paper.tex

国外免费地震资料处理软件包

\title{Imaging in shot-geophone space}
\author{Jon Claerbout}
\maketitle

Till now, we have limited our data processing to midpoint-offset space.
We have not analyzed reflection data directly in shot-geophone space.
In practice this is often satisfactory.
Sometimes it is not.
The principal factor that drives us away from $(y,h)$-space into $(s,g)$-space
is lateral velocity variation $v(x,z) \ne v(z)$.
In this chapter,
we will see how migration can be performed
in the presence of $v(x,z)$
by going to $(s,g)$-space.
\par
Unfortunately, this chapter has no prescription for finding $v(x,z)$,
although we will see how the problem manifests itself
even in apparently stratified regions.
We will also see why, in practice, amplitudes are dangerous.



\section{TOMOGRAPY OF REFLECTION DATA}
\sx{tomography}
\par
Sometimes the earth strata lie horizontally with little irregularity.
There we may hope to ignore the effects of migration.
Seismic rays should fit a simple model with large reflection
angles occurring at wide offsets.
Such data should be ideal for the measurement of reflection
coefficient as a function of angle,
or for the measurement of the earth acoustic absorptivity  $1/Q$.
In his doctoral dissertation, Einar
\bx{Kjartansson}
reported such a study.
The results were so instructive that the study will be thoroughly reviewed here.
I don't know to what extent the Grand Isle gas field typifies
the rest of the earth,
but it is an excellent place to begin learning about the
meaning of shot-geophone offset.
\subsection{The grand isle gas field: a classic bright spot}
\par
The dataset \bx{Kjartansson} studied was a seismic line across the Grand Isle
gas field, off the shore of Louisiana.
The data contain several classic ``bright spots'' (strong reflections)
on some rather flat undisturbed bedding.
Of interest are the lateral variations in amplitude
on reflections at a time depth of about 2.3 seconds on Figure~\ref{fig:kjcos}.
It is widely believed that such bright spots
arise from gas-bearing sands.

\par
Theory predicts that reflection coefficient should
be a function of angle.
For an anomalous physical situation
like gas-saturated sands, the function should be distinctive.
Evidence should be found on common-midpoint gathers
like those shown in Figure~\ref{fig:kjcmg}.
\plot{kjcmg}{height=7.5in}{
	Top left is shot point 210; top right is shot point 220.
	No processing has been applied to the data except
	for a display gain proportional to time.
	Bottom shows shot points 305 and 315.  (Kjartansson)
	}
Looking at any one of these gathers you will note that the reflection
strength versus offset seems to be a smooth,
sensibly behaved function, apparently quite measurable.
Using layered media theory, however, it was determined that only
the most improbably bizarre medium could exhibit such strong
variation of reflection coefficient with angle,
particularly at small angles of incidence.
(The reflection angle of the energy arriving at wide offset at time 2.5 seconds
is not a large angle.
Assuming constant velocity, ${\rm arccos} (2.3/2.6)\,=\,28$$^\circ$).
Compounding the puzzle, each common-midpoint gather shows a
{\em  different}
smooth, sensibly behaved, measurable function. 
Furthermore, these midpoints are near one another,
ten shot points 
spanning a horizontal distance of 820 feet.
\subsection{Kjartansson's model for lateral variation in amplitude}
\inputdir{XFig}
\par
The Grand Isle data is incomprehensible in terms of the
model based on layered media theory.
Kjartansson proposed an alternative model.
Figure~\ref{fig:kjidea} illustrates a geometry in which rays travel
in straight lines from any source to a flat horizontal reflector,
and thence to the receivers.
%\activeplot{kjidea}{height=5.5in}{NR}{
\plot{kjidea}{height=4.5in}{
	Kjartansson's model.  The model on the top 
	produces the disturbed data space sketched below it. 
	Anomalous material in pods A, B, and C may be detected
	by its effect on reflections from a deeper layer.
	}
The only complications are ``pods'' of some material
that is presumed to disturb seismic rays in some anomalous way.
Initially you may imagine that the pods absorb wave energy.
(In the end it will be unclear whether the disturbance results from
energy focusing or absorbing).

\inputdir{.}
\par
Pod A is near the surface.
The seismic survey is affected by it twice---once
when the pod is traversed by the shot and once when it is
traversed by the geophone.
Pod C is near the reflector and encompasses a small area of it.
Pod C is seen at all offsets  $h$  but only at one midpoint,  $y_0$.
The raypath depicted on the top of Figure~\ref{fig:kjidea} is one that is
affected by all pods.
It is at midpoint  $y_0$  and at the widest offset  $h_{\rm max}$.
Find the raypath on the lower diagram in Figure~\ref{fig:kjidea}.
\par
Pod B is part way between A and C.
The slope of affected points in the $(y,h)$-plane is part way between
the slope of A and the slope of C.
\par
Figure~\ref{fig:kjcos} shows a common-offset section across the gas field.
The offset shown is the fifth trace from the near offset,
1070 feet from the shot point.
Don't be tricked into thinking the water was deep.
The first break at about .33 seconds is wide-angle propagation.
\plot{kjcos}{height=8.5in}{
	A constant-offset section across the Grand Isle gas field.
	The offset shown is the fifth from the near trace.  (Kjartansson, Gulf)
	}
\plot{kja}{height=8.5in}{
	(a) amplitude (h,y), (b) timing (h,y)
	(c) amplitude (z,y), (d) timing (d,y)
	}
\par
The power in each seismogram was computed in the interval from
1.5 to 3 seconds.
The logarithm of the power is plotted in Figure~\ref{fig:kja}a as a function
of midpoint and offset.
Notice streaks of energy slicing across the  $(y,h)$-plane
at about a 45$^\circ$ angle.
The strongest streak crosses at exactly 45$^\circ$ 
through the near offset at shot point 170.
This is a missing shot, as is clearly visible in Figure~\ref{fig:kjcos}.
Next, think about the gas sand described as pod C in the model.
Any gas-sand effect in the data should show up as a streak
across all offsets at the midpoint of the gas sand---that is,
horizontally across the page.
I don't see such streaks in Figure~\ref{fig:kja}a.
Careful study of the figure shows that
the rest of the many clearly visible streaks cut the plane at
an angle noticeably
{\em  less}
than $\pm$45$^\circ$.
The explanation for the angle of the streaks in the figure 
is that they are like pod B.
They are part way between the surface and the reflector.
The angle determines the depth.
Being closer to 45$^\circ$ than to 0$^\circ$, the pods are
closer to the surface than to the reflector.
\par
Figure \ref{fig:kja}b shows timing information in the same form that
Figure~\ref{fig:kja}a shows amplitude.
A CDP stack was computed, and each field
seismogram was compared to it.
A residual time shift for each trace was determined
and plotted in Figure~\ref{fig:kja}b.
The timing residuals on one of the common-midpoint
gathers is shown in Figure~\ref{fig:kjmid}.
\sideplot{kjmid}{width=3.0in}{
	Midpoint gather 220 (same as timing of (h,y) in
	Figure~\protect\ref{fig:kja}b) after moveout.
	Shown is a one-second window centered at 2.3 seconds, time shifted
	according to an NMO for an event at 2.3 seconds, using a velocity
	of 7000 feet/sec.  (Kjartansson)
	}
\par
The results resemble the amplitudes, except that the
results become noisy when the amplitude is low or
where a ``leg jump'' has confounded the measurement.
Figure \ref{fig:kja}b clearly shows that the disturbing influence on timing
occurs at the same depth as that which disturbs amplitudes.

\par
The process of 
{\em  inverse \bx{slant stack}}
(not described in this book)
enables one to determine the depth distribution of the pods.
This distribution is displayed
in figures \ref{fig:kja}c and \ref{fig:kja}d.
\subsection{Rotten alligators}
\inputdir{meander}
\par
The sediments carried by the Mississippi River are dropped at the delta.
There are sand bars, point bars, old river bows now silted in,
a crow's foot of sandy distributary channels,
and between channels, swampy flood plains 
are filled with decaying organic material.
The landscape is clearly laterally variable,
and eventually it will all sink of its own weight,
aided by growth faults and the weight of later sedimentation.
After it is buried and out of sight the lateral variations will remain
as pods that will be observable by the seismologists of the future.
These seismologists may see something like Figure~\ref{fig:meander}.
Figure~\ref{fig:meander} shows a %
{\em  three-dimensional %
} seismic survey,
that is, the ship sails many parallel lines about 70 meters apart.
The top plane, a slice at constant time, shows buried river meanders.
%The data shown in Figure~\ref{fig:meander} is described more fully by its donors, Dahm and Graebner [1982].
\plot{meander}{width=4.5in}{
	Three-dimensional seismic data from the Gulf of Thailand.
	Data planes from within the cube are displayed on the faces of the cube.
	The top plane shows ancient river meanders now submerged.
	(Dahm and Graebner)
	}

\subsection{Focusing or absorption?}
\par
Highly absorptive rocks usually have low velocity.
Behind a low velocity pod, waves should be weakened by absorption.
They should also be strengthened by focusing.
Which effect dominates?
How does the phenomenon depend on spatial wavelength?
Maybe you can figure it out
knowing that black on Figure~\ref{fig:kja}c denotes
low amplitude or high absorption, and
black on Figure~\ref{fig:kja}d denotes low velocities.
\par
I'm inclined to believe the issue is focusing, not absorption.
Even with that assumption, however, a reconstruction of the velocity
$v(x,z)$
for this data has never been done.
This falls within the realm of ``reflection \bx{tomography}'',
a topic too difficult to cover here.
Tomography generally reconstructs a velocity model $v(x,z)$
from travel time anomalies.
It is worth noticing that with this data, however,
the amplitude anomalies seem to give more reliable information.


\begin{exer}

\item
Consider waves converted from pressure $P$ waves
to shear $S$ waves.
Assume an $S$-wave speed of about half the $P$-wave speed.
What would Figure~\ref{fig:kjidea} look like for these waves?
\end{exer}



\section{SEISMIC RECIPROCITY IN PRINCIPLE AND IN PRACTICE}
\inputdir{toldi}
\par
The principle of \bx{reciprocity} says
that the same seismogram should be recorded if the 
locations of the source and geophone are exchanged.
A physical reason for the validity of reciprocity is that no matter how
complicated a geometrical arrangement,
the speed of sound along a ray is the same in either direction.

\par
Mathematically, the reciprocity principle arises because
symmetric matrices arise.
The final result is that very complicated electromechanical systems mixing
elastic and electromagnetic waves generally fulfill the reciprocal principle.
To break the reciprocal principle,
you need something like a windy atmosphere so that sound
going upwind has a different velocity than sound going downwind.

\par
Anyway, since the
impulse-response matrix is symmetric,
elements across the matrix diagonal are equal to one another.
Each element of any pair is a response to an impulsive source.
The opposite element of the pair refers to
an experiment where the source and receiver
have had their locations interchanged.

\par
A tricky thing about the reciprocity principle
is the way antenna patterns must be handled.
For example, a single vertical geophone has a natural antenna pattern.
It cannot see horizontally propagating pressure waves nor vertically
propagating shear waves.
For reciprocity to be applicable,
antenna patterns must not be interchanged
when source and receiver are interchanged.
The antenna pattern must be regarded as attached to the medium.

\par
I searched our data library for split-spread land data that
would illustrate reciprocity under field conditions.
The constant-offset section in Figure~\ref{fig:toldi} was recorded
by vertical vibrators into vertical geophones.
\plot{toldi}{height=4.0in}{
	Constant-offset section from the Central Valley of California. 
	(Chevron)
	}

The survey was not intended to be a test of reciprocity,
so there likely was a slight lateral offset of the source line
from the receiver line.
Likewise the sender and receiver arrays (clusters)
may have a slightly different geometry.
The earth dips in Figure~\ref{fig:toldi} happen to be quite small
although lateral velocity variation
is known to be a problem in this area.
\par
In Figure~\ref{fig:reciptrace},
three seismograms were plotted on top of their reciprocals.
\plot{reciptrace}{width=6in,height=2.5in}{
	Overlain reciprocal seismograms.
	}
Pairs were chosen at near offset, at mid range,