Fourier Transform:
Applications
• Seismograms
• Eigenmodes of the Earth
• Time derivatives of seismograms
• The pseudo-spectral
method for acoustic wave
propagation
Fourier: Space and Time
Space x space variable
L spatial wavelength k=2/ spatial wavenumber F(k) wavenumber spectrum
Space x space variable
L spatial wavelength k=2/ spatial wavenumber F(k) wavenumber spectrum
t Time variable Time
T period
f frequency
=2 f angular frequency t Time variable Time
T period
f frequency
=2 f angular frequency Fourier integrals
Fourier integrals
With the complex representation of sinusoidal functions e
ikx(or (e
iwt) the Fourier transformation can be written as:
With the complex representation of sinusoidal functions e
ikx(or (e
iwt) the Fourier transformation can be written as:
dx e
x f k
F
dx e
k F x
f
ikx ikx
) 2 (
) 1 (
) 2 (
) 1 (
The Fourier Transform
discrete vs. continuous
dx e
x f k
F
dx e
k F x
f
ikx ikx
) 2 (
) 1 (
) 2 (
) 1 (
1 ,...,
1 , 0 ,
1 ,...,
1 , 0 1 ,
/ 1 2
0
/ 1 2
0
N k
e F f
N k
e N f
F
N N ikj
j
j k
N N ikj
j
j k
discrete
continuous
Whatever we do on the computer with data will be based on the discrete Fourier transform
Whatever we do on the
computer with data will
be based on the discrete
Fourier transform
The Fast Fourier Transform
... the latter approach became interesting with the introduction of the Fast Fourier Transform (FFT). What’s so fast about it ?
The FFT originates from a paper by Cooley and Tukey (1965, Math.
Comp. vol 19 297-301) which revolutionised all fields where Fourier transforms where essential to progress.
The discrete Fourier Transform can be written as
1 ,...,
1 , 0 ˆ ,
1 ,...,
1 , 0 1 ,
ˆ
/ 1 2
0
/ 1 2
0
N k
e u u
N k
e N u
u
N N ikj
j
j k
N N ikj
j
j k
The Fast Fourier Transform
... this can be written as matrix-vector products ...
for example the inverse transform yields ...
1 2 1 0
1 2 1 0
) 1 ( 1
2 2 6
4 2
1 3
2
ˆ ˆ
ˆ ˆ
1 1 1
1 1
1 1
1
2