% Authors { Mihai Popovici and Biondo Biondi}
\lefthead{Popovici and Biondi}
\righthead{PSPM kinematics}
\footer{SEP 61}
\title{Kinematics of Prestack Partial Migration 
in a variable velocity medium}
\author{Alexander M. Popovici and Biondo Biondi}
\ABS{
Prestack partial migration (PSPM)
is a well known 
process for transforming to zero offset the prestack data,
when the velocity function
is slowly varying.
To adapt PSPM to media in which the velocity {\bf v(z)} is
a rapid function of {\bf z}  
the generalized PSPM impulse response is computed using raytracing
and is applied to the data as an integral operator.
The impulse response of the PSPM operator 
can be computed by considering the PSPM process
as the combination of
full prestack migration followed by
zero-offset modeling.
\parab
The kinematics of the proposed PSPM operator and the ones
of the constant velocity PSPM
differ significantly  
in some realistic cases, such as in presence of a low velocity layer or
when a hard sea-bottom causes
a jump in velocity.
}
\INT
Prestack data are usually transformed to zero offset by
the application of normal moveout (NMO) followed by dip moveout (DMO).
Different algorithms 
have been proposed 
to perform DMO assuming
a constant 
velocity medium
(Deregowski and Rocca, 1981; Hale, 1983; Jakubowitz, 1984;
Biondi and Ronen, 1987).  
These conventional DMO operators are
kinematically correct when velocity is constant and they are 
approximately correct when  velocity linearly increases
with depth (Hale, 1983),
but they cannot be applied when velocity varies rapidly.
In this case the prestack partial migration (PSPM) process
cannot even be split in NMO followed by DMO,
but must be applied as a single process.
\par
A PSPM algorithm that transforms the prestack data to zero offset, even
when velocity is a general 2-D function, is an attractive alternative
to full prestack migration
because it requires less computer resources.
In 2-D the spatial aperture of the PSPM operator in constant offset sections
is limited by the offset, 
while the corresponding full prestack migration operator
has unlimited aperture.
In 3-D the gain in efficiency is even more important; 
the impulse response of full prestack
migration is a a 3-D ellipsoid, while the impulse
response of the PSPM operator is a simple 2-D ellipse 
lying in a vertical plane passing through the source
and the receiver.
Another application where PSPM has potential
advantages over full prestack migration is velocity estimation.
The velocity function assumed for applying PSPM influences 
the alignment in time of the reflections over offset but not their
absolute position. On the contrary, the choice of the migration velocity 
influences the absolute position of the reflections (Fowler, 1988).
\par
The determination of the correct amplitudes is
important for many applications.
In particular when DMO is used for 
Amplitude Versus Offset (AVO) analysis of the prestack data
(Gardner and Forel, 1988; Black and Egan, 1988;
Beasley and Mobley, 1988).
\par
We propose in this paper 
an algorithm for computing the impulse 
response of the PSPM operator for any {\bf v(z)}  velocity model.
The method can be easily adapted for {\bf v(x,z)}.
We apply the proposed algorithm
for computing the kinematics of the impulse response
when velocity is constant and for different velocity functions of depth.
We show that the impulse response in variable velocity media
not only differs from 
the constant velocity one by a time shift but in certain
cases triplications appear in the impulse response.
The proposed algorithm can be applied to partially migrate
the prestack data using the impulse response as an integral
operator. 
\mhead {PRESTACK PARTIAL MIGRATION}
Prestack depth migration of reflectors is a two-step
process: focusing followed by mapping. 
In conventional processing the focusing is accomplished by
normal moveout (NMO) followed by stacking, while
the mapping is provided by poststack depth migration.
In presence of dipping reflectors the NMO step must be
 followed by dip-moveout to properly focus the 
reflections from dipping beds. 
Migrating the data in two steps has some advantages with
respect to migrating it in only one step.
One theoretical advantage is that focusing produces
a result in time and not in depth, therefore focusing
enhances the reflectors in a zero offset
section but does not affect their position.
This property of focusing satisfies both
the conservatives, who think that velocity cannot be
estimated well enough to rely on the result of 
mapping, and the anisotropic people,
who believe that horizontal (focusing) velocities and
vertical (mapping) velocities are different. 
Efficiency is the other important advantage in splitting
the migration process.
Only the stacked data is mapped to depth and the 
whole prestack data is focused instead of 
being fully migrated.

\shead {DMO}
DMO is much cheaper than full prestack migration because
the spatial aperture of the
DMO operator in constant offset
sections is limited by the offset, while the 
corresponding full prestack 
migration operator has unlimited width.
The gain in efficiency is important in a 2-D medium, 
but it is dramatic in a 3-D medium.
\par
DMO is usually applied as a velocity independent processing.
A velocity independent DMO is correct when the velocity is constant; 
it is approximately correct when the velocity varies with depth, 
but it produces wrong the results in the presence of strong
lateral velocity variations.
In these cases the standard processing of NMO followed by DMO
cannot be used because the moveouts in common midpoint gathers 
are not any more hyperbolic, but the focusing 
can still be accomplished in the single step 
procedure which is the prestack partial 
migration (Yilmaz and Claerbout, 1980).

\mhead {THE PSPM OPERATOR IN CONSTANT VELOCITY MEDIA}
Prestack partial migration is a mapping of prestack data
to zero-offset data. This mapping can be subdivided in two steps: 
\begin{enumerate}
  \item Full prestack migration to a depth model.
An input data point in a constant offset section, is migrated into
an ellipse.  In other words a single impulse on a trace can be generated
by a reflection in any point on the reflecting elliptic surface.
  \item Zero-offset modeling.
Each point on the ellipse can be modeled into a hyperbola, which
is obtained by considering the chosen point as a point diffractor.
\end{enumerate}
\par
As a result, an impulse in a constant offset section will be moved 
in a zero offset section accounting also for the effect of the
dipping reflector.
The element which fixes the position of the migrated point on the ellipse is 
the dipping angle $\alpha$.  The coordinates of the points on the ellipse are 
functions of the dip angle $\alpha$.

The mapping of the input point $(t_h,x_h)$
from a constant offset section into
the point $(t_0,x_0)$ in a zero-offset section can be expressed as 
$$ (t_h,x_h) = migration(\bf {v},h,\alpha) 
\Longrightarrow (z_m,x_m;\alpha) $$
$$ (z_m,x_m;\alpha) = modeling(\bf {v},\alpha) 
\Longrightarrow (t_0,x_0;\alpha). $$

\begin{figure}[t]
\vspace{4.in}
\caption{Mapping ellipse parametrized by the dip angle $\alpha$}
\end{figure}

Given the geometry in Figure 1, the coordinates of the point P can be 
expressed as a function of the semiaxes of 
the ellipse ({\bf a} and {\bf b} ) and the dip angle $\alpha$.
The semiaxis are {\bf a} $= t_hV $ and {\bf b} 
$=\sqrt{t^2_hV^2 - h^2}$, where V is half the earth velocity.
The equation of the ellipse is:

\begin{equation}
{x^2_m\over a^2}+{z^2_m\over b^2} = 1
\end{equation}
Which can be rewritten as

$$ z^2_ma^2 = a^2b^2 - x^2_mb^2 $$

The equation of the tangent to the ellipse is $z_m={\bf m}x_m+{\bf c}$ where 
${\bf{m}}=\tan\alpha$
and ${\bf c}=\sqrt{a^2\tan^2{\alpha} + b^2}$ . 

\begin{equation}
z_m = x_m \tan\alpha + \sqrt{a^2\tan^2{\alpha} + b^2}
\end{equation}

Squaring equation 2 and multiplying both sides with ${\bf a^2}$ we obtain:

$$z^2_ma^2 = a^2(x^2_m\tan^2{\alpha} + 2x_m\tan{\alpha}
\sqrt{a^2\tan^2{\alpha}+b^2}
+ a^2\tan^2{\alpha} + b^2)$$
As the intersection of the tangent to the ellipse should satisfy both
equation (1) and (2), by equalizing both sides of $z_m^2a^2$ we obtain:
\begin{equation}
x^2_m(b^2 + a^2\tan^2{\alpha}) + 2x_ma^2\tan{\alpha}\sqrt{a^2\tan^2{\alpha}+b^2}
+ a^4\tan^2{\alpha} + a^2b^2 = a^2b^2
\end{equation}

\begin{equation}
(x_m\sqrt{a^2\tan^2{\alpha}+b^2} + a^2\tan{\alpha})^2 = 0
\end{equation}

From the previous equation (4) we obtain the value for $x_m$, 
and introducing this value in equation 1, we obtain $z_m$.

\begin{equation}
x_m = -{a^2\tan{\alpha}\over \sqrt{a^2\tan^2{\alpha}+b^2} }
\end{equation}
\begin{equation}
z_m = {b^2 \over \sqrt{a^2\tan^2{\alpha}+b^2}}
\end{equation}

\begin{figure}
\vspace{3.in}
\caption{ (a) Modeling hyperbolas and (b) the PSPM operator}
\end{figure}

Each point on the ellipse corresponding to a 
certain parameter $\alpha$ (dipping angle), can be considered a point
diffractor and therefore generate a hyperbola.  
For a zero-offset section the point will be 
situated on the modeling hyperbola with the following coordinates:
\begin{equation}
t_0 = {z_m \over{V\cos{\alpha} }}
\end{equation}
\begin{equation}
x_0 = x_m + z_m\tan{\alpha}
\end{equation}

Inserting equations (5,6) in (7,8) we obtain:

\begin{equation}
t_0 = {b^2 \over V} {\sqrt{1\over{a^2\sin^2{\alpha}+b^2\cos^2{\alpha} }}}
\end{equation}

\begin{equation}
x_0 = {-h^2\sin{\alpha}\sqrt{1\over{a^2\sin^2{\alpha}+b^2\cos^2{\alpha} }}}
\end{equation}
which are the parametric equations (function of the dip angle $\alpha$)
of the well known DMO ellipse.

In the end the DMO operator contoured for only half space will look like
in Figure 3.  We demonstrate in Appendix A that the operator 
follows the envelope of the hyperbolas.

\mhead {PSPM IMPULSE RESPONSE IN MEDIA WITH VARIABLE VELOCITY}
The above formulation of the DMO operator allows us to
obtain the impulse response of the DMO in a medium with
variable velocity in depth.
The mappings can be evaluated using 
ray-tracing for an arbitrary velocity
model.  The migration depends on the half offset $h$
and thus must be computed for each common-offset section, while the
modeling is independent from the offset of the data and must be  
evaluated only once.
The algorithm used to investigate the DMO operator follows
the definition of the PSPM method and can be divided
in two parts:
\begin{enumerate}
  \item
  Construct the full migration depth model.  For a constant 
  velocity medium this is equivalent to constructing the 
  migration ellipse for a constant offset section.  For a 
  variable velocity medium, the loci of the points with 
  equal travel time from source to receiver is a curve resembling
  an ellipse or a superposition of several ellipses. See Figures
  ?? for an example of migration depth models.
  \item
  Zero-offset modeling.  Given the depth model (the reflector
  which generates the same impulse in the constant offset section)
  we raytrace back at 90 degree from the reflector, simulating
  the zero-offset case. The intersection of the ray with the 
  surface will give the x coordinate of the DMO operator, 
  while the travel time along the raypath will provide 
  the time coordinate.  See Figures ?? for an example of
  DMO impulse responses.
\end{enumerate}
The raytracing algorithm is based on solving the acoustic
wave equation in the high frequency approximation (eikonal
equation).  The velocity model has only a depth variation,
though the algorithm can be easily modified to handle lateral velocity
variation as well.  The velocity is assumed to be a continuous 
function of depth, and the smoothness of the variation can be
controlled by a B spline interpolation subroutine.


For a medium with linear increase of the velocity with depth 
$v(z)=v_0+az$,
an impulse will map in a elongated ellipse, the elongation being
proportional with the coefficient {\em a}.  The coefficient {\em a}
varies from 0.3 to 1.3 according to Telford.  The deviation from
the constant velocity case is negligible for an coefficient
{\em a} smaller then 1, when the DMO velocity is chosen correctly.
\begin{figure}
\vspace{3.in}
\caption{Reflector in medium with linear increase of the velocity}
\end{figure}
\begin{figure}
\vspace{3.in}
\caption{DMO operator in medium with linear increase of the velocity}
\end{figure}

\par
For media where velocity is not a linear function of depth, the
DMO impulse response may have serious variations from the constant
velocity case.  Triplications can occur in the normally 
ellipse-shaped DMO impulse response.  Figures ?? show an example of 
atypical DMO operators.

More here on DMO differences!!!


\mhead{Integral DMO with raytracing}

Describe here algorithm for DMO to real data

\par
One common objection to an exact PSPM is the presence of blind
zones when lateral velocity gradients are large (Deregowski, 1986; Hale, 1988).
A simple remedy of this problem that we would like to investigate 
is using PSPM for transforming data
not to zero-offset, but to a sufficiently large offset without blind zones,
or, if needed, to two different offsets.


\REF
\reference{Beasley, C. J., and Mobley, E., 1988, 
Amplitude and antialiasing treatment in (x,t) domain  DMO:
paper presented at the 58th Annual International SEG Meeting, Anaheim, CA.}
\reference{Black, J. L., and Egan, M. S., 1988, 
True amplitude DMO in 3-D:
paper presented at the 58th Annual International SEG Meeting, Anaheim, CA.}
\reference{Dellinger, J., and Muir, F., 1988, Migration = focus + map: 
SEP -- {\bf 57}, 307-312.}
\reference{Deregowski, S. M., and Rocca, F., 1981, 
Geometrical optics and wave theory of constant-offset sections 
in layered media: Geophysical Prospecting }
\reference{Deregowski, S. M.,
An integral implementation of dip moveout: 
paper presented at the 47th Annual International EAEG Meeting, Budapest,
Hungary.}
\reference{Deregowski, S. M., 1986, What is DMO?: 
First Break, {\bf 4}, 6-24.}
\reference{Etgen, J., 1988, 
Prestack depth migration velocity analysis: linear theory:
SEP -- {\bf 59}, 121-140.}
\reference{Fowler, P., 1988, Seismic velocity estimation using prestack time migration: Ph. D. dissertation, Stanford University.}
\reference{Gardner, G. H. F., and Forel, G., 1988, 
Amplitude preservation equations for DMO:
paper presented at the 58th Annual International SEG Meeting, Anaheim, CA.}
\reference{Gardner, G. H. F., and Uren, N. F., 1988, 
Dip moveout in anisotropic media:
paper presented at the 58th Annual International SEG Meeting, Anaheim, CA.}
\reference{Hale, I. D., 1988, Dip-moveout processing: 
Course notes from SEG continuing education course, SEG.}
\reference{Yilmaz, O., and Claerbout, J. F., 1980, 
Prestack partial migration:
Geophysics, {\bf 45}, 1753-1779.}



\newpage
\shead{Appendix A}
The impulse response of the DMO operator is the envelope to the 
family of hyperbolas obtained by considering each point on the 
migration ellipse, a point diffractor.  This is a rather peculiar 
property of the DMO operator and we determine here the 
equation of the envelope of the hyperbolas, which
coincides with the DMO operator. While the equation of the envelope is 
the equation of an ellipse with the horizontal semiaxis equal with
the offset, the impulse response of the DMO is only a part of the 
total ellipse, the maximum {\bf x} coordinate being $$x_{max}=h^2
\over a$$;  where {\bf h} is the offset and {\bf a} is the 
semiaxis of the full constant offset migration ellipse $$a= {\bf v}
t_h \over 2$$.

The equation of the family of hyperbolas in Fig.A.1 is:

\begin{figure}[b]
\vspace{3.in}
\caption{Envelope of the Hyperbolas}
\end{figure}

\begin{equation}
z^2 = z_m^2 + (x_m - x)^2
\end{equation}
The points of coordinates $(x_m,z_m)$ are situated on the ellipse, 
representing a parametrized form of the ellipse equation.

$$x_m = -{a^2\tan{\alpha}\over \sqrt{a^2\tan^2{\alpha}+b^2} }$$
$$z_m = {b^2\over\sqrt{a^2\tan^2{\alpha}+b^2}}$$

In expanded form the function $F(x,z,t)$ where $t=\tan\alpha$ will be:

\begin{equation}
F(x,z,t)=z_m^2+(x_m-x)^2-z^2= {b^4\over {a^2t^2+b^2}} + ({-a^2t \over 
\sqrt{a^2t^2+b^2}} - x)^2 -z^2 = 0
\end{equation}
The envelope of the family of curves represented in equation ?? is 
obtained by eliminating the parameter $\bf t$ from the two equations:
$$F(x,z,t)=0$$
$${\partial {F(x,z,t)} \over {t} } = 0$$
We have then the equations:

\begin{equation}
F(x,z,t)={ {b^4+a^4t^2} \over {a^2t^2+b^2} } + { {2a^2tx} \over 
\sqrt{a^2t^2+b^2} } + x^2 -z^2 = 0
\end{equation}

\begin{equation}
F'(x,z,t)={ {2a^2b^2t(a^2-b^2)} \over {(a^2t^2+b^2)^2} } +
{ {2a^2b^2x} \over {(a^2t^2+b^2)^{3 \over 2} } }
\end{equation}

$$F'(x,z,t)={ {2a^2b^2(t(a^2-b^2)+x \sqrt{a^2t^2+b^2})} \over 
{(a^2t^2+b^2)^2} } = 0 $$

Which gives for $\bf t$:
$$t(b^2-a^2)=x \sqrt{a^2t^2+b^2}$$
\begin{equation}
t^2={ {b^2x^2} \over {(b^2-a^2)^2-a^2x^2} }
\end{equation}

Introducing $\bf t$ in ?? we obtain:
\begin{equation}
z^2=b^2-{ {x^2b^2} \over {(b^2-a^2} }
\end{equation}

Because $\bf a$ and $\bf b$ represent the semiaxis of the ellipse, 
the offset $\bf h$ is given by $h^2 = a^2 - b^2$.  The equation 
?? can be rewritten as:
\begin{equation}
z^2=b^2-{ {x^2b^2} \over {h^2} }
\end{equation}
$$ { {z^2} \over {b^2} } + {{x^2} \over {h^2} } = 1$$
The equation of the envelope is the equation of an ellipse, with
the vertical semiaxis equal to $\bf b$ and the horizontal semiaxis
equal to the offset $\bf h$.
As the depth {\bf z} is given by $$z={\bf v} t_dmo$$ and the 
semiaxis {\bf b} is given by $$b={\bf v} t_n$$ where $$t_n$$ is
the time impusle after the NMO correction, we can rewrite the 
equation as
\begin{equation}
 { {{t_dmo}^2} \over {{t_n}^2} } + {{x^2} \over {h^2} } = 1
\end{equation}
which is the well known equation of the full DMO ellipse.

\end{document}

