\title{Kinematic residual prestack migration}
\author{John T. Etgen}
\lefthead{Etgen}
\footer{SEP--61}
\righthead{Residual prestack migration}
\ABS{Using a kinematic approach, it is possible to derive a residual
migration operator that
converts a constant-offset section migrated with one slowness
to a constant-offset section migrated with another slowness. 
Residual constant-offset migration can be decomposed into
the following sequence of processes: 
1) residual common-reflection-point gathering or DMO,
2) residual NMO, and 3) residual zero-offset migration.
If the residual
zero-offset migration part of the operator is ``subtracted" from
residual constant-offset migration we are left with a 
residual NMO+DMO operator that corrects a migrated constant-offset section
for the residual NMO and residual common-reflection-point smear associated
with a change in migration slowness without performing residual zero-offset
migration. This ability to change the migration slowness of the data
without moving events around is useful for velocity analysis.
}
\INT
In a classic paper,
Rothman et al., (1985) described residual zero-offset migration
providing a method
for changing the migration velocity of a zero-offset section
without remigrating the data from scratch. 
Residual
zero-offset migration (also known as cascaded migration) 
proved to be very useful. Larner et al., (1986) 
used residual migration 
to enhance the accuracy of 15 degree
migrations extending them to higher dips. Beasley et al., (1988) use
cascaded migration to extend Stolt (constant velocity $f$--$k$)
migration to variable velocity media.
Residual migration seemed so useful,
that one would hope the concept of residual migration
would extend to prestack migration.
It might be useful for reducing sensitivity
to the velocity model used for prestack migration 
because errors in the migration could be corrected by
doing residual migration.
Residual migration might also be useful for migration velocity analysis
because many migrations could be obtained for little cost. It might
also be possible to do prestack depth migration by first performing
inexpensive constant-velocity migration, followed by
a residual migration to account for $v(x,z)$.
\par
Because of the potential rewards,
Fowler and Al-Yahya (1986) tried to formulate residual prestack migration. 
They concluded
that it is possible to construct an 
operator that converts constant-offset
or shot-profile
data, migrated with one velocity, to data migrated at another velocity. 
However,
the
operator is not equivalent to applying constant-offset 
migration or shot profile migration with some other velocity
which is the case for zero-offset residual migration. Because
the operator they obtained was not a ``migration" in the classical
sense, it seemed that residual prestack migration would not be as
useful as residual zero-offset migration.
\par
My velocity analysis project (Etgen, 1988, 1989) requires the ability to
find changes in migrated images due to perturbations
in the velocity model used for prestack depth migration.
In the past, approximated
residual migration to derive residual moveout corrections
that can be applied to migrated constant-offset sections.
The residual migration operators of this paper supercede the previous residual
moveout operators. 
To obtain an inexpensive residual migration
operator I make two approximations. First, I only use the
stationary phase path of the full residual migration operator.
Second, I build the operators only for the constant-velocity
case.  The residual migration is only
a residual time migration; ray bending is ignored
in the construction of the migration operator.
\par
Subtracting the kinematics of residual zero-offset migration from the
kinematics of residual
constant-offset migration, leaves an operator that
performs only residual DMO and residual NMO. This
converts migrated constant-offset sections to zero-offset
sections for a change in the migration velocity.
Note that if the velocity doesn't change, migrated constant-offset
sections are already converted to migrated zero-offset sections. 
Residual constant-offset NMO+DMO is useful
for velocity analysis. It provides a velocity analysis method
analogous to stacking velocity analysis,
that works in arbitrary structure and
after prestack depth migration.
%
%
\mhead{RESIDUAL CONSTANT-OFFSET MIGRATION}
Define residual constant-offset migration as the process that converts
a constant-offset section migrated with slowness $w$ to
a constant-offset section migrated with slowness $w_n$.
Building a kinematic residual constant-offset
migration operator simply requires finding where the
image of a point on a dipping reflector in a migrated constant-offset
section moves as the migration velocity changes.
Doing this for all points and all dips in the image
defines a kinematic
residual migration operator.  
A wave-theoretic residual migration operator
can be found by cascading constant-offset modeling at the
original migration slowness and constant-offset migration
with a new slowness. A kinematic development will
give a stationary phase approximation to this
wave theoretic residual migration operator. Since
my main application of this operator will be velocity
analysis, the errors of the stationary
phase
approximation should not be important. 
I further approximate, by building residual constant-offset migration
operators 
using
constant velocity prestack migration. Although
the kinematic arguments will be strictly valid only in
the constant slowness case, the residual migration operator
should still be useful, even when velocity varies. 
Constant-offset
depth migration followed by constant-velocity constant-offset residual 
migration will still give an approximation to constant-offset
depth migration with a new interval velocity model. The approximation
gets better as the amount residual migration gets smaller.
%
\par
The impulse response of constant-velocity
constant-offset
migration is an ellipse with foci at source and receiver. 
Each point on the impulse response ellipse represents
a different time dip in the data. 
A small segment of a dipping reflection event assumed
to contain only a single traveltime dip, after migration becomes 
the tangent to
the impulse response ellipse for the appropriate traveltime.
The traveltime determines the size of the ellipse, and the traveltime dip
determines the position of the image along the impulse
response ellipse.
Seen in reverse,
given an image point and its depth dip, we can solve for the position
of the source and receiver that have an impulse
response that goes through our point
and is tangent to the dipping reflector segment. 
\par
Starting at some point in depth with an assumed dip
and solving for the source and receiver points that contain a specular
reflection from the point,
we can write an equation in polar coordinates
that describes the migration ellipse that goes through the point.
%
$$
r(\theta , w)=
\sqrt{ {{\displaystyle t^2 \over \displaystyle 4 \displaystyle w^2}-h^2 
\over 1-{\displaystyle 4 \displaystyle w^2 \over \displaystyle t^2}
h^2 \cos \theta}}\ \ .
\eqno (1)$$
%
The origin is shifted to
be the center of the ellipse.
Equation (1) describes all image points that have specular reflections
recorded at this shot receiver pair. Specify a point of
interest as $r(\theta_0)$.
Write the traveltime of the reflection from the dipping
reflector 
as 
$$
t=w\biggl[ \sqrt{(x-h)^2+z^2}+\sqrt{(x+h)^2+z^2} \biggr] \ \ ,
\eqno (2)$$
where $x=r(\theta) \cos \theta $ and $z=r(\theta) \sin \theta $.
Also write the time dip of any reflection from a tangent to the ellipse as
$$
{\partial t \over \partial y}=w \biggl[{r(\theta) \cos \theta -h \over \sqrt
+{r(\theta) \cos \theta + h \over \sqrt
{r^2(\theta)+h^2-2r(\theta)h\cos \theta }}
{r^2(\theta)+h^2+2r(\theta)h \cos \theta}}\biggr] \ \ .
\eqno (3)$$
Figure 1 shows a selected point ``$P$" with its tangent impulse
response ellipse and specular reflection rays for a specified dip.
%
%
\plottop{}{3.1in}{/r2/john/figs/61/fig1}
{Choosing a depth point $P$ and a dip determines the shot-receiver pair
that contains the specular reflection from the point.}
%
%
\par
After choosing an initial depth and depth dip and solving for the
shot receiver pair,
equations (1)--(3) give the traveltime and 
traveltime dip of
the specular reflection from that point.
When the migration slowness changes,
the migration ellipse will
move to a new position.
The key to finding the position of the
reflector as the slowness changes is to remember that the traveltime,
midpoint, and traveltime dip of the reflector segment in
the constant-offset section is fixed.
Only its image in depth changes as the migration slowness
changes. 
First, find a new impulse response after the migration slowness
changes for the original
shot--receiver pair that has the same traveltime as
the previous impulse response at the old migration
velocity. Then find 
a point on the new impulse response that has the same
traveltime dip as the original point using the original migration slowness.
This procedure gives the new location and dip of the event.
Keeping $t$ fixed as the
traveltime of a reflection from our original point, 
the equation of the new impulse response ellipse at
slowness $w_n$ can be written as
$$
r(\theta , w_n)=
\sqrt{ {{\displaystyle t^2 \over \displaystyle 4 \displaystyle w_n^2}-h^2 
\over 1-{\displaystyle 4 \displaystyle w_n^2 \over \displaystyle t^2}
h^2 \cos \theta}} \ \ .
\eqno (4)$$
Identify $d=t/w$ as the shot to image point to geophone
distance of the reflector at the original migration slowness.
Also substitute $\gamma=w/w_n$ (Al-Yahya, 1987). 
$\gamma$ can be called 
the residual slowness or slowness scale factor.
When $\gamma=1$, residual migration leaves the image unchanged.
With these changes, rewrite equation 4 as:
$$
r(\theta , \gamma)=
\sqrt{ {{\displaystyle \gamma^2 \displaystyle d^2 \over \displaystyle 4 }-h^2 
\over 1-{\displaystyle 4 \over \displaystyle \gamma^2  \displaystyle d^2}
h^2 \cos \theta}} \ \ .
\eqno (5)$$
\par
The time dip imaged at any point on the new impulse response ellipse
is likewise given by:
$$
{\partial t \over \partial y}=w_n \biggl[{r \cos \theta -h \over \sqrt
{r^2+h^2-2rh\cos \theta }}+{r \cos \theta + h \over \sqrt
{r^2+h^2+2rh \cos \theta}}
\biggr]\ \ .
\eqno (6)$$
\par
Equating traveltime dip of the reflector at the
new and original slownesses and plugging in the
new relation for $r(\theta,\gamma)$ which equates the traveltimes 
gives
a single equation to solve for
$\theta_n$ as a function
of $\gamma$ and the position of the original point.
$$
\gamma \biggl[{r(\theta_0,d) \cos \theta_0 -h \over \sqrt
{r^2(\theta_0, d)+h^2-2r(\theta_0, d)h\cos \theta_0 }}
+{r(\theta_0, d) \cos \theta_0 + h \over \sqrt
{r^2(\theta_0, d)+h^2+2r(\theta_0, d)h \cos \theta_0}}
\biggr]
\eqno (7)$$
$$
=\biggl[{r(\theta_n , \gamma) \cos \theta_n -h \over \sqrt
{r^2(\theta_n , \gamma)+h^2-2r(\theta_n , \gamma)h\cos \theta_n }}
+{r(\theta_n , \gamma) \cos \theta_n + h \over \sqrt
{r^2(\theta_n , \gamma)+h^2+2r(\theta_n , \gamma)h \cos \theta_n}}
\biggr]\ .
$$
\par
Unfortunately, equation (7) is very difficult to solve analytically
except for the trivial cases of zero dip or zero offset.
To compute the operator I solve equation (7) numerically
using Newton's method for finding roots.
%
%
\plottop{}{3.1in}{/r2/john/figs/61/fig2}
{Traveltime, traveltime dip, and midpoint are fixed, so changing
the migration slowness moves the image of the
reflector from point $P$ to point $P'$.
}
%
%
Converting the solution of equation (7) back to Cartesian
coordinates gives $x(r,\theta_n)$ and $z(r,\theta_n)$,
the new position of the dipping reflector segment as shown in
Figure 2. 
\par
Solving equation (7) for an initial depth and all initial dips
will trace out the ``spraying" operator for residual constant-offset
migration. The operator traces out the new
positions of events for a range of dips 
after the migration slowness changes 
for a fixed original point in the image.
This curve
is also the summation path if we change the role of
starting and final points by redefining $\gamma=w_n/w$.
It is often more convenient to write the computer
code in terms of a summation operator. The
equations are symmetric so the summation operator for
$\gamma$ is the ``spraying" operator for $1/\gamma$.
%
%
\plottop{}{4.0in}{/r2/john/figs/61/fig3}
{Residual constant-offset (offset=1 000 m) migration operators for
a series of depth points. Slowness scale $\gamma$=1.2. }
%
%
Figure 3 shows an example of the summation operator when
the slowness increases. Depending
on the change in migration slowness and the initial depth
of the reflector, the operator can triplicate.
%
%
\plottop{}{4.0in}{/r2/john/figs/61/fig4}
{Residual constant-offset (offset=1 000 m) migration operators for
a series of depth points. Slowness scale $\gamma$=.8. }
%
%
This happens for image points
with large offset/depth ratio. Figure 4 shows an examples of
the residual migration operators for $\gamma$ less than
one, when slowness decreases. 
\par
To get the correct amplitudes along the summation operator,
compute the points on the summation operators in equal
dip angle increments in the original image. To prevent
operator aliasing I resample the summation operators in
equal arclength along the summation trajectory. To recover
equal dip weighting of the image, 
the amplitude along the summation path
can be taken as the jacobian of the change of variables from
arclength to dip angle.
%
%
\mhead{RESIDUAL NMO+DMO}
%
%
\plottop{}{4.5in}{/r2/john/figs/61/fig5}
{Events on three different constant-offset sections at a fixed original point 
$P_{orig}$ follow 
different trajectories when migration slowness changes.
The top points are the positions of the image points when
the slowness increases; the lower points are the positions
of the image points when slowness decreases. $P_0$ are
the positions of the zero-offset image; $P_h$ are the
positions of the image points for an offset one half
the offset of the points labeled $P_H$.}
%
%
Figure 5 shows the positions of a point ``$P_{orig}$" for several
different values of residual slowness $\gamma$ for one initial 
dip and 3 different
offsets. The new positions $P_0$, $P_h$, and $P_H$ 
for different offsets move along different trajectories as
migration slowness changes. If, for a fixed $\gamma$, the
new points
were above one another vertically, a residual NMO correction
could convert constant-offset sections to zero-offset
sections for a change in migration slowness. 
Residual constant-offset migration
would be equivalent to residual NMO followed by
residual zero-offset migration.
For zero dip, as expected this is the case. Residual
constant-offset migration is just residual NMO
followed by residual time to depth conversion. For
nonzero dip (as shown in Figure 5), the new positions
do not line up vertically. Furthermore, the amount
of the vertical movement as a function of offset
is not the same as it is for zero dip. Using analogy to full prestack
migration,
call residual DMO the movement required by
residual constant-offset migration and not described by residual NMO
or residual zero-offset migration.
Figure 6 shows how residual constant-offset migration
can be built from three processes: residual DMO, residual NMO,
and residual zero-offset migration.
%
%
\plottop{}{3.3in}{/r2/john/figs/61/fig6}
{Decomposing residual constant-offset migration (RCOM) into
residual zero-offset migration (RZOM), residual normal moveout (RNMO), and
residual dip moveout (RDMO).}
%
%
\par
The previous section gave the necessary
equations to compute residual constant-offset
migration operators; 
subtracting the residual zero-offset migration part 
of residual constant-offset migration,
obtained from the same equations,
gives residual NMO+DMO. Amplitudes of the residual NMO+DMO
operators are obtained with the same method used to obtain
the amplitudes of residual constant-offset migration operators.
Figure 7 shows the residual NMO+DMO operators for a series
of depth points for a large
offset for $\gamma=1.2$ and $\gamma=.833$.
%
%
\plottop{}{5.7in}{/r2/john/figs/61/fig7}
{Residual NMO+DMO operators for
a series of depth points; offset=$1 000 m$.
Residual slowness for the left plot $\gamma=1.2$.
Residual slowness for the right plot $\gamma$=.8. }
%
%
\par
Residual constant-offset migration has the desired property that
it performs residual common-reflection-point gathering.
For any residual slowness, any point on the
output images of different constant-offset sections all correspond to
the same reflection point. This is a necessary step for
correct migration
velocity analysis. Velocity analysis 
uses the traveltime of reflections
from a single point in the earth; thus, when performing
migration velocity analysis by comparing
traveltimes versus offset, the relevant
part of residual constant-offset migration is 
residual NMO+DMO. The residual zero-offset migration part
of residual constant-offset migration confuses velocity
analysis by moving the image of a fixed reflection event 
around the image as the migration slowness changes
(Fowler, 1988; Etgen, 1989).
It is preferable to ignore the residual zero-offset migration
term and keep a fixed reflection event at a fixed location
in the image. This is analogous to conventional velocity
analysis where NMO and stacking are performed at a
fixed $t_0$. The true-depth position of the reflector can be
calculated and stored, and residual zero-offset migration
can be applied later, after velocity analysis. The 
kinematic residual zero-offset migration is needed
by tomographic velocity analysis methods.
\mhead{EXAMPLES}
To test the residual constant-offset migration
and residual NMO+DMO operators, I made a synthetic
constant-offset survey over a series of point diffractors.
The constant-offset data were migrated with a slowness
equal to .8333 times the correct slowness.
I applied residual constant-offset migration at
$\gamma=1.2$ and successfully recovered the images of the point
diffractors as shown in Figure 8. I also applied residual
NMO+DMO to the migrated constant-offset section
to obtain the lower plot in Figure 9, which should
differ from the upper plot in Figure 8 by only
a residual zero-offset migration. The upper
plot in Figure 9 is the result of applying residual
zero-offset migration to the lower plot. As hoped,
the images of the point diffractors are recovered.
\par
There are some differences between the top plots
in Figures 8 and 9. The most important difference
is the reduction in amplitude of the top point
diffractor in the NMO+DMO+residual zero-offset migrated image.
Near the cusps of the operators (see Figure 7), 
the stationary phase approximation
breaks down because there is a discontinuity in the
derivative of the slope of the summation curve. The
kinematic summation operators have amplitudes
that drop to zero
immediately at the edge of the
cusp, while in reality, energy spreads out away from
the cusp. Wave-theoretic residual NMO+DMO would not have
this problem, but it is more expensive to apply.
\par
One advantage of residual constant-offset
migration over residual NMO+DMO followed by 
residual zero-offset migration
is that residual constant-offset migration
has cusps only for shallow depths and large
offsets. The residual NMO+DMO operator has cusps
for all offsets (except zero-offset) and all depths.
For large depth-offset ratios, the
cusps aren't a problem however, because the operator
effectively reduces to a point, namely residual NMO.
This is where residual NMO+DMO has a great advantage
over residual constant-offset migration. The size of the aperture 
of the residual constant-offset migration operator
grows without bound for increasing depths, while the residual
NMO+DMO operator shrinks to a point; so residual
NMO+DMO is much more economical to apply.
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig9}
{Point diffractor model. 
The lower plot is a migrated constant-offset section, migrated
with a slowness equal to .8333 times the correct slowness.
The upper plot shows the result of applying residual
constant-offset migration at $\gamma=1/.8333=1.2$.}
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig10}
{Point diffractor model.
The lower plot is the migrated constant-offset section of
Figure 8, after residual NMO+DMO applied with $\gamma=1.2$.
The upper plot shows the result of applying residual
zero-offset migration at $\gamma=1.2$ after residual NMO+DMO.
}
%
%
\par
To test the residual constant-offset migration and
residual NMO+DMO methods on 
real data and to see if residual NMO+DMO can be used
as a velocity analysis tool after migration, I migrated 180 constant-offset
sections from a survey in the Gulf of Mexico.
Figure 10 shows a constant-offset section from the
data; this offset was at about one-half the total cable length.
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig11}
{A constant-offset section from a Gulf of Mexico data set
used as input to constant-offset prestack migration.}
%
%
I chose
a velocity model that varies only in depth and made
the velocities increase with depth but purposefully
too slow. 
After prestack migration, 
I stacked together every 12 migrated constant-offset
sections to increase signal to noise, remove aliasing, and
reduce the data volume.
Figure 11 shows one partial stack of nearby migrated constant-offset
sections. 
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig12}
{Partial stack of 12 migrated constant-offset sections. The 
interval velocity model was purposely chosen to
have velocities too slow so events
are undermigrated.}
%
%
After migration and partial stacking, residual NMO+DMO was applied
to the migrated constant-offset sections for the range
$0.645 < \gamma < 1.15$. Figures 12, 13, and 14
show stacked sections after residual NMO+DMO. 
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig13}
{Stacked section after prestack migration and 
residual NMO+DMO using $\gamma=.745$.
Deeper events stack better because this $\gamma$ corresponds
to a large increase in the migration velocity.}
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig14}
{Stacked section after prestack migration and
residual NMO+DMO using $\gamma=.835$.
Events in the middle of the section 
stack better, corresponding to an intermediate increase in
migration velocity.} 
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig15}
{Stacked section after prestack migration and
residual NMO+DMO using $\gamma=.94$.
Events near the water bottom stack best at this value of $\gamma$,
corresponding to a small change in the migration
velocity.}
%
%
At the lower
values of $\gamma$, corresponding to the largest increase in velocity,
the deeper events stack well. At the higher values of $\gamma$,
closer to the original velocity, the shallower events stack well.
Note that events of all dips at a given depth stack well
at the same value of $\gamma$. Also note that events stay
fixed as their migration slowness changes. It is easier
to analyze velocities using residual NMO+DMO than by
scanning migration velocity with many full prestack migrations
which forces events to move as the migration slowness changes.
\par
If we wish to see the reflectors in their proper
spatial locations, we can apply residual zero-offset
migration to the stacks of Figures 12--14.
Figures 15--17 show the result of residual
zero-offset migration applied to each stack at its appropriate
value of $\gamma$.
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig18}
{Residual zero-offset migration applied at $\gamma=.745$ to the
residual NMO+DMO corrected stack of Figure 12. Only the
deepest part of the image is correctly migrated.}
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig17}
{Residual zero-offset migration applied at $\gamma=.835$ to the
residual NMO+DMO corrected stack of Figure 13. The middle
depths of the image are correctly migrated.}
%
%
\plot{}{8.0in}{/r2/john/figs/61/fig16}
{Residual zero-offset migration applied at $\gamma=.94$ to the
residual NMO+DMO corrected stack of Figure 14. The shallow
reflectors are correctly migrated.}
%
%
\par
Figure 18 shows sample velocity analysis panels.
Since residual NMO+DMO correctly accounts for
residual common-reflection-point smear, these
panels can be interpreted just as residual stacking
velocity panels are interpreted for horizontal
reflectors. Strictly speaking, these panels allow one
to pick the
best residual prestack time migration after
prestack depth migration. 
\par
I plan to use residual NMO+DMO
to analyze errors in interval velocity models used for prestack
depth migration. My previous papers (Etgen, 1988, 1989) describe
an operator that relates changes in a residual slowness
measure to changes in the interval velocity model. The
residual-slowness  semblance panels obtained using
residual NMO+DMO described in this paper are ideal
for measuring residual slowness
after prestack depth migration. Events don't
move as residual slowness changes, the points in the
images correspond to true common-reflection points
and the residual zero-offset migration can be calculated
and applied after velocity analysis.
%
%
\plot{}{5.0in}{/r2/john/figs/61/fig19}
{Residual NMO+DMO and stack semblance spectra for 3 midpoints.
The peaks of the stacking semblance tell what 
residual prestack time migration should
be applied after the original prestack depth migration to
account for traveltime versus offset at each depth point.}
%
%
\mhead{CONCLUSIONS}
Although the residual migration operators come 
from kinematic considerations, they work despite troubles
near their triplications. It is possible to build the 
operators more closely based on the wave equation, but
the resulting algorithm would resemble a phase shift migration
and would be costly to apply for a broad range of 
residual slownesses which is needed for velocity analysis. 
My application for
residual NMO+DMO is velocity analysis, and it is ideally
suited to this task. Residual NMO+DMO can find the
best residual prestack time migration slowness for fixed
events which is different from finding the best migration
slowness for a fixed location in the Earth.
It is not confused by events moving
around the image as migration slowness changes. 
When a high-quality final
wave-equation-migration output is needed, it is probably easier
to migrate the data from scratch with a new interval velocity model
or use wave-theoretic residual NMO+DMO.
\ACK
I thank Chevron for releasing the data set to SEP.
\REF
\reference{Al-Yahya, K., 1987, Velocity analysis by profile migration:
Ph.D. Thesis, Stanford University; also SEP--{\bf 53}, 11-30.}
%
\reference{Al-Yahya, K., and Fowler, P., 1986, Prestack residual migration:
SEP--{\bf 50}, 219--229.}
%
\reference{Beasley, C., Lynn, W., Larner, K., and Nguyen, H., 1988,
Cascaded $f-k$ migration: Removing the restrictions of depth-varying
velocity: Geophysics, {\bf 53}, 881--893.}
%
\reference{Etgen, J., 1989, Velocity analysis using prestack depth migration: 
applying the linear theory:
 SEP--{\bf 60}, 25--40.}
%
\reference{Etgen, J., 1988, Velocity analysis by prestack depth migration: 
linear theory revised:
 SEP--{\bf 59}, 121--140.}
%
\reference{Fowler, P., 1988, Seismic velocity estimation using prestack 
migration: Ph.D. Thesis, Stanford University; also SEP--{\bf 58}, 1--185. }
%
\reference{Larner, K., and Beasley, C., 1987, Cascaded migrations: 
improving the accuracy of finite-difference migration: Geophysics, {\bf 52},
618--643.}
%
\reference{Rothman, D.H., Levin, S.A., and Rocca, F., 1985, Residual migration:
applications and limitations: Geophysics, {\bf 50}, 110--126.}
