Characterization and source-receiver continuation of seismic reflection data

Transcription

1 Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Characterization and source-receiver continuation of seismic reflection data Maarten V. de Hoop 1, Gunther Uhlmann 2 1 Center for Wave Phenomena, Colorado School of Mines, Golden CO , USA. mdehoop@mines.edu 2 Department of Mathematics, University of Washington, Seattle WA , USA. g@math.washington.edu Received: / Accepted: Abstract: In reflection seismology one places sources and receivers on the Earth s surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. In the field, often, there are obstructions to collect seismic data for all source-receiver pairs desirable or needed for data processing and application of inverse scattering methods. Typically, data are measured on the Earth s surface; the location of the receiver relative to the source can be coordinated by offset and azimuth. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Seismic data are commonly modeled by a scattering operator developed in a high-frequency, single scattering approximation. We initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects a space of distributions on the data acquisition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. We illustrate the data continuation by a numerical example. 1. Introduction In reflection seismology one places sources and receivers on the Earth s surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. Seismic data collected in the field are often not ideal for data processing and application of inverse scattering methods. Typically, data are measured on the Earth s (two-dimensional) surface; the location Partly supported by NSF and a John Simon Guggenheim fellowship.

2 2 M. V. de Hoop and G. Uhlmann of the receiver relative to the source can be coordinated by offset and azimuth. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Special cases of data continuation are the so-called transformation to zero offset (derived from what seismologists call Dip MoveOut [9] to generate data at zero offsets, and transformation to common azimuth (derived from what seismologists call Azimuth MoveOut [2]) to generate data at a fixed, prescribed azimuth. Data continuation can also play the role of forward extrapolation [11] in a data regularization scheme. Seismic data are commonly modeled by a scattering operator developed in a highfrequency single scattering approximation. In this approximation one assumes that the medium is described by a singular contrast superimposed on a smooth background. Under geological constraints, often, the contrast is a conormal distribution. Initially, we focus on the determination of the range of the forward scattering operator that models the singular part of the data in the single scattering approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects the space of distributions on the acquisition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. Through characterization of features in the data, applications of data continuation extend to survey design (i.e. the design of the acquisition geometry describing the locations of the source-receiver pairs). The range of the scattering operator can also be used as a criterion for muting the data for features that are undesirable for the purpose of imaging the data (such as multiple scattered waves). The notion of data continuation has been introduced in exploration seismology quite some time ago. As early as in 1982, Bolondi et al. [3] came up with the idea of describing data offset continuation and Dip MoveOut in the form of solving a partial differential equation. Their approach, built on the approach of Deregowski and Rocca [9], is valid in homogeneous media for acoustic waves while their partial differential operator is approximate only. An exact partial differential equation for space dimension n = 2 that addresses mentioned offset continuation was later derived by Goldin [13]. In this application it is implicitly used that the kernel of the associated partial differential operator determines the range of the operator that models the singular part of seismic data in the single scattering approximation. The operator can be written in the form of a generalized Radon transform. Heuristically, the procedure and analysis presented in this paper can be thought of as a generalization from two to higher dimensions, from acoustic to elastic, and from homogeneous to heterogeneous media, of Goldin s offset continuation equation. Let the data be denoted by d = d(s, r, t) where s denotes source position, r receiver position, t the time, while (s, r, t) Y and Y denotes the acquisition manifold. Let Y R 2n 1. We introduce the map κ : (s, r, t) (z, t n, h), z = 1 2 (r+s), h = 1 2 (r s), t n = t n (h, t) = t 2 4h2 v 2, where, v is the acoustic wave speed. Let r be the pull back of d by the inverse of this map, r = (κ 1 ) d. The singular support of r can be parametrized by (z, h) according to (z, T n (z, h), h) with T n (z, h) = t n (h, T (z, h)), in which the function T (z, h) denotes the traveltime of a particular reflection in the data; in seismological terms, the function

3 Characterization of seismic reflection data 3 T n is the traveltime after Normal MoveOut correction. Goldin s equation is of the form (n = 2) P r = 0, P 2 ( ) 2 := t n t n h + h h 2 2 z 2. (1) This equation is supplemented with the initial conditions r(z, t n, h) h=h0, r h (z, t n, h) h=h0. The first initial condition represents what seismologists call a post-normal -moveout constant-offset section at half offset h 0 ; the second initial condition the firstorder derivative of post-normal-moveout section at half offset h 0. Goldin s equation is not exact in the sense that it does not account for the symbols of the reflection operators associated with the reflectors in the subsurface. The notion of data continuation has also been introduced and exploited in helical x-ray transmission tomography (CT) [25]. Consider a flat area detector, which is contained in the plane described by Cartesian coordinates (u, v). Let R denoted the radius of the helix and 2πh its pitch. Let λ denote the angle describing rotation of the cone vertex. The axial shift of the assembly of the x-ray source and the detector is denoted by ζ. The data are denoted by g = g(u, v, λ, ζ). In this case, John s equation [22] describing the range of the Radon transform is used. John s equation for the x-ray transform in dimension 3 is given by R 2 2 g u ζ 2u g v + (R2 + u 2 ) 2 g u v = R 2 g λ v Rh 2 g v ζ + uv 2 g v 2. (2) This equation is supplemented with the initial conditions g(u, v, λ, ζ) ζ=0 that are measured. (The standard form of John s equation is much simpler than (2).) Gel fand and Graev [12] have generalized John s result to k-planes in R n. John s (and Goldin s) partial differential equation in higher space dimension is second order and of ultrahyperbolic type. The seismic forward scattering operator is a Fourier integral operator and can be identified with a generalized Radon transform [1, 5, 26]. We characterize seismic data by analyzing the range of the forward scattering operator. This range coincides with the kernel of a self-adjoint, second-order pseudodifferential operator, P, derived from annihilators, P i, of the data, d, P i d = 0, P = i P 2 i. (3) Let Q denote the global parametrix of P. The mentioned elliptic minimal projector then follows to be π = I QP + smoothing operator (4) and provides the Fourier integral operator for continuing the singular part of the data. The annihilators are functionally dependent on the background medium, and hence can be used to form a criterion to estimate it. This estimation is known to seismologists as

4 4 M. V. de Hoop and G. Uhlmann velocity analysis and can be formulated as a reflection tomography problem. Thus, data continuation and reflection tomography, and imaging, are intimately connected. The results presented in this paper are based on the work by Guillemin and Uhlmann [17]. Here, we speak of data continuation rather than offset continuation, because our approach continues data in sources and receivers and not only in offset due to the heterogeneity of the subsurface we can allow. 2. Modeling of seismic data in the single scattering approximation The propagation and scattering of seismic waves is governed by the elastic wave equation, which is written in the form W il u l = f i, (5) where u l = ρ(x)(displacement) l, f i = 1 ρ(x) (volume force density) i, (6) and 2 W il = δ il t 2 c ijkl (x) x j ρ(x) x k + l.o.t.. (7) Here, x R n and the subscripts i, j, k, l {1,..., n}. The system of partial differential equations is assumed to be of principal type. It supports different wave types (modes), one compressional and n 1 shear Decoupling the modes. First, we consider a smoothly varying medium. Decoupling of the modes is then accomplished by diagonalizing the system. We assume that ρ is smooth and bounded away from zero. System (5) is real, time reversal invariant, and its solutions satisfy reciprocity. We describe how the system (5) can be decoupled by transforming it with appropriate pseudodifferential operators, see Taylor [27], Ivrii [21] and Dencker [8]. The goal is to transform the operator W il by conjugation with a matrix-valued pseudodifferential operator Q(x, D x ) im, D x = i x, to an operator that is of diagonal form, modulo a regularizing part. Since the time derivative in W il is already in diagonal form, it remains only to diagonalize its spatial part, A il = c ijkl (x) + l.o.t.. x j ρ(x) x k The goal becomes finding Q im and A M such that Q(x, D x ) 1 Mi A il(x, D x ) Q(x, D x ) ln = diag(a M (x, D x ) ; M = 1,..., n) MN. (8) The indices M, N denote the mode of propagation. Then u M = Q(x, D x ) 1 Mi u i, satisfy the uncoupled equations f M = Q(x, D x ) 1 Mi f i (9) W M (x, D x, D t )u M = f M, (10)

5 Characterization of seismic reflection data 5 where D t = i t, and in which W M (x, D x, D t ) = 2 t 2 + A M (x, D x ). Remark 1. For the construction of Fourier integral operator solutions developed in the scalar wave case, it is sufficient to transform the partial differential operator W il to block-diagonal form, where each of the blocks W M = W M (x, D x, D t ) has scalar principal part proportional to the identity matrix. In this case we will use the indices M, N to denote the block, and we will omit indices for the components within each block. 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 A prin il (x, ξ) of A il (x, D x ) is a positive symmetric matrix. Hence, it can be diagonalized by an orthogonal matrix. On the level of principal symbols, composition of pseudodifferential operators reduces to multiplication. Therefore, we let Q prin im (x, ξ) be this orthogonal matrix, and we let A prin (x, ξ) be the eigenvalues of Aprin(x, ξ), so that Q prin M Mi (x, ξ) 1 A prin il (x, ξ)q prin ln (x, ξ) = diag(aprin M (x, ξ)) MN. (11) The principal symbol Q prin im (x, ξ) is the matrix that has as its columns the orthonormalized polarization vectors associated with the modes of propagation. Having assumed that W il is of principal type, the multiplicities of the eigenvalues A prin M (x, ξ) are constant, whence the principal symbol Qprin im (x, ξ) depends smoothly on (x, ξ) and microlocally equation (11) carries over to an operator equation. Taylor [27] has shown that if this condition is satisfied, then decoupling can be accomplished to all orders, where each block corresponds to a different eigenvalue. The second-order equations (10) describe the decoupling of the original system into different elastic modes. These equations inherit the symmetries of the original system, such as time-reversal invariance and reciprocity. Time-reversal invariance follows because the operators Q im (x, D x ), A M (x, D x ) can be chosen in such a way that Q im (x, ξ) = Q im (x, ξ), A M (x, ξ) = A M (x, ξ). Then Q im, A M are real-valued. Reciprocity for the causal Green s function G ij (x, x 0, t t 0 ) means that G ij (x, x 0, t t 0 ) = G ji (x 0, x, t t 0 ). Such a relationship also holds (modulo smoothing operators) for the Green s function G M (x, x 0, t t 0 ) associated with (10). il 2.2. The Green s function. To evaluate the Green s function, we use the first-order system for u M that is equivalent to (10), ( ) ( ) ( ) ( ) um 0 1 um 0 = +. (12) t A M (x, D x ) 0 f M u M t This system can be decoupled also, namely by the matrix-valued pseudodifferential operator ( ) 1 1, ib M (x, D x ) ib M (x, D x ) u M t

6 6 M. V. de Hoop and G. Uhlmann where B M (x, D x ) = A M (x, D x ), which is a pseudodifferential operator of order 1 that exists because A M (x, D x ) is positive definite. The principal symbol of B M (x, D x ) is given by B prin M (x, ξ) = A prin (x, ξ). Then M u M,± = 1 2 u M ± 1 2 ib M (x, D x ) 1 u M t satisfy the two first-order equations, f M,± = ± 1 2 ib M (x, D x ) 1 f M (13) W M,± (x, D x, D t )u M,± = f M,±, (14) where W M,± (x, D x, D t ) = t ± ib M (x, D x ), W M,+ W M, = W M. (15) We construct operators G M,± with Lagrangian distribution kernels G M,± (x, x 0, t) that solve the initial value problem for (14). The operators G M,± are Fourier integral operators. Their construction is well known, see for example Duistermaat [10], Chapter 5. Singularities are propagated along the bicharacteristics, that are determined by Hamilton s equations generated by the principal symbol (factor i divided out) τ ± B prin M (x, ξ) of (14), x λ = ± ξ Bprin M (x, ξ), t λ = 1, (16) ξ λ = x Bprin M (x, ξ), τ λ = 0. Clearly, the solution may be parameterized by t. We denote the solution of (16) with the + sign and initial values (x 0, ξ 0 ) at t = 0 by (x M (x 0, ξ 0, t), ξ M (x 0, ξ 0, t)). The solution with the sign is found upon reversing the time direction and is given by (x M (x 0, ξ 0, t), ξ M (x 0, ξ 0, t)). A complete view of the propagation of singularities is provided by the canonical relation of the operator G M,±, given by C M,± = {(x M (x 0, ξ 0, ±t), t, ξ M (x 0, ξ 0, ±t), τ = B M,± (x 0, ξ 0 ); x 0, ξ 0 )}. (17) A convenient choice of phase function for the oscillatory integral representation for the kernel of G M,± is described in Maslov and Fedoriuk [24]. They state that one can always use a subset of the cotangent vector components as phase variables. Let us choose coordinates for C M,+ of the form (x I, x 0, ξ J, τ), (18) where I J is a partition of {1,..., n}. It follows from Theorem 4.21 in Maslov and Fedoriuk [24] that there is a function S M (x I, x 0, ξ J, τ), such that locally C M,+ is given by x J = S M ξ J, t = S M τ, ξ I = S M x I, ξ 0 = S M x 0. (19)

7 Characterization of seismic reflection data 7 Here we take into account the fact that C M,+ is a canonical relation, which introduces a minus sign for ξ 0. A nondegenerate phase function for C M,+ is then found to be φ M,+ (x, x 0, t, ξ J, τ) = S M (x I, x 0, ξ J, τ) + ξ J, x J + τt. (20) In case J =, the generating function S M reduces to frequency, τ, times the negative of traveltime, which we denote by T M = T M (x, x 0 ). On the other hand, the canonical relation C M, is given by C M, = {(x, t, ξ, τ; x 0, ξ 0 ) (x, t, ξ, τ; x 0, ξ 0 ) C M,+ }. Thus a phase function for C M, is φ M, (x, x 0, t, ξ J, τ) = φ M,+ (x, x 0, t, ξ J, τ). A theorem of Hörmander (see also Theorem of Duistermaat [10]) implies that the operator G M,± is microlocally a Fourier integral operator of order 1 4. Hence, microlocally we have an expression for G M,± in the form of an oscillatory integral G M,± (x, x 0, t) (21) = (2π) J n+1 4 a M,± (x I, x 0, ξ J, τ) exp[iφ M,± (x, x 0, t, ξ J, τ)] dξ J dτ. The amplitude a M,± (x I, x 0, ξ J, τ) satisfies a transport equation along the bicharacteristics (x M (x 0, ξ 0, ±t), ξ M (x 0, ξ 0, ±t)). We may define the canonical relation for G M as C M = C M,+ C M, and a phase function { φm, if τ > 0, φ M = φ M,+ if τ < 0. It follows from (13) that the Green s function for the second-order decoupled equation is given by G M (x, x 0, t) = 1 2 i (G M,+(x, x 0, t) G M, (x, x 0, t)) B M (x 0, D x0 ) 1, (22) where B M (x 0, D x0 ) 1 is of order 1. Lemma 1. The solution u M to (10) is given microlocally, away from WF(f M ), by u M (x, t) = G M (x, x 0, t t 0 )f M (x 0, t 0 ) dx 0 dt 0, (23) where G M (x, x 0, t) is the kernel of a Fourier integral operator with canonical relation C M and order 1 1 4, mapping functions of x 0 to functions of (x, t). It can be written as G M (x, x 0, t) (24) = (2π) J n+1 4 a M (x I, x 0, ξ J, τ) exp[iφ M (x, x 0, t, ξ J, τ)] dξ J dτ. For the amplitude a M (x I, x 0, ξ J, τ) we have to highest order a M (x I, x 0, ξ J, τ) = (2π) 1/4 (x 0, ξ 0, t) det (x I, x 0, ξ J, τ) 1/2 1 2 τ. (25)

8 8 M. V. de Hoop and G. Uhlmann The amplitude is an element of M CM Ω 1/2 (C M ), the tensor product of the Keller- Maslov bundle M CM and the half-densities on the canonical relation C M. If the subprincipal part of A M (x, D x ) is a matrix, then the amplitude is also a matrix. The Keller- Maslov bundle gives a factor i k, where k is an index, which we will absorb in the amplitude. The index keeps track of the passage through caustics. Remark 2. It is possible to weaken the assumption that the wave operator in (5) is of principal type. In fact, the assumption required in Lemma 1 can be formulated as follows: On a neighborhood of any bicharacteristic through WF(f M ) Char(W M ), the multiplicity of the eigenvalue A prin (x, ξ) in (11) is constant. M The amplitude in (25) can be numerically evaluated by solving the system of equations for the paraxial rays in ray-centered (Fermi) coordinates The single scattering approximation. In the contrast formulation the total value of the medium parameters ρ, c ijkl is written as the sum of a smooth background constituent ρ(x), c ijkl (x) and a singular perturbation δρ(x), δc ijkl (x), viz. ρ + δρ, c ijkl + δc ijkl. This decomposition induces a perturbation of W il (cf. (7)), δw il = δ il δρ(x) ρ(x) 2 t 2 δc ijkl (x). x j ρ(x) x k We denote the causal solution operator associated with (5) by G il and its distribution kernel by G il (x, x 0, t t 0 ). The first-order perturbation, δg il, of G il is derived by requiring that the leading-order term in (W ij + δw ij )(G jk + δg jk ) vanishes. This results in the representation δg jk ( x, x, t) = t 0 X G ji ( x, x 0, t t 0 ) δw il (x 0, D x0, D t0 ) G lk (x 0, x, t 0 ) dx 0 dt 0, (26) which is the Born approximation. Here, x denotes a source location, x a receiver location, and x 0 a scattering point location. We restrict our time window (of observation) to (0, T ) for some 0 < T <. Because the background model is smooth the operator δg il contains only the single scattered field. We apply reciprocity in ( x, x 0 ) to the integrand of the right-hand side of (26) and introduce the MN contribution, δg MN, to δg jk as follows, δg jk ( x, x, t) = δg MN ( x, x, t) Q( x, D x ) 1 1 Mj Q( x, D x) Nk.

9 Characterization of seismic reflection data 9 We have δg MN ( x, x, t) = = = t 0 t 0 X t 0 X Q(x 0, D x0 ) im G M (x 0, x, t t 0 ) δw il (x 0, D x0, D t0 ) Q(x 0, D x0 ) ln G N (x 0, x, t 0 ) dx 0 dt 0 ( ) δρ(x0 ) Q(x 0, D x0 ) im G M (x 0, x, t t 0 ) X (t 0, x 0,j ) ρ(x 0 ), δc ijkl(x 0 ) ρ(x 0 ) (t 0, x 0,k ) Q(x 0, D x0 ) ln G N (x 0, x, t 0 ) dx 0 dt 0 ( ) (t 0, x 0,j ) Q(x δρ(x0 ) 0, D x0 ) im G M (x 0, x, t t 0 ) ρ(x 0 ), δc ijkl(x 0 ) ρ(x 0 ) (t 0, x 0,k ) Q(x 0, D x0 ) ln G N (x 0, x, t 0 ) dx 0 dt 0 (27) upon integration by parts. Reciprocity implies that G M (x 0, x, t t 0 ) = G M ( x, x 0, t t 0 ) while G N (x 0, x, t 0 ) = G N ( x, x 0, t 0 ). Microlocally, we can substitute (24) for G M, with appropriate substitutions for its arguments: ( x, ξĵ) for (x, ξ J ) while Î Ĵ = {1,..., n}; for G N we have the substitutions ( x, ξ J) for (x, ξ J ) while Ĩ J = {1,..., n}. Using Egorov s theorem, it follows that the composition (t 0, x 0,j ) Q(x 0, D x0 ) im G M is a Fourier integral operator with the same phase as G M, and amplitude that to highest order equals the product a M ( xî, x 0, ξĵ, τ)q(x 0, ξ 0 ) im i (τ, ξ 0,j ), subject to the substitution ξ 0 = ξ 0 ( xî, x 0, ξĵ, τ) (cf. (19)). We write (27) in the form of an oscillatory integral. To this end, we introduce the radiation patterns (w MN;0, w MN;ijkl ) as w MN;0 ( xî, xĩ, x 0, ξĵ, ξ J, τ) = Q im (x 0, ξ 0 )Q in (x 0, ξ 0 ) τ 2, (28) w MN;ijkl ( xî, xĩ, x 0, ξĵ, ξ J, τ) = Q im (x 0, ξ 0 )Q ln (x 0, ξ 0 ) ξ 0,j ξ0,k, (29) subject to the substitutions ξ 0 = ξ 0 ( xî, x 0, ξĵ, τ), ξ 0 = ξ 0 ( xĩ, x 0, ξ J, τ) (cf. (19)). We introduce the amplitude, b MN ( xî, xĩ, x 0, ξĵ, ξ J, τ) = (2π) n 1 4 a M ( xî, x 0, ξĵ, τ) a N ( xĩ, x 0, ξ J, τ) (30) and the phase function Φ MN ( x, x, t, x 0, ξĵ, ξ J, τ) = φ M ( x, x 0, t, ξĵ, τ) + φ N ( x, x 0, t, ξ J, τ) τt. (31) Then δg MN ( x, x, t) = (2π) Ĵ + J n+1 4 b MN ( xî, xĩ, x 0, ξĵ, ξ J, τ) X ( w MN;0 ( xî, xĩ, x 0, ξĵ, ξ J, τ) δρ(x 0) ρ(x 0 ) + w MN;ijkl( xî, xĩ, x 0, ξĵ, ξ J, τ) δc ) ijkl(x 0 ) ρ(x 0 ) exp[iφ MN ( x, x, t, x 0, ξĵ, ξ J, τ)] dx 0 d ξĵd ξ Jdτ (32)

10 10 M. V. de Hoop and G. Uhlmann 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 (27): They are assumed to be representable by δg MN ( x, x, t) for ( x, x, t) in some acquisition manifold, which contains the source and receiver points and time. Let y ( x(y), x(y), t(y)) be a coordinate transformation, such that y = (y, y ) and the acquisition manifold, Y say, is given by y = 0. We assume that the dimension of y is 2 + c, where c is the codimension of the acquisition geometry. For example, for marine acquisition in seismic reflection data, c = 1, while also in global seismology for many, but not all regions c = 1 seismologists recognize this as lack of azimuthal coverage. An example of c = 0 is provided by the common- offset acquisition geometry. In this framework, the data are modeled by ( δρ(x) ρ(x), δc ) ijkl(x) δg MN ( x(y, 0), x(y, 0), t(y, 0)). (33) ρ(x) We investigate the propagation of singularities by this mapping. Let τ = B M (x 0, ξ 0 ), and x = x M (x 0, ξ 0, ± t), x = x N (x 0, ξ 0, ± t), t = t + t, ξ = ξ M (x 0, ξ 0, ± t), ξ = ξ N (x 0, ξ 0, ± t). We then obtain (y(x 0, ξ 0, ξ 0, t, t), η(x 0, ξ 0, ξ 0, t, t)) by transforming ( x, x, t+ t, ξ, ξ, τ) to (y, η) coordinates. We invoke the following assumptions that concern scattering over π and rays grazing the acquisition manifold, Assumption 1 There are no elements (y, 0, η, η ) with (y, η ) T Y \0 such that there is a direct bicharacteristic from ( x(y, 0), ξ(y, 0, η, η )) to ( x(y, 0), ξ(y, 0, η, η )) with arrival time t(y, 0). Assumption 2 The matrix y (x 0, ξ 0, ξ has maximal rank. (34) 0, t, t) The propagation of singularities by (33) is governed by the canonical relation Λ MN = {(y (x 0, ξ 0, ξ 0, t, t), η (x 0, ξ 0, ξ 0, t, t); x 0, ξ 0 + ξ 0 ) (35) B M (x 0, ξ 0 ) = B N (x 0, ξ 0 ) = τ, y (x 0, ξ 0, ξ 0, t, t) = 0} T Y \0 T X\0. The condition y (x 0, ξ 0, ξ 0, t, t) = 0 determines the traveltimes t for given (x 0, ξ 0 ) and t for given (x 0, ξ 0 ). Following, as before, Maslov and Fedoriuk [24], we choose coordinates for Λ MN of the form (y I, x 0, η J ), (36) where I J is a partition of {1,..., 2n 1 c}, with associated generating function S MN = S MN (y I, x 0, η J ). The phase function in these coordinates becomes Φ MN = Φ MN (y, x 0, η J ).

11 Characterization of seismic reflection data 11 We derive an expression for the leading-order contribution to b MN (y I, x 0, η J ). To this end, we define τ = 1 2 ( τ + τ) and τ = τ τ so that τ = τ τ and τ = τ 1 2 τ; we then substitute (25) into the amplitude given in (30). We find that b MN ( xî, xĩ, x 0, ξĵ, ξ J, τ) = (2π) n 1 4 det (x 0, ξ 0, ξ 0, t, t) 1/2 1 ( xî, xĩ, x 0, ξĵ, ξ J, τ, τ) 4τ 2. τ=0 (37) The amplitude transforms as a half-density on the canonical relation Λ MN, yielding a factor det (y I, y, η J ) 1/2 ( xî, xĩ, ξĵ, ξ ; (38) J, τ) the coordinate transformation from ( x, x, t) to y = (y, y ) in the Fourier integral induces a factor ( x, x, t) 1/2 det (y, y ). (39) Including a normalization factor (2π) 2+c 4 then leads to the expression b MN (y I, x 0, η J) (40) = (2π) n+1+c 4 ( x, x, t) det (x 0, (y, y ) 1/2 det ξ 0, ξ 1/2 0, t, t) (x 0, y I, y, η J, τ) 1 4τ 2. y =0, τ=0 The map (x 0, ξ 0, ξ 0, t, t) (x 0, y I, y, η J, τ) (41) is bijective. Thus, for y = 0 and τ = 0 we can express ( ξ 0, ξ 0 ) as functions of (y I, x 0, η J ). Subject to such a substitution, we transform the radiation patterns using (28)-(29), w MN;0 (y I, x 0, η J) = Q im (x 0, ξ 0 )Q in (x 0, ξ 0 ) τ 2, w MN;ijkl (y I, x 0, η J) = Q im (x 0, ξ 0 )Q ln (x 0, ξ 0 ) ξ 0,j ξ0,k. We simplify the notation, and write x for x 0 and (y, η) for (y, η ) in Theorem 1. [26] Suppose Assumptions 1, and 2 are satisfied microlocally for the relevant part of the data. Let Φ MN (y, x, η J ) and b MN (y I, x, η J ) be the phase function and amplitude introduced above. Then the data are, microlocally, modeled by G refl MN(y) = (2π) J 2 3n 1 c 4 (b MN (y I, x, η J ) + l.o.t.) (42) X ( w MN;0 (y I, x, η J ) δρ(x) ρ(x) + w MN;ijkl(y I, x, η J ) δc ) ijkl(x) ρ(x) exp[iφ MN (y, x, η J )] dxdη J, i.e. by ( a Fourier) integral operator with canonical relation Λ MN acting on the distribution δρ ρ, δc ijkl ρ.

12 12 M. V. de Hoop and G. Uhlmann ( ) We now assume that δρ ρ, δc ijkl ρ are described by conormal distributions. We consider the case of a single interface, and a jump discontinuity in (δρ, δc ijkl ) across this interface. Let κ : R n R n, x z be a coordinate transformation such that the interface is given by z n = 0. The corresponding cotangent vector is denoted by ζ, and transforms according to ζ i (x, ξ) = (( κ x ) 1 ) t ij ξ j; the z form coordinates on the manifold X and we write z = (z, z n ). We introduce the distributions ( δρ, δc ijkl ) by pull back with κ: Then δρ(κ(x)) = δρ(x), δc ijkl (κ(x)) = δc ijkl (x). (43) x δρ = z n x ρ + l.o.t., ρ = δρ, z n where ρ contains a factor δ(z n (x)), and similarly for δc ijkl x. Substituting (43 into the integral over X in (42), and integrating by parts, then yields an oscillatory integral representation of the type (42) in which w MN;0 (y I, x, η J ) δρ(x) ρ(x) + w MN;ijkl(y I, x, η J ) δc ijkl(x) ρ(x) has been replaced by 2iτR MN (y I, x, η J ) z n x δ(z n(x)). Here, we use that τ will be one of the components of η J. Also, ( ) z n x δ(z n (x)) dx = z ( ) n=0 det x z n z x dz becomes the Euclidean surface integral over the surface or manifold z n = 0. Corollary 1. Suppose the assumptions in Theorem 1 are satisfied. Let Φ MN (y, x, η J ) and b MN (y I, x, η J ) be the phase function and amplitude as in Theorem 1. Then the mapping F : z n x δ(z n(x)) G refl MN (y), where G refl J MN (y) = (2π) 2 3n 1 c 4 (2iτ(η J ) b MN (y I, x, η J )R MN (y I, x, η J ) + l.o.t.) X z n x δ(z n(x)) exp[iφ MN (y, x, η J )] dxdη J, (44) defines a Fourier integral operator with canonical relation Λ MN and of order n 1+c 4 1. In the Kirchhoff approximation, one can identify the principal part of R MN with the plane-wave reflection coefficient: Using (41) we find the (x, ξ 0, ξ 0 ) associated with (y I, x, η J ). A reflection from an incident N-mode with covector ξ 0 into a scattered M- mode with covector ξ 0 takes place, at x, if the frequencies are equal and ξ 0 + ξ 0 is in the wavefront set of δ(z n (x)). Given ξ 0, ξ 0 one can identify the down- and upgoing modes

13 Characterization of seismic reflection data 13 µ(m), ν(n) relative to the interface, and define (at least to highest order) the reflection coefficient at x, R MN = R µ(m),ν(n) (z (x), ζ (x, ξ 0 ), τ) if z n (x) = 0, (45) see De Hoop and Bleistein [4] and Stolk and De Hoop [26]. The Kirchhoff approximation requires the following assumption Assumption 3 There are no rays tangent to the interface z n = 0, i.e. elements in Λ MN associated with (x(z, 0), ξ 0 (z, 0, ζ, 0)) or with (x(z, 0), ξ 0 (z, 0, ζ, 0)) (cf. (35)). For a treatment of reflection and transmission of waves in the elastic case, using microlocal analysis, see Taylor [27]; for the acoustic case, see also Hansen [18]. Examples of conormal distributions, z n x δ(zn (x)), in the Earth sciences the reflections off which are observed include the core-mantle boundary, thermal and chemical boundary layers in the deep mantle, fault zones, and geological interfaces in sedimentary basins. 3. Extension of the scattering operator For simplicity of notation, from here on, we drop the subscripts MN and consider a single mode pair The wavefront set of seismic data. The wavefront set of the modeled data is not arbitrary. This is a consequence of the fact that data consist of multiple experiments designed to provide a degree of redundancy, which we explain here. In the single scattering approximation, subject to restriction to the acquisition manifold Y, the singular part of the medium parameters is a function of n variables, while the data are a function of 2n 1 c variables. Assumption 4 (Guillemin [15]) The projection π Y of Λ on T Y \0 is an embedding. This assumption is known as the Bolker condition. It admits the presence of caustics. Because Λ is a canonical relation that projects submersively on the subsurface variables (x, ξ) (using that the operator W il is of principal type), the projection of (35) on T Y \0 is immersive [19, Lemma and (25.3.4)]. In fact, only the injectivity part of the Bolker condition needs to be verified. The image L of π Y is locally a coisotropic submanifold of T Y \0. Hence, for each (y, η) L, (T (y,η) L) T (y,η) L. Setting V (y,η) = (T (y,η) L), the vector bundle V L whose fiber at (y, η) is (T (y,η) L), is an integrable subbundle of T L. Applying [16, Prop. 8.1], from the Bolker condition it follows that L satisfies their Axiom F: the foliation of L associated with V is fibrating, i.e. there exists a C Hausdorff manifold X and a smooth fiber map L X whose fibers are the connected leaves of the foliation defined by V. We choose coordinates revealing the mentioned fibration. Since the projection π X of Λ on T X\0 is submersive, we can choose (x, ξ) as the first 2n local coordinates on Λ; the remaining dim Y n = n 1 c coordinates are denoted by e E, E being a manifold itself. The sets X (x, ξ) = const. are the isotropic fibers of the fibration of Hörmander [20], Theorem , see also Theorem Duistermaat [10] calls them characteristic strips (see Theorem 3.6.2). The wavefront set of the data is contained in L and is a union of such fibers. The map π X π 1 Y : L X is a canonical isotropic fibration, known to seismologists as map migration.

14 14 M. V. de Hoop and G. Uhlmann We consider again the canonical relation Λ and suppose that Assumption 4 is satisfied. We define Ω as the mapping π Y π 1 X, Ω : (x, ξ, e) (y(x, ξ, e), η(x, ξ, e)) : T X\0 E T Y \0, which is known to seismologists as map demigration. This map conserves the symplectic form of T X\0. Indeed, let σ Y denote the fundamental symplectic form on T Y \0. We consider the vector fields over an open subset of L with components w xi = (y,η) x i and similarly for w ξi and w ei. Then σ Y (w xi, w xj ) = σ Y (w ξi, w ξj ) = 0, σ Y (w ξi, w xj ) = δ ij, σ Y (w ei, w xj ) = σ Y (w ei, w ξj ) = σ Y (w ei, w ej ) = 0. (46) The (x, ξ, e) are symplectic coordinates on the projection L of Λ on T Y \0. In the following lemma, we extend these coordinates to symplectic coordinates on an open neighborhood of L, which is a manifestation of Darboux s theorem stating that T Y can be covered with symplectic local charts. Lemma 2. Let L be an embedded coisotropic submanifold of T Y \0, with coordinates (x, ξ, e) such that (46) holds. Denote L (y, η) = Ω(x, ξ, e). We can find a homogeneous canonical map G from an open part of T (X E)\0 to an open neighborhood of L in T Y \0, such that G(x, e, ξ, ε = 0) = Ω(x, ξ, e). Proof. The e i, 1 i m n with m = dim Y = 2n c 1, can be viewed as (coordinate) functions on L. We first extend them to functions on the whole T Y \0 such that the Poisson brackets {e i, e j } satisfy {e i, e j } = 0, 1 i, j m n. (47) This can be accomplished successively for e 1,..., e m n by the construction that we describe in the sequel, see for example Treves [28], Chapter 7, the proof of Theorem 3.3. Suppose we have extended e 1,..., e l ; we extend e l+1. In order to satisfy (47), e l+1 has to be a solution u of H ei u = 0, 1 i l, (48) where H ei is the Hamilton field associated with the function e i, with initial condition on some manifold transversal to the H ei. For any (y, η) L the covectors de i, 1 i l restricted to T (y,η) L are linearly independent, so the H ei are transversal to L and they are linearly independent modulo L. So we can supplement (48) with the initial condition u L = e l+1, and even prescribe u on a larger manifold, which leads to nonuniqueness of the extensions e i. We now have m n commuting vector fields H ei that are transversal to L and linearly independent on some open neighborhood of L. The Hamilton systems implied by these fields, with parameters ε i, are y j ε i = e i η j (y, η), η j ε i = e i y j (y, η), 1 i, j m n. Let G(x, e, ξ, ε) be the solution for (y, η) of these Hamilton systems supplemented with initial value (y, η) = Ω(x, ξ, e), with flowout parameters ε. This gives a diffeomorphic map from a neigborhood of the set ε = 0 in T (X E)\0 to a neighborhood of L in T Y \0. One can check from the Hamilton systems that this map is homogeneous.

15 Characterization of seismic reflection data 15 It remains to check the commutation relations. The relations (46) are valid for any ε, because the Hamilton flow conserves the symplectic form on T Y \0. The commutation relations for w εi follow using that (y,η) ε i = H ei. = (y,η) ε i 3.2. A generalized Radon transform used in seismology. The operator F can be identified with a generalized Radon transform. Concerning the oscillatory integral representation of its kernel, the minimum number of phase variables is given by the corank of the projection Dπ : T Λ T (T Y T X) at (x, ξ, e), which is > 1 when s or r is in a caustic point relative to x. Let Λ = Λ\{closed neighborhood of {λ Λ corank D λ π > 1}}. The subset Λ can be described by phase functions of the traveltime form, Φ (m) (s, r, t, x, τ) = τt τt (m) (s, r, x) }{{}}{{} y +S(s,r,x,τ) (49) with the only phase variable τ ( J = 1); see (35)-(36). Here, T (m) is the value of the time variable in y in (35) following from t+ t. Each T (m) is defined on a subset D (m) (s, r, x); note that D (m) R\0 (s, r, x, τ) is a domain for the coordinates on Λ in (36). The associated amplitude is 2iτ b (m) (s, r, x, τ)r (m) (s, r, x, τ) up to leading order (cf. (44)). This amplitude, together with the phase function Φ (m) define the contribution F (m) to F. Changing coordinates, D (m) R\0 (s, r, x, τ) (x, ξ, e), on Λ, we obtain mappings or e (m) = e (m) (s, r, x) and ξ (m) = ξ (m) (s, r, x, τ) ν (m) = ν (m) (s, r, x) = ξ(m) (s, r, x, τ) ξ (m) (s, r, x, τ). The action of the operator F (m) on a distribution in x is associated with an integration over isochrones, T (m) (s, r, x) = t, for given (s, r, t) (cf. (49)). Imaging based on the contribution from F (m) to the data, can be written in the following form, { ((F (m) ) d)(x) = (2π) 1 2 3n 1 c 4 (2iτ b (m) (s, r, x, τ)r (m) (s, r, x, τ) + l.o.t.) Y } exp[ iφ (m) (s, r, t, x, τ)] d(s, r, t) dτ δ(e e (m) (s, r, x)) dsdrdt de.(50) The expression in between braces defines the generalized Radon transform used by Beylkin [1] in acoustic media in the absence of caustics, and later by De Hoop et al. [7] in anisotropic elastic media in the presence of caustics (this generalized Radon transform replaces what is known to seismologists as common-offset Kirchhoff migration where e (m) (s, r, x) = r s): If D (m) R\0 (s, r, x, τ) (x, ξ (m), e (m) ) is locally diffeomorphic, the integration variables (s, r) can be changed to (ν, e) for given x and

16 16 M. V. de Hoop and G. Uhlmann τ. Equation (50) admits an oscillatory integral representation with phase variables (τ, ε) and with phase Ψ (m), Ψ (m) (s, r, t, x, e, τ, ε) = τt τt (m) (s, r, x) + ε, e (m) (s, r, x) e, from which we extract the function S (m) = S (m) (s, r, x, τ, ε) = τt (m) (s, r, x) + ε, e (m) (s, r, x). We introduce the operator F (m), E (X E) D (Y ), defined by F (m) c = (2π) 1 2 3n 1 c 4 (2iτ b (m) (s, r, x, τ)r (m) (s, r, x, τ) + l.o.t.) X exp[iψ (m) (s, r, t, x, e, τ, ε)] c(x) dτdε dxde, identifying c(x) with z n x δ(zn (x)). The amplitude b (m) (s, r, x, τ) will not have the correct decay of derivatives on a conic neighborhood of τ = 0 and ε 0. To ensure that F (m) is a Fourier integral operator, we compose it by an appropriate pseudodifferential cutoff ψ G. Let K denote the closed conic set determined by (σ, ρ) 1 ɛ τ for some small ɛ > 0; let ψ G be a cutoff in S 0 that is 0 on K, while ψ G = 1 on a conic neighborhood of K. Then ψ G F (m) is a Fourier integral operator. Its canonical relation is given by { Λ (m) = (s, r, T (m) (s, r, x), σ, ρ, τ; x, e (m) (s, r, x), ξ, ε) (s, r, x, τ) D (m) R\0, ε R n 1, σ = S(m) s }, ρ = S(m), ξ = S(m) r x T Y \0 T (X E)\0. (51) We suppress ψ G in our notation, write (50) in the form ((F (m) ) d)(x) = (( F (m) ) d)(x, e) de, and use this result in Section An invertible Fourier integral operator. Let M be the canonical relation associated with the map G we constructed in Lemma 2, i.e. M = {(G(x, e, ξ, ε); x, e, ξ, ε)} T Y \0 T (X E)\0. We now construct a Maslov-type phase function for M that is directly related to a phase function for Λ. Suppose (y I, x, η J ) are suitable coordinates for Λ, at ε = 0. For ε small, the constant-ε subset of M allows the same set of coordinates, thus we can use coordinates (y I, η J, x, ε) on M. Now there is (see Theorem 4.21 in Maslov and Fedoriuk [24]) a function S(y I, x, η J, ε), called the generating function, such that M is given by y J = S, η I = S, η J y I ξ = S x, e = S ε. Thus a phase function for M is given by (52) Ψ(y, x, e, η J, ε) = S(y I, x, η J, ε) + η J, y J ε, e. (53)

17 Characterization of seismic reflection data 17 A phase function for Λ then follows as Φ(y, x, η J ) = Ψ(y, x, S ε ε=0, η J, 0) = S(y I, x, η J, 0) + η J, y J (cf. (31)). We introduce the amplitude b(y I, x, η J, ε) on M such that b(y I, x, η J, ε = 0) is as given in Theorem 1. To leading order, ε b = 0 because the coordinates ε i are in involution. We construct a mapping from the reflectivity function to seismic data. This is done by applying the results of Section 3.1 to the Kirchhoff modeling formula (44), and its equivalent under the Born approximation (42). We apply the change of coordinates on Λ from (y I, x, η J ) to (x, ξ, e) to the symbol R MN and write now R MN = R(x, ξ, e). To highest order, R does not depend on ξ and is simply a function of (x, e). Theorem 2. [26] Suppose microlocally that Assumptions 3 (no grazing rays at any interface), 1 (no scattering over π), 2 (transversality), and 4 (Bolker condition) are satisfied. Let H be the Fourier integral operator, H : E (X E) D (Y ), with canonical relation given by the extended map G : (x, ξ, e, ε) (y, η) constructed in Section 3.1, and with amplitude to highest order given by (2π) n/2 2iτ(η J )b(y I, x, η J, ε) expressible in terms of the coordinates (x, e, ξ, ε). Then the data, in both Born and Kirchhoff approximations, can be modeled by H acting on a distribution r(x, e) of the form r(x, e) = R(x, D x, e) c(x), (54) where R stands for a smooth e-family of pseudodifferential operators and c E (X). For the Kirchhoff approximation the distribution c equals zn x δ(z n(x)), while the principal symbol of the pseudodifferential operator R equals R(x, e), so to highest order r(x, e) = R(x, e) z n x δ(z n(x)). (55) For the Born approximation the function r(x, e) is given by a pseudodifferential operator R with principal symbol (2iτ(x, ξ, e)) ) 1 (w MN;0 (x, ξ, e), w MN;ijkl (x, ξ, e)), act-, so to highest order ( δcijkl ing on a distribution c given by ρ, δρ ρ [ r(x, e) = (2iτ(x, D x, e)) 1 w MN;0 (x, D x, e) δρ(x) ρ(x) +w MN;ijkl(x, D x, e) δc ijkl(x) ρ(x) see (30). The operator H is invertible. Remark 3. Microlocally, we have obtained the following diagram (suggested by Symes, personal communication) E (X E) R(x, D x, e) E (X) H D (Y ) Id F D (Y ) We note that R(x, D x, e) is of order 0. H 1 maps data into what seismologists call common-image-point gathers (the integral over ε replaces the notion of beamforming; e plays the role of scattering angle and azimuth). (56) ],

18 18 M. V. de Hoop and G. Uhlmann 4. A procedure for data continuation 4.1. The range of the scattering operator. If n 1 c > 0, there is a redundancy in the data parametrized by the variable e. The redundancy in the data manifests itself as a redundancy in images of the subsurface from these data. A smooth background (Subsection 2.3) is considered acceptable if the data are contained in the range of F (or H). If a smooth background is acceptable, then applying the operator H 1 of Theorem 2 to the data results in a reflectivity distribution r(x, e), the singular support (in x) of which does not depend on e. One way to measure the agreement in singular supports between images of reflectivity r(x, e) parametrized by e is by taking a derivative with respect to e. Taking (54) as the point of departure, we find that ( R(x, D x, e) e R e (x, D x, e) ) r(x, e) = [ R(x, D x, e), R ] e (x, D x, e) c(x). Hence, microlocally where R(x, D x, e) is elliptic, ( R(x, D x, e) e R [ e (x, D x, e) R(x, D x, e), R ] ) e (x, D x, e) R(x, D x, e) 1 (57) r(x, e) = 0 (58) to all orders. We observe that the first operator acting on distributions in (x, e) in the sum is of order 1, the second operator is of order 0, while the third operator is of order 1. Falling back on (55) we exploit that, up to leading order, the operator R acts as a multiplication by R(x, e). Clearly, [ R(x, e), R (x, e) e ] z n x δ(z n(x)) = 0 cf. (57). Substituting (54) into (58) reveals that the operator in between parentheses on the left-hand side equals (R(x, e) e R e (x, e)) up to the leading two orders. Hence, ( R(x, e) e R ) (x, e) r(x, e) = 0 (59) e up to the highest two orders. Conjugating the operator in between parentheses in (59), or in (58), with the invertible Fourier integral operator H, we obtain a pseudodifferential operator on D (Y ) [26] Lemma 3. Let the pseudodifferential operators P i (y, D y ) : D (Y ) D (Y ) of order 1 be given by the composition ( P i (y, D y ) = H R(x, e) R ) (x, e) H 1, i = 1,..., n 1 c e i e i }{{}. r Then for Kirchhoff data d(y) modeled by F, we have to the highest two orders P i (y, D y )d(y) = 0, i = 1,..., n 1 c. (60) Microlocally, for values of e where R(x, e) 0, the operator P i (y, D y ) can be modeled after (58) such that (60) is valid to all orders.

19 Characterization of seismic reflection data 19 The principal part of the symbol of P i is denoted by p i, while the next order term in the symbol s polyhomogeneous expansion is denoted by p i;0. The subprincipal symbols (which show up naturally in the Weyl calculus of symbols), c i, of the annihilators are then given by c i := p i;0 + i 2 j 2 p i y j η j. Remark 4. The wavefront set of the data is contained in L = π Y (Λ), which, in analogy with the eikonal equation, is also the submanifold of T Y \0 defined by p i (y, η) = 0, i = 1,..., n 1 c, (61) where p i is the principal symbol of P i as before, and is of codimension 2 [2(n 1) c + 1] (3n 1 c) = n 1 c, which is also the dimension of the covector ε. The operator F in Corollary 1 is continuous H n 1 2 (X) L 2 (Y ). We now define the operator projection, π : L 2 (Y ) L 2 (Y ), onto the range of the scattering operator F. Microlocally, π 2 = π. Since Assumption 4 is satisfied, using [16, Proposition 8.3], π is an elliptic minimal projector. By [16, Theorem 6.6], the kernel of P = P P 2 n 1 c (62) is identical with the range of π. More precisely, let Q denote the global parametrix of P, then, by [16, Theorem 6.7], π = I QP + smoothing operator. (63) 4.2. A global parametrix. The construction of a global parametrix, Q, for an operator of the type P is given by Guillemin and Uhlmann [17]. Let Λ 0 and Λ 1 denote Lagrangian submanifolds of T Y \0 T Y \0 intersecting cleanly in a submanifold of given codimension. Guillemin and Uhlmann s construction relies on the introduction of the space of distributional half densities, I p,l (Y Y ; Λ 0, Λ 1 ), defining a class of Fourier integral operators with singular symbols, with the properties l I p,l (Y Y ; Λ 0, Λ 1 ) = I p (Y Y, Λ 1 ) (defining standard Fourier integral operators with canonical relation Λ 1 ) for p fixed, and p I p,l (Y Y ; Λ 0, Λ 1 ) = C 0 (Y Y ) for l fixed. In our case, Λ 0 is the diagonal diag(t Y \0) in T Y \0 T Y \0, while Λ 1 is the fiber product L X L. Viewing the Schwartz kernel of the identity (I) as an element of I 0,0 (Y Y ; Λ 0, Λ 1 ), Guilleman and Uhlmann s recursive construction results in QP = I π + R, where the kernel of π belongs to l I p,l (Y Y ; Λ 0, Λ 1 ), and R is a smoothing operator with kernel in p I p,l (Y Y ; Λ 0, Λ 1 ). Here, we discuss the properties of Q. The Lagrangian submanifold Λ 1 T Y \0 T Y \0 precisely consists of points on the joint flowout from diag(t Y \0) (Λ 0 Λ 1 ) by H p1,..., H pn 1 c, where H pi denotes the Hamiltonian field associated with the function p i. The flowout is described by the solution to the Hamilton systems with parameters e i, y j e i = p i η j (y, η), η j e i = p i y j (y, η), 1 i, j n 1 c. (64) The Lagrangian submanifolds Λ 0 and Λ 1 intersect cleanly in a submanifold of codimension n 1 c, see Remark 4. We observe that Λ 1 Λ 1 = Λ 1. The elliptic minimal

20 20 M. V. de Hoop and G. Uhlmann projector π, introduced in the previous subsection, is a Fourier integral operator with canonical relation Λ 1, L (64) L (65) X X The wavefront set of Q is contained in Λ 0 Λ 1. Q is a pseudodifferential operator 1 on Λ 0 \(Λ 0 Λ 1 ) and its principal symbol there is given by σ 0 = up to p p2 n 1 c Maslov factors and half densities. Q is a Fourier integral operator on Λ 1 \(Λ 0 Λ 1 ). Its principal symbol, σ 1, solves the transport equation n 1 c i=1 (ih pi c i ) 2 σ 1 = σ π on Λ 1 \(Λ 0 Λ 1 ), where σ π denotes the symbol of π. (We return to the evaluation of σ π in Section 6.) The expression between parentheses is an elliptic differential operator of order 2 on each fiber of L. The equation is Laplace s equation in every leaf of the foliation generated by the commuting vector fields H pi, i = 1,..., n 1 c. The principal symbol σ 1 has a conormal singularity at Λ 0 Λ 1, which is the Fourier transform of the singularity of σ 0, see [17, (5.14)] Data continuation. We apply the results of the previous subsections to the problem of source-receiver continuation of seismic data: Seismic data are commonly measured on an open subset of the manifold of all possible observations. Continuation of these data from the open subset to the full acquisition manifold is desired for various data processing procedures, including imaging in the seismic literature this continuation is referred to as the forward extrapolation step within data regularization [11], or data healing. Theorem 3. Suppose u is a distribution belonging to the range of the scattering operator F. Let χ = χ(y, D y ) be a pseudodifferential operator of order 0 that acts as a cutoff in phase space T Y \0. Assume that we observe u 0 = χu in accordance with the constraints of the acquisition geometry. Suppose χ is elliptic on a leaf of the foliation of L, then WF(πu 0 ) intersected with this leaf is equal to WF(u) intersected with the same leaf. In this case, π heals the data on this leaf. Proof. We observe that u = πv for some v. Then Because π 2 u = u, it is natural to investigate πu 0, i.e. u 0 = (χπ) v. (66) πu 0 = (πχπ) v. (67) If χ is elliptic on a leaf of the foliation of L, then (π πχπ) v = 0, or u πu 0 = 0, microlocally on this leaf. This implies the statement in the theorem. Remark 5. The transformation to zero offset (TZO) of seismic data, which is derived from Dip MoveOut, can be expressed in the form R 0 π, where R 0 is the restriction of distributions on Y to an acquisition manifold with coinciding sources and receivers: In this case, y = (y, y ) with y = ( 1 2 (s + r), t) and y = 1 2 (r s) if (s, r, t) are the original local coordinates on Y. Assumption 2, subject to this substitution, guarantees that the composition, R 0 π, is again a Fourier integral operator.

21 Characterization of seismic reflection data Goldin s equation revisited In context of the simplest seismic scattering theory, in a background that essentially is constant, the following simplications are made. To begin with, the source, s, and receiver, r, points in Y are assumed to be contained in a flat surface, R n 1, while e is initially replaced by half source-receiver offset h = 1 2 (r s) Rn 1. Thus y is replaced by (s, r, t); we write η = (σ, ρ, τ). Essentially, we assume that the rays between reflector and acquisition surface are straight, see figure 1. We repeat the NMO correction, κ : (s, r, t) (z, t n, h), z = 1 2 (r + s), h = 1 2 (r s), t n = t 2 4h2 v 2, of the introduction. (The subscript n refers in this section to normal moveout.) Here, v could be thought of as the so-called NMO velocity, which can be introduced for pure mode scattering (i.e. M = N), even in the anisotropic media under consideration here [14]; Goldin, however, restricts his analysis to an isotropic medium and compressional waves. We will reproduce Goldin s result here in the context of our analysis subject to the substitutions n = 2 and c = 0 (and m = 1). NMO correction applied to the data yields (κ 1 ) d. Including a so-called geometrical spreading correction, a multiplication by time t, then leads to the map d(s, r, t) ((κ 1 ) (t d))(z, t n, h) (68) that replaces (H 1 d)(x, e); the point x has attained coordinates (z, t n ). The outcome is of the form r(z, t n, h) = R n (z, h) c n (t n T n (z, h)). (69) Equation (69) replaces (55). The reflection time, T (s, r, x) (there is only one branch, cf. (49)), maps under κ according to T (s, r, x) T n (z, h) = (T (z h, z + h, x)) 2 4h2 v 2. We observe that, in the simplication considered, R n is independent of t n, while c n is not only a function of (z, t n ) but also of h. Hence, a simple derivative of r with respect to h, motivated by (59), would not yield a vanishing outcome up to leading order. Instead, it is possible to construct a candidate operator P 1, acting on the data, directly. In figure 1 we introduce angles α and γ; in fact, (x, α, γ) can be identified with (x, ν, e) in Section 3.2. Using simple trigonometric identities (including the law of sines) and the geometry in figure 1 (observing that the total length of the reflected ray is vt = (r s) cos α sin γ with 2γ denoting the scattering angle and α denoting the incidence angle of the zero-offset ray at the surface), it follows that p 1 (s, r, t, σ, ρ, τ) = (t 2 + (r s)2 v 2 ) ( τ ) (σ ρ) 2 (r s) t v 2 τ 1 σρ = 0 (70) defines the points in L (Remark 4) in the simplification under consideration; p 1 is v dependent. Applying the coordinate transformation implied by κ to this symbol, and multiplying the result by frequency τ, yields p (z, t n, h, ζ, τ n, ε) = (τp 1 )(κ(s, r, t), ((κ ) 1 ) t (σ, ρ, τ)) (71)

22 22 M. V. de Hoop and G. Uhlmann or p (z, t n, h, ζ, τ n, ε) = t n τ n ε + h (ζ 2 ε 2 ), (72) which defines the principal symbol of an operator P ; we observe that p is v independent. We recover Goldin s equation, P (z, t n, h, D z, D tn, D h )r = 0, P 2 ( ) 2 (z, t n, h, D z, D tn, D h ) = t n t n h h z 2 2 h 2, (73) which is valid up to highest order. It would be valid up to the next order if we had not applied the geometrical spreading correction in (68). Accounting for this correction leads to a subprincipal symbol contribution: P (z, t n, h, D z, D tn, D h ) := t n 2 ( 2 t n h h z 2 2 h 2 ) h. Through the coordinate transformation implied by κ, we obtain a subprincipal symbol contribution to the operator with principal symbol p 1 (cf. (70)). Note that the operator P is of second order unlike the operator annihilating r in (59) which is of first order. However, in (71) we introduced a multiplication by τ, raising the order by one. A firstorder operator derived from P in (73) follows to be P (z, t n, h, D z, D tn, D h ) = ( ) 1 h h t n ( ) 2 h 2 2 z 2 +. t n We write down the Hamiltonian system describing the flowout as in (64); there is only one such system since n c 1 = 1. We use the first-order symbol, p (z, t n, h, ζ, τ n, ε) = τ n h εt n (ζ 2 ε 2 ) (we omitted a factor i), so that z e = 2hζ εt n, ζ e = 0, t n e = 1, τ n e = h εt 2 (ζ 2 ε 2 ), n (74) h e = h ζ2 ε 2 + 2h, t n t n ε e = 1 (ζ 2 ε 2 ). εt n We set e = t n, and eliminate τ n. To this end, we introduce the slowness vectors ζ and ε according to ζ = τ n ζ and ε = τn ε. Substituting τ n = h εt n (ζ 2 ε 2 ) (using that p = 0), and the equation for τn e, then yields the system z = 2h ζ, t n ε t n h t n = 1 ε + h t n, ζ t n = ζ t n, ε = ζ 2. t n ε t n (75)

23 Characterization of seismic reflection data 23 We note that t n and h are directly related to one another. Indeed, let the zero-offset reflecton time be given by T 0 = T (z 0, z 0, x) = T n (z 0, 0). Then, for given z, T n h = T 0 (h 2 (z z 0 ) 2 ). 1/2 Along bicharacteristics, h = v2 ε t n 4 sin 2 (76) α is invariant (i.e. its derivative with respect to e is zero). We can convert t n to half scattering angle γ keeping (x, α) fixed according to the relation T n cos α cos γ = T 0 (cos 2 α sin 2 γ), T 1/2 0 = 2V v. (77) We discuss in as much the kernel of P determines the range of F under the simplication ( straight rays ) under consideration. The range is described by wavefields of the form ( ) 1 cos2 γ + V C d(s, r, t) = vt r(x, γ) δ(t T ), T = T (s, r, x), (78) cos γ obtained after preprocessing d for time signature (a 2.5D correction) and source or receiver radiation characteristics, applying an appropriate pseudodifferential operator. In (78), V = 1 2vT (z, z, x) denotes the length of the zero-offset ray, and C = x 2 (x 1) cos 3 α denotes the curvature of the reflector, see figure 1. We have assumed that the reflecting interface can be described by the graph (x 1, x 2 (x 1 )). (If the interface is the zero level set of z 2 = z 2 (x 1, x 2 ) then we assume that z2 x 2 0.) Equation (78) is the outcome of a stationary phase calculation of the scattered field in the Kirchhoff approximation. We set r(x, γ) 1 and apply (κ 1 ) to d(s, r, t) and obtain r(z, t n, h) = A n (z, h) δ(t n T n ), T n = T n (z, h), A n = ( vt cos2 γ + V C cos γ ) 1 t t n, (79) because dtn dt = t t n. Expression (79) is indeed of the form (69). To verify whether the wavefields in (78), via (79), coincide with functions in the kernel of P, up to leading order, we first notice that by derivation p T (z, T n, h, iτ n n z, τ n, T iτ n n h ) = 0. Secondly, we consider the transport equation derived from (73), which is given by ( T n 2h T ) ( n An h h + 2h T n A n 2 ) z z + ha T n n z 2 2 T n h 2 = 0. The velocity vector associated with a ray or characteristic is given by ( Tn along a characteristic, the transport equation becomes 1 ( ) 1 ( da n T n 2 T n + h T n A n dt n h z 2 z, Tn h ). Thus, ) 2 T n h 2 = 0, (80)

24 24 M. V. de Hoop and G. Uhlmann x 2 x 1 s σ T = t + ˆt α r ρ vt 0 = D v t vˆt γ x C x 2 (x 1 ) Fig. 1. Geometry underlying the annihilator symbol for constant coefficients. where we made use of (75). We change variables according to (77), with 1 dt n T n dγ = sin 2 α sin γ (cos 2 α sin 2 γ) cos γ. Furthermore, using the ray geometry, we find the identity ( 2 ) ( ) T n T n z 2 2 T n h 2 = 4 T 2 T s r + cos2 γ v 2 = 4 cos2 γ v 2 sin 2 α + V C cos 2 γ + V C. (81) Substituting identity (81) and invariant (76) into (80), applying the change of variables (77), leads to the equation, 1 ( ) da n A n dγ + 1 (cos 2 α sin 2 γ) 1 cos 2 sin γ cos γ = 0. (82) γ + V C This equation can be directly integrated to yield solutions for A n of the type (79). We conclude that the kernel of P generates wavefields of the type (78) which comprise the range of the scattering operator subject to processing for time signature and setting r(x, γ) Numerical example Since the minimal elliptic projector π is a Fourier integral operator it is directly implementable and applicable to data. This is the subject of this section.

25 Characterization of seismic reflection data The canonical relation of the minimal elliptic projector. In view of the Bolker condition, Assumption 4, the composition F F is an elliptic pseudodifferential operator of order n 1. Let Ξ denote the parametrix for F F. The operator F Ξ F belongs to Guillemin and Sternberg s algebra R L [16] of Fourier integral operators with canonical relation Λ 1 [6]. Clearly, (F Ξ F ) 2 = F Ξ F microlocally, while I Ξ F F = I F F Ξ is the orthogonal projection onto the kernel of F F. Indeed, F Ξ F is precisely an elliptic minimal projector [16, proof of Thm. 8.3] of the type introduced in Section 4. The symbol of this operator follows by the standard composition calculus. Following the composition of F with F in F Ξ F, we represent the canonical relation Λ 1 as the composition of canonical relations Λ with Λ. In seismic processing one uses (50) and implements F Ξ F u 0 = F Ξ F u 0 de, cf. Theorem 3. Remark 6. The significance of the insights in this subsection and Section 4 is the following. In this subsection we showed that, given a smooth background model, we can construct a minimal elliptic projector for data continuation by operator composition, F Ξ F. On the other hand, in Section 4 we showed that, given the annihilators of the data (in practice, just their principal parts), we can construct the global parametrix Q, from which the elliptic minimal projector follows. This procedure is related to what Guillemin and Sternberg call relative geometrical quantization Data continuation. In our example, n = 2 and χ is replace by a smooth cutoff ψ Y ; the cutoff restricts the data to the set {(s, r, t) s, r R n 1, r s > h 0, t (0, T )}. The goal is to continue the data to an acquisition manifold with the constraint r s > h 0 removed. Elastic-wave data were simulated over a model illustrated in figure 2 (top). The P-wave velocities are shown in grey scale; a low-velocity Gaussian lens was inserted (white-to-grey). The continuation is illustrated for the P-wave constituents even though the simulated data contained S waves as well: By selecting the vertical displacement component, most energy in the wavefield can be attributed to P waves. For one value of s the synthetic data as a function of r and t are shown in figure 2 (bottom, right). We set T = 3s and h 0 = 500m. The input to the continuation (u 0 ) were the data with r values in between the black vertical lines removed figure 2 (bottom, left). The result of the continuation is plotted in between the black vertical lines of the same figure and should be compared with figure 2 (bottom, right). The algorithm used is designed and explained in [23]. References 1. G. Beylkin. The inversion problem and applications of the generalized Radon transform. Comm. Pure Appl. Math., XXXVII: , B. Biondi, S. Fomel, and N. Chemingui. Azimuth moveout for 3-D prestack imaging. Geophysics, 63: , G. Bolondi, E. Loinger, and F. Rocca. Offset continuation of seismic sections. Geoph. Prosp., 30: , M.V. De Hoop and N. Bleistein. Generalized radon transform inversions for reflectivity in anisotropic elastic media. Inverse Problems, 13: , M.V. De Hoop and S. Brandsberg-Dahl. Maslov asymptotic extension of generalized Radon transform inversion in anisotropic elastic media: A least-squares approach. Inverse Problems, 16: , M.V. De Hoop, A.E. Malcolm, and J.H. Le Rousseau. Seismic wavefield continuation in the single scattering approximation: A framework for Dip and Azimuth MoveOut. Can. Appl. Math. Q., 10: , 2002.

26 26 M. V. de Hoop and G. Uhlmann depth (km) 0 2 distance (km) s r: geophone position (m) 5000 r: geophone position (m) time (s) t: t: time (s) Fig. 2. A numerical example of data continuation: The top figure is the isotropic P-wave velocity model used in the reflection data simulation, containing a Gaussian (low velocity) lens (in white). The bottom two figures both show a shot record (receiver location versus time) with source location at the black vertical line in top figure; the left record shows the outcome of continuation (the data in between the two vertical lines was missing) while the right record shows the original simulated data. 7. M.V. De Hoop, C. Spencer, and R. Burridge. The resolving power of seismic amplitude data: An anisotropic inversion/migration approach. Geophysics, 64: , N. Dencker. On the propagation of polarization sets for systems of real principal type. Journal of Functional Analysis, 46: , S.G. Deregowski and F. Rocca. Geometrical optics and wave theory of constant offset sections in layered media. Geoph. Prosp., 29: , J. J. Duistermaat. Fourier integral operators. Birkh äuser, Boston, S. Fomel. Theory of differential offset continuation. Geophysics, 68: , I.M. Gel fand and M.I. Graev. Complexes of straight lines in the space C n. Functional Analysis Appl., 2:39 52, S. Goldin. Superposition and continuation of transformations used in seismic migration. Russian Geology and Geophysics, 35: , V. Grechka, I. Tsvankin, and J.K. Cohen. Generalized Dix equation and analytic treatment of normalmoveout velocity for anisotropic media. Geoph. Prosp., 47: , V. Guillemin. Pseudodifferential operators and applications (Notre Dame, Ind., 1984), chapter On some results of Gel fand in integral geometry, pages Amer. Math. Soc., Providence, RI, V. Guillemin and S. Sternberg. Some problems in integral geometry and some related problems in microlocal analysis. Amer. J. of Math., 101: , V. Guillemin and G. Uhlmann. Oscillatory integrals with singular symbols. Duke Math. J., 48: , S. Hansen. Solution of a hyperbolic inverse problem by linearization. Communications in Partial Differential Equations, 16: , 1991.