Environmental and Exploration Geophysics II
tom.h.wilson
tom.wilson@mail.wvu.edu
Tom Wilson, Department of Geology and Geography
Mathematical operation used to
compute the synthetic seismogram

C (t )   a( )b(t   )d
The convolution integral

Seismic Analog

S (t )   w( )r (t   )d

where S is the seismic signal or
trace, w is the seismic wavelet,
and r is the reflectivity sequence
Tom Wilson, Department of Geology and Geography
Physical nature of the seismic response
Tom Wilson, Department of Geology and Geography

Convolutional model
S (t )   w( )r (t   )d

The output is a superposition of reflections from all acoustic interfaces and the
convolution integral is a statement of the superposition principle.
Tom Wilson, Department of Geology and Geography
Discrete form of the Convolution Integral
t
St   w rt 
 0
As defined by this equation, the process of convolution
consists of 4 simple mathematical operations
1) Folding
2) Shifting
3) Multiplication
4) Summation
Tom Wilson, Department of Geology and Geography
Complex numbers
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
Given the in-phase and quadrature components, it is
easy to calculate the amplitude and phase or vice versa.
Tom Wilson, Department of Geology and Geography
The seismic trace is
the “real” or in-phase
component of the
complex trace
How do we find the
quadrature component?
Tom Wilson, Department of Geology and Geography
In the time domain the Hilbert Transform consists of a series
of values that are asymmetrical in shape: positive to one side
and negative to the other. Values in the series are located at
odd sample points relative to the middle of the series and
diminish in magnitude with odd divisors: 1,3,5, etc.
Tom Wilson, Department of Geology and Geography
Tanner, Koehler, and Sheriff, 1979
View of Hilbert transform operator in
relation to the samples in a seismic trace
From Marfurt, 2006, SEG Short Course
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
Recall Frequency Domain versus Time Domain Relationships
Tom Wilson, Department of Geology and Geography
Amplitude spectrum
Fourier
Transform
of a time
series
Phase spectrum
Individual
frequency
components
Time-domain wavelets
Tom Wilson, Department of Geology and Geography
Zero Phase
Minimum Phase
Seismic Trace
6000
4000
Amplitude
The seismic
response is
a “real” time
series
2000
0
-2000
-4000
-6000
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Time (seconds)
Amplitude Spectrum
This is its
amplitude
spectrum
Amplitude
16000
12000
8000
4000
0
0
50
100
150
Frequency
Tom Wilson, Department of Geology and Geography
200
250
The Fourier Transform of a real function, like a seismic
trace, is complex, i.e., it has real and imaginary parts.
Real Part of the Fourier Transform
5
2x10
The real
part is
even
Amplitude
5
2x10
5
1x10
Symmetrical
4
5x10
0
4
-5x10
5
-1x10
-2x10
5
-20
-15
-10
-5
0
5
10
15
20
Frequency (Hz)
Imaginary Part of the Fourier Transform
5
2x10
5
Asymmetrical
1x10
Amplitude
The
imaginary
part is odd
4
5x10
0
-5x10
-1x10
-2x10
4
5
5
-20
-15
-10
-5
0
5
Frequency (Hz)
Tom Wilson, Department of Geology and Geography
10
15
20
Real Part
Amplitude
1000
Line 49, CrossLine 30
500
0
150
Spectrum
-500
100
100
Amplitude
Amplitude
-1000
50
0
80
-1500
-100
60
-50
0
50
100
Frequency
40
-50
20
Imaginary Part
-100
0
0.8
1.0
1.2
1.4
Time (sec)
Seismic response
of the channel
1.6
-50
0
50
100
Frequency (Hz)
Amplitude
-100
-150
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-100
-50
0
Frequency (Hz)
Tom Wilson, Department of Geology and Geography
50
100
Line 49, CrossLine 30
150
Amplitude
100
50
0
-50
-100
-150
0.8
1.0
1.2
Time (sec)
Tom Wilson, Department of Geology and Geography
1.4
1.6
Tom Wilson, Department of Geology and Geography
Tom Wilson, Department of Geology and Geography
Computer exercise: generating attributes and evaluating their ability to
enhance the view of the channel observed in the Gulf Coast 3D volume
Tom Wilson, Department of Geology and Geography