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An exasperating problem in seismology is the ``ghost'' problem,
in which a waveform is replicated a moment after it occurs
because of a strong nearby reflection.
In marine seismology the nearby reflector is the sea surface.
Because the sea surface is near both the airgun and the hydrophones,
it creates two ghosts.
Upgoing and downgoing waves at the sea surface
have opposite polarity because their pressures combine
to zero at the surface.
Thus waves seen in the hydrophone encounter
the ghost operator twice,
once for the surface near the source and once for
the surface near the hydrophone.
The number of zeros is typically small,
depending on the depth of the device.
The sound receivers can be kept away from surfacewater wave noise
by positioning them deeper,
but that extends the ghost delay;
and as we will see, this particular ghost is
very hard to eliminate by processing.
For simplicity, let us analyze just one of the two ghosts.
Take it to be G(Z)=1Z^{2}.
Theoretically, the inverse is of infinite duration,
namely, .
Since an infinitely long operator is not satisfactory,
I used the program shaper() above
to solve a leastsquares problem for an antighost operator of finite duration.
Since we know that the leastsquares method abhors large errors
and thus tends to equalize them,
we should be able to guess the result.
The filter
(.9, .0, .8, .0,.7 ,.0, .6, .0, .5, .0, .4, .0, .3, .0, .2, .0, .1),
when convolved with (1,0,1), produces
the desired spike (impulse) along with
equal squared errors
of .01 at each output time.
Thus, the leastsquares filter has the same problem as the analytical oneit
is very long.
This disappointment can be described in the Fourier domain
by the many zeros in the spectrum of (1,0,1).
Since we cannot divide by zero, we should not try to divide
by 1Z^{n}, which has zeros uniformly distributed on the unit circle.
The method of least squares prevents disaster, but it cannot perform miracles.
I consider ghosts to be a problem in search of a different solution.
Ghosts also arise when seismograms are
recorded in a shallow borehole.
As mentioned,
the total problem generally includes many waveforms
propagating in more than one direction;
thus it is not as onedimensional
as it may appear in Figures 3
and 1, in which I did not display the wideoffset signals.
EXERCISES:

What inputs to subroutine shaper() give
the filter
mentioned above?

Figure 1 shows many seismograms that resemble each other
but differ in the x location of the receiver.
Sketch the overdetermined simultaneous equations
that can be used to find the bestfitting source function S(Z),
where for various x.

Continue solving the previous problem by
defining a contranx() subroutine
that includes several signals going through the same filter.
In order to substitute your
contranx() into
shaper() to replace contran() ,
you will need to be sure that the output and the filter are adjoint
(not the output and the input).
Suggestion: define real xx(nt,nx), etc.
Next: SYNTHETIC DATA FROM FILTERED
Up: SHAPING FILTER
Previous: Source waveform and multiple
Stanford Exploration Project
10/21/1998