paper.tex

国外免费地震资料处理软件包

% copyright (c) 1997 Jon Claerbout

\title{Plane waves in three dimensions}
\author{Jon Claerbout}
\label{paper:lmn}
\maketitle

\inputdir{XFig}
In this chapter we seek a deeper understanding
of \bx{plane wave}s in {\it three} dimensions,
where the examples and theory typically refer to functions
of time $t$ and two space coordinates $(x,y)$,
or to 3-D migration images where the $t$
coordinate is depth or traveltime depth.
As in Chapter \ref{pch/paper:pch},
we need to decompose data volumes into subcubes,
shown in Figure \ref{fig:rayab3D}.
\sideplot{rayab3D}{width=3.00in}{
  Left is space of inputs and outputs.
  Right is their separation during analysis.
}

\par
In this chapter we will see that
the wave model implies the 3-D whitener is not a cube filter
but two planar filters.
The wave model allows us to determine
the scale factor of a signal,
even where signals fluctuate in strength because of interference.
Finally, we examine the local-monoplane concept
that uses the superposition principle
to distinguish a sedimentary model cube from a data \bx{cube}.

%\begin{notforlecture}
\section{THE LEVELER:  A VOLUME OR TWO PLANES?}
\sx{leveler}
\par
In {\it two} dimensions,
levelers were taken to be PEFs,
small rectangular planes of numbers in which the time axis
included enough points to include reasonable stepouts were included
and the space axis contained one level plus another space level,
for each plane-wave slope supposed to be present.
\par
We saw that a whitening filter in {\it three} dimensions
is a small volume with shape defined by subroutine 
\texttt{createhelix()}.
%\vpageref{/prog:createhelix_mod}.  % unresolved reference
It might seem natural that the number of points on the $x$-
and $y$-axes be related to the number of plane waves present.
Instead, I assert that if the volume contains plane waves,
we don't want a {\it volume} filter to whiten it;
we can use a {\it pair} of {\it planar} filters to do so
and the order of those filters is the number of planes
thought to be simultaneously present.
I have no firm mathematical proofs,
but I offer you some interesting discussions, examples,
and computer tools for you to experiment with.
It seems that some applications call for the volume filter
while others call for the two planes.
Because two planes of numbers generally contain
many fewer adjustable values than a volume,
statistical-estimation reasons also favor the planes.

\par\noindent
\boxit{
        What is the lowest-order filter that,
        when applied to a volume,
        will destroy
        one and only one slope of plane wave?
        }

\par
First we seek the answer to the question,
``What is the lowest order filter that will destroy
one and only one plane?''
To begin with, we consider that plane to be horizontal
so the volume of numbers is
$f(t,x,y) = b(t)$
where $b(t)$ is an arbitrary function of time.
One filter that has zero-valued output (destroys the plane)
is 
$\partial_x \equiv \partial / \partial x$.
Another is the operator
$\partial_y \equiv \partial / \partial y$.
Still another is the \bx{Laplacian operator} which is
$\partial_{xx}+\partial_{yy}\equiv\partial^2/\partial x^2+\partial^2/\partial y^2$.

\par
The problem with $\partial / \partial x$ is that
although it destroys the required plane,
it also destroys
$f(t,x,y) = a(t,y)$ where $a(t,y)$ is an {\it arbitrary} function of $(t,y)$
such as a cylinder with axis parallel to the $x$-axis.
The operator $\partial / \partial y$ has the same problem
but with the axes rotated.
The Laplacian operator not only destroys our desired plane,
but it also destroys the well known nonplanar function
$e^{ax}\cos(ay)$,
which is just one example of the many other interesting shapes
that constitute solutions to Laplace's equation.
\par
I remind you of a basic fact:
When we set up the fitting goal $\bold 0\approx \bold A \bold f$,
the \bx{quadratic form} minimized is $\bold f'\bold A' \bold A \bold f$,
which by differentiation with respect to $\bold f'$
gives us (in a constraint-free region) $\bold A'\bold A\bold f = \bold 0$.
So, minimizing the volume integral (actually the sum)
of the squares of the components of the gradient
implies that Laplace's equation is satisfied.

\par
In any volume,
the lowest-order filter
that will destroy level planes and no other wave slope
is a filter that has one input and {\it two outputs}.
That filter is the gradient,
$(\partial / \partial x, \partial / \partial y)$.
Both outputs vanish if and only if
the plane has the proper horizontal orientation.
Other objects and functions are not extinguished
(except for the non-wave-like function $f(t,x,y)= {\rm const}$).
It is annoying that we must deal with {\it two} outputs
and that will be the topic of further discussion.

\par
A wavefield of tilted parallel planes is
$f(t, x,y)= g(\tau -p_x x -p_y y)$, where $g()$ is
an arbitrary one-dimensional function.
The operator that destroys these tilted planes
is the two-component operator
$(\partial_x + p_x \partial_t,\   \partial_y + p_y \partial_t)$.

\par\noindent
\boxit{
\noindent
The operator that destroys a family of dipping planes
$$f(t, x,y) \eq g(\tau -p_x x -p_y y)$$
is
$$
\left[
\begin{array}{c}
        {\partial \over \partial x} \ +\  p_x \,{\partial \over \partial t} \\
        \ \\
        {\partial \over \partial y} \ +\  p_y \,{\partial \over \partial t} 
\end{array}
\right]
$$
}

\subsection{PEFs overcome spatial aliasing of difference operators}
The problem I found with
finite-difference representations of differential operators
is that they are susceptible to \bx{spatial aliasing}.
Even before they encounter spatial aliasing,
they are susceptible to accuracy problems known
in finite-difference wave propagation as ``frequency dispersion.''
The aliasing problem can be avoided by the use of spatial prediction operators
such as
\begin{equation}
   \begin{array}{cc}
      \cdot &a     \\
      \cdot &b     \\
      1     &c     \\
      \cdot &d     \\
      \cdot &e     \end{array}
\label{eqn:spatialpred}
\end{equation}
where the vertical axis is time;
the horizontal axis is space; and
the ``$\cdot$''s are zeros.
Another possibility is the 2-D whitening filter
\begin{equation}
   \begin{array}{cc}
      f     &a     \\
      g     &b     \\
      1     &c     \\
      \cdot &d     \\
      \cdot &e
      \end{array}
\label{eqn:twoDwhite}
\end{equation}
Imagine all the coefficients vanished but $d=-1$ and the given 1.
Such filters would annihilate the appropriately sloping plane wave.
Slopes that are not exact integers are also approximately extinguishable,
because the adjustable filter coefficients can interpolate in time.
Filters like (\ref{eqn:twoDwhite})
do the operation $\partial_x + p_x \partial_t$,
which is a component of the gradient in the plane of the wavefront,
{\it and}
they include a temporal deconvolution aspect
and a spatial coherency aspect.
My experience shows that the operators (\ref{eqn:spatialpred}) and
(\ref{eqn:twoDwhite})
behave significantly differently in practice,
and I am not prepared to fully explain the difference,
but it seems to be similar to the gapping of one-dimensional filters.

\par
You might find it alarming
that your teacher is not fully prepared
to explain the difference between a volume and two planes,
but please remember that we are talking about
the factorization of the volumetric spectrum.
Spectral matrices are well known to have structure,
but books on theory typically handle them as simply $\lambda \bold I$.
Anyway, wherever you see an $\bold A$ in a three-dimensional context,
you may wonder whether it should be interpreted as a cubic filter
that takes one volume to another,
or as two planar filters
that take one volume to two volumes
as shown in Figure~\ref{fig:rayab3Doper}.
\sideplot{rayab3Doper}{width=1.50in}{
  An inline 2-D PEF and a crossline 2-D PEF
  both applied throughout the volume.
  To find each filter,
  minimize each output power independently.
}

\subsection{My view of the world}
\par
I start from the idea that the four-dimensional world $(t,x,y,z)$
is filled with expanding spherical waves and with quasispherical waves
that result from reflection from quasiplanar objects
and refraction through quasihomogeneous materials.
We rarely, if ever see
in an observational data cube,
an entire expanding spherical wave,
but we normally have a two- or three-dimensional slice
with some wavefront curvature.
We analyze data subcubes that I call \bx{bricks}.
In any brick we see only local patches of apparent plane waves.
I call them \bx{platelets}.
From the microview of this brick,
the platelets come from the ``great random-point-generator in the sky,''
which then somehow convolves the random points
with a platelike impulse response.
If we could deconvolve these platelets back to their random source points,
there would be nothing left inside the brick
because the energy would have gone outside.
We would have destroyed the energy inside the brick.
If the platelets were coin shaped,
then the gradient magnitude would convert each coin to its circular rim.
The plate sizes and shapes are all different
and they damp with distance from their centers, as do Gaussian beams.
If we observed {\it rays} instead of wavefront platelets
then we might think of the world as being filled with noodles,
and then.\ .\ .\ .
\par
How is it possible that in a small brick we can do something realistic
about deconvolving a spheroidal impulse response
that is much bigger than the brick?
The same way as in one dimension,
where
in a small time interval we can estimate the correct deconvolution filter
of a long resonant signal.
A three-point filter destroys a sinusoid.
\par
The inverse filter to the expanding spherical wave might be a huge cube.
Good approximations to this inverse at the brick level
might be two small planes.
Their time extent would be chosen to encompass the slowest waves,
and their spatial extent could be two or three points,
representing
the idea that normally we can listen to only one person at a time,
occasionally we can listen to two,
and we can never listen to three people talking at the same time.

\section{WAVE INTERFERENCE AND TRACE SCALING}
\sx{interference}
\sx{scaling a trace}
Although neighboring seismometers tend to show equal powers,
the energy on one seismometer can differ greatly
from that of a neighbor for both theoretical reasons
and practical ones.
Should a trace ever be rescaled
to give it the same energy as its neighbors?
Here we review the strong theoretical arguments against rescaling.
In practice,
however, especially on land where coupling is irregular,
scaling seems a necessity.
The question is,
what can go wrong if we scale traces to have equal energy,
and more basically,
where the proper \bx{scale factor} cannot be recorded,
what should we do to get the best scale factor?
A related question is how to make good measurements of amplitude versus offset.
To understand these issues we review
the fundamentals of wave interference.

\inputdir{scale}

\par
Theoretically, a scale-factor problem arises
because {\it locally,} wavefields, not energies, add.
Nodes on standing waves are familiar from theory,
but they could give you the wrong idea that the concept of node