HAL Id: hal-01653927
https://hal.archives-ouvertes.fr/hal-01653927
Submitted on 5 Dec 2017
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
medium containing remote localized perturbations: a
numerical solution to the scattering problem
Yder J. Masson, Barbara Romanowicz
To cite this version:
Yder J. Masson, Barbara Romanowicz.
Fast computation of synthetic seismograms within a
medium containing remote localized perturbations: a numerical solution to the scattering
problem. Geophysical Journal International, Oxford University Press (OUP), 2016, 208 (2), pp.674
-692. �10.1093/gji/ggw412�. �hal-01653927�
Geophysical Journal International
Geophys. J. Int. (2017)208, 674–692 doi: 10.1093/gji/ggw412
Advance Access publication 2016 November 4 GJI Seismology
Fast computation of synthetic seismograms within a medium
containing remote localized perturbations: a numerical solution to
the scattering problem
Yder Masson
1and Barbara Romanowicz
1,2,31Institut de Physique du Globe, 1 Rue Jussieu, F-75005 Paris, France. E-mail:masson@ipgp.fr 2Coll`ege de France, 11 Place Marcelin Berthelot, F-75231 Paris Cedex 05, France
3Berkeley Seismological Laboratory, University of California, Berkeley, CA 94720, USA
Accepted 2016 November 21. Received 2016 October 30; in original form 2016 August 23
S U M M A R Y
We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave prop-agation modelling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the subvolume containing the perturbations or scatter-ers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modelled numerically, while in the surrounding volume, where the medium isn’t perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhof–Helmholtz integrals that naturally adapt to the numerical scheme employed for modelling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered ‘exact’ as the total numerical error is comparable to that of the method employed for modelling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localized sensitivity ker-nels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarized in a dedicated section.
Key words: Seismic tomography; Theoretical seismology; Wave scattering and diffraction;
Wave propagation. 1 I N T R O D U C T I O N
1.1 Objectives
The main objective of this paper is to develop an efficient method for computing synthetic seismograms in an arbitrary complex ref-erence medium that has been perturbed locally. To do so, we need to solve the so called scattering problem, that is, to determine the wavefield generated by the interaction of waves travelling through the medium with the scatterers, that is, the perturbations with re-spect to the reference medium. For the sake of efficiency, we search for a solution where the numerical modelling of wave propagation is confined to the smallest computational volume possible, that is, the limited subvolume that encompasses the scatterers. Furthermore, for the sake of completeness, the desired solution should accommodate
arbitrary data sets, that is, provide us with synthetic seismograms for arbitrary sources/receivers configurations. For a source/receiver pair, since the source and the receiver can either lie inside or out-side the computational volume, there are only four canonical setups that need to be addressed as illustrated in Fig.1. More complicated setups involving multiple sources and receivers can be treated as combinations of these four simple geometries. Throughout this pa-per, we derive general formulae that allow us to calculate the seismic wavefield recorded with arbitrary acquisition setup, for propagation through a medium with localized buried scatterers.
1.2 Motivation
Fast and accurate calculation of elastic wavefields within heteroge-neous media is key for developing and improving accurate imaging
Figure 1. The four canonical setups that need to be considered when constructing a general solution to the scattering problem. More complicated acquisition setups can be treated as combinations of the simple geometries in (a)–(d). With the solution derived in this paper, synthetic seismograms can be computed at arbitrary receiver locations (green triangles) for arbitrary source locations (red explosions), this, by modelling wave propagation in a small subvolume (in pale yellow) that contains the scatterers of interest (δc). In our solution, the interaction between the waves induced by the presence of scatterers and the background/reference velocity model outside the computational domain is handled through wavefield extrapolation.
techniques relying on full-waveform analysis. As opposed to ray to-mography, waveform tomography or full-waveform inversion (e.g. Tarantola 1984) uses complete recordings of the wavefield (e.g. seismograms) to infer the elastic structure of the medium through which it propagates. These methods require massive computation and are used in various fields, for example, in medical imaging (e.g. Arnal et al.2013), in exploration geophysics (e.g. Virieux & Operto2009) and in global seismology (e.g. French et al.2013). Typically, obtaining an image using full-waveform inversion re-quires thousands of numerical simulations where wave propagation is modelled through the body to be imaged. To obtain better images, faster, it is of importance to lighten this computational burden and to develop fast methods for computing synthetic seismograms in 3D heterogeneous media.
When using standard full waveform inversion methods (see e.g. Bamberger et al. 1982; Chen et al.2007; Tape et al. 2009; Zhu
et al.2012; Fichtner et al.2013; Yuan et al.2014), the computa-tional domain (i.e. volume in which wave propagation is modelled) must be chosen large enough so it contains all the sources that are used to illuminate the medium as well as the all the receivers at which seismograms are recorded. This corresponds to the geometry in Fig.1(a). This constraint is not so dramatic so far the sources and the receivers are located in the vicinity of the region to be imaged. However, when using remote data, that is, sources or receivers that are far away from the target to be imaged, as in Figs1(b)–(d), the computational effort quickly becomes unreasonably large as wave propagation is needlessly modelled within large regions that are left unperturbed. To overcome this problem, numerous methods, often referred to as hybrid modelling methods have been proposed (see e.g. Moczo et al. 1997; Wen & Helmberger 1998; Bielak & Christiano 1984; Bielak et al. 2003; Capdeville et al. 2003; Yoshimura et al. 2003; To et al. 2005; Bouchon & S´anchez-Sesma2007; Zhao et al.2008; Godinho et al.2009). Hybrid meth-ods usually employ exact wave propagation modelling inside the imaged volume combined with fast approximate solutions for mod-elling wave propagation outside that volume, that is, to extrapolate
the wavefield from distant sources and toward distant receivers. This permits efficient computation of the wavefield generated by scatter-ers that are localized within the imaged region, at the expense of some approximations.
A specific class of hybrid methods called injection methods have gained interest in recent years with the increasing computational power available. The principle of injection methods is to first model wave propagation through a reference velocity model in a large volume that encompasses all the sources and receivers. During this operation, the wavefield is recorded at the surface of a smaller sub-volume of interest (e.g. the region in which we would like to perform imaging). In subsequent simulations, wave propagation is modelled locally and the original wavefield is regenerated within the sub-volume thanks to secondary sources that are constructed from the recorded wavefield. The big advantage of this approach is that many smaller scale simulations can be performed to efficiently quantify the effect on the wavefield induced by velocity perturbations within the subvolume. Injection methods can be traced back to Alterman & Karal (1968) and have been developed in different studies (e.g. Taflove & Brodwin1975; Robertsson & Chapman2000; Takeuchi & Geller2003; Oprˇsal et al.2009; Masson et al.2014). Although these methods are often associated with finite difference modelling, they can be implemented efficiently using other numerical schemes such as the spectral element method (Monteiller et al.2013). A uni-fied theory for injection methods has been introduced by Masson
et al. (2014). This theoretical framework allows for easy imple-mentation of injection techniques with arbitrary numerical schemes including the finite difference method and finite/spectral element methods. Injection methods have been employed to compute syn-thetic seismograms at distant receivers (Robertsson et al.2000) and for imaging (e.g. Borisov et al. 2015; Monteiller et al. 2015; Willemsen et al.2016). Van Manen et al. (2007) used injection techniques to construct an exact boundary condition for the scatter-ing problem.
With the exception of Robertsson et al. (2000) and Willemsen
distant sources (located outside the subvolume in which the simu-lations are performed) and local receivers (located inside the sub-volume in which the simulations are performed), as pictured in Fig.1(b). Further, to the exception of Van Manen et al. (2007) and Willemsen et al. (2016), all studies to date use absorbing boundary condition to truncate the computational domain at the surface of the subvolume in which the numerical modelling is done. In this case, some higher order multiple scattering effects are not taken into account.
This motivates the present study, which aims to derive a complete numerical solution to the scattering problem that is not limited to finite difference schemes but is appropriate for an arbitrary numeri-cal method, that accounts for all higher order scattering effects, and that allows us to compute exact synthetic seismograms for arbitrary sources’ and receivers’ positions.
1.3 Content
The first section of this paper is dedicated to the derivation of our numerical solution to the scattering problem. We first derive mixed representation theorems that are used for domain decomposition. These mixed representations allow us to express the wavefield ei-ther implicitly (i.e. as the solution of the wave equation), or explic-itly (i.e. as a function of the Green’s function of the propagating medium) in different partitions of space. We then specialize to the scattering problem and obtain a dedicated representation. In this representation, the wavefield is expressed implicitly within the sub-volume containing the scatterers and explicitly in the remaining volume. This allows us to confine wave propagation modelling to the subvolume enclosing the scatterers and to obtain the wavefield with extrapolation techniques elsewhere. Finally, we discretize this representation to obtain formulae for computing the wavefield at arbitrary locations. At the end of this section, we detail the different steps involved in the computation of synthetic seismograms.
In the second section of this paper, we present numerical exam-ples. We detail a practical implementation of our general solution to the scattering problem using the finite difference method. We demonstrate the accuracy of the method by computing synthetic seismograms for the four canonical setups in Fig.1, and, we discuss the effects of neglecting some higher order multiple scattering ef-fects in our solution. Finally, to illustrate possible applications, we present time-reversal experiments, and we compute localized sen-sitivity kernels that can be used for the imaging of remote targets.
2 T H E O RY
2.1 The scattering problem
For the sake of clarity, our reasoning is carried out by consider-ing a scalar acoustic pressure wavefield p(x, t) that satisfies the Helmholtz eq. (1), however, our final result in eq. (32) easily adapts to elastic wavefields as explained in Section 2.7.
Knowing the response or Green’s function of a reference medium with velocity c0(x), we search for an efficient way to compute the
acoustic response p(x, t) for a locally perturbed medium with veloc-ity c(x)= c0(x)+ δc(x) where the perturbation δc(x) is localized
inside a limited volume as pictured in Fig.2. In this situation, it is well known that the acoustic response outside the volume containing the heterogeneityδc(x) may be obtained through wavefield extrapo-lation (e.g. Fokkema & Van Den Berg2013), that is, by convolving the acoustic response measured at the surface of a volume
contain-Figure 2. We decompose the volume V into two subvolumes, a computa-tional volume Vcand an extrapolation volume Ve. We express the sound speed c(x)= c0(x)+ δc(x) as the sum of a background velocity
distribu-tion c0(x) and some velocity perturbationδc(x). Further, we assume that
the velocity perturbationδc(x) is non-zero only inside Vc. Within the inner volume Vc, the wavefield p(x)= pc(x) is obtained by solving the wave equa-tion numerically. Within the outer volume Ve, the wavefield p(x)= pe(x) is obtained through wavefield extrapolation.
ing the perturbation or scatterersδc(x) with the Green’s function of the reference medium (i.e. with velocity c0(x)). Therefore, when
solving the scattering problem numerically, it is not needed to re-compute wave propagation in the entire volume V to obtain the acoustic response outside the region containing the velocity pertur-bation.
In the next paragraphs, we derive a general formula to compute the response p(x, t) at arbitrary locations without modelling wave propagation in the entire volume V but only within a subvolume containing the scatterers.
2.2 Domain decomposition
We first derive a mixed representation for the wavefield p(x, t) that is used for domain decomposition. The objective is to obtain a general formula where the wavefield p(x, t) can be expressed as a solution of the wave equation only within the small subvolume containing scatterers.
The spatiotemporal evolution of the acoustic pressure wavefield
p(x, t), generated by a source distribution s(x, t), within an arbitrary
volume V having smoothly varying sound velocity c(x) may be expressed implicitly as the solution of the scalar Helmholtz wave equation 1 c(x)2 ∂p(x, t) ∂t2 − ∇ 2p(x, t) = s(x, t). (1)
Alternatively, by convolving the source term on the right hand side of eq. (1) with the Green’s function of the propagating medium and integrating over volume, one may explicit p(x, t) as
p(x, t) = t 0 V g(x, x, t − τ)s(x, τ)dVdτ (2)
where the Green’s function g(x, x, t) is by definition the response
p(x, t) generated by an impulsive source at position xand satisfies
1
c(x)2
∂g(x, x, t)
∂t2 − ∇
2g(x, x, t) = δ(x − x)δ(t). (3)
Depending on the situation, one can use either one of the two rep-resentations in eq. (1) or eq. (2) to evaluate p(x, t) in V. When the Green’s function g(x, x, t) is known, using eq. (2) is usually more practical and faster (e.g. Van Driel et al.2015), while, when one only has access to the velocity distribution c(x), eq. (1) is more adequate. Note that, for the sake of clarity, we omitted to define the bound-ary conditions on∂V (the surface of V) that are needed to obtain a unique solution for eqs (1)–(3). This is because these boundary conditions do not appear explicitly in our final expressions and will be accounted for when computing the Green’s functions within a reference velocity model defined on V.
We now divide the volume V into two subvolumes named Vcand
Ve, as pictured in Fig.2, that will later correspond to our
computa-tional domain and extrapolation domain, respectively. Accordingly, we define pc(x, t) as the response p(x, t) observed within Vcand
pe(x, t) as the response p(x, t) observed within Ve. Our objective
is to obtain explicit and implicit representations for pc(x, t) and
pe(x, t) that can be combined to obtain mixed representations of
p(x, t) within V.
Classic explicit representations for pc(x, t) and pe(x, t) are given
by the Helmholtz–Kirchhoff representation theorem (e.g. De Hoop
1958; Aki & Richards1980; Snieder2002; Masson et al.2014) as
pc(x, t) = t 0 Vc g(x, x, t − τ)s(x, τ)dVdτ, + t 0 ∂Vc p(x, τ)∇g(x, x, t − τ)dScdτ − t 0 ∂Vc g(x, x, t − τ)∇ p(x, τ)dS cdτ (4) pe(x, t) = t 0 Ve g(x, x, t − τ)s(x, τ)dVdτ, + t 0 ∂Ve p(x, τ)∇g(x, x, t − τ)dScdτ − t 0 ∂Ve g(x, x, t − τ)∇ p(x, τ)dS cdτ (5)
where∂Vcis the surface of Vc,∂Veis the surface of Ve , dScis
the outward normal vector of∂Vcand d Se is the outward normal
vector of∂Ve. Masson et al. (2014) proposed the equivalent implicit
representations: 1 c(x)2 ∂pc(x, t) ∂t2 − ∇ 2p c(x, t) = fc(x, t). (6) 1 c(x)2 ∂pe(x, t) ∂t2 − ∇ 2p e(x, t) = fe(x, t). (7)
for pc(x, t) and pe(x, t). The source terms fcand fein eqs (6) and
(7) are given by fc(x, t) = w(x)∇2p(x, t) − ∇2{w(x)p(x, t)} + w(x)s(x, t) (8) fe(x, t) = ∇2{w(x)p(x, t)} − w(x)∇2p(x, t) + (1 − w(x))s(x, t) (9) where w(x) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 ∀ x ∈ Vc 1 2 ∀ x ∈ ∂Vc 0 ∀ x /∈ Vc (10)
is a spatial window function. Note that it is important to setw(x) = 1/2 on ∂Vcso that eqs (4) and (6) give the same response pc, and
eqs (5) and (7) give the same response pe, on the boundary between
Vcand Ve. Alternative explicit representations for pcand pecan be
obtained by substituting the source term s in eq. (2) with fcand fe
in eqs (8) and (9). This gives
pc(x, t) = t 0 V g(x, x, t − τ) fc(x, τ)dVdτ (11) pe(x, t) = t 0 V g(x, x, t − τ) fe (x, τ)dVdτ. (12) Withw as defined in eqs (10), (4) and (5) are exactly equivalent to eqs (11) and (12), respectively. However, from now on, we will systematically use eqs (11) and (12) which are easier to handle numerically. This is because, in these expressions, the surface∂Vc
is defined implicitly through the window functionw (see Masson
et al.2014).
Mixed representations for p(x, t) within V can now be obtained by combining either one of eqs (4), (6) and (11) for pcwith either
one of eqs (5), (7) and (12) for pe. We have
p(x, t) = pc(x, t) + pe(x, t). (13)
Equation (13) is valid for any propagating medium with velocity distribution c(x) and allows us to employ different representations for the wavefield p(x, t) in different partitions of space.
2.3 A mixed representation for the scattering problem
We now adapt the representation in eq. (13) to the scattering problem where the velocity distribution c(x)= c0(x)+ δc(x) corresponds to
a reference or background velocity distribution c0(x) with local
velocity perturbationδc(x) inside Vc. Keeping in mind that our
ob-jective is to obtain a fast solution, we proceed as follow: First we express the wavefield p(x, t) implicitly as a solution of the wave equation within the small volume containing the scatterers in or-der to reduce the size of the simulations. Second, in the volume surrounding the scatterers, we use an explicit representation for
p(x, t) that involves Green’s functions of the reference unperturbed
medium. This way, the Green’s functions needed for wavefield ex-trapolation can be pre-computed, stored and reused in subsequent simulations.
By construction, the solution of eq. (7), that is the wavefield
pe(x, t) generated by the source term fe in eq. (9), is equal to
p(x, t) within Ve and to zero within Vc. Physically, this means
that the wavefield pe does not interact with the velocity structure
inside Vc. Consequently, eq. (7) has the same solution pefor any
velocity distribution c(x)= c0(x)+ δc(x) so far the perturbation
δc(x) is equal to zero inside Ve. It follows, as a corollary, that
the Green’s function g associated with any velocity distribution
c(x)= c0(x)+ δc(x) inside Vcand c(x)= c0(x) outside Vccan be
used with the representations in eq. (5) and eq. (12) to obtain pe. In
particular, one can use g0 the Green’s function taken in the
refer-ence model with velocity distribution c(x)= c0(x). By combining
eq. (6) with eq. (12) where g is replaced with g0, we obtain the mixed
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ p(x, t) = pc(x, t) + pe(x, t) (14a) 1 c2(x) ∂pc(x, t) ∂t2 − ∇ 2p c(x, t) = fc(x, t) (14b) fc(x, t) = w(x)∇2p(x, t) − ∇2w(x)p(x, t) + w(x)s(x, t) (14c) pe(x, t) = t 0 V g0(x, x, t − τ) fe(x, τ)dVdτ (14d) fe(x, t) = ∇2w(x)p(x, t) − w(x)∇2p(x, t) + (1 − w(x))s(x, t) (14e) that holds for any velocity distribution that satisfies
c(x)= ⎧ ⎪ ⎨ ⎪ ⎩ c0(x)+ δc(x) ∀ x ∈ Vc c0(x) ∀ x ∈ ∂Vc c0(x) ∀ x /∈ Vc . (15)
Eq. (14) gives the response p(x, t) for arbitrary heterogeneous me-dia with velocity perturbations inside Vc. As desired, within the
volume Vccontaining the perturbationδc(x), the acoustic response
is now expressed implicitly as the solution of the wave equation (eq. 14b), while, in the external volume Ve, where the velocity
distribution is unperturbed, the acoustic response is expressed ex-plicitly (eq. 14d) in terms of the Green’s function g0 of a
refer-ence medium with velocity distribution c0(x). The advantage of
the representation in eq. (14) is that wave propagation modelling only needs to be performed within the inner volume Vc to
ob-tain the acoustic response at arbitrary locations within the total volume V, provided that the Green’s function g0 has been
pre-computed. In the next two sections, we derive a discrete equivalent of eq. (14).
2.4 Discrete representation theorems
We first derive a discrete equivalent of the representation theorem in eq. (4), (5), (11) and (12).
In order to model wave propagation numerically, one usually transforms the wave eq. (1) into a linear system
M ¨¯p+ K¯p = ¯s (16)
where M and K are the mass matrix and stiffness matrix, respec-tively, and the vector ¯p contains the discrete values of the pressure wavefield evaluated at the grid points or some related expansion coefficients. Most popular methods for modelling wave propaga-tion, such as, for example, finite difference methods, finite/spectral element methods and Discontinuous Galerkin methods lead to the generic discrete wave eq. (16) and can be used to implement the domain decomposition method introduced in this study.
The discrete counterparts of eqs (6) and (7) are
M ¨¯pc+ K¯pc = ¯fc (17)
M ¨¯pe+ K¯pe = ¯fe (18)
where ¯pc and ¯pecontain the discrete values or expansion
coeffi-cients of pcand pe, respectively. We have ¯p= ¯pc+ ¯pe. Following
Masson et al. (2014), and taking the few steps as shown in Appendix A, the source terms ¯fc in eq. (17) and ¯fe in eq. (18)
may be computed using ¯fc = ¯
W¯s+ ¯WK ¯W ¯p− ¯WK ¯W¯p (19)
¯fe= ¯W¯s+ ¯
WK ¯W¯p− ¯WK ¯W ¯p. (20) where ¯W is a general matrix that acts as the window functionw(x)
in eq. (10) and ¯Wis a general matrix that acts as the complementary window function w (x)= 1 − w(x). Formally, ¯W and ¯Ware defined through the expressions ¯pc= ¯W ¯p and ¯pe= ¯
W
¯p, respectively. The advantage of eqs (A9) and (A10) is that the grid nodes at which ¯fc
and ¯feare non-zero and need to be evaluated are the same as those
where ¯p needs to be known to evaluate ¯fcand ¯fe, as we shall see
later.
Let the discrete Green’s function ¯gxcorrespond to the numerical
wavefield generated by an impulsive source ¯δxlocated at position x
and satisfies
M¨¯gx+ K¯gx= ¯δx. (21)
Explicit representations that are equivalent to eq. (4) and (5) and to eqs (11) and (12) can be obtained by convolving ¯gxwith the source
terms ¯fcand ¯fe. For example, when working in the time domain and
¯gx( j, n − l) contains the discrete values of the wavefield recorded at
grid nodes with spatial indices j evaluated at times tn− l, we obtain
¯pc(x, n) = n l=0 j ¯gx( j, n − l) ¯fc( j, l) (22) ¯pe(x, n) = n l=0 j ¯gx( j, n − l) ¯fe( j, l), (23)
which can be viewed as exact discrete equivalents of the Helmholtz– Kirchhoff integral representation theorems in eqs (4), (5), (11), and (12).
2.5 Numerical solution to the scattering problem
At this point, we express our discrete solution to the scattering problem, we then adapt it to obtain our final formula in eq. (32) that allows us to compute synthetic seismograms at arbitrary locations by running global simulations, once and for all, in the reference medium and local simulations in the perturbed medium.
By substituting the continuous representations in eq. (14) with the discrete representation in eqs (17), (18), (22) and (23) we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯p= ¯pc+ ¯pe (24a) M ¨¯pc+ K¯pc= ¯fc (24b) ¯fc= ¯ W¯s+ ¯WK ¯W ¯p− ¯WK ¯W¯p (24c) ¯pe(x, n) = n l=0 j ¯g0x( j, n − l) ¯fe( j, l) (24d) ¯fe= ¯W ¯s+ ¯WK ¯W¯p− ¯WK ¯W ¯p (24e) where ¯p in eq. (24a) is a discrete solution to the scattering problem. Eq. (24), however, is not very practical because it requires to know the global wavefield ¯p (i.e. computed in the perturbed medium) in order to solve the local wave propagation problem in eq. (24b). A simple solution to that problem is to split the interface between the computational volume Vcand the extrapolation volume Ve as
pictured in Fig. 3. This way, the wavefield measured on ∂Ve
Figure 3. We split the interface between the computational volume Vcand an extrapolation volume Veso that the wavefield may be recorded at the inner interface∂Veand extrapolated to obtain the wavefield at the outer boundary
∂Vcof the computational domain. Note that the volumes Vcand Venow overlap, we havewc(x)= 1 inside Vc,wc(x)= 0 outside Vc,we(x)= 0 inside Veandwe(x)= 1 outside Ve. The complementary window functions are wc= 1 − wc(x) and we= 1 − we(x).
wavefield at the boundary∂Vc of the computational domain. By
substituting ¯W with ¯Wc, ¯
W
with ¯Wcand ¯p with ¯pe in eq. (24c),
and, by substituting ¯W with ¯We, ¯
W
with ¯We and ¯p with ¯pc in
eq. (24e), we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯p= ¯pc+ ¯ W c¯pe (25a) M ¨¯pc+ K¯pc= ¯fc (25b) ¯fc= ¯W c¯s+ ¯ W cK ¯Wc¯pe− ¯WcK ¯ W c¯pe (25c) ¯pe(x, n) = n l=0 j ¯g0 x( j, n − l) ¯f e( j, l) (25d) ¯fe= ¯W e¯s+ ¯WeK ¯ W e¯pc− ¯ W eK ¯We¯pc (25e) where ¯pe is multiplied by ¯ W
c in eq. (25a) because the volumes
Vcand Venow overlap and ¯p= ¯pc+ ¯pe= 2¯p in the small volume
between ∂Vc and ∂Ve. As opposed to eq. (24b), eq. (25b) is
ex-plicit as its left hand side does not depend on the values of the global wavefield ¯p. Practically, this means that we do not need to re-model wave propagation in the total volume V when adding a ve-locity perturbation inside Vc. To get a more comprehensive form for
eq. (25) we express the wavefield
¯pe= ¯p0+ ¯px (26)
as the sum of the reference wavefield ¯p0that is the wavefield
com-puted in the reference medium and satisfies
M0¨¯p 0 + K0¯p0= ¯se, (27) where ¯se= ¯W e¯s. (28)
and the extrapolated wavefield ¯pxthat is the locally computed
wave-field ¯pcmeasured on∂V
eand extrapolated in the outer volume Ve.
We have: ¯px(x, n) = n l=0 j ¯g0x( j, n − l) ¯fx( j, l) (29) were ¯fx= ¯W eK ¯ W e¯pc− ¯ W eK ¯We¯pc. (30)
Finally, using eq. (26) and rewriting eq. (25c) as
¯fc= ¯si + ¯sv+ ¯feb (31) we obtain ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¯p(xr)= ¯pc(xr)+ ¯wc¯p0(xr)+ ¯wc¯px(xr) (32a) M ¨¯pc+ K¯pc= ¯si+ ¯sv+ ¯feb (32b) ¯si = ¯ Wc¯s (32c) ¯sv= ¯WcK ¯Wc¯p0− ¯WcK ¯ W c¯p0 (32d) ¯feb= ¯W cK ¯Wc¯px− ¯WcK ¯ W c¯px (32e) M0¨¯p 0 + K0¯p0= ¯se (32f) ¯se= ¯W e¯s (32g) ¯px(x, n) = n l=0 j ¯g0 x( j, n − l) ¯f x( j, l) (32h) ¯fx= ¯ WeK ¯ W e¯pc− ¯ W eK ¯We¯pc (32i)
where ¯p gives the solution to the scattering problem at any position within a given volume V having arbitrary velocity perturbations inside the computational subvolume Vc.
Practically, ¯p is obtained by computing wave propagation locally, that is, by solving eq. (32b) in the computational volume Vc. The
source term ¯si to the right hand side of eq. (32b) corresponds to
internal sources located within Vc which are treated as standard
sources by the numerical solver. The source term ¯sv to the right hand side of eq. (32b) corresponds to virtual sources, that is, sources that are physically located outside Vc and that have been moved
to the boundary∂Vc of the computational domain. These virtual
sources ¯svcan be pre-computed by modelling wave propagation in the reference model with velocity distribution c0. They do not need
to be recomputed when modifying the velocity structure inside the computational domain Vc. The source term ¯febto the right hand side
of eq. (32b) is imposing an exact boundary condition that accounts for all high order scattering effects and effectively confines the local wavefield ¯pcinside the computational domain Vc. This source term
¯febneeds to be updated in real time, that is, at each time step, by
extrapolating the locally computed wavefield ¯pc measured on the
extrapolating surface∂Ve(i.e. by evaluating successively eqs (32i),
(32h), and (32e)).
Within the computational domain Vc, we have ¯p= ¯pcand ¯p is
obtained directly by taking the value of the wavefield computed locally. Outside the computational domain Vc, when no sources are
present there, we have ¯p= ¯Wc¯pxand ¯pxis obtained by
extrapolat-ing the locally computed wavefield ¯pcmeasured on the extrapolation
surface∂Ve. If some sources are present outside the computational
domain Vc, we have ¯p= ¯
W
c¯p0se+ ¯
W
c¯px and we need to add the
reference wavefield ¯p0
se to the extrapolated wavefield ¯p
x. The
ref-erence wavefield ¯p0
se is obtained by modelling wave propagation in
the reference medium with velocity distribution c0and by applying
the external sources that are located outside Vc.
2.6 Numerical recipe for solving the scattering problem
We now detail the practical steps for computing synthetic seismo-grams using the general solution in eq. (32). We then show how to adapt the method to model time-reversal experiments and adjoint simulations.
2.6.1 Forward modelling: computing synthetic seismograms
(i) Define the window functions and the associated window
ma-trices. In order to perform domain decomposition, one first needs
to define two window functionswc(x) andwe(x) and their
com-plementary window functions wc(x)= 1 − wc(x) and we(x)=
1− we(x). We havewc(x)= 1 within the computational domain
Vc andwc(x)= 0 outside Vc, and, we havewe(x)= 1 within the
extrapolation domain Veandwe(x)= 0 outside Ve, as pictured in
Fig.3. In theory, the distance between the surface∂Vcof the
com-putational domain and the surface Veof the extrapolation domain
should be large enough so that the shortest time needed by waves to travel from any point on∂Vcto any point on∂Veis larger than the
time stept. This is because no information can travel faster than the waves. Practically, it is sufficient to ensure that the grid points where ¯fx= 0 are different from the grid points where ¯feb= 0. Once
the window functions have been defined, the window matrices can be constructed, for example, for ¯Wc, we have ¯Wc(i, j) = 0 when i =
j, ¯Wc(i, i) = 1 when wc(xi)= 1 and ¯Wc(i, i) = 0 when wc(xi)= 0,
where xiis the position of the grid node with index i.
(ii) Find the grid nodes on∂Vcand on∂Veat which one needs
to record the wavefield to compute ¯sv ¯feband ¯fx. Two lists of grid
nodes need to be constructed. First, the list of nodes ic(1, . . . , N c),
that is a 1-D array with dimension Nc containing the indices of
the grid nodes at which ¯sv and ¯febare non-zero, that is, we have
¯sv(ic( j ))= 0 and ¯feb(ic( j ))= 0 for all 1 < j < N
c. Second, the
list of nodes ie(1, . . . , N
e), that is a 1D array with dimension Ne
containing the indices of the grid nodes at which ¯fx is non-zero,
that is, we have ¯fx(ie( j ))= 0 for all 1 < j < N
e. The list ic can
be obtained by setting ¯px= ¯px(i )= 1 for all i in eq. (32e) and by
evaluating the right hand side. All the grid nodes i where ¯feb(i )= 0
must be added to the list. The list iecan be obtained the same way
by setting ¯pc= ¯pc(i )= 1 for all i in eq. (32i) and checking where
¯fx(i )= 0.
(iii) Pre-compute the virtual sources ¯svand the reference seismo-grams ¯p0(x
r). This is done by running one global simulation in the
entire volume V where one uses the reference model with velocity distribution c0and we apply the external sources ¯sethat are located
outside the computational domain Vc, that is, we solve
M0¨¯p 0
+ K0¯p0= ¯
W
e¯s= ¯se. (33)
Knowing the wavefield ¯p0, we can evaluate and record the virtual
source ¯svat grid nodes with indices i= ic(j) where 1< j < N c. Also,
we record and store the reference recordings ¯p0(x
r) for all receivers
located outside the computational domain Vc.
(iv) Pre-compute the Green’s functions needed to extrapolate the
wavefield from the surface∂Veto the external receivers. Thanks to
the reciprocity principle, this can be done by running one simulation per receiver, that is, for all receivers r with positions xr located
outside Vc, we solve M0¨¯g 0 xr+ K0¯g 0 xr= ¯δxr, (34)
and record the wavefield ¯g0
xr(i
e
j, n) at grid nodes with indices i e j =
ie( j ) where 1< j < N
eand at time steps 1< n < Nt.
(v) Pre-compute the Green’s functions needed to implement the
exact boundary condition ¯feb. For all grid nodes with indices ic i =
ic(i ) where 1< i < N
c, run a simulation that solves
M0¨¯g 0 ic i + K0¯g 0 ici = ¯δici, (35)
and record the wavefield ¯g0
ic i(i
e
j, n) at grid nodes with indices iej =
ie( j ) where 1< j < N
eand at time steps 1< n < Nt.
(vi) Run the local simulation. Here we solve eq. (32b) in the time domain. The following steps need to be repeated at each time step n:
(a) Knowing the wavefield ¯pcat time step n, compute the
extrap-olating force field ¯fxusing eq. (32i) and store its values ¯fx(ie j, n) at
grid nodes with indices ie
j = ie( j ) where 1< j < Ne.
(b) Knowing the present and past values of the extrapolating force field ¯fx, compute the extrapolated wavefield
¯px(ic i, n) = n l=0 Ne j=1 ¯g0 ic i ie j, n − l ¯fx ie j, l (36)
at time step n for all grid nodes with indices ic
i = ic(i ) where 1<
i< Nc.
(c) Knowing the extrapolated wavefield ¯pxat time step n,
com-pute the exact boundary source term ¯febat time step n, that is, using
eq. (32e).
(d) Advance to the next time step, for example when using a second order centred finite difference operator to approximate ¨¯pcin
eq. (32b) we have ¯pc(n+ 1) = 2¯pc(n)− ¯pc(n− 1) + t2 M−1 ¯si(n)+ ¯sv(n) + ¯feb (n)− K¯pc(n) (37) (e) Record the wavefield values ¯pc(xr) at receivers located inside
Vc.
(vii) Compute the seismograms at external receivers. First, eval-uate the extrapolated wavefield
¯px(x r, n) = n l=0 Ne j=1 ¯g0 xr ie j, n − l ¯fx ie j, l (38)
at all receivers located outside Vc. This operation can be
imple-mented efficiently using a fast Fourier transform to compute the convolution product. Finally, add the reference seismograms ¯p0(x
r)
computed in Step (iii) to the extrapolated seismograms ¯px(x r, n) to
obtain the complete seismograms ¯p(xr)= ¯wc¯p0(xr)+ ¯wc¯px(xr) at
external receivers.
2.6.2 Time-reversal and adjoint modelling
Imagine that we computed synthetic seismograms at Nr receivers
with positions xr where 1< r < Nrby following the steps in the
adjoint modelling, we need to run additional simulations where the sources are moved to the receivers position xr and constructed
using the seismograms ¯p(xr) obtained in the first simulation. The
new source ¯s consists of multiple point sources with source time functions Sr(t) and positions xir. Distinguishing between the
in-ternal contribution ¯si of the sources located inside V
c and the
external contribution ¯se of the sources located outside V c, we have ¯s= ¯si+ ¯se ¯si = Nri r=1 Sri(t)· ¯δxi r ¯se= Ne r r=1 Se r(t)· ¯δxer (39)
where the ¯δxer are spatial delta functions at the positions x
e r of the
receivers located outside Vc, the ¯δxir are spatial delta functions at
the positions xi
r of the receivers located inside Vc, Nri is the
num-ber of receivers inside Vc, Nreis the number of receivers outside
Vc, and, Sri(t) and Sre(t) are the source time functions of the
re-ceivers inside and outside Vc, respectively. In a time reversal
exper-iment, the source time functions Srα(t) are simply the time reversed versions ¯p(xαr, t ← −t) of the seismograms obtained in the first
simulation. In the context of adjoint modelling, the source time functions Srα(t) are computed according to a misfit functional, for
example, by taking the difference between observed seismograms and computed seismograms (e.g. Fichtner2010). In order to model the time-reversed or adjoint wavefields generated by these sources, one simply proceeds as explained in the previous paragraph except that in step 3 eq. (32f) is replaced by
¯p0 iic, n = n l=0 Nre i=1 ¯g0 xer iic, n − l ¯Se r(l). (40)
The only difference with the forward modelling is that the virtual sources corresponding to the time-reversal or adjoint sources are now obtained through wavefield extrapolation. This way, there is no need to run additional global simulations (i.e. as in eq. (32f)) to obtain ¯svwhen adding velocity perturbations in Vcor modifying
the source time functions Srα(t). As for the forward modelling prob-lem, the Green’s functions ¯g0
xe r(i
c
i, n − l) need to be pre-computed
prior to running the local simulations. Note that, to avoid additional simulations, we could define the virtual sources ¯sv on∂Ve as the
Green’s functions ¯g0
xe r(i
e
i, n − l) are readily available. In this case,
one substitutes ic i with i e i in eq. (40), ¯ W cand ¯Wcwith ¯ W eand ¯We
in eq. (32d), and distinguishes between receivers inside and outside
Vein eq. (39).
Knowing the source terms ¯si, ¯sv and ¯seb, we can run local
sim-ulations (i.e. solve eq. (32b)) to access time-reversed or adjoint wavefields within Vc for a perturbed medium. Getting access to
these wavefields is important when addressing the inverse scatter-ing or imagscatter-ing problem, for example, to compute sensitivity kernels or adjoint gradients.
2.7 Extension to the elastic case
To end our theoretical section, we now briefly explain how to modify the acoustic solution in eq. (32) to obtain the corresponding elastic solution to the scattering problem.
Wave propagation through generally anisotropic media can be described using the wave equation
ρ ¨un(x, t) − ∂j
Cn j kl(x)∂kul(x, t)
= sn(x, t) (41)
where un(x, t) denotes the n ∈ [1, 2, 3] component of the
displace-ment vector field u= [ux, uy, uz] in the three directions of space
[nx, ny, nz], sn(x, t) denotes the n component of the source, and
Cn j kl(x) is the stiffness tensor. Once discretized, eq. (41) transforms
to the linear system
M ¨¯u+ K¯u = ¯s (42)
where M and K are the elastic mass matrix and stiffness matrix, respectively, and the vector ¯u usually contains the discrete values of the three component displacement vector wavefield evaluated at the grid nodes.
To obtain the elastic equivalent of eq. (32) we may substitute eq. (1) with eq. (41) and follow the subsequent steps. However, since eq. (42) has the same form as eq. (16), we can directly substitute ¯p, ¯pc, ¯p0 and ¯px with ¯u, ¯uc, ¯u0 and ¯ux, respectively, in eq. (32a),
(32b), (32c), (32d), (32e), (32f), (32g) and (32i), and, use M and
K as defined in eq. (42). In this case, the vectors ¯si, ¯sv, ¯feb, ¯seand
¯fx contain the three component discrete values of the respective
force fields. The new window matrices, ¯Wc, ¯
W
c, ¯We and ¯
W
eare
defined using ¯uc= ¯Wc¯u, ¯ue= ¯
W e¯u, ¯ W c= I − ¯Wcand ¯We= I − ¯ W
e. In the elastic case, the Green’s functions needed for wavefield
extrapolation transform to nine-component tensor fields that satisfy the wave equation
ρ ¨gi n(x, x, t) − ∂j
Cn j kl∂kgil(x, x, t)
= δi nδ(x − x)δ(t) (43)
where ginis the n∈ [1, 2, 3] component of the displacement vector
field generated by an impulsive unit force source applied in direction
i, and,δindenotes Kronecker delta that is equal to 1 when i= n and
to 0 when i= n. To compute the elastic Green’s tensor gil, one needs
to run three simulations solving
M¨¯gx,ni+ K¯gx,ni = ¯δx,ni (44)
where ¯gx,ni corresponds to the numerical wavefield generated by
an impulsive force source ¯δx,ni in direction ni located at position
x. When working with Cartesian coordinates, the three simulations
correspond to the cases where ni = nx, ni= nyand ni = nz.
Fi-nally, knowing the Green’s tensors ¯gx,ni, the extrapolated
displace-ment vector field ¯ux
k(x, n) can be obtained by substituting eq. (32h)
with the discrete elastic representation theorem
¯ukx(x, n) = 3 i=1 n l=0 j ¯gk0x,ni( j, n − l) ¯f x i ( j, l). (45) 3 E X A M P L E S
3.1 Exact solution to the scattering problem
In order to test our numerical solution to the scattering problem, we implemented eq. (32) using the finite difference method as detailed in Appendix B. Also, we constructed two random media pictured in Fig. 4, using the procedure presented in Appendix C. These two media have similar velocity distributions except in a central heart shaped region where the medium in Fig.4(b) has velocity perturbations with respect to the reference medium in Fig.4(a).
As a reference, we first ran global simulations (i.e. where we solved eq. (B1) in the entire volume) in both media. In all simula-tions, we used the Morlet wavelet plotted in Fig.5for the source
Figure 4. The velocity models employed for the numerical simulations. All the models are defined on a 2-D domain where x∈ [−5, 5](m) and y ∈ [−5, 5](m). Only the central part of the domains is pictured. The reference model with velocity distribution c(x)= c0(x) is pictured in (a), the perturbed model with
velocity distribution c(x)= c0(x)+ δc(x) is pictured in (b), and, the velocity perturbation δc(x) is pictured in (d). The heart shaped solid line in (a) and (b)
corresponds to the surface where x2+ (y −√|x|)2= 1 that is implicitly defined by the window function in eq. (B6). Outside the heart shaped solid line, (a) and (b) are identical, while, inside the heart shaped solid line, (b) has velocity perturbation with respect to (a). The black pixels forming the outer heart shaped solid line in (c) shows the grid nodes labelled m with indices (i, j) = (imc, jmc) where, ¯sv|imc, jmc = 0 and ¯f
eb|ic
m, jmc = 0 in eq. (B3), and follow the surface ∂Vc
of the computational domain Vc. The black pixels forming the outer heart shaped solid line in (c) show the grid nodes labelled k with indices (i, j) = (ike, jke) where, ¯fx|ie
k, jke= 0 in eq. (B3), and follow the surface ∂Veof the extrapolation domain Ve. The discrete velocity distributions ¯c0(i, j) and ¯c(i, j) used for the
numerical simulations have dimensions 400× 400 and have been obtained by evaluating c(x) and c0(x) at the grid nodes with indices (i, j) and coordinates
(x0+ ix, y0+ y), where, x0= y0= −5 and x = y = 2.5 × 10−2.
Figure 5. The source time function employed in the numerical simulations.
time function ¯s to the right hand side of eq. (B1). We computed syn-thetic seismograms for the four canonical configurations sketched in Fig.1. The waveforms obtained for the reference medium (in Fig.4a) and for the perturbed medium (in Fig.4b) are pictured as dashed and solid lines in the left panels of Fig.6, respectively. The differences between the waveforms computed in the reference and in the perturbed media are shown in the right panels of Fig.6. Note that the differences between the two sets of waveforms have roughly the same amplitude as the waveforms themselves.
To test the exactness of our solution to the scattering problem, we reproduced the waveforms computed using global simulations in the perturbed medium. This time, the computations involve no global simulations in the perturbed medium, that is, we performed
local simulations in the perturbed medium and global simulations in the reference medium. The results are presented in Fig.7. The solid lines in the left panels of Fig.7are the same as those in the left panels of Fig.6and were obtained using global simulations in the perturbed medium, that is, by solving eq. (B1) in the entire volume. The dashed lines in the left panels of Fig.7have been obtained using our finite difference solution to the scattering problem in eq. (B3) that involves only local simulations in the perturbed medium. The right panels of Fig.7present the differences between the two sets of waveforms. We observed that the error in the waveforms obtained with our solution to the scattering problem is about 14 order of magnitude smaller than the differences between the waveforms computed in the reference medium and those computed in the perturbed medium (pictured in the left panels of Fig.7). This error is negligible when compared to the error inherent to the finite difference approximation of the wave equation which usually is on the order of a few per cent of the magnitude of the signals (i.e. when compared to analytical solutions). In fact, since the Kirchhoff extrapolation formulas in eq. (B3) are discretized according to the numerical method em-ployed to model wave propagation (i.e. the finite difference method here), no additional error is introduced (i.e. that would add up to the error of the numerical scheme itself) and the result ob-tained with our general solution in eq. (32) can be considered exact.
These results clearly illustrate that our solution to the scattering problem in eq. (32) allows us to compute synthetic seismograms at arbitrary locations and for any acquisition setup built out of the four canonical configurations in Fig.1.
3.2 Approximate solution to the scattering problem
In the previous paragraph, we showed that our solution to the scatter-ing problem in eq. (32) permits the exact computation of synthetic waveforms at arbitrary locations using only local simulations in the
Figure 6. Comparison between the waveforms obtained for the reference model in Fig.4(a) and the waveforms obtained for the perturbed model in Fig.4(b). All the recordings have been obtained by modelling wave propagation in the entire domain using eq. (B1). The recordings obtained for the reference model are shown on the right panels as dashed lines. The recordings obtained for the perturbed model are shown on the right panels as solid lines. The left panels show the difference between the recordings obtained for the reference model and the recordings obtained for the perturbed model. The vertical arrows in the bottom left panel point towards the times at which the sensitivity kernels in Fig.12have been evaluated.
perturbed medium. We now discuss the effect of neglecting some higher order scattering effects in our solution.
To construct the source term imposing the exact boundary con-dition in eq. (32e) or eq. (B3e), a large number of simulations need to be performed to obtain the Green’s functions in the reference medium. These need to be computed prior to running the local simulations, as detailed in Section 2.6 and Appendix B. In some applications, the computational cost of these prior simulations may become very large and limit the efficiency of the proposed approach. In this case, neglecting the source term in eq. (32e) greatly reduces the computational effort needed to obtain synthetic waveforms. The effect of neglecting this source term is illustrated in Fig.8and can be observed by comparing the snapshot in Fig.8(b) where the exact boundary source term is accounted for and the snapshot in Fig.8(c) where this source term is ignored. Observe that, when the exact
boundary is turned off, waves leak out and a residual wavefield escapes the subvolume containing the perturbations. This residual wavefield is generated by the interaction between the waves and the velocity perturbations introduced in the heart-shaped region. When adding this wavefield to the reference wavefield (i.e. computed in the reference medium), we obtain the same wavefield as if we were modelling wave propagation in the entire perturbed medium (i.e. as pictured in Fig.8a). In order to keep the modelling local, when ignoring the exact boundary source term in eq. (32e), one needs to truncate the computational domain using a different method, for example, using absorbing boundaries (e.g. Berenger1994). This is illustrated in Fig.8(d) that is similar to Fig.8(c) except that ab-sorbing boundaries (PML) are used to truncate the computational domain around the region containing the velocity perturbations. In this case, the higher order scattering effects (that correspond to
Figure 7. Comparison between the waveforms obtained using the finite difference solution to the scattering problem in eq. (B3a) (pictured as dotted lines in the right panels and corresponding to the wavefield pictured in Fig.8b) and the waveforms obtained by modelling wave propagation in the entire domain using eq. (B1) (pictured as solid lines in the right panels and corresponding to the wavefield pictured in Fig.8a). All recordings were computed using the perturbed model in Fig.4(b). The left panels show the difference between the two sets of recordings. The vertical arrows in the bottom left panel point towards the times at which the sensitivity kernels in Fig.12have been evaluated.
waves that escape the local modelling domain, interact with the structure outside the modelling domain, and get scattered back in-side the modelling domain) will not be present or visible in the computed waveforms.
To investigate the error introduced in the waveforms when ne-glecting these higher order scattering effects, we recomputed the waveforms in the perturbed medium by replacing the exact bound-ary source term in eq. (32e) with an absorbing boundbound-ary. These waveforms are plotted in Fig.9. The solid lines in the left panels are the same as those in the left panels of Fig.6and are exact solutions obtained using global simulations in the perturbed model, that is, by solving eq. (B1) in the entire volume. The dashed lines in the left panels have been obtained using an approximate finite difference solution to the scattering problem where the exact boundary source
term in eq. (B3e) has been substituted with an absorbing bound-ary. The right panels of Fig.9present the differences between the approximate solutions and the exact solutions. Note that the y-axis scales on the right panels of Fig.9and Fig.6are similar for easier comparison. When looking at the first arrivals (i.e. the beginning of the waveforms) we observe a perfect match between the exact and the approximate waveforms. This is because the energy recorded at these times corresponds to waves travelling around the minimum time ray path that are not scattered off this path. At later times, in the coda, the difference between the exact and the approximate waveforms becomes larger as some scattered waves get lost due to the presence of the absorbing boundary. However, the error present in the approximate waveforms is much smaller than the amplitude of the signals generated by the scatterers present in the perturbed
Figure 8. Snapshots showing the acoustic pressure wavefield at times t= 3.58 × 10−3(s) (top panels) and t= 7.45 × 10−3(s) (bottom panels) generated by a point source inA. All simulation were performed on a grid with dimensions 400 × 400 and grid nodes coordinates (x0+ ix, y0+ y), where, x0= y0
= −5 and x = y = 2.5 × 10−2. The dimension of the computational domain are x∈ [−5 m, 5 m] and y ∈ [−5 m, 5 m], and, only the central part where
x∈ [−2 m, 2 m] and y ∈ [−2 m, 3 m] is represented in the figures. All simulations where performed using the perturbed velocity model pictured in Fig.4(b). The source time function employed for the simulations is plotted in Fig.5. In (a), wave propagation is modelled in the entire domain by solving eq. (B1). In (b), wave propagation is modelled locally by solving eq. (B3b). (c) is similar to (b) but the source term ¯febin eq. (B3b), that is imposing an exact boundary condition, has been set to zero. (d) is similar to (c) but an absorbing boundary layer (PML) has been employed to truncate the computational domain.
medium (shown in the right panels of Fig.6). It is remarkable that, even late in the coda, the approximate waveforms are still accurate. When comparing the observed error in the approximate waveforms for the different combinations of source and receiver positions (in the right panels of Fig.9), we see that the maximum error is ob-served when both the source and the receiver are located within the subvolume which contains the velocity perturbations (i.e. the configuration Source B–Receiver C in Fig.9). This is an important observation as this setup is used in most imaging studies. This is however not very surprising because, in this case, the source and the receiver are located closer to the scatterers located just out-side the modelling region that consequently have a larger influence on the wavefield. Interestingly, when the source and the receiver are located further away from the region containing the perturba-tion (e.g. Source A–Receiver D in Fig.9) the error observed in the approximate waveforms is smaller.
These results suggest that replacing the exact boundary with an absorbing boundary is a reasonable approximation even when the sources and the receivers are located far away from the region containing the scatterers of interest.
3.3 A time reversal experiment
To further investigate the consequences of neglecting higher order multiple scattering effects (i.e. when substituting an exact boundary condition with an absorbing boundary condition), we performed some time reversal experiments. Time reversal as introduced by Fink (1992) is a two-step process: First, waves propagating through a medium are recorded with an array of transducers. Then, the records are reversed in time and re-emitted by the transducers back into the medium, so that the wave energy is refocused in time and space at the position of the source. In the situation where a single transducer is used to re-emit or back-propagate the waves through the medium, spatial refocusing at the source point is achievable only in heavily scattering media. This is because the waves emit-ted at the receiver point will bounce on the scatterers, which will act as secondary sources and coherently emit energy toward the source point. This aspect is key in our tests. Indeed, if some of these scattering effects are not accounted for in the modelling, time reversal may not be completely doable. In Fig.10, we present the spatial refocusing of the wavefields obtained in three different time
Figure 9. Waveforms showing the effect of neglecting the source term ¯febthat is imposing the exact boundary condition in eq. (B3b) and accounts for the multiple scattering effects. The waveforms pictured as dashed lines in the right panels have been obtained using the finite difference solution to the scattering problem in eq. (B3a) where the source term ¯febin eq. (B3b) has been set to zero and an absorbing boundary have been used to truncate computational the domain, the corresponding wavefield is pictured in Fig.8(d). The acoustic recordings pictured as solid lines in the right panels have been obtained by modelling wave propagation in the entire domain using eq. (B1), the corresponding wavefield is pictured in Fig.8(a). The left panels shows the difference between the two sets of recordings. The vertical arrows in the bottom left panel point towards the times at which the sensitivity kernels in Fig.12have been evaluated.
reversal experiments. In all three experiments, we used the same procedure. First we ran a forward simulation using a point source at point B where we recorded the wavefield at point D. Then, we ran a backward simulation where the signal recorded in the forward simulation was time reversed and used as source at point D. Dur-ing the backward simulation, we recorded the wavefield at point B. In the first experiment, pictured in Fig.10(a), we modelled wave propagation in the entire domain in both simulations. In the second experiment, pictured in Fig.10(b), we modelled wave propaga-tion locally in the heart-shaped region and we used eq. (B3b) to obtain the recordings at points B and D in the backward and the forward simulations, respectively. In the third experiment, pictured in Fig.10(c), we proceeded as for the second experiment but the
exact boundary condition in eq. (B3e) was substituted with an ab-sorbing boundary condition (PML). In the first experiment, both global simulations have been performed in the perturbed medium pictured in Fig.4(a). In the second and the third experiments, all global simulations involved in the calculations where performed in the reference medium in Fig.4(a), while the local simulations were performed using the perturbed medium in Fig.4(b). As expected, when using the exact solution to the scattering problem in eq. (B3), the wavefield and the refocusing of the energy at the source point (in Fig.10b) are indistinguishable from those obtained when mod-elling wave propagation in the entire domain (in Fig.10a). What is more remarkable is that the wavefield obtained using the approx-imate solution (in Fig.10c) is almost identical to those obtained
