paper.tex

国外免费地震资料处理软件包

%\def\SEPCLASSLIB{../../../../sepclasslib}
\def\CAKEDIR{.}

\title{Zero-offset migration}
\author{Jon Claerbout}
\maketitle
%\chapter{Kirchhoff migration}
\label{paper:krch}

\sx{Kirchhoff migration}
\sx{migration!Kirchhoff}

\par
In chapter~\ref{vela/paper:vela} we discussed
methods of imaging horizontal reflectors
and of estimating velocity $v(z)$
from the offset dependence of seismic recordings.
In this chapter,
we turn our attention to imaging methods for {\em dipping} reflectors.
These imaging methods
are usually referred to as ``migration'' techniques.   

\par
Offset is a geometrical nuisance when reflectors have dip.
For this reason,
we develop migration methods here and in the next chapter
for forming images from hypothetical 
{\em zero-offset} seismic experiments.  
Although there is usually ample data recorded near zero-offset,
we never record purely zero-offset seismic data.
However, when we 
consider offset and dip together in chapter~\ref{dpmv/paper:dpmv} we will encounter
a widely-used technique (dip-moveout) that often 
converts finite-offset data into a useful estimate of the equivalent 
zero-offset data.
For this reason,
\bx{zero-offset migration}
methods are widely used today in industrial practice.
Furthermore the concepts of zero-offset migration
are the simplest starting point for approaching
the complications of finite-offset migration.

\section{MIGRATION DEFINED}
\par
The term ``migration'' probably got its name
from some association with movement.
A casual inspection of migrated and unmigrated sections
shows that migration causes many reflection events
to shift their positions.
These shifts are necessary because the {\em apparent} positions
of reflection events on unmigrated sections are generally not the {\em true}
positions of the reflectors in the earth.
It is not difficult to visualize why such ``acoustic illusions'' occur.
An analysis of a zero-offset section
shot above a dipping reflector illustrates most of the key concepts.

\subsection{A dipping reflector}
\inputdir{vplot}
\par
Consider the zero-offset seismic survey shown in Figure~\ref{fig:reflexpt}.
This survey  uses one source-receiver pair,
and the receiver is always at the same location as the source.
At each position, denoted by 
$S_1, S_2, \mbox{and} S_3$ in the figure,
the source emits waves and the receiver records the echoes
as a single seismic trace.
After each trace is recorded,
the source-receiver pair is moved a small distance
and the experiment is repeated.
%
\sideplot{reflexpt}{width=3.5in}{
  Raypaths and wavefronts for a zero-offset seismic line shot
  above a dipping reflector.
  The earth's propagation velocity is constant.
}
%
\par
As shown in the figure,
the source at $S_2$ emits a spherically-spreading wave
that bounces off the reflector
and then returns to the receiver at $S_2$.
The raypaths drawn between $S_i$ and $R_i$ are orthogonal
to the reflector and hence are called {\em normal rays}.
\sx{normal rays}
\sx{rays, normal}
These rays reveal how the zero-offset section misrepresents the truth. 
For example,
the trace recorded at $S_2$ is dominated by the reflectivity 
near reflection point $R_2$,
where the normal ray from $S_2$ hits the reflector.   
If the zero-offset section corresponding to Figure~\ref{fig:reflexpt} is displayed, 
the reflectivity at $R_2$ will be falsely displayed
as though it were directly beneath $S_2$,
which it certainly is not.
This lateral mispositioning is the first part of the illusion.
The second part is vertical:
if converted to depth,
the zero-offset section will show $R_2$ to be deeper than it really is.
The reason is that the slant path of the normal ray
is longer than a vertical shaft drilled from the surface down to $R_2$.

\subsection{Dipping-reflector shifts}
\par
A little geometry gives simple expressions
for the horizontal and vertical position errors on the zero-offset section,
which are to be corrected by migration.
Figure~\ref{fig:reflkine} defines the
required quantities for a reflection event recorded at $S$ corresponding
to the reflectivity at $R$.  
%
\sideplot{reflkine}{width=3.5in}{
  Geometry of the normal ray of length $d$ and the vertical ``shaft'' 
  of length $z$ for
  a zero-offset experiment above a dipping reflector.
}
%
The two-way travel time for the event is
related to the length $d$ of the normal ray by
\begin{equation}
t \eq {2\,d \over v}
\label{eqn:dipt0} \ \ ,
\end{equation}
where $v$ is the constant propagation velocity.
Geometry of the triangle $CRS$ shows
that the true depth of the reflector at $R$ is given by
\begin{equation}
z \eq d\ \cos\theta \ \ ,
\label{eqn:dipz}
\end{equation}
and the lateral shift between true position $C$ and false position $S$
is given by
\begin{equation}
\Delta x \eq d\ \sin\theta  \eq {v\,t \over 2}\ \sin\theta \ \ .
\label{eqn:dipdx}
\end{equation}
It is conventional to rewrite equation~(\ref{eqn:dipz}) in terms of two-way
{\em vertical} traveltime $\tau$:
\begin{equation}
\tau \eq {2\,z \over v} \eq t\, \cos\theta  \ \ .
\label{eqn:diptau}
\end{equation}
Thus both the vertical shift $t - \tau$ and the horizontal shift $\Delta x$
are seen to vanish when the dip angle $\theta$ is zero.  

\subsection{Hand migration}
\sx{hand migration}
\sx{migration!hand}
\par
Geophysicists recognized the need to correct these positioning errors
on zero-offset sections long before it was practical to use computers
to make the corrections.
Thus a number of hand-migration techniques arose.
It is instructive to see how one such scheme works.  
Equations~(\ref{eqn:dipdx}) and (\ref{eqn:diptau}) require knowledge of three quantities:
$t$, $v$, and $\theta$.
Of these, the event time $t$ is readily measured on the zero-offset section.
The velocity $v$ is usually {\em not} measurable on the zero offset section
and must be estimated from finite-offset data,
as was shown in chapter~\ref{vela/paper:vela}.
That leaves the dip angle $\theta$.
This can be related to the reflection slope $p$ of the observed event,
which is measurable on the zero-offset section: 
\begin{equation}
p_0 \eq {\partial t \over \partial y}  \ \ ,
\label{eqn:pdefn}
\end{equation}
where $y$ (the midpoint coordinate) is the location of the 
source-receiver pair.
The slope $p_0$ is sometimes called the {\em ``time-dip of the event''} or 
\sx{time dip}
more loosely 
as the {\em ``dip of the event''}.
It is obviously closely related to Snell's parameter,
which we discussed in chapter~\ref{wvs/paper:wvs}.   
The relationship between the measurable time-dip $p_0$ 
and the dip angle $\theta$ is called
``\bx{Tuchel's law}'':
\begin{equation}
\sin\theta \eq {v\,p_0 \over 2}   \ \ .
\label{eqn:tuchel}
\end{equation}
This equation is clearly just another version
of equation~(\ref{wvs/eqn:5iei4a}),
in which a factor of $2$ has been inserted to account for the 
two-way traveltime
of the zero-offset section.

%

\par
Rewriting the migration shift equations in terms of the measurable
quantities $t$ and $p$ yields usable ``hand-migration'' formulas:
\begin{eqnarray}
\Delta x &\eq& {v^2\ p\ t \over 4}
\label{eqn:laydxp}
\\
\tau &\eq& t\ \sqrt{1\ -\ {v^2 p^2 \over 4} }  \ \  .
\label{eqn:laytaup}
\end{eqnarray}
Hand migration divides each observed reflection event
into a set of small segments for which $p$ has been measured.
This is necessary because $p$ is generally 
not constant along real seismic events.
But we can consider more general
events to be the union of a large number of very small dipping reflectors.
Each such segment is then mapped from its unmigrated $(y,t)$ location
to its migrated $(y,\tau)$ location based on the equations above.  
Such a procedure is sometimes also known as ``map migration.''  

\par
Equations~(\ref{eqn:laydxp}) and (\ref{eqn:laytaup}) 
are useful for giving an idea of what goes on in zero-offset migration.
But using these equations directly for practical seismic migration 
can be tedious and error-prone 
because of the need to provide the time dip $p$ as a separate
set of input data values as a function of $y$ and $t$.  
One nasty complication is that it is quite common
to see {\em crossing events} on zero-offset sections.
This happens whenever reflection energy coming from two different reflectors
arrives at a receiver at the same time.  
When this happens the time dip $p$
becomes a {\em multi-valued} function of the $(y,t)$ coordinates.
Furthermore, the recorded wavefield is now the sum of two different events.
It is then difficult to figure out which part of summed amplitude
to move in one direction and which part to move in the other direction.

\par
For the above reasons,
the seismic industry has generally turned away from hand-migration techniques
in favor of more automatic methods.
These methods require as inputs nothing more than 
\begin{itemize}
\item The zero-offset section 
\item The velocity $v$  \ \ .
\end{itemize}
There is no need to separately estimate a $p(y,t)$ field.
The automatic migration program somehow ``figures out''
which way to move the events,
even if they cross one another.
Such automatic methods are generally referred to as
{\em ``wave-equation migration''}  techniques,
and are the subject of the remainder of this chapter.
But before we introduce the automatic migration methods,
we need to introduce one additional concept
that greatly simplifies the migration of zero-offset sections.

\subsection{A powerful analogy}
\inputdir{XFig}
\par
Figure~\ref{fig:expref} shows two wave-propagation situations. %
\plot{expref}{width=5.0in}{
  Echoes collected with a source-receiver pair
  moved to all points on the earth's surface (left)
  and the ``exploding-reflectors'' conceptual model (right).
}%
The first is realistic field sounding.
The second is a thought experiment in which
the reflectors in the earth suddenly explode.
Waves from the hypothetical explosion propagate
up to the earth's surface where they are
observed by a hypothetical string of geophones.
\par
Notice in the figure that the ray paths in the
field-recording case seem to be the
same as those in the \bxbx{exploding-reflector}{exploding reflector} case.
It is a great conceptual advantage to imagine that the two wavefields,
the observed and the hypothetical, are indeed the same.
If they are the same,
the many thousands of experiments
that have really been done can be ignored,
and attention can be focused on the one hypothetical experiment.
One obvious difference between the two cases is that in the
field geometry waves must first go down
and then return upward along the same path,
whereas in the hypothetical experiment they just go up.
Travel time in field experiments could be divided by two.
In practice, the data of the field experiments (two-way time)
is analyzed assuming the sound velocity to be half its true value.

\subsection{Limitations of the exploding-reflector concept}
\par
The exploding-reflector concept is a powerful and fortunate analogy.
It enables us to think of the data of many experiments
as though it were a single experiment.
Unfortunately,
the exploding-reflector concept has a serious shortcoming.
No one has yet figured out how to extend the concept
to apply to data recorded at nonzero offset.
Furthermore, most data is recorded at rather large offsets.
In a modern marine prospecting survey,
there is not one hydrophone,
but hundreds, which are strung out in a cable towed behind the ship.
The recording cable
is typically 2-3 kilometers long.
Drilling may be about 3 kilometers deep.
So in practice the angles are big.
Therein lie both new problems and new opportunities,
none of which will be considered until 
chapter~\ref{dpmv/paper:dpmv}.
\par
Furthermore, even at zero offset,
the exploding-reflector concept is not quantitatively correct.
For the moment, note three obvious failings:
First, Figure~\ref{fig:fail} shows rays that are not predicted
by the exploding-reflector model.
\plot{fail}{}{
  Two rays, not predicted by the exploding-reflector model,
  that would nevertheless be found on a zero-offset section.
}
%was \activeplot{fail}{height=1.6in}{NR}{}
These rays will be present in a zero-offset section.
Lateral velocity variation is required for this
situation to exist.  
\par
Second, the exploding-reflector concept fails with \bx{multiple reflection}s.
For a flat sea floor with a two-way travel time  $t_1$, multiple reflections
are predicted at times  $2t_1$,  3$t_1$,  4$t_1$, etc.
In the exploding-reflector geometry the first multiple
goes from reflector to surface,
then from surface to reflector,
then from reflector to surface, for a total time  3$t_1$.
Subsequent multiples occur at times  5$t_1$,  7$t_1$, etc.
Clearly the multiple reflections generated on the zero-offset section
differ from those of the exploding-reflector model.
\par
The third failing of the exploding-reflector model
is where we are able to see waves
bounced from both sides of an interface.
The exploding-reflector model predicts