Fourier Transforms and Waves.pdf

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FOURIER TRANSFORMS AND WAVES:
in four lectures
Jon F. Clærbout
Cecil and Ida Green Professor of Geophysics
Stanford University
c January 18, 1999
Contents
1 Convolution and Spectra
1.1
1.2
1.3
1.4
SAMPLED DATA AND Z-TRANSFORMS . . . . . . . . . . . . . . . . .
FOURIER SUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FOURIER AND Z-TRANSFORM . . . . . . . . . . . . . . . . . . . . . .
1
1
5
8
CORRELATION AND SPECTRA . . . . . . . . . . . . . . . . . . . . . . 11
17
2 Discrete Fourier transform
2.1
2.2
2.3
2.4
FT AS AN INVERTIBLE MATRIX . . . . . . . . . . . . . . . . . . . . . 17
INVERTIBLE SLOW FT PROGRAM . . . . . . . . . . . . . . . . . . . . 20
SYMMETRIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
TWO-DIMENSIONAL FT . . . . . . . . . . . . . . . . . . . . . . . . . . 23
29
3 Downward continuation of waves
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
DIPPING WAVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
DOWNWARD CONTINUATION . . . . . . . . . . . . . . . . . . . . . . 32
A matlab program for downward continuation . . . . . . . . . . . . . . . . 36
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
CONTENTS
3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Index
39
Why Geophysics uses Fourier Analysis
When earth material properties are constant in any of the cartesian variables
it is useful to Fourier transform (FT) that variable.
then
In seismology, the earth does not change with time (the ocean does!) so for the earth, we
can generally gain by Fourier transforming the time axis thereby converting time-dependent
differential equations (hard) to algebraic equations (easier) in frequency (temporal fre-
quency).
In seismology, the earth generally changes rather strongly with depth, so we cannot
usefully Fourier transform the depth axis and we are stuck with differential equations in
. On the other hand, we can model a layered earth where each layer has material properties
that are constant in . Then we get analytic solutions in layers and we need to patch them
together.
Thirty years ago, computers were so weak that we always Fourier transformed the
and coordinates. That meant that their analyses were limited to earth models in which
velocity was horizontally layered. Today we still often Fourier transform
but not ,
so we reduce the partial differential equations of physics to ordinary differential equations
(ODEs). A big advantage of knowing FT theory is that it enables us to visualize physical
behavior without us needing to use a computer.
The Fourier transform variables are called frequencies. For each axis
have a corresponding frequency
. The ’s are spatial frequencies,
temporal frequency.
we
is the
The frequency is inverse to the wavelength. Question: A seismic wave from the fast
earth goes into the slow ocean. The temporal frequency stays the same. What happens to
the spatial frequency (inverse spatial wavelength)?
In a layered earth, the horizonal spatial frequency is a constant function of depth. We
will find this to be Snell’s law.
In a spherical coordinate system or a cylindrical coordinate system, Fourier transforms
are useless but they are closely related to “spherical harmonic functions” and Bessel trans-
formations which play a role similar to FT.
Our goal for these four lectures is to develop Fourier transform insights and use them
to take observations made on the earth’s surface and “downward continue” them, to extrap-
i

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