paper.tex

国外免费地震资料处理软件包

on the earth's surface,
there are many different philosophies
about designing mutes.
Some mute programs use a data dependent weighting function
(such as automatic gain control).
Subroutine \texttt{mutter()} \vpageref{lst:mutter},
however, operates on a simpler idea: 
the user supplies trajectories defining the mute zone.
\moddex{mutter}{mute}{44}{71}{filt/imag}

\par
Figure~\ref{fig:mutter} shows an example of use
of the routine \texttt{mutter()} \vpageref{lst:mutter} on the shallow water data
shown in Figure~\ref{fig:wglmo}.
\plot{mutter}{width=6.00in,height=2.7in}{
  Jim's first gather before and after muting.
}

\section{DIPPING WAVES}
%\sectionlabel{dipping}

Above we considered waves going vertically
and waves going horizontally.
Now let us consider waves propagating at the intermediate angles.  
For the sake of definiteness, I have chosen
to consider only downgoing waves in this section.
We will later use the concepts developed
here to handle both downgoing and upcoming waves.

\subsection{Rays and fronts}
\inputdir{XFig}
\par
It is natural to begin studies of waves
with equations that describe plane waves
in a medium of constant velocity.

Figure~\ref{fig:front} depicts a ray moving down into the earth
at an angle $ \theta $ from the vertical.
\sideplot{front}{width=3.3in}{
  Downgoing ray and wavefront.
}
%was \activesideplot{front}{width=3.0in}{NR}{
Perpendicular to the ray is a wavefront.
By elementary geometry the angle between the wave\bx{front}
and the earth's surface
is also  $ \theta $.
The \bx{ray} increases its length at a speed  $v$.
The speed that is observable on the earth's surface is the intercept
of the wavefront with the earth's surface.
This speed, namely  $ v / \sin \theta $,  is faster than  $v$.
Likewise, the speed of the intercept of the wavefront and
the vertical axis is  $ v / \cos \theta $.
A mathematical expression for a straight line
like that shown to be the wavefront in Figure~\ref{fig:front} is
\begin{equation}
z \ \ =\ \  z_0 \ -\  x \  \tan \, \theta
\label{eqn:2.1}
\end{equation}
\par
In this expression  $ z_0 $  is the intercept between the wavefront
and the vertical axis.
To make the intercept move downward, replace it by the
appropriate velocity times time:
\begin{equation}
z \ \ =\ \  {v \, t \over  \cos \, \theta } \ -\  x \  \tan \, \theta
\label{eqn:2.2}
\end{equation}
Solving for time gives
\begin{equation}
t(x,z) \ \ =\ \  {z\over v }\ \cos\,\theta \ +\  {x \over v }\  \sin \, \theta
\label{eqn:txz}
\end{equation}
Equation~(\ref{eqn:txz}) tells the time that the wavefront will pass any
particular location  $(x , z)$.
The expression for a shifted waveform
of arbitrary shape is  $ f(t - t_0 ) $.
Using (\ref{eqn:txz}) to define the time shift $ t_0 $ gives an expression for
a wavefield that is some waveform moving on a \bx{ray}.
\begin{equation}
\hbox{moving wavefield} \ \ =\ \ 
f\left( \ t\ -\  {x \over v}\ \sin\,\theta \ -\ {z\over v}\ \cos\,\theta\right)
\label{eqn:mvwv}
\end{equation}

\subsection{Snell waves}
\par
In reflection seismic surveys the velocity
contrast between shallowest and deepest
reflectors ordinarily exceeds a factor of two.
Thus depth variation of velocity is almost always included
in the analysis of field data.
Seismological theory needs to consider waves
that are just like plane waves except that they bend
to accommodate the velocity stratification  $v(z)$.
Figure~\ref{fig:airplane} shows this in an idealized geometry:
waves radiated from the horizontal flight of a supersonic airplane.
\plot{airplane}{width=6.0in}{
  Fast airplane radiating a sound wave into the earth.
  From the figure you can deduce that
  the horizontal speed of the wavefront
  is the same at depth  $z_1$  as it is at depth  $z_2$.
  This leads (in isotropic media) to Snell's law.
}
% was \activeplot{airplane}{height=2.5in}{NR}{
The airplane passes location $x$ at time $t_0(x)$
flying horizontally at a constant speed.
Imagine an earth of horizontal plane layers.
In this model there is nothing to distinguish any point
on the $x$-axis from any other point on the $x$-axis.
But the seismic velocity varies from layer to layer.
There may be reflections, head waves, shear waves, converted waves,
anisotropy, and multiple reflections.
Whatever the picture is, it moves along with the airplane.
A picture of the wavefronts near the airplane moves along with the airplane.
The top of the picture and the bottom of the picture both move laterally at
the same speed even if the earth velocity increases with depth.
If the top and bottom didn't go at the same speed,
the picture would become distorted,
contradicting the presumed symmetry of translation.
This horizontal speed, or rather its inverse  ${\partial t_0}/{\partial x}$,
has several names.
In practical work it is called the
{\em  \bx{stepout}.}
In theoretical work it is called the
{\em  \bx{ray parameter}}.
It is very important to note that  ${\partial t_0}/{\partial x}$
does not change with depth,
even though the seismic velocity does change with depth.
In a constant-velocity medium, the angle of a wave
does not change with depth.
In a stratified medium,
${\partial t_0}/{\partial x}$  does not change with depth.
\par
Figure~\ref{fig:frontz} illustrates the differential geometry of the wave.
Notice that triangles have their
hypotenuse on the $x$-axis and the $z$-axis
but not along the ray.
That's because this figure refers to wave fronts.
(If you were thinking the hypotenuse would measure $v\Delta t$,
it could be you were thinking of the tip of a ray
and its projection onto the $x$ and $z$ axes.)
\plot{frontz}{height=1.8in}{
  Downgoing fronts and rays in stratified medium  $v(z)$.
  The wavefronts are horizontal translations of one another.
}
The diagram shows that
\begin{eqnarray}
{\partial t_0 \over \partial x} \ \ \ &=&\ \ \  { \sin \, \theta  \over v }
\label{eqn:5iei4a}
\\
{\partial t_0 \over \partial z} \ \ \ &=&\ \ \  { \cos \, \theta  \over v }
\label{eqn:5iei4b}
\end{eqnarray}
These two equations define two (inverse) speeds.
The first is a horizontal speed,
measured along the earth's surface,
called the
{\em 
horizontal \bx{phase velocity}.
}
The second is a vertical speed, measurable in a borehole, called the
{\em 
vertical phase velocity.
}
Notice that both these speeds
{\em  exceed}
the velocity  $v$  of wave propagation in the medium.
Projection of wave
{\em  fronts}
onto coordinate axes gives speeds larger than  $v$,
whereas projection of
{\em  rays}
onto coordinate axes gives speeds smaller than $v$.
The inverse of the phase velocities is called the
{\em  \bx{stepout}}
or the 
{\em  \bx{slowness}.}
\par
\bx{Snell's law} relates the angle of a wave
in one layer with the angle in another.
The constancy of equation (\ref{eqn:5iei4a}) in depth is really just
the statement of Snell's law.
Indeed, we have just derived Snell's law.
All waves in seismology propagate in a
velocity-stratified medium.  So they cannot be called
plane waves.  But we need a name for waves that are
near to plane waves.  A %
{\em  \bx{Snell wave} %
} will be defined to be the generalization of a plane wave
to a stratified medium  $v(z)$.
A plane wave that happens to enter a medium
of depth-variable velocity  $v(z)$  gets changed into a Snell wave.
While a plane wave has an angle of propagation, a
Snell wave has instead a %
{\em  \bx{Snell parameter} %
} $p\,=\,{\partial t_0}/{\partial x}$.
\par
It is noteworthy that
Snell's parameter  $p\,=\,{\partial t_0}/{\partial x}$  is directly
observable at the surface,
whereas neither  $v$  nor  $\theta$  is directly observable.
Since  $p\,=\,{\partial t_0}/{\partial x}$  is not only observable,
but constant in depth, it is customary to use it
to eliminate  $\theta$  from equations (\ref{eqn:5iei4a}) and (\ref{eqn:5iei4b}):
\begin{eqnarray}
{\partial t_0  \over \partial x} \ \ \ &=&\ \ \ {\sin\,\theta  \over v }\eq p
\label{eqn:5iei5a}
\\
{\partial t_0  \over \partial z}\ \ \ &=&\ \ \ 
{\cos\,\theta  \over v }\eq \sqrt{
 {1 \over v (z)^2}\ -\ p^2  }
\label{eqn:5iei5b}
\end{eqnarray}

\subsection{Evanescent waves}
Suppose the velocity increases to infinity at infinite depth.
Then equation (\ref{eqn:5iei5b}) tells us that something
strange happens when we reach the depth for which
$p^2$ equals $1/v(z)^2$.
That is the depth at which the ray turns horizontal.
We will see in a later chapter that below this critical depth
the seismic wavefield damps exponentially with increasing depth.
Such waves are called \bx{evanescent}.
For a physical example of an evanescent wave,
forget the airplane and think about a moving bicycle.
For a bicyclist, the slowness $p$ is so large that it dominates $1/v(z)^2$
for all earth materials.
The bicyclist does not radiate a wave,
but produces a ground deformation
that decreases exponentially into the earth.
To radiate a wave,
a source must move faster than the material velocity.

\subsection{Solution to kinematic equations}
\par
The above differential equations will often reoccur in later analysis,
so they are very important.
Interestingly, these differential equations have a simple solution.
Taking the Snell wave to go through the origin at time zero,
an expression for the 
arrival time of the Snell wave at any other location
is given by
\begin{eqnarray}
t_0(x,z) \ \ \ &=&\ \ \ {\sin\,\theta  \over v }\  x\ +\ \int_0^z\ 
{\cos\,\theta  \over v }\ d z 
\label{eqn:5iei6a}
\\
t_0(x,z)\ \ \ &=&\ \ \ p\,x\ +\ \int_0^z\ 
\sqrt{ {1 \over v ( z ) ^2 }\ -\ 
p^2 } \ \ d z 
\label{eqn:5iei6b}
\end{eqnarray}
The validity of equations~(\ref{eqn:5iei6a}) and~ (\ref{eqn:5iei6b})
is readily checked by
computing  $\partial t_0 / \partial x$  and  $\partial t_0 / \partial z $,
then comparing with (\ref{eqn:5iei5a}) and (\ref{eqn:5iei5b}).

\par
An arbitrary waveform  $f(t)$  may be carried by the Snell wave.
Use (\ref{eqn:5iei6a}) and (\ref{eqn:5iei6b}) to {\em  define} the time $t_0$ for
a delayed wave  $f[t-t_0 (x,z)]$  at the location  $(x,z)$.
\begin{equation}
\hbox{SnellWave}(t,x,z)\eq f \, \left( \  t\ -\ 
p\,x\ -\  \int_0^z\ 
\sqrt{ {1 \over v ( z )^2}\ -\ p^2 } \ \ dz \  \right)
\label{eqn:5iei7}
\end{equation}
Equation~(\ref{eqn:5iei7})
carries an arbitrary signal throughout the whole medium.
Interestingly, it does not agree with wave propagation theory
or real life because
equation~(\ref{eqn:5iei7}) does not correctly account for amplitude
changes that result from
velocity changes and reflections.
Thus it is said that
Equation~(\ref{eqn:5iei7})
is ``kinematically'' correct but ``dynamically'' incorrect.
It happens that most industrial data processing only requires
things to be kinematically correct,
so this expression is a usable one.

\section{CURVED WAVEFRONTS}
The simplest waves are expanding circles.
An equation for a circle expanding with velocity  $v$
is
\begin{equation}
v^2 \, t^2  \eq  x^2 \ \ +\ \ z^2 
\label{eqn:circeqn}
\end{equation}
Considering  $t$  to be a constant,
i.e.~taking a snapshot, equation~(\ref{eqn:circeqn}) is that of a circle.
Considering  $z$  to be a constant,
it is an equation in the $(x , t)$-plane for a hyperbola.
Considered in the $(t , x , z)$-volume,
equation~(\ref{eqn:circeqn}) is that of a cone.
Slices at various values of  $t$  show circles of various sizes.
Slices of various values of  $z$  show various hyperbolas.
\par
Converting equation~(\ref{eqn:circeqn})
to traveltime depth $\tau$ we get
\begin{eqnarray}
v^2 \, t^2 &=& z^2 \ +\  x^2 \\
t^2        &=& \tau^2 \ +\ { x^2   \over  v^2 } 
\label{eqn:hyper}
\end{eqnarray}
The earth's velocity typically increases
by more than a factor of two between the earth's surface,
and reflectors of interest.
Thus we might expect that equation~(\ref{eqn:hyper}) would have little
practical use.
Luckily, this simple equation will solve many problems for us
if we know how to interpret the velocity as an average velocity.

\subsection{Root-mean-square velocity}
\sx{root-mean-square}
\sx{RMS velocity}
When a ray travels in a depth-stratified medium,
Snell's parameter $p=v^{-1}\sin\theta$ is constant along the ray.
If the ray emerges at the surface,
we can measure the distance $x$ that it has traveled,
the time $t$ it took, and its apparent speed $dx/dt=1/p$.
A well-known estimate $\hat v$
for the earth velocity contains this apparent speed.
\begin{equation}
\hat v \eq \sqrt{ {x\over t} \ {dx\over dt} }
\label{eqn:vrmsobs}
\end{equation}
To see where this velocity estimate comes from,
first notice that the stratified velocity $v(z)$ can be parameterized
as a function of time and take-off angle of a ray from the surface.
\begin{equation}
v(z) \eq v(x,z) \eq v'(p,t)
\end{equation}
The $x$ coordinate of the tip of a ray with Snell parameter $p$ is
the horizontal component of velocity integrated over time.
\begin{equation}
x(p,t) \eq \int_0^t \ v'(p,t) \ \sin\theta(p,t)\ dt
       \eq p\ \int_0^t v'(p,t)^2\ dt \ 
\end{equation}
Inserting this into equation~(\ref{eqn:vrmsobs})
and canceling $p=dt/dx$ we have
\begin{equation}
\hat v \eq
v_\RMS \eq \sqrt{ {1\over t} \ \int_0^t v'(p,t)^2\ dt\ \ }
\label{eqn:vrmsdefine}
\end{equation}
which shows that the observed velocity is the ``root-mean-square'' velocity.

\par
When velocity varies with depth,
the traveltime curve is only roughly a hyperbola.
If we break the event into many short line segments where the
$i$-th segment has a slope $p_i$ and a midpoint $(t_i,x_i)$
each segment gives a different $\hat v(p_i,t_i)$
and we have the unwelcome chore of assembling the best model.
Instead, we can fit the observational data to the best fitting hyperbola
using a different velocity hyperbola for each apex,
in other words,
find $V(\tau )$ so this equation
will best flatten the data in $(\tau,x)$-space.
\begin{equation}
t^2 \eq \tau^2 + x^2/V(\tau)^2
\end{equation}
Differentiate with respect to $x$ at constant $\tau$ getting
\begin{equation}
2t\, dt/dx \eq 2x/V(\tau)^2
\end{equation}
which confirms that the observed velocity
$\hat v$ in equation (\ref{eqn:vrmsobs}),
is the same as $V(\tau )$ no matter where you measure
$\hat v$ on a hyperbola.