CWP-717

Focusing the wavefield inside an unknown 1D medium:

Beyond seismic interferometry

Filippo Broggini1, Roel Snieder1 & Kees Wapenaar21 Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA2 Department of Geotechnology, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands

ABSTRACT

With seismic interferometry one can retrieve the response to a virtual sourceinside an unknown medium, assuming there is a receiver at the position of thevirtual source. In this paper, we demonstrate that for a 1D medium the re-quirement of having an actual receiver inside the medium can be circumvented,going beyond seismic interferometry. This is accomplished using inverse scat-tering theory. We show that the wavefield can be focused inside an unknownmedium with independent variations in velocity and density using reflectiondata only.

Key words: acoustic, scattering, wave propagation, crosscorrelation, reflec-tivity

1 INTRODUCTION

There are different ways to reconstruct the wave fieldexcited by a hypothetical source in the interior of anunknown medium. First, with seismic interferometry itis possible to retrieve the response to a virtual sourceinside the medium, assuming there is a receiver at theposition where the virtual source is to be created andassuming the medium is surrounded by sources. Themedium parameters need not be known. Second, in thispaper we show that, with one-dimensional inverse scat-tering theory, the response to a virtual source inside themedium can be obtained from reflected waves recordedat one side of the medium. We demonstrate that thesecond approach is possible for the one-dimensional sit-uation without knowing the medium parameters. Thisis fascinating because it allows one to obtain the samevirtual source response as with seismic interferometry(including all scattering effects), but without the needof having a receiver at the virtual source location. Anessential element of this approach is to use a compli-cated incident wave that is designed to collapse onto apoint inside the medium at a specified time.

Seismic interferometry, inverse scattering, and fo-cusing are subjects usually studied in different researchareas such as seismology (Aki and Richards, 2002),quantum mechanics (Rodberg and Thaler, 1967), op-tics (Born and Wolf, 1999), non-destructive evalua-

tion of material (Shull, 2002), and medical di-agnostics (Epstein, 2003). Seismic interferometry

(Wapenaar et al., 2005; Curtis et al., 2006; Schuster,2009) is a technique that allows one to reconstruct theresponse between two receivers from the cross-correla-tion of the wavefields measured at these two receiversthat are excited by uncorrelated sources surroundingthe studied system. This technique is also known as thevirtual source method (Bakulin and Calvert, 2006); thisrefers to the fact that the new response is reconstructedas if one receiver had recorded the response due to avirtual source located at the other receiver position. In-verse scattering (Chadan and Sabatier, 1989; Gladwell,1993; Colton and Kress, 1998) is the problem of deter-mining the perturbation of a medium (e.g., of a constantvelocity medium) from the field scattered by this per-turbation. In other words, one wants to reconstruct theproperties of the perturbation from a set of measureddata. Exact inverse scattering takes into account thenonlinearity of the inverse problem, but it also presentssome drawbacks: it may be improperly posed from thepoint of view of numerical computations (Dorren et al.,1994), and it requires data recorded at locations usu-ally not accessible due to practical limitations. In thispaper, the term focusing (Rose, 2001, 2002b) refers tothe technique of finding an incident wave that collapsesto a spatial delta function δ(z − z0) at the location z0

170 F. Broggini, R. Snieder & K. Wapenaar

0 1 2 3 40

1

2

3

t (s

)

z (km)

Figure 1. Scattering experiment with a source located atz = 2.44 km. The traces are recorded by receivers locatedat each location in the model (shown in Figure 2) with aspacing of 40 m.

and at a prescribed time t0 (i.e., the wavefield is focusedat z0 at t0). In a one-dimensional medium, we deal witha one-sided problem when observations from only oneside of the perturbation are available (e.g., due to thepractical consideration that we can only record reflectedwaves); otherwise, we call it a two-sided problem whenwe have access to both sides of the medium and accountfor both reflected and transmitted waves.

2 WAVEFIELD FOCUSING

Figure 1 illustrates a numerical scattering experimentin a one-dimensional acoustic medium where an impul-sive source is placed at the position z = 2.44 km in themodel shown in Figure 2. Note that velocity and densityvary independently in depth. The acoustic wave equa-tion is LG(z, z0, t) = −δ(z−z0)

ddtδ(t), with the differen-

tial operator L ≡ ρ(z) ddz(ρ(z)−1 d

dz) − c(z)−2 d2

dt2. Here

z0 = 2.44 km and the initial condition is G(z, z0, t <

0) = 0. The incident wavefield propagates toward thediscontinuities in the model, interacts with them, andgenerates scattered waves. We use a time-space finite-difference code with absorbing boundary condition tosimulate the propagation of the one-dimensional wavesand to produce the numerical examples shown in thissection. For a computational purpose, the source func-tion −δ(z − z0)

ddtδ(t) is convolved with a band-limited

time wavelet S(t). The computed wavefield is shown inFigure 1 and it represents the causal Green’s functionof the system, G. Causality ensures that the wavefield isnon zero only in the region delimited by the first arrival(i.e., the direct waves). The slope of the lines, repre-senting the first arrival, depends on the velocity modelof Figure 2.

Due to practical limitations, we usually are not ableto place a source inside the medium we want to probe,which raises the following question: Is it possible to cre-ate the wavefield illustrated in Figure 1 without having

0 1 2 3 40

0.5

1

1.5

2

Velo

city (

km

/s)

z (km)

0 1 2 3 40

0.5

1

1.5

2

De

nsity (

g/c

m3)

z (km)

Figure 2. Velocity and density profiles of the one-dimensional model. The perturbation in the velocity is lo-cated between z = 1.3 − 3.5 km and c0 = 1 km/s. The

perturbation in the density is located between z = 2.0− 3.5km and ρ0 = 1 g/cm3.

a real source at the position z = 2.44 km? An initial an-swer to such a question is given by seismic interferome-try. This technique allows one to reconstruct the wave-field that propagates between a virtual source and otherreceivers located inside the medium (Wapenaar et al.,2005). We remind the reader that this technique yieldsa combination of the causal wavefield G and its time-reversed version Ga (i.e., anti-causal). This is due tothe fact that the reconstructed wavefield is propagat-ing between a receiver and a virtual source. Concep-tually speaking, without a real (physical) source, onemust have non-zero incident waves on a receiver to cre-ate waves that emanate from that receiver. The fun-damental steps to reconstruct the Green’s function are(Curtis et al., 2006)

Focusing the wavefield 171

0 1 2 3 4

0

1

2

Ve

locity (

km

/s)

z (km)

zsl

zsr

zvs

Figure 3. Diagram showing the locations of the real andvirtual sources for seismic interferometry. zsl and zsr indicatethe two real sources. zvs shows the virtual source location.

0 1 2 3 40

1

2

3

t (s

)

z (km)

Figure 4. Causal part of the wavefield estimated by theGreen’s function reconstruction technique when the receiverlocated at z = 2.44 km acts as a virtual source.

1. measure the wavefields G(z, zsl, t) andG(z, zsr, t) at a receiver located at z (z variesfrom 0.5 km to 4 km) excited by impulsivesources located at zsl and zsr at both sides ofthe perturbation (a total of two sources in 1D) asshown in Figure 3;2. cross-correlate G(z, zsl, t) with G(zvs, zsl, t),where zvs = 2.44 km and vs stands for virtual

source;3. cross-correlate G(z, zsr, t) with G(zvs, zsr, t);4. sum the results computed at the two previouspoints to obtain G(z, zvs, t) +Ga(z, zvs, t);5. repeat steps 1 to 4 for a receiver located at adifferent z.

The causal part of wavefield estimated by the Green’sfunction reconstruction technique is shown in Figure4 and it is consistent with the result of the scatter-ing experiment produced with a real source located atz = 2.44 km, shown in Figure 1.

We thus have two different ways to reconstruct thesame wavefield, but often we are not able to access acertain portion of the medium we want to study andhence we cannot place any sources or receivers inside it.We next assume that we only have access to reflectedwaves R(t) measured above of the perturbation, i.e. thereflected impulse response measured at z = 0 km due

to an impulsive source placed at z = 0 km. This furtherlimitation raises another question: Can we reconstructthe same wavefield shown in Figure 1 having knowl-edge only of the reflected waves R(t)? Since there areno sources present inside the perturbation, we speculatethat the reconstructed wavefield consists of a causal andan anti-causal part.

For this one-dimensional problem, the answer tothis question is given by Rose (2001, 2002a). He showsthat there exists a particular incident wave that col-lapses the wavefield to a spatial delta function at thedesired location after it interacts with the medium, andthat this incident wave consists of a spatial delta func-tion followed by the solution of the Marchenko equation(Chadan and Sabatier, 1989; Lamb, 1980).

The Marchenko integral equation is a fundamentalrelation of one-dimensional inverse scattering theory. Itis an integral equation that relates the reflected wavesR(t) to the incident wavefield u(t, tf ) which will create afocus in the interior of the medium and ultimately givesthe perturbation of the medium. The one-dimensionalform of this equation is

0 = R(t+ tf )+u(t, tf )+

∫ tf

−∞

R(t+ t′)u(t′, tf )dt

′

, (1)

where tf is the one-way travel time from z = 0 to thefocusing location. We numerically solve the Marchenkoequation and construct the particular incident wave thatfocuses at t = 0 s at a location specified by tf = 3s. Any appropriate numerical method to solve integralequations can be used to compute u(t, tf ). Note thatsolving Equation 1 does not require any knowledge ofthe medium: all that is needed is the reflection responseR(t) and the one-way travel time tf . Next, we injectthe particular incident wave δ(t + tf ) + u(−t, tf ), asshown in Figure 5, at z = 0 km and compute the time-space diagram shown in the top panel of Figure 6: itshows the evolution in time of the wavefield when theincident wave is injected at z = 0 into the model. Weemphasize that this method is data driven, hence thetrue model is not needed to build the particular incidentwavefield. The time-space diagram shown in Figure 6 iscomputed using the true model, but this is done onlyto illustrate the physics of the focusing process. Thebottom panel of Figure 6 shows a cross-section of thewavefield at time t = 0 s: the wavefield vanishes exceptat location z = 2.44 km. Note, however, that the timederivative of the wavefield in Figure 6 (i.e., the particlevelocity) is not focused at z = 2.44 km, hence the energyis not focused at this location. For this particular model,tf = 3 s corresponds to spatial focusing at the samelocation where we placed the virtual source in Figure 4.In order to focus the wavefield at a prescribed locationzf (and not at the prescribed one-way travel time tf ), weneed to know an estimate of the travel times of the firstarrivals, but no information about the density would berequired.

172 F. Broggini, R. Snieder & K. Wapenaar

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5−4

−2

0

2

4

6

Am

plit

ud

e

t (s)

Figure 5. Particular incident wave at z = 0 km that fo-

cuses at t = 0 s at z = 2.44 km. It is composed by a deltafunction (at t = −3 s) plus the time-reversed solution of theMarchenko equation, u(−t, tf ), where tf = 3 s. This wavehas an approximate duration of 3.5 s.

0 1 2 3 4

−3

−2

−1

0

1

2

3

t (s

)

0 1 2 3 40

1

2

3

Am

plit

ud

e

z (km)

Figure 6. Top: At z = 0 km, we inject the particular incident

wave in the model and compute the time-space diagram byforward modeling. We denote this wavefield as K(z, t). Notethat the waves continue to propagate after 3 s. Bottom: cross-section of the wavefield at t = 0 s.

Figure 6 does not yet resemble the wavefield shownin Figures 1 and 4. However, if we denote the wave-field in Figure 6 as K(z, t) and its time-reversed versionas K(z,−t), we obtain the wavefield shown in Figure 7by adding K(z, t) and K(z,−t). With this process, weeffectively go from one-sided to two-sided illuminationbecause in Figure 7 waves are incident on the scattererfrom both sides for t < 0 s. The incident waves are nonzero for −6s < t < −3s, but to facilitate a comparisonwith Figures 1 and 4 this time interval is not completelyincluded in the figure. According to Figure 7, we createa focus at a location inside the perturbation withouthaving a source or a receiver at such a location and with-out any knowledge of the medium properties; we onlyhave access to the reflected impulse response measuredabove of the perturbation. With an appropriate choiceof sources and receivers, this experiment can be done inpractice, e.g., in an acoustic laboratory (Rose, 2002a).Burridge (1980) shows diagrams similar to Figures 6and 7 and explains how to combine such diagrams us-ing causality and symmetry properties. The upper conein Figure 7 corresponds to the causal Green’s functionG and the lower cone represents the anti-causal Green’sfunction Ga. Note that a small amount of energy is out-side the two cones. This is due to numerical inaccuraciesin our solution of the Marchenko equation. The causalpart of the trace at z = 0 km is the virtual source re-sponse that we obtained without using the model.

The anti-causal Green’s function Ga follows fromG by time-reversal, hence it satisfies LGa = −δ(z −

z0)d

d(−t)δ(−t) = δ(z − z0)

ddtδ(t), where we used that

L is invariant to time-reversal. Adding the differentialequations for G and Ga shows that G+Ga satisfies thehomogeneous equation: L(G+Ga) = −δ(z−z0)

ddtδ(t)+

δ(z− z0)ddtδ(t) = 0. The term homogeneous means that

the sum of G and Ga is source free. Hence, to focusthe wavefield at the virtual source location, there mustbe a particular incident wavefield coming from anotherlocation. In fact, the knowledge that G+Ga satisfies ahomogeneous equation suggests that a combination ofthe causal and anti-causal Green’s functions is neededto focus the wavefield at a location where there is no realsource (i.e., source free), as shown in Figure 7. Oristaglio(1989) shows a similar result, although he derives thedifference (instead of the sum) of the causal and anti-causal Green’s functions due to a different definition ofthe Green’s functions.

The extension of the iterative process in twoand three dimensions still needs to be investigated(Wapenaar et al., 2011). We speculate that focusing thewavefield at a prescribed location in two- and three- di-mensional media requires an estimate of the primarytraveltimes from the virtual source location. ? give afirst mathematical proof for a two-dimensional mediumwith density variations only.

Focusing the wavefield 173

0 1 2 3 4

−3

−2

−1

0

1

2

3

t (s

)

z (km)

Figure 7. Wavefield that focuses at z = 2.44 km at t = 0 swithout a source or a receiver at this location. This wavefield

corresponds to K(z, t) +K(z,−t). This wavefield consists ofa causal (t > 0) and an anti-causal (t < 0) part.

3 CONCLUSIONS

There are three distinct ways to reconstruct the samephysical wave state. A physical source, seismic interfer-ometry, and inverse scattering theory allow one to createthe same wave state (see Figures 1, 4, and 7) that fo-cuses at a certain location zs (2.44 km in our examples).Seismic interferometry tells us how to build an estimateof the wavefield without knowing the medium proper-ties, if we have a receiver at the same location zs of thereal source in the scattering experiment of Figure 1 andsources surrounding the medium. Inverse scattering goesbeyond this as it allows us to focus the wavefield insidethe medium without knowing its properties, using onlyreflected waves recorded at one side of the medium. Thiscan be done also when density and velocity vary inde-pendently. This fact is a new contribution because theinverse scattering theory of Rose (2001, 2002b) and oth-ers (Aktosun and Rose, 2002) did not deal with simulta-neous changes in density and velocity, because one can-not retrieve two independent quantities from one timeseries of reflected waves. We also add another step, i.e.,K(z, t)+K(z,−t), which creates the response of the vir-tual source. This is an important insight that does notappear in any of the papers by Rose. We show that theinteraction between causal and anti-causal wavefields isa key element to focus the wavefield where there is noreal source.

We speculate that many of the insights gained inour one-dimensional framework are still valid in a two-and three-dimensional framework. An extension of this

work to two or three dimensions would give us the theo-retical tools for many useful practical applications. Forexample, if we knew how to create the three-dimensionalversion of the particular incident wavefield, we could fo-cus the wavefield to a point in the subsurface to simu-late a source at depth and to record data at the surface.This kind of application will be helpful for full waveforminversion (Brenders and Pratt, 2007) and subsalt imag-ing (Sava and Biondi, 2004), where waves that have tra-versed a strongly inhomogeneous overburden are of ex-treme importance.

ACKNOWLEDGMENTS

The authors would like to thank the members of theCenter for Wave Phenomena, Kasper van Wijk, An-drew Curtis, and two anonymous reviewers for theirconstructive comments. This work was supported by thesponsors of the Consortium Project on Seismic InverseMethods for Complex Structures at the Center for WavePhenomena.

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