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First we will review the calculus of step functions
and power functions and their Fourier transforms
leading up to the half-order differential operator.
With this background we can answer the question,
what is the difference between a line
and the sequence of short dashes that make up the line?
Likewise, what is the difference between a plane
and a dense array of tiny patches that make up the plane?
Is each tiny patch the same as a point source?
These questions will be answered from the viewpoint
of wave propagation where it is conceptually and computationally
convenient to make plane waves by superposing an array
of spherical waves.
We will find that a plane wave with an impulsive waveform
can be made from a superposition of circles (or hyperbolas)
or spheres (or hyperboloids of revolution)
but that in three dimensions the hyperbolid carries
not an impulsive waveform, but a time derivative, *d*/*dt*,
and in two dimensions the hyperbola carries the half-order derivative waveform.
These waveforms are generally clear on synthetic data
but on field data
they are usually obscured by waveforms from other sources.

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** Up:** Claerbout: Hankel tail
** Previous:** Claerbout: Hankel tail
Stanford Exploration Project

11/17/1997