paper.tex

国外免费地震资料处理软件包

The collection of all such traces is called the
{\em 
CDP-stacked section.
}
In practice the CDP-stacked section is often interpreted and migrated
as though it were a zero-offset section.
In this chapter we will learn to avoid this popular,
oversimplified assumption.
\par
The next simplest environment is to have a planar reflector
that is oriented vertically rather than horizontally.
This might not seem typical,
but the essential feature (that the rays run horizontally)
really is common in practice
(see for example Figure~\ref{fig:shell}.)
Also, the effect of dip, while generally complicated,
becomes particularly simple in the extreme case.
If you wish to avoid thinking of 
wave propagation along the air-earth interface
you can take the reflector to be inclined a slight angle from the vertical,
as in Figure~\ref{fig:vertlay}.
\plot{vertlay}{height=2.5in}{
	Near-vertical reflector, a gather, and a section.
	}
%\newslide
\par
Figure~\ref{fig:vertlay} shows that the travel time
does not change as the offset changes.
It may seem paradoxical that the travel time does not increase as
the shot and geophone get further apart.
The key to the paradox is that midpoint is held constant, not shotpoint.
As offset increases,
the shot gets further from the reflector
while the geophone gets closer.
Time lost on one path is gained on the other.
\par
A planar reflector may have any dip between horizontal and vertical.
Then the common-midpoint gather lies between
the common-midpoint gather of Figure~\ref{fig:simple}
and that of Figure~\ref{fig:vertlay}.
The zero-offset section in Figure~\ref{fig:vertlay} is a straight line,
which turns out to be the asymptote of a family of hyperbolas.
The slope of the asymptote is the inverse of the velocity  $v_1$.

\par
It is interesting to notice that at small dips,
information about the earth velocity is essentially carried
on the offset axis whereas at large dips,
the velocity information is essentially on the midpoint axis.

\subsection{The response of two points}
\par
Another simple geometry is a reflecting point within the earth.
A wave incident on the point from any direction
reflects waves in all directions.
This geometry is particularly important because
any model is a superposition of such point scatterers.
Figure~\ref{fig:twopoint} shows an example.
\plot{twopoint}{height=2.4in}{
	Response of two point scatterers.
	Note the flat spots.
	}
%\newslide
The curves in Figure~\ref{fig:twopoint}
include flat spots for the same reasons
that some of the curves in Figures~\ref{fig:simple} and \ref{fig:vertlay}
were straight lines.

\inputdir{.}
\plot{shell}{height=6.5in}{
%	TO SEE THIS FIGURE, RENAME IT shell. (instead of unshell)
	Undocumented data from a recruitment brochure.
	This data may be assumed to be of textbook quality.
	The speed of sound in water is about 1500 m/sec.
	Identify the events at A, B, and C.
	Is this a common-shotpoint gather or a
	common-midpoint gather?  (Shell Oil Company)
	}
%\newslide

\subsection{The dipping bed}
\sx{dipping bed}
\inputdir{XFig}
\par
While the traveltime curves resulting from a dipping bed are simple,
they are not simple to derive.
Before the derivation, the result will be stated:
for a bed dipping at angle  $\alpha$  from the horizontal,
the traveltime curve is
\begin{equation}
t^2 \, v^2  \eq 
4 \, (y - y_0 )^2 \, \sin^2  \alpha
\ +\ 
4 \, h^2 \, \cos^2  \alpha
\label{eqn:shuki}
\end{equation}
For $\alpha \,=\,$ 45$^\circ$,
equation (\ref{eqn:shuki}) is the familiar Pythagoras cone---it
is just like  $t^2 = z^2 \,+\, x^2 $.
For other values of  $\alpha$,  the equation is still a cone,
but a less familiar one because of the stretched axes.
\par
For a common-midpoint gather at  $y\,=\, y_1$  in $(h,t)$-space,
equation (\ref{eqn:shuki}) looks
like  $t^2 \,=\, t_0^2 \,+$
$4h^2 / v_{\rm apparent}^2$.
Thus the common-midpoint gather contains an
{\em  exact}
hyperbola, regardless of the earth dip angle  $\alpha$.
The effect of dip is to change the asymptote of the hyperbola,
thus changing the apparent velocity.
The result has great significance in applied work and is
known as Levin's dip correction [1971]:
\begin{equation}
v_{\rm apparent}  \eq  {v_{\rm earth}  \over  \cos ( \alpha ) }
\label{eqn:2.2}
\end{equation}
(See also Slotnick [1959]).
In summary, dip increases the stacking velocity.
\sx{velocity!dip dependent}
\par
Figure~\ref{fig:dipray} depicts some rays from a common-midpoint gather.
%\activesideplot{dipray}{width=6in}{NR}{  I think this printed OK
%\activesideplot{dipray}{width=3.8in}{NR}{	 SPARC printer error.
\sideplot{dipray}{width=3in}{
	Rays from a common-midpoint gather.
	}
Notice that each ray strikes the dipping bed at a different place.
So a common-%
{\em  midpoint %
} gather is not a common-%
{\em  depth-point %
} gather.
To realize why the reflection point moves on the reflector,
recall the basic geometrical fact that an
angle bisector in a triangle generally doesn't bisect the opposite side.
The reflection point moves
{\em  up}
dip with increasing offset.
\par
Finally, equation (\ref{eqn:shuki}) will be proved.
Figure~\ref{fig:lawcos}
shows the basic geometry along with an ``image'' source
on another reflector of twice the dip.
%\activesideplot{lawcos}{width=3.8in}{NR}{ SPARC printer error
%\activesideplot{lawcos}{width=3in}{NR}{	I think this was OK.
\sideplot{lawcos}{width=3in}{
	Travel time from image source at  $s'$  to  $g$  may be
	expressed by the law of cosines.
	}
%\newslide
For convenience, the bed intercepts the surface at  $y_0 \,=\, 0$.
The length of the line $s'  g$ in Figure~\ref{fig:lawcos} is determined by
the trigonometric Law of Cosines to be
\begin{eqnarray*}
t^2 \, v^2 \ \ \ &=&\ \ \ 
s^2 \ +\  g^2 \ -\  2\,s\,g\,\cos\,2 \alpha
\\
t^2 \, v^2 \ \ \ &=&\ \ \ 
(y\,-\,h)^2 \ +\  (y\,+\,h)^2 \ -\ 2\,(y\,-\,h)(y\,+\,h)\, \cos \, 2 \alpha
\\
t^2 \, v^2 \ \ \ &=&\ \ \ 
2\,( y^2 \,+\,h^2 ) \ -\  2\,(y^{2\,-\,} h^2 ) \,
( \cos^2 \alpha \,-\, \sin^2 \alpha )
\\
t^2 \, v^2 \ \ \ &=&\ \ \ 
4\, y^2 \, \sin^2 \alpha \ \ +\ \  4\, h^2 \, \cos^2 \alpha
\end{eqnarray*}
which is equation (\ref{eqn:shuki}).
\par
Another facet of equation (\ref{eqn:shuki}) is that it describes
the constant-offset section.
Surprisingly, the travel time of a dipping planar bed becomes curved
at nonzero offset---it too becomes hyperbolic.




\subsection{Randomly dipping layers}
\inputdir{randip}
On a horizontally layered earth,
a common shotpoint gather looks like
a common midpoint gather.
For an earth model of random dipping planes
the two kinds of gathers have quite different
traveltime curves as we see in
Figure \ref{fig:randip}.
\plot{randip}{width=6in,height=3in}{
	Seismic arrival times on an earth of random dipping planes.
	Left is for CSP.  Right is for CMP.
	}
%\newslide

\par
The common-shot gather is more easily understood.
Although a reflector is dipping,
a spherical wave incident remains a spherical wave after reflection.
The center of the reflected wave sphere is called the image point.
The traveltime equation is again a cone centered
at the image point.
The traveltime curves are simply hyperbolas
topped above the image point having the usual asymptotic slope.
The new feature introduced by dip
is that the hyperbola is laterally shifted
which implies arrivals before the fastest possible straight-line arrivals
at $vt = |g|$.
Such arrivals cannot happen.
These hyperbolas must be truncated where $vt = |g|$.
This discontinuity has the physical meaning
of a dipping bed hitting the surface
at geophone location
$|g|=vt$.
Beyond the truncation, either the shot or the receiver
has gone beyond the intersection.
Eventually both are beyond.
When either is beyond the intersection, there are no reflections.

\par
On the common-midpoint gather we see 
hyperbolas all topping at zero offset,
but with asymptotic velocities
higher (by the Levin cosine of dip)
than the earth velocity.
Hyperbolas truncate, now at $|h| = tv/2 $,
again where a dipping bed hits the surface at a geophone.


\par
On a CMP gather, some hyperbolas may seem high velocity,
but it is the dip, not the earth velocity itself that causes it.
Imagine Figure \ref{fig:randip} with all layers at $90^\circ$ dip
(abandon curves and keep straight lines).
Such dip is like the backscattered groundroll
seen on the common-shot gather of Figure \ref{fig:shell}.
The backscattered groundroll
becomes a ``flat top'' on
the CMP gather in Figure \ref{fig:randip}.

\par
\inputdir{.}
Such strong horizontal events near zero offset
will match any velocity,
particularly higher velocities such as primaries.
Unfortunately such noise events
thus make a strong contribution to a CMP stack.
Let us see how these flat-tops in offset create
the diagonal streaks you see in midpoint in Figure \ref{fig:shelikof}.
\plot{shelikof}{height=3.3in}{
        CDP stack with water noise from the Shelikof Strait, Alaska.
        (by permission from
        {\em  Geophysics,}
        Larner et al.{[1983]})   % Don't remove the braces!  Latex bug!
        }
%\newslide

\par
Consider 360 rocks of random sizes scattered in an exact
circle of 2 km diameter on the ocean floor.
The rocks are distributed along one degree intervals.
Our survey ship sails from south to north
towing a streamer across the exact center of the circle,
coincidentally crossing directly over rock number 180 and number 0.
Let us consider the common midpoint gather corresponding
to the midpoint in the center of the circle.
Rocks 0 and 180 produce flat-top hyperbolas.
The top is flat for $0<|h|<1$ km.
Rocks 90 and 270 are $90^\circ$ out of the plane of the survey.
Rays to those rocks propagate entirely within the water layer.
Since this is a homogeneous media,
the travel time expression of these rocks
is a simple hyperbola of water velocity.
Now our CMP gather at the circle center has
a ``flat top'' and a simple hyperbola
both going through zero offset at time $t=2/v$ (diameter 2 km, water velocity).
Both curves have the same water velocity asymptote
and of course the curves are tangent at zero offset.

\par
Now consider all the other rocks.
They give curves inbetween the simple water hyperbola and the flat top.
Near zero offset, these curves range in apparent velocity
between water velocity and infinity.
One of these curves will have an apparent velocity that matches
that of sediment velocity.
This rock (and all those near the same azimuth)
will have velocities that are near the sediment velocity.
This noise will stack very well.
The CDP stack at sediment velocity will stack
in a lot of water borne noise.
This noise is propagating somewhat off the survey line
but not very far off it.

\par
Now let us think about the appearance of the CDP stack.
We turn attention from offset to midpoint.
The easiest way to imagine the CDP stack
is to imagine the zero-offset section.
Every rock has a water velocity asymptote.
These asymptotes are evident on
the CDP stack in Figure \ref{fig:shelikof}.
This result was first recognized by Ken Larner.

\par
Thus, backscattered low-velocity noises have
a way of showing up on higher-velocity stacked data.
There are two approaches
to suppressing this noise:
(1) mute the inner traces,
and as we will see,
(2) dip moveout processing.


\section{TROUBLE WITH DIPPING REFLECTORS}
\par
The ``standard process'' is NMO, stack, and zero-offset migration.
Its major shortcoming is the failure of NMO and stack
to produce a section that resembles the true zero-offset section.
In chapter~\ref{vela/paper:vela} we derived
the NMO equations for a stratified earth,
but then applied them to seismic field data 
that was not really stratified.
That this works at all is a little surprising,
but it turns out that NMO hyperbolas
apply to dipping reflectors as well as horizontal ones.
%One of the first results of this section is the establishment of this result.
When people try to put this result into practice,
however,
they run into a nasty conflict:
reflectors generally require a {\em dip-dependent}
NMO velocity in order to produce a ``good'' stack.
Which NMO velocity are we to apply
when a dipping event is near (or even crosses) a horizontal event?  
Using conventional NMO/stack techniques
generally forces velocity analysts to choose
which events they wish to preserve on the stack.
This inability to simultaneously produce a good stack
for events with {\em all} dips is a serious shortcoming,
which we now wish to understand more quantitatively.

\subsection{Gulf of Mexico example}
\inputdir{krchdmo}
\par
Recall the Gulf of Mexico dataset presented in chapter~\ref{vela/paper:vela}.
We did a reasonably careful job of NMO velocity analysis
in order to produce the stack shown in Figure~\ref{vela/fig:agcstack}.
But is this the best possible stack?
To begin to answer this question, Figure~\ref{fig:cvstacks} shows
some constant-velocity stacks of this dataset done with subroutine
\texttt{velsimp()} \vpageref{/prog:velsimp}.
This figure clearly shows that
there are some very steeply-dipping reflections