propagator matrices and marchenko-type focusing arxiv:2110

Wave-field representations with Green’s functions,

propagator matrices and Marchenko-type focusing

functions

Kees Wapenaar

Department of Geoscience and Engineering, Delft University of Technology,

Stevinweg 1, 2628 CN Delft, The Netherlands

SUMMARY

Classical acoustic wave-field representations consist of volume and boundary integrals, of which

the integrands contain specific combinations of Green’s functions, source distributions and wave

fields. Using a unified matrix-vector wave equation for different wave phenomena, these represen-

tations can be reformulated in terms of Green’s matrices, source vectors and wave vectors. The

matrix-vector formalism also allows the formulation of representations in which propagator matri-

ces replace the Green’s matrices. These propagator matrices, in turn, can be expressed in terms of

Marchenko-type focusing functions. An advantage of the representations with propagator matrices

and focusing functions is that the boundary integrals in these representations are limited to a single

open boundary. This makes these representations a suited basis for developing advanced inverse

scattering, imaging and monitoring methods for wave fields acquired on a single boundary.

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2 Wapenaar

1 INTRODUCTION

The aim of this paper is to give a systematic treatment of different types of wave-field represen-

tation (with Green’s functions, propagator matrices and Marchenko-type focusing functions),

to discuss their mutual relations, and indicate some new applications.

• Representations with Green’s functions. A Green’s function is the response of a medium

to an impulsive point source. It is named after George Green, who, in a privately published

essay (Green 1828), introduced the use of impulse responses in field representations (Challis &

Sheard 2003). Wave-field representations with Green’s functions have been formulated, among

others, for optics (Born & Wolf 1965), acoustics (Rayleigh 1878; Bleistein 1984), elastodynam-

ics (Knopoff 1956; de Hoop 1958; Gangi 1970; Pao & Varatharajulu 1976) and electromagnet-

ics (Kong 1986; Altman & Suchy 1991; de Hoop 1995). They find numerous applications in

forward modeling problems (Hilterman 1970; Frazer & Sen 1985; Mansuripur 2021), inverse

source problems (Porter & Devaney 1982; de Hoop 1995), inverse scattering problems (De-

vaney 1982; Bojarski 1983; Bleistein 1984; Oristaglio 1989), imaging (Porter 1970; Schneider

1978; Berkhout 1982; Maynard et al. 1985; Esmersoy & Oristaglio 1988; Lindsey & Braun

2004), time-reversal acoustics (Fink & Prada 2001), and Green’s function retrieval from am-

bient noise (Derode et al. 2003; Wapenaar 2003; Weaver & Lobkis 2004).

• Representations with propagator matrices. In elastodynamic wave theory, a matrix for-

malism has been introduced to describe the propagation of waves in laterally invariant layered

media (Thomson 1950; Haskell 1953) . This formalism was refined by Gilbert & Backus (1966),

who coined the name propagator matrix. In essence, a propagator matrix ‘propagates’ a wave

field (represented as a vectorial quantity) from one plane in space to another. Using pertur-

bation theory, the connection between wave-field representations with Green’s functions and

the propagator matrix formalism was discussed (Kennett 1972a). The propagator matrix has

been extended for laterally varying isotropic and anisotropic layered media (Kennett 1972b;

Woodhouse 1974). Propagation invariants for laterally varying layered media have been in-

troduced and the propagator matrix concept has been proposed for the modeling of reflection

and transmission responses (Haines 1988; Kennett et al. 1990; Koketsu et al. 1991; Takenaka

Wave-field representations 3

et al. 1993). The propagator matrix has also been used in a seismic imaging method that

accounts for multiple scattering in a model-driven way (Wapenaar et al. 1987).

• Representations with Marchenko-type focusing functions. Building on a 1D acoustic aut-

ofocusing method, it has been shown that the wave field inside a laterally invariant layered

medium can be retrieved with the Marchenko method from the single-sided reflection response

at the surface of the medium (Rose 2001, 2002; Broggini & Snieder 2012). This concept was

extended to a 3D wave-field retrieval method for laterally varying media (Wapenaar et al.

2014). Central in the 3D Marchenko method are wave-field representations containing focus-

ing functions. These representations have found applications in imaging methods (Ravasi et al.

2016; Staring et al. 2018; Jia et al. 2018) and inverse source problems (Van der Neut et al.

2017) that account for multiple scattering in a data-driven way.

The setup of this paper is as follows. In section 2 we briefly review the matrix-vector

wave equation for laterally varying media (Kennett 1972b; Woodhouse 1974), generalized

for different wave phenomena, and we briefly discuss the concept of the Green’s matrix, the

propagator matrix and the Marchenko-type focusing function. The symmetry properties of

the matrix-vector wave equation allow the formulation of unified matrix-vector wave-field

reciprocity theorems (Wapenaar 1996; Haines & de Hoop 1996), which are reviewed in section

3. These reciprocity theorems form the basis for a systematic treatment of the different types

of wave-field representation mentioned above. Traditionally, a wave-field representation is

obtained by replacing one the states in a reciprocity theorem by a Green’s state. In section

4 we follow this approach for the matrix-vector reciprocity theorems. By replacing one of the

wave-field vectors by the Green’s matrix (and the source vector by a unit source matrix) we

obtain wave-field representations with Green’s matrices. Analogous to this, in section 5 we

replace one of the wave-field vectors in the reciprocity theorems by the propagator matrix

and thus obtain wave-field representations with propagator matrices. In section 6 we discuss

a mixed form, by replacing one of the wave-field vectors by the Green’s matrix and the

other by the propagator matrix. In section 7 we discuss the relation between the propagator

matrix and the Marchenko-type focusing functions and use this relation to derive wave-field

representations with focusing functions. We end with conclusions in section 8.

4 Wapenaar

2 THE UNIFIED MATRIX-VECTOR WAVE EQUATION, THE GREEN’S

MATRIX, THE PROPAGATOR MATRIX AND THE FOCUSING

FUNCTION

2.1 The matrix-vector wave equation

We use a unified matrix-vector wave equation as the basis for the derivations in this paper.

In the space-frequency domain it has the following form (Gilbert & Backus 1966; Kennett

1972a,b; Woodhouse 1974; Haines 1988)

∂3q−Aq = d, (1)

with

q =

q1

q2

, d =

d1

d2

, A =

A11 A12

A21 A22

. (2)

Here q(x, ω) is a space- and frequency-dependent N×1 wave-field vector, where x denotes the

Cartesian coordinate vector (x1, x2, x3) (with the positive x3-axis pointing downward) and ω

is the angular frequency. The N/2 × 1 sub-vectors q1(x, ω) and q2(x, ω) contain wave-field

quantities, which are specified for different wave phenomena in Table 1. Operator ∂3 stands for

the differential operator ∂/∂x3. Matrix A(x, ω) is an N ×N operator matrix; it contains the

space- and frequency-dependent anisotropic medium parameters and the horizontal differential

operators ∂1 and ∂2. Definitions of this operator matrix for different wave phenomena can

be found in many of the references mentioned in the introduction. N × 1 vector d(x, ω)

contains the space- and frequency-dependent source functions. A comprehensive overview of

the operator matrices and source vectors for the wave phenomena considered in Table 1 (with

some minor modifications) is given by Wapenaar (2019).

For all wave phenomena considered in Table 1, operator A obeys the following symmetry

properties

AtN = −NA, (3)

A†K = −KA, (4)

A∗J = JA, (5)


with

N =

O I

−I O

, K =

O I

I O

, J =

I O

O −I

, (6)

where O and I are N/2 × N/2 zero and identity matrices. Superscript t denotes transposition

(meaning that the matrix is transposed and the operators in the matrix are also transposed,

with ∂t1 = −∂1 and ∂t2 = −∂2), ∗ denotes complex conjugation, and † transposition and com-

plex conjugation. In general, the medium parameters in A are complex-valued and frequency-

dependent, accounting for losses. The bar above a quantity means that this quantity is defined

in the adjoint medium. Hence, when A is defined in a lossy medium, then A is defined in

an effectual medium and vice versa (a wave propagating through an effectual medium gains

energy (Bojarski 1983; de Hoop 1988)). For lossless media the bar can be dropped. For all

wave phenomena considered in Table 1, the power-flux density j in the x3-direction is related

to the sub-vectors q1 and q2 according to

j =1

4(q†1q2 + q†2q1). (7)

As a special case we consider acoustic waves in an inhomogeneous medium with complex-

valued and frequency-dependent compressibility κ(x, ω) and mass density ρjk(x, ω). The latter

is defined as a tensor, to account for effective anisotropy, for example due to fine layering

at the micro scale (Schoenberg & Sen 1983). The mass density tensor is symmetric, that

is, ρjk(x, ω) = ρkj(x, ω). We introduce the inverse of the mass density tensor, the specific

volume tensor ϑij(x, ω), via ϑijρjk = δik. Einstein’s summation convention holds for repeated

subscripts (unless otherwise noted); Latin subscripts run from 1 to 3 and Greek subscripts

from 1 to 2. For the acoustic situation the vectors and matrix in equation (1) are given by

q =

p

v3

, d =

ϑ−133 ϑ3ifi

1iω∂α(bαβfβ) + q

, (8)

A =

−ϑ−133 ϑ3β∂β iωϑ−133

iωκ− 1iω∂αbαβ∂β −∂αϑα3ϑ−133

, (9)

6 Wapenaar

Table 1. Wave-field sub-vectors q1(x, ω) and q2(x, ω) for different wave phenomena. For acousticwaves, p and v3 stand for the acoustic pressure and the vertical component of the particle velocity,respectively. For electromagnetic waves, Eα and Hα (α = 1, 2) are the electric and magnetic fieldstrength components, respectively. For elastodynamic waves, vk and τk3 (k = 1, 2, 3) are the particlevelocity and traction components, respectively. The same quantities appear in the vectors for the otherwave phenomena, where, for poroelastodynamic and seismoelectric waves, the quantities are averagedin the bulk, fluid or solid, as indicated by the superscripts b, f and s, respectively. Finally, φ denotesthe porosity.

N q1 q2

Acoustic 2 p v3

Electromagnetic 4 E0 =

(E1E2

)H0 =

(H2−H1

)

Elastodynamic 6 v =

(v1v2v3

)−τ 3 = −

(τ13τ23τ33

)

Poroelastodynamic 8

(vs

φ(vf3 − vs3)

) (−τ b3pf

)Piezoelectric 10

(v

H0

) (−τ 3E0

)

Seismoelectric 12

(vs

φ(vf3 − vs3)H0

) (−τ b3pf

E0

)

with

bαβ = ϑαβ − ϑα3ϑ−133 ϑ3β, (10)

where i is the imaginary unit and q(x, ω) and fi(x, ω) are sources in terms of volume-injection

rate density and external force density, respectively.

Finally, for an isotropic medium, using ϑij = ρ−1δij , we obtain for the source vector and

operator matrix (Corones 1975; Ursin 1983; Fishman & McCoy 1984; Wapenaar & Berkhout

1989; de Hoop 1996)

d =

f3

1iω∂α(1ρfα) + q

,A =

0 iωρ

iωκ− 1iω∂α

1ρ∂α 0

. (11)

2.2 The Green’s matrix

In the space-time domain, a Green’s function is the response to an impulsive point source,

with the impulse defined as δ(t). The Fourier transform of δ(t) equals 1, hence, in the space-

frequency domain the Green’s function is the response to a point source with unit amplitude


for all frequencies. We introduce the N × N Green’s matrix G(x,xA, ω) for an arbitrary

inhomogeneous anisotropic medium as the solution of

∂3G−AG = Iδ(x− xA), (12)

where xA = (x1,A, x2,A, x3,A) defines the position of the point source and I is a N×N identity

matrix. Here I has a size different from that in equation (6). For simplicity we use one notation

for differently sized identity matrices (the size always follows from the context). Also for the

zero matrix O we use a single notation for differently sized matrices. Similar to operator

matrix A, the Green’s matrix is partitioned as

G(x,xA, ω) =

G11 G12

G21 G22

(x,xA, ω). (13)

Equation (12) does not have a unique solution. To make the solution unique, we demand that

the time-domain Green’s function G(x,xA, t) is causal, hence

G(x,xA, t < 0) = O. (14)

This condition implies that G is outward propagating for |x− xA| → ∞.

The simplest representation involving the Green’s matrix is obtained when q and G reside

in the same medium throughout space and both are outward propagating for |x− xA| → ∞.

Whereas q(x, ω) is the response to a source distribution d(x, ω) (equation (1)), G(x,xA, ω)

is the response to a point source Iδ(x − xA) for an arbitray source position xA (equation

(12)). Since both equations are linear, q(x, ω) follows by applying the superposition principle,

according to

q(x, ω) =

∫R3

G(x,xA, ω)d(xA, ω)dxA, (15)

where R is the set of real numbers. This representation is a special case of more general

representations with Green’s matrices, derived in a more formal way in section 4.

Next, we discuss the 2× 2 acoustic Green’s matrix as a special case of the N ×N Green’s

8 Wapenaar

matrix. For this situation G is partitioned as

G(x,xA, ω) =

Gp,f Gp,q

Gv,f Gv,q

(x,xA, ω). (16)

Here the first superscript (p or v) refers to the observed wave-field quantity at x (acoustic

pressure or vertical component of particle velocity), whereas the second superscript (f or q)

refers to the source type at xA (vertical component of force or volume injection rate). Note that

the upper-right element, Gp,q(x,xA, ω), corresponds to the common scalar Green’s function.

For an isotropic medium, the other elements can be expressed in terms of the upper-right

element, as follows

Gv,q(x,xA, ω) =1

iωρ(x, ω)∂3G

p,q(x,xA, ω), (17)

Gp,f (x,xA, ω) = − 1

iωρ(xA, ω)∂3,AG

p,q(x,xA, ω), (18)

Gv,f (x,xA, ω) =1

iωρ(x, ω)

(∂3G

p,f (x,xA, ω)− δ(x− xA)). (19)

Here ∂3,A stands for differentiation with respect to the source coordinate x3,A. Equations

(17) and (19) follow directly from equations (11), (12) and (16). Equation (18) follows from

equation (17) and a source-receiver reciprocity relation, which is derived in section 4.1.

We illustrateGp,q, decomposed into plane waves, for a horizontally layered lossless isotropic

medium. To this end, we first define the spatial Fourier transform of a space- and frequency-

dependent function u(x, ω) along the horizontal coordinates xH = (x1, x2), according to

u(s, x3, ω) =

∫R2

exp{−iωs · xH}u(xH, x3, ω)dxH, (20)

with s = (s1, s2), where s1 and s2 are horizontal slownesses. This transform decomposes the

function u(x, ω) into monochromatic plane-wave components. Next, we define the inverse

temporal Fourier transform, per slowness value s, as

u(s, x3, τ) =1

π<∫ ∞0

u(s, x3, ω) exp{−iωτ)dω, (21)

where < denotes the real part and τ is the intercept time (Stoffa 1989). We apply these

transforms to the Green’s function Gp,q(x,xA, ω) and the source function δ(x−xA), choosing


(a)x3,0 = 0m

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c1 = 1600m/s<latexit sha1_base64="GsFlWCl9OuMsdK0w8fCEOBNEXw8=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCC6lJFe1GKLhxWcE+oAlhMp20Q2cmYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlHefbKiwtr6yuFddLG5tb2zv27l5LxanEpIljFstOiBRhVJCmppqRTiIJ4iEj7XB4M/HbD0QqGot7PUqIz1Ff0IhipI0U2Ac4cK/dS8fxTmHmSQ75mRoHdtmpOFPAReLmpAxyNAL7y+vFOOVEaMyQUl3XSbSfIakpZmRc8lJFEoSHqE+6hgrEifKz6fVjeGyUHoxiaUpoOFV/T2SIKzXioenkSA/UvDcR//O6qY5qfkZFkmoi8GxRlDKoYziJAvaoJFizkSEIS2puhXiAJMLaBFYyIbjzLy+SVrXinleqdxflei2PowgOwRE4AS64AnVwCxqgCTB4BM/gFbxZT9aL9W59zFoLVj6zD/7A+vwBs5eTbg==</latexit>

c0 = 1600m/s<latexit sha1_base64="Gysqjrs0gNpMhVLRyGMA18AsjUY=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCC6lJFe1GKLhxWcE+oAlhMp20Q2cmYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlHefbKiwtr6yuFddLG5tb2zv27l5LxanEpIljFstOiBRhVJCmppqRTiIJ4iEj7XB4M/HbD0QqGot7PUqIz1Ff0IhipI0U2Ac4cK7dS8fxTmHmSQ75mRoHdtmpOFPAReLmpAxyNAL7y+vFOOVEaMyQUl3XSbSfIakpZmRc8lJFEoSHqE+6hgrEifKz6fVjeGyUHoxiaUpoOFV/T2SIKzXioenkSA/UvDcR//O6qY5qfkZFkmoi8GxRlDKoYziJAvaoJFizkSEIS2puhXiAJMLaBFYyIbjzLy+SVrXinleqdxflei2PowgOwRE4AS64AnVwCxqgCTB4BM/gFbxZT9aL9W59zFoLVj6zD/7A+vwBsgKTbQ==</latexit>

x3,1 = 400m<latexit sha1_base64="zNwVs6U4eGwWoQpLUvtuGO/ZU/I=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCi1KStmA3QsGNywr2AU0Ik+mkHTqThJmJWEPwV9y4UMSt/+HOv3HaZqGtBy4czrmXe+/xY0alsqxvo7C2vrG5Vdwu7ezu7R+Yh0ddGSUCkw6OWCT6PpKE0ZB0FFWM9GNBEPcZ6fmT65nfuydC0ii8U9OYuByNQhpQjJSWPPPkwUvrFTu7aliWU0kdwSHPPLNsVa054Cqxc1IGOdqe+eUMI5xwEirMkJQD24qVmyKhKGYkKzmJJDHCEzQiA01DxIl00/n1GTzXyhAGkdAVKjhXf0+kiEs55b7u5EiN5bI3E//zBokKmm5KwzhRJMSLRUHCoIrgLAo4pIJgxaaaICyovhXiMRIIKx1YSYdgL7+8Srq1ql2v1m4b5VYzj6MITsEZuAA2uAQtcAPaoAMweATP4BW8GU/Gi/FufCxaC0Y+cwz+wPj8AW+Mk+U=</latexit>

x3,2 = 760m<latexit sha1_base64="8iXj8c7A+Bvqil0DnnMNgiFWa60=">AAAB/XicbVDLSsNAFJ34rPUVHzs3g0VwUUrSiu1GKLhxWcE+oAlhMp20Q2eSMDMRawj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1+zKhUlvVtrKyurW9sFraK2zu7e/vmwWFHRonApI0jFomejyRhNCRtRRUjvVgQxH1Guv74eup374mQNArv1CQmLkfDkAYUI6Ulzzx+8NJauZpd1S8tp5w6gkOeeWbJqlgzwGVi56QEcrQ888sZRDjhJFSYISn7thUrN0VCUcxIVnQSSWKEx2hI+pqGiBPpprPrM3imlQEMIqErVHCm/p5IEZdywn3dyZEayUVvKv7n9RMVNNyUhnGiSIjni4KEQRXBaRRwQAXBik00QVhQfSvEIyQQVjqwog7BXnx5mXSqFbtWqd5elJqNPI4COAGn4BzYoA6a4Aa0QBtg8AiewSt4M56MF+Pd+Ji3rhj5zBH4A+PzB38fk+8=</latexit>

x3,3 = 800m<latexit sha1_base64="dmNs2F7LTn0lIzI25Fcbq91ady8=">AAAB/XicbVDLSsNAFJ34rPUVHzs3g0VwUUrSCnYjFNy4rGAf0IQwmU7aoTOTMDMRayj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1hwqjSjvNtrayurW9sFraK2zu7e/v2wWFbxanEpIVjFstuiBRhVJCWppqRbiIJ4iEjnXB0PfU790QqGos7PU6Iz9FA0IhipI0U2McPQVYr1yZXdcfxypknOeSTwC45FWcGuEzcnJRAjmZgf3n9GKecCI0ZUqrnOon2MyQ1xYxMil6qSILwCA1Iz1CBOFF+Nrt+As+M0odRLE0JDWfq74kMcaXGPDSdHOmhWvSm4n9eL9VR3c+oSFJNBJ4vilIGdQynUcA+lQRrNjYEYUnNrRAPkURYm8CKJgR38eVl0q5W3FqlentRatTzOArgBJyCc+CCS9AAN6AJWgCDR/AMXsGb9WS9WO/Wx7x1xcpnjsAfWJ8/eOyT6w==</latexit>

x3,4 = 1200m<latexit sha1_base64="v4/r0ZS1bB2YXY4kxFQeyDUx/0o=">AAAB/nicbVDLSsNAFJ34rPUVFVduBovgopSkLdiNUHDjsoJ9QBPCZDpph85MwsxELKHgr7hxoYhbv8Odf+O0zUJbD1w4nHMv994TJowq7Tjf1tr6xubWdmGnuLu3f3BoHx13VJxKTNo4ZrHshUgRRgVpa6oZ6SWSIB4y0g3HNzO/+0CkorG415OE+BwNBY0oRtpIgX36GGS1cn167VYdxytnnuSQTwO75FScOeAqcXNSAjlagf3lDWKcciI0Zkipvusk2s+Q1BQzMi16qSIJwmM0JH1DBeJE+dn8/Cm8MMoARrE0JTScq78nMsSVmvDQdHKkR2rZm4n/ef1URw0/oyJJNRF4sShKGdQxnGUBB1QSrNnEEIQlNbdCPEISYW0SK5oQ3OWXV0mnWnFrlepdvdRs5HEUwBk4B5fABVegCW5BC7QBBhl4Bq/gzXqyXqx362PRumblMyfgD6zPH+V8lCE=</latexit>

c2 = 2400m/s<latexit sha1_base64="2WdXIADeAsODdJGEC1ekAlzsoxM=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCC6lJLNiNUHDjsoJ9QBPCZDpph85MwsxEqKH4K25cKOLW/3Dn3zh9LLT1wIXDOfdy7z1RyqjSjvNtFVZW19Y3ipulre2d3T17/6Clkkxi0sQJS2QnQoowKkhTU81IJ5UE8YiRdjS8mfjtByIVTcS9HqUk4KgvaEwx0kYK7SMcetde1XH8c5j7kkN+ocahXXYqzhRwmbhzUgZzNEL7y+8lOONEaMyQUl3XSXWQI6kpZmRc8jNFUoSHqE+6hgrEiQry6fVjeGqUHowTaUpoOFV/T+SIKzXikenkSA/UojcR//O6mY5rQU5Fmmki8GxRnDGoEziJAvaoJFizkSEIS2puhXiAJMLaBFYyIbiLLy+TlldxLyveXbVcr83jKIJjcALOgAuuQB3cggZoAgwewTN4BW/Wk/VivVsfs9aCNZ85BH9gff4As5uTbg==</latexit>

c3 = 3600m/s<latexit sha1_base64="a0/ZHQnBaGYvv8gNKd3BumSNYLw=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCC6lJK9qNUHDjsoJ9QBPCZDpph85MwsxEqKH4K25cKOLW/3Dn3zhts9DqgQuHc+7l3nvChFGlHefLKiwtr6yuFddLG5tb2zv27l5bxanEpIVjFstuiBRhVJCWppqRbiIJ4iEjnXB0PfU790QqGos7PU6Iz9FA0IhipI0U2Ac4qF3VLhzHO4WZJznkZ2oS2GWn4swA/xI3J2WQoxnYn14/xiknQmOGlOq5TqL9DElNMSOTkpcqkiA8QgPSM1QgTpSfza6fwGOj9GEUS1NCw5n6cyJDXKkxD00nR3qoFr2p+J/XS3VU9zMqklQTgeeLopRBHcNpFLBPJcGajQ1BWFJzK8RDJBHWJrCSCcFdfPkvaVcrbq1SvT0vN+p5HEVwCI7ACXDBJWiAG9AELYDBA3gCL+DVerSerTfrfd5asPKZffAL1sc3ueeTcg==</latexit>

c4 = 2200m/s<latexit sha1_base64="bXp+vG60Lp0ce7AZhWfahob2uFk=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GSyCC6lJLNiNUHDjsoJ9QBPCZDpph85MwsxEqKH4K25cKOLW/3Dn3zh9LLT1wIXDOfdy7z1RyqjSjvNtFVZW19Y3ipulre2d3T17/6Clkkxi0sQJS2QnQoowKkhTU81IJ5UE8YiRdjS8mfjtByIVTcS9HqUk4KgvaEwx0kYK7SMcVq89z3H8c5j7kkN+ocahXXYqzhRwmbhzUgZzNEL7y+8lOONEaMyQUl3XSXWQI6kpZmRc8jNFUoSHqE+6hgrEiQry6fVjeGqUHowTaUpoOFV/T+SIKzXikenkSA/UojcR//O6mY5rQU5Fmmki8GxRnDGoEziJAvaoJFizkSEIS2puhXiAJMLaBFYyIbiLLy+TlldxLyveXbVcr83jKIJjcALOgAuuQB3cggZoAgwewTN4BW/Wk/VivVsfs9aCNZ85BH9gff4As6GTbg==</latexit>

0 0.2 0.4 0.6 0.8

x3,0 = 0m<latexit sha1_base64="6Zw9msJbAtcWM/xRHVHZzpwkrS0=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgopSkFexGKLhxWcE+oAlhMp22Q2cmYWYiLSG/4saFIm79EXf+jdM2C209cOFwzr3ce08YM6q043xbha3tnd294n7p4PDo+MQ+LXdVlEhMOjhikeyHSBFGBeloqhnpx5IgHjLSC6d3C7/3RKSikXjU85j4HI0FHVGMtJECuzwL0kbVyW4dr5p6kkOeBXbFqTlLwE3i5qQCcrQD+8sbRjjhRGjMkFID14m1nyKpKWYkK3mJIjHCUzQmA0MF4kT56fL2DF4aZQhHkTQlNFyqvydSxJWa89B0cqQnat1biP95g0SPmn5KRZxoIvBq0ShhUEdwEQQcUkmwZnNDEJbU3ArxBEmEtYmrZEJw11/eJN16zW3U6g/XlVYzj6MIzsEFuAIuuAEtcA/aoAMwmIFn8ArerMx6sd6tj1VrwcpnzsAfWJ8/gr2TbA==</latexit>


x3,2 = 760m<latexit sha1_base64="8iXj8c7A+Bvqil0DnnMNgiFWa60=">AAAB/XicbVDLSsNAFJ34rPUVHzs3g0VwUUrSiu1GKLhxWcE+oAlhMp20Q2eSMDMRawj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1+zKhUlvVtrKyurW9sFraK2zu7e/vmwWFHRonApI0jFomejyRhNCRtRRUjvVgQxH1Guv74eup374mQNArv1CQmLkfDkAYUI6Ulzzx+8NJauZpd1S8tp5w6gkOeeWbJqlgzwGVi56QEcrQ888sZRDjhJFSYISn7thUrN0VCUcxIVnQSSWKEx2hI+pqGiBPpprPrM3imlQEMIqErVHCm/p5IEZdywn3dyZEayUVvKv7n9RMVNNyUhnGiSIjni4KEQRXBaRRwQAXBik00QVhQfSvEIyQQVjqwog7BXnx5mXSqFbtWqd5elJqNPI4COAGn4BzYoA6a4Aa0QBtg8AiewSt4M56MF+Pd+Ji3rhj5zBH4A+PzB38fk+8=</latexit>x3,3 = 800m<latexit sha1_base64="dmNs2F7LTn0lIzI25Fcbq91ady8=">AAAB/XicbVDLSsNAFJ34rPUVHzs3g0VwUUrSCnYjFNy4rGAf0IQwmU7aoTOTMDMRayj+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1hwqjSjvNtrayurW9sFraK2zu7e/v2wWFbxanEpIVjFstuiBRhVJCWppqRbiIJ4iEjnXB0PfU790QqGos7PU6Iz9FA0IhipI0U2McPQVYr1yZXdcfxypknOeSTwC45FWcGuEzcnJRAjmZgf3n9GKecCI0ZUqrnOon2MyQ1xYxMil6qSILwCA1Iz1CBOFF+Nrt+As+M0odRLE0JDWfq74kMcaXGPDSdHOmhWvSm4n9eL9VR3c+oSFJNBJ4vilIGdQynUcA+lQRrNjYEYUnNrRAPkURYm8CKJgR38eVl0q5W3FqlentRatTzOArgBJyCc+CCS9AAN6AJWgCDR/AMXsGb9WS9WO/Wx7x1xcpnjsAfWJ8/eOyT6w==</latexit>


causa

lity

condit

ion

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tunnelling<latexit sha1_base64="ano/I2sKWoaVE3KcoCBbVW71EUQ=">AAAB+HicbVDLSgMxFL3js9ZHR126CRbBVZmpgl0W3LisYB/QDiWTpm1okhnyEOrQL3HjQhG3foo7/8a0nYW2HrhwOOfe5N4Tp5xpEwTf3sbm1vbObmGvuH9weFTyj09aOrGK0CZJeKI6MdaUM0mbhhlOO6miWMSctuPJ7dxvP1KlWSIfzDSlkcAjyYaMYOOkvl/qKZEZKyXl7onRrO+Xg0qwAFonYU7KkKPR9796g4RYQaUhHGvdDYPURBlWhhFOZ8We1TTFZIJHtOuoxILqKFssPkMXThmgYaJcSYMW6u+JDAutpyJ2nQKbsV715uJ/XteaYS3KmEytoZIsPxpajkyC5imgAVOUGD51BBPF3K6IjLHCxLisii6EcPXkddKqVsKrSvX+ulyv5XEU4AzO4RJCuIE63EEDmkDAwjO8wpv35L14797HsnXDy2dO4Q+8zx+ffpOu</latexit>

(b)

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⌧ (s)<latexit sha1_base64="8HsrKzOrkA3ZzdApvu7beLkHN44=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSxCBSlJFXThouDGZQX7gCaUyXTSDp2ZhHkINfRL3LhQxK2f4s6/cdpmoa0HLhzOuZd774lSRpX2vG+nsLa+sblV3C7t7O7tl92Dw7ZKjMSkhROWyG6EFGFUkJammpFuKgniESOdaHw78zuPRCqaiAc9SUnI0VDQmGKkrdR3y4FGJjivZoHkUE3P+m7Fq3lzwFXi56QCcjT77lcwSLDhRGjMkFI930t1mCGpKWZkWgqMIinCYzQkPUsF4kSF2fzwKTy1ygDGibQlNJyrvycyxJWa8Mh2cqRHatmbif95PaPj6zCjIjWaCLxYFBsGdQJnKcABlQRrNrEEYUntrRCPkERY26xKNgR/+eVV0q7X/Ita/f6y0rjJ4yiCY3ACqsAHV6AB7kATtAAGBjyDV/DmPDkvzrvzsWgtOPnMEfgD5/MHz/+ShA==</latexit>

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Figure 1. (a) Horizontally layered medium. (b) Green’s function Gp,q(s1, x3, x3,0, τ) (fixed s1), con-volved with a wavelet.

xA = (0, 0, x3,0) and setting s2 = 0 (the field is cylindrically symmetric in the considered

horizontally layered isotropic medium). We thus obtain Gp,q(s1, x3, x3,0, τ), which is the plane-

wave response (as a function of x3 and τ) to a source function δ(x3 − x3,0)δ(τ).

An example horizontally layered medium is shown in Figure 1(a). The propagation ve-

locities in the layers are indicated by ck (with c = 1/√κρ). The depth level of the source is

chosen as x3,0 = 0 m, see Figure 1(b). The vertical green line in this figure is the line τ = 0,

left of which the field is zero due to the causality condition (similar as in equation (14)).

Figure 1(b) further shows the numerically modelled Green’s function Gp,q(s1, x3, x3,0, τ) as a

function of x3 and τ , for a single horizontal slowness s1 = 1/3500 s/m (red and blue arrows

indicate downgoing and upgoing waves). This is the wave field that would be measured by

a series of acoustic pressure receivers, vertically above and below the volume-injection rate

source at x3,0 = 0 m. Each trace shows Gp,q(s1, x3, x3,0, τ) for a specific depth x3 as a function

10 Wapenaar

of τ (actually, at each depth the Green’s function has been convolved with a time-symmetric

wavelet, with a central frequency of 50 Hz, to get a nicer display).

The horizontal slowness s1 is related to the propagation angle αk via sinαk = s1ck, hence,

in layer 1 (with c1 = 1600 m/s) the propagation angle is α1 = 27.2o. In the thin layer, with

c3 = 3600 m/s, we obtain sinα3 = 1.03, meaning that α3 is complex-valued. This implies that

the wave is evanescent in this layer. Since the layer is thin, the wave tunnels through the layer

and continues with a lower amplitude as a downgoing wave in the lower half-space.

2.3 The homogeneous Green’s matrix

For a lossless medium, a homogeneous Green’s function is the superposition of a Green’s

function and its complex conjugate (or, in the time domain, its time-reversed version). The

superposition is chosen such that the source terms of the two functions cancel each other,

hence, the homogeneous Green’s function obeys a wave equation without a source term (Porter

1970; Oristaglio 1989). Here we extend this concept for the matrix-vector wave equation for

a medium with losses.

Let G(x,xA, ω) be again the outward propagating solution of equation (12) for an ar-

bitrary inhomogeneous anisotropic medium. We introduce the Green’s matrix of the adjoint

medium, G(x,xA, ω), as the outward propagating solution of

∂3G− AG = Iδ(x− xA). (22)

Pre- and post multiplying all terms by J and subsequently using equation (5) and JJ = I

gives

∂3JGJ−A∗JGJ = Iδ(x− xA). (23)

Subtracting the complex conjugate of all terms in this equation from the corresponding terms

in equation (12) yields

∂3Gh −AGh = O, (24)


with

Gh(x,xA, ω) = G(x,xA, ω)− JG∗(x,xA, ω)J. (25)

Since Gh(x,xA, ω) obeys a wave equation without a source term, we call it the homogeneous

Green’s matrix. According to equations (6), (13) and (25), it is partitioned as

Gh(x,xA, ω) =

{G11 − G∗11} {G12 + G∗12}

{G21 + G∗21} {G22 − G∗22}

(x,xA, ω). (26)

2.4 The propagator matrix

We introduce the N × N propagator matrix W(x,xA, ω) for an arbitrary inhomogeneous

anisotropic medium as the solution of the unified matrix-vector wave equation (1), but without

the source vector d. Hence,

∂3W −AW = O. (27)

Similar to operator matrix A and Green’s matrix G, the propagator matrix is partitioned as

W(x,xA, ω) =

W11 W12

W21 W22

(x,xA, ω). (28)

Equation (27) does not have a unique solution. To make the solution unique, we impose the

boundary condition

W(x,xA, ω)|x3=x3,A = Iδ(xH − xH,A), (29)

where xH,A denotes the horizontal coordinates of xA, hence, xH,A = (x1,A, x2,A). Since equa-

tion (27) is first order in ∂3, a single-boundary condition suffices. Note that W(x,xA, ω) only

depends on the medium parameters between depth levels x3,A and x3. This is different from

the Green’s matrix G(x,xA, ω), which, for an arbitrary inhomogeneous medium, depends on

the entire medium (this is easily understood from the illustration in Figure 1(b)).

The simplest representation involving the propagator matrix is obtained when q and W

reside in the same medium in the region between x3,A and x3 and both have no sources in this

region. Whereas q(x, ω) obeys no boundary conditions in this region, W(x,xA, ω) collapses

12 Wapenaar

to Iδ(xH−xH,A) at depth level x3,A for an arbitrary horizontal position xH,A (equation (29)).

Applying the superposition principle again yields

q(x, ω) =

∫∂DA

W(x,xA, ω)q(xA, ω)dxA, (30)

where ∂DA is the horizontal boundary defined as x3 = x3,A (Gilbert & Backus 1966; Kennett

1972a,b; Woodhouse 1974; Haines 1988). Note that W(x,xA, ω) ‘propagates’ the field vector

q from depth level x3,A to x3, hence the name ‘propagator matrix’. This representation is

a special case of more general representations with propagator matrices, derived in a more

formal way in section 5.

When we replace q(xA, ω) by a Green’s matrix G(xA,xB, ω), with xB outside the region

between x3,A and x3, we obtain

G(x,xB, ω) =

∫∂DA

W(x,xA, ω)G(xA,xB, ω)dxA. (31)

This is the simplest relation between the Green’s matrix and the propagator matrix. It is a

special case of more general representations with Green’s matrices and propagator matrices,

derived in a more formal way in section 6.1. Alternatively, we may replace G(x,xB, ω) by

the homogeneous Green’s matrix Gh(x,xB, ω), where xB may be located anywhere, since the

homogeneous Green’s function has no source at xB, hence

Gh(x,xB, ω) =

∫∂DA

W(x,xA, ω)Gh(xA,xB, ω)dxA. (32)

This relation will be derived in a more formal way in section 6.2.

Next, we discuss the 2 × 2 acoustic propagator matrix as a special case of the N × N

propagator matrix. For this situation W is partitioned as

W(x,xA, ω) =

W p,p W p,v

W v,p W v,v

(x,xA, ω). (33)

The first and second superscripts refer to the wave-field quantities at x and xA, respectively

(with superscript p again standing for acoustic pressure and v for the vertical component of

particle velocity). For an isotropic medium, the elements can be expressed in terms of the


-0.4 -0.2 0.0 0.2 0.4

(a)





boundary condition<latexit sha1_base64="FjDCV+L0lid+eAbPgiwUNAh63F0=">AAACAXicbVDLSsNAFJ3UV62vqBvBTbAILqQkdWGXBTcuK9gHNKFMJpN26DzCzEQIoW78FTcuFHHrX7jzb5y0WWjrgQuHc+7lck6YUKK0635blbX1jc2t6nZtZ3dv/8A+POopkUqEu0hQIQchVJgSjruaaIoHicSQhRT3w+lN4fcfsFRE8HudJThgcMxJTBDURhrZJ75keShSHkGZ+ZdI8IgUzmxk192GO4ezSryS1EGJzsj+8iOBUoa5RhQqNfTcRAc5lJogimc1P1U4gWgKx3hoKIcMqyCfJ5g550aJnFhIM1w7c/X3RQ6ZUhkLzSaDeqKWvUL8zxumOm4FOeFJqjFHi0dxSh0tnKIOJyISI00zQyCSJjpy0ARKiLQprWZK8JYjr5Jes+FdNZp3zXq7VdZRBafgDFwAD1yDNrgFHdAFCDyCZ/AK3qwn68V6tz4WqxWrvDkGf2B9/gBsT5d5</latexit>


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amplitu

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<latexit sha1_base64="lAHdKVIj3JQzEWBkM2O/rApMA0E=">AAAB+HicbVBNS8NAEN34WetHox69LBbBU0mqYI8FLx4r2A9oQ9lsNu3S3STszgo19Jd48aCIV3+KN/+N2zYHbX0w8Hhvhpl5YSa4Bs/7djY2t7Z3dkt75f2Dw6OKe3zS0alRlLVpKlLVC4lmgiesDRwE62WKERkK1g0nt3O/+8iU5mnyANOMBZKMEh5zSsBKQ7eSD5TERNpdYCI2G7pVr+YtgNeJX5AqKtAaul+DKKVGsgSoIFr3fS+DICcKOBVsVh4YzTJCJ2TE+pYmRDId5IvDZ/jCKhGOU2UrAbxQf0/kRGo9laHtlATGetWbi/95fQNxI8h5khlgCV0uio3AkOJ5CjjiilEQU0sIVdzeiumYKELBZlW2IfirL6+TTr3mX9Xq99fVZqOIo4TO0Dm6RD66QU10h1qojSgy6Bm9ojfnyXlx3p2PZeuGU8ycoj9wPn8AEhuTUw==</latexit>

(b)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-6

-4

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0

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6 106

-0.4 -0.2 0.0 0.2 0.4 ⌧ (s)

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Figure 2. (a) Propagator element W p,p(s1, x3, x3,0, τ) (fixed s1), convolved with a wavelet. (b) Lasttrace of (a).

upper-right element, as follows

W v,v(x,xA, ω) =1

iωρ(x, ω)∂3W

p,v(x,xA, ω), (34)

W p,p(x,xA, ω) = − 1

iωρ(xA, ω)∂3,AW

p,v(x,xA, ω), (35)

W v,p(x,xA, ω) =1

iωρ(x, ω)∂3W

p,p(x,xA, ω). (36)

Equations (34) and (36) follow directly from equations (11), (27) and (33). Equation (35)

follows from equations (34) and a source-receiver reciprocity relation, which is derived in

section 5.1.

We illustrate the elements W p,p and W p,v, decomposed into plane waves, for the hori-

zontally layered medium of Figure 1(a). We choose again xA = (0, 0, x3,0) and set s2 = 0.

Usually the propagator matrix is considered in the frequency domain, but to facilitate the

comparison with the Green’s function in Figure 1(b), we consider the time-domain functions

W p,p(s1, x3, x3,0, τ) and W p,v(s1, x3, x3,0, τ) (obtained via the transforms of equations (20)

and (21)), again for a single horizontal slowness s1 = 1/3500 s/m, see Figures 2(a) and

3(a). At x3 = x3,0 (the horizontal green lines in these figures) the boundary conditions for

14 Wapenaar

-0.4 -0.2 0.0 0.2 0.4

(a)





boundary condition<latexit sha1_base64="FjDCV+L0lid+eAbPgiwUNAh63F0=">AAACAXicbVDLSsNAFJ3UV62vqBvBTbAILqQkdWGXBTcuK9gHNKFMJpN26DzCzEQIoW78FTcuFHHrX7jzb5y0WWjrgQuHc+7lck6YUKK0635blbX1jc2t6nZtZ3dv/8A+POopkUqEu0hQIQchVJgSjruaaIoHicSQhRT3w+lN4fcfsFRE8HudJThgcMxJTBDURhrZJ75keShSHkGZ+ZdI8IgUzmxk192GO4ezSryS1EGJzsj+8iOBUoa5RhQqNfTcRAc5lJogimc1P1U4gWgKx3hoKIcMqyCfJ5g550aJnFhIM1w7c/X3RQ6ZUhkLzSaDeqKWvUL8zxumOm4FOeFJqjFHi0dxSh0tnKIOJyISI00zQyCSJjpy0ARKiLQprWZK8JYjr5Jes+FdNZp3zXq7VdZRBafgDFwAD1yDNrgFHdAFCDyCZ/AK3qwn68V6tz4WqxWrvDkGf2B9/gBsT5d5</latexit>

0<latexit sha1_base64="8SPvTV/z8oIhjlRiKd59oqPvbVY=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KkkV7LHgxWML9gPaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dHLa1nGqGLZYLGLVDahGwSW2DDcCu4lCGgUCO8Hkbu53nlBpHssHM03Qj+hI8pAzaqzUdAelsltxFyDrxMtJGXI0BqWv/jBmaYTSMEG17nluYvyMKsOZwFmxn2pMKJvQEfYslTRC7WeLQ2fk0ipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmrDmZ1wmqUHJlovCVBATk/nXZMgVMiOmllCmuL2VsDFVlBmbTdGG4K2+vE7a1Yp3Xak2b8r1Wh5HAc7hAq7Ag1uowz00oAUMEJ7hFd6cR+fFeXc+lq0bTj5zBn/gfP4Ad2WMrg==</latexit>


amplitu

de


(b)

-0.4 -0.2 0.0 0.2 0.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-6

-4

-2

0

2

4

6 106


Figure 3. (a) Propagator element W p,v(s1, x3, x3,0, τ) (fixed s1), convolved with a wavelet. (b) Lasttrace of (a).

these functions are W p,p(s1, x3,0, x3,0, τ) = δ(τ) and W p,v(s1, x3,0, x3,0, τ) = 0, respectively

(this follows from applying the transforms of equations (20) and (21) to equation (29), with

xH,A = (0, 0)). Following these functions along the depth coordinate, starting at x3 = x3,0,

we observe an acausal upgoing wave and a causal downgoing wave until we reach the first

interface. Here both events split into upgoing and downgoing waves below the interface. The

waves tunnel through the high-velocity layer, split again, and continue with higher amplitudes

in the next layer. This illustrates that evanescent waves may lead to unstable behaviour of

the propagator matrix and should be handled with care. Note that W p,p(s1, x3, x3,0, τ) and

W p,v(s1, x3, x3,0, τ) are symmetric and asymmetric in time, respectively. This is best seen in

Figures 2(b) and 3(b), which show the last trace of both elements. The same holds for elements

W v,v and W v,p, which are not shown.


2.5 The Marchenko-type focusing function

For an arbitrary inhomogeneous anisotropic medium, the time-domain version of boundary

condition (29) reads

W(x,xA, t)|x3=x3,A = Iδ(xH − xH,A)δ(t). (37)

This boundary condition has a similar form as the focusing condition for the focusing functions

appearing in the multidimensional Marchenko method (Wapenaar et al. 2014). In section 7.1

we discuss the general relations between the propagator matrix and the Marchenko-type

focusing functions. Here we present a short preview of these relations by considering the

acoustic propagator matrix in the horizontally layered lossless isotropic medium of Figure

1(a). We combine the elements W p,p and W p,v, decomposed into plane waves (Figures 2 and

3), as follows (Wapenaar & de Ridder 2021)

F p(s1, x3, x3,0, τ) = W p,p(s1, x3, x3,0, τ)− s3,0ρ0

W p,v(s1, x3, x3,0, τ), (38)

where the vertical slowness s3,0 is defined as

s3,0 =√

1/c20 − s21, (39)

with c0 and ρ0 being the propagation velocity and mass density of the upper half-space. Due

to the symmetric form of W p,p and the asymmetric form of W p,v, half of the events double

in amplitude and the other half of the events cancel. The result is shown in Figure 4. For

the interpretation of this focusing function we start at the bottom of Figure 4(a). The blue

arrows indicate upgoing waves, which are tuned such that, after interaction with the tunneling

waves in the thin layer and the downgoing wave just above the thin layer (indicated by the

red arrow), they continue as a single upgoing wave, which finally focuses at depth level x3,0

as a temporal delta function δ(τ) and continues as an upgoing wave into the homogeneous

upper half-space.

Note that, although we interpret the focusing function here in terms of upgoing and

downgoing waves, it is derived from the propagator matrix (which does not rely on up/down

decomposition) and the vertical slowness s3,0 and mass density ρ0 of the upper half-space.

16 Wapenaar

-0.4 -0.2 0.0 0.2 0.4

(a)





tunnelling<latexit sha1_base64="ano/I2sKWoaVE3KcoCBbVW71EUQ=">AAAB+HicbVDLSgMxFL3js9ZHR126CRbBVZmpgl0W3LisYB/QDiWTpm1okhnyEOrQL3HjQhG3foo7/8a0nYW2HrhwOOfe5N4Tp5xpEwTf3sbm1vbObmGvuH9weFTyj09aOrGK0CZJeKI6MdaUM0mbhhlOO6miWMSctuPJ7dxvP1KlWSIfzDSlkcAjyYaMYOOkvl/qKZEZKyXl7onRrO+Xg0qwAFonYU7KkKPR9796g4RYQaUhHGvdDYPURBlWhhFOZ8We1TTFZIJHtOuoxILqKFssPkMXThmgYaJcSYMW6u+JDAutpyJ2nQKbsV715uJ/XteaYS3KmEytoZIsPxpajkyC5imgAVOUGD51BBPF3K6IjLHCxLisii6EcPXkddKqVsKrSvX+ulyv5XEU4AzO4RJCuIE63EEDmkDAwjO8wpv35L14797HsnXDy2dO4Q+8zx+ffpOu</latexit>


focus �(⌧)<latexit sha1_base64="M88kn6AzIX07cHUilY7TpLfRcuc=">AAACAnicbVDLSsNAFJ34rPUVdSVuBotQQUpSBbssuHFZwT6gCWUynbRDZ5IwcyOUUNz4K25cKOLWr3Dn3zhts9DWAxcO59w7c+8JEsE1OM63tbK6tr6xWdgqbu/s7u3bB4ctHaeKsiaNRaw6AdFM8Ig1gYNgnUQxIgPB2sHoZuq3H5jSPI7uYZwwX5JBxENOCRipZx97SmZhTFM98S6w12cCSNkDkp737JJTcWbAy8TNSQnlaPTsL69vHpIsAiqI1l3XScDPiAJOBZsUvVSzhNARGbCuoRGRTPvZ7IQJPjNKH4exMhUBnqm/JzIitR7LwHRKAkO96E3F/7xuCmHNz3iUpMAiOv8oTAWGGE/zwH2uGAUxNoRQxc2umA6JIhRMakUTgrt48jJpVSvuZaV6d1Wq1/I4CugEnaIyctE1qqNb1EBNRNEjekav6M16sl6sd+tj3rpi5TNH6A+szx+/Fpb3</latexit>

amplitu

de


(b)

-0.4 -0.2 0.0 0.2 0.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-6

-4

-2

0

2

4

6 106


Figure 4. (a) Focusing function F p(s1, x3, x3,0, τ) (fixed s1), convolved with a wavelet. (b) Last traceof (a).

Moreover, the focusing function is defined in the actual medium rather than in a truncated

version of the actual medium, as is usually the case for Marchenko-type focusing functions

(Wapenaar et al. 2014). Hence, the only assumption is that a real-valued vertical slowness

s3,0 exists in the upper half-space. Other than that, the focusing function F p(s1, x3, x3,0, τ),

as defined in equation (38), does not require a truncated medium, does not rely on up/down

decomposition inside the medium, and does (at least in principle) not break down when waves

become evanescent inside the medium. These properties also hold for the more general version

of the Marchenko-type focusing functions defined in section 7.

3 UNIFIED MATRIX-VECTOR WAVE-FIELD RECIPROCITY

THEOREMS

As the basis for the derivation of the general representations with Green’s matrices (section

4), propagator matrices (section 5), or a combination thereof (section 6), we introduce unified

matrix-vector wave-field reciprocity theorems. In general, a wave-field reciprocity theorem

interrelates two wave states (sources, wave fields and medium parameters) in the same spatial


domain (de Hoop 1995). Reciprocity theorems have been formulated for acoustic (Rayleigh

1878), electromagnetic (Lorentz 1895), elastodynamic (Knopoff & Gangi 1959; de Hoop 1966),

poroelastodynamic (Flekkøy & Pride 1999), piezoelectric (Auld 1979; Achenbach 2003) and

seismoelectric waves (Pride & Haartsen 1996). The matrix-vector equation discussed in section

2 allows a unified formulation of the reciprocity theorems for these different wave phenomena

(Wapenaar 1996; Haines & de Hoop 1996), which extends the theory of propagation invariants

(Haines 1988; Kennett et al. 1990; Koketsu et al. 1991; Takenaka et al. 1993). Using the

symmetry properties of operator matrix A, formulated in equations (3) and (4), the following

matrix-vector reciprocity theorems can be derived (Wapenaar 1996; Haines & de Hoop 1996)∫D

(dtANqB + qtANdB

)dx =

∫∂D0∪∂DM

qtANqBn3dx +

∫D

qtAN(AA −AB)qBdx (40)

and∫D

(d†AKqB + q†AKdB

)dx =

∫∂D0∪∂DM

q†AKqBn3dx +

∫D

q†AK(AA −AB)qBdx. (41)

Here D denotes a domain enclosed by two infinite horizontal boundaries ∂D0 and ∂DM at

depth levels x3,0 and x3,M with outward pointing normals n3 = −1 and n3 = 1, respectively,

see Figure 5. Subscripts A and B refer to two independent states. These theorems hold for

lossless media and for media with losses (Wapenaar 2019). Equation (40) is a convolution-

type reciprocity theorem, since products like qtANqB in the frequency domain correspond

to convolutions in the time domain (like in reference (de Hoop 1966)). Equation (41) is a

correlation-type reciprocity theorem, since products like q†AKqB in the frequency domain

correspond to correlations in the time domain (like in reference (Bojarski 1983)).

A special case is obtained when the sources, wave fields and medium parameters are

identical in both states. We may then drop the subscripts A and B and equation (41) simplifies

to ∫D

1

4

(d†Kq + q†Kd

)dx =

∫∂D0∪∂DM

1

4q†Kqn3dx +

∫D

1

4q†K(A−A)qdx. (42)

Since 14q†Kq = 1

4(q†1q2 + q†2q1) = j (see equation (7)), equation (42) formulates the unified

power balance. The term on the left-hand side is the power generated by the sources in D.

The first term on the right-hand side is the power flux through the boundary ∂D0∪∂DM (i.e.,

18 Wapenaar

D

n3 = �1

n3 = +1

x1

x2x3

@D0

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xB<latexit sha1_base64="CKEVBVgVebcono6E6UztW1rzybY=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgj0WvXisYD+kXUo2zbahSXZJsmJZ+iu8eFDEqz/Hm//GdLsHbX0w8Hhvhpl5QcyZNq777RTW1jc2t4rbpZ3dvf2D8uFRW0eJIrRFIh6pboA15UzSlmGG026sKBYBp51gcjP3O49UaRbJezONqS/wSLKQEWys9JD2gxA9zQbXg3LFrboZ0CrxclKBHM1B+as/jEgiqDSEY617nhsbP8XKMMLprNRPNI0xmeAR7VkqsaDaT7ODZ+jMKkMURsqWNChTf0+kWGg9FYHtFNiM9bI3F//zeokJ637KZJwYKsliUZhwZCI0/x4NmaLE8KklmChmb0VkjBUmxmZUsiF4yy+vknat6l1Ua3eXlUY9j6MIJ3AK5+DBFTTgFprQAgICnuEV3hzlvDjvzseiteDkM8fwB87nD3ESkCM=</latexit>

@DM<latexit sha1_base64="8ls+Oy+TN5lzDSkrC27BPJrD5O4=">AAAB/XicbVDLSgMxFL1TX7W+xsfOTbAIrspMFeyyoAs3QgX7gM4wZNJMG5p5kGSEOhR/xY0LRdz6H+78GzPtLLT1QOBwzr3JyfETzqSyrG+jtLK6tr5R3qxsbe/s7pn7Bx0Zp4LQNol5LHo+lpSziLYVU5z2EkFx6HPa9cdXud99oEKyOLpXk4S6IR5GLGAEKy155pGTYKEY5k6I1cj3s+upd+uZVatmzYCWiV2QKhRoeeaXM4hJGtJIEY6l7NtWotwsv5lwOq04qaQJJmM8pH1NIxxS6Waz9FN0qpUBCmKhT6TQTP29keFQykno68k8o1z0cvE/r5+qoOFmLEpSRSMyfyhIOVIxyqtAAyYoUXyiCSaC6ayIjLDAROnCKroEe/HLy6RTr9nntfrdRbXZKOoowzGcwBnYcAlNuIEWtIHAIzzDK7wZT8aL8W58zEdLRrFzCH9gfP4Ay66VaA==</latexit>

Figure 5. Configuration for the matrix-vector reciprocity theorems, equations (40) and (41), and forthe representations with Green’s matrices.

the power leaving the domain D) and the second term on the right-hand side is the dissipated

power in D.

4 REPRESENTATIONS WITH GREEN’S MATRICES

A wave-field representation is obtained by replacing one of the states in a reciprocity theorem

by a Green’s state (Knopoff 1956; de Hoop 1958; Gangi 1970; Pao & Varatharajulu 1976). In

this section we follow this approach to derive wave-field representations with Green’s matrices

from the matrix-vector reciprocity theorems discussed in section 3.

4.1 Symmetry property of the Green’s matrix

Before we derive wave-field representations, we first derive a symmetry property of the Green’s

matrix. To this end, we replace both wave-field vectors qA and qB in reciprocity theorem (40)

by Green’s matrices G(x,xA, ω) and G(x,xB, ω), respectively. Accordingly, we replace the

source vectors dA and dB by Iδ(x−xA) and Iδ(x−xB), respectively. Furthermore, we replace

D by R3, so that the boundary integral in equation (40) vanishes (Sommerfeld radiation

condition). Both Green’s matrices are defined in the same medium, hence, AA = AB. This

implies that the second integral on the right-hand side of equation (40) also vanishes. From

the remaining integral we thus obtain the following symmetry property of the Green’s matrix

Gt(xB,xA, ω)N = −NG(xA,xB, ω). (43)


The Green’s matrix on the left-hand side is the response to a source at xA, observed by

a receiver at xB. Similarly, the Green’s matrix on the right-hand side is the response to a

source at xB, observed by a receiver at xA. Hence, equation (43) is a unified source-receiver

reciprocity relation.

Using this relation and JN = −NJ, we find for the homogeneous Green’s matrix defined

in equation (25)

Gth(xB,xA, ω)N = −NGh(xA,xB, ω). (44)

4.2 Representations of the convolution type with the Green’s matrix

We derive a representation of the convolution type for the actual wave-field vector q(x, ω),

emitted by the actual source distribution d(x, ω) in the actual medium; the operator matrix

in the actual medium is defined as A(x, ω). We let state B in reciprocity theorem (40) be

this actual state, hence, we drop subscript B from qB, dB and AB. For state A we choose

the Green’s state. Hence, we replace qA(x, ω) in reciprocity theorem (40) by G(x,xA, ω) and

dA(x, ω) by Iδ(x − xA). We keep the subscript A in AA(x, ω), to account for the fact that

in general this operator matrix is defined in a medium that may be different from the actual

medium. We thus obtain

χD(xA)Nq(xA, ω) = −∫D

Gt(x,xA, ω)Nd(x, ω)dx +

∫∂D0∪∂DM

Gt(x,xA, ω)Nq(x, ω)n3dx

+

∫D

Gt(x,xA, ω)N{AA −A}q(x, ω)dx, (45)

where χD(x) is the characteristic function for domain D, defined as

χD(x) =

1, for x inside D,

12 , for x on ∂D0 ∪ ∂DM ,

0, for x outside D.

(46)

20 Wapenaar

Using the symmetry property of the Green’s matrix, formulated by equation (43), we obtain

χD(xA)q(xA, ω) =

∫D

G(xA,x, ω)d(x, ω)dx−∫∂D0∪∂DM

G(xA,x, ω)q(x, ω)n3dx

−∫D

G(xA,x, ω){AA −A}q(x, ω)dx. (47)

This is the unified wave-field representation of the convolution type with the Green’s ma-

trix. The left-hand side is the wave-field vector q at a specific point xA, multiplied with the

characteristic function. According to the right-hand side, this field consists of a contribution

from the source distribution in D (the first integral), a contribution from the wave field on

the boundary of D (the second integral), and a contribution caused by the contrasts between

the operator matrices in the Green’s and the actual state (the third integral). In its general

form, equation (47) is a basis for analysing forward problems.

Note that equation (15) follows as a special case of equation (47) if we choose the same

medium parameters in both states and replace D by R3 (except that the roles of x and xA

are interchanged since we used the symmetry property of equation (43) in the derivation of

equation (47)).

Next we consider another special case. We choose again the same medium parameters in

both states, but this time we replace D by the entire half-space below ∂D0, which we choose

to be source-free in the actual state. Assuming xA lies in the lower half-space, we thus obtain

q(xA, ω) =

∫∂D0

G(xA,x, ω)q(x, ω)dx, forx3,A > x3,0. (48)

Using equations (8), (16) and (18) for the isotropic acoustic situation, equation (48) yields for

the upper element of q(xA, ω)

p(xA, ω) =

∫∂D0

(− 1

iωρ(x, ω){∂3Gp,q(xA,x, ω)}p(x, ω) +Gp,q(xA,x, ω)v3(x, ω)

)dx, (49)

which is the well-known acoustic Kirchhoff-Helmholtz integral. Hence, the representation of

equation (48) is the unified Kirchhoff-Helmholtz integral. It can be used for forward wave-field

extrapolation of the wave vector q(x, ω) from ∂D0 to any point xA below ∂D0. Note that the

Green’s matrix G(xA,x, ω) depends on the medium parameters of the entire half-space below

∂D0. In section 5.2 we derive a relation similar to equation (48), but with the Green’s matrix


replaced by the propagator matrix, which depends only on the medium parameters between

x3,0 and x3,A.

Finally, we replace q(x, ω) by G(x,xB, ω) in equation (48). Since we assumed that the

lower half-space is source-free for q, we choose xB in the upper half-space. We thus obtain

G(xA,xB, ω) =

∫∂D0

G(xA,x, ω)G(x,xB, ω)dx, forx3,A > x3,0 > x3,B. (50)

This expression shows how the Green’s matrix can be composed from a cascade of Green’s

matrices, assuming a specific order of the depth levels at which the Green’s sources and

receivers are situated. In section 5.2 we derive a similar relation for propagator matrices, for

an arbitrary order of depth levels.

Equation (50) can be seen as a generalization of the representation that underlies Green’s

function retrieval by cross-convolution (where G(xA,xB, ω) is the unknown (Slob & Wapenaar

2007)) or by multidimensional deconvolution (where G(xA,x, ω) is the unknown (Wapenaar

& van der Neut 2010)).

4.3 Representations of the correlation type with the Green’s matrix

We derive a representation of the correlation type for the actual wave-field vector q(x, ω),

emitted by the actual source distribution d(x, ω) in the actual medium. Similar as in section

4.2, we let state B be this actual state, and state A the Green’s state. Hence, making the

same substitutions as in section 4.2, this time in reciprocity theorem (41), using equation (43)

and N−1K = −J, yields

χD(xA)q(xA, ω) =

∫D

JG∗(xA,x, ω)Jd(x, ω)dx−∫∂D0∪∂DM

JG∗(xA,x, ω)Jq(x, ω)n3dx

−∫D

JG∗(xA,x, ω)J{AA −A}q(x, ω)dx. (51)

This is the unified wave-field representation of the correlation type with the Green’s matrix.

In its general form, equation (51) is a basis for analysing inverse problems.

As a special case we derive a representation for the homogeneous Green’s matrix Gh. To

this end, for state B we choose the Green’s matrix in the actual medium, hence, we replace

q(x, ω) by G(x,xB, ω), and d(x, ω) by Iδ(x−xB). For state A we replace the Green’s matrix

22 Wapenaar

by that in the adjoint of the actual medium, hence, we replace G(xA,x, ω) by G(xA,x, ω),

and AA by A. With this choice the contrast operator AA −A = ¯A −A vanishes. Making

these substitutions in equation (51), taking xA and xB both inside D, using equation (25), we

obtain

Gh(xA,xB, ω) = −∫∂D0∪∂DM

JG∗(xA,x, ω)JG(x,xB, ω)n3dx,

for x3,M > {x3,A, x3,B} > x3,0. (52)

This is the unified homogeneous Green’s matrix representation. It finds applications in optical,

acoustic and seismic holography (Porter 1970; Lindsey & Braun 2004), inverse source problems

(Porter & Devaney 1982), inverse scattering methods (Oristaglio 1989) and Green’s function

retrieval by cross-correlation (Derode et al. 2003; Wapenaar 2003; Weaver & Lobkis 2004;

Wapenaar et al. 2006; Snieder et al. 2007). A disadvantage is that the integral is taken along

two boundaries ∂D0 and ∂DM , whereas in many practical situations, measurements are only

available on a single boundary. Using the propagator matrix, in section 6.2 we present a

single-sided unified homogeneous Green’s function representation.

Finally, using equations (16), (17) and (18) for the isotropic acoustic situation, equation

(52) yields for the upper-right element of Gh(xA,xB, ω)

Gp,qh (xA,xB, ω) = − 1

iω

∫∂D0∪∂DM

(1

ρ∗(x, ω){∂3Gp,q(xA,x, ω)}∗Gp,q(x,xB, ω)

− 1

ρ(x, ω){Gp,q(xA,x, ω)}∗∂3Gp,q(x,xB, ω)

)n3dx, (53)

with, according to equations (16) and (26),

Gp,qh (xA,xB, ω) = Gp,q(xA,xB, ω) + {Gp,q(xA,xB, ω)}∗. (54)

For a lossless constant density medium, using Gp,q(x,xB, ω) = −iωρG(x,xB, ω), equation (53)

is the well-known scalar homogeneous Green’s function representation (Porter 1970; Oristaglio

1989), applied to the configuration of Figure 5.


n3 = �1

n3 = +1

x1

x2x3

@D0

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Figure 6. Configuration for the representations with propagator matrices.

5 REPRESENTATIONS WITH PROPAGATOR MATRICES

We follow a similar approach as in section 4 to derive wave-field representations. However,

this time we replace one of the states in the reciprocity theorems by a propagator-matrix state

(instead of a Green’s state). Hence, this leads to wave-field representations with propagator

matrices.

5.1 Symmetry properties of the propagator matrix

We start with deriving symmetry properties of the propagator matrix. To this end, we re-

place both wave-field vectors qA and qB in reciprocity theorem (40) by propagator matrices

W(x,xA, ω) and W(x,xB, ω), respectively. We replace ∂D0 and ∂DM in equation (40) by hor-

izontal boundaries ∂DA and ∂DB (and D by the region enclosed by these boundaries). Bound-

aries ∂DA and ∂DB are chosen such that they contain the points xA and xB, respectively,

see Figure 6. The arrangement of ∂DA and ∂DB is arbitrary. Note that boundary condition

(29) implies W(x,xA, ω) = Iδ(xH−xH,A) for x at ∂DA and W(x,xB, ω) = Iδ(xH−xH,B) for

x at ∂DB. The propagator obeys wave equation (27) without sources, hence, we set dA and

dB to O. This implies that the integral on the left-hand side of equation (40) vanishes. Both

propagator matrices are defined in the same medium, hence, AA = AB. This implies that the

second integral on the right-hand side vanishes. Evaluating the remaining integral along the

boundary ∂DA ∪ ∂DB yields

Wt(xB,xA, ω)N = NW(xA,xB, ω). (55)

This is the first unified symmetry relation for the propagator matrix.

A second symmetry relation can be derived from reciprocity theorem (41). To this end we

24 Wapenaar

replace qA and qB by W(x,xA, ω) and W(x,xB, ω), respectively. We replace ∂D0 ∪ ∂DM in

equation (41) by ∂DA ∪ ∂DB (and D by the enclosed region). We set dA and dB to O, hence,

the integral on the left-hand side vanishes. The propagator matrices are defined in mutually

adjoint media, hence, AA = AB. This implies that the second integral on the right-hand side

of equation (41) vanishes. From the remaining integral we thus obtain

W†(xB,xA, ω)K = KW(xA,xB, ω). (56)

From equations (55) and (56), using KN−1 = J, we find

W∗(xB,xA, ω)J = JW(xB,xA, ω). (57)

Note that in this last equation xB and xA appear in the same order on the left- and right-hand

sides.

5.2 Representations of the convolution type with the propagator matrix

We derive a representation of the convolution type for the actual wave-field vector q(x, ω),

emitted by the actual source distribution d(x, ω) in the actual medium; the operator ma-

trix in the actual medium is defined as A(x, ω). We let state B in reciprocity theorem (40)

be this actual state, hence, we drop subscript B from qB, dB and AB. For state A we

choose the propagator-matrix state. Hence, we replace qA(x, ω) in reciprocity theorem (40)

by W(x,xA, ω) and dA(x, ω) by O. We keep the subscript A in AA(x, ω), to account for

the fact that in general this operator matrix is defined in a medium that may be different

from the actual medium. We replace ∂D0 ∪ ∂DM in reciprocity theorem (40) by ∂D0 ∪ ∂DA,

where ∂DA is the boundary containing xA. Here and in the following ∂DA is below ∂D0, hence

x3,A > x3,0. The domain enclosed by this boundary is called DA, see Figure 6. Unlike in the

classical, decomposition-based derivations of the Marchenko method (Wapenaar et al. 2014;

Slob et al. 2014), the domain DA does not define a truncated medium; in general the medium

below ∂DA is inhomogeneous. Applying the mentioned substitutions in equation (40), using


boundary condition (29) and symmetry relation (55) (with xB replaced by x) yields

q(xA, ω) =

∫DA

W(xA,x, ω)d(x, ω)dx +

∫∂D0

W(xA,x, ω)q(x, ω)dx

−∫DA

W(xA,x, ω){AA −A}q(x, ω)dx. (58)

This is the unified wave-field representation of the convolution type with the propagator

matrix. Note the analogy with the representation of the convolution type with the Green’s

matrix, equation (47). An important difference is that the boundary integral in equation (58)

is single-sided.

We consider a special case by choosing the same medium parameters in DA in both states,

and choosing DA to be source-free. We thus obtain

q(xA, ω) =

∫∂D0

W(xA,x, ω)q(x, ω)dx. (59)

This is the special case that was already presented in equation (30) (except that the roles

of x and xA are interchanged since we used the symmetry property of equation (55) in the

derivation of equation (58)).

Note the analogy of equation (59) with equation (48), which contains a Green’s matrix

instead of the propagator matrix. The propagator matrix in equation (59) depends only on

the medium parameters between x3,0 and x3,A. Equation (59) is used in section 7 as the basis

for deriving Marchenko-type representations.

Next, we consider again equation (58), in which we replace q(x, ω) by W(x,xB, ω), and

hence d(x, ω) by O and A by AA. This implies that the first and the third integral on the

right-hand side vanish. Moreover, we replace ∂D0 by ∂DC at a newly chosen depth level x3,C .

This yields

W(xA,xB, ω) =

∫∂DC

W(xA,x, ω)W(x,xB, ω)dx. (60)

This expression shows how the propagator matrix can be composed from a cascade of propa-

gator matrices (Gilbert & Backus 1966; Kennett 1972a,b; Woodhouse 1974; Haines 1988). It

is similar to equation (50) with the cascade of Green’s matrices, but unlike in equation (50),

26 Wapenaar

where x3,A > x3,0 > x3,B, the arrangement of x3,A, x3,B and x3,C in equation (60) is arbitrary

(since there are no sources in both states).

Finally, we replace xB in equation (60) by x′A = (x′H,A, x3,A). Applying boundary condition

(29) to the left-hand side, we obtain

Iδ(xH,A − x′H,A) =

∫∂DC

W(xA,x, ω)W(x,x′A, ω)dx. (61)

This equation defines W(xA,x, ω) as the inverse of W(x,xA, ω) (Gilbert & Backus 1966;

Kennett 1972a,b; Woodhouse 1974; Haines 1988).

5.3 Representations of the correlation type with the propagator matrix

We aim to derive a representation of the correlation type for the actual wave-field vector

q(x, ω), emitted by the actual source distribution d(x, ω) in the actual medium. Similar as in

section 5.2, we let state B be this actual state. For state A we choose the adjoint propagator-

matrix state. Hence, we replace qA(x, ω), AA(x, ω) and dA(x, ω) by W(x,xA, ω), AA(x, ω)

and O, respectively. Furthermore, we replace ∂D0 ∪ ∂DM again by ∂D0 ∪ ∂DA. Applying

these substitutions in reciprocity theorem (41), using boundary condition (29) and symmetry

relation (56) (with xB replaced by x) yields again equation (58). Hence, due to the additional

symmetry property of the propagator matrix (equation (56)), the correlation-type reciprocity

theorem does not lead to a new representation.

6 REPRESENTATIONS WITH GREEN’S MATRICES AND PROPAGATOR

MATRICES

We follow a similar approach as in sections 4 and 5 to derive wave-field representations.

However, this time we replace one of the states in the reciprocity theorem of the convolution

type by a Green’s state and the other by a propagator-matrix state. This leads to wave-field

representations with Green’s matrices and propagator matrices.


6.1 Single-sided Green’s matrix representation

In reciprocity theorem (40) we choose for state A the propagator-matrix state and for state

B the Green’s state. Hence, in state A we replace qA(x, ω) and dA(x, ω) by W(x,xA, ω) and

O, respectively. In state B we replace qB(x, ω) and dB(x, ω) by G(x,xB, ω) and Iδ(x− xB),

respectively. The medium parameters in states A and B may be different, hence, we keep

the subscripts in AA(x, ω) and AB(x, ω). Last but not least, we replace ∂D0 ∪ ∂DM by

∂D0 ∪ ∂DA and D by the enclosed region DA, see Figure 6. With these substitutions, using

boundary condition (29) and symmetry relation (55), reciprocity theorem (40) yields

G(xA,xB, ω)− χDA(xB)W(xA,xB, ω) =∫

∂D0

W(xA,x, ω)G(x,xB, ω)dx−∫DA

W(xA,x, ω){AA −AB}G(x,xB, ω)dx, (62)

where χDA(x) is the characteristic function for domain DA, defined as

χDA(x) =

1, for x inside DA,

12 , for x on ∂D0 ∪ ∂DA,

0, for x outside DA.

(63)

Equation (62) is a representation for G −W (when xB is inside DA) in terms of integrals

containing G and W. When G and W are defined in the same medium, this representation

simplifies to

G(xA,xB, ω)− χDA(xB)W(xA,xB, ω) =

∫∂D0

W(xA,x, ω)G(x,xB, ω)dx. (64)

Finally, when xB (the position of the source of the Green’s function) lies outside DA (i.e.,

above ∂D0 or below ∂DA), then the latter expression simplifies to

G(xA,xB, ω) =

∫∂D0

W(xA,x, ω)G(x,xB, ω)dx, forx3,B < x3,0 orx3,B > x3,A. (65)

This is a single-sided representation of the Green’s matrix that uses the propagator matrix.

This special case was already presented in equation (31) (except that the roles of x and xA

are interchanged since we used the symmetry property of equation (55) in the derivation of

equation (65)).

28 Wapenaar

Note the analogy of equation (65) with equation (50), which contains a Green’s matrix

instead of the propagator matrix. The propagator matrix in equation (65) depends only on the

medium parameters between x3,0 and x3,A. Moreover, unlike the representation of equation

(50), which only holds for x3,B < x3,0, equation (65) also holds for x3,B > x3,A.

6.2 Single-sided homogeneous Green’s matrix representation

We follow the same procedure as in section 6.1, except in state B we choose the homoge-

neous Green’s function, hence, we replace qB(x, ω) and dB(x, ω) by Gh(x,xB, ω) and O,

respectively. We thus obtain

Gh(xA,xB, ω) =

∫∂D0

W(xA,x, ω)Gh(x,xB, ω)dx

−∫DA

W(xA,x, ω){AA −AB}Gh(x,xB, ω)dx. (66)

When Gh and W are defined in the same medium, this representation simplifies to

Gh(xA,xB, ω) =

∫∂D0

W(xA,x, ω)Gh(x,xB, ω)dx. (67)

This is a single-sided representation of the homogeneous Green’s matrix that uses the propa-

gator matrix. This special case was already presented in equation (32) (except that the roles

of x and xA are interchanged since we used the symmetry property of equation (55) in the

derivation of equation (67)). Note that, unlike the Green’s matrix representation of equation

(65), the homogeneous Green’s matrix representation of equation (67) has no restrictions for

the position of xB (since there is no source at xB).

Equation (67) is the single-sided counterpart of the unified homogeneous Green’s matrix

representation of equation (52). Whereas in equation (52) the integral is taken along two

boundaries ∂D0 and ∂DM , in equation (67) the integral is taken only along ∂D0. This is an

important advantage for practical situations, where measurements are often available only on

a single boundary. In section 6.3 we indicate how equation (67) can be used in a process called

source and receiver redatuming.

Finally, using equations (16), (17), (26), (33) and (35) for the isotropic acoustic situation,


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xB<latexit sha1_base64="Tkfhzj1LiWSS+oXCwiCWQtaB+8w=">AAAB8HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygvmQ5Ah7m71kye7esbsnhiO/wsZCEVt/jp3/xs3lCk18MPB4b4aZeUHMmTau++0UVlbX1jeKm6Wt7Z3dvfL+QUtHiSK0SSIeqU6ANeVM0qZhhtNOrCgWAaftYHwz89uPVGkWyXsziakv8FCykBFsrPSQ9oIQPU371/1yxa26GdAy8XJSgRyNfvmrN4hIIqg0hGOtu54bGz/FyjDC6bTUSzSNMRnjIe1aKrGg2k+zg6foxCoDFEbKljQoU39PpFhoPRGB7RTYjPSiNxP/87qJCS/9lMk4MVSS+aIw4chEaPY9GjBFieETSzBRzN6KyAgrTIzNqGRD8BZfXiatWtU7q9buziv1qzyOIhzBMZyCBxdQh1toQBMICHiGV3hzlPPivDsf89aCk88cwh84nz9yRpAn</latexit>

W(xS ,xB ,!)<latexit sha1_base64="bxt9QD1OnVLe5vZ/zXMRAubtKQI=">AAACDHicbVDNSgMxGMz6W+tf1aOXYBEqlLJbBT14KHrxWNH+QHcp2TTbhibZJcmKZekDePFVvHhQxKsP4M23Md2uoK0DIcPMfCTf+BGjStv2l7WwuLS8sppby69vbG5tF3Z2myqMJSYNHLJQtn2kCKOCNDTVjLQjSRD3GWn5w8uJ37ojUtFQ3OpRRDyO+oIGFCNtpG6hmLh+AFvjUnrfj7s35R92UXZDTvroyKTsip0CzhMnI0WQod4tfLq9EMecCI0ZUqrj2JH2EiQ1xYyM826sSITwEPVJx1CBOFFeki4zhodG6cEglOYIDVP190SCuFIj7pskR3qgZr2J+J/XiXVw5iVURLEmAk8fCmIGdQgnzcAelQRrNjIEYUnNXyEeIImwNv3lTQnO7MrzpFmtOMeV6vVJsXae1ZED++AAlIADTkENXIE6aAAMHsATeAGv1qP1bL1Z79PogpXN7IE/sD6+ASX9mmM=</latexit>

. . . . . .xS . . . . . .<latexit sha1_base64="d4QudPiCZJ+O8eUb5jzyDSaWGcQ=">AAACBnicbVDLSgMxFM34rPU16lKEYBFclZkq6MJFwY3LivYB7VAyaaYNzWSG5I5Yhq7c+CtuXCji1m9w59+YTgfU1gMJh3PuIbnHjwXX4Dhf1sLi0vLKamGtuL6xubVt7+w2dJQoyuo0EpFq+UQzwSWrAwfBWrFiJPQFa/rDy4nfvGNK80jewihmXkj6kgecEjBS1z7o9CLQ2ZV2/ADfj7s3P1LXLjllJwOeJ25OSihHrWt/mhxNQiaBCqJ123Vi8FKigFPBxsVOollM6JD0WdtQSUKmvTRbY4yPjNLDQaTMkYAz9XciJaHWo9A3kyGBgZ71JuJ/XjuB4NxLuYwTYJJOHwoSgSHCk05wjytGQYwMIVRx81dMB0QRCqa5oinBnV15njQqZfekXLk+LVUv8joKaB8domPkojNURVeohuqIogf0hF7Qq/VoPVtv1vt0dMHKM3voD6yPbyYwmYk=</latexit>

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(c)


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W(xS ,xB ,!)<latexit sha1_base64="bxt9QD1OnVLe5vZ/zXMRAubtKQI=">AAACDHicbVDNSgMxGMz6W+tf1aOXYBEqlLJbBT14KHrxWNH+QHcp2TTbhibZJcmKZekDePFVvHhQxKsP4M23Md2uoK0DIcPMfCTf+BGjStv2l7WwuLS8sppby69vbG5tF3Z2myqMJSYNHLJQtn2kCKOCNDTVjLQjSRD3GWn5w8uJ37ojUtFQ3OpRRDyO+oIGFCNtpG6hmLh+AFvjUnrfj7s35R92UXZDTvroyKTsip0CzhMnI0WQod4tfLq9EMecCI0ZUqrj2JH2EiQ1xYyM826sSITwEPVJx1CBOFFeki4zhodG6cEglOYIDVP190SCuFIj7pskR3qgZr2J+J/XiXVw5iVURLEmAk8fCmIGdQgnzcAelQRrNjIEYUnNXyEeIImwNv3lTQnO7MrzpFmtOMeV6vVJsXae1ZED++AAlIADTkENXIE6aAAMHsATeAGv1qP1bL1Z79PogpXN7IE/sD6+ASX9mmM=</latexit>

xA<latexit sha1_base64="TDsRQAs2KT49b9WGg/tL2oSeaNw=">AAAB8HicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVFxMYygvmQ5Ah7m71kye7esbsnhiO/wsZCEVt/jp3/xs3lCk18MPB4b4aZeUHMmTau++0UVlbX1jeKm6Wt7Z3dvfL+QUtHiSK0SSIeqU6ANeVM0qZhhtNOrCgWAaftYHwz89uPVGkWyXsziakv8FCykBFsrPSQ9oIQPU371/1yxa26GdAy8XJSgRyNfvmrN4hIIqg0hGOtu54bGz/FyjDC6bTUSzSNMRnjIe1aKrGg2k+zg6foxCoDFEbKljQoU39PpFhoPRGB7RTYjPSiNxP/87qJCS/9lMk4MVSS+aIw4chEaPY9GjBFieETSzBRzN6KyAgrTIzNqGRD8BZfXiatWtU7q9buziv1qzyOIhzBMZyCBxdQh1toQBMICHiGV3hzlPPivDsf89aCk88cwh84nz9wwpAm</latexit>

Gh(xA,xB ,!)<latexit sha1_base64="mrLFcLkxfNadOqLjEJr+UkgHZgg=">AAACFHicbZBNS8MwGMdTX+d8q3r0EhzCxDHaKejBw9SDHie4F1hHSbN0C0vakqTiKP0QXvwqXjwo4tWDN7+NWTdBN/8Q+OX/PA/J8/ciRqWyrC9jbn5hcWk5t5JfXVvf2DS3thsyjAUmdRyyULQ8JAmjAakrqhhpRYIg7jHS9AaXo3rzjghJw+BWDSPS4agXUJ9ipLTlmoeJ4/nwKnUTR3DYT4vZ/T51z0s/dFFyQk566MA1C1bZygRnwZ5AAUxUc81PpxvimJNAYYakbNtWpDoJEopiRtK8E0sSITxAPdLWGCBOZCfJlkrhvna60A+FPoGCmft7IkFcyiH3dCdHqi+nayPzv1o7Vv5pJ6FBFCsS4PFDfsygCuEoIdilgmDFhhoQFlT/FeI+EggrnWNeh2BPrzwLjUrZPipXbo4L1bNJHDmwC/ZAEdjgBFTBNaiBOsDgATyBF/BqPBrPxpvxPm6dMyYzO+CPjI9vQ4Gdqw==</latexit>

W(xA,xR,!)<latexit sha1_base64="vTMB9rYw1UG5tZFsW0tPuj0hWnw=">AAACDHicbVDNSgMxGMz6W+tf1aOXYBEqlLJbBT14qHjxWMX+QHcp2TTbhibZJcmKZekDePFVvHhQxKsP4M23Md2uoK0DIcPMfCTf+BGjStv2l7WwuLS8sppby69vbG5tF3Z2myqMJSYNHLJQtn2kCKOCNDTVjLQjSRD3GWn5w8uJ37ojUtFQ3OpRRDyO+oIGFCNtpG6hmLh+AFvjUnrfj7sX5R92U3ZDTvroyKTsip0CzhMnI0WQod4tfLq9EMecCI0ZUqrj2JH2EiQ1xYyM826sSITwEPVJx1CBOFFeki4zhodG6cEglOYIDVP190SCuFIj7pskR3qgZr2J+J/XiXVw5iVURLEmAk8fCmIGdQgnzcAelQRrNjIEYUnNXyEeIImwNv3lTQnO7MrzpFmtOMeV6vVJsXae1ZED++AAlIADTkENXIE6aAAMHsATeAGv1qP1bL1Z79PogpXN7IE/sD6+ASIxmmE=</latexit>

Figure 7. Illustration of source and receiver redatuming. All responses in this figure are representedby simple rays, but in reality these are multi-component wave fields, including primaries, multiples,converted, refracted and evanescent waves. (a) The response G(xR,xS , ω) at the surface. The homo-geneous Green’s matrix Gh(xR,xS , ω) is obtained from this response with equation (25). (b) Sourceredatuming. Using equation (70), the homogeneous Green’s matrix Gh(xR,xB , ω) is obtained for avirtual source at xB . (c) Receiver redatuming. Using equation (71), the homogeneous Green’s matrixGh(xA,xB , ω) is obtained for a virtual receiver at xA.

assuming real-valued ρ(x), equation (67) yields for the upper-right element of Gh(xA,xB, ω)

Gp,qh (xA,xB, ω) = − 1

iω

∫∂D0

1

ρ(x)

({∂3W p,v(xA,x, ω)}Gp,qh (x,xB, ω)

− W p,v(xA,x, ω)∂3Gp,qh (x,xB, ω)

)dx, (68)

with Gp,qh (xA,xB, ω) defined in equation (54).

30 Wapenaar

6.3 Source and receiver redatuming

Redatuming is the process of moving sources and/or receivers from the acquisition boundary

to positions inside the medium (Berkhout 1982; Berryhill 1984; Wapenaar & Berkhout 1989;

Hokstad 2000). Traditionally this is done with Kirchhoff-Helmholtz integrals with Green’s

functions defined in a smooth background medium, thus ignoring multiple scattering. Here we

derive a unified redatuming method that accounts for multiple scattering, using the single-

sided homogeneous Green’s matrix representation of section 6.2 as the starting point.

Let xS and xR denote source and receiver coordinates, respectively, at the acquisition

boundary ∂D0. Then the Green’s matrix G(xR,xS , ω) represents the medium’s response,

measured with sources and receivers at the surface, see Figure 7(a). Assuming the medium is

lossless, the homogeneous Green’s matrix Gh(xR,xS , ω) is obtained from this Green’s matrix

via equation (25).

First we discuss a method for source redatuming. We rename some variables in equation

(67) (xB → xR, xA → xB, x→ xS) and transpose all matrices, which gives

Gth(xB,xR, ω) =

∫∂D0

Gth(xS ,xR, ω)Wt(xB,xS , ω)dxS . (69)

Using equations (44) and (55) we obtain

Gh(xR,xB, ω) =

∫∂D0

Gh(xR,xS , ω)W(xS ,xB, ω)dxS . (70)

The latter expression describes redatuming of the sources from xS at the acquisition boundary

to a virtual-source position xB inside the medium, see Figure 7(b).

Next, we discuss receiver redatuming. We replace x by xR in equation (67), which gives

Gh(xA,xB, ω) =

∫∂D0

W(xA,xR, ω)Gh(xR,xB, ω)dxR. (71)

This expression describes redatuming of the receivers from xR at the acquisition boundary to

a virtual-receiver position xA inside the medium, see Figure 7(c). The Green’s matrix under

the integral is the output of the source redatuming method, described by equation (70).

Combining equation (70) with equation (71) gives the following expression for combined


source and receiver redatuming

Gh(xA,xB, ω) =

∫∂D0

∫∂D0

W(xA,xR, ω)Gh(xR,xS , ω)W(xS ,xB, ω)dxSdxR. (72)

This expression redatums the actual sources and receivers from the acquisition boundary

∂D0 to virtual sources and receivers inside the medium. It makes use of propagator matrices.

When the medium is known, these matrices can be numerically modelled. Alternatively, the

propagator matrices can be expressed in terms of focusing functions which, at least for the

acoustic situation, can be retrieved via the Marchenko method from the reflection response at

the acquisition boundary. In the next section we introduce relations between the propagator

matrix and focusing functions (section 7.1) and derive Marchenko-type representations which

relate the focusing functions to the reflection response at the acquisition boundary (section

7.2).

7 REPRESENTATIONS WITH FOCUSING FUNCTIONS

7.1 Relation between propagator matrix and focusing functions

In section 2.5 we indicated that there is a relation between the propagator matrix and the

Marchenko-type focusing functions. Here we derive this relation for the propagator matrix of

the unified matrix-vector wave equation, assuming the medium is lossless. Our starting point

is equation (59), which is repeated here for convenience

q(xA, ω) =

∫∂D0

W(xA,x, ω)q(x, ω)dx, (73)

with boundary condition

W(xA,x, ω)|x3,A=x3,0 = Iδ(xH,A − xH), (74)

for x at ∂D0. From here onward we assume that the half-space above ∂D0 (including ∂D0) is

homogeneous and isotropic; the medium below ∂D0 is arbitrary inhomogeneous, anisotropic

and source free, and xA is chosen at or below ∂D0 (hence x3,A ≥ x3,0).

Without loss of generality, we can decompose q(x, ω) in the upper half-space into downgo-

ing and upgoing plane waves. To this end, we defined the spatial Fourier transform u(s, x3, ω)

32 Wapenaar

of a space- and frequency-dependent function u(x, ω) in equation (20). For a function of

two space variables, u(xA,x, ω), we define the spatial Fourier transform along the horizontal

components of the second space variable as

u(xA, s, x3, ω) =

∫R2

u(xA,xH, x3, ω) exp{iωs · xH}dxH. (75)

Using these definitions and Parseval’s theorem, we rewrite equation (73) as

q(xA, ω) =ω2

4π2

∫R2

W(xA, s, x3,0, ω)q(s, x3,0, ω)ds, (76)

with boundary condition

W(xA, s, x3,0, ω)|x3,A=x3,0 = I exp{iωs · xH,A}. (77)

In the upper half-space we relate the wave vector q to downgoing and upgoing wave vectors

p+ and p−, respectively, via

q =

q1

q2

=

L+1 L−1

L+2 L−2

p+

p−

, for x3 ≤ x3,0 (78)

(Kennett et al. 1978; Fishman & McCoy 1984; Ursin 1983; Wapenaar & Berkhout 1989).

Next, we renormalize the downgoing and upgoing wave vectors. To this end, we define the

downgoing and upgoing parts of q1 as

q±1 = L±1 p± (79)

and rewrite equation (78) as

q =

q1

q2

=

I I

D+1 D−1

︸︷︷︸

D1

q+1

q−1

︸︷︷︸

b1

, for x3 ≤ x3,0, (80)

with

D±1 = L±2 (L±1 )−1. (81)

In Appendix A we give explicit expressions for D±1 for the acoustic, electromagnetic and


elastodynamic situation. Substitution of equation (80) into equation (76) gives

q(xA, ω) =ω2

4π2

∫R2

Y1(xA, s, x3,0, ω)b1(s, x3,0, ω)ds, (82)

with

Y1(xA, s, x3,0, ω) = W(xA, s, x3,0, ω)D1(s). (83)

Let F1(xA, s, x3,0, ω) be the upper-right N/2 × N/2 matrix of block matrix Y1(xA, s, x3,0, ω).

According to equations (28), (83) and the definition of D1(s) in equation (80), it is defined as

F1(xA, s, x3,0, ω) = (W11 + W12D−1 )(xA, s, x3,0, ω). (84)

This is a generalization of the definition of the acoustic focusing function for a horizon-

tally layered medium, defined in equation (38). From equations (28) and (77) it follows that

F1(xA, s, x3,0, ω) obeys the boundary condition

F1(xA, s, x3,0, ω)|x3,A=x3,0 = I exp{iωs · xH,A}, (85)

or, applying an inverse spatial and temporal Fourier transform,

F1(xA,x, t)|x3,A=x3,0 = Iδ(xH,A − xH)δ(t), (86)

for x at ∂D0. Hence, F1(xA,x, t) is indeed a focusing function (with xA being a variable and

x being the focal point at ∂D0).

To analyse the upper-left N/2 × N/2 matrix of block matrix Y1(xA, s, x3,0, ω), we first

establish some symmetry properties. The symmetry of W follows from the Fourier transform

of equation (57) for the lossless situation, hence

W(xA, s, x3,0, ω) = JW∗(xA,−s, x3,0, ω)J. (87)

For the sub-matrices of W this implies

Wαβ(xA, s, x3,0, ω) = JααW∗αβ(xA,−s, x3,0, ω)Jββ (88)

(no summation convention) with J11 = −J22 = I. The symmetry of D±1 follows from its

34 Wapenaar

explicit definitions in Appendix A. In all cases, the following symmetry holds

D+1 (s) = −D−1 (−s). (89)

When we ignore evanescent waves at and above ∂D0, then D±1 (s) is real-valued, see Appendix

A. Hence, given that F1(xA, s, x3,0, ω) is the upper-rightN/2×N/2 matrix of Y1(xA, s, x3,0, ω),

we derive from equations (28), (83), (88) and (89) and the structure of D1 indicated in equa-

tion (80), that the upper-left N/2 × N/2 matrix is equal to F∗1(xA,−s, x3,0, ω). Using this in

equation (82) we obtain for the upper N/2 × 1 vector of q(xA, ω)

q1(xA, ω) =ω2

4π2

∫R2

F∗1(xA,−s, x3,0, ω)q+1 (s, x3,0, ω)ds

+ω2

4π2

∫R2

F1(xA, s, x3,0, ω)q−1 (s, x3,0, ω)ds. (90)

Applying Parseval’s theorem again, we obtain

q1(xA, ω) =

∫∂D0

F∗1(xA,x, ω)q+1 (x, ω)dx +

∫∂D0

F1(xA,x, ω)q−1 (x, ω)dx. (91)

This relation has previously been derived via another route for the situations of acoustic waves

(q1 = p) and elastodynamic waves (q1 = v) (Wapenaar et al. 2021). In section 7.2 we use

equation (91) as the basis for deriving Marchenko-type Green’s matrix representations.

Here we derive a representation similar to equation (91) for q2(xA, ω). To this end, we

define the downgoing and upgoing parts of q2 as

q±2 = L±2 p± (92)

and rewrite equation (78), analogous to equation (80), as

q =

q1

q2

=

D+2 D−2

I I

︸︷︷︸

D2

q+2

q−2

︸︷︷︸

b2

, for x3 ≤ x3,0, (93)

with

D±2 = L±1 (L±2 )−1 = (D±1 )−1. (94)


Substitution of equation (93) into equation (76) gives

q(xA, ω) =ω2

4π2

∫R2

Y2(xA, s, x3,0, ω)b2(s, x3,0, ω)ds, (95)

with

Y2(xA, s, x3,0, ω) = W(xA, s, x3,0, ω)D2(s). (96)

Let F2(xA, s, x3,0, ω) be the lower-right N/2 × N/2 matrix of block-matrix Y2(xA, s, x3,0, ω).

According to equations (28), (96) and the definition of D2(s) in equation (93), it is defined as

F2(xA, s, x3,0, ω) = (W21D−2 + W22)(xA, s, x3,0, ω). (97)

F2 is a focusing function, with similar focusing properties as F1, expressed in equations (85)

and (86).

To analyse the lower-left N/2 × N/2 matrix of block matrix Y2(xA, s, x3,0, ω), we first

derive from equations (89) and (94)

D+2 (s) = −D−2 (−s). (98)

When we ignore evanescent waves at and above ∂D0, then D±2 (s) is real-valued. Hence, given

that F2(xA, s, x3,0, ω) is the lower-right N/2 × N/2 matrix of Y2(xA, s, x3,0, ω), we derive

from equations (28), (88), (96) and (98) and the structure of D2 indicated in equation (93),

that the lower-left N/2 × N/2 matrix is equal to F∗2(xA,−s, x3,0, ω). Using this in equation

(95) we obtain for the lower N/2 × 1 vector of q(xA, ω)

q2(xA, ω) =ω2

4π2

∫R2

F∗2(xA,−s, x3,0, ω)q+2 (s, x3,0, ω)ds

+ω2

4π2

∫R2

F2(xA, s, x3,0, ω)q−2 (s, x3,0, ω)ds. (99)

Applying Parseval’s theorem again, we obtain

q2(xA, ω) =

∫∂D0

F∗2(xA,x, ω)q+2 (x, ω)dx +

∫∂D0

F2(xA,x, ω)q−2 (x, ω)dx. (100)

From equations (79), (81) and (92) we obtain q±2 = D±1 q±1 . Substituting this into equation

(99) we obtain a representation for q2(xA, ω) in terms of q+1 and q−1 . Combining this with

36 Wapenaar

equation (90) into a single equation, we obtain equation (82), with (using equation (89))

Y1(xA, s, x3,0, ω) =

F∗1(xA,−s, x3,0, ω) F1(xA, s, x3,0, ω)

−F∗2(xA,−s, x3,0, ω)D−1 (−s) F2(xA, s, x3,0, ω)D−1 (s)

. (101)

Once Y1 is known, the propagator matrix W follows from inverting equation (83), according

to

W(xA, s, x3,0, ω) = Y1(xA, s, x3,0, ω){D1(s)}−1. (102)

Equations (101) and (102), with D1(s) defined in equation (80), express the propagator matrix

explicitly in terms of focusing functions.

For the acoustic situation, using equation (A.16), equations (84) and (97) become

F p(xA, s, x3,0, ω) =(W p,p − s3,0

ρ0W p,v

)(xA, s, x3,0, ω), (103)

F v(xA, s, x3,0, ω) =(− ρ0s3,0

W v,p + W v,v)

(xA, s, x3,0, ω), (104)

with s3 defined in equation (A.15). Equation (101) yields for the acoustic situation

Y1(xA, s, x3,0, ω) =

{F p(xA,−s, x3,0, ω)}∗ F p(xA, s, x3,0, ω)

s3,0ρ0{F v(xA,−s, x3,0, ω)}∗ − s3,0

ρ0F v(xA, s, x3,0, ω)

. (105)

Using this in equation (102), together with

{D1(s)}−1 =

12

ρ02s3,0

12 − ρ0

2s3,0

, (106)

we obtain an explicit expression for W(xA, s, x3,0, ω) in terms of acoustic focusing functions.

Transforming this to the space-frequency domain, replacing s3,0 by 1iω∂3, yields

W(xA,x, ω) =

<{F p(xA,x, ω)} −iωρ0<{∂−13 F p(xA,x, ω)}−1iωρ0<{∂3F v(xA,x, ω)} <{F v(xA,x, ω)}

, (107)

for x at ∂D0, where ∂−13 stands for indefinite integration in the x3-direction with zero inte-

gration constant.


7.2 Marchenko-type Green’s matrix representations with focusing functions

We use equations (91) and (100) as the starting point for deriving two Marchenko-type Green’s

function representations. We follow a similar procedure as reference (Wapenaar et al. 2021,

Appendix B), generalized for the different wave phenomena considered in this paper. We

replace the N/2 × 1 vector q1(x, ω) by a modified version Γ12(x,xS , ω) of the N/2 × N/2

Green’s matrix G12(x,xS , ω). Here G12(x,xS , ω) stands for the q1-type field observed at x, in

response to a unit d2-type source at xS . We choose xS = (xH,S , x3,S) in the upper half-space,

at a vanishing distance ε above ∂D0, hence, x3,S = x3,0− ε. Our aim is to modify this Green’s

matrix such that for x at ∂D0 (i.e., just below the source) its downgoing part simplifies to

Γ+12(x,xS , ω)|x3=x3,0 = Iδ(xH − xH,S). (108)

First we derive the properties of the downgoing part of G12(x,xS , ω) for x at ∂D0. To this

end, consider the inverse of equation (80), which readsq+1

q−1

=

−(∆1)−1D−1 (∆1)

−1

(∆1)−1D+

1 −(∆1)−1

︸︷︷︸

(D1)−1

q1

q2

, (109)

for x3 ≤ x3,0, with

∆1 = D+1 − D−1 . (110)

The upper-right N/2 × N/2 matrix (∆1)−1 transforms q2 into a downgoing field vector q+

1 .

In a similar way, this matrix transforms a unit d2-type source in a homogeneous half-space

into the downgoing part of the Green’s matrix G12 just below this source, according to

limx3↓x3,S

G+12(s, x3,0, x3,S , ω) = {∆1(s)}−1. (111)

Explicit expressions for (∆1)−1 are given in Appendix A. In equation (111) the source is

located at (0, x3,S). Next, we consider G12(x,xS , ω) for a laterally shifted source position

(xH,S , x3,S). Applying a spatial Fourier transform along the horizontal source coordinate xH,S ,

using equation (75) with xH replaced by xH,S , yields G12(x, s, x3,S , ω). For the downgoing part

just below the source we obtain a phase-shifted version of the Green’s function of equation

38 Wapenaar

(111), according to

limx3↓x3,S

G+12(x, s, x3,S , ω) = {∆1(s)}−1 exp{iωs · xH}. (112)

This suggests to define the modified Green’s matrix as

Γ12(x, s, x3,S , ω) = G12(x, s, x3,S , ω)∆1(s), (113)

such that

limx3↓x3,S

Γ+12(x, s, x3,S , ω) = I exp{iωs · xH}. (114)

Transforming this back to the space domain yields indeed equation (108). Next, we define

the reflection response R12(x,xS , ω) of the medium below ∂D0 as the upgoing part of the

modified Green’s function Γ12(x,xS , ω), with x at ∂D0, hence

R12(x,xS , ω) = Γ−12(x,xS , ω). (115)

Substituting q1(xA, ω) = Γ12(xA,xS , ω) and q±1 (x, ω) = Γ±12(x,xS , ω) into equation (91),

using equations (108) and (115), gives

Γ12(xA,xS , ω) =

∫∂D0

F1(xA,x, ω)R12(x,xS , ω)dx + F∗1(xA,xS , ω), (116)

for x3,A ≥ x3,0. This is the first Marchenko-type representation. In a similar way, we derive

a second Marchenko-type representation, for a modified version Γ22(x,xS , ω) of the Green’s

matrix G22(x,xS , ω). We derive the properties of the downgoing part of G22(x,xS , ω) for x

at ∂D0. Consider the inverse of equation (93), which readsq+2

q−2

=

(∆2)−1 −(∆2)

−1D−2

−(∆2)−1 (∆2)

−1D+2

︸︷︷︸

(D2)−1

q1

q2

, (117)

for x3 ≤ x3,0, with

∆2 = D+2 − D−2 . (118)


The upper-right N/2 × N/2 matrix −(∆2)−1D−2 transforms q2 into a downgoing field vector

q+2 . In a similar way, this matrix transforms a unit d2-type source in a homogeneous half-space

into the downgoing part of the Green’s matrix G22 just below this source, according to

limx3↓x3,S

G+22(s, x3,0, x3,S , ω) = −{∆2(s)}−1D−2 (s). (119)

Explicit expressions for −(∆2)−1D−2 are given in Appendix A. Similar steps as below equation

(111) lead to

limx3↓x3,S

G+22(x, s, x3,S , ω) = −{∆2(s)}−1D−2 (s) exp{iωs · xH}. (120)

This suggests to define the modified Green’s matrix as

Γ22(x, s, x3,S , ω) = −G22(x, s, x3,S , ω){D−2 (s)}−1∆2(s), (121)

such that

Γ+22(x,xS , ω)|x3=x3,0 = Iδ(xH − xH,S). (122)

We define the reflection response R22(x,xS , ω) as

R22(x,xS , ω) = Γ−22(x,xS , ω) (123)

for x at ∂D0. Substituting q2(xA, ω) = Γ22(xA,xS , ω) and q±2 (x, ω) = Γ±22(x,xS , ω) into

equation (100), using equations (122) and (123), gives

Γ22(xA,xS , ω) =

∫∂D0

F2(xA,x, ω)R22(x,xS , ω)dx + F∗2(xA,xS , ω), (124)

for x3,A ≥ x3,0. This is the second Marchenko-type representation.

For the acoustic case, using equations (A.16) and (A.17), equations (113) and (121) become

Γ12(x, s, x3,S , ω) =2s3,0ρ0

Gp,q(x, s, x3,S , ω), (125)

= − 2

iωρ0∂3,SG

p,q(x, s, x3,S , ω)

Γ22(x, s, x3,S , ω) = 2Gv,q(x, s, x3,S , ω), (126)

40 Wapenaar

or, in the space-frequency domain (using equation (18)),

Γ12(x,xS , ω) = 2Gp,f (x,xS , ω), (127)

Γ22(x,xS , ω) = 2Gv,q(x,xS , ω). (128)

For this situation, R12(x,xS , ω) and R22(x,xS , ω) in the representations of equations (116)

and (124) are the upgoing parts of 2Gp,f (x,xS , ω) and 2Gv,q(x,xS , ω), respectively, for x at

∂D0. Note that, according to equation (43), Gv,q(x,xS , ω) = −Gp,f (xS ,x, ω).

In their general form, the representations of equations (116) and (124) are generalizations

of previously derived representations for the 3D Marchenko method for acoustic (Wapenaar

et al. 2013; Slob et al. 2014; Van der Neut et al. 2015) and elastodynamic wave fields (Wape-

naar & Slob 2014; da Costa Filho et al. 2014) (but note that the subscripts 1 and 2 of the

focusing functions have a different meaning than in those papers; here they refer to the wave-

field components q1 and q2). In most previous work on the 3D Marchenko method, one of

the underlying assumptions is that the wave field can be decomposed into downgoing and

upgoing waves in the interior of the medium. Only recently several authors proposed to avoid

decomposition inside the medium (Kiraz et al. 2021; Diekmann & Vasconcelos 2021; Wape-

naar et al. 2021). The representations discussed in this section expand on this. Only in the

homogeneous isotropic upper half-space the waves are decomposed into downgoing and up-

going waves and evanescent waves are neglected. However, inside the medium no wave-field

decomposition takes place. Moreover, evanescent waves inside the medium (for example in

high-velocity layers) are accounted for by the representations of equations (116) and (124).

These representations, transformed to the time domain, form the basis for the development of

Marchenko schemes, aiming at resolving the focusing functions F1 and F2 from the reflection

responses R12 and R22 at the acquisition boundary ∂D0. Such schemes have been successfully

developed for precritical acoustic data (Ravasi et al. 2016; Staring et al. 2018; Jia et al. 2018;

Brackenhoff et al. 2019; Pereira et al. 2019; Zhang & Slob 2020; Staring & Wapenaar 2020).

For more complex situations, research on retrieving the focusing functions from reflection re-

sponses is ongoing (for example (Reinicke et al. 2020)). Once the focusing functions are found,

they can be used to define the propagator matrix via equations (101) and (102), which can

subsequently be used in the representations of sections 5 and 6.


8 CONCLUSIONS

We have discussed different types of wave-field representation in a systematic way. Classical

wave-field representations contain Green’s functions. Starting with a unified matrix-vector

wave equation, we have formulated wave-field representations with Green’s matrices, analo-

gous to the classical representations. For example, the classical Kirchhoff-Helmholtz integral

follows as a special case of the unified representation with the Green’s matrix. Another special

case is the classical homogeneous Green’s function representation.

Using the same matrix-vector formalism, we formulated wave-field representations with

propagator matrices. Unlike a Green’s matrix, a propagator matrix depends only on the

medium parameters between the two depth levels for which this matrix is defined. The rep-

resentations with the propagator matrices have a similar form as those with the Green’s ma-

trices. An important difference is that the boundary integrals in the representations with the

propagator matrices are single-sided. We also formulated representations containing Green’s

matrices and propagator matrices. A special case is the single-sided homogeneous Green’s func-

tion representation, as the counterpart of the classical closed-boundary homogeneous Green’s

function representation.

We have shown that the propagator matrix is related to Marchenko-type focusing func-

tions. We have used this relation to reformulate the representations with the propagator

matrix into representations with focusing functions. For the acoustic situation, these focusing

functions can be retrieved from the single-sided reflection response of the medium by applying

the Marchenko method. For more complex situations, research on retrieving these focusing

functions from the reflection response is ongoing. Once the focusing functions are known, they

can be used to construct the propagator matrix. Subsequently, the propagator matrix can be

used in the representations to obtain the wave field inside the medium which, in turn, can

be used for example for imaging or monitoring. Unlike earlier imaging methods that use the

propagator matrix, this is a data-driven approach (since the propagator matrix is retrieved

from the reflection response) and hence it does not require a detailed model of the medium.

42 Wapenaar

ACKNOWLEDGMENTS

The author thanks Roel Snieder, Sjoerd de Ridder and Evert Slob for fruitful discussions. This

research is funded by the European Research Council (ERC) under the European Union’s

Horizon 2020 research and innovation programme (grant agreement No: 742703).

APPENDIX A: EXPLICIT EXPRESSIONS FOR SOME MATRICES IN THE

HOMOGENEOUS ISOTROPIC UPPER HALF-SPACE

We derive explicit expressions for D±1 , (∆1)−1 and −(∆2)

−1D−2 in the homogeneous isotropic

upper half-space x3 ≤ x3,0 for the acoustic, electromagnetic and elastodynamic situation.

UNIFIED EXPRESSIONS IN HORIZONTAL SLOWNESS DOMAIN

For the lossless homogeneous isotropic upper half-space x3 ≤ x3,0, we transform the unified

matrix-vector wave equation (1) to the horizontal slowness domain, using the spatial Fourier

transform of equation (20). This yields

∂3q− Aq = d, (A.1)

with

q =

q1

q2

, d =

d1

d2

, A =

A11 A12

A21 A22

. (A.2)

The symmetry property defined by equation (3) transforms to

{A(−s, ω)}tN = −NA(s, ω). (A.3)

We decompose matrix A as follows

A = LΛL−1, (A.4)

with

L =

L+1 L−1

L+2 L−2

, Λ =

Λ+

O

O Λ−

. (A.5)


With a proper scaling of the columns of matrix L, these matrices obey the following symmetry

properties

{L(−s)}tN = −N{L(s)}−1, (A.6)

{Λ(−s, ω)}tN = −NΛ(s, ω). (A.7)

Substitution of equation (A.4) into equation (A.1) and premultiplying all terms with L−1

yields

∂3p− Λp = L−1d, (A.8)

with

p = L−1q =

p+

p−

, (A.9)

where p+ and p− are wave vectors containing downgoing (+) and upgoing (−) waves, respec-

tively. Following equations (81), (94), (110) and (118) we define

D±1 (s) = L±2 (L±1 )−1, {∆1(s)}−1 = (D+1 − D−1 )−1, (A.10)

D±2 (s) = L±1 (L±2 )−1, {∆2(s)}−1 = (D+2 − D−2 )−1. (A.11)

Acoustic wave equation

For the acoustic wave equation the 1× 1 sub-matrices of A are

A12 = iωρ, A21 = iω(κ− s2r/ρ), A11 = A22 = 0, (A.12)

with

s2r = s21 + s22. (A.13)

Here κ and ρ are the compressibility and mass density of the upper half-space. For convenience,

here and in the remainder of this appendix we do not use subscripts 0 to indicate parameters

44 Wapenaar

of the upper half-space. The 1× 1 sub-matrices of L and Λ are

L±1 =( ρ

2s3

)1/2, L±2 = ±

( s32ρ

)1/2, Λ

±= ±iωs3, (A.14)

with the vertical slowness s3 defined as

s3 =

√

1/c2 − s2r , for s2r ≤ 1/c2,

i√s2r − 1/c2, for s2r > 1/c2,

(A.15)

with propagation velocity c defined as c = 1/√κρ. The two expressions in equation (A.15) rep-

resent propagating and evanescent waves, respectively. Upon substitution of equation (A.14)

into equations (A.10) and (A.11) we obtain

D±1 (s) = ±s3ρ, {∆1(s)}−1 =

ρ

2s3, (A.16)

D±2 (s) = ± ρ

s3, −{∆2(s)}−1D−2 (s) =

1

2. (A.17)

Electromagnetic wave equation

For the electromagnetic wave equation the 2× 2 sub-matrices of A are

A12 = iω

µ− s21ε − s1s2

ε

− s1s2ε µ− s22

ε

, A21 = iω

ε− s22µ

s1s2µ

s1s2µ ε− s21

µ

, (A.18)

A11 = A22 = O. (A.19)

Here ε and µ are the permittivity and permeability of the upper half-space. The 2 × 2 sub-

matrices of L and Λ are

L±1 =i√2sr

s1( s3ε )1/2 −s2( µs3

)1/2s2(s3ε

)1/2s1( µs3

)1/2 , (A.20)

L±2 = ± i√2sr

s1( εs3 )1/2 −s2(s3µ

)1/2s2(εs3

)1/2s1(s3µ

)1/2 , (A.21)

Λ±

= ±iω

s3 0

0 s3

, (A.22)


with sr and s3 defined in equations (A.13) and (A.15), respectively, but this time with prop-

agation velocity c defined as c = 1/√εµ. Upon substitution of equations (A.20) and (A.21)


D±1 (s) = ± 1

µs3

c−20 − s22 s1s2

s1s2 c−20 − s21

, (A.23)

{∆1(s)}−1 =1

2εs3

c−20 − s21 −s1s2−s1s2 c−20 − s22

, (A.24)

D±2 (s) = ± 1

εs3

c−20 − s21 −s1s2−s1s2 c−20 − s22

, (A.25)

−(∆2)−1D−2 =

12 0

0 12

. (A.26)

Elastodynamic wave equation

For the elastodynamic wave equation the 3× 3 sub-matrices of A are

A11 = iω

0 0 −s10 0 −s2

− λλ+2µs1 − λ

λ+2µs2 0

, (A.27)

A12 = iω

1µ 0 0

0 1µ 0

0 0 1λ+2µ

, (A.28)

A21 = iω

ρ− ν1s21 − µs22 −(ν2 + µ)s1s2 0

−(ν2 + µ)s1s2 ρ− µs21 − ν1s22 0

0 0 ρ

, (A.29)

A22 = (A11)t, (A.30)

with

ν1 = 4µ( λ+ µ

λ+ 2µ

), ν2 = 2µ

( λ

λ+ 2µ

), (A.31)

46 Wapenaar

where λ and µ are the Lame parameters and ρ the mass density of the upper half-space. The

3× 3 sub-matrices of L and Λ are

L±1 =1

(2ρ0)1/2

s1

(sP3 )1/2− s1(sS3 )

1/2

sr− s2cSsr(s

S3 )

1/2

s2(sP3 )1/2

− s2(sS3 )1/2

srs1

cSsr(sS3 )

1/2

±(sP3 )1/2 ± sr(sS3 )

1/2 0

, (A.32)

L±2 =(ρ0

2

)1/2c2S

±2s1(s

P3 )1/2 ∓ s1(c

−2S −2s

2r)

sr(sS3 )1/2 ∓ s2(sS3 )

1/2

cSsr

±2s2(sP3 )1/2 ∓ s2(c

−2S −2s

2r)

sr(sS3 )1/2 ± s1(sS3 )

1/2

cSsr

(c−2S −2s

2r)

(sP3 )1/22sr(s

S3 )1/2 0

, (A.33)

Λ±

= ±iω

sP3 0 0

0 sS3 0

0 0 sS3

, (A.34)

with sr defined in equation (A.13), and the vertical slownesses sP3 and sS3 defined as

sP,S3 =

√

1/c2P,S − s2r , for s2r ≤ 1/c2P,S ,

i√s2r − 1/c2P,S , for s2r > 1/c2P,S .

(A.35)

Here cP and cS are the P - and S-wave velocities of the upper half-space, defined as cP =√(λ+ 2µ)/ρ and cS =

√µ/ρ, respectively. Upon substitution of equations (A.32) and (A.33)


D±1 (s) =ρc2S

sP3 sS3 + s2r

±((c−2S − s22)sP3 + s22s

S3 ) ±s1s2(sP3 − sS3 ) −s1(c−2S − 2S)

±s1s2(sP3 − sS3 ) ±((c−2S − s21)sP3 + s21sS3 ) −s2(c−2S − 2S)

s1(c−2S − 2S) s2(c

−2S − 2S) ±sS3 c−2S

,

(A.36)

{∆1(s)}−1 =1

2ρ

s21sP3

+(

1c2S− s21

)1sS3

(1sP3− 1

sS3

)s1s2 0(

1sP3− 1

sS3

)s1s2

s22sP3

+(

1c2S− s22

)1sS3

0

0 0 sP3 + s2rsS3

, (A.37)

−{∆2(s)}−1D−2 (s) =

12 0

s1(Sc2S−

12

)sS3

0 12

s2(Sc2S−

12

)sS3

− s1(Sc2S−

12

)sP3

− s2(Sc2S−

12

)sP3

12

. (A.38)


with S = (sP3 sS3 + s2r).

REFERENCES

Achenbach, J. D., 2003. Reciprocity in elastodynamics, Cambridge University Press, Cambridge.

Altman, C. & Suchy, K., 1991. Reciprocity, spatial mapping and time reversal in electromagnetics,

Kluwer, Dordrecht.

Auld, B. A., 1979. General electromechanical reciprocity relations applied to the calculation of elastic

wave scattering coefficients, Wave Motion, 1, 3–10.

Berkhout, A. J., 1982. Seismic Migration. Imaging of acoustic energy by wave field extrapolation. A.

Theoretical aspects, Elsevier.

Berryhill, J. R., 1984. Wave-equation datuming before stack, Geophysics, 49, 2064–2066.

Bleistein, N., 1984. Mathematical methods for wave phenomena, Academic Press, Inc., Orlando.

Bojarski, N. N., 1983. Generalized reaction principles and reciprocity theorems for the wave equations,

and the relationship between the time-advanced and time-retarded fields, Journal of the Acoustical

Society of America, 74, 281–285.

Born, M. & Wolf, E., 1965. Principles of optics, Pergamon Press, London.

Brackenhoff, J., Thorbecke, J., & Wapenaar, K., 2019. Virtual sources and receivers in the real Earth:

Considerations for practical applications, Journal of Geophysical Research, 124, 11,802–11,821.

Broggini, F. & Snieder, R., 2012. Connection of scattering principles: a visual and mathematical tour,

European Journal of Physics, 33, 593–613.

Challis, L. & Sheard, F., 2003. The Green of Green functions, Physics Today , 56(12), 41–46.

Corones, J. P., 1975. Bremmer series that correct parabolic approximations, J. Math. Anal. Appl.,

50, 361–372.

da Costa Filho, C. A., Ravasi, M., Curtis, A., & Meles, G. A., 2014. Elastodynamic Green’s function

retrieval through single-sided Marchenko inverse scattering, Physical Review E , 90, 063201.

de Hoop, A. T., 1958. Representation theorems for the displacement in an elastic solid and their

applications to elastodynamic diffraction theory , Ph.D. thesis, Delft University of Technology, Delft.

de Hoop, A. T., 1966. An elastodynamic reciprocity theorem for linear, viscoelastic media, Applied

Scientific Research, 16, 39–45.

de Hoop, A. T., 1988. Time-domain reciprocity theorems for acoustic wave fields in fluids with

relaxation, Journal of the Acoustical Society of America, 84, 1877–1882.

de Hoop, A. T., 1995. Handbook of radiation and scattering of waves, Academic Press, London.

48 Wapenaar

de Hoop, M. V., 1996. Generalization of the Bremmer coupling series, J. Math. Phys., 37, 3246–3282.

Derode, A., Larose, E., Tanter, M., de Rosny, J., Tourin, A., Campillo, M., & Fink, M., 2003. Recov-

ering the Green’s function from field-field correlations in an open scattering medium (L), Journal

of the Acoustical Society of America, 113(6), 2973–2976.

Devaney, A. J., 1982. A filtered backpropagation algorithm for diffraction tomography, Ultrasonic

Imaging , 4, 336–350.

Diekmann, L. & Vasconcelos, I., 2021. Focusing and Green’s function retrieval in three-dimensional

inverse scattering revisited: A single-sided Marchenko integral for the full wave field, Physical Review

Research, 3, 013206.

Esmersoy, C. & Oristaglio, M., 1988. Reverse-time wave-field extrapolation, imaging, and inversion,

Geophysics, 53, 920–931.

Fink, M. & Prada, C., 2001. Acoustic time-reversal mirrors, Inverse Problems, 17, R1–R38.

Fishman, L. & McCoy, J. J., 1984. Derivation and application of extended parabolic wave theories.

I. The factorized Helmholtz equation, J. Math. Phys., 25(2), 285–296.

Flekkøy, E. G. & Pride, S. R., 1999. Reciprocity and cross coupling of two-phase flow in porous media

from Onsager theory, Physical Review E , 60, 4130–4137.

Frazer, L. N. & Sen, M. K., 1985. Kirchhoff-Helmholtz reflection seismograms in a laterally inho-

mogeneous multi-layered elastic medium, I, Theory, Geophysical Journal of the Royal Astronomical

Society , 80, 121–147.

Gangi, A. F., 1970. A derivation of the seismic representation theorem using seismic reciprocity,

Journal of Geophysical Research, 75, 2088–2095.

Gilbert, F. & Backus, G. E., 1966. Propagator matrices in elastic wave and vibration problems,

Geophysics, 31(2), 326–332.

Green, G., 1828. An essay on the application of mathematical analysis to the theories of electricity and

magnetism, Originally published as book in Nottingham, 1828. Reprinted in three parts in Journal

fur die reine und angewandte Mathematik Vol. 39, 1 (1850) p. 73–89, Vol. 44, 4 (1852) p. 356–74,

and Vol. 47, 3 (1854) p. 161–221.

Haines, A. J., 1988. Multi-source, multi-receiver synthetic seismograms for laterally heterogeneous

media using F-K domain propagators, Geophysical Journal International , 95, 237–260.

Haines, A. J. & de Hoop, M. V., 1996. An invariant imbedding analysis of general wave scattering

problems, J. Math. Phys., 37, 3854–3881.

Haskell, N. A., 1953. The dispersion of surface waves on multilayered media, Bulletin of the Seismo-

logical Society of America, 43, 17–34.


Hilterman, F. J., 1970. Three-dimensional seismic modeling, Geophysics, 35, 1020–1037.

Hokstad, K., 2000. Multicomponent Kirchhoff migration, Geophysics, 65(3), 861–873.

Jia, X., Guitton, A., & Snieder, R., 2018. A practical implementation of subsalt Marchenko imaging

with a Gulf of Mexico data set, Geophysics, 83(5), S409–S419.

Kennett, B. L. N., 1972a. The connection between elastodynamic representation theorems and prop-

agator matrices, Bulletin of the Seismological Society of America, 62, 973–983.

Kennett, B. L. N., 1972b. Seismic waves in laterally inhomogeneous media, Geophysical Journal of

the Royal Astronomical Society , 27, 301–325.

Kennett, B. L. N., Kerry, N. J., & Woodhouse, J. H., 1978. Symmetries in the reflection and trans-

mission of elastic waves, Geophysical Journal of the Royal Astronomical Society , 52, 215–229.

Kennett, B. L. N., Koketsu, K., & Haines, A. J., 1990. Propagation invariants, reflection and transmis-

sion in anisotropic, laterally heterogeneous media, Geophysical Journal International , 103, 95–101.

Kiraz, M. S. R., Snieder, R., & Wapenaar, K., 2021. Focusing waves in an unknown medium without

wavefield decomposition, JASA Express Letters, 1(5), 055602.

Knopoff, L., 1956. Diffraction of elastic waves, Journal of the Acoustical Society of America, 28,

217–229.

Knopoff, L. & Gangi, A. F., 1959. Seismic reciprocity, Geophysics, 24, 681–691.

Koketsu, K., Kennett, B. L. N., & Takenaka, H., 1991. 2-D reflectivity method and synthetic seis-

mograms for irregularly layered structures - II. Invariant embedding approach, Geophysical Journal

International , 105, 119–130.

Kong, J. A., 1986. Electromagnetic wave theory , Wiley Interscience.

Lindsey, C. & Braun, D. C., 2004. Principles of seismic holography for diagnostics of the shallow

subphotosphere, The Astrophysical Journal Supplement Series, 155(1), 209–225.

Lorentz, H. A., 1895. The theorem of Poynting concerning the energy in the electromagnetic field and

two general propositions concerning the propagation of light, Verslagen der Afdeeling Natuurkunde

van de Koninklijke Akademie van Wetenschappen, 4, 176–187.

Mansuripur, M., 2021. A tutorial on the classical theories of electromagnetic scattering and diffraction,

Nanophotonics, 10, 315–342.

Maynard, J. D., Williams, E. G., & Lee, Y., 1985. Nearfield acoustic holography: I. Theory of

generalized holography and the development of NAH, Journal of the Acoustical Society of America,

78(4), 1395–1413.

Oristaglio, M. L., 1989. An inverse scattering formula that uses all the data, Inverse Problems, 5,

1097–1105.

50 Wapenaar

Pao, Y. H. & Varatharajulu, V., 1976. Huygens’ principle, radiation conditions, and integral for-

mulations for the scattering of elastic waves, Journal of the Acoustical Society of America, 59,

1361–1371.

Pereira, R., Ramzy, M., Griscenco, P., Huard, B., Huang, H., Cypriano, L., & Khalil, A., 2019.

Internal multiple attenuation for OBN data with overburden/target separation, in SEG, Expanded

Abstracts, pp. 4520–4524.

Porter, R. P., 1970. Diffraction-limited, scalar image formation with holograms of arbitrary shape,

Journal of the Optical Society of America, 60, 1051–1059.

Porter, R. P. & Devaney, A. J., 1982. Holography and the inverse source problem, Journal of the

Optical Society of America, 72, 327–330.

Pride, S. R. & Haartsen, M. W., 1996. Electroseismic wave properties, Journal of the Acoustical

Society of America, 100, 1301–1315.

Ravasi, M., Vasconcelos, I., Kritski, A., Curtis, A., da Costa Filho, C. A., & Meles, G. A., 2016.

Target-oriented Marchenko imaging of a North Sea field, Geophysical Journal International , 205,

99–104.

Rayleigh, J. W. S., 1878. The theory of sound. Volume II , Dover Publications, Inc. (Reprint 1945).

Reinicke, C., Dukalski, M., & Wapenaar, K., 2020. Comparison of monotonicity challenges encoun-

tered by the inverse scattering series and the Marchenko demultiple method for elastic waves, Geo-

physics, 85(5), Q11–Q26.

Rose, J. H., 2001. “Single-sided” focusing of the time-dependent Schrodinger equation, Physical

Review A, 65, 012707.

Rose, J. H., 2002. ‘Single-sided’ autofocusing of sound in layered materials, Inverse Problems, 18,

1923–1934.

Schneider, W. A., 1978. Integral formulation for migration in two and three dimensions, Geophysics,

43, 49–76.

Schoenberg, M. & Sen, P. N., 1983. Properties of a periodically stratified acoustic half-space and its

relation to a Biot fluid, Journal of the Acoustical Society of America, 73, 61–67.

Slob, E. & Wapenaar, K., 2007. Electromagnetic Green’s functions retrieval by cross-correlation and

cross-convolution in media with losses, Geophysical Research Letters, 34, L05307.

Slob, E., Wapenaar, K., Broggini, F., & Snieder, R., 2014. Seismic reflector imaging using internal

multiples with Marchenko-type equations, Geophysics, 79(2), S63–S76.

Snieder, R., Wapenaar, K., & Wegler, U., 2007. Unified Green’s function retrieval by cross-correlation;

connection with energy principles, Physical Review E , 75, 036103.


Staring, M. & Wapenaar, K., 2020. Three-dimensional Marchenko internal multiple attenuation on

narrow azimuth streamer data of the Santos Basin, Brazil, Geophysical Prospecting , 68, 1864–1877.

Staring, M., Pereira, R., Douma, H., van der Neut, J., & Wapenaar, K., 2018. Source-receiver

Marchenko redatuming on field data using an adaptive double-focusing method, Geophysics, 83(6),

S579–S590.

Stoffa, P. L., 1989. Tau-p - A plane wave approach to the analysis of seismic data, Kluwer Academic

Publishers, Dordrecht.

Takenaka, H., Kennett, B. L. N., & Koketsu, K., 1993. The integral operator representation of

propagation invariants for elastic waves in irregularly layered media, Wave Motion, 17, 299–317.

Thomson, W. T., 1950. Transmission of elastic waves through a stratified solid medium, Journal of

Applied Physics, 21, 89–93.

Ursin, B., 1983. Review of elastic and electromagnetic wave propagation in horizontally layered media,

Geophysics, 48, 1063–1081.

Van der Neut, J., Vasconcelos, I., & Wapenaar, K., 2015. On Green’s function retrieval by iterative

substitution of the coupled Marchenko equations, Geophysical Journal International , 203, 792–813.

Van der Neut, J., Johnson, J. L., van Wijk, K., Singh, S., Slob, E., & Wapenaar, K., 2017. A Marchenko

equation for acoustic inverse source problems, Journal of the Acoustical Society of America, 141(6),

4332–4346.

Wapenaar, C. P. A., 1996. Reciprocity theorems for two-way and one-way wave vectors: a comparison,

Journal of the Acoustical Society of America, 100, 3508–3518.

Wapenaar, C. P. A. & Berkhout, A. J., 1989. Elastic wave field extrapolation, Elsevier, Amsterdam.

Wapenaar, C. P. A., Kinneging, N. A., & Berkhout, A. J., 1987. Principle of prestack migration based

on the full elastic two-way wave equation, Geophysics, 52(2), 151–173.

Wapenaar, K., 2003. Synthesis of an inhomogeneous medium from its acoustic transmission response,

Geophysics, 68, 1756–1759.

Wapenaar, K., 2019. Unified matrix-vector wave equation, reciprocity and representations, Geophys-

ical Journal International , 216, 560–583.

Wapenaar, K. & de Ridder, S., 2021. On the relation between the propagator matrix and the

Marchenko focusing function, Geophysics (under review).

Wapenaar, K. & Slob, E., 2014. On the Marchenko equation for multicomponent single-sided reflection

data, Geophysical Journal International , 199, 1367–1371.

Wapenaar, K. & van der Neut, J., 2010. A representation for Green’s function retrieval by multidi-

mensional deconvolution, Journal of the Acoustical Society of America, 128(6), EL366–EL371.

52 Wapenaar

Wapenaar, K., Slob, E., & Snieder, R., 2006. Unified Green’s function retrieval by cross correlation,

Physical Review Letters, 97, 234301.

Wapenaar, K., Broggini, F., Slob, E., & Snieder, R., 2013. Three-dimensional single-sided Marchenko

inverse scattering, data-driven focusing, Green’s function retrieval, and their mutual relations, Phys-

ical Review Letters, 110, 084301.

Wapenaar, K., Thorbecke, J., van der Neut, J., Broggini, F., Slob, E., & Snieder, R., 2014. Green’s

function retrieval from reflection data, in absence of a receiver at the virtual source position, Journal

of the Acoustical Society of America, 135(5), 2847–2861.

Wapenaar, K., Snieder, R., de Ridder, S., & Slob, E., 2021. Green’s function representations for

Marchenko imaging without up/down decomposition, Geophysical Journal International , 227, 184–

203.

Weaver, R. L. & Lobkis, O. I., 2004. Diffuse fields in open systems and the emergence of the Green’s

function (L), Journal of the Acoustical Society of America, 116(5), 2731–2734.

Woodhouse, J. H., 1974. Surface waves in a laterally varying layered structure, Geophysical Journal

of the Royal Astronomical Society , 37, 461–490.

Zhang, L. & Slob, E., 2020. A field data example of Marchenko multiple elimination, Geophysics,

85(2), S65–S70.

propagator matrices and marchenko-type focusing arxiv:2110

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