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Reconstructing the earth model with the conjugate option in kirchfast()
yields the result in Figure 2.
skmig
Figure 2
Left is the original model
Right is the reconstruction.
The reconstruction generally succeeds
but is imperfect in a number of interesting ways.
Near the bottom and right side, the reconstruction fades away
especially where the dips are steeper.
Bottom fading results because in modeling the data
we abandoned arrivals after a certain maximum time.
Thus energy needed to reconstruct dipping beds near the bottom
was abandoned.
Likewise along the side we abandoned rays shooting off the frame.
Difficult migrations are well known for producing semicircular reflectors.
Here we have controlled everything fairly well so none are obvious,
but on a video screen I see some semicircles.
Next is the problem of the spectrum.
Notice in Figure 2 that the reconstruction
lacks the sharp crispness of the original.
It can be shown mathematically
that the spectrum of our reconstruction
loses high frequencies by a scale of about .Philosophically, we can think of the integration as boosting
low frequencies.
Figure 3 shows the average over x
of the relevant spectra.
kirspec
Figure 3
Top is the spectrum of the left side of
Figure 2.
Bottom is the spectrum of the right side of
Figure 2.
Middle is bottom times frequency.

 
First, notice the high frequencies are weak because
there is little high frequency energy in the original model.
Then notice that our cavalier approach to interpolation
created more high frequency energy.
Finally, notice that multiplying the spectrum of our
migrated model by frequency, f, brought the important
part of the spectral bands into agreement.
This suggest applying an filter to our reconstruction,
or operator to both the modeling and the
reconstruction.
We will take that up somewhere else.
Neither of these Kirchhoff codes addresses the issue of spatial aliasing.
Spatial aliasing is a vexing issue of numerical analysis.
The Kirchhoff codes shown here do not work as expected
unless the space mesh size is suitably more refined than the time mesh.
Figure 4 shows an example of forward modeling
with a spatial mesh of 50 and 100 points.
skmod
Figure 4
Left is model.
Right is synthetic data from the model.
Top has 50 points on the xaxis,
bottom has 100.
(Previous figures used 200 points on space.
All use 200 mesh points on the time.)
Subroutine kirchfast() does interpolation by moving values
to the nearest neighbor of the theoretical location.
Had we taken the trouble to interpolate the two nearest points,
our results would have been a little better,
but the basic problem (resolved in jon/res/trimo) would remain.
Next: REFERENCES
Up: Claerbout: Kirchhoff introduction
Previous: FAST KIRCHHOFF CODE
Stanford Exploration Project
11/18/1997