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The solution to each partial differential equation, given the
amplitude values along an initial curve, will generate a surface.
The surface represents the correct distribution of amplitudes
across the 2-D operator. The problem of finding the
characteristic curves is formulated in an overdetermined form.
I expected to find both families of curves as solutions to
the partial differential equation. However each PDE
offers a single family of curves which coincides with the
required characteristic, while the other associated family
of curves is different from the form presented in Figure 1.
The answer to this problem could be found in either using a higher
order of differential equations to have as solutions the
two families of curves, or to require a single family of
curves to be described by the characteristics associated
with the PDE.
Using a single family of curves will not generate the cutoff
in the DMO ellipse.
While the approach to find the partial differential equations
was an inductive approach, a deductive analysis of the amplitudes
of the operator could generate a better understanding of the
problem.

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Stanford Exploration Project

1/13/1998