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![]() | An algorithm for interpolation using Ronen's pyramid | ![]() |
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Let us review Ronen's pyramid transform (Hung et al., 2005).
Fourier transform data from to
.
Ronen then shrinks and stretches the space axis to transform from
to
.
All the information on a trace (constant
),
in the
plane,
lands on the line
in the the
-plane.
This is a line through the origin.
Ordinary traces have become radial traces.
Look ahead to Figure 1 and you will see an example of the basic property
of Ronen's space, the property we will prove next, namely:
Whereas dips in space are curves in
space,
they return to straight lines in
space.
Dips in physical
space become frequencies on Ronen's
-axis,
the same
-space frequency for all
.
To track the three events in Figure 1 into Ronen's space,
it's helpful to notice that each has a different
spectrum.
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synth3
Figure 1. LEFT: Input data ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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To establish the basic property of pyramid space we
consider a dipping signal in -space, say
.
Transform time
to frequency
.
Now we have a plane of complex valued information say
.
What is it in Ronen's
space?
Say
fourier transforms to
.
Then
transforms to
.
Take the time shift
to be distance
times slope
.
Then
transforms to
.
Add some extra plane waves.
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What happens in 3-D?
Instead of -axes we have
axes.
It is not immediately clear what applications will bring us to do in the
-plane.
Some applications may call for a 2-D PEF there,
while others call for two 1-D PEFs.
What are we expecting to describe in
-space?
Globally, there may be a bounding range in a slowness circle
corresponding to a slowest material velocity.
Locally things will look more like plane waves.
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![]() | An algorithm for interpolation using Ronen's pyramid | ![]() |
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