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

is to find analytic expressions for the travel time
in the Rocca operator.
This we do now.

\par
The Rocca operator $ \bold R = \bold C'_0 \bold C_h$
says to spray out an ellipse and then sum over a circle.
This approach,
associated with Gerry \bx{Gardner},
says that we are interested in all circles
that are inside and tangent to an ellipse,
since only the ones that are tangent
will have a constructive interference.

\par
The Gardner formulation answers this question:
Given a single nonzero offset impulse,
which events on the zero-offset section
will result in the same migrated subsurface picture?
Since we know the migration response
of a zero and nonzero offset impulses (circle and ellipse)
we can rephrase this question:
Given an ellipse corresponding
to a nonzero offset impulse,
what are the circles tangent to it
that have their centers at the earth's surface?
These circles if superposed will yield the ellipse.
Furthermore,
each of these circles corresponds
to an impulse on the zero-offset section.
The set of these impulses in the zero offset section
is the \bx{DMO}+NMO impulse response for a given nonzero offset event.

\plot{ell}{width=4.in,height=3.in}{
	The nonzero offset migration impulse response is an ellipse.
	This ellipse can be mapped as a superposition
	of tangential circles with centers along the survey line.
	These circles correspond to zero offset migration impulse responses.
	}
%\newslide

\subsection{Restatement of ellipse equations}
Recall equation~(\ref{eqn:canellipse})
for an ellipse centered at the origin.
\begin{equation}
        \label{eqn:ellipse}
	0 \eq {y^{2}\over{A^{2}}} + {z^{2}\over{B^{2}}} -1 .
\end{equation}
where
\begin{equation}
        \label{eqn:ellipseA}
	A \eq  \vhalf\, t_h ,
\end{equation}
\begin{equation}
        \label{eqn:ellipseB}
	B^2 \eq A^2 - h^2 .
\end{equation}

The ray goes from the shot at one focus of the ellipse
to anywhere on the ellipse,
and then to the receiver in traveltime $t_h$.
The equation for a circle of radius $R=t_0 \vhalf$
with center on the surface
at the source-receiver pair coordinate $x=b$ is
\begin{equation}
        \label{eqn:circle}
	R^2 \eq (y - b)^2 + z^{2} ,
\end{equation}
\noindent
where
\begin{equation}
	\label{eqn:radius}
        R \eq t_0 \, \vhalf .
\end{equation}

To get the circle and ellipse tangent to each other,
their slopes must match.
Implicit differentiation of equation (\ref{eqn:ellipse}) and (\ref{eqn:circle})
with respect to $y$ yields:
\begin{equation}
        \label{eqn:slope.ell}
	0 \eq {y \over{A^2}} + {z \over{B^2}}
	\ {dz \over dy}
\end{equation}
\begin{equation}
        \label{eqn:slope.cir}
	0 \eq (y-b) + z
	\ {dz \over dy}
\end{equation}
\noindent
Eliminating $dz/dy$ from equations~(\ref{eqn:slope.ell}) and (\ref{eqn:slope.cir}) yields:
\begin{equation}
        \label{eqn:b}
	y \eq {b\over 1 - {B^{2}\over{A^{2}}}}   .
\end{equation}
At the point of tangency the circle and the ellipse should coincide.
Thus we need to combine equations to eliminate $x$ and $z$.
We eliminate $z$ from equation (\ref{eqn:ellipse}) and (\ref{eqn:circle})
to get an equation only dependent on the $y$ variable. 
This $y$ variable can be eliminated by inserting equation~(\ref{eqn:b}).
\begin{equation}
        \label{eqn:geodmo}
	R^2 \eq B^2 \left( {A^2 - B^2 - b^2 \over{A^2 - B^2}} \right).
\end{equation}
\par
Substituting the definitions (\ref{eqn:ellipseA}), (\ref{eqn:ellipseB}), 
(\ref{eqn:radius}) of various parameter gives the
relation between zero-offset traveltime $t_0$ and nonzero traveltime
$t_h$:
\begin{equation}
        \label{eqn:opdmo}
	t_0^2\eq
	\left(t_h^2-{h^{2}\over\vhalf^2}\right)
	\left(1-{b^2\over h^2}\right).
\end{equation}

As with the Rocca operator, equation (\ref{eqn:opdmo})
includes both dip moveout \bx{DMO} and NMO.
%For a more careful derivation,
%please see Forel and Gardner's paper \shortcite{GEO53.05.06040610}.

\section{DMO IN THE PROCESSING FLOW}
\inputdir{matt}
\par
Instead of implementing equation (\ref{eqn:opdmo})
in one step we can split it into two steps. 
The first step converts raw data at time $t_h$
to NMOed data at time $t_n$.

\begin{equation}
	\label{eqn:nmoeq}
	t_n^2 \eq t_h^2 - {h^2 \over{\vhalf^2}}
\end{equation}

The second step is the \bx{DMO} step which
like Kirchhoff migration itself
is a convolution over the $x$-axis (or $b$-axis) with
\begin{equation}
	\label{eqn:dmoeq}
	t_0^2 \eq t_n^2 \left( 1 - {b^2 \over h^2} \right)
\end{equation}
and it converts time $t_n$ to time $t_0$.
Substituting (\ref{eqn:nmoeq}) into (\ref{eqn:dmoeq}) leads back to (\ref{eqn:opdmo}).
As equation (\ref{eqn:dmoeq}) clearly states,
the \bx{DMO} step itself is essentially velocity independent, 
but the NMO step naturally is not. 

Now the program.
Backsolving equation~(\ref{eqn:dmoeq}) for $t_n$ gives

\begin{equation}
        \label{eqn:dmoprog}
	t_n^2 \eq
	{t_0^2 \over 1-b^2/h^2 } .
\end{equation}

%Like subroutine~\texttt{flathyp()} \vpageref{/prog:flathyp},
%our \bx{DMO} subroutine~\texttt{dmokirch()} \vpageref{/prog:dmokirch} is based on
%subroutine~\texttt{kirchfast()} \vpageref{/prog:kirchfast}.
%It is just the same,
%except where {\tt kirchfast()} has a hyperbola
%we put equation~(\ref{eqn:dmoprog}).
%In the program,
%    the variable $t_0$ is called {\tt z}
%and the variable $t_n$ is called {\tt t}.
%Note, that the velocity {\tt velhalf} does exclusively
%occur in the break condition
%(which we have failed to derive,
%but which is where the circle and ellipse touch at $z=0$).
%\progdex{dmokirch}{fast Kirchhoff dip-moveout}

In figures ~\ref{fig:dmatt} and ~\ref{fig:coffs},
%were made with subroutine~\texttt{dmokirch()} \vpageref{/prog:dmokirch}. 
notice the big noise reduction over Figure~\ref{fig:frocca}.
\sideplot{dmatt}{width=3.in}{
	Impulse response of DMO and NMO
	}
%\newslide
\sideplot{coffs}{width=3.in}{
	Synthetic Cheop's pyramid
	}
%\newslide

\subsection{Residual NMO}
\sx{residual NMO}
Unfortunately, the theory above shows
that \bx{DMO} should be performed {\em  after} NMO.
\bx{DMO} is a convolutional operator,
and significantly more costly than NMO.
This is an annoyance because it would
be much nicer if it could be done
once and for all, and not need to be redone for each new NMO velocity.

%A deeper problem is that it is not obvious how
%we should process data with a depth variable velocity $v(z)$
%since the theory here
%seems so strongly dependent on mathematics of constant velocity media.
%As has often been demonstrated in practice,
%it is easy to do better with \bx{DMO} than without.
%We can imagine two approaches.
%\begin{enumerate}
%
%\item
%	\begin{enumerate}
%	\item
%		Do NMO with a regional constant velocity $\bar v$.
%	\item
%		Do DMO  with a regional constant velocity $\bar v$.
%		(Alternately, you could do NMO and DMO together).
%	\item
%		Do adjoint NMO with velocity $\bar v$.
%	\item
%		Treat the result as input to ``standard processing,''
%		i.e.~pick the velocity $v(\tau)$,
%		do moveout, stack, and do zero-offset migration.
%	\end{enumerate}
%
%\item
%	\begin{enumerate}
%	\item
%	Do NMO as always for the best velocity model $v(\tau)$.
%	\item
%	Then do DMO at some constant regional velocity  $\bar v$.
%	\item
%	Then do zero-offset migration using $v(\tau)$.
%	\end{enumerate}
%
%\end{enumerate}

\par
Much practical work is done with using constant velocity for the DMO process.
This is roughly valid since DMO, unlike NMO, does little
to the data so the error of using the wrong velocity is much less.

\par
It is not easy to find a theoretical impulse response
for the DMO operator in $v(z)$ media,
but you can easily compute the impulse response in $v(z)$
by using $\bold R=\bold H_0\bold H_h'$ from
equation (\ref{eqn:fourR}).

\subsection{Results of our DMO program}
\inputdir{krchdmo}
\par
We now return to the field data from the Gulf of Mexico,
which we have processed earlier
in this chapter and in chapter~\ref{vela/paper:vela}.  

%Figure~\ref{fig:cvdmostks} is a repeat of Figure~\ref{fig:cvstacks} except that
%the data has now been processed
%with \bx{DMO} prior to the final NMO and stack operation.
%Notice that the amount of conflict in velocity choice is greatly diminished,
%although not completely absent.
%\activeplot{cvdmostks}{width=6.0in}{ER}{
%	Stacks of Gulf of Mexico data
%	with two different constant NMO velocities.
%	This figure is the same as Figure~\protect\ref{fig:cvstacks}
%	except that dip-moveout has been applied to the data.
%	}\newslide

\plot{wgdmostk}{width=6.20in,height=7.8in}{
	Stack after the dip-moveout correction.
	Compare this result
	with Figure~\protect\ref{vela/fig:agcstack}.
	This one has fault plane reflections to the right of the faults.
	}
%\newslide

\plot{wgdmomig}{width=6.20in,height=7.8in}{
	Kirchhoff migration of the previous figure.
	Now the fault plane reflections jump to the fault.
	}
%\newslide

%HEY, MAYBE THEY WOULD LIKE TO SEE SOME CONSTANT OFFSET MIGRATIONS
%FOLLOWED BY STACK?
%I think I will postpone that until I
%can verify quality of existing results and
%get some timings of individual figures.

%\todo{	 % comment out Black DMO geometry and miscellaneous IEI stuff
%
%\section{THE END}
%The Figure~\FIG{dmospike1} demonstrates the processing 
%sequence. The left frame shows the input, a column of
%spikes in a constant-offset section. The middle frame
%shows the same constant-offset gather after Normal moveout with the 
%corresponding velocity $vhalf$. All the events are moved up in time.
%The rightmost frame displays the gather after DMO was applied. The
%Events are smeared into neighboring midpoints along the DMO smile.
%\activeplot{dmospike1}{height=3.0in}{}{
%	Illustration of the DMO impulse response.
%	The left panel is the common-offset input data
%	at an offset of $1.0$ km.
%	The center panel show the result of applying normal moveout
%	with a velocity of $2.0$ km/s.
%	The right panel is after the application of dip-moveout
%	with the program {\tt kdmoslow()}
%	}
%\subsection{Ottolini's radial trace dip moveout}
%\par
%Ordinarily we regard a common-midpoint gather as a collection of seismic traces,
%that is, a collection of time functions,
%each one for some particular offset  $h$.
%But this  $(h,t)$  data space could be represented
%in a different coordinate system.
%A system with some nice attributes is the radial-trace
%system introduced by Turhan Taner.
%In this system the traces are not taken at constant  $h$,
%but at constant angle.
%The idea is illustrated in Figure~\FIG{otto}.
%\activesideplot{otto}{height=2.5in}{}{
%	Inside the data volume of a reflection seismic traverse
%	are planes called {\it radial-trace sections.}
%	A point scatterer inside the earth puts a hyperbola on a
%	radial-trace section.
%	}
%\par
%Besides having some theoretical advantages, which will become apparent,
%this system also has some practical advantages, notably:
%(1) the traces neatly fill the space where data is nonzero;
%(2) the traces are close together at early times where wavelengths are short,
%and wider apart where wavelengths are long; and
%(3) the energy on a given trace tends to represent
%wave propagation at a fixed angle.
%The last characteristic is especially important
%with multiple reflections.
%But for our purposes the best attribute of radial traces is still 
%another one.
%\par
%Richard Ottolini noticed that a point scatterer in the earth
%appears on a radial-trace section as an
%{\it exact}
%hyperbola, not a flat-topped hyperboloid.
%The traveltime curve for a point scatterer, Cheops' pyramid,
%can be written as a ``string length'' equation,
%or a stretched-circle equation.
%Making the definition
%\begin{equation}
%\sin \, \psi  \eq  { 2\,h  \over  v\,t}
%\EQNLABEL{rickdef}
%\end{equation}
%and substituting
%it and equation~\EQN{mymajor} into
%equation~\EQN{torick} yields
%\begin{equation}
%v\,t  \eq 
%2\  \sqrt{ { z^2    \over  \cos^2 \psi } \ +\  (y\ -\ y_0 )^2 }
%\EQNLABEL{6.8}
%\end{equation}
%Scaling the $z$-axis by  $ \cos \, \psi $  gives the circle and