paper.tex

来自「国外免费地震资料处理软件包」· TEX 代码 · 共 1,753 行 · 第 1/5 页

that are missing in Figure~\ref{vela/fig:agcstack}.
These steep reflections appear only when the NMO velocity
is quite high compared with the velocity
that does a good job on the horizontal reflectors.
This phenomenon is consistent with 
the predictions of equation~(\ref{eqn:shuki}),
which says that dipping events 
require a {\em higher} NMO velocity than nearby horizontal events.
\plot{cvstacks}{width=6.0in}{
	Stacks of Gulf of Mexico data
	with two different constant NMO velocities.
	Press button to see a \bx{movie} in which each frame
	is a stack with a different constant velocity.
	}
%\newslide
%
\par
Another way of seeing the same conflict in the data
is to look at a velocity-analysis panel
at a single common-midpoint location
such as the panel shown in Figure~\ref{fig:velscan}
made by subroutine~\texttt{velsimp()} \vpageref{/prog:velsimp}.
In this figure it is easy to see that the velocity
which is good for the dipping event at 1.5 sec is too high
for the horizontal events in its vicinity.
%
\sideplot{velscan}{width=4.0in,height=3.3in}{
	Velocity analysis panel of one of the panels
	in Figure~\protect\ref{fig:cvstacks}
	before (left) and after (right) DMO.
	Before DMO, at 2.2 sec you can notice two values of slowness,
	the main branch at .5 sec/km, and another at .4 sec/km.
	The faster velocity $s=.4$ is a fault-plane reflection.
	}
%\newslide

\section{SHERWOOD'S DEVILISH}
\inputdir{.}
The migration process should be thought of as being interwoven with the
velocity estimation process.
J.W.C. \bx{Sherwood} [1976] indicated how the two processes,
migration and velocity estimation, should be interwoven.
The moveout correction should be considered in two parts,
one depending on offset, the NMO, and the other depending on dip.
This latter process was conceptually new.
Sherwood described the process as a kind of filtering,
but he did not provide implementation details.
He called his process
{\em  Devilish,}
an acronym for ``dipping-event velocity inequalities licked.''
The process was later described more functionally by \bx{Yilmaz} as
{\em 
\bx{prestack partial migration},
}
\sx{migration!prestack partial}
and now
the process is usually called
{\em 
dip moveout
}
(\bx{DMO})
although some call it MZO, migration to zero offset.
We will first see Sherwood's results,
then Rocca's conceptual model of the \bx{DMO} process,
and finally two conceptually distinct, quantitative specifications
of the process.
\par
Figure~\ref{fig:digicon} contains a panel from a stacked section.
\plot{digicon}{height=3.8in}{
	Conventional stacks with varying velocity.
	(distributed by Digicon, Inc.)
	}
%\newslide
The panel is shown several times;
each time the stacking velocity is different.  
It should be noted that at the low velocities,
the horizontal events dominate,
whereas at the high velocities,
the steeply dipping events dominate.
After the
{\em  Devilish}
correction was applied, the data was restacked as before.
Figure~\ref{fig:devlish} shows that the stacking velocity
no longer depends on the dip.
\plot{devlish}{height=3.8in}{
	{\em  Devilish} 
	stacks with varying velocity.
	(distributed by Digicon, Inc.)
	}
%\newslide
This means that after 
{\em  Devilish,}
the velocity may be determined without regard to dip.
In other words,
events with all dips contribute to the same consistent velocity
rather than each dipping event predicting a different velocity.
So the 
{\em  Devilish} 
process should provide better velocities for data with conflicting dips.
And we can expect a better final stack as well.

\section{ROCCA'S SMEAR OPERATOR}
\inputdir{matt}
Fabio \bx{Rocca} developed
a clear conceptual model for Sherwood's dip corrections.
Start with an impulse on a common offset section,
and migrate it getting ellipses like in Figure~\ref{fig:Cos1}.
%We did this with subroutine~\texttt{flathyp()} \vpageref{/prog:flathyp}
%using some constant-offset  {\tt h}.
Although the result is an ellipsoidal curve,
think of it as a row of many points along an ellipsoidal curve.
Then diffract the image
thus turning each of the many points into a hyperbola.
%We do this with the return path of the same subroutine {\tt flathyp()},
%however the path back is taken with {\tt h=0}.
The result is shown in Figure~\ref{fig:frocca}.
To enhance the appearance of the figure,
I injected an intermediate step of converting the ellipsoid
curve into a trajectory of dots on the ellipse.
Notice that the hyperbola tops
are not on the strong smear function that results
from the superposition.

\par
The strong smear function that you see in Figure~\ref{fig:frocca}
is Rocca's \bx{DMO}+NMO operator,
the operator that converts a point on a constant-offset section
to a zero-offset section.
The important feature of this operator is that
the bulk of the energy is in a much narrower region
than the big ellipse of migration.
The narrowness of the Rocca operator is important
since it means that energies will not move far,
so the operator will not have a drastic effect
and be unduly affected by remote data.
(Being a small operator also makes it cheaper to apply).
The little signals you see away from the central burst
in Figure~\ref{fig:frocca} result mainly from
my modulating the ellipse curve into a sequence of dots.
However, noises from sampling and nearest-neighbor interpolation
also yield a figure much like Figure~\ref{fig:frocca}.
This warrants a more careful theoretical study
to see how to represent the Rocca operator
directly (rather than as a sequence of two nearly opposite operators).
\sideplot{frocca}{width=3in,height=2in}{
	Rocca's prestack partial-migration operator is
	a superposition of hyperbolas, each with its top on an ellipse.
	}
%\newslide
\par
\inputdir{Math}
To get a sharper, more theoretical view of the Rocca operator,
Figure~\ref{fig:rocca2} shows line drawings of the curves in
a Rocca construction.
It happens, and we will later show,
that the Rocca operator lies along an ellipse
that passes through $\pm h$
(and hence is independent of velocity!)
Curiously,
we see something we could not see on Figure~\ref{fig:frocca},
that the Rocca curve ends part way up the ellipse
and it does not reach the surface.
The place where the Rocca operator ends
and the velocity independent ellipse continues is, however,
velocity dependent as we will see.
The Rocca operator is along the curve of osculation in Figure~\ref{fig:rocca2},
i.e.,~the smile-shaped curve where the hyperbolas reinforce one another.
\plot{rocca2}{width=5in}{
	Rocca's smile. (Ronen)
	}
%\newslide

\subsection{Push and pull}
\inputdir{matt}
%A program is a ``pull'' program if the loop creating the output
%covers each location in the output and gathers the input from wherever
%it may be.
%A program is a ``push'' program if it takes each input and
%pushes it to wherever it belongs.
%The moveout and stack and Kirchhoff migration programs
%we have been writing are
%``pull'' programs
%for doing the model building (data processing),
%and they are ``push'' programs for the data building.
%We could have written programs that worked the other way around,
%namely, a loop over $t$ with $z$ found
%by the semicircle calculation $z=\sqrt{t^2/v^2-x^2}$.

\par
Migration and diffraction operators can be conceived
and programmed in two different ways.
Let $\vec {\bold t}$ denote data and $\vec {\bold z}$ denote the depth image.
We have
\begin{eqnarray}
\vec {\bold z} &=& \bold C_h\ \vec {\bold t}
        \quad\quad {\rm spray\ or\ push\ an\ ellipse\ into\ the\ output}
 \\
\vec {\bold t} &=& \bold H_h\ \vec {\bold z}
\quad\quad {\rm spray\ or\ push\ a\ flattened\ hyperbola\ into\ the\ output}
\end{eqnarray}
where $h$ is half the shot-geophone offset.
The adjoints are
\begin{eqnarray}
\vec {\bold t} &=& \bold C_h' \; \vec {\bold z} \quad\quad
	{\rm sum\ or\ pull\ a\ semiCircle\ from\ the\ input}
	\\
\vec {\bold z} &=& \bold H_h' \; \vec {\bold t} \quad\quad
	{\rm sum\ or\ pull\ a\ flattened\ Hyperbola\ from\ the\ input}
\end{eqnarray}
In practice we can choose either of $\bold C \approx \bold H'$.
A natural question is which is more correct or better.
The question of ``more correct''
applies to modeling and is best answered by theoreticians
(who will find more than simply a hyperbola;
they will find its waveform including its amplitude and phase
as a function of frequency).
The question of ``better'' is something else.
An important practical issue is that the transformation
should not leave miscellaneous holes in the output.
It is typically desirable to write programs that
loop over all positions in the {\em  output} space,
``pulling'' in whatever inputs are required.
It is usually less desirable
to loop over all positions in the {\em  input} space,
``pushing'' or ``spraying'' each input value to the appropriate
location in the output space.
Programs that push the input data to the output space
might leave the output
too sparsely distributed.
Also, because of gridding, the output data might be irregularly positioned.
Thus, to produce smooth outputs, we usually
{\em  prefer the summation operators}
$\bold H'$ for migration and $\bold C'$ for diffraction modeling.
Since one could always force smooth outputs by lowpass filtering,
what we really seek is the highest possible resolution.

\par
Given a nonzero-offset section,
we seek to convert it to a zero-offset section.
Rocca's concept is to first migrate the constant offset data
with an ellipsoid push operator $\bold C_h$
and then take each point on the ellipsoid
and diffract it out to a zero-offset hyperbola
with a push operator $\bold H_0$.
The product of push operators
$\bold R = \bold H_0 \bold C_h$
is known as Rocca's smile.
This smile operator includes both normal moveout and dip moveout.
(We could say that dip moveout is defined by Rocca's smile
after restoring the normal moveout.)

\par
Because of the approximation $\bold H \approx \bold C'$,
we have four different ways to express the Rocca smile:
\begin{equation}
\bold R \eq
		 \bold H_0 \bold C_h
\quad\approx\quad
		 \bold H_0 \bold H'_h
\quad\approx\quad
		 \bold C'_0 \bold H'_h
\quad\approx\quad
		 \bold C'_0 \bold C_h
\label{eqn:fourR}
\end{equation}
$\bold H_0 \bold H'_h$ says sum over a flat-top and then spray
a regular hyperbola.
\par
The operator $\bold C'_0 \bold H'_h$,
having two pull operators should have smoothest output.
Sergey Fomel suggests an interesting illustration of it:
Its adjoint is two push operators,
$ \bold R' = \bold H_h \bold C_0$.
$ \bold R'$ takes us from zero offset to nonzero offset
first by pushing a data point to a semicircle
and then by pushing points on the semicircle to flat-topped hyperbolas.
As before, to make the hyperbolas more distinct,
I broke the circle into dots along the circle
and show the result in Figure \ref{fig:sergey}.
The whole truth is a little more complicated.
Subroutine %\texttt{flathyp()} \vpageref{/prog:flathyp} 
implements $\bold H$ and $\bold H'$.
Since I had no subroutine for $\bold C$,
figures \ref{fig:frocca} and \ref{fig:sergey}
were actually made with only $\bold H$ and $\bold H'$.
\sideplot{sergey}{width=3in,height=2in}{
	The adjoint of Rocca's smile
	is a superposition of flattened hyperbolas,
	each with its top on a circle.
        }
%\newslide
We discuss the
$\bold C'_0 \bold C_h$
representation of $\bold R$ in the next section.

\subsection{Dip moveout with $v(z)$}
\inputdir{yalei}
It is worth noticing that the concepts in this section
are not limited to constant velocity
but apply as well to $v(z)$.
However,
the circle operator $\bold C$ presents some difficulties.
Let us see why.
Starting from the Dix moveout approximation,
$t^2 = \tau^2 + x^2/v(\tau )^2$,
we can directly solve for $t(\tau ,x)$
but finding $\tau (t,x)$ is an iterative process at best.
Even worse, at wide offsets, hyperbolas cross one another
which means that $\tau (t,x)$ is multivalued.
The spray (push)
operators $\bold C$ and $\bold H$ loop over inputs
and compute the location of their outputs.
Thus
$ \vec {\bold z} = \bold C_h\ \vec {\bold t}$
requires we compute $\tau$ from $t$ so it is
one of the troublesome cases.
Likewise, the sum (pull)
operators $\bold C'$ and $\bold H'$ loop over outputs.
Thus
$ \vec {\bold t} = {\bold C'}_h\ \vec {\bold z}$
causes us the same trouble.
In both cases, the circle operator
turns out to be the troublesome one.
As a consequence, most practical work is done with
the hyperbola operator.
\par
A summary of the meaning of the Rocca smile
and its adjoint is found in Figures
\ref{fig:yalei2} and \ref{fig:yalei1}.
%which were computed using subroutine
%\texttt{flathyp()} \vpageref{/prog:flathyp}.

\plot{yalei2}{width=6.in,height=1.5in}{ 
	Impulses on a zero-offset section migrate
	to semicircles.
	The corresponding constant-offset section
	contains the adjoint of the Rocca smile.
	}
%\newslide

\plot{yalei1}{width=6.in,height=1.5in}{ 
	Impulses on a constant-offset section
	become ellipses in depth and
	Rocca smiles on the zero-offset section.
	}
%\newslide


\section{GARDNER'S SMEAR OPERATOR}
\inputdir{XFig}
A task, even in constant velocity media,