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One of the main ideas in Fourier analysis
is that an impulse function
(a delta function)
can be constructed by the superposition of sinusoids
(or complex exponentials).
In the study of time series this construction is used for the
impulse response
of a filter.
In the study of functions of space,
it is used to make a physical point source
that can manufacture the downgoing waves
that initialize the reflection seismic experiment.
Likewise observed upcoming waves can be Fourier transformed over and
.
Recall in chapter , a plane wave carrying
an arbitrary waveform, specified by
equation (
).
Specializing the arbitrary function to be
the real part of the function
gives
Using Fourier integrals on time functions we encounter the
Fourier kernel
.
To use Fourier integrals on the
space-axis
the spatial angular frequency must be defined.
Since we will ultimately encounter many space axes
(three for shot, three for geophone, also the midpoint and offset),
the convention will be to use a
subscript on the letter
to denote the
axis being Fourier transformed.
So
is the angular spatial frequency on
the
-axis and
is
its Fourier kernel.
For each axis and Fourier kernel there is the question of the
sign before the
.
The sign convention used here is the one used in most physics books,
namely, the one that agrees with equation (7.8).
Reasons for the choice are given in chapter
.
With this convention, a wave moves in the
positive
direction along the space axes.
Thus the Fourier kernel for
-space
will be taken to be
Now for the whistles, bells, and trumpets.
Equating (7.8) to the real part of (7.9),
physical angles and velocity are related to Fourier components.
The Fourier kernel has the form of a plane wave.
These relations should be memorized!
A point in
-space is a plane wave.
The one-dimensional Fourier kernel extracts frequencies.
The multi-dimensional Fourier kernel extracts (monochromatic) plane waves.
Equally important is what comes next.
Insert the angle definitions into the familiar
relation
.
This gives a most important relationship:
Equation (7.11) also achieves fame as the ``dispersion relation of the scalar wave equation,'' a topic developed more fully in IEI.
Given any and its Fourier transform
we can shift
by
if we multiply
by
.
This also works on the
-axis.
If we were given
we could shift it from the earth surface
down to any
by multiplying by
.
Nobody ever gives us
,
but from measurements on the earth surface
and double Fourier transform, we can compute
.
If we assert/assume that we have measured a wavefield, then we have
,
so knowing
means we know
.
Actually, we know
.
Technically, we also know
, but
we are not going to use it in this book.
We are almost ready to extrapolate waves from the surface into the earth
but we need to know one more thing -- which square root do
we take for ?
That choice amounts to the assumption/assertion of upcoming or
downgoing waves.
With the exploding reflector model we have no downgoing waves.
A more correct analysis has two downgoing waves to think about:
First is the spherical wave expanding about the shot.
Second arises when upcoming waves hit the surface and reflect back down.
The study of multiple reflections requires these waves.
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![]() | Downward continuation | ![]() |
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