paper.tex

国外免费地震资料处理软件包

The output panel tends to be small where there is only a single dip present.
Where two dips cross, they tend to be equal in magnitude.
Studying the output more carefully,
we notice that of the two dips,
the one that is strongest on the input
becomes irregular and noisy on the output,
whereas the other dip tends to remain phase-coherent.
\par
I could rebuild Figure~\ref{fig:conflict} 
to do a better job of suppressing monodip areas
if I passed the image through a lowpass filter,
and then designed a gapped deconvolution operator.
Instead, I preferred to show you high-frequency noise
in the place of an attenuated wavefront.

\par
The residual of prediction-error deconvolution
tends to have a white spectrum in time.
This aspect of deconvolution is somewhat irritating
and in practice it requires us to postfilter for display,
to regain continuity of signals.
As is well known (PVI, for example),
an alternative to postfiltering is to put a gap in the filter.
A gapped filter should work with 2-D filters too,
but it is too early to describe how
experimenters will ultimately choose
to arrange gaps, if any,  in 2-D filters.
There are some interesting possibilities.
(Inserting a gap also reduces the required number of CD iterations.)

\subsection{Tests of 2-D LOMOPLAN on field data}
Although the LOMOPLAN concept was developed for geophysical {\it models},
not raw {\it data},
initial experience showed that the LOMOPLAN program
is effective for quality testing data
and data interpretation.
\par
Some field-data examples are in Figures~\ref{fig:dgulf} and \ref{fig:yc27}.
These results are not surprising.
A dominant local plane is removed,
and noise or the second-from-strongest local plane is left.
These data sets fit the local plane model so well
that subtracting the residual noise from the data made little improvement.
These figures are clearer on a video screen.
To facilitate examination of the residual on Figure~\ref{fig:dgulf} on paper
(which has a lesser dynamic range than video),
I recolored the white residual with a short triangle filter on the time axis.

\plot{dgulf}{width=6.00in,height=3.5in}{
  Data section from the Gulf of Mexico (left)
  and after LOMOPLAN (right)
  Press button for movie.
}

The residual in Figure~\ref{fig:yc27} is large at the dead trace
and wherever the data contains crossing events.
Also, closer examination showed that the strong residual trace
near 1.1 km offset is apparently slightly time-shifted,
almost certainly a cable problem,
perhaps resulting from a combination of the stepout and a few dead pickups.
Overall, the local-plane residual shows a low-frequency water-velocity wave
seeming to originate from the ship.

\plot{yc27}{width=6.00in,height=3.5in}{
  Portion of Yilmaz and Cumro data set 27 (left)
  and after LOMOPLAN (right).
  Press button for movie.
}

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\section{GRADIENT ALONG THE BEDDING PLANE}
\par
The LOMOPLAN (LOcal MOnoPLane ANnihilator) filter
in three dimensions is a deconvolution filter that takes a volume in
and produces two volumes out.
The $x$-output volume results from a first order
prediction-error filter on the $x$-axis,
and the $y$-output volume is likewise on the $y$-axis.

\par
Although
I conceived of 2-D LOMOPLAN as the ``ultimate'' optimization criterion
for inversion problems in reflection seismology of sedimentary sections,
it turned out that it was more useful in data interpretation
and in data-quality inspection.
In this study, I sought to evaluate usefulness with
{\it three}-dimensional data
such as 3-D stacks or migrated volumes, or 2-D prestack data.

\par
In experimenting with 3-D LOMOPLAN,
I came upon a conceptual oversimplification,
which although it is not precisely correct,
gives a suitable feeling of the meaning of the operator.
Imagine that the earth was flat horizontal layers, except for occasional faults.
Then, to find the faults you might invoke the horizontal
gradient of the 3-D continuum of data.
The horizontal components of gradient vanish except at a fault,
where their relative magnitudes tell you the orientation of the fault.
Instead of using the gradient vector,
you could use prediction-error filters of first order (two components)
along $x$ and $y$ directions.
3-D LOMOPLAN is like this,
but the flat horizontal \bx{bedding} may be dipping or curved.
No output is produced (ideally) except at faults.
The 3-D LOMOPLAN is like the gradient
{\it along the plane of the bedding}.
It is nonzero where the bedding has an intrinsic change.

\par\noindent
\boxit{LOMOPLAN flags the bedding where there is an intrinsic change.}

\subsection{Definition of LOMOPLAN in 3-D}
Three-dimensional LOMOPLAN is somewhat like multiple passes
of two-dimensional LOMOPLAN;
i.e., we first LOMOPLAN the $(t,x)$-plane for each $y$,
and then we
LOMOPLAN the $(t,y)$-plane for each $x$.
Actually, 3-D LOMOPLAN is
a little more complicated than this.
Each LOMOPLAN filter is designed on all the data in a small $(t,x,y)$ volume.
\par
%I began from my earlier two-dimensional code
%and made the obvious extensions to three dimensions.
%For example,
%converting subroutine \texttt{icaf2()} \vpageref{/prog:icaf2} to the 3-D version
%{\tt icaf3()} gives a subroutine that convolves
%a volume over a volume to get a volume.
%When we need a 2-D filter,
%we pick {\tt a3=1} and with it,
%subroutine {\tt icaf3()}
%convolves the planar filter throughout the input volume
%and thus gives an output volume.
%When we want a monoplane filter orthogonally oriented,
%we take {\tt a2=1} but {\tt a3=2}.

\par
To put the LOcal in LOMOPLAN we use subcubes (bricks).
Recall that we can do 2-D LOMOPLAN with
the prediction-error subroutine 
\texttt{find\_lopef()} \vpageref{/prog:lopef}.
To do 3-D LOMOPLAN we need to make two calls to subroutine
{\tt find\_lopef()},
one for the $x$-axis in-line planar filters
and one for the $y$-axis crossline filters.
That is what I will try next time
I install this book on a computer with a bigger memory.

\subsection{The quarterdome  3-D synthetic (qdome)}
\sx{quarterdome 3-D synthetic (qdome)}
\sx{qdome}
\inputdir{sep77}
Figure~\ref{fig:sigmoid}
used a model called ``\bx{Sigmoid}.''
Using the same modeling concepts,
I set out to make a three-dimensional model.
The model has horizontal layers near the top,
a Gaussian appearance in the middle, and dipping layers on the bottom,
with horizontal unconformities between the three regions.
Figure~\ref{fig:qdomesico} shows
a vertical slice through the 3-D ``qdome'' model
and components of its LOMOPLAN.
There is also a fault that will be described later.
\plot{qdomesico}{width=6.00in,height=3.5in}{
  Left is a vertical slice through the 3-D ``qdome'' model.
  Center is the in-line component of the LOMOPLAN.
  Right is the cross-line component of the LOMOPLAN.
}
The most interesting part of the qdome model is the Gaussian center.
I started from the equation of a \bx{Gaussian}
\begin{equation}
z(x,y,t) \quad = \quad e^{-(x^2+y^2)/ t^2}
\end{equation}
and backsolved for $t$
\begin{equation}
t(x,y,z) \quad = \quad \sqrt{x^2+y^2 \over -\ln z}
\end{equation}
Then I used a random-number generator
to make a blocky one-dimensional impedance function of $t$.
At each $(x,y,z)$ location in the model
I used the impedance at time $t(x,y,z)$,
and finally defined reflectivity as the logarithmic derivative
of the impedance.
Without careful interpolation (particularly where the beds pinch out)
a variety of curious artifacts appear.
I hope to find time to
use the experience of making the qdome model
to make a tutorial lesson on interpolation.
A refinement to the model is that within a certain subvolume
the time $t(x,y,z)$ is given a small additive constant.
This gives a fault along the edge of the subvolume.
Ray \bx{Abma} defined the subvolume for me in the qdome model.
The fault looks quite realistic,
and it is easy to make faults of any shape,
though I wonder how they would relate to realistic fault dynamics.
Figure~\ref{fig:qdometoco} shows
        a top view of
        the 3-D qdome model
        and components of its LOMOPLAN.
\noindent
Notice that the cross-line spacing
has been chosen to be double the in-line spacing.
Evidently a consequence of this,
in both
Figure~\ref{fig:qdomesico} and
Figure~\ref{fig:qdometoco}, is that the Gaussian dome is not so well suppressed
on the crossline cut as on the in-line cut.
By comparison, notice that the horizontal bedding above the dome
is perfectly suppressed,
whereas the dipping bedding below the dome is imperfectly suppressed.
\plot{qdometoco}{width=6.00in,height=2.1in}{
  Left is a horizontal slice through the 3-D qdome model.
  Center is the in-line component of the LOMOPLAN.
  Right is the cross-line component of the LOMOPLAN.
%  Press button for volume view.
}

\par
Finally, I became irritated at the need to look at {\it two} output volumes.
Because I rarely if ever interpreted the polarity of the LOMOPLAN components,
I formed their sum of squares and show the single
square root volume in Figure~\ref{fig:qdometora}.

\plot{qdometora}{width=6.00in,height=3.2in}{
  Left is the model.
  Right is the magnitude of the LOMOPLAN
  components in Figure \protect\ref{fig:qdometoco}.
%  Press button for volume view.
}

\section{3-D SPECTRAL FACTORIZATION}
Hi Sergey, Matt, and Sean,
Here are my latest speculations, plans:

The 3-D Lomoplan resembles a gradient, one field in, two or three out.
Lomoplan times its adjoint is like a generalized laplacian.
Factorizing it yields a lomoplan generalization of the helix derivative,
i.e. a one-to-one operator with the same spectral charactoristic
as the original lomoplan.
It will probably not come out to be a juxtaposition of planes,
will be more cube like.

The advantage of being one-to-one is
that it can be used as a preconditioner.
The application, naturally enough,
is estimating things with a prescribed dip spectrum.
Things like missing data and velocities.

Why use multiplanar lomoplan estimates if they will then be
converted by this complicated process into a cube?
Why not estimate the cube directly?  Maybe to impose
the ``pancake" model instead of the noodle model of covariance.
Maybe to reduce the number of coefficients to estimate.

I haven't figured out yet how to convert this speculation
into an example leading to some figures.
If you like the idea, feel free to beat me to it :)


%\subsection{Attempted field test of 3-D LOMOPLAN}
%I experimented with some 3-D data to see if the LOMOPLAN
%view would prove interesting.
%Preliminary views were not,
%so I omit them here.
%First I looked at the Gulf of Thailand GSI migrations.
%Unfortunately all I could find was the pixel byte form,
%so I converted that back to floats.
%Second, I examined Mihai's unmigrated stack from
%Halliburton's High Island 3D spec survey.
%I experimented first with the shallow faults,
%but these were confounded by multiples
%and the little dip present made the LOMOPLAN time slices in no way
%more interesting than those of the input data.
%A problem I had with the early times on this survey
%is that the crosslines had noticeable
%(but inexplicable) timeshifts from one line to another.
%Thus the crosslines, and hence the 3-D nature of the LOMOPLAN
%was not exemplified.
%The crossline problem was absent at later times.
%Because the computations were fairly demanding however,
%before results could be obtained
%I turned to learning the parallel computer,
%and got caught up with the good results
%I obtained elsewhere in this monograph with steep-dip deconvolution.
%\par
%Prestack two-dimensional data offers another interesting area
%for investigation.

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