Transcript No Slide Title

Environmental and Exploration Geophysics II
tom.h.wilson
[email protected]
Seismic Trace Attributes
Complex numbers
Real and Imaginary
Interrelationships
Given the in-phase and quadrature components, it is
easy to calculate the amplitude and phase or vice versa.
Seismic Data
The seismic trace is
the “real” or in-phase
component of the
complex trace
How do we find the
quadrature component?
Recall Frequency Domain versus Time Domain Relationships
Amplitude spectrum
Amplitude and Phase Spectra
Phase spectrum
Fourier
Transform
of a time
series
Individual
frequency
components
Time-domain wavelets
Zero Phase
Minimum Phase
Seismic Trace and its Amplitude Spectrum
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
T ime (seconds)
Amplitude Spectrum
This is its
amplitude
spectrum
Amplitude
16000
12000
8000
4000
0
0
50
100
150
Frequency
200
250
The Fourier Transform of a real function, like a seismic
trace, is complex, i.e., it has real and imaginary parts.
Real and Imaginary Parts of the Fourier Transform
Real Part of the Fourier Transform
5
2x10
The real part
is even
Amplitude
5
Symmetrical
2x10
5
1x10
4
5x10
0
4
-5x10
5
-1x10
5
-2x10
-20
-15
-10
-5
0
5
10
15
20
Frequency (Hz)
Imaginary Part of the Fourier Transform
5
2x10
Asymmetrical
5
1x10
The imaginary
part is odd
Amplitude
4
5x10
0
4
-5x10
5
-1x10
5
-2x10
-20
-15
-10
-5
0
5
Frequency (Hz)
10
15
20
Creating the quadrature Component
We assume that the complex time series
s(t) = sr(t) + isi(t)
exists and that sr(t) is the recorded seismic signal.
For this to be so, the Fourier transform of s, defined
as S(f), must have a real part (Sr) equal to the
Fourier transform of sr and an imaginary part (Si)
equal to the Fourier transform of si.
Amplitude and Phase Representation
of the Real and Quadrature Traces
The Spectrum of the Complex Trace
Real Trace
Complex Trace
Tanner, Koehler, and Sheriff, 1979
That relationship between Si and Sr is written as
The time-domain Hilbert Transform
Si(f) = H(f)Sr(f)
where H(f) is a step function having value -i for 0
and i for - 0.
That function is referred to as a Hilbert transform, and
the inverse Fourier transform of this imaginary step
function in the frequency domain yields the real
function h(t) in the time domain.
Tanner, Koehler, and Sheriff, 1979
Fourier Transforms and Convolution
Multiplication in the frequency domain
equals convolution in the time domain

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
Seismic Response
Physical nature of the seismic response

Convolutional model
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.
Discrete form of the convolution integral
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
Simple digital components
Folding and Shifting
Multiply and Sum
Output sample 0
Output Sample 1
Output Sample 2
Computing the Quadrature Trace
Generating Attributes in Kingdom Suite
Instantaneous Frequency