0, ∂twq;s|t=0 = 0,

the adjoint state equation

Pip(x,Dx, Dt)u∗

p =

∫dr di(r, s, T − t)δ(x− x̂(r, 0)), u

∗

p|t=0 = 0, ∂tu∗

p|t=0 = 0,

and cross correlation

∫ T

0

∂

∂x0,jPMip (x0, Dx0)u

∗

p(x0, T − t; s)∂

∂x0,kPNlq (x0, Dx0)wq;s(x0, t; s)dt = I

MNijkl (x0; s),

where PMji = QjMQ−1Mi (no sum) is the projection onto mode M and P

Nji = QjNQ

−1Ni (no sum)

is the projection onto mode N [25, 26, 36]. The adjoint state formulation of / RTM approach toimaging has been used to compensate for the normal operator via Least-Squares optimization; see,for example, [18].

Σ x̃

x̃

Σ x̃

x̃

Fig. 2. Array receiver functions: Detecting the incident field, and scattered field. Top: distinct arrays; bottom:single array (teleseismic situation).

4. Array receiver functions.

4.1. Backpropagating the incident field. In this section, we will remove the knowledge ofthe source by obtaining the incident field, Q(x0, Dx0)lN

∫GN (x0, x̃

′, t0)Q−1Nk′(x̃

′, s)ek′ dx̃′ in (3.1),

which corresponds with wl;s(x0, t0; s), from the data.

Backward extrapolation yields (Σs is contained in a flat surface)

wq;s(x0, t0; s) = −2

∫ ∫

Σs

cnk;il(x̂(r′, 0))

∂Gql

∂x̂k(x0, x̂(r

′, 0), t′ − t0) di(r′, t′; s) dr′dt′,

restricting the wave field to a particularly polarized upgoing constituent (cf. (2.26); the integrand


is related to Nν (cf. (2.28)) via N = N(ν)); upon decomposition into polarizations, we obtain

wq;s(x0, t0; s) = −2Q(x0, Dx0)qN

∫ ∫

Σs

cnk;il(x̂(r′, 0))

∂

∂x̂kGN (x0, x̂(r

′, 0), t′ − t0)(Q(x̂, Dx̂)−1Nldi)(r

′, t′; s) dr′dt′

so that

(4.1) wN ;s(x0, t0; s) =

∫ ∫

Σs

(−2)∂

∂x̂kGN (x0, x̂(r

′, 0), t′ − t0)

(cnk;il(x̂(r′, 0))Q(x̂, Dx̂)

−1NlQ(x̂, Dx̂)iNdN )(r

′, t′; s) dr′dt′

(no sum over N). Following the propagation of singularities, it becomes clear that the singularitiesin the wave field are recovered at x0 if the ray connecting s with x0 intersects the boundary at apoint in Σs, see Fig. 3.

Σ x̃

x0

x̃

stationarypath

Fig. 3. Backward extrapolation and propagation of singularities.

4.2. Cross correlation formulation. Upon substituting (4.1) into (3.1), we obtain

(4.2) IMNijkl (x0; s) =

∫∂

∂x0,jQ(x0, Dx0)iM

∫ T

t0

∫

Σs

GM (x0, x̂(r, 0), t− t0)

dMN (r, t; s)drdt

∂

∂x0,kQ(x0, Dx0)lN

∫ T

t0

∫

Σs

(−2)∂

∂x̂k′GN (x0, x̂(r

′, 0), t′ − t0)

(cnk′;i′l′(x̂(r′, 0))Q(x̂, Dx̂)

−1Nl′Q(x̂, Dx̂)i′NdN )(r

′, t′; s) dr′dt′ dt0.

Interchanging orders of integration leads to the introduction of

(4.3) Kijkl;MNk′(x0; r, t, r′) =

∫ t

0

∂

∂x0,jQ(x0, Dx0)iMGM (x0, x̂(r, 0), t− t0)

∂

∂x0,kQ(x0, Dx0)lN (−2)

∂

∂x̂k′GN (x0, x̂(r

′, 0),−t0) dt0


defining an integral operator KMN , say. The propagation of singularities by KMN is described bythe Lagrangian submanifold,

ΛKMN = {(x0, ξ̂0 − ξ̃0; ŷ

′(x0, ξ̂0, t̂), ŷ′(x0,−ξ̃0, t̃), t̂− t̃,

η̂′(x0, ξ̂0, t̂),−η̂′(x0,−ξ̃0, t̃),ω) |BM (x0, ξ̂0) = BN (x0, ξ̃0) = ∓ω,

ŷ′′(x0, ξ̂0, t̂) = 0, ŷ′′(x0,−ξ̃0, t̃) = 0} ⊂ T

∗X\0× T ∗Z\0,

where Z = Σs × Σs × (0, T ), see Fig. 4.

Assumption 4. The matrix

(4.4)∂(ŷ′′(x0, ξ̂0, t̂), ŷ

′′(x0,−ξ̃0, t̃))

∂(x0, ξ̂0, ξ̃0, t̂, t̃)has rank 2.

With this assumption, following the proof of [33, Theorem 4.2], we find

Lemma 4.1. Let M 1= N . KMN : E′(Z) → D′(X) is a Fourier integral operator with canonical

relation ΛKMN .

Σ x̃

Fig. 4. The canonical relation, ΛKMN

, of KMN (isotropic case).

Definition 4.2. For M 1= N , the array receiver function (ARF), RMN , is defined as thebilinear map, d → RMNk′ , with

(4.5) (RMNk′(d(., .; s)))(r, t, r′) =

∫dMN (r, t+ t

′; s)

(cnk′;i′l′(x̂(r′, 0))Q(x̂, Dx̂)

−1Nl′Q(x̂, Dx̂)i′NdN )(r

′, t′; s) dt′ (no sum over N).

The observational assumption is that dMN = dM , whence RMNk′(d(., .; s)) can be obtained fromthe multi-component data. Using (4.2) it now follows that


Lemma 4.3. (Equivalence) It holds true that, microlocally,

(4.6) (KMN (RMN (d(., .; s)))(., ., .))ijkl(x0) = (F∗

MNdMN (., .; s))ijkl(x0).

Following the propagation of singularities, ΛKMN ◦WF(RMN (d(., ., ; s))), substituting FMNδcijklfor dMN , reveals how the geometry of imaging δcijkl reduces to the geometry depicted in Fig. 1 bycombining Fig. 4 with Fig. 3.

5. Flat, translationally invariant models: Propagation of singularities, receiver

functions. In view of translational invariance, (2.34) attains the form,

(5.1) δGMN (x̂, x̃, t) =

∫

[0,Z]

{ ∫ t

0

∫∂

∂(t0, x0,j)Q(x0, Dx0)iMGM (x0, x̂, t− t0)

∂

∂(t0, x0,k)Q(x0, Dx0)lNGN (x0, x̃, t0) dx

′

0dt0

}

(δil

δρ(z0)

ρ(z0),−

δcijkl(z0)

ρ(z0)

)dz0,

writing x0 = (x′

0, z0). Upon restriction to x̂ = x̂(r, 0), x̃ = s, the expression in between braces onthe right-hand side defines the kernel, F0MN ;ijkl(r, t; z0) say, of a single scattering operator F

0MN :

(5.2) F0MN ;ijkl(r, t; z0) =

∫ ∫ t

0

∫∂

∂(t0, x0,j)Q(x0, Dx0)iMGM (x0, x̂(r, 0), t− t0)

∂

∂(t0, x0,k)Q(x0, Dx0)lNGN (x0, x̃

′, t0) dx′

0dt0 Q−1Nk′(x̃

′, s)ek′ dx̃′,

cf. (2.40). The associated imaging operator, (F 0MN )∗, maps the (conversion) data to an image as a

function of z0 (and s): IMNijkl (z0; s) = ((F

0MN )

∗dMN )ijkl(z0), cf. (3.1).We introduce so-called midpoint-offset coordinates r′ = m−h and r = m+h (drdr′ = 2dmdh).

Following the construction that leads to Lemma 4.3, we find that

(F 0MN )∗dMN (., .; s) = K

0MN (RMN (d(., .; s))(., ., .)),

where K0MN is an operator with kernel

(5.3) K0ijkl;MNk′(z0;m+ h, t,m− h)

= 2

∫ t

0

∫∂

∂x0,jQ(x0, Dx0)iMGM (x

′

0, z0, x̂(m+ h, 0), t− t0)

∂

∂x0,kQ(x0, Dx0)lN (−2)

∂

∂x̂k′GN (x

′

0, z0, x̂(m− h, 0),−t0) dx′

0dt0,

cf. (4.3). To study the propagation of singularities by this operator, we substitute the WKBJapproximations for GM and GN (cf. (2.31)) in this expression.

K0MN , imaging. We focus on the propagation of singularities and, hence, the relevant phasefunctions; the amplitudes follow from standard stationary phase arguments. The WKBJ phasefunction associated with K0ijkl;MNk′ becomes

ω

[− τµ(0, z0, p̂) + τν(0, z0, p̂

′) +n−1∑

j=1

(p̂− p̂′)j(m− x′

0)j −n−1∑

j=1

(p̂+ p̂′)jhj + t

].


Carrying out the integrations over x′0 and p̂′ leads toK0ijkl;MNk′(z0;m+h, t,m−h) ≈ K̇

0ijkl;MNk′(z0;h, t),

which admits an integral representation with WKBJ phase function

ω

[− τµ(0, z0, p̂) + τν(0, z0, p̂)− 2

n−1∑

j=1

p̂jhj + t

].

We get

K0MN (RMN (d(., .; s))(., ., .)) ≈ K̇0MN (R

0MN (d(., .; s))(., ., .)),

where

(5.4) (R0MNk′(d(., .; s)))(h, t) =

∫(RMNk′(d(., .; s)))(m+ h, t,m− h) dm.

Applying the method of stationary phase to the integral representation forK̇0ijkl;MNk′(z0;h, t) in p̂, yields stationary points p̂ = p̂

0(z0, h) satisfying

(5.5) −

[∂τµ(0, z0, p̂)

∂p̂−

∂τν(0, z0, p̂)

∂p̂

]= 2h,

revealing the propagation of singularities: This equation defines a pair of rays sharing the samehorizontal slowness p̂, originating at (image) depth z0, and reaching the acquisiton surface at r =∂τµ(0,z0,p̂)

∂p̂ , r′ = ∂τν(0,z0,p̂)

∂p̂ , respectively; in the imaging point of view, the rays intersect at depth

z0, whence r′ − r = 2h. The corresponding differential travel time is given by τµ(0, z0, p̂

0(z0, h)) −

τν(0, z0, p̂0(z0, h))+2

∑n−1j=1 p̂

0j (z0, h)hj (we note that p̂

0j (z0, h) is the negative of the usual geometric

ray parameter in view of our Fourier transform convention). The geometry is illustrated in Fig. 5(pair of solid rays).

2h

0

z0

Fig. 5. Propagation of singularities by K0MN

. The double arrows relate to (scattered) mode µ, while the singlearrow relates to (incident) mode ν. Translational invariance yields alternative ray pairs (dashed) for which the phaseof K0

ijkl;MNk′is stationary.

R0MN , modelling. Using (5.1), substituting the WKBJ approximations for GM and GN in(R0MNk′(d(., .; s)))(h, t), and carrying out the integrations over x

′

0 and p̃′, we obtain a linear integral

operator representation (acting on (δilδρ(z′0)ρ(z′0)

,−δcijkl(z

′

0)ρ(z′0)

)) for the integrand in (5.4): For fixed s =


(s′, zs), zs > z0, the WKBJ phase function of its kernel representation is

ω

[− τµ(0, z

′

0, p̂)− τν(z′

0, zs, p̂) + τν(0, zs, p̃)

+

n−1∑

j=1

p̂j(m− h− s′)j −

n−1∑

j=1

p̃j(m+ h− s′)j + t

].

Carrying out the integrations over m and then p̂ leads to∫(RMNk′(d(., .; s)))(m+h, t,m−h) dm ≈

(Ṙ0MNk′(d(., .; s)))(h, t), which admits an integral representation with WKBJ phase function

ω

[− τµ(0, z

′

0, p̃) + τν(0, z′

0, p̃)− 2n−1∑

j=1

p̃jhj + t

];

the integration over p̃ signifies a “plane-wave” decomposition of the source, while the explicit de-pendence on (s′, zs) has disappeared. Thus

K̇0MN (R0MN (d(., .; s))(., .)) ≈ K̇

0MN (Ṙ

0MN (d(., .; s))(., .)),

yielding a resolution analysis in depth. The stationary phase analysis in, and following (5.5) applies,leading to the introduction of p̃0(z′0, h) and associated ray geometry if there is a non-vanishingcontrast (horizontal reflector) at depth z′0.

For the singularities to appear in (R0MNk′(d(., .; s)))(h, t), it is necessary that p̃0(z′0, h) coincides

with the stationary value, p̃s say, of p̃ associated with the WKBJ approximation of the incidentfield (dN ), determined by s and m+ h.

Receiver Functions, plane-wave synthesis. Let d̃(., .; p̃s) denote the frequency-domaindata obtained by synthesizing a source plane wave with parameter p̃s (as in plane-wave Kirch-hoff migration) including an appropriate amplitude weighting function derived from the WKBJapproximation (here, we depart from the single source acquisition). We correlate these data us-ing (4.5) subjected to a Fourier transform in time, with d(., .; s) replaced by d̃(., .; p̃s). We obtain

R̂MNk′(d̃(., .; p̃s))(m+ h,ω,m− h), with the property

(5.6) Ṙ0MNk′(d(., .; s))(h, t)

∼

∫R̂MNk′(d̃(., .; p̃s))(m0,ω,m0) exp

(iω

[− 2

n−1∑

j=1

p̃sjhj + t

] )dp̃sdω

for any (m0, 0) contained in Σs. The quantity R̂MNk′(d̃(., .; p̃s))(m0,ω,m0) is what seismologistscall a receiver function. The phase shift and receiver function are illustrated by the dashed ray(single array) paired with the ray indicated by double arrows in Fig. 5.

6. Discussion. We presented a program for elastic-wave inverse scattering of reflection seismicdata generated by an active (known) source or a passive (unknown) source, focussed on modeconversions. We derived the normal operator for, and an adaption of the elastic-wave reverse-time migration (RTM) approach to partially incorporate the parametix of the normal operator andsuppress smooth artifacts. We introduced array receiver functions (ARFs), an extension of thenotion of receiver functions, which can be used for inverse scattering of passive source data with aresolution comparable to RTM. In principle, the ARFs can be used also for imaging with certain(mode-converted) multiple scattered waves. Microlocally, the RTM and the generalized Radontransform (GRT) based inverse scattering are the same, the key restriction being the absence ofsource caustics. We established how ARFs reduce to receiver functions in the case of flat, planarlylayered earth models.


Acknowledgments. This research was supported in part by the members, BP, ConocoPhillips,ExxonMobil, Statoil,Total, of the Geo-Mathematical Imaging Group at Purdue University.

Appendix A. Diagonalization of Ail with a unitary operator.

If the AprinM (x, ξ) are all different, Ail can be diagonalized with a unitary operator, that is,

Q(x,Dx)−1 = Q(x,Dx)

∗. We write Q̃0lN (x,Dx) = QprinlN (x,Dx); then (Q̃

0)∗Mi(x,Dx) has pricipalsymbol (Qprin)tMi(x, ξ). We have

(Q̃0)∗Mi(x,Dx)Q̃0iN (x,Dx) = δMN +R

0MN (x,Dx)

where R0MN (x,Dx) is self adjoint and of order −1. Then Q0iN (x,Dx) = (Q̃

0(I +R0)−1/2)iN (x,Dx)is, microlocally, unitary. We write

A01(x,Dx) =12 [A

prin1 (x,Dx) + (A

prin1 )

∗(x,Dx)],

A02(x,Dx) = diag(12 [A

prinM (x,Dx) + (A

prinM )

∗(x,Dx)];M = 2, . . . , n),

so that

(Q0)∗AQ0 =

(A01 00 A02

)+

(B11 B12B21 B22

),

where B11 and the elements of B12 (a 1×(n−1) matrix), B21 (a (n−1)×1 matrix of pseudodifferentialoperators), and B22 (a (n−1)× (n1) matrix of pseudodifferential operators), are of order 1; B mustbe self adjoint, whence B12 = B

∗

21.

Next, we seek an operator, Q̃1lN (x,Dx) = δlN + r1lN (x,Dx), assuming that

r1 =

(0 −(r121)

∗

r121 0

),

whence (r1)∗ = −r1, such that

((Q̃1)∗((

A01 00 A02

)+

(B11 B12B21 B22

))Q̃1)21 = 0,

((Q̃1)∗((

A01 00 A02

)+

(B11 B12B21 B22

))Q̃1)12 = 0,

modulo terms of order 0. This holds true if

r121A01 −A

02r

121 = B21;

r121 must be of order −1. Up to principal parts, this is a matrix equation for the symbol of r121,

given the principal symbols of A01 and A02; we note that the principals of A

prin1 and A

01, and of

diag(AprinM ;M = 2, . . . , n) and A02, coincide. Because the eigenvalues of A

02(x, ξ) all differ from

A01(x, ξ), it follows that this system of algebraic equations has a solution. With the solution we formthe unitary operator Q1iN (x,Dx) = ((I + r

1)(I − (r1)2)−1/2)iN (x,Dx). Then

(Q1)∗(Q0)∗AQ0Q1 =

(A11 00 A12

)+

(C11 C12C21 C22

),

where A11 and A12 are self adjoint. C11 and the elements of C12 (a 1× (n−1) matrix of pseudodiffer-

ential operators), C21 (a (n−1)×1 matrix of pseudodifferential operators), and C22 (a (n−1)×(n1)matrix of pseudodifferential operators), are of order 0; C must be self adjoint, whence C12 = C

∗

21.This procedure is continued to find Q0Q1 · · ·Qk which is microlocally unitary and brings A in blockdiagonal form modulo terms of order 1− k. Next, we repeat the procedure for Ak2 .


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elastic-wave inverse scattering with active ...scalar waves [39, 24, 1] to elastic waves. we make...

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