CvE()PHYSl('S, VOL. 49, N0. 3 (MARCH l9R4|: P. 237-249, 14 FIGS.
`
`The migration of common midpoint slant stacks
`
`Richard Otto|ini* and Jon F. C|aerbout*
`
`ABSTRACT
`
`Reflection seismic data can be imaged by migrating
`common midpoint slant stacks. The basic method is to
`assemble slant stack sections from the slant stack of
`
`each common midpoint gather at the same ray parame-
`ter. Earlier investigators have described migration meth-
`ods for slant stacked shot profiles or common receiver
`gathers instead of common midpoint gathers. However.
`common midpoint slant stacks enjoy the practical ad-
`vantages of midpoint coordinates. In addition, the mi-
`gration equation makes no approximation for steep
`dips, wide offsets, or vertical velocity variations. A theo-
`retical disadvantage is that there is no exact treatment
`of lateral velocity variations.
`Slant stack migration is a method of “migration
`before stack." It solves the dip selectivity problem of
`conventional stacking, particularly when horizontal re-
`flectors intersect steep dipping reflectors. The correct
`handling of all dips also improves lateral resolution in
`the image. Slant stack migration provides a straightfor-
`ward method of measuring interval velocity after migra-
`tion has improved the seismic data.
`The kinematics (traveltime treatment) of slant stack
`migration is also accurate for postcritical reflections and
`refractions. These events transform into a p-‘E surface
`with the additional dimension of midpoint. The slant
`stack migration equation converts the [)-‘C surface into a
`depth-midpoint velocity surface. As with migration in
`general, the effects of dip are automatically accounted
`for during velocity inversion.
`
`INTRODUCTION
`
`The goal of migration processing is to obtain a picture of the
`reflectivity and velocity beneath the earth’s surface from seismic
`data recorded at
`the earth‘s surface. Specifically, migration
`converts the temporal pattern of reflections into the spatial
`geometry of reflectors. At the same time it focuses redundant
`and scattered reflections, thereby increasing signal over noise.
`These actions are interwoven with the estimation and use of a
`subsurface velocity model.
`
`Conventional migration processing includes common mid-
`point (CM P) stacking and migration of stacked sections. (The
`term common midpoint is a more accurate description than the
`more often used term common depth point, especially con-
`cerning dipping reflectors.) CMP stacking is considered a part
`of a migration processing because it behaves like a migration
`process. Previous authors (Rockwell. 1971; Stolt, 1978; Yilmaz
`and Claerbout, 1980) have mathematically demonstrated that
`CMP stacking is but a part of a more general migration process
`for imaging unstacked data.
`Conventional stacking has several problems. First, the hy-
`perbolic stacking trajectory is approximate for data sets where
`there are both dipping events and variable velocities (Brown,
`1969). Attempts to find exact and simple closed formed equa-
`tions for
`the stacking trajectory have not been successful.
`Second, the widely used hyperbolic formula contains the dip-
`dependent term v/cos 6 where 8 is the in-line dip. In regions
`where events of different dips intersect one must select one of
`the dips. Also,
`there is no exact migration equation for a
`conventional stack. Most poststack migration implementations
`assume that a conventional stack is a zero ofl‘set.
`
`Figure 1 illustrates the problems of migrating conventional
`stacks. Figure la is a migrated common depth stack of a data
`set from the Texas Gulf. The geology consists of growth faults
`dipping as steeply as 50 degrees. Figure 1b is an unmigrated
`constant-offset section from the same data set. It clearly shows
`fault plane refiections which were greatly attenuated during
`conventional stacking.
`One alternative is to restack the data for each of the dips
`present in the seismic data and then composite these into a
`single stacked section. For each stack, data outside of the dip
`range should be excluded. However, this solution is compli-
`cated and does not help the accuracy problem.
`Another alternative is to migrate unstacked data. There are
`several ways to do this. They may be classified by the coordi-
`nate system in which the data is processed.
`Field coordinate methods operate on seismic data that re-
`mains organized as it was collected: a set of shot profiles.
`Figure 2a shows this geometry in a three-dimensional (3-D)
`coordinate system of shot coordinate, geophone coordinate,
`and time. Although both the shot and geophone coordinates
`lay in the survey direction, plotting them as orthogonal coordi-
`nates is mathematically useful.
`
`Presented in part at the 48th Annual International SEG Meeting October 31, 1978, in San Francisco; also presented in part at the 1981 and 1982
`Fall Meetings of AGU, in San Francisco. Manuscript received by the Editor February 24, 1983; revised manuscript received August 8, 1983.
`‘Department of Geophysics. Stanford University, Stanford, CA 94305.
`Q 1984 Society of Exploration Geophysicists. All rights reserved.
`
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`
`

`
`238
`
`Ottolini and Claerbout
`
`Various authors have migrated field coordinate data all at
`once (Rockwell, 1971; Sattlegger and Stiller, 1974; Stolt, 1978),
`or as sets ofprofiles, alone (Jacobs, 1982), or combined with sets
`of common receiver gathers (Schultz and Sherwood, 1980).
`Field coordinate methods are accurate for steep dips, wide
`offsets, and depth and lateral variable velocity. Theoretically,
`processing data in their original coordinate system is straight-
`forward, especially when trying to deal with lateral velocity
`variations.
`
`The problems of field coordinate methods seem to be of a
`practical nature. Edge efiects tend to dominate the relatively
`narrow width of shot profiles or common receiver gathers.
`
`Reflections on shot profiles are often dip aliased, particularly
`shallow flat events at wide offsets. Dipping events complicate
`velocity estimation because the hyperbola apex is at nonzero
`ofTset.
`
`A more practical alternative to field coordinate methods is
`the migration of constant offset sections in midpoint coordi-
`nates. Figure 2b shows the relation of field coordinates to
`midpoint coordinates. The orthogonal axes of midpoint and
`offset are defined at a 45 degree angle to the field coordinate
`axes. Constant—offset sections run parallel to the midpoint axis.
`They are typically much wider than shot profiles.
`Midpoint coordinate methods include the migration of
`
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`FIG. 1. (a) Common midpoint stack migration of Texas Gulf Coast data set. (b) Unmigrated constant—offset section from the same
`data set. Dipping fault plane refieetions appearing on the unprocessed data have disappeared as a result of conventional processing.
`
`WesternGeco Ex. 1014, pg. 2
`
`

`
`Migratlon of Mldpolnl Slant Stacks
`
`constant—ofTset sections (Claerbout and Doherty, 1972; Dere-
`gowski and Rocca, 1981) or partial migration before stack
`(Yilmaz and Claerbout, 1980; Bolondi et al, 1982; Hale, 1983).
`Because constant-offset sections are longer than shot profiles,
`edge effects are minimized. The more geologically common flat
`events are not dip aliased in constant-offset sections as they can
`be on shot profiles. Velocity estimation is easier on CMP
`gathers, because the apex of a dipping reflector lies at zero
`offset when there are no lateral velocity variations. It is easier to
`interpret geology on constant—offset sections than on shot pro-
`files. The main difficulty with migrating constant-offset sections
`is in deriving a migration equation that is accurate for steep
`dips, wide offsets, and velocity variations (Deregowski and
`Rocca, 198]), though Hale (1983) has made good progress.
`This paper introduces the method of migrating common
`midpoint slant stack sections. A slant stack section is formed by
`slant stacking each CMP gather for the same slant angle, i.e.,
`ray parameter (Figure 3). Earlier investigators (Ryabinkin et al,
`1962; Taner, 1977; Phinney and Jurdy, 1979; Schultz and
`Claerbout, 1979; Garotta, 1980; Treitel et al, 1982, Zavalishen,
`1982) have described migration methods for slant stacked shot
`profiles or common receiver gathers instead of CMP gathers.
`However, CMP slant stacks enjoy the practical advantages of
`midpoint coordinates. In addition, a migration equation accu-
`rate for steep dips, wide ofl'sets, and any vertical velocity func-
`tion can be derived for slant stack sections. A theoretical disad-
`
`vantage is that there is no exact treatment of lateral velocity
`variations.
`
`SLANT STACKING
`
`Mathematical description of slant stacking
`
`Slant stacking may be thought ofas a two-step process. First,
`we apply a time shift to each trace of a CMP gather. The
`amount of the time shift is directly proportional to the offset.
`Second, we horizontally sum the time shifted traces across
`offset. This two-step process is equivalent to summing along
`tilted linear trajectories across the gather.
`The coordinate transformation for the time shift is defined by
`
`h’ = h
`
`t’ = z — Zph,
`
`(1)
`
`where h is half offset, t is traveltime, and p is the slope of the
`time shift. (A factor of 2 here simplifies the migration equa-
`tions.)
`Next we find the frequency domain equivalent of equation
`(1). We first note that the seismic wave field P is identical in
`either coordinate system,
`
`P(h. 1) = P(h’, I’).
`
`(2)
`
`Then we find the derivatives of one coordinate system in terms
`of the other:
`
`FIG. 2. (a) A seismic survey represented as a set of shot profiles
`in field coordinates. (b) The same seismic survey represented as
`a set of constant-offset sections in common midpoint coordi-
`nates. Note the relative dimensions of the planes of seismic data
`in each coordinate system.
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`In the frequency domain these relations are
`
`03’ = 0)
`
`k;, = k,, + Zpto,
`
`2F EQ
`
`where 0) is the traveltime wavenumber and k,, the offset wave-
`number.
`The second step to slant stacking is to sum across offset. In
`the frequency domain this is equivalent to selecting the zeroth
`
`FIG. 3. A CMP slant stack trace is formed by summing along
`inclined linear trajectories (dashed lines) across a CMP gather.
`Hyperbolic reflection moveouts from a depth increasing veloci-
`ty model are drawn in solid lines. The dark patches point out a
`typical Fresnel zone width.
`
`WesternGeco Ex. 1014, pg. 3
`
`

`
`Ottolini and Claerboul
`
`QLIQSED FIND TRUNCQTED GRTHER
`hog parameter
`
`(3)
`
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`U1
`
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`FIG. 4. (a) Slant stack of an aliased and truncated CMP gather. (b) Slant stack of the same gather with extrapolated offsets. All
`missing near ofisets were extrapolated and the offset density increased by a factor of four.
`
`kl, frequency:
`
`0 = k,, + 2pm.
`
`(5)
`
`In a slightly different form equation (5)
`frequency domain statement of Snell’s law
`
`is the well-known
`
`(Q
`
`This equation says that slant stacking decomposes the CMP
`gather into portions with the same ray parameter p. We use the
`term ray parameter in the mathematical sense that it
`is the
`invariant in Snell’s law and the time/offset slope of the data.
`However, the ray parameter concept does not represent ray-
`paths or wavefronts in CMP coordinates. In other words, one
`cannot design a field experiment to record a common midpoint
`slant stack.
`
`High-quality slant stack images
`
`We have found the construction of high—quality slant stack
`images to be the most crucial step in the slant stack migration
`
`process. Figure 4a is a slant stack of a CMP gather from the
`field data set of Figure 1 without any special pre-processing.
`Figure 4b is a much improved slant stack image of the same
`CMP gather using the methods described below. Before pro-
`cessing the entire data set, we study the quality of a slant stack
`image on a CMP gather or two. We slant stack for a large
`number of ray parameters and assemble them into a slant stack
`gather such as in Figure 4.
`The causes of the poor quality of Figure 4a are (1) too widely
`separated oflsets and (2) missing oflsets beyond the ends of the
`geophone cable. Most of the diagonal line artifacts are aliases
`of the desired slant stack arcs. On a slant stack section they
`may resemble horizontal reflectors. They occur where the offset
`separation is too large, that is, where the hyperbolic moveout
`relative to the slant stack trajectory is more than half the
`dominant wavelength. Missing near and wide offsets also cause
`diagonal artifacts,
`though fewer, on a slant stack. More
`seriously, they distort the event times at the highest and lowest
`ray parameters. The event arcs in Figure 4 should intersect the
`zero ray parameter axis orthogonally if the event times are
`correct. Bad event times result in poorer images and erroneous
`velocity estimates.
`
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`WesternGeco Ex. 1014, pg. 4
`
`

`
`Migration ol Midpoint Slant Stacks
`ERRTH MODEL FOR SYNYHEYICS
`l
`g.,...—l‘)..\.-4
`recei vet“
`
`midpoint
`l00
`
`iso
`
`—
`source
`
`VB,°c;,g
`1
`
`2
`
`FIG. 6. Earth model for synthetic data set. Velocity increases
`linearly with depth. Time scale units are unity and midpoint
`scale units are 0.5. 100 offsets were modeled at 1.8 units apart.
`Traveltimes were modeled based upon Slotnick (1962). There is
`no amplitude modeling.
`
`receivers. For a horizontal plane wave, CD is inverse velocity.
`For a set of sources and receivers everywhere on the surface of
`the earth along a straight line, equation (7) becomes
`
`2: —i9[,/1-(Y+H)2+./1—(Y—H)2]P.
`oz
`1:
`
`(3)
`
`Equation (8) was derived by Yilmaz and Claerbout (1980) and
`Stolt (1978). The notation is that of Yilmaz and Claerbout. This
`equation has been Fourier transformed over midpoint y, offset
`h, and time I. These coordinates are shown in Figure 5. Basi-
`cally, one square root is due to the sources and the other due to
`the receivers.
`
`To make the notation more compact, the terms Y and H
`represent the ratios
`
`_ vky
`Y——,
`203
`
`H _ uk,_
`_—-.
`2a)
`
`(9)
`
`These ratios have simple interpretations in terms of seismic
`data and earth angles. Y is the slope of data on a midpoint
`section and H is the slope on a CMP gather. In the earth Y is
`approximately the refiector dip and H is approximately the
`offset angle.
`Common midpoint slant stacks are an obvious way to im-
`plement equation (8). They directly decompose the seismic data
`into portions of fixed H. A migration equation is obtained by
`substituting the formula (6) for slant stacking into equation (8).
`
`-1" 51:3 [\/1 —(Y + pv)2 + \/1—(Y — pv)2]P.
`
`(10)
`
`We have made no assumptions about dip, offset, or vertical
`velocity variations in deriving this formula, so it is exact for
`migrating seismic data without
`lateral velocity variations.
`Equation (10) is easier to implement than the more general
`form of equation (8). k,, implies a coupling between offsets,
`making equation (8) a 3-D computation, whereas p is a fixed
`parameter, making equation (10) a 2-D computation.
`Equation (10) can be implemented in the computer using any
`of several migration algorithms such as the phase shift method,
`
`reflector
`
`FIG. 5. Relation of mathematical quantities to reflection seismic
`geometry. Midpoint is y and half-offset is h.
`
`We extrapolate the missing offsets in order to produce better
`slant stacks. We tested the remedies proposed by Schultz and
`Claerbout (1978) and Stoffa et al (1981), but found them inad-
`equate. Schultz and Claerbout mute out events with aliased
`moveouts. However, this method requires a velocity model and
`is dip selective. Stoffa et al weight the slant stack according to
`the semblence statistic along the same slant stack sum. How-
`ever,
`it didn’t work very well for the small-fold data set (24
`offsets) used in this paper. Another method is to collect data
`with better offset sampling. This can certainly help increase
`offset density, but not necessarily obtain the crucial near offsets.
`We contend that it is not necessary to resort to an acquisition
`based solution which is likely to be more costly than a com-
`puter based solution.
`We extrapolate the missing between, near, and wide offsets
`on a CMP gather before slant stacking. We use velocity insen-
`sitive extrapolation methods to avoid biasing the extrapolation.
`Hale (personal communication) suggested using linear interpo-
`lation titled at the approximate moveout slope to extrapolate
`between offsets. In practice, this method is not very velocity
`sensitive. We extrapolated 12.5 m separations between original
`50 m offset separations for the CMP gather of Figure 4. Clay-
`ton (personal communication) suggested using Burg’s method
`(1976) to extrapolate amplitude trends beyond the ends of
`geophone cables. The data are first Fourier transformed in
`time. This method works best on straight line events, so normal
`moveout is applied to the near offsets and an offset-squared
`time-squared stretch is applied to wide offsets.
`
`MIGRATION
`
`Migration equation for CMP slant stacks
`
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`Our imaging principle is that a subsurface image of the earth
`is what would be recorded at zero traveltime if the seismic
`sources and receivers were located directly above the reflector.
`However, we are really recording traveltimes at the surface of
`the earth. An equation to downward continue these traveltimes
`into the earth is ofthc form
`
`2F E0
`
`8P
`5}’ __
`EJDE.
`
`(7)
`
`The expression (1) contains the geometry of the sources and
`
`WesternGeco Ex. 1014, pg. 5
`
`

`
`242
`
`Ottolinl and Claerbout
`
`the wavenumber-frequency method,
`dilltaction summation,
`and finite-differences. We use the phase shift method (Gazdag,
`1978) for the examples in this article because it is the most
`accurate of the four in depth velocity media (see the Appendix).
`A finite-difference implementation (Claerbout, 1976; Jacobs,
`1982) would give control over lateral velocity variations with a
`loss of steep dip accuracy.
`
`Lateral velocity variations
`
`Although CMP slant stack migration was not designed for
`lateral velocity variations, there is strong reason to believe that
`the effect of lateral velocity variations is not as severe as for
`conventional processing. Conventional stacking averages later-
`al velocity variations within a geophone cable length. This
`leads to imaging and velocity estimation errors (Lynn and
`Claerbout, 1982). On the other hand, slant stacking averages
`lateral velocity variations within a smaller offset range. A slant
`
`stack selects data from a CMP gather at offsets where the linear
`stacking trajectory is tangent to the data (Figure 3). This state-
`ment
`is the physical
`interpretation of equation (5). Because
`seismic events are not impulses but possess a certain duration
`and frequency bandwidth, this tangcncy is smeared out across
`several ofisets. Called the Fresnel zone, this width extends to
`both sides ofthe tangency point where the event has curved less
`than halfa wavelength at the dominant frequency.
`
`Migration procedure and results
`
`There are two main steps in the migration procedure—
`construction of the slant stacks and their migration. Slant
`stacking details were discussed in a previous section. The role of
`velocity estimation is discussed in the next section.
`Figures 7 through 9 show various slant stack migration steps
`using the synthetic data set of Figure 6. The earth model
`consists of two flat reflectors and four diagonal reflectors dip-
`
`(p=0.0J
`
`midpoint
`
`[p=.36)
`
`midpoint
`100
`
`200
`
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`FIG. 7. Unmigrated slant stack sections of synthetic data. The ray parameter is given above each section. Notice the decreasing
`moveout of event A. Event B IS a truncation artifact.
`
`2F ED
`
`WesternGeco Ex. 1014, pg. 6
`
`

`
`Migration of Midpoint slant Stacks
`
`243
`
`ping up to 80 degrees. The velocity increases linearly by a factor
`of 3 from the top to the bottom of the section. This model
`contains more severe dip angle, offset angle, and depth velocity
`variations than found in typical seismic data. The traveltimes
`were computed analytically based upon Slotnick (1962).
`Figure 7 contains four unmigrated slant stack sections from
`this model. Notice that the traveltime of each event decreases as
`the ray parameter increases (events Al—A4). Note that
`the
`steepest dipping events disappear at high ray parameters. This
`is because the slopes of the moveout curves on the CMP
`gathers are relativity shallow for the steepest events.
`Figure 8 shows these CMP slant stack sections after migra-
`tion. These have not yet been converted to depth. Migration
`converts each slant stack section into an earth image. Therefore
`events moveout to their zero-ofl"set traveltimes (events Al—A4).
`Events B, C. and D are numerical artifacts. The real seismic
`
`events appear in the same locations on different ray parameter
`sections. while the artifacts appear in different places. There-
`fore, summing various migrated ray parameter sections will
`attenuate the artifacts and strengthen the actual events (Figure
`9).
`
`Figure 9 compares optimal, conventional. and CMP slant
`stack migration of the synthetic data set. Migrated time has
`been converted to depth in all of these examples. Figure 9a is
`the zero-offset section migration, the best possible migration for
`comparison purposes. Most seismic surveys are unable to
`record good zero-offset sections. Figure 9b is the result of
`conventional processing with missing near and wide offsets. It
`was tuned for flat dips, thereby degrading the steep dipping
`events. The low time reflections are missing because these only
`occur on the missing near ofi“sets. Figure 9c is the slant stack
`migration using all offsets. Figure 9d is a slant stack migration
`
`lp=0.0)
`
`midpoint
`
`(p=.36l
`
`midpoint
`
`(p=.72)
`
`midpoint
`
`(p=.l.08l
`
`midpoint
`
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`FIG. 8. Time migrations of the same sections from Figure 7. The migration implementation was the Fourier domain phase shift
`algorithm (Gazdag, 1978). Note that real reflection events have the same traveltime on each migrated ray parameter section. Events
`B are wraparound artifacts. Events C are midpoint truncation artifacts. Event D is an offset truncation artifact.
`
`2F E0
`
`WesternGeco Ex. 1014, pg. 7
`
`

`
`Otiolinl and Claerbout
`
`(H)
`
`ZERO OFFSET SECTION
`midpoint
`100
`
`200
`
`CHP STRCK [MISSING OFFSETS)
`midpoint
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`SLQNT smcx (FILL OFFSETSJ
`midpoint
`100
`
`SLQNT STFICK (MISSING OFFSETSJ
`midpoint
`100
`
`FIG. 9. Comparison of (a) optimal, (b) conventional, and (c,d) slant stack migration results of the synthetic data set. The convention-
`al stack is a sum of offsets 10 to 50 with no cosine velocity correction. The slant stack migrations are a sum of 50 slant stack
`separ.ations.wit.h.a. ray parameter. separ.at.ion.of. 0036 units.
`
`TEXIIIS GULF: SLFINT STQCK MICRFITION
`
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`
`5
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`8'13CV
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`2F5oD
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`FIG. 10. Slant stack migration of field data set from Figure 1. Fault plane reflections are now visible and lateral resolution
`improved. The migration is a sum of 20 ray parameter sections of ray parameters 0.015 + 0.004 x msec/m.
`
`WesternGeco Ex. 1014, pg. 8
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`

`
`Migration of Midpoint Slant Stacks
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`245
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`of the same offset range as in the conventional processing
`example. Both slant stack migrations