An Outlook on the Future of Seismic Imaging, Part I_Berkout

Geophysical Prospecting, 2014, 62, 911930 doi: 10.1111/1365-2478.12161

Review Paper: An outlook on the future of seismic imaging, Part I:forward and reverse modelling

A.J. (Guus) BerkhoutDelft University of Technology, CiTG, Stevinweg 1, 2628 CN Delft, The Netherlands

Received November 2013, revision accepted April 2014

ABSTRACTThe next generation of seismic imaging algorithms will use full wavefield migration,which regards multiple scattering as indispensable information. These algorithmswill also include autonomous velocity-updating in the migration process, called jointmigration inversion. Full wavefield migration and joint migration inversion addressindustrial requirements to improve the images of highly complex reservoirs as well asthe industrial ambition to produce these images more automatically (automation inseismic processing).

In these vision papers on seismic imaging, full wavefield migration and joint mi-gration inversion are formulated in terms of a closed-loop, estimation algorithm thatcan be physically explained by an iterative double-focusing process (full wavefieldCommon Focus Point technology). A critical module in this formulation is forwardmodelling, allowing feedback from the migrated output to the unmigrated input(closing the loop). For this purpose, a full wavefield modelling module has beendeveloped, which uses an operator description of complex geology. Full wavefieldmodelling is pre-eminently suited to function in the feedback path of a closed-loopmigration algorithm.

The Future of Seismic Imaging is presented as a coherent trilogy of papers thatpropose the migration framework of the future. In Part I, the theory of full wavefieldmodelling is explained, showing the fundamental distinction with the finite-differenceapproach. Full wavefield modelling allows the computation of complex shot recordswithout the specification of velocity and density models. Instead, an operator descrip-tion of the subsurface is used. The capability of full wavefield modelling is illustratedwith examples. Finally, the theory of full wavefield modelling is extended to full wave-field reverse modelling (FWMod1), which allows accurate estimation of (blended)source properties from (blended) shot records.

INTRODUCTION

In standard migration practice, we have little informationabout the inconsistency between output and input: migrationis implemented as an open-loop process. In particular, if wewant to use the information in multiple scattering, a simpleopen-loop approach is no longer acceptable. By taking theopen-loop seismic image as the input in a forward modellingalgorithm, we are able to close the loop in migration, so thatwe generate numerically simulated measurements in the feed-

E-mail: [email protected]

back path. Next, iterative minimization of the difference be-tween simulated and real measurements allows us to optimizethe seismic image (see the basic diagram in Fig. 1). Multiplesare an integral part of this process. As I will explain later,multiples are a blended wavefield phenomenon, the blendedsources being natural. Full wavefield migration (FWM), there-fore, is also the obvious solution to migrate manmade blendedshot records.

In closed-loop FWM, forward modelling is a critical pro-cess. Errors in the modelling result must be avoided becausethey are transferred to errors in the residue (i.e., the difference

911C 2014 European Association of Geoscientists & Engineers

912 A.J. (Guus) Berkhout

Figure 1 Migration is formulated as a closed-loop process, so that output and input are connected via a feedback loop with a forward modellingmodule that transforms subsurface reflectivities into simulated measurements. The residue steers the next iteration.

Figure 2 Wavefields in the same subsurface, but with two fundamentally different descriptions. Although the operator description is differentfrom the property description, FWMod modelling and FinDif modelling generate the same wavefields. In closed-loop migration, the FinDif-related property description must be replaced by the FWMod-related operator description.

between measured and simulated data), and therefore, er-rors may be introduced in the migration output. Today,finite-difference modelling yields excellent results, but thesubsurface must be described in terms of detailed elastic prop-erties (Moczo et al. 2007). Such a description is outside themigration framework and, therefore, should not be used atthis stage of seismic processing. In this seismic trilogy on thefuture of migration, the usual property description of the sub-surface in terms of a detailed velocity and density model isreplaced by an alternative description that makes use of lo-cal propagation and scattering operators. The consequenceis that property-driven modelling (in which the algorithm isbased on a differential equation for seismic wavefields) can

be replaced by a new type of operator-driven modelling (inwhich the algorithm is based on an integral equation for seis-mic wavefields). Figure 2 illustrates this with an example: thesubsurface is described in terms of local operators (a) and interms of properties (b).

Figure 2(a) shows the full wavefield forward modelling(FWMod) result (using the operator description as input),Fig. 2(b) shows the FinDif output (using the property descrip-tion as input), and Fig. 2(c) shows the difference. As expected,both modelling methods generate the same response (differ-ences are the subject of current research), but the detailedvelocitydensity description of the subsurface is not suitablefor closed-loop migration. To move to the next generation

C 2014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 62, 911930

An outlook on the future of seismic imaging, Part I 913

of migration algorithms, it is critical that we abandon thetraditional thinking about the role of velocity models. I willshow that accurate depth migration can be achieved withoutthe specification of accurate velocity information. In fact, themore complex the geology for instance, deep reservoirs be-low a strongly inhomogeneous and anisotropic overburden the more doubtful it is that our traditional concepts can beused to implement migration velocity estimation outside themigration loop.

In this seismic imaging trilogy, the author gives a re-view of his work on the future of migration. In the presentpaper (Part I of this migration trilogy), I derive the theoryof FWMod by extending the Huygens principle. The theo-retical results are illustrated with examples. Next, the the-ory of FWMod is extended to full wavefield reverse mod-elling (FWMod1), allowing iterative estimation of unknownsource properties. In the second paper (Part II), I will showhow primary wavefield migration (PWM) can be extended toFWM, including both surface multiples and internal multi-ples. Finally, in the third paper (Part III), I will explain howjoint migration inversion (JMI) can create accurate depthimages without specifying the migration velocity model. InJMI, anisotropic velocities and inelastic absorption coeffi-cients are not specified by the user, but they are seen asimage attributes and are therefore part of the migrationoutput.

FORWARD MODEL FOR PRIMARYWAVEFIELD MIGRATION

In PWM, first-order reflections only are addressed (pri-maries), and events with multiple bounces are considered asshot-generated noise, so that the forward model is linear interms of reflectivity. Using the operator notation in the tem-poral frequency domain (Berkhout 1982), the discrete linearwavefield model for PWM can be summarized by the follow-ing two monochromatic expressions for PP-reflections (seeFig. 3):a. for the downgoing incident wavefields (m = 1, 2, . . . ,M):P+j (zm; z0) = W+ (zm, z0) S+j (z0) ; (1a)b. for the upgoing reflected wavefields (m = 0, 1, . . . , M 1):

Pj (zm; z0) =M

n=m+1W (zm, zn)R

(zn, zn) P+j (zn; z0). (1b)

Vector-matrix equations (1a) and (1b) formulate the sim-plest version of the discrete scattering integral (first-orderWRW-model). In equations (1a) and (1b), the elements of

Figure 3 In primary wavefield migration, there is no gridpoint in-teraction, so that i) scattering is upward only (no multiples) and ii)propagating wavefields are not influenced by the scattering process(no transmission effects). Because of these simplifications, the scat-tered wavefields are discontinuous in primary wavefield migration,leading to artefacts in the seismic image.

vector S+j (z0) represent the (blended) source array with iden-tification label j at the surface z0, matrix W+ (zm, z0) rep-resents the downward propagation operator between depthlevels z0 and zm, matrix R (zn, zn) represents the angle-dependent reflection operator at zn (operator R (zn, zn) trans-forms downgoing wavefield P+j (zn; z0) into upgoing wavefieldR (zn, zn) P+j (zn; z0) by an elastic reflection process), and ma-trix W (zm, zn) represents the upward propagation operatorbetween depth levels zn and zm (n>m).

Looking at the wavefields in a single gridpoint, the ar-chitecture of the above operator notation can be further ex-plained: scalar P+kj (zn; z0) represents the downgoing wavefieldincident to gridpoint k at depth level zn (where zn = zn ) thatwas generated by source array j at depth level z0, and scalarPkj (zn; z0) represents the upgoing wavefield incident to grid-point k at depth level zn (where zn = z+n ) that was generatedby source array j at depth level z0. Note that source vectorS+j (z0) includes the influence of the stress-free surface at z0.Of course, expressions (1a) and (1b) may be extended by in-cluding man-made sources at any depth level (zs).

Finally, in forward model equations (1a) and (1b), thepropagation matrices W can be represented by a recursiveexpression (Berkhout 1982):

W (z0, zm) =m

n=1W

(zn1, zn

)and (1c)

W+ (zm, z0) =1

n=mW+

(zn, zn1

), (1d)

where the columns of W(zn1, zn

)and W+

(zn, zn1

)are de-

termined by the local velocities.



Including transmission operators

A fundamental shortcoming of the PWM model is that theprimary wavefields are discontinuous at the reflectors: the to-tal wavefield just above a reflecting depth level is given bythe composition P+j (zn; z0) + R (zn, zn) P+j (zn; z0), which isthe superposition of the incoming and reflected wavefield,whereas the total wavefield below this reflecting depth levelis assumed to be the incident wavefield only, P+j (zn; z0). Thisviolation of wavefield continuity can be repaired by multi-plying the transmitted wavefield by the transmission operator[I + T+ (zn, zn)

], where T+ (zn, zn) = R (zn, zn). Similarly,

the total wavefield just below a reflecting depth level is givenby Pj (zn; z0) + R (zn, zn) Pj (zn; z0), which is the superposi-tion of the incoming and reflected wavefield, whereas the totalwavefield above this reflecting depth level is assumed to be theincident wavefield only, Pj (zn; z0). Hence, for an upward-travelling wavefield, the continuity property is guaranteedby introducing the transmission operator

[I + T (zn, zn)

],

where T (zn, zn) = R (zn, zn). Note that R (zn, zn) repre-sents the reflection operator at zn that transforms the up-going wavefield Pj (zn; z0) into the downgoing wavefieldat zn: R (zn, zn) Pj (zn; z0). We will see that R (zn, zn) =R (zn, zn) if we neglect wave conversion, meaning that weassume a small shear contrast at zn.

If we include these transmission operators in the defi-nition of the recursive propagation operators, then the pri-mary forward modelling equations (1a) and (1b) can be up-dated to the model of PP-reflections that do obey wavefieldcontinuity:a. for the downgoing incident wavefields (m = 1, 2, . . . ,M):P+j (zm; z0) = W+ (zm, z0) S+j (z0) ; (2a)b. for the upgoing reflected wavefields (m = 0, 1, . . . , M 1):

Pj (zm; z0) =M

n=m+1W(zm, zn)R

(zn, zn) P+j (zn; z0), (2b)

where the expressions for the hybrid propagation operatorsare given by

W (z0, zm) = W (z0, z1)m1n=1

[I + T (zn, zn)

]W(zn, zn+1) (3a)

andW+ (zm, z0) = W+

(zm, zm1

) 1n=m1

[I + T+ (zn, zn)

]

W+(zn, zn1

). (3b)

Compare expressions (3a) and (3b) with expressions (1c)and (1d): hybrid operators W represent a mixture of propa-gation (W) and transmission (T) effects. In the following,we will see that the inclusion of transmission effects in thepropagation operators (from W to W) is not the appropri-ate approach to take. We will aim at a strict separation ofpropagation and scattering.

Including surface-related multiples

If we want to include the strong surface-related multiples inthe primary forward model, then equation (2a) must be ex-tended to (see also Fig. 4):

P+j (zm; z0) = W+ (zm, z0) Q+j (z0; z0) , (4a)with

Q+j (z0; z0) = S+j (z0) + R (z0, z0) Pj (z0; z0) , (4b)where matrix R (z0, z0) is the reflectivity operator at the sur-face that transforms upgoing wavefield Pj (z0; z0) into down-going wavefield R (z0, z0) Pj (z0; z0).

Using equations (4a) and (4b) , the following family ofextended forward models for PWM can be formulated:a. for the total response (primaries + surface multiples):Pj (z0, z0) = X0 (z0, z0) Q+j (z0; z0) ; (5a)b. for the primaries only:

P 0, j (z0; z0) = X0 (z0, z0) S+j (z0) ; (5b)c. for the surface multiples only:

M0, j (z0; z0) = X0 (z0, z0) R (z0, z0) Pj (z0; z0) . (5c)In expressions (5a)(5c), transfer operator X0 has been

approximated by

X0 (z0, z0) =M

m=1W (z0, zm)R

(zm, zm)W+ (zm, z0), (6a)

which is the WRW-model without transmission effects andwithout internal multiples. If we include transmission effects,equation (6a) must be updated to

X0 (z0, z0) =M

m=1W (z0, zm)R

(zm, zm)W+ (zm, z0). (6b)

Comparing equations (5b) and (5c), we can easily verifythat surface multiples can be migrated by an extended ver-sion of the PWM algorithm (Berkhout and Verschuur 1994;Verschuur and Berkhout 2011; Lu et al. 2011). The recipe is



Figure 4 The feedback model, showing the up- and downgoing wavefields ( Pj and Q+j ) at the surface z0. In primary wavefield migration, itis assumed that the surface-related multiples have been removed from the input, so that Q+j = S+j . Additionally, for linearization purposes,internal multiples are neglected.

simple: replace downgoing source wavefield S+j (z0) by down-going reflected wavefield R (z0, z0) Pj (z0; z0), and replaceprimary response P0, j (z0; z0) by surface multiple responseM0, j (z0; z0) .

Importantly, in surface multiple migration, the sourcewavelet does not need to be known: both input and output aregiven by the measured data (compare equations (5b) and (5c)).In Part II, it will be shown that FWM can be used to migrateprimaries, surface multiples as well as internal multiples. Thisall multiple option in migration demonstrates the problemswith traditional approaches in making large investments toremoving multiples. In fact, as will be explained in Part III,the best primarymultiple separation is achieved by the fullwavefield migration algorithm.

FORWARD MODEL FOR FULL WAVEFIELDMIGRATION

Using the operator formulation of seismic wave theory, theforward model for PWM can be easily extended to the for-ward model for FWM, leading to the following vector-matrixexpressions for the total PP-response (see Fig. 5):

a. for the downgoing wavefields (m = 1, 2, . . . , M):

Figure 5 In full wavefield migration, reflective gridpoints scatter bothupward and downward, causing full gridpoint interaction. This meansthat the propagating wavefields are modified by the two-way scatter-ing process at each gridpoint. In full wavefield migration, all wave-fields are continuous.

P+j (zm; z0) = W+ (zm, z0) S+j (z0)

+m1n=0

W+ (zm, zn)R (zn, zn) Pj (zn; z0); (7a)

b. for the upgoing wavefields (m = 0, 1, . . . , M 1):

Pj (zm; z0) = W (zm, zM) Pj (zM; z0)

+M

n=m+1W (zm, zn)R

(zn, zn) P+j (zn; z0), (7b)



where the hybrid propagation operatorsW include the trans-mission effects; see recursive expressions (3a) and (3b). Bycomparing full wavefield model equations (7a) and (7b) withprimary model equations (2a) and (2b), we see the additionof a second term in each equation.

The second term in expression (7a) has large conse-quences: it introduces at each depth level an extra reflectionprocess (quantified by reflection operator R), which gener-ates the surface (n = 0) and the internal multiples (n > 0). Thefirst term in expression (7b) includes the response of the lowerhalf-space z > zM. If in (7b) we choosem = 0, then we obtainthe upgoing wavefield in the reflection measurements at z0. Ifin (7a) we choose m = M, then we obtain the downgoingwavefield in the transmission measurements at zM.

In the following, we will not make use of full wavefieldequations (7a) and (7b). Instead, we will take the transmissionoperators [I + T] outside the hybrid propagation opera-tors W, and we assign to T+ and T the role of forwardscattering operators (compare these with backward scatter-ing operators R and R). By doing this, we reintroduce inour full wavefield model the scatter-free propagation opera-tors W (expressions (1c) and (1d)), and we use both T+,T and R, R as scattering operators. This leads to the pre-ferred scattering formulation that will be the basis of all fullwavefield algorithms in Part II and Part III (Berkhout 2012):

a. for the downgoing wavefields (m = 1, 2, . . . , M):

P+j(zm; z0

) = W+ (zm, z0) S+j (z0)+

m1n=0

W+(zm, z

+n

)S+j

(z+n ; z0

); (8a)

b. for the upgoing wavefields (m = 0, 1, . . . , M 1):

Pj(z+m; z0

) = W (z+m, zM) Pj (zM; z0)+

Mn=m+1

W(z+m, z

n

)Sj

(zn ; z0

), (8b)

where vectors Sj represent the two-way secondary sourcesin the inhomogeneous gridpoints at depth level zn (gener-ating the physical or A-scattering), and matrices W definethe scatter-free wavefield propagation operators between twodepth levels (see expressions (1c) and (1d)). Equations (8a) and(8b) formulate the full wavefield version of theWRW-model,describing the two-way scattering process of one-way wave-fields. In (8a) and (8b), the secondary source vectors Sj are

given by a weighted superposition of the up- and downgoingincident wavefields (see also Appendix A):

S j(zn ; z0

) = R (zn , zn ) P+j (zn ; z0)+ T (zn , z+n ) P j (z+n ; z0) (9a)

and

S+j(z+n ; z0

) = R (z+n , z+n ) P j (z+n ; z0)+ T+ (z+n , zn ) P +j (zn ; z0) , (9b)

where S+j = Sj and T = R, T+ = R if we can ne-glect wave conversion at zn. Equations (9a) and (9b) representthe combined reflection (first term) and transmission process(second term) at depth level zn (see Fig. 6).

We will see that the proposed scattering formulation ofthe FWMod equations (8a) and (8b) (in which propagationoperators W are free of scattering, and scattering operators(T, R) are free of propagation) is critical to migrate accu-rately the seismic response from complex geology. Later inthis paper, it will be shown that equations (8a) and (8b) canbe easily extended to the multi-mode situation, so that bothP- and S-waves are simultaneously taken into account to com-plement the PP-image with the SP-, PS-, and SS-images (fromone to four subsurface images). Again, in this multi-mode sit-uation, propagation will be scatter-free, and scattering will bepropagation-free.

Extension of the Huygens principle

Based on full wavefield forward model equation (8a), we canwrite

P+j (zm; z0) =k

W+k(zm, zm1

)P+kj

(zm1; z0

)

+k

W+k(zm, zm1

)S+kj

(zm1; z0

). (10a)

In equation (10a), the first term describes wave propaga-tion according to the classical Huygens principle. In gridpointk at depth level zm1, wavefield sample P

+kj acts as a one-way

secondary point source. Its wavefield is propagated by oper-ator W+k to depth level zm. Similarly, in the second term ofequation (10a), wavefield sample S+kj also acts as a one-waysecondary point source. I will refer to S+kj as an extendedHuygens source. Both Huygens sources radiate downward.For a homogeneous gridpoint, the extended Huygens sourceis zero. Using full wavefield forward model equation (8b), wecan write



Figure 6 The elastic PP-scattering process at gridpoint k, showing the scattered wavefields Sj in full wavefield migration. Note that in primarywavefield migration, the reflection operator R is set to zero (no multiples) and the transmission operators T are deleted (transmission effectsare ignored).

Pj (zm; z0) =k

Wk(zm, zm+1

)Pkj

(zm+1; z0

)

+k

Wk(zm, zm+1

)Skj

(zm+1; z0

). (10b)

Again, in the first term, wavefield sample Pkj functionsas a secondary point source according to the classical Huy-gens principle; the second term Skj represents an extendedHuygens source. Both Huygens sources radiate upward.The extended Huygens source is zero for a homogeneousgridpoint.

In Part II, we will see that the extended Huygens sourcescan be found in the CFP-gathers, each of which is obtained byfull wavefield focusing at detection. By a minimization pro-cess on the CFP-gathers, the Huygens sources are transformedinto the scattering operators, which represent full wavefieldfocusing at emission.

Wavefield propagation as a natural blending process

The secondary source vectors Sj can be interpreted as dualblended source arrays at each depth level (the blending codebeing natural):

Sj(zn ; z0

) = k

[Rk

(zn , z

n

)P+kj

(zn ; z0

)+ Tk

(zn , z

+n

)Pkj

(z+n ; z0

)](10c)

(upward radiating, see equation (9a))

and

S+j(z+n ; z0

) = k

[Rk

(z+n , z

+n

)Pkj

(z+n ; z0

)+ T+k

(z+n , z

n

)P+kj

(zn ; z0

)], (10d)

(downward radiating, see equation (9b))

where vectors ( Rk , Tk ) represent the uncoded upward ra-diating source elements of the blended array at zn , vectors( Rk , T+k ) represent the uncoded downward-radiating sourceelements of the blended array at z+n , and scalars (P

+kj , P

kj )

function as natural blending codes at gridpoint k.The blending codes (P+kj , P

kj ) are, respectively, the down-

going and upgoing incident wavefield at gridpoint k; un-coded source elements ( Rk , Rk ) are the kth column ofreflectivity matrices R and R, respectively (representingangle-dependent backscattering at gridpoint k); and uncodedsource elements ( Tk , T+k ) are the kth column of differen-tial transmissivity matrices T and T+, respectively (rep-resenting angle-dependent forward scattering at gridpoint k).The blended arrays have the unique property that S+j Sj , T+k Rk , Tk Rk , and Rk Rk for small offsetsand/or low shear contrasts. To illustrate this for reflectivity,Fig. 7 clearly shows that offset and shear contrast determinethe difference between Rk and Rk .

If we combine this interpretation of the secondary sourcearrays with equation (8b), we can conclude that seismic shotrecords consist of blended wavefields, in which the blendingprocess is natural: blended source array Sj consists of codedsource elements at each subsurface gridpoint, and shot recordPj (z0; z0) is the superposition of the responses of all these



Cp =2000Cs =1500 =1800

Cp =2500Cs =1500 =2300

Cp =2000 Cs =0.000 =1800

Cp=2500 Cs=0.000 =2300

Rk

Rk

Rk

Rk

RkRk

-1000

(a) (b)

(c) (d)

+1000lateral position (m)

amplitude

ray parameter (s/m)

depth (m)

amplitude

depth (m)

lateral position (m) +1000-1000

ray parameter (s/m)

Rk

Rk

Rk

Rk

RkRk

Cp=2500Cs=1500 =2300

Rk

Rk

RkRk

RkRk

-1000 +1000lateral position (m)

depth (m)

amplitude

ray parameter (s/m)

Cp=2000 Cs= 800 =1800

Cp=2500Cs=1500 =2300

Rk

Rk

RkRk

RkRk

-1000 +1000lateral position (m)

amplitude

ray parameter (s/m)

Cp =2000Cs =1200 =1800 depth (m

)

Figure 7 The difference between reflectivity matrices R and R at one reflector gridpoint. For small offsets and small shear contrasts,R R, but for large shear contrasts and/or large offsets, the difference becomes significant. As expected, around the critical angle, thedifference is always large.

coded source elements, measured at the surface (z0). We willrefer to the response of one coded source element of arraySj as the gridpoint response (GPR), and we will refer tothe superposition process of all GPRs as the natural blendingprocess for reflection measurements.

Similarly, if we combine the interpretation of the sec-ondary source arrays with equation (8a), we can concludethat seismic shot records at zM consist of blended wave-fields, in which the blending is natural: blended source ar-ray S+j consists of coded source elements at each subsurface



gridpoint, and shot record P+j (zM; z0) is the superposition ofthe responses of all these source elements, measured at zM.Again, we will refer to the response of one coded source ele-ment of array S+j as the GPR and to the superposition processof all GPRs as the natural blending process for transmissionmeasurements.

The natural blending property of seismic data is illus-trated in Fig. 8 for one reflecting boundary. Interestingly, themore irregular the reflector, the more the individual GPRsare visible as a separate phenomenon. In reflector-based raytheory, this property is an awkward complication, but forthe proposed gridpoint-related wave theory, this causes noextra complication. Note that perfect spatial sampling ofthe subsurface depends on the aliasing criterion. For effi-ciency reasons, this may lead to an algorithm in the tem-poral frequency domain with a frequency-dependent spatialgrid.

FULL WAVEFIELD MODELLING IN FULLWAVEFIELD MIGRATION

If we make the forward model recursive in depth, expres-sions (8a) and (8b) can be rewritten in terms of addinga source term, followed by wavefield extrapolation (seeFig. 9):

a. for the downgoing wavefields (m = 1, 2, . . . .., M):

1. Q+j(z+m1; z0

) = P+j (zm1; z0) + S+j (z+m1; z0) (11a)2. P+j

(zm; z0

) = W+ (zm, z+m1) Q+j (z+m1; z0) (11b)b. for the upgoing wavefields (m = M 1, M 2, . . . .., 0):

1. Qj(zm+1; z0

) = Pj (z+m+1; z0) + Sj (zm+1; z0) (12a)2. Pj

(z+m; z0

) = W (z+m, zm+1) Qj (zm+1; z0) , (12b)where Sj and S+j are given by equations (9a) and(9b), respectively. At the acquisition surface, we can writeP+j (z0 , z0) = S+j (z0) and S+j (z+0 ; z0) = R(z+0 , z+0 ) Pj (z+0 ; z0).Note that at the last boundary, the response from half-space z > zM, Pj (zM; z0), is given or assumed to be zero, andP+j (zM; z0) is the downgoing wavefield that illuminates half-space z zM. Based on equations (11a) and (12a), for thetotal wavefield, which is the sum of up and down, at zm (m =1, 2, . . . , M), we can write:

Q+j(z+m, z0

)+ Pj (z+m, z0) = Qj (zm, z0)+ P+j (zm, z0) , (13)

which confirms the continuity of the PP-wavefields for thesituation of negligible wave conversion at zm. Note that in theall-elastic case (wavefields represent both P and S), expression(13) is valid for all situations.

Figure 10 shows a computational diagram of FW-Mod, which consists of a recursive downward extrapola-tion process (right-hand side, increasing m), according toequations (11a) and (11b), and a recursive upward ex-trapolation process (left-hand side, decreasing m), accord-ing to full wavefield equations (12a) and (12b). Notethat each roundtrip adds one order of two-way scatteringto the modelling result, starting with order 1 (primariesonly).

Physically, one roundtrip of the FWMod process canbe described by an increase of the scattering order in re-sponse Pj of one, so that after a roundtrip, the (blended)source vectors are transformed into an estimate of the first-order response (primaries), the first-order response is trans-formed into an estimate of the second-order response, andso on.

This can be easily illustrated by using the feedback modelat z0 (Fig. 4):

Pj = X0S+j +X0R Pj at z0. (14a)

Application of the first roundtrip in FWMod involvesmultiplication by a first estimate of X0, giving the primariesonly:

[Pj](1)

= [X0](1) S+j at z0 (14b)and application of the second roundtrip:

[Pj](2)

= [X0](2) S+j + [X0](2) R [ Pj ](1) at z0, (14c)adding the first-order multiples, and so on. The end resultyields a full wavefield response that is fully consistent withthe source vector and the subsurface properties.

Examples of full wavefield modelling in the feedback path offull wavefield migration

An example of the forward modelling module FWMod isshown in Fig. 11. In practice, FWMod is active in the feedbackpath of FWM, so that the reflectivities are provided by FWM.After each roundtrip, one extra order of scattering is gener-ated. By showing the difference, the extra data is visualized tobe used in the next iteration of FWM. We will see that this



Figure 8 Illustration of the forward modelling concept in full wavefield migration and joint migration inversion, showing that the response ofa single reflector represents an interference pattern of gridpoint responses, where each gridpoint response is generated by one coded secondarysource element ( Rk P+kj ). Note that the directivity of this source element is given by the angle-dependent reflection property in gridpoint k.

is an essential property of FWM: during the iteration process,internal multiples help the migration algorithm to convergeto the correct minimum. Using full wavefield technology, weexpect that multiple-rich areas will provide better images thanmultiple-poor areas: the more multiple scattering in the data,the more information is available about the reflecting bound-aries and, therefore, the better the FWM results. This is par-ticularly true for deep reservoirs with complex overburdens.In those situations, the reservoir may be situated in a pri-

mary shadow zone, so that multiples provide the only usefulillumination.

Hierarchical full wavefield modelling in full wavefieldmigration

The mass interference of events in seismic data is a fundamen-tal problem in high-resolution seismic imaging and is knownas internal crosstalk. The assumption of sparsity will not



Figure 9 Forward extrapolation of the upgoing wavefields (a) and the downgoing wavefields (b) in depth layer (zm1, zm), where Sj is thephysical scattering at the layer boundaries, and the columns of W are the local propagation operators inside the layer.

Figure 10 Computational diagram for the full wavefieldmodelling algorithm FWMod,which transforms scattering operators into full wavefields.Each roundtrip adds one order of scattering, starting with order 1 (primaries only).

provide a desirable solution, because sparsity does not rep-resent the property of a real Earth. Dynamic thresholding inFWMod is an interesting approach to dealing with crosstalkin FWM and JMI without making any assumption on asparse end result. By using an automatic thresholding processon the reflectivities, we can obtain a first iteration that showsthe primary GPRs with only the largest reflectivities. In thenext iterations, the threshold is lowered step-by-step, so

that new primary responses are included and a higher orderof multiple scattering is generated by the gridpoints fromthe previous iterations, and so on. In the final iteration,the smallest reflectivities are taken into account. I call thishierarchical forward modelling. Figure 12 illustrates thehierarchical version of FWMod with an example.

The hierarchical FWMod example shows that, in eachroundtrip, new primaries are included and a new order of



Figure 11 Example of full wavefield modelling (FWMod). In each roundtrip, a new order of multiple scattering is generated. This extra data isused in the next full wavefield migration (and joint migration inversion) iteration, helping to steer the solution to the correct minimum.

multiples is generated. Hence, the strongest reflectors auto-matically generate the highest-order multiples. Note the in-teresting property of the hierarchical modelling strategy: thestrongest GPRs, together with their surface and internal mul-tiples, automatically have priority in the migration process. In

FWM and JMI, their responses are subtracted before estimat-ing the weaker ones, and so on. This strategy acts as a typeof L1 constraint in L2-minimization (Daubechies, Defrise andde Mol 2004). However, hierarchical FWMod brings param-eter selection outside the constrained L2-minimization box,



Figure 12 Example of hierarchical full wavefield modelling. In each roundtrip, new primaries are included, and a new order of multiple scatteringis generated (compare with Fig. 11). Note that response 1 is not shown (which is only the shallowest reflection).

Figure 13 The coda of a reflective overburden is seriously masking the reservoir response. A hierarchical imaging strategy in full wavefieldmigration (and joint migration inversion) solves this problem.

giving the user full control, and any smart selection processcan then be implemented. For instance, in addition to reflectorstrength, priority may also be given to shallow versus deep.This is of particular importance if we are dealing with a highlyreflective overburden, which causes a strong coda on top of

the deeper response. Figure 13 illustrates this notorious prob-lem. The example shows that the response of the reservoiris completely masked by the coda of the overburden. By giv-ing higher priority to the strong reflectors of the overburden,the crosstalk between the overburden coda and the reservoir



response is removed prior to migrating the deeper part. Fordeep reservoirs, this is the appropriate approach to take.

DESCRIPTION OF THE REVERSEMODELLING ALGORITHM

In FWMod, the source vectors (S+j for each j), propagation op-erators (W, W+), and scattering operators (R, T) are given,and seismic measurements ( Pj for each j) must be computed(measurement simulation process). The process is iterative andconsists of several roundtrips, starting at the source positionsand ending at the detector locations. In full wavefield reversemodelling (FWMod1), it is the other way around: measure-ments are provided, and the source vectors must be computed(source estimation process). This process is similarly iterative,consisting of several roundtrips, but in this case, it starts atthe receiver locations and ends at the source positions. Ineach roundtrip, the updated source wavefield is used to steerthe estimation process. Finally, the end result yields the sourcevector S+j that explains all primaries and multiples in the mea-surement vector Pj for each j in the data volume.

Reverse modelling equations

We start by considering the nonrecursive forward model forthe upgoing wavefield at the surface z0 (see equation (8b)):

Pj(z+0 ; z0

) = W (z+0 , zm) I (zm, z+m) Pj (z+m; z0)+

mn=1

W(z+0 , z

n

)Sj

(zn ; z0

), (15a)

where I is the unity matrix and Pj(z+m; z0

)is the response

of the lower half-space (z > zm). From a physics point ofview, this equation describes forward propagation as a mas-sive upward-moving defocusing process at the detector side,starting at depth level zm and ending at the surface z0 (m = 1,2, . . . ,M). If we multiply equation (15a) by scatter-free focaloperator F+

(zm, z

+0

) = [W+ (zm, z+0 )], then we can write:Pj

(z+m; z0

) = I (z+m, zm) F+ (zm, z+0 ) Pj (z+0 ; z0)

mn=1

F+(z+m, z

n

)Sj

(zn ; z0

), (15b)

where Pj(z+0 ; z0

)is known. Expression (15b) shows that re-

verse modelling is a massive downward-moving focusing pro-cess at the detector side, starting at the surface z0 and endingat depth level zm (m = 1, 2, . . . ,M). Note that the summation

removes the transmission effects and the multiple scatteringfrom the response.

Similarly, the nonrecursive forward model of the down-going wavefield at maximum depth level zM (see equation(8a)) is given by

P+j(zM; z0

) = W+ (zM, z+m) I (z+m, zm) P+j (zm; z0)

+M1n=m

W+(zM, z

+n

)S+j

(z+n ; z0

). (16a)

Again, from a physics point of view, this equation de-scribes forward propagation as a massive downward-movingdefocusing process, now at the source side, starting at depthlevel zm and ending at zM (m = M 1, M 2, . . . , 0).If we multiply equation (16a) by scatter-free focal operatorF

(z+m, z

M

) = [W (z+m, zM)], then we can write (m = M 1,M 2, . . . , 0):P+j

(zm; z0

) = I (zm, z+m)F (z+m, zM) P+j (zM; z0)

M1n=m

F(zm, z

+n

)S+j

(z+n ; z0

), (16b)

where P+j(zM; z0

)is known. Expression (16b) shows again

that reverse modelling is a massive focusing process, now atthe source side and moving upward, starting at maximumdepth level zM and ending at depth level zm; compare this with(15b). Note that when we turn around at zM, the startingwavefield is given by P+j

(zM; z0

), and when we turn around

at z0, the starting wavefield is again Pj(z+0 ; z0

). The next

roundtrip makes use of improved estimates of the secondarysources Sj at each depth level zm (similar to FWMod) andsource vector S+j (z0) = P+j

(z0 ; z0

).

If we compare equation (15a) with (15b) and (16a) with(16b), we see that the full wavefield forwardmodelling processin each roundtrip of FWMod is replaced by the full wavefieldreverse modelling process in FWMod1: application of Wfollowed by addition is replaced by application of F = Wfollowed by subtraction. Figure 14 shows the computationaldiagram of FWMod1 (the combination of focusing and re-moval processes at the detector and source side as described inequations (15b) and (16b), respectively). Similar to FWMod,the algorithm is recursive (compare Fig. 14 with Fig. 10, seealso Appendix B).

Physically, one roundtrip of the FWMod1 process can bedescribed by a decrease of the scattering order in Pj by one,so that after a roundtrip, the first-order response (primaries)is transformed into an estimate of the (blended) source,the second-order response is transformed into the first-order



Figure 14 Computational diagram for the full wavefield reverse modelling algorithm FWMod1, which applies focusing by reverse extrapolationand removes physical scattering by subtraction at both the detector and the source side. Each roundtrip transforms one order of scattering intoan update of S+j

(z0), starting with order 1 (primaries) in the first roundtrip. Compare with Fig. 10.

response, and so on. Similar to that shown for the forwardmodelling algorithm (FWMod), this can also be easily illus-trated for FWMod1 by again using the feedback model at z0(see equation 14a):

S+j = X10 Pj R Pj at z0 (17a)

Application of the first roundtrip in FWMod1 involvesmultiplication by the first estimate of X10 , yielding[S+j](1)

= [X10 ](1) Pj A(1)R Pj at z0 (17b)and in the second roundtrip[S+j](2)

= [X10 ](2) Pj A(2)R Pj at z0, (17c)where A represents a diagonal matrix of scaling factorsthat minimizes the subtraction result (line search), makingFWMod1 robust. The end result consists of a causal sourcewavefield including directivity and tail that is fully consis-tent with the subsurface properties (X0) and the measurements( Pj ). Note the similarity between expressions (17a)(17c) and(14a)(14c).

The reverse modelling algorithm can be generalizedfor any (blended) source vector at depth level zm, repre-

sented by the combination of primary and secondary sourcesSj (zm) + Sj (zm; z0) at that depth level. Such an extension ofthe algorithm is most interesting for the detection and char-acterization of (micro) seismic sources, Sj (zm), using bothprimaries and multiples.

COMBINED FORWARD AND REVERSEMODELLING

Finally, let us look at the situation in which the wavefieldvectors at z0, Pj

(z+0 ; z0

)and S+j

(z0), are known and the

wavefield vectors at zM, Pj(z+M; z0

)and P+j

(zM; z0

), must

be computed. Note that this is the task of full wavefield re-datuming (from z0 to zM) in a known subsurface. For thesolution, we need to combine equations (15b) and (16a), sothat reverse modelling at the detector side is combined withforwardmodelling at the source side. Figure 15 shows the hy-brid computational diagram. At each depth level the internalwavefields Pj , Q+j and secondary sources Sj are iterativelycomputed.

Of course, source estimation (Fig. 14) and shot recordredatuming (Fig. 15) can be integrated by combining bothcomputational diagrams. In Part II and Part III of this trilogy



Figure 15 Computational diagram for the full wavefield redatuming process from z0 to zM, which is a combination of reverse modelling at thedetector side and forward modelling at the source side.

of papers, we will see that the combination of full wavefieldredatuming and full wavefield source estimation is an integralpart of FWM and JMI.

INCLUDING WAVE CONVERSION INFWMOD AND FWMOD1

Generally, the Earth response is measured by P-sensors (ma-rine) or Vz-sensors (land). These measurements are the resultof a linear superposition of PP-, PS-, SP-, and SS-wavefields inthe subsurface. By using the operator presentation of seismicwave theory, we can easily extend the forward and reversemodelling algorithm to include wave conversion. By assigningnot only Rpp but also Rsp, Rps, and Rss to each gridpoint, wecan extend the secondary sources Sp (representing two-wayP-scattering) to Ss (representing two-way S-scattering). Addi-tionally, by specifying not only a P-wave propagation velocitydistribution (yielding Wpp) but also an S-wave propagationvelocity distribution (yielding Wss), the converted gridpointscattering can also be migrated. Using the matrix expressionfor all shot records, we can write for the multi-mode modelequations in FWMod:

a. for the downgoing wavefields, starting at the surface (m =1, 2, . . . , M):

P+(zm; z0

) = W+ (zm, z0) S+ (z0)+

m1n=0

W+(zm, z

+n

)S+

(z+n ; z0

)at the source side; (18a)

b. for the upgoing wavefields, starting at the deepest level (m= M 1, M 2, . . . , 0):

P(z+m; z0

) = W (z+m, zM)P (zM; z0)+

Mn=m+1

W(z+m, z

n

)S

(zn ; z0

)at the detector side. (18b)

And we can write for the multi-mode model equations inFWMod1:c. for the downgoing wavefields, starting at the deepest level(m = M 1, M 2, . . . , 0):

P+(zm; z0

) = F (zm, zM)P+ (zM; z0)

M1n=m

F(zm, z

+n

)S+

(z+n ; z0

)at the source side; (18c)



d. for the upgoing wavefields, starting at the surface (m = 1,2, . . . , M):

P(z+m; z0

) = F+ (z+m, z0) P (z0; z0)

mn=1

F+(z+m, z

n

)S

(zn ; z0

)at the detector side, (18d)

where

P =[PpPs

], W =

[Wpp 00 Wss

]and F = [W] (19a)

S =[

SpSs

]=

[Rpp RpsRsp Rss

][P+pP+s

]

+[

Tpp TpsTsp Tss

][PpPs

], (19b)

S+ =[

S+pS+s

]=

[Rpp RpsRsp Rss

][PpPs

]

+[

T+pp T+psT+sp T+ss

][P+pP+s

]. (19c)

When we use the above expressions in the recursiveFWMod and FWMod1 algorithm, every possible conver-sion (PS and SP) at every depth level is automatically in-cluded (there is no user involvement). Note that wave con-version is only of practical importance for the gridpoints withlarger reflectivities (due to large P- and/or S-contrasts) andfor the wavefields with larger incident angles (due to largesourcereceiver offsets). For these situations, T = R is nolonger a valid approximation, and therefore ||S+ S||is a measure of the degree of wavefield conversion. Notealso that FWMod is not designed as a stand-alone algo-rithm, but it functions as an integral part of the closed-loop,iterative, full wavefield process. This means that the fourtypes of reflectivities (Rpp,Rsp,Rps,Rss) and transmissivities(Tpp, Tsp, Tps, Tss) are provided by the estimation processof FWM and JMI. In JMI, the propagation operatorsWpp andWss are also estimated, yielding the P-wave and S-wave prop-agation velocities by applying a separate inversion process tothese propagation operators.

CONCLUSIONS

In this paper, I have proposed a new way of forward mod-elling, called FWMod, which functions in the feedback pathof the closed-loop, iterative algorithms FWM and JMI. FW-Mod transforms the output of FWM and JMI into seismicshot records that include surface and internal multiples (fullwavefield modelling). Optionally, converted waves may beincluded in the modelling result.

The traditional FinDif-related presentation of the sub-surface in terms of detailed velocity and density is outsidethe framework of migration and, therefore, cannot be used inclosed-loop imaging. Instead, FWMod describes the subsur-face in terms of propagation and scattering operators. Theseoperators are not specified by the user, but they are estimatedby FWM and JMI.

In FWMod, each inhomogeneous gridpoint (labelled k)functions as a two-way secondary source (extended Huygenssource), the source properties being determined by the scat-tering operators [ Rk, Tk] and the one-way incident wave-fields [Pkj , P

+kj ]. The combination of the classical and ex-

tended Huygens sources generates wavefields that properlyrepresent the nonlinear scattering properties of any complexgeology.

The phase relationship between gridpoint k and its neigh-bouring gridpoints is determined by the recursive propaga-tion operators [ W+k , Wk ]. A direct relationship exists be-tween these recursive operators and the local velocity proper-ties around gridpoint k. By keeping the scattering operatorsoutside the propagation operators (the big decoupling), weensure that W+k and Wk are scattering-free operators witha unit spatial amplitude spectrum if we neglect inelastic ab-sorption. This choice has far-reaching consequences for mi-gration and inversion algorithms: scattering operators aredetermined only by the amplitude properties, and propaga-tion operators are determined only by the phase propertiesof the seismic data. The result of this orthogonal property isthat the nonlinearity in migration and inversion is decreasedsignificantly.

By using the concept of secondary sources (Sj ), eachshot record can be considered to be the response of a blendedarray of coded sources (Sj ), at each depth level the uncodedsources being given by the scattering operators in each grid-point [ Rk, Tk], and the blending codes being given by theincident wavefields (Pkj , P

+kj ) at these gridpoints.

As explained in this paper, FWMod offers great mod-elling flexibility and opens up new opportunities in closed-loop seismic imaging. For instance, to minimize interference



effects in full wavefield migration, the FWMod algorithmmaybe started with gridpoints that generate the largest contribu-tion (the output of a selection process). In subsequent itera-tions, gridpoints are brought in with smaller contributions. Inthis selection process, shallow may have higher priority thandeep. Finally, all gridpoints (from strong to weak, and fromshallow to deep) are included. This capability is referred to ashierarchical modelling, which is considered a critical strategiccomponent of the next generation migration algorithms.

It is expected that FWMod will play an important rolein future time-lapse applications. By using the full wavefieldimage of the previous survey and by taking the acquisitiongeometry of the current survey as the input for FWMod,a residue matrix (P) is computed that functions as in-put data for JMI. The result is a full wavefield differentialimage.

By using the operator formulation of wave theory, itis straightforward for us to extend the FWMod algorithmfor PP-reflections to wave conversion at each gridpoint(Rsp and Tsp) and to shear wave propagation (Wss and W+ss)in each layer, allowing the generation of converted waves(SP and PS) and the generation of SS-scattering by including(Rss and Tss) .

Finally, we have seen how the theory of FWMod canbe extended to full wavefield reverse modelling (FWMod1),facilitating the estimation of complex source propertiesfrom (blended) shot records. This reverse modelling algo-rithm opens up new opportunities for the design of multi-depth, blended source arrays as well as the detection andcharacterization of micro-seismic sources. The combinationof FWMod and FWMod1 allows the formulation of afull wavefield redatuming process. FWMod and FWMod1

are therefore the basic algorithmic modules in FWMand JMI.

EP ILOGUE

In the proposed reformulation of seismic wave theory, the sub-surface is not represented by the usual elastic parameters, butis described in terms of decoupled propagation and scatter-ing operators. The propagation operators are scattering-freeand determine the traveltime properties of the seismic mea-surements. The scattering operators are propagation-free anddetermine the amplitude properties of the seismic measure-ments. Data-driven propagation operators contain anisotropyand inelastic absorption; data-driven scattering operators con-tain angle-dependency and conversion losses.

The dual operator description (propagation, scattering)allows an extension of the well-known Huygens principle by

introducing extra secondary sources that represent the elasticscattering properties in each inhomogeneous gridpoint. Thismeans that wave propagation involves moving through thesubsurface and collecting the total contribution of all theseHuygens sources. It leads to a new forward modelling algo-rithm for seismic reflection data that computes all types ofmultiple scattering (surface-related and internal) and takesinto account complex medium properties such as anisotropyand critical angle effects. Additionally, because of the decou-pling in propagation and scattering, the forward modellingalgorithm (from source properties to measurements) can beeasily modified to a reverse modelling algorithm (from mea-surements to sources properties).

Today, innovations take place by making a connectionbetween different disciplines. We see this bridging principlealso in the seismic imaging trilogy: modelling, migration andinversion are interconnected to strengthen each other. This oc-curs in closed-loop architecture and leads to new capabilities.For instance, the operator-based forwardmodelling algorithmfunctions in the feedback loop of the iterative FWM process(see Fig. 1), and therefore, the modelling operators are notprovided by the user but by the migration output. Hence, themodelling algorithm is indirectly driven by the seismic mea-surements. This approach is self-learning and allows us to dothings we have never done before, such as depth imaging with-out specifying the velocities. It also allows us to critically re-consider our current modelling tools. Today, finite differenceis the modelling tool in our industry, but it requires a descrip-tion of the subsurface in terms of elastic parameters. These pa-rameters, however, are outside the framework of migration.For instance, in the practice of reverse time migration, we seea paradox in that velocity distributions are required that mustaccurately address the traveltimes but that should not gener-ate a user-induced set of false transmission effects and false(multiple) reflections in the modelled wavefields. This prob-lem is difficult to solve because finite-difference algorithmsuse hybrid operators that address phase and amplitude si-multaneously at each gridpoint. This strong interdependencyintroduces huge nonlinearities, as is well known from classi-cal inversion theory. Such complex coupling problems can beavoided by abandoning finite-difference modelling in migra-tion. I call this the big decoupling in full wavefield migrationand inversion.

In conclusion, the next big step in seismic modelling, mi-gration and inversion will not be a mathematical one but afundamental one in terms of physics. It is not a matter ofboundary integral methods versus differential equation meth-ods, but is a matter of how we describe the subsurface in eachof our closed-loop processing phases. This description is not



given in terms of elastic parameters but in terms of wavefieldoperators, requiring a forward modelling algorithm that ac-cepts this description. This automatically leads to a recursiveintegral-basedmodelling algorithm inwhich themodelling op-erators are continuously updated by the residue-driven imag-ing process (tight integration betweenmodelling and imaging).By doing this, we move away from todays open-loop mind-set that everything should be (almost) fully correct from thebeginning onward. In Part II and Part III, we will see thatthis tight integration allows us to start with simple operators.The initial propagation operators may be full-bandwidth, lo-cal phase-shift operators that follow from some user-specifiedinitial velocity distribution. For the initial scattering operators,it is even simpler because zero turns out to be a good start.Unlike classical inversion, the output of closed-loop migrationappears to be very insensitive to the initial values. The residue-driven updating process automatically applies any correctionand refinement that is required to explain the data. In nearfuture, we will see that this tight integration can be extendedto elastic inversion. By using the output from the imaging pro-cess that is, the wavefields in each gridpoint the inversionalgorithm becomes significantly more linear than that used incurrent inversion processes. The latter can be better under-stood by acknowledging that, to date, inversion has had tocarry out the difficult task of estimating both the wavefieldsand the elastic parameters at each subsurface gridpoint at thesame time. Migration is the ideal pre-processor for inversion.

ACKNOWLEDGEMENTS

The author would like to thank Eric Verschuur for the inspir-ing discussions on integrating the theory of FWMod in thefeedback loop of the Delphi migration package. Thanks arealso due to Alok Soni for his help in the generation of the nu-merical modelling examples. Last but not least, I would liketo thank the Delphi sponsors for the stimulating discussionson industry requirements during the Delphi meetings and fortheir continuing financial support.

REFERENCES

Berkhout A.J. 1982. Seismic Migration, Imaging of Acoustic Energyby Wavefield Extrapolation, A: Theoretical Aspects, 2nd edn. El-sevier.

Berkhout A.J. 2012. Combining full wavefield migration and fullwaveform inversion, a glance into the future of seismic imaging.Geophysics 77, S43S50.

Berkhout A.J. and Verschuur D.J. 1994. Multiple technology, part2: migration of multiple reflections. 64th SEG meeting, ExpandedAbstracts, 14971500.

Daubechies I., Defrise M. and de Mol C. 2004. An iterative thresh-olding algorithm for linear inverse problems with a sparsity con-straint. Communications on Pure and Applied Mathematics 57,14131457.

Lu S., Whitmore N.D., Valenciano A. and Chemingui N. 2011. Imag-ing of primaries and multiples with 3D SEAM synthetic. 81st SEGmeeting, Expanded Abstracts, 32173221.

Moczo, P., Robertsson, J.O.A. and Eisner L. 2007. The finite-difference time-domain method for modeling of seismic wave prop-agation. Advances in Geophysics 48, 421516.

Verschuur D.J. and Berkhout A.J. 2011. Seismic migration of blendedshot records with surface-related multiple scattering. Geophysics76, A7A13.

APPENDIX A

TWO-WAY EXPRESS ION OF EXTENDEDHUYGENS SOURCES

The expression of the extended Huygens sources in terms ofone-way wavefields is given by

Sj(zn ; z0

) = R (zn , zn ) P+j (zn ; z0)+ T (zn , z+n ) P j (z+n ; z0) , (A1a)

or, if we omit the depth level indication,

Sj = R P+j + T Pj . (A1b)

It can be easily verified that this expression can be rewrit-ten in terms of

Sj =12

(R + T) ( P+j + Pj )+ 12

(R T) ( P+j Pj ) ,

= R +(P+j + Pj

)+ R

(P+j Pj

), (A2a)

where ( P+j + Pj ) is the two-way wavefield, and ( P+j Pj )is related to the vertical derivative of the two-way wavefield(vertical depending on the coordinate system). Similarly, forthe downward-radiating version, we can write

S+j =12

(R + T+) ( Pj + P+j )+ 12

(R T+) ( Pj P+j ) ,

= R+(Pj + P+j

)+ R

(Pj P+j

). (A2b)

Note that if we neglect wave conversion (for instance, atsmall offsets) both expressions simplify significantly:

Sj = R( P+j Pj ) (A3a)

and

S+j = R( Pj P+j ), (A3b)

so that S+j = Sj .



APPENDIX B

RECURSIVE FULL WAVEFIELDEXTRAPOLATION IN FWMOD-1

Using equation (15b) its recursive version can be written as(m = 1,2, M) :Qj

(zm; z0

) = F+ (zm, z+m1) Pj (z+m1; z0) (B1a)with

Pj(z+m1; z0

) = Qj (zm1; z0) Sj (zm1; z0) . (B1b)

Using equation (16b) its recursive version can be written as(m= M 1, M 2, 0) :

Q+j(z+m; z0

) = F (z+m, zm+1) P+j (zm+1; z0) . (B2a)with

P+j(zm+1; z0

) = Q+j (z+m+1; z0) S+ (z+m+1; z0) . (B2b)Note F = [W]. In full wavefield migration and joint mi-gration inversion, FWMod and FWmod-1 are the basic com-putational modules (see Figs 10 and 14).


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An Outlook on the Future of Seismic Imaging, Part I_Berkout