Focusing the wavefield inside an unknown 1D medium: Beyond
seismic interferometry
Filippo Broggini
1
, Roel Snieder
1
, and Kees Wapenaar
2
ABSTRACT
With seismic interferometry one can retrieve the response
to a virtual source inside an unknown medium, if there is a
receiver at the position of the virtual source. Using inverse
scattering theory, we demonstrate that, for a 1D medium, the
requirement of having an actual receiver inside the medium
can be circumvented, going beyond seismic interferometry.
In this case, the wavefield can be focused inside an unknown
medium with independent variations in velocity and density
using reflection data only.
INTRODUCTION
There are different ways to reconstruct the wavefield excited by a
hypothetical source in the interior of an unknown medium. First,
with seismic interferometry (Weaver and Lobkis, 2001; Wapenaar
et al., 2005; Curtis et al., 2006; Schuster, 2009) it is possible to
retrieve the response to a virtual source inside the medium with
a receiver at the position where the virtual source is to be created,
assuming the medium is surrounded by uncorrelated sources. The
medium parameters need not be known. Second, in this paper, we
show that, with 1D inverse scattering theory (Chadan and Sabatier,
1989; Gladwell, 1993; Colton and Kress, 1998), the response to a
virtual source inside the medium can be obtained from reflected
waves recorded at one side of the medium. We demonstrate that,
in 1D media, this is possible without knowing the medium para-
meters. This is fascinating because it allows one to obtain the same
virtual source response as with seismic interferometry (including all
multiples), but without the need to have a receiver at the virtual
source location. An essential element of this approach is to build
an incident wave that is designed to collapse onto a point inside
the medium at a specified time. The reconstructed wavefield can
be used to illuminate the medium under a complicated overburden,
and the extracted Green’s function can be used for imaging.
In this paper, the term focusing (Rose, 2001, 2002b) refers to the
technique of finding an incident wave that collapses to a spatial del-
ta function δðz − z
0
Þ at the location z
0
and at a prescribed time t
0
(i.e., the wavefield is focused at z
0
at t
0
). In a 1D medium, we deal
with a one-sided problem when observations from only one side of
the perturbation are available (e.g., due to the practical considera-
tion that we can only record reflected waves); otherwise, we call it
a two-sided problem when we have access to both sides of the
medium and account for reflected and transmitted waves.
WAVEFIELD FOCUSING
Figure 1 shows the velocity and density profiles of a 1D acoustic
medium. Note that velocity and density vary independently in
depth. We simulate a numerical scattering experiment where an im-
pulsive source is placed at the position z ¼ 2.44 km . The acoustic
wave equation is LGðz; z
VS
;tÞ¼−δðz − z
VS
Þ
d
dt
δðtÞ, with the
differential operator L ≡ ρðzÞ
d
dz
ðρðzÞ
−1
d
dz
Þ − cðzÞ
−2
d
2
dt
2
. Here,
z
VS
¼ 2.44 km and the initial condition is Gðz; z
VS
;t < 0Þ¼0.
The incident wavefield propagates toward the discontinuities in
the model, interacts with them, and generates scattered waves.
We use a time-space finite-difference code with absorbing boundary
conditions to simulate the propagation of the 1D waves and to pro-
duce the numerical examples shown in this section. For computa-
tional purposes, the source function −δðz − z
VS
Þ
d
dt
δðtÞ is convolved
with a band-limited wavelet sðtÞ. The computed wavefield shown in
Figure 2 represents the causal Green’s function of the system G
convolved with sðtÞ. Causality ensures that the wavefield is nonzero
only in the region delimited by the first arrivals (i.e., the direct
waves) and we refer to it as the “causal region.”
Due to practical limitations in field experiments, we usually are
not able to place a source inside the medium we want to probe.
However, when there are receivers inside the medium, seismic in-
terferometry allows us to determine the wavefield as if there was a
source at the position of any of the receivers, e.g., at z ¼ 2.44 km .
Manuscript received by the Editor 17 February 2012; revised manuscript received 16 April 2012; published online 8 August 2012.
1
Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, USA. E-mail: fbroggin@mines.edu; rsnieder@mines.edu.
2
Delft University of Technology, Department of Geotechnology, Delft, The Netherlands. E-mail: c.p.a.wapenaar@tudelft.nl.
© 2012 Society of Exploration Geophysicists. All rights reserved.
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GEOPHYSICS, VOL. 77, NO. 5 (SEPTEMBER-OCTOBER 2012); P. A25–A28, 4 FIGS.
10.1190/GEO2012-0060.1
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This technique allows one to reconstruct the wavefield that propa-
gates between a virtual source and other receivers located inside the
medium (Wapenaar et al., 2005). This technique yields a combina-
tion of the causal wavefield G and its time-reversed version G
a
(i.e.,
anticausal). This is due to the fact that the reconstructed wavefield
propagates between a receiver and a virtual source. Conceptually
speaking, without a real (physical) source, one must have nonzero
incident waves on a receiver to create waves that emanate from
that receiver. The fundamental equation to reconstruct the Green’s
function is (Wapenaar and Fokkema, 2006)
Gðz; z
VS
;tÞþGðz; z
VS
; −tÞ
∝
X
z
0
¼z
S1
;z
S2
Gðz; z
0
;tÞGðz
VS
;z
0
; −tÞ;
(1)
where z
VS
¼ 2.44 km, and z
S1
and z
S2
are the coordinates of
impulsive sources located at both sides of the perturbation (a total
of two sources in 1D) as shown at the bottom of Figure 1. Between
z
S1
and z
S2
, the causal part of the wavefield estimated by this
Green’s function reconstruction technique is consistent with the
wavefield shown in Figure 2.
We thus have two different ways to obtain the same wavefield,
but often we cannot place any sources or receivers inside the med-
ium. We next assume that we only have access to reflected waves
RðtÞ measured above the perturbation, i.e., the reflected impulse
response measured at z ¼ 0kmdue to an impulsive source placed
at z ¼ 0km. This limitation raises another question: Can we recon-
struct the same wavefield shown in Figure 2 having knowledge only
of the reflected waves RðtÞ? For this 1D problem, the answer to this
question is given by the Marchenko equation (Lamb, 1980; Chadan
and Sabatier, 1989). Its solution provides a particular incident wave
that collapses the wavefield to a spatial delta function at the desired
location after it interacts with the medium, and this incident wave
consists of a delta function added to the solution of the Marchenko
equation (Rose, 2001, 2002a).
The Marchenko integral equation is a fundamental relation of 1D
inverse scattering theory. It is an integral equation that relates the
reflected waves RðtÞ to the incident wavefield uðt; t
f
Þ, which cre-
ates a focus in the interior of the medium and ultimately gives the
perturbation of the medium. The 1D form of this equation is
0 ¼ Rð t þ t
f
Þþuðt; t
f
Þþ
Z
t
f
−∞
Rðt þ t
0
Þuðt
0
;t
f
Þdt
0
; (2)
where t
f
is the one-way traveltime from z ¼ 0 to the focusing loca-
tion. We numerically solve the Marchenko equation and construct
the particular incident wave that focuses at t ¼ 0sat a location spe-
cified by t
f
¼ 3s. Any appropriate numerical method to solve in-
tegral equations can be used to compute uðt; t
f
Þ (e.g., an iterative
method). Note that solving equation 2 does not require any knowl-
edge of the medium: All that is needed is the reflection response
RðtÞ and the one-way traveltime t
f
, which specifies the location
of the focus. Next, we inject the particular incident wave
δðt þ t
f
Þþuð−t; t
f
Þ at z ¼ 0kmand compute the time-space dia-
gram shown in the top panel of Figure 3. This shows the wavefield
when the incident wave is injected at z ¼ 0 into the model. We
define this wavefield as Kðz; tÞ. The time-space diagram shown
in Figure 3 is computed using the true model, but this is done only
to illustrate the physics of the focusing process. The bottom panel of
Figure 3 shows a cross section of the wavefield at time t ¼ 0s: The
wavefield vanishes except at location z ¼ 2.44 km. For this parti-
cular model, t
f
¼ 3scorresponds to spatial focusing at the same
location where we placed the virtual source in Figure 1. We empha-
size that this method is data driven, hence the true model is not
needed to build the particular incident wavefield.
Figure 3 does not yet resemble the wavefield shown in Figure 2.
In fact, in Figure 3 waves cross the solid line at t ¼ 0sfor locations
z ≤ 2.44 km, whereas in Figure 2 waves cross the same solid line
only at the virtual source location z ¼ 2.44 km. However, denoting
as Kðz; −tÞ the time-reversed version of Kðz; tÞ, we obtain the wa-
vefield shown in Figure 4 by adding Kðz; tÞ and Kðz; −tÞ. With this
summation, we create the response to a virtual source located at
z
VS
¼ 2.44 km, namely Gðz; z
VS
;tÞþGðz; z
VS
; −tÞ (convolved
with sðtÞ). This step is the main result of this paper. Note that
the trace at z ¼ 0, Kð0;tÞþKð0; −tÞ, has been obtained without
any information about the model. As in Figure 3, the remainder
of Figure 4 is based on the true model and is only shown to explain
the physics of the focusing process. In Figure 4, the wavefield out-
side the causal region is zero because the portion of Kðz; tÞ outside
0 0.5 1 1.5 2 2.5 3 3.5 4
0.5
1
1.5
2
Velocity (km/s)
Depth (km)
z
S1
z
S2
z
VS
0 0.5 1 1.5 2 2.5 3 3.5 4
0.5
1
1.5
Density (g/cm
3
)
Figure 1. Top: Velocity profile of the 1D model (solid line). The
perturbation in the velocity is located between z ¼ 1.3–3.5 km
and c
0
¼ 1km∕s. Middle: Density profile of the 1D model (dashed
line). The perturbation in the density is located between
z ¼ 2.0–3.5 km and ρ
0
¼ 1g∕cm
3
. Bottom: Locations of the real
and virtual sources for seismic interferometry; z
S1
and z
S2
indicate
the two real sources and z
VS
shows the virtual source location.
01234
0
1
2
3
Time (s)
Depth (km)
Figure 2. Response to a source located at z ¼ 2.44 km. The traces
are recorded by receivers located at each location in the model
(shown in Figure 1) with a spacing of 40 m. Waves are emanating
from the line t ¼ 0sonly at z ¼ 2.44 km.
A26 Broggini et al.
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the causal region is antisymmetric in time and hence cancels in the
sum Kðz; tÞþKðz; −tÞ. With this process, we effectively go from
one-sided to two-sided illumination because in Figure 4 waves are
incident onto the focusing location from both sides for t < 0 s. The
incident waves are non zero for −6s< t < −3s, but to facilitate a
comparison with Figure 2 this time interval is not completely in-
cluded in the figure. According to Figure 4, we create a focus at
a location inside the inhomogeneous medium without having a
source or a receiver at such a location and without any knowledge
of the medium properties; we only have access to the reflected im-
pulse response measured above the perturbation. With an appropri-
ate choice of sources and receivers, this experiment can be done in
practice, e.g., in an acoustics laboratory (Rose, 2002a). Burridge
(1980) shows diagrams similar to Figures 3 and 4 and explains
how to combine such diagrams using causality and symmetry prop-
erties. The wavefield for positive times t>0 (causal region) in
Figure 4 corresponds to the causal Green’s function G and the
wavefield for negative times t < 0 represents the anticausal Green’s
function G
a
(defining the anticausal region). A small amount of en-
ergy is outside of the causal and anticausal regions due to numerical
inaccuracies in our solution of the Marchenko equation (this is also
visible in the bottom panel of Figures 3 and 4). The causal part of
the trace at z ¼ 0kmis the virtual source response Gð0;z
VS
;tÞ that
we obtained without using the model.
The anticausal Green’s function G
a
follows from G by
time-reversal, hence it satisfies LG
a
¼ −δðz − z
VS
Þ
d
dð−tÞ
δð−tÞ¼
δðz − z
VS
Þ
d
dt
δðtÞ, where we used that L is invariant to time-reversal.
Adding the differential equations for G and G
a
shows that G þ G
a
satisfies the homogeneous equation: LðG þ G
a
Þ¼−δðz − z
VS
Þ
d
dt
δðtÞþδðz − z
VS
Þ
d
dt
δðtÞ¼0. The term “homogeneous” suggests
that the sum of G and G
a
is source-free. Hence, to focus the wave-
field at the virtual source location, there must be a particular
incident wavefield coming from another location. In fact, the
knowledge that G þ G
a
satisfies a homogeneous equation suggests
that a combination of the causal and anticausal Green’s functions is
needed to focus the wavefield at a location where there is no real
source (i.e., source-free), as shown in Figure 4. Oristaglio (1989)
shows a similar result, although he derives the difference (instead
of the sum) of the causal and anticausal Green’s functions due to a
different definition of the Green’s functions.
DISCUSSION
In the previous section, we use the one-way traveltime t
f
to de-
termine the depth z
VS
of the virtual source. In other words, the wa-
vefield focuses at the virtual source location z
VS
after it has
propagated inside the medium for a length of time equal to t
f
.
To directly choose a prescribed focusing location z
VS
(and not a
prescribed one-way traveltime t
f
), we need to know the average
velocity of the medium between the surface and the depth of the
focusing location. However, no information about either the density
or the details of the velocity profile is required.
This method also works when density and velocity vary indepen-
dently. This fact is a ne w contribution because the previous in verse
scattering theory of (Rose 2001, 2002b)andothers(Aktosun and
Rose, 2002) does not deal with simultaneous changes in density
and velocity because one cannot retrie v e two independent quantities
from one time series of reflected waves. We also add another step
beyond the work of Rose by forming the sum Kðz; tÞþKðz; −tÞ,
which ensures that the wavefield vanishes outside the causal and
anticausal regions and creates the response of the virtual source.
We show that the interaction between causal and anticausal wavefields
is a key element to focus the wav ef ield where there is no real source.
We are currently investigating the application of the central ideas
of this work to wave propagation in two and three dimensions
01234
−3
−2
−1
0
1
2
3
Time (s)
01234
0
2
4
6
Amplitude
Depth (km)
Figure 4. Top: Wavefield that focuses at z ¼ 2.44 km at t ¼ 0s
without a source or a receiver at this location. This wavefield cor-
responds to Kðz; tÞþKðz; −tÞ and consists of a causal (t>0) and
an anticausal (t < 0) region. Bottom: Cross section of the wavefield
at t ¼ 0s.
01234
−3
−2
−1
0
1
2
3
Time (s)
01234
0
1
2
3
Amplitude
Depth (km)
Figure 3. Top: At z ¼ 0km, we inject the particular incident wave
in the model and compute the time-space diagram by forward mod-
eling. We denote this wavefield as Kðz; tÞ. Waves cross the solid line
at t ¼ 0sfor locations z ≤ 2.44 km. Note that the waves continue to
propagate after 3 s. Bottom: Cross section of the wavefield at t ¼ 0s.
Wavefield focusing
A27
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(Wapenaar et al., 2011). This extension allows us to focus the wave-
field to a point in the subsurface to simulate a source at depth and to
record data at the surface. This kind of application will be helpful
for full-waveform inversion (Brenders and Pratt, 2007) and subsalt
imaging (Sava and Biondi, 2004), where waves that have traversed a
strongly inhomogeneous overburden are of extreme importance. We
speculate that focusing the wavefield at a prescribed location in 2D
and 3D media requires an estimate of the primary traveltimes from
the virtual source location to the receivers (e.g., using a macro mod-
el). Wapenaar et al. (2012) give a first mathematical proof for a 2D
medium with density variations only.
CONCLUSIONS
There are three distinct ways to reconstruct the same physical
wave state. A physical source, seismic interferometry, and inverse
scattering theory allow one to create the same wave state that focuses
at a certain location z
VS
. Seismic interferometry tells us how to build
an estimate of the wavefield without knowing the medium properties,
if we have a receiver at the same location z
VS
of the real source in the
scattering experiment of Figure 2 and sources surrounding the med-
ium. Inverse scattering goes beyond this as it allows us to focus the
wavefield inside the medium without knowing its properties, using
only reflected waves RðtÞ recorded at one side of the medium.
ACKNOWLEDGMENTS
The authors thank the members of the Center for Wave Phenom-
ena, Kasper van Wijk, Andrew Curtis, and two anonymous reviewers
for their constructive comments. This work was supported by the
sponsors of the Consortium Project on Seismic Inverse Methods
for Complex Structures at the Center for Wave Phenomena.
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