Incorporating free-surface multiples in Marchenko imaging · 2019. 8. 31. · Incorporating...

CWP-827

Incorporating free-surface multiples in Marchenkoimaging

Satyan Singh1, Roel Snieder1, Joost van der Neut2, Evert Slob2,Jan Thorbecke2 and Kees Wapenaar2

(1) Center for Wave Phenomena, Department of Geophysics, Colorado School of Mines, Golden, Colorado, USA

(2) Department of Geoscience and Engineering, Delft University of Technology, GA Delft, The Netherlands

ABSTRACTImagine placing a receiver at any location in the Earth and recording the re-sponse at that location to sources on the surface. In such a world, we could placereceivers around our reservoir to better image the reservoir and understand itsproperties. Realistically, this is not a feasible approach for understanding thesubsurface. Here, we present an alternative and realizable approach to obtain-ing the response of a buried virtual receiver for sources at the surface. Thismethod is capable of retrieving the Green’s function for a virtual point in thesubsurface to the acquisition surface. In our case, a physical receiver is not re-quired at the subsurface point; instead, we require the reflection measurementsfor sources and receivers at the surface of the Earth and a macro-model (nosmall-scale details of the model are necessary). We can interpret the retrievedGreen’s function as the response to sources at the surface for a virtual receiverin the subsurface. We obtain this Green’s function by solving the Marchenkoequation, an integral equation pertinent to inverse scattering problems. Ourderivation of the Marchenko equation for the Green’s function retrieval takesinto account the free-surface reflections. We decompose the Marchenko equationinto up- and down-going fields and solve for these fields iteratively. We use theseup- and down-going fields, which includes the free-surface multiples, to obtain a2D image of our area of interest, in this case, below a synclinal structure. Thisimaging is called Marchenko imaging.

1 INTRODUCTION

Traditionally, to image the subsurface using standardimaging methods like reverse time migration (RTM)or Kirchhoff migration, one assumes the first-orderBorn approximation. This assumption only allows us touse primary reflections in conventional imaging (single-scattered waves). However, the assumption of the firstBorn approximation leads to artifacts in the presence ofmultiples. In order to implement conventional imagingand to ensure the assumption of single scattering holds,one has to remove multiply reflected waves. Multiplesconsist of internal and free-surface multiples. The re-moval of free-surface multiples is generally a priority inthe recorded reflection response since free surface mul-tiples are, in general, stronger than internal multiples.

To remove surface multiples from the reflection re-sponse, there are model-based methods (Wiggins, 1988;Lokshtanov, 1999), inverse-scattering based methods(Weglein et al., 1997), data-driven methods (Verschuuret al., 1992; Berkhout and Verschuur, 1997) and re-cently, inversion methods (van Groenestijn and Ver-

schuur, 2009; Ypma and Verschuur, 2013). The data-driven technique proposed by Verschuur et al. (1992),called surface-related multiple elimination (SRME), isthe most popular method for attenuating multiples be-cause it has been proven to be effective on many realdata examples. Although internal multiples are weaker,there are data driven methods (Berkhout and Ver-schuur, 1997; Verschuur and Berkhout, 2005) and in-verse scattering methods (Ramırez et al., 2005) that re-move them from the reflection response. Removing themultiples is not always a simple task; in addition, re-moval does not allow us to use the valuable informationprovided by these multiples. Multiples provide redun-dant as well as new information that is still useful toimprove our image. Using multiples can increase the il-lumination and lead to better vertical resolution in theimage (Schuster et al., 2003; Jiang et al., 2007; Muijset al., 2007a,b).

A method to use the information embedded in mul-tiples is proposed by Reiter et al. (1991), who use a ray-equation Kirchhoff depth migration to image with free-surface multiple reflections and primaries. In the final

2 S. Singh, R. Snieder, J. van der Neut, E. Slob, J. Thorbecke, K. Wapenaar

image, they achieve extended lateral coverage and an in-creased signal-to-noise ratio compared to imaging withprimaries. However, their method requires reliable sep-aration of free-surface multiples and primaries. In addi-tion, ray-based algorithms, such as that given in Reiteret al. (1991), might fail in complex geologic structuresdue to multipathing.

One-way wave equation migration of multiples isproposed by Guitton et al. (2002), Muijs et al. (2007a),and Malcolm et al. (2009) to overcome the shortfallsof ray-based methods. One-way wave equation migra-tion limits imaging of steep angle reflectors. Berkhoutand Verschuur (2006) modify the principle of SRME totransform multiples into primaries. Accordingly, thesenew primaries can be subjected to the same imagingcriteria as normal primaries. Ong et al. (2002) incorpo-rate RTM (two-way wave equation) into imaging mul-tiples by using the source and receiver wavefield as theprimary and multiple response, respectively. Althoughthe subsurface image produced by the modified RTM ofmultiples gives better illumination and spatial resolu-tion, there are imaging artifacts caused by high-ordermultiples correlating with the primaries, which placespurious reflectors incorrectly deeper (Ong et al., 2002).

We propose to use an inverse scattering approachfor imaging multiples. The physical basis for exact in-verse scattering is focusing and time reversal (Rose,2002b,a), which yield the Marchenko equation. Thisequation is an integral equation that determines thewavefield for a (virtual) source at any point x, i.e., theretrieved Green’s function, given the impulse responsefunction.

Broggini et al. (2012) extend the work of Rose(2002a) to geophysics for retrieving the Green’s func-tion from reflected waves at the surface. These Green’sfunctions include only primaries and internal multiples(Broggini et al., 2012, 2014). They use the Green’sfunction to image the subsurface (Marchenko imaging),whereby they minimize the artifacts produced by in-ternal multiples. Marchenko imaging uses the up- anddown-going Green’s function for imaging. We have in-corporated the free-surface multiples in the Green’sfunction retrieval algorithm (Singh et al., 2015); there-fore our retrieved Green’s functions also include free-surface multiples with the internal multiples and pri-maries. The major differences between our previouswork (Singh et al., 2015) and this work are: (1) weuse pressure-normalized wavefields compared to flux-normalized wavefields to obtain the Marchenko-typeequations, (2) we solve the Marchenko equations usingthe f1 focusing functions (more details on normalizedwavefields and focusing functions are given in the the-ory section), and (3) we show 2D imaging examples. Thenew focusing functions, f1 directly solve for the up- anddown-going Green’s functions; and these Green’s func-tions are used in our imaging scheme.

There is another approach to imaging using inverse

scattering proposed by Weglein et al. (2003), who usesa non-closed or series solution called the inverse scat-tering series. Unlike Weglein et al. (2003), our inversesolution to the wave equation is in the form of Fredholmintegral equations of the second kind (Marchenko-typeequations).

In this paper, we derive the retrieval of the Green’sfunction by solving Marchenko-type equations usingpressure-normalized wavefields. The reason for usingpressure-normalized fields are given in the theory sec-tion. We show numerical examples of imaging the sub-surface using the Green’s functions at different depths.Note that the Green’s function includes primaries, in-ternal multiples, and free-surface multiples, so we areusing all the scattered events in the imaging. We callimaging with these Green’s functions Marchenko imag-ing. The distinction with our work and the previouspapers Wapenaar et al. (2014a), Slob et al. (2014) andWapenaar et al. (2014b) is that we include free-surfacemultiples in the imaging.

2 THEORY

Retrieving the Green’s function in the presence of afree surface, using Marchenko-type equations, is derivedin multi-dimensions by Singh et al. (2015), but theirnumerical examples are one dimensional. The reflec-tion response R that Singh et al. (2015) uses to re-trieve these functions is flux-normalized, which facil-itates the derivation of the 3D Marchenko equations(Wapenaar et al., 2014a). Similarly, the retrieval of theGreen’s function without a free surface also uses flux-normalized wavefields, (Broggini et al., 2012; Wapenaaret al., 2013). However, the Green’s function retrievalis not restricted to flux-normalized fields and can bemodified to pressure-normalized fields. Wapenaar et al.(2014a) derive the retrieval of the Green’s function us-ing pressure-normalized fields in the absence of a freesurface.

In this paper, we demonstrate an alternative ap-proach by using pressure-normalized fields to retrievethe Green’s function in the presence of a free sur-face. Like previous papers on Green’s function retrieval,we accomplish this retrieval by solving Marchenko-typeequations (Wapenaar et al., 2014b; Slob et al., 2014;Wapenaar et al., 2014a). We show 2D numerical ex-amples of the retrieval and its application to imag-ing the subsurface. More details on flux- and pressure-normalized wavefields can be obtained from Wapenaarand Grimbergen (1996) and Wapenaar (1998).

Acoustic pressure p and vertical particle veloc-ity vz are related to any type of one-way normalizedfields (down-going p+ and up-going p−) in the space-frequency domain according to(

pvz

)=

(L1 L1

L2 −L2

)(p+

p−

), (1)

Incorporating free-surface multiples in Marchenko imaging 3

conversely, the p+ and p− are related to p and vz by(p+

p−

)=

1

2

(L−1

1 L−12

L−11 −L−1

2

)(pvz

). (2)

Here L1, L2 and their inverses are psuedo-differentialoperators (Wapenaar, 1998). For pressure normaliza-tion, L1 = I (Identity) , while for flux-normalization,equation 2 becomes(

p+

p−

)=

(Lt2 Lt1Lt2 −Lt1

)(pvz

), (3)

where superscript t denotes operator transposition (formore details see Wapenaar (1998) or Wapenaar et al.(2001)).

In a laterally invariant medium, equations 1 and 2becomes, in the wavenumber-frequency domain,(

pvz

)=

(L1 L1

L2 −L2

)(p+

p−

), (4)

and (p+

p−

)=

1

2

(L−1

1 L−12

L−11 −L−1

2

)(pvz

), (5)

respecitively. Here L1, L2 and their inverses are scalarfunctions (no operators). Equations 4 and 5 hold forany type of normalization. For pressure normaliza-tion we have L1 = 1 and L2 = kz/ωρ, wherekz =

√ω2/c2 − |k|2, with k = (kx, ky). For flux-

normalization we have L1 =√ωρ/2kz and L2 =√

kz/2ωρ.Pressure normalization is computationally faster to

implement compared to flux-normalization since the op-erator L1 is required to scale the acoustic pressure forflux-normalized fields but is not required in pressurenormalization (as L1 = 1). Hence, applying decomposi-tion on the reflection response, the computational speedfor pressure normalization is faster than for flux normal-ization. The increase in speed is not as significant withsources and receivers at the surface since the pressurefield is zero at this datum. Another advantage of usingpressure-normalized wavefields is that the relationshipbetween the two-way Green’s function and the pressure-normalized one-way Green’s functions is much simplerthan with flux normalization. The flux-normalized up-and down-going Green’s functions are related in thespace domain to the two-way Green’s function by (equa-tion 1)

G = L1(x3,i)L1(x3,0)(G

++G

−), (6)

where L1(x3,0) and L1(x3,i) are the operators at thedepth level x3 = x3,0 and x3 = x3,i, respectively, and

G+

and G−

are flux-normalized Green’s functions.Therefore, in order to obtain the two-way Green’s

function of the pressure recording for a source of volume-injection type using flux-normalized wavefields one mustapply L1 at x3 = x3,0 and x3 = x3,i to the sum of

G+

and G−

(Wapenaar et al., 2014a). However, to ob-tain this same two-way Green’s function using pressure-normalized wavefields, we simply add the up- and down-going retrieved Green’s functions.

Although the pressure-normalized wavefields aresimpler to obtain compared to flux-normalized wave-fields, their use in the derivation of the retrieval of theGreen’s function is more involved.

We begin the retrieval of the Green’s functionderivation with the frequency-domain one-way reci-procity theorems of the convolution and correlation type(Wapenaar et al., 2014a), which hold for lossless mediabetween ∂D0 (acquisition surface) and ∂Di (arbitrarydepth level):∫

∂D0

ρ−1(x)[(∂3p+A)P−B + (∂3p

−A)p+B ]dx0 =

−∫∂Di

ρ−1(x)[p+A(∂3p−B) + p−A(∂3p

+B ]dxi,

(7)

∫∂D0

ρ−1(x)[(∂3p+A)∗p+B + (∂3p

−A)∗p−B ]dx0 =

−∫∂Di

ρ−1(x)[(p+A)∗(∂3p+B) + (p−A)∗(∂3p

−B)]dxi.

(8)

The asterisk * denotes complex conjugation, and thesubscripts A ans B are two wave states. Equations 7 and8 are the reciprocity theorems for pressure-normalizedone-way wavefields. Equation 8 does not account forevanescent waves. The spatial coordinates are definedby their horizontal and depth components, for instancex0 = (xH,0, x3,0), where xH,0 are the horizontal coor-dinates at a depth x3,0. These one-way reciprocity the-orems hold for up- and down-going pressure-normalizedfields.

2.1 One-way wavefields

The reciprocity theorems are used to solve for theGreen’s function. We define the Green’s function asthe response to an impulsive point source at x′′0 justabove ∂D0 of volume injection rate. This Green’s func-tion obeys the scalar wave equation

ρ5 .

(1

ρ5G

)− 1

c2∂2G

∂t2= −ρδ(x− x′′0)

∂δ(t)

∂t. (9)

We include the time derivative on the right hand sidebecause we consider the source to be of volume in-jection rate. Using the Fourier convention, p(x, ω) =∫∞−∞ p(x, t) exp(−jωt)dt, in the frequency domain, equa-

tion 9 becomes

ρ5 .

(1

ρ5G

)+ω2

c2G = −jωρδ(x− x′′0). (10)

Since we are using one-way reciprocity theorems, equa-tions 7 and 8, we define our Green’s function (two-way)


as a sum of the up- and down-going pressure-normalizedone-way Green’s functions:

G(x,x′′0, ω) = G+,q(x,x′′0, ω) +G−,q(x,x′′0, ω), (11)

where x is the observation point. Defined this way, theone-way Green’s functions are decomposed at the obser-vation point x denoted by the first superscript + or −.We consider downwards to be positive, hence the super-script + represents down-going waves and − up-goingwaves. The second superscript (q) refers to the volume-rate injection source at x′′0. For instance, G−,q(x,x′′0, ω)is the pressure-normalized up-going Green’s function atx due to a volume injection source at x′′0 in the frequencydomain.

Similar to equation A11 in Wapenaar et al. (2014a),we define the vertical derivative of the up-going Green’sfunction at the acquisition surface ∂D0 as

∂3G−,q(x,x′′0, ω)|x3=x3,0 =

1

2jωρ(x0)R(x′′0,x0, ω).

However in our case, both ∂3G−,q and R include the

free-surface multiples. Considering the downward com-ponent of the source and the surface-reflected waves, wedefine

∂3G+,q(x′i,x

′′0, ω)|x3=x3,0 =− 1

2[jωρ(x0)δ(x− x′′0)

+ jωρ(x0)rR(x′′0 ,x0, ω)],

where r denotes the reflection coefficient of the freesurface. For the down-going field ∂3G

+,q, at and be-low ∂D0, we consider both the downward component of

the source −1

2jωρ(x0)δ(xH − x′′H) and the reflections

from the free surface −1

2jωρ(x0)rR(x′′0,x0, ω), similar

to the Marchenko derivation with flux-normalized fieldsfrom Singh et al. (2015). At ∂Di, the up- and down-going waves are G−,q and G+,q, respectively. We defineState A, shown in Figure 1, as the one-way pressure-normalized wavefields in the actual medium p±A at ∂D0

and ∂Di, as shown in Table 1.Similar to previous papers that derive Marchenko-

type equations (Wapenaar et al., 2013, 2014a; Slobet al., 2014; Singh et al., 2015), we also define focus-ing functions, see Figure 2, as State B. The focusingfunction f1 is a solution for the waves that focus at apoint just below the bottom of the truncated medium.The truncated medium is called the reference mediumas it is reflection free above and below ∂D0 and ∂Di,respectively, but is the same as the actual medium be-tween ∂D0 and ∂Di (see Figure 2). The f1 function isdefined as waves that focus at x′i at a defined depth level(∂Di) for incoming f+

1 and outgoing f−1 waves at theacquisition surface (∂D0) x0 (Figure 2).

The one-way wavefields for the f1 function at thedepth levels ∂D0 and ∂Di, which we define as State B,are shown in Figure 2 and Table 2. The one-way focus-ing function f+

1 (x,x′i, t) is shaped such that f1(x,x′i, t)

∂ D0

∂ Di

Actual inhomogeneous half-space

D

G+ , q

δ

Free surface

rG+ , q

G+

Actual inhomogeneous medium

G−

Figure 1. The one-way Green’s functions in the actual in-

homogeneous medium in the presence of a free surface at theacquisition surface ∂D0 and the arbitrary surface ∂Di. The

tree indicates the presence of the free surface.

∂ D0

∂ Di

DActual inhomogeneous medium

Reflection-free reference half-space

Homogeneous half-space

f 1+(xi , x i

' , t )

f 1+(x0 , x i

' , t ) f 1−

(x0 , x i' , t)

x i'

Figure 2. Focusing function f1 that focuses at x′i in the ref-

erence medium, where above ∂D0 is homogeneous and below∂Di is reflection-free.

focuses at x′i at t = 0. At the focusing point x′i of f1,

we define ∂3f1(x,x′i, t) as −1

2ρ(x′i)δ(xH −x′H)∂δ(t)/∂t,

a two-dimensional (2D) and 1D Dirac delta function inspace and time, respectively (see Figure 2 and Table 2).After the focusing point, f1(x,x′i, t) continues to divergeas a down-going field f+

1 (x,x′i, t) into the reflection-freereference half-space (Wapenaar et al., 2014a).


On ∂D0: ∂3p+A = ∂3G+,q(x0,x′′0 , ω) = −

1

2

(jωρ(x0)δ(xH − x′′H) + jωρ(x0)rR(x

′′0 ,x0, ω)

),

∂3p−A = ∂3G−,q(x0,x′′0 , ω) =

1

2jωρ(x0)R(x

′′0 ,x0, ω), ‘

On ∂Di: p+A = G+,q(xi,x′′0 , ω),

p−A = G−,q(xi,x′′0 , ω).

Table 1. The pressure-normalized one-way wavefields in the actual inhomogeneous medium in the presence of a free surface at

the depth level ∂D0 and ∂Di. p±A symbolizes one-way wavefields at arbitrary depth levels in the inhomogeneous medium, while

r is the reflection coefficient of the free surface.

On ∂D0: p+B = f+1 (x0,x′i, ω),

p−B = f−1 (x0,x′i, ω).

On ∂Di: ∂3p+B = ∂3f

+1 (xi,x

′i, ω) = −

1

2jωρ(x′i)δ(xH − x′H),

∂3p−B = ∂3f

−1 (xi,x

′i, ω) = 0.

Table 2. The one-way wavefields of the focusing function f1 at the acquisition surface ∂D0 and the level where f1 focuses,∂Di. p

±B symbolizes one-way wavefields in the frequency domain, at arbitrary depth levels in the reference medium, see Figure

2.

By substituting the one-way wavefields given in Table 1 (State A) and Table 2 (State B) into the convolutionreciprocity theorem, 7, we get the up-going Green’s function

G−,q(x′i,x′′0, ω) =

∫∂D0

[f+1 (x0,x

′i, ω)R(x′′0,x0, ω)− rf−1 (x0,x

′i, ω)R(x′′0,x0, ω)]dx0

− f−1 (x′′0,x′i, ω).

(12)

Likewise, substituting the one-way wavefields in Tables 1 and 2 into the correlation reciprocity theorem, 8, we getthe down-going Green’s function

G+,q(x′i,x′′0, ω) =−

∫∂D0

[f−1 (x0,x′i, ω)∗R(x′′0,x0, ω)− rf+

1 (x0,x′i, ω)∗R(x′′0,x0, ω)]dx0

+ f+1 (x′′0,x

′i, ω)∗.

(13)

Equations 12 and 13 are identical to the equations for G− and G+ of Singh et al. (2015), however our Green’sfunctions are pressure normalized. In addition, unlike Singh et al. (2015), there is no need to use equation 6 to obtainthe two-way Green’s function; one can simply use equation 11 to get G. The equations in 12 and 13 are the startingpoint for deriving the 3D Marchenko-type equations.

2.2 Marchenko Equations

Equations 12 and 13 are two equations for four unknowns (G+,q, G−,q, f+1 , and f−1 ). After an inverse Fourier transform,

we can separate these equations into two temporal parts: time less than the first arrival and time more than the firstarrival of the Green’s function at the virtual receiver location. We consider td(x

′i,x′′0) to be the first-arrival time of

the Green’s function. Hence, we can separate equations 12 and 13 for t > td and t < td. These temporal componentsgive rise to four equations and four unknowns.

An estimate of the first arrival time td(x′i,x′′0) is, for example, obtained by using finite-difference modeling of the

waveforms in a smooth velocity model that acts as a macro-model. The time before td gives rise to

f−1 (x′′0,x′i, t) =

∫∂D0

dx0

∫ t

−∞[f+

1 (x0,x′i, t′)R(x′′0,x0, t− t′)− rf−1 (x0,x

′i, t′)R(x′′0,x0, t− t′)]dt′, (14)


∂ D0

∂ Di

DActual inhomogeneous medium

Reflection-free reference half-space

Homogeneous half-space

T (x0 , x i' , t)

xi'

Figure 3. The transmission response T (x0,x′i, t) in the reference configuration.

On ∂D0: ∂3p+C = 0,

∂3p−C = ∂3T (x0,x′i, ω),

On ∂Di: p+C = 0,

p−C = δ(xH − x′H).

Table 3. The one-way wavefields in the reference medium at the acquisition surface ∂D0 and the level where f1 focuses, ∂Di.

p±C symbolizes one-way wavefields in the frequency domain, at arbitrary depth levels in the reference medium, see Figure 3.

The source location is just below ∂Di.

f+1 (x′′0,x

′i,−t) =

∫∂D0

dx0

∫ t

−∞[f−1 (x0,x

′i,−t′)R(x′′0,x0, t− t′)− rf+

1 (x0,x′i,−t′)R(x′′0,x0, t− t′)]dt′ (15)

because causality dictates that G± vanish for t < td(x′i,x′′0).

In the reference medium where the focusing functions exist, we can define up- and down-going waves with respectto transmission responses T (x0,x

′i, t) at arbitrary depth levels (State C), as shown in Figure 3. Hence, T (x0,x

′i, t)

is the transmission in the reference medium; which is the actual inhomogeneous medium, between ∂D0 and ∂Di,but homogeneous above and below ∂D0 and ∂Di. The up- and down-going waves in Figure 3 are defined in Table 3according to the reciprocity relations:

Substituting the one-way wavefields represented in Tables 2 and 3 into the one-way convolution reciprocitytheorem, 7, yields

δ(x′′H − x′H) =

∫∂D0

∂3T (x0,x′i, ω)

−1

2jωρ(x0)

f+1 (x0,x

′′i , ω)dx0, (16)

where we represent the source positions of the focusing function f+1 with double primes instead of single. For simplicity,

we define T (x0,x′i, ω) =

∂3T (x0,x′i, ω)

−1

2jωρ(x0)

; hence in the time domain (from equation 16), f+1 is the inverse of the


transmission response:

f+1 (x0,x

′i , t) = T inv(x0,x

′i, t). (17)

Analogous to Wapenaar et al. (2014b), Slob et al. (2014), and Singh et al. (2015), we consider the assumptionfor the pressure-normalized version of f+

1 to be

f+1 (x0,x

′i , t) = T invd (x0,x

′i, t) +M(x0,x

′i, t), (18)

where T invd is the inverse of the direct arrival of the transmission response, and M is the coda following T invd . We canapproximate T invd as the time-reversed direct arrival of the pressure-normalized Green’s function (hence the need fora smooth velocity model as previously mentioned).

Substituting assumption 18 into the time-domain representation of equations 14 and 15 yields the followingMarchenko equations for t < td(x

′i,x′′0):

f−1 (x′′0,x′i, t) =

∫∂D0

dx0

∫ −tεd(x′i,x0)

−∞T invd (x0,x

′i, t′)R(x′′0,x0, t− t′)dt′ +∫

∂D0

dx0

∫ t

−tεd(x′

i,x0)

M(x0,x′i, t′)R(x0,x

′′0, t− t′)dt′ −

r

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

f−1 (x0,x′i, t′)R(x0,x

′′0, t− t′)]dt′,

(19)

M(x′′0,x′i,−t) =

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

f−1 (x0,x′i,−t′)R(x′′0,x0, t− t′)dt′ −

r

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

M(x0,x′i, t′)R(x′′0,x0, t+ t′)]dt′ −

r

∫∂D0

dx0


−∞T invd (x0,x

′i, t′)R(x′′0,x0, t+ t′)]dt′,

(20)

with tεd(x′i,x′′0)= td(x

′i,x′′0)− ε, where ε is a small positive constant to include the direct arrival in the integrals. We

choose to solve the Marchenko equations (19 and 20) iteratively as follows:

f−1,k(x′′0,x′i, t) =−

∫∂D0

dx0


−∞T invd (x0,x

′i, t′)R(x′′0,x0, t− t′)dt′ +∫

∂D0

dx0

∫ t

−tεd(x′

i,x0)

Mk−1(x0,x′i, t′)R(x′′0,x0, t− t′)dt′ −

r

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

f−1,k−1(x0,x′i, t′)R(x′′0,x0, t− t′)]dt′,

(21)

Mk(x′′0,x′i,−t) =

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

f−1,k(x0,x′i,−t′)R(x′′0,x0, t− t′)dt′ −

r

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

Mk−1(x0,x′i, t′)R(x′′0,x0, t+ t′)]dt′ −

r

∫∂D0

dx0


−∞T invd (x0,x

′i, t′)R(x′′0,x0, t+ t′)]dt′.

(22)

Note that we are not limited to solving the Marchenko equations iteratively; one can use a preferred integral solversuch as conjugate gradients or least-squares inversion. The corresponding focusing function f+

1 for each iterationreads (from equation 17)

f+1,k(x0,x

′i, t) = T invd (x0,x

′i, t) +Mk−1(x0,x

′i, t). (23)


2.3 Marchenko iterative scheme

We initialize the Marchenko iterative scheme by obtaining the direct arrival of the Green’s function. The time-reversedversion of this direct arrival can be used as an approximation for T invd which takes into account travel times andgeometric spreading but ignores transmission losses at the interfaces (Wapenaar et al., 2014a, 1989).

With this initialization, the iterative scheme for k = 0 is as follows:

f−1,0(x′′0,x′i,−t) =−

∫∂D0

dx0


−∞T invd (x0,x

′i, t′)R(x′′0,x0, t− t

′)dt

′, (24)

M0(x′′0,x′i,−t) =

∫∂D0

dx0

∫ t

−tεd(x′

i,x0)

f−1,0(x0,x′i,−t′)R(x′′0,x0, t− t′)dt′ −

r

∫∂D0

dx0


−∞T invd (x0,x

′i, t′)R(x′′0,x0, t+ t′)]dt′.

(25)

Now the iterative scheme described in equations 21-23 can be initiated with equations 24 and 25 to solve for f+1 and

f−1 . These focusing functions can then be substituted in equations 11, 12 and 13 to obtain the retrieved two-waypressure-normalized Green’s function, and the up- and down-going one-way pressure-normalized Green’s function,respectively.

2.4 Marchenko imaging

Broggini et al. (2012, 2014); Wapenaar et al. (2011); Slob et al. (2014); Wapenaar et al. (2014b); Singh et al. (2015)have all used the retrieved one-way Green’s functions to produce an image. Marchenko imaging is built on the conceptof obtaining the reflection response from the up- and down-going wavefields at an arbitrary depth level. The use ofup- and down-going wavefield for imaging is not a new principle. Claerbout (1971), Wapenaar et al. (2000) andAmundsen (2001) have shown that one can get the reflection response below an arbitrary depth level once the up-and down-going wavefields are available.

The governing equation for imaging with such one-way wavefields is (Wapenaar et al., 2008)

G−,q(x′i,x′′0, t) =

∫∂Di

dxi

∫ ∞−∞

G+,q(xi,x′′0, t− t′)R0(x′i,xi, t

′)dt′, (26)

where ∂Di is an arbitrary depth level and R0 is the reflection response of the medium below ∂Di. Note that equation26 holds for out- and in-going wavefields normal to the surface ∂Di. However, the retrieved Green’s functions (currentmethods) are strictly up- and down-going wavefields at arbitrary depth levels, which corresponds to a flat surface∂Di. The reflection response R0, in equation 26, is the response as if everything above ∂Di is transparent. Therefore,R0 is a virtual reflection response as if there were receivers and sources at ∂Di, in the absence of a free-surface at∂Di. Significantly, the response R0 is blind to the overburden above ∂Di. Wapenaar et al. (2014b) have shown theretrieval of this virtual reflection below a complex overburden. In this paper, any variable with a subscript 0 (e.g.,R0) indicates that no free-surface is present.

We choose to solve for R0 in equation 26 by multidimensional deconvolution (MDD) (van der Neut et al.,2011). Details of solving equation 26 using retrieved Green’s functions are given in Wapenaar et al. (2014b). Thesignificant difference between our work and the previous Marchenko imaging papers is that our Green’s functionsincludes information of the actual medium with the free-surface and include all (free-surface and internal) multiples.This corresponds to using the free-surface multiples in the imaging. Once we obtain R0 at each image point, oursubsurface image is the contribution of R0 at zero offset and zero time, i.e., R0(xi,xi, 0).

3 NUMERICAL EXAMPLES

Our numerical model has a constant velocity of 2.5 km/s with variable density, as shown in Figure 4, however,constant velocity is not a restriction of the our algorithm. The density is a 2D inhomogeneous subsurface model witha syncline structure. The horizontal range of the model is −3000 m to 3000 m. Our goal is to show: (1) the retrievalof the Green’s function G(x′i,x

′′0, t) for a virtual receiver at xi = (0, 1100) m and the corresponding variable source

locations at x′′0 and (2) the subsurface image below the syncline structure. To obtain the Green’s function, we needthe pressure-normalized reflection response R(x′′0,x0, ω) and a macro-model (no small-scale details of the model arenecessary). The reflection response is computed by finite differences Thorbecke and Draganov (2011) with


Figure 5. The reflection response corresponding to a homo-

geneous velocity of 2.5 km/s and the inhomogeneous density

model in Figure 4.

vertical-force sources and particle-velocity receiverrecordings, both at the surface. The receiver spacing is10 m and the source is a Ricker wavelet with a centralfrequency of 20 Hz. We use this finite-difference responseand equation 2 to get the pressure-normalized reflectionresponse R(x′′0,x0, ω) (see Figure 5 for an example of asingle shot at x′′0 = (0, 0) with the direct arrivals fromsource to receivers removed), which we deconvolve withthe source wavelet. The macro-model is a smooth ver-sion of the velocity model, in this case we just need theconstant velocity model. No information of the densityis required. In the situation where the velocity model isvarying, the macro-model will be a smooth version ofthe velocity model, since we only need the macro-modelto compute the travel times of the direct arrival.

We use the macro-model to obtain the first-arrivalfrom the virtual source at xi = (0, 1100) m to the surface(by finite differences). This first arrival is time-reversedto initialize the iterative scheme T invd as well as to guideus in choosing the time windows for equations 12-15.Figure 6 shows T invd , which is equivalent to f+

1,0.

3.1 Focusing functions

We build the focusing functions f+1,k and f−1,k using the

iterative scheme in equations 21-23. Figure 7 shows thefunctions f+

1,k and f−1,k for iteration index k = 0, 1, 5.Note that these one-way focusing functions reside in thetime-window −td < t < td.

The integrals that we use to solve for the focus-ing function, equations 14 and 15, have spatial limitsbetween −∞ and ∞, which means we require infiniteaperture. In our implementation we truncate the spa-

Figure 6. The time-reversed first arrival for a virtual sourceat xi = (0, 1100) m and receivers at the surface. This event

is used to initialize the Marchenko iterative scheme (T invd ).

tial integral. This truncation requires tapering at theedges of the reflection response, which corresponds tothe reduced amplitudes of the focusing functions at thefar offsets.

From iteration index k = 0 to k = 1, new events aregenerated in the focusing function. Even the focusingfunction f−1,k at k = 0 already has the main features thatare obtained after five iterations. In iteration k = 1 tok = 5, the focusing functions look kinematically similar.Higher-order iterations generally correct the amplitudeerrors in the focusing functions.

3.2 Green’s function retrieval

By substituting the focusing functions in equations12 and 13 we obtain the one-way pressure-normalizedGreen’s functions, as shown in Figure 8. These up- anddown-going Green’s functions are the response for a re-ceiver at x′i = (0, 1100) and variable source locationsx′′0. To see the internal multiples and the free-surfacemultiples in Figure 8, we display the Green’s functionswith a time-dependent gain of exp(1.5 ∗ t(s)).

The two-way Green’s function is given as the sum-mation of the up- and down-going Green’s function. Acomparison of this retrieved two-way Green’s functionwith the modeled Green’s function (modeled with theexact small-scale variations in the density) is shown inFigure 9. For display, we apply a gain of exp(1.5 ∗ t(s))to the Green’s functions in Figure 9 to better see theinternal multiples and free-surface multiples at depth.The retrieved and modeled Green’s function match al-most perfectly, as shown in Figure 9. As expected, thefar-offsets do not provide a good match because we are


Figure 4. The density model ranging from densities 1 to 3.5 g/cm3 as shown in the color bar.

Figure 9. The retrieved two-way Green’s function [in red]superimposed on the modeled Green’s function (computed by

finite differences with the small-scale details in the densitymodel included)[in blue].

truncating the spatial integrals in the Marchenko equa-tions.

3.3 Comparison of Green’s functions with andwithout the free surface

In previous formulations of Green’s function retrieval,mentioned in the introduction of this paper, they re-

quired the reflection response without free-surface mul-tiples. This means that an additional processing stepto remove the surface reflections is required before im-plementing their Green’s function retrieval algorithm.For such an implementation, the Green’s function doesnot include free-surface multiples, hence the imagingprocedure does not take them into account. Figure 10shows the up- and down-going one-way Green’s functionwithout free surface multiples. These Green’s functionsin Figure10 are retrieved using the Marchenko methodthat does not take into account free-surface multiples(Broggini et al., 2014; Wapenaar et al., 2014a), hence,we removed the free-surface multiples from the reflec-tion response before retrieving these Green’s functions.

The Green’s functions in Figure 10 are the responsefor a virtual receiver position x′i = (0, 1100) m and vari-

able source positions x′′0 . For display we also applied a

time-dependent gain of exp(1.5 ∗ t(s)) to the Green’sfunctions. As expected the Green’s functions with thefree-surface, G+ and G−, have greater waveform com-plexity and higher amplitudes than the Green’s func-tions in the absence of the free surface, G+

0 and G−0 . Thisis obvious for times larger than 1.5 seconds for both Fig-ures 8 and 10. For the shallower reflectors it seems thatthe one-way Green’s functions G±0 may have strongerevents than G±. This is because the free-surface multi-ples interfere destructively with the primaries and inter-nal multiples. Figure 11 best illustrates the interferencejust below the first arriving event in G+ (approximately0.1-0.2 s below the first arrival). However, for time largerthan 1 second, the free-surface multiples (in blue) dom-inate in Figure 11. Most likely, G should be better forimaging than G0 because of the additional information


(a) f+1,0 (b) f−1,0

(c) f+1,1 (d) f−1,1

(e) f+1,5 (f) f−1,5

Figure 7. One-way focusing functions f+1,k and f−1,k that focus at x′i for iteration index k = 1, 2, 5.


(a) G+ (b) G−

Figure 8. One-way pressure-normalized Green’s functions G+ (down-going) and G− (up-going) for a virtual receiver positionx′i = (0, 1100) m and a range of source positions x′′0 .These Green’s functions include free-surface multiples

(a) G+0 (b) G−0

Figure 10. One-way pressure-normalized Green’s functions G+0 (down-going) and G−0 (up-going) for a virtual receiver position

x′i = (0, 1100) m and a range of source positions x′′0 . These Green’s functions do not include free-surface multiples.


Figure 11. Green’s function G0 without the free surface(red) and the Green’s function G with the free surface (blue)

for a virtual receiver at x′i = 90, 1100) m and a range of

source positions x′′0 .

gained from the free-surface multiples in G. In addition,we avoid SRME on the reflection response by using theMarchenko equations for Green’s function retrieval thatincludes free-surface multiples (our work). It remains tobe investigated to what extent these retrieved multipleswill improve the image quality.

3.4 Marchenko imaging -Target oriented

Target-oriented Marchenko imaging entails retrievingthe up- and down-going Green’s functions in the tar-get area and using them to construct the image. Figure12 shows the Marchenko imaging of the model in Figure4. To compute this image we retrieve the up- and down-going Green’s function G±,q(x′i,x

′′0, t) at the virtual re-

ceiver locations x′i = (xH, x3,i) ranging from xH,i = −2to 2 km and x3,i = 1 to 1.6 km. We sampled xH,i andx3,i every 0.040 km and 0.05 km, respectively, to re-trieve the Green’s function. These functions are used tocompute R0(xi,x

′i, t) as explained in the theory section.

The contribution to the image is R0(xi,xi, 0), which isR0 at zero-offset and zero time for the range of xi.

The target-oriented Marchenko image, Figure 12,is free of artifacts caused by the internal multiples andfree-surface multiples in the overburden. This is be-cause Marchenko imaging correctly migrates the pri-maries and all multiples to the correct reflector loca-tion. If the free-surface multiples were not handled cor-rectly by Marchenko imaging then the associated mul-tiples caused by the syncline and the layers within thesyncline would be present in our image. This is be-

Figure 12. Target-oriented Marchenko imaging of the model

in Figure 4 below the syncline structure. The image isR0(xi,xi, 0) for xi ranging from xH,i = −2 to 2 km and

x3,i = 1 to 1.6 km.

cause Marchenko imaging correctly migrates the pri-maries and all multiples to the correct reflector location.

4 CONCLUSION

We have shown that we can retrieve the Green’s functionat any location in the model without any knowledgeof the small-scale variations of the subsurface once wehave sufficient aperture coverage on the surface over thevirtual source location. To retrieve the Green’s function,we require the reflection response at the surface anda macro-model of the subsurface overburden (at leastbetween the surface and the virtual source depth level).

The reflection response is required to be well sam-pled at the surface. The more densely sampled our re-flection response, the more accurately we can solve theMarchenko equations. The accuracy of our Green’s func-tion retrieval is also dependent on the kinematic ac-curacy of the macro-model. Another limitation of theGreen’s function retrieval scheme is that we assumeall waves can be decomposed into up- and down-goingevents; hence, horizontally propagating waves are notincluded in our current method. This retrieval is cur-rently an acoustic algorithm; however, work has alreadybegun on making the procedure elastodynamic. In theelastodynamic formulation, the causality assumptionsare more complex (da Costa et al., 2014; Wapenaaret al., 2014b). Our next step is to overcome and/or min-imize these limitations.

On the positive side, our Green’s functions includeprimaries and all multiples (internal and free-surface).Once we know the Green’s function at the surface andthe virtual receiver locations, we should be able to infer


what is inside the medium (volume). Hence, the Green’sfunction is our redatuming operator. We can form theimage in two ways (1) downward continuation to a givenreference level, and do conventional imaging, (2) targetoriented imaging. In this paper we construct a targetoriented image by deconvolution of the up- and down-going Green’s function, evaluated at zero offset and zerotime.

In the numerical examples, we observe no signifi-cant artifacts in the Marchenko image, due to misplacedmultiples, even though the reflection response includesmultiples (no preprocessing is done to remove the multi-ples). How the multiples improve the image is yet to beinvestigated; however, for certain, Marchenko imagingnaturally migrates the multiples (and primaries) to thecorrect reflector location.

Significantly, the inputs for Marchenko imaging andfor the current state-of-the-art imaging techniques arethe same: the reflection response and a macro-model.However, in Marchenko imaging, we accurately handlenot only the primaries but also the multiples.

5 ACKNOWLEDGMENTS

This work was funded by the sponsor companies ofthe Consortium Project on Seismic Inverse Methods forComplex Structures. We are grateful to Diane Wittersfor her help in preparing this manuscript. The numericexamples in this paper are generated with the Madagas-car open-source software package freely available fromhttp://www.ahay.org. We would also like to thank Es-teban Diaz (CWP) and Vladmir Li (CWP) for fruitfuldiscussions.

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