paper.tex

国外免费地震资料处理软件包

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

\title{Waves in strata}
\author{Jon Claerbout}
\label{paper:wvs}
\maketitle

\def\RMS{{{\sc rms}}}   % Global RMS.  Always type this one, even in equations.
                        % Then we can change it to suit final editor.
                        % This definition dies in equations.

\def\RMS{{\rm RMS}}     % Global RMS.  Always type this one, even in equations.

The waves of practical interest in reflection seismology are usually complicated
because the propagation velocities are generally complex.
In this book, we have chosen to build up the complexity of the waves we
consider, chapter by chapter.  The simplest waves to understand are simple
plane waves and spherical waves propagating through a constant-velocity medium.
In seismology however, the earth's velocity is almost never well approximated
by a constant.
A good first approximation is to assume that the earth's 
velocity increases with depth.
In this situation, the simple planar and circular wavefronts 
are modified by the effects of $v(z)$.
In this chapter we study the basic equations describing plane-like 
and spherical-like waves propagating 
in media where the velocity $v(z)$ is a function only of depth.
This is a reasonable starting point,
even though it neglects the even more complicated distortions that occur
when there are lateral velocity variations.
We will also examine data that shows plane-like waves
and spherical-like waves resulting when waves
from a point source bounce back from a planar reflector.

\section{TRAVEL-TIME DEPTH}
\par
Echo soundings give us a picture of the earth.
A zero-offest section, for example,
is a planar display of traces where the horizontal axis runs along 
the earth's surface
and the vertical axis, running down, seems to measure depth,
but actually measures the two-way echo delay time.
Thus, in practice the vertical axis is almost never depth $z$;
it is the
{\em 
vertical travel time
}
$ \tau $.
In a constant-velocity earth
the time and the depth are related
by a simple scale factor, the speed of sound.  This is analogous to the
way that astronomers measure distances in light-years, always referencing
the speed of light.
The meaning of the scale factor in seismic imaging 
is that the  $(x, \tau$)-plane
has a vertical exaggeration compared to the  $(x,z)$-plane.
In reconnaissance work,
the vertical is often exaggerated by about a factor of five.
By the time prospects have been sufficiently narrowed for a drill
site to be selected,
the vertical exaggeration factor in use is likely to be about unity
(no exaggeration).

\todo{
[JON, THIS IS PROBABLY TRUE BUT IT IS NOT CONVINCING
UNLESS WE SPECIFY THE
DISPLAY SCALES THAT ARE TYPICALLY USED: KM/INCH, SEC/IN.  SEE MY 
RELATED COMMENTS BELOW THE NEXT PARAGRAPH]  [IF WE WISH TO
RETAIN THIS ``EXAGGERATION'' MATERIAL, WHICH I AM INCLINED TO DO, THEN
WE REALLY NEED A FIGURE THAT GRAPHICALLY ILLUSTRATES THE PHENOMENON WITH
ACTUAL DISPLAY SCALES AND AN ACTUAL FUNCTION v(z)].  THE READER SHOULD
BE ABLE TO ``PLAY WITH'' THE DISPLAY PARAMETERS AND WITH v(z) AND SEE
HOW THE EXAGGERATION CHANGES.  WE COULD GENERATE A SET OF SYNTHETIC HORIZONTAL
REFLECTORS AT DEPTHS OF 1.0, 2.0, 3.0 KM, FOR EXAMPLE
}

\par
In seismic reflection imaging, the waves go down and then up,
so the \bx{traveltime depth} $\tau$
is defined as two-way vertical travel time.
\begin{equation}
\label{eqn:twowaytau}
                \tau \eq  {2\,z  \over  v }  \ \ .
\end{equation}
This is the convention that I have chosen to use throughout this book.

\subsection{Vertical exaggeration}
\par
The first task in interpretation of seismic data
is to figure out the approximate numerical value
of the \bx{vertical exaggeration}.
The vertical exaggeration is $2/v$ because it is the ratio of
the apparent slope
$\Delta \tau / \Delta x$
to the actual slope
$\Delta z/ \Delta x$
where $\Delta \tau = 2\ \Delta z / v$.
Since
the velocity generally
{\em  increases} with depth,
the \bx{vertical exaggeration} generally
{\em  decreases} with depth.

\par
For velocity-stratified media,
the time-to-depth conversion formula is
\begin{equation}
\tau (z) \ \ =\ \  \int_0^z \  {2\ dz \over v (z) }
\ \ \ \ \ \ \ \ \  {\rm or} \ \ \ \ \ \ \ \ 
{ d \tau   \over dz }\ \ =\ \  {2 \over v }
\label{eqn:3.2}
\end{equation}

\todo { [JON, IF WE WANT TO
RETAIN THIS ``EXAGGERATION'' MATERIAL, WHICH I REALLY THINK WE SHOULD, THEN
WE MUST HAVE A FIGURE THAT GRAPHICALLY ILLUSTRATES THE PHENOMENON WITH
ACTUAL DISPLAY SCALES AND AN ACTUAL FUNCTION v(z)].  THE READER SHOULD
BE ABLE TO ``PLAY WITH'' THE DISPLAY PARAMETERS AND WITH v(z) AND SEE
HOW THE EXAGGERATION CHANGES.  WE COULD GENERATE A SET OF SYNTHETIC HORIZONTAL
REFLECTORS AT DEPTHS OF 1.0, 2.0, 3.0 KM, FOR EXAMPLE
}

\section{HORIZONTALLY MOVING WAVES}
\inputdir{headray}
In practice, horizontally going waves are easy to recognize
because their travel time is a linear function
of the offset distance between shot and receiver.
There are two kinds of horizontally going waves,
one where the traveltime line goes through the origin,
and the other where it does not.
When the line goes through the origin,
it means the ray path is always near the earth's surface
where the sound source and the receivers are located.
(Such waves are called ``\bx{ground roll}'' on land
or ``\bx{guided wave}s'' at sea;
sometimes they are just called ``\bx{direct arrivals}''.)

\par
When the traveltime line does not pass through the origin
it means parts of the ray path plunge into the earth.
This is usually explained by
the unlikely looking rays shown in Figure~\ref{fig:headray}
which frequently occur in practice.%
\sideplot{headray}{width=3in,height=.8in}{
  Rays associated with \bx{head wave}s.
} %
Later in this chapter we will see that Snell's law
predicts these rays in a model of the earth with two layers,
where the deeper layer is faster and the ray bottom
is along the interface between the slow medium and the fast medium.
Meanwhile, however, notice that these ray paths
imply data with a linear travel time versus distance
corresponding to increasing ray length along the ray bottom.
Where the ray is horizontal in the lower medium,
its wavefronts are vertical.
These waves are called ``\bx{head wave}s,''
perhaps because they are typically fast
and arrive {\em  ahead} of other waves.

\subsection{Amplitudes}
\par
The nearly vertically-propagating waves (reflections)
spread out essentially in three dimensions,
whereas the nearly horizontally-going waves
never get deep into the earth because,
as we will see,
they are deflected back upward by the velocity gradient.
Thus horizontal waves spread out in essentially two dimensions, 
so that energy conservation suggests
that their amplitudes should dominate the amplitudes of reflections
on raw data.
This is often true for \bx{ground roll}.
Head waves, on the other hand,
are often much weaker, often being visible only because
they often arrive before more energetic waves.
The weakness of \bx{head wave}s
is explained by
the small percentage of solid angle occupied by the waves leaving a source
that eventually happen to match up with layer boundaries and propagate as
head waves.
I %\footnote{
  %      In this chapter, and perhaps throughout the book,
  %      the word ``I'' refers to the two authors working
  %      in consultation, and the word ``we'' refers
  %      to readers and authors.
  %      }
selected the examples below because of the strong headwaves.
They are nearly as strong as the guided waves.
To compensate for diminishing energy with distance,
I scaled data displays by multiplying by the offset distance
between the shot and the receiver.
\par
In data display, the slowness (slope of the time-distance curve)
is often called the \bx{stepout}  $p$.  Other commonly-used names for
this slope are \bx{time dip} and \bx{reflection slope}.
The best way to view waves with \bx{linear moveout}
is after time shifting to remove a standard linear moveout
such as that of water.
An equation for the shifted time is
\begin{equation}
\tau \eq t - p x
\label{eqn:lmo}
\end{equation}
where $p$ is often chosen to be the inverse of the velocity of water,
namely, about 1.5 km/s, or $p=.66 {\rm s/km}$
and $x=2h$ is the horizontal separation between
the sound source and receiver, usually referred to as the \bx{offset}.

\inputdir{head}
\par
\bxbx{Ground roll}{ground roll}
and \bx{guided wave}s are typically slow
because materials near the earth's surface typically are slow.
Slow waves are steeply sloped on a time-versus-offset display.
It is not surprising that marine guided waves
typically have speeds comparable to water waves
(near 1.47 km/s approximately 1.5 km/s).
It is perhaps surprising that \bx{ground roll}
also often has the speed of sound in water.
Indeed, the depth to underground water is often determined
by seismology before drilling for water.
Ground roll also often has a speed
comparable to the speed of sound in air,
0.3 km/sec, though, much to my annoyance I could not find
a good example of it today.
%(Record wz.25 seems to have both water speed and air speed head waves,
%but the given parameters imply two speeds exactly twice as fast
%as those two speeds and I am suspicious.)
Figure~\ref{fig:wzl-34} is an example of energetic \bx{ground roll} (land)
that happens to have a speed close to that of water.
\plot{wzl-34}{width=6.00in,height=3.0in}{
  Land shot profile (Yilmaz and Cumro) \#39 from the Middle East
  before (left) and after (right)
  linear moveout at water velocity.
}

\par
The speed of a ray traveling along a layer interface
is the rock speed in the faster layer (nearly always the lower layer).
It is not an average of the layer above and the layer below.

% The discussion below is based on a JLB fantasy figure.
%  I doubt this discussion belongs here.  Meanwhile, I commented it out. -Jon
%Rays are not directly observable; they are theoretical.
%To better understand what is happening, think of 
%the wavefronts for the \bx{head wave} in Figure~\FIG{horzrays}.
%A ray along an interface is perpendicular
%to a vertical wavefront (horizontal ray) in the lower medium
%and a wavefront dipping at angle $\theta_c$ in the upper medium.
%So the horizontal slowness in the upper medium,
%$\sin\theta_c /v_1$, matches the horizontal slowness $1/v_2$
%of the front in the lower medium.  This fact, known as \bx{Snell's law}
%will be proven in Section~{dipping} of this chapter.

\par
Figures~\ref{fig:wzl-20} and~\ref{fig:wzl-32}
are examples of energetic marine guided waves.
In Figure~\ref{fig:wzl-20}
at $\tau=0$ (designated {\tt t-t\_water}) at small offset
is the wave that travels directly from the shot to the receivers.
This wave dies out rapidly with offset
(because it interferes with a wave of opposite polarity
reflected from the water surface).
At near offset slightly later than $\tau=0$ is the water bottom reflection.
At wide offset, the water bottom reflection is quickly followed 
by multiple reflections from the bottom.
Critical angle reflection is defined as where the \bx{head wave}
comes tangent to the reflected wave.
Before (above) $\tau=0$ are the \bx{head wave}s.
There are two obvious slopes,
hence two obvious layer interfaces.
Figure \ref{fig:wzl-32} is much like Figure \ref{fig:wzl-20}
but the water bottom is shallower.

\plot{wzl-20}{width=6.00in,height=3.0in}{
  Marine shot profile (Yilmaz and Cumro) \#20 from the Aleutian Islands.
}
\plot{wzl-32}{width=6.00in,height=3.0in}{
  Marine shot profile (Yilmaz and Cumro) \#32 from the North Sea.
}

\par
Figure~\ref{fig:wglmo} shows data where the first arriving energy
is not along a few straight line segments,
but is along a curve.
This means the velocity increases smoothly with depth
as soft sediments compress.
\plot{wglmo}{width=6.00in,height=2.7in}{
  A common midpoint gather from the Gulf of Mexico
  before (left) and after (right) linear moveout
  at water velocity.
  Later I hope to estimate
  velocity with depth
  in shallow strata.
  Press button for \bx{movie} over midpoint.
}

\subsection{LMO by nearest-neighbor interpolation}
To do \bx{linear moveout} (\bx{LMO}) correction, we need to time-shift data.
Shifting data requires us to interpolate it.
The easiest interpolation method is the nearest-neighbor method.
We begin with a signal given at times {\tt t = t0+dt*(it-1)}
where {\tt it} is an integer.
Then we can use equation~(\ref{eqn:lmo}),
namely $\tau=t-px$.
Given the location {\tt tau} of the desired value
we backsolve for an integer, say {\tt itau}.
In Fortran, conversion of a real value to an integer is done by
truncating the fractional part of the real value.
To get rounding up as well as down,
we add
{\tt 0.5}
before conversion to an integer,
namely {\tt itau=int(1.5+(tau-tau0)/dt)}.
This gives the nearest neighbor.
The
way the program works is to identify two points,
one in $(t,x)$-space and one in $(\tau,x)$-space.
Then
the data value at one point in one space is carried to the other.
The adjoint operation copies $\tau$ space back to $t$ space.
%The subroutine used in the illustrations above is
%\texttt{lmo()} \vpageref{/prog:lmo}
%with {\tt adj=1}.
%\progdex{lmo}{linear moveout}

\par
Nearest neighbor rounding is crude but ordinarily very reliable.
I discovered a very rare numerical roundoff problem
peculiar to signal time-shifting, a problem
which arises in the linear moveout application
when the water velocity, about 1.48km/sec is approximated by 1.5=3/2.
The problem arises only where the amount of the time shift
is a numerical value (like 12.5000001 or 12.499999)
and the fractional part should be exactly 1/2 but
numerical rounding pushes it randomly in either direction.
We would not care if an entire signal was shifted
by either 12 units or by 13 units.
What is troublesome, however, is if some random portion
of the signal shifts 12 units while the rest of it shifts 13 units.
Then the output signal has places which are empty while
adjacent places contain the sum of two values.
Linear moveout is the only application
where I have ever encountered this difficulty.
A simple fix here was to modify the
\texttt{lmo()} \vpageref{lst:lmo}
subroutine changing
the ``1.5'' to ``1.5001''.
The problem disappears if we use a more accurate sound velocity
or if we switch from nearest-neighbor interpolation
to linear interpolation.

\subsection{Muting}
\inputdir{vscan}
Surface waves are a mathematician's delight
because they exhibit many complex phenomena.
Since these waves are often extremely strong,
and since the information they contain about the earth
refers only to the shallowest layers,
typically,
considerable effort is applied to array design in field recording
to suppress these waves.
Nevertheless, in many areas of the earth,
these pesky waves may totally dominate the data.

\par
A simple method to suppress \bx{ground roll} in data processing
is to multiply a strip of data by a near-zero weight (the mute).
To reduce truncation artifacts,
the mute should taper smoothly to zero (or some small value).
Because of the extreme variability from place to place