Seismic Reflection Data Processing
and Interpretation
Lecturer Title:
1D Fourier Transform
A Workshop in Cairo
28 Oct. – 9 Nov. 2006
Cairo University, Egypt
Dr. Sherif Mohamed Hanafy
1D Fourier Transform
Theory and Practice
Real Data Examples Using
SU and MatLab
Simple Harmonic Motion


A simple harmonic motion is fully described in
time domain by its amplitude, frequency, and
phase difference.
A simple harmonic motion is fully described in
space domain by its amplitude, wavelength, and
phase shift.
A wave in time domain
6
Period
4
2
Amplitude
Time
0
-2
-4
-6
Frequency = 1/Period
A wave in space domain
6
Wave length
4
2
0
-2
-4
-6
Amplitude
Distance
The connection between time and
space domains is
Velocity
Velocity  Frequency *Wave _ Length
Simple Harmonic Motion
Y (t )  A sin( 2t )
Where
A is the amplitude
w is the angular frequency
t is the time
Consider the following simple
harmonic motions
6
Amp. = 5 & Freq. = 10 Hz
4
2
0
-2
-4
-6
Time
8
Amp. = 8 & Freq. = 18 Hz
4
0
-4
-8
Time
15
Amp. = 11 & Freq. = 5 Hz
10
5
0
-5
-10
-15
Time
2
Amp. = 2 & Freq. = 14 Hz
1
0
-1
-2
Time
12
Amp. = 9 & Freq. = 21 Hz
8
4
0
-4
-8
-12
Time
Adding some of these simple harmonic
motions together will give us a more
complex harmonic motions
20
Amp. = 5 & Freq. = 10 Hz
Amp. = 9 & Freq. = 21 Hz
10
0
-10
-20
Time
20
Amp. = 5 & Freq. = 10 Hz
Amp. = 11 & Freq. = 5 Hz
10
0
-10
-20
Time
12
8
4
0
-4
-8
-12
Amp. = 2 & Freq. = 14 Hz
Amp. = 8 & Freq. = 18 Hz
Time
30
20
10
0
-10
-20
-30
Amp. = 5 & Freq. = 10 Hz
Amp. = 11 & Freq. = 5 Hz
Amp. = 9 & Freq. = 21 Hz
Time
20
Amp. = 5, 8, 11, 2 & 9 &
Freq. = 10, 18, 5, 14 & 21Hz
10
0
-10
-20
-30
Time
If we have the sinusoidal wave in
time domain, could we know the
frequencies making it?
Yes, using Fourier transform
What dose “transform” means?
Transform is taking a group of data as input to give
another group of data as output. The output results can
not be calculated unless all the input is available and used
at once
Jean Baptiste Joseph Fourier (17681830), a French mathematician and
physicist said that; “Any signal in the
time domain is the summation of a
specific number of simple sinusoidal
waves”
Fourier Transform is given by :
FT [h(t )]  H ( f ) 
 2 ift
h
(
t
)
e
dt


Inverse Fourier Transform is given by :
IFT [ H ( f )]  h(t ) 
 H ( f )e

2 ift
df
Discrete Fourier Transform
H (n f ) 
N t 1
 h ( n )e
nt  0
 2 int n f / N t
t
Discrete Inverse Fourier Transform
N t 1
1
 2int n f / N t
h(nt )   H (n f )e
N n f 0
Note
1
f 
N t t
e
2ift
1
t 
N f f
 cos(2ft)  i sin( 2ft)
DFT
Fast Fourier Transform
If number of data is = 2n, where n is a positive integer
number. Then we can use fast Fourier transform
FFT is incredibly more efficient, often reducing the
computation time by hundreds
Practical solution of FFT
1. Transform the 1 signal of N points into N
signals of 1 point
With the help of binary numbers, things are
much easier
2. Find the frequency spectra of the 1 point
time domain signals
Nothing could be easier; the frequency
spectrum of a 1 point signal is equal to itself
3. Combine the N frequency spectra in the
exact reverse order that the time domain
decomposition took place
Unfortunately, the bit reversal shortcut is not applicable, and we
must go back one stage at a time. In the first stage, 16
frequency spectra (1 point each) are synthesized into 8
frequency spectra (2 points each). In the second stage, the 8
frequency spectra (2 points each) are synthesized into 4
frequency spectra (4 points each), and so on. The last stage
results in the output of the FFT, a 16 point frequency spectrum
Synthetic Examples
1.2
A=1
F=1
Phi = 0
Amplitude
0.8
0.4
0
-0.4
-0.8
-1.2
0
Amplitude
300
0.2
0.4
0.6
0.8
1
Time (sec)
200
100
0
0
20
40
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
6
A=5
F=3
Phi = 0
Amplitude
4
2
0
-2
-4
0
-6
0.2
0.4
0.6
0.8
1
Time (sec)
1600
Amplitude
1200
800
400
0
0
20
40
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
4
A=3
F=2
Phi = 35
Amplitude
2
0
-2
-4
0
0.2
0.4
0.6
0.8
1
Time (sec)
800
Amplitude
600
400
200
0
0
20
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
A = 1, 3, 3, 2, 5, 1, 3, 2, 3, 4
F = 3, 7, 11, 15, 19, 23, 27, 31, 35, 40
Phi = 0, 30, 12, 45, 90, 13, 0, 23, 21, 40
20
Amplitude
40
0
-20
0
0.2
0.4
0.6
0.8
1
Time (sec)
1600
Amplitude
1200
800
400
0
0
20
40
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
800
A = Random, 5<= A <=10
F = 10 to 20 step 0.1
Phi = Random 0<= Phi <=90
Amplitude
400
0
-400
-800
0
0.2
0.4
0.6
0.8
1
Time (sec)
16000
Amplitude
12000
8000
4000
0
0
20
40
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
800
A = Random, 5<= A <=10
F = 10 to 12 step 0.05 + 30 to 36 step 0.1
Phi = Random 0<= Phi <=90
Amplitude
400
0
-400
-800
0
0.2
0.4
0.6
0.8
1
Time (sec)
20000
Amplitude
16000
12000
8000
4000
0
0
20
40
60
80
100
120
140
Frequancy (Hz)
160
180
200
220
240
Real Examples
Real Examples
Real Examples
End of this lecture
Thank You for you attention
All examples on this lecture is based on my work