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An example of a
twodimensional Fourier transform
of a pulse is shown in Figure 5.
ft2dofpulse
Figure 5
A broadened pulse (left) and the real part of its FT (right).
Notice the location of the pulse.
It is closer to the time axis than the frequency axis.
This will affect the real part of the FT in a certain way
(see exercises).
Notice the broadening of the pulse.
It was an impulse smoothed over time (vertically) by convolution
with (1,1) and over space (horizontally) with (1,4,6,4,1).
This will affect the real part of the FT in another way.
Another example of a twodimensional Fourier transform
is given in Figure 6.
This example simulates an impulsive air wave originating at a point
on the xaxis.
We see a wave propagating in each direction
from the location of the source of the wave.
In Fourier space there are also two lines, one for each wave.
Notice that there are other lines which do not go through the origin;
these lines are called ``spatial aliases.''
Each actually goes through the origin
of another square plane that is not shown,
but which we can imagine alongside the one shown.
These other planes are periodic replicas of the one shown.
airwave
Figure 6
A simulated air wave (left) and the amplitude of its FT (right).
EXERCISES:

Most time functions are real.
Their imaginary part is zero.
Show that this means that can
be determined from .

What would change in Figure 5
if the pulse were moved
(a) earlier on the taxis, and
(b) further on the xaxis?
What would change in Figure 5
if instead
the time axis were smoothed with (1,4,6,4,1)
and the space axis with (1,1)?

What would Figure 6
look like on an earth with half the earth velocity?

Numerically (or theoretically)
compute the twodimensional spectrum
of a plane wave [], where
the plane wave has a randomly fluctuating amplitude:
say, rand(x) is a random number between ,and the randomly modulated plane wave is
[].

Explain the horizontal ``layering'' in Figure 4
in the plot of .What determines the ``layer'' separation?
What determines the ``layer'' slope?
Next: THE HALFORDER DERIVATIVE WAVEFORM
Up: SETTING UP 2D FT
Previous: Signs in Fourier transforms
Stanford Exploration Project
12/26/2000