Summary Methods for removal of free-surface and internal multiples have been developed from bath a feedback model approach and inverse scatterin g theory. White these two formulations derive from different mathematica) viewpoints, the resulting algorithm s for free-surface multiple are very similar. By contrast , the feedback and inverse scattering method for internal multiple are totally different and have different requirements for sub surface information or interpretive intervention . The former removes all multiple related to a certain boundary with the a of a surface integral along this boundary ; the alter wilt predict and attenuate a ll internal multiple a t the same time . In this paper, we continue our comparison study of these internal multiple attenuation method ; specifically , we examine two different realizations of the feedback method and the inverse scattering technique .

1. 1-1 4
Summary
Methods for removal of free-surface and internat muhiples have been developed Erom bath a feedback model ap-
proach and i nverse scattering theory. White these two formulation s derive Erom different mathematica) viewpoints,
the resulting algorithms forfree-surface multiplee are very similar. By contrast, [he feedback and inverse scattering
methode for intergul multiplee are totally different and have d i fferent requiremenis for subsurface information or
interpretive intervention . The former removes all multiplee related to a certain boundary with the afd of a surface
integral along this boundary; [he )alter wilt predict and attenuate all internat multiplee at the same time. In this pa-
per, we continue our compari son study of these internat mu ltiple attenuation methode; specifically, we eiamine two
different realizations of the feedback method and the inverse scatterin g technique.
Introduction
Free-surface and internat multiplee are defined as multiply reflected events that experience two or more upward
reflections in the subsurface. The former consists of all multiplee that have experienced one or more reflections at the
earth 's surface. The laffer are events that have all of their downward reflection .points below the free-surface.
There are c urrently two comprehen sive approaches for attenuating free-surface and internat m ultiplee that are specif-
ically designed to accomodate a multi-dimensional Barth. The first is the feed-back method that derives Erom a view
of the generati on and inversion of data in terme of a free-surface, interfaces, a nd actual propagation between these
reflectors. The second is the inverse scatteri ng method that derives Erom a view of the generation and roversion of
data in terms of a free-swface, water propagation, and point scatterers whose strength depends on the local differente
between Barth proporties and water. The first method derives from Berkhout's feedback model (Berkhout , 1982) that
removes all free surface multiplee and internat multiplee (sec Verschuur, 199 1 ; Berkhout and Verschuur, 1 997 ; Ver-
schuur and Werkhout, 1 997) . The second appr oach derives from a ta sk-separated inverse stalt ering series (sec e.g.
Wegsein et al ., 1997; Carvalho, 1992; Araujo et al., 1994; Matson and Weglein, 1996). These twó comprehensive
methode provide similar algorithms for free-surface multiple removal hut totally different methode for internat mul-
tiple attenuation . These two methode for internat multiplee veere compared in a recent series of papers (Verschuur
et al ., 1998 ; Matso n et al., 1998; Weglein, 1999b,a; Berkhout, 1999). These papers describe and illu strate how th ese
two method e have different requirements for a-priori subsurface information or interpretive intervention as veel) as
different computational demands. In this paper, this study continues with a comparison of two realizations of the
feed-back model for interaal multiplee and the inverse scattering method .
Surface-related multiple remova l
We will consider two layer-related interaal multiple r emoval scheures. Both of these scheures are extensions of the
surface-related multiple approache s of Kennett (1979), W erkhout (1982) and Verschuur et al . (1992). In particular,
following Werkhout and Verschuur (1997), surface-related multiplee can be removed using a fast-converging iterative
method which, using Berkhout's notation, can be written as :
Pop+') = P - Ao(w)P
ó'
) P . (1)
The physical interpretation is that multiple reflections can be considered a s a combination of primary events that are
connected at the interface where the downward reflection takes pace . For example, the first order surface multiple
o i' Figure 1 a (SR) can be constructed by laktag two primary wavefield components (SR ' and S'R). With equation
(1 ) reflection point A at the surface is considered as a secondary source point where the wave field SR ' is used as
illuminating source wave field in S' to obtain the multiple SR. Although the mathematica] starttag points for the
inverse stallering approach are different, the prediction reduces to integrale along the surface, and similar expresslons
are found (see e. g. Carvalho et al ., 1 99 1 ; Weglein et al ., 1 997).
Interface-related interaal multiple removal method 1 : operator driven
Similarly, internat multiples can be described as combinations of primary w ave field components. However, in this
case the downward reflectioa(s) take place below the surface (e .g. in Figure tb this is point B). Hence, the equivalent
EA GE Gist Conference and Tec h n i cal E x h ibi tion -He l sink i, Finland , 7 - 11 June 199 9
D.J. VERSCH UUR', K.H . MATSON2, A.J. BERKHOUTI,
A.B. WEGLEIN2, C.Y. YOUNG2 and H. JAKUBOWfCZ3
Delft University of Technology, Laboratory of Seismics and Acoustics,
PO Box 5046, 2600 GA Delft, The Netherlands
2 Arco EPT
3 Verftas DGC Ltd

2. S R' S' R
Figure . L• a) A surface multiple can be considered as tbc combination of two .primaries. b) An internal multiplas can
be considered as the combination of two primary wave fields, being connected at interace k.
prediction requires measureme nts on the corresponding interface. One way in which the feedback model lor wavelet
estimation and surface-related multiple removal can be extended to trest internal multiplas is baserf on the work
of (Berkhout, 1982, p. 188). In partic ular, after downward-extrapolatin g the shot records to a reHecting boundary, all
multiplas related to thi s internat interface can be removed by the surface-related algorithm. In Werkhout and Verschuur
(1997) the iterative version has been proposed. For the a boundary k the expresion is given by :
P(~+`) = Po - Po~(2)
where F- and F' contain tbc inverse extrapolation operators from the surface up t o multiple-generatin.- interface
k. In Pact the [wo operators F~ and F~ describe the proces of applying the two dashed arrow s in Fi gure lb as a
ereprocessing step to the surFace data. The data with multiplas remov ed•from all reflectors down to k is now written
,►s Pk, the subscript meaning [he level up to which multipl e removal bas been applied . Additionaly , all primary
reflections relaied to all int erface above and inclduing k Weed be muted in the surface data, indicated by the bars: Pk
and Po . Nota that for the situation o f the surface-related multiplas we h ave a comparab le proces: the direct wave
Weed be muted prior to predi ction . Thus, tbc operators F ~ and F~ take (he surface data and bring it to multiple-
,generating interface k at (he source or [he receiver side, such that a convolution along this interfase can be npplied.
Examples of this proces on synthetic and field data have already been shown in Verschuur and Werkhout (1996),
Hadidi and Verschuur (1997) and Verschuur et al . (1998).
The practical implementation of this procedure consists of four steps : 1) Obtain tbc redatuming operators for [he
interface of interest. Thi s can be very accurately done usin g tbc CFP operator updating technique (sec Scrkhout,
1997 ; Thorbecke, 1997 ; Botte and Verschuur, 1998). In this way accurate operators can be obtained without precise
knowledge of the vetocity depth structures above the multiple-generating interface; 2) apply the redatuming to the
shot records and mat e the reflections above the reflector; 3) calculate the interface-consistent convolutions to predict
the multiplas and 4) subtract them Erom the original surface data.
Interface-related internat multiple removal method II : data driven
Jakubowicz (1998) recognized that the combination of the two inverse propa ;ation matrices F - and F+ in equation(2)
can be constructed from the time-reversé of the primary reflection from that interface . In ferms of Figure 1 h it means
that the two dashed arrows constitute (he primary reflection S'BR', which is also present in the data . These geometri-
cal considerations ware also observed by Keydar et al . (1997). By extracting the required primary reflection trom the
data, the interbed multiple prediction proces can then be rewritten in a full data-drieen approach :
k- 1
where the * means complex conjugation . Again the bar above the quantities indicates the mate of all events related
to reflectors above and including multiple-generating interface k. In a practical implementation, the following steps
Weed be taken: 1) Select primary reflection related to tee multiple-generating boundary from the shots ; 2) apply two
surface consistent convolutions of this óperator with the (muted) surface data and 3) subtract the predicted multiplas
from [he original input .
Inverse scattering internat multiple elimination
For the approach baserf on the inverse staftering series, as described by Weglein et al. ( 1 997), Araujo et al. (1994) and
Coates and Weglein (1996), one procedure predicts all possible internat multiplas which can tien be subtracted from
the input data. This procedure is derived Erom an inverse scattering subseries and can be interpretec3 as a sum over
all possible combinations of three ima ged data points that have a constrained lower-higher-lower relationship. These
imaged data are the original pre-stack data migrated using the water vetoci ty. The multiple uttenuatior7 subseries
1 -1 4
b) !►rternul nuultiplea) Barface multiple

3. is a first order approximation to an internap multiple reneovaf series. Al[hou~h approximate , this subseries properly
predicts the iravel time of all internat multiplee, even converted phases (Coates and Weglein, 1996). For typical
seismic evenre, the amplitudes of the predicted multipl ee are 80-95% of the actual multiple .
Internat multiple attenuation is performed by using the origina l prestack data to calculatie a series of terme each of
which attenuates a different order of multiple . The equation for first order multipees in 2-D marine data is (Weglein
et al., 1997 : Araujo et al ., 1994)
I 1 X47 ~'g +- s)
l 2_) - -oo
1 _ ~ -i~91 t4 2 ~=~ 4n~= E=s~
d.a6l( ~2,~s, :a~el I
9 2 + 9 .~ )= a (4)
. 2+ e
Here, ky, k_G are the Fourier transform variables over geophone and source locations respectively, qy and qs are vertical
wavenumbers biven by qg = f (W/eo)' - k9)) and qs ks }) where co is the water vetocity and w is
the angular temporal frequency. The source and receiver depths are given by z9 and zs respectively . The parameter
e ensures that zl is always greater than and not equal to z2 and simitarly for z3 . The quantity 61(kg, kl, zi) is an
uncollapsed migration which has been transformed to pseudo-depth using the water vetocity (sec, e .g. Wegsein et al.,
1997). To compute the multiple estimate, bi is input into an algorithm for equation 4 which then outputs the estimated
internat multiplee, b3l Af . When added to 61, b3lnf suppresses all first order internat multiplee . Notie that the l imits in
the integrale over zl, z2, and z3 limit the different portions of the data that are combined together. These limits are
such that zi > zz and z -, < z3 which is a direct consequence of the way mu ltiplee are cata loged according to the
number of reflections the multiple has experienced and the re lative geometry of chose reflections. The first term in
the internat mul tiple attenuation series attenuates a 11 first order internat multiples for all depths and omes and altere
higher order i nternat multipree. Successively higher order ferms in the series attenuate higher order internat multiplee
(see, e.g. Weglein et al., 1997) .
In practise, (he main strength of this procedure is that al t internat multiplee are handled at the same time, witho ut the
Weed for any a-priori subsurface information, i .e ., no description of the reflecting boundaries is required .
lateral Bistance (m) Synthetic data example
2000 3000 4000
o In Figure 2 a subsurface model is shown that has been used as
input for an acoustic finite differente modeling experiment . 101
shots have been modeled with the shot positions rangmg from
1800 to 4800 m with steps of 30 m . The surface has been cho-
iooo- F sen
non-reflective, so that only primary and i nternat multiple re-
ftections have been be simulated . To test the three methode on
Ê
this dataset, first a stack of the input data is produced . As this
- - - "' test was done by three different groups, three slightly different. ..
stacking procedures have been followed . In this paper, effort bas
2000 been put to display them in (roughly) the same amplitude lev-
els, as shown in Figure 3a. The last primary reflection (Erom the
reflector at about 2700 m depth in the mode l) is visible at 2.0
seconde. It is noticeable that especially in the lower half (below
3000- 2
.0 s) internap multiplee are present, especially below the lateral
position of the sync linal sfructure. The two verslons of the inter-
Figure 2: Subsurface vetocity model . face method have been app lied for the first interfáce o nly.
1 2 3 2 3 2 3
0 0 0
,~r f
u ~r!+" ' .'_ ? ~~ ira I • '` -~ .
' s p u 'tFFI ~u~ .th41 t$~~f, : t 'h~ ' 1~ _
Ry ~! ~PtAM +Wy~H(i
1 .0 r y 1.0
40
1 . 5 1 . 5
~r ~ . ry / ~ qH j ep4+1 ~4 r
2 .0
r~~
~C ~ ~ e• ~y~?~" ~~¢'* ~Y a~~t
'
2.0 F fi7j~' 5~ ~¢j lf uki• ~`~ 'J ~ J , ~ ~ ~ .fit ~4'~~
' ` .^ } ~ ~.iM "
. !"~ ~ n i,,~~~t~r AMY A .1
T ,`~^ ti• I~,~N~ ~ n~1~ ,'
.yr; . ;
.~ ► : . . . ~ •v ~ ' ~ . .T ..~~' . _
~,. ,4,~.y w r : .rom ~ !' .
a) Input stacks b) Output stacks c) Differente
Figure 3 : Comparison of three internal multiple removal methode at stack level . a) Three input sections . b) Ouptut
stacks after internal m ultiple removal. e) Stacks of multiplee only.
SAGE 61 et Conference and Technical Exhibition - Helsinki, Finland, 7 - 11 June 1999

4. Of course, the inverse sacattering method (numbered 3 in Figure 3) wilt predict all possible internat multiples . After
stacking the effect of the multiple removál can be obeserved in Figure 3 for all three methods . To further analyse the
results, the different plots are produced (Figure 3c), all in the same amplitude level . The first two interface-related
methods (numbered 1 and 2 in Figure 3) have strongly related implementations, which is confirmed by the first two
difference plots in Figure 3e. Although exact amplitudes of removed multiple vary between tho two, cleárly the
same multiple events are addressed. When inspecting the inverse scattering result (labeled with 3) it appears that the
character of the removed multiplas is different : the method has predicted alt possible internat multiple reflections and
is not lied to any particular interface .
ConCluslon
s Method Interface Interface Inverse
The inverse scattenng method and two realizations Feature method 1 method II scanering
of the feedback method for attenuating internat
multiplas are compa Ced using synthetic data ex- Type of internat One interface, all One interface, all All interfaces, one
amples to illustrate theirrelative stengths and waak- multiplas orders at a time orders at a time order at a time
nessen. In FiLure 4 some specific features of (he Frameworcl data driee n and
three algorithms are summarized. Both the feed- subsurface implicit model rata driven tata drieen
back and inverse scatteing methods operala by pre- information (CFP operators
)
dicting and subtracting the internat multiplas . How- idemificasion and identification!
User interaction operator updating isolation reftection none
ever, the two approaches are distinclly different in on time sectio
n
the way multiplas are cataloged and their requiré-
ments for a-priori information . The feed-back method Cost 2 SMA - 2 SMA - 10 SMA
catalogs multipÍes according to a subsurface inter- per inter
face per interface all interfaces
face which is responsible for the existence of [he Figure 4
: Comparison of the characteristics for three types of
internat mulitples
. To predict thern, operators are internal multiple removal schames ,
required teat describe the propagation from th e
surface to the interface method . The two feedback algorithms differ in the way these operators are achieved, ei-
ther by antimaling these propagation operators Erom traveltimes (model drieen) or by selecting primary reflections
from the seisrnic data itsel€ (data-drieen). The inverse scattering method provides a procedure that does not require
any subsurface information i .e., it requires no avant picking, interpretation or description of the reflection boundaries .
The method attenuates all internat multiplas for a given order for all subsurface reflectors, one order at a time . The
feedback and inverse scattering methods could complement Bach other : the choice degending on the type of multiple
and the complexity of the interface and its overborden.
Separate field data tests of internat multiple attenuation using bots the feedback interface method and inverse scatter-
ing method are encouraging. Our fotore plans include field data comparisons of these methods .
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