1. science

Full PrestackVersusShot RecordMigration
s 15.7
C p A. Wapenaarand A. 1 Berkhout,Delf University
of 2chnology Netherlands
SUMMARY
is that true amplitude depth sections may be obtained
by shot record migration plus stacking, thus
avoiding cumbersomedata reordering at each depth stel
Full prc-stack migration can be replaced by shot
record migration plus stacking without loss of
This conclusion is particularly
accuracy. To prove this,
applications.
it is most convenient to
important for 3-D
describe forward seismic modeling in terms of matrix
Imultiplications.
Then full
pre-stack migration can be
MATRIXREPRESENTATION
OF WAVEPROPAGATION
described in terms of matrix inversions. By using some
We introduce a matrix representation of wave field
simple rules of matrix algebra it can then be shown
that the full
pre-stack migration result can be wrftte
as the sumof shot record migration results.
,conclusion
is particularly
This
important for the 3-D case
full
For convenience we consider the 2-D
chromatic one-way wave field
extrapolation
for the
following four cases.
Ibecause the data managementin 3-D shot record
Imigration
extrapolation.
case. With reference to Figure 1, we describe mono-
is dramatically more convenient than in 3-D
Forward extrapolatlon
pre-stack migration.
of downgoingwaves (Figure
la):
INTROOUCTION
P+(zi)=W+(zi,Lo)
In sefsmfc literature
,concerning
full
P+(Zo).
(la)
there is muchconfusion
the similarities
Forward extrapolation
and differences between
of upgoing waves (Figure
lb):
pre-stack migration (or double downwardcontinuat
ion migration) on one hand and migration per shot
Irecord
(or profile
Icommonly
accepted
Iof
that full
(lb)
iii.
Inverse extrapolation
of downgoingwaves (Figure
and accurately positioned --true amplitude imagl
complicated subsurface. Unfortunate-
lc):
pre-stack migration is not very attractive
P+(zo)= _F+(z,.z,) P+(q).
a geologically
ly, full
from a practical
I:umbersomedata
(2a)
point of view because it requires
iv.
reordering (from commonshot records
Inverse extrapolation
of upgoing waves (Figure
Id):
into commondetector records or vice versa) at each
,depth
step. Migration per shot record, followed by
P-h,).
stacking, is conmionlyconsidered as a practical
alternative
(
I:apacftfes
pzi,zo)
P-(z,).
(2b)
with good resolution and positioning
lere the data vector &z,,)
but with incorrect amplitude handling. For
Inany existing
shot record migration schemesthis is
Iproperly designed shot record migration schemeyields
4exactly the same results as a full
pre-stack migration
_
;cheme. The practical
contains the pressure
falues of a downgoing(t) or upgoing f-1 wave at all
indeed true. However, it is shownin this paper that a
4
P-h,).
pre-stack (depth) migratfoi
sefsmfc data is the proper way to obtain a well
Iresolved
I3f
P(z,)=y(2,,2,1
migration1 on the other hand. It is
significance of this observation
i
761
ateral
positions at depth level tn. The forward and
nverse extrapolation
matrfces$and
E+ contain
ireen's functions. We define a Green's matrix
~zo,zfl,
the e'th column of which contains the mono-
2
Prestsck vs. shot record migration
:hromatic
pressure
:he surface
zo,
response
at all
lateral
positions
at
due to a monopole in the subsurface
x 2.)
see also Figure 2.
l'
1 '
:he principle
of reciprocity
In this
matrix
at
reflector
the
With the Born-approximation
at zi.
would be a diagonal
representing
notation,
position
reads
matrix,
a reflection
at depth
each diagonal
coefficient
use of the Born approximation,
(3)
5
represents
there T denotes
hanged.
If
that
rows and columns should be inter-
we consider
an acoustic
loss-less,
variable
presenting
clocity,
constant density medium between z. and zi,
-:hen the forward wave field
extrapolation
operators
are
,elated
to this
Green's
matrix,
according
one frequency
upgoing wave at z.
(61,
component of the detected
each element
re-
position.
Base
at one detector
the forward
model of a seismic
line
reads
(4a)
Source matrix
waves for
the spatially
extrapolation
matrix
P,(z,)
(7)
Ind
'ield
general
vector
to
W+(‘*Zo)‘
j dz _C(Zi,Z’Z,)
'urthermore,
Finally,
(one shot record),
the response
on relation
we do not make
hence,in
a band-structure.
may have
element
at one lateral
In the following
zi.
it
band-limited
operators
inverse
are given
wave
source vectorSi(
detector
frequency
(5al
column
e(z,)
component of all
contains
which
matrix?-(zo)
seismic
dataset
shot
The m'th
We included
matrix
Finally,
downgoing source
records.
shot
optional
complete
all
contains
contains
patterns.
by
$(z,)
the different
an
represents
represents
detected
column
field
one
upgoing waves in a
(one seismic
line).
The m'th
Pi(zo).
record
Ind
Tm(Zi ,Zo) p
fhere
A series
on of these
ossy,
(5b)
I
denotes
*
:aken.
[~m(Zo,Zi)]sl=
dzs*(zi
,z=zo)
that
the complex conjugated
of papers
relations
arbitrarily
being prepared
for
dealing
with
acoustic
should
-FULL PRE-STACK MIGRATION
be
Based on forward
the generalizat-
or full
according
model (7)
we define
a matrix
gzj)
to
(8)
elastic,
inhomogeneous media is currently
(‘z,)l-‘.
by the authors.
Note that
:HE FORWARDMODEL OF A SEISMIC DATASET
(9a)
With the forward
the previous
single
extrapolation
section,
reflector
the
matrices,
(monochromatic)
Figure
response
at depth zi can be written
p~(zO)‘~-(zo~zi)
~(Zi)~+(Zi,Zo)
Ice alS0
defined
3. Here S,f[z,)
of a
as
source
at Zo'
iatrix
In case of a single
vector
gzi)
contains
one
describes
source at lateral
one non-zero
the reflectivity
the seismic
and detector
dataset,
properties.
deconvolved
for
element
[with
1'
deconvolved
position
zj.
only.
properties
the
Simllarly,
(9b)
Frequency component of the downgoing source wave field
cm this
represents
(6)
S:(Z,),
represents
in
Y$zj)
of
762
and
E-
defined
seismic
By substituting
consists
in
(511,
represents
the
datasct,
redatumed to depth level
relation
(7)
of two terms
into
(9)
we find
that
Prestack vs. shot record migration
3
I
$Zj)
=lJzj) +
response'
(10)
of the medium is defined
upgoing
and downgoing waves at zj.
vectors
Pi(zj)
for
m=l...M,
columns of matrix
hence,
the redatumed
reflectivity
matrix_R(zjl
tributions
other
for
thus
plus
contains
all
according
place
definitions,
at
term
practical
applications
recursively
elements
(9b)
are computed.
should be carried
in the following
records
ordering
records
for
deconvolving
coannon shot
common shot records
into
compensation
place
according
deconvolving
(14b)
where we assumed for
simplicity
Similarly,
to (5al
according
(12b),
records
Hence,
full
record
loss
the shot-
pre-stack
CONCLUSIONS
redatumed
data
we may write
for
plus
nigration
(profile
(13b)
may be replaced
@+(zo).
handled
stacking.
A similar
importance
(13c)
)e properly
Relation
(13a)
states
that
at zj
the
'spatial
(14a),
involves
pre-stack
763
by migration
followed
for
re-
proof
pre-stack
applied
without
loss
and phase may be
migration
can be given
in the 3-D case it
that
pre-stack
per shot record
by stacking,
by shot record
for
is of great
migration
per shot record,
:umbersome data handling.
impulse
without
the same remark is true
Hence, both amplitude
sorrectly
,articularly
and
~+(zj)‘~+(Zjszo)
(which
by (13),
per shot
imaging).
migration),
Jf accuracy.
(13a)
where
_P~(Zj)‘C~(z~)~-(z~~Zj)l~l~-(z~)
extrapolation
It has been shown for 2-D media that full
the
set
~(Zj)=,P~(Zj)[S+(zj)l-l,
S_+(Z,)=S(W)I_.
as described
by 'stacking'
Of course
migration
MIGRATION PER SHOT RECORD
(8)
according
(14c)
that
redatuming,
(14b,c), followed
datuming
to relation
simplicity
by wave field
of accuracy.
coordinates.
According
.
vectors
F+(Z,,Zj),
pre-stack
may be replaced
by spatial.
along
and (13~1,
to the source vectors,
where we assumed for
re-
downgoing propagat-
to relation
thatlJ!(z,)=D(w)l
IIslw)11*
upgoing
After
pm(zo)'
to
common detector
for
coimnon detector
-
lID(w)$
compensation
(13b), vectors
and
~-(zj,zo)
way
the detector-coordinates.
(x-XT),
ion takes
ly
by spatially
along
to (5b)
in Figure
to the measured shot records,
[z:~zjllT~~s~~zo~l*T
propagation
is visualized
EgzoH*T
P$zj)=
out
(12b)
describes
as
to
L+$,(zj) are related
(12a)
can be rewritten
relation
according
In
(12a)
Relation
the
With these
(14a)
this
are related
according
the diagonal
for
Note that,
q(tj)
(11)
Real
only
[St(zj)]-'.
(13a)
we define
which represents
ifjlI*..l,
4.
Optionally
matrix
relation
The 'proof'
=*
Futhermore,
m=I . ..M.
the
X(Zj)‘~p-(Z’)[Z:(zj)lT.
m=lm J
bandwidth,
to
<RJzjb
NOWwe define
which represent
I-_.
for
of
by summing
w in the seismic
the phase distorted
Z+,(zj)
rows of inverse
con-
matrices$zitj)
Imaging takes
frequencies
suppressing
vectors
the
phase-distorted
from the reflectivity
depth levels.
I
datasetz(zj)
as the ratio
(or
plus
3-D media.
practical
redatuming)
thus avoiding
can
4
Prestack vs. shot record migration
\\1=
1 !
I I
\-_I=
a-_
-
-Q_-_-
--F---
\
\
a
----0
a_-
-
-*_
-
-
--“;---
\\
=
\
\
---Q,
FIG. 1. Forward and inverse extrapolationof downgoingand
upgoing waves.
a--
-
_@--__
--a___
\
x
x
x
=
\
\
\
a‘
- _ -_
a---a__
--“;--
'i
-
(000----o)
000----o
ooo----0
I
\
-‘o
FIG. 2. The Pth column of the Green’s matrix.
i: :j
I
000----o
I
I
FIG.4. Visualizationof relation (14a), top, and (13a) bottom.
FIG. 3. Basic model for the seismic response from depth
level zi.
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