Need help with assignment 7 as you can see in the attachment files you can see the assignment and the lecture if needed.
Geophys 6001: Advanced Geophysical Data Analysis
Assignment 7; 10 points
Is the following time sequence minimum phase? If not, find the minimum phase
sequence with the same amplitude spectra.
{2, 5 , -3}
Note:
Study Lecture 18
GEOPHYS 6001
Advanced Geophysical Data Analysis
Lecture 18
Contents
Phase
Kelly Liu
Constant phase shift
Linear phase shift
Zero-phase filter
Minimum phase, maximum phase, mixed phase
Reference: Yilmaz, “Seismic data analysis”
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Phase
Rn=|Rn|e iφn
To construct a time sequence, we must also specify the phase,
then take the inverse Fourier transform.
The simplest choice is to make the phase zero and simply invert
the amplitude spectrum.
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Yilmaz “Seismic Data Analysis”
Phase
Zero-phase
A zero-phase wavelet is
symmetric with respect
to zero time and peaks
at zero time.
Zero-phase filters
obviously do not change
the phase of the original
time sequence, which
also means that they will
not in general shift
peaks of the original
signal.
Frequency 1-32 Hz
Phase angle=0o
Amplitude=constant
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Yilmaz “Seismic Data Analysis”
Phase
900 shift is applied to
each of the sinusoids
compared to the zero
phase waveform.
Constant phase shift
A: amplitude
Frequency 1-32 Hz
Phase angle=900
Amplitude=constant
……
A0
1
Same amplitude Spectra.
32 (hz)
f: frequency
Φ: phase
The difference in wavelet
shape is a result of the
difference in their phase
spectra.
……
π/2
1
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f
4
Yilmaz “Seismic Data Analysis”
Phase
Constant phase shift on a zero-phase wavelet
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Yilmaz “Seismic Data Analysis”
Phase
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Yilmaz “Seismic Data Analysis”
Phase
Linear phase shift
phase=constant x frequency
A linear phase shift is
equivalent to a constant time
shift.
The slope of the line describing
the phase spectrum is
proportional to the time shift.
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Yilmaz “Seismic Data Analysis”
Phase
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Yilmaz “Seismic Data Analysis”
Phase
The wavelet can be shifted by any amount of time simply by
changing the slope of the line that describes the phase spectrum.
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Yilmaz “Seismic Data Analysis”
Phase
What kind of
phase change?
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Yilmaz “Seismic Data Analysis”
Phase
Linear phase shift
combined with a
constant phase shift
A time-shifted
asymmetric wavelet
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Yilmaz “Seismic Data Analysis”
Phase
Variations in phase spectrum
By keeping the amplitude spectrum unchanged, the wavelet shape
can be changed by modifying the phase spectrum.
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Yilmaz “Seismic Data Analysis”
Phase
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Yilmaz “Seismic Data Analysis”
Phase
Zero
phase
900
phase
Which polarity?
Wavelet
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Brown, “Interpretation of Three-Dimensional Seismic Data”
Phase & Filter
How to design a zero-phase frequency filter and apply the filter in the time domain?
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Yilmaz, “Seismic Data Analysis”
Filter & Phase
Y=|Y|e iφy =|X|e iφx|F|e iφf
=|X|.|F| ei(φx+φf)
f-filter
x-input
y-output
|F|,|X|, |Y|-amplitude spectra
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Filter & Phase
What will happen if the filter is not zero-phase?
-The filtered output will have a phase shift.
To solve the problem, we need to apply a reversed filter.
x(t)
Input
Filter System f(t)
Y=|Y|e iφy =|X|e iφx|F|e iφf
=|X|.|F| ei(φx+φf)
y(t)
Output
Zero-phase filtering:
Run the filter forward and backward over the data, instead of
just forward over the data.
Two-pass operation results in a filter magnitude response
which is the square of the original magnitude response.
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Phase
Minimum phase: giving a time sequence with energy concentrated at
the front. A minimum-phase filter has the desirable property of being the
‘shortest’ filter with the given amplitude spectrum.
Minimum-phase
Zero-phase
Liu’s Geophys 6001, Summer 2020
A minimum phase
signal is causal. it is
zero for negative
times.
A zero-phase is
acausal. It is
symmetric about
t=0. The wavelet
has been shift by 2
s to show the
shape.
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Yilmaz, “Seismic Data Analysis”
Phase
Minimum-phase (minimum-delay) filter may be constructed by factorizing the Z-transform:
N −1
F ( z ) = f N −1 ∏ ((Z-Z
z − zi)i )
i =0
Each factor (Z-Zi) has a phase.
Each factor of the Z-transform is therefore either minimum or maximum phase,
depending on whether Zi>1 or Zi 1minimum phase.
The whole Z-transform is minimum phase only if all its constituent factors are
minimum phase.
We can therefore form a minimum delay sequence by changing all the maximum
phase factors to minimum. The amplitude spectrum is unchanged.
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Yilmaz, “Seismic Data Analysis”
Phase
Exercise
Q: Is the sequence (2, -5, -4, 3) a minimum phase? If not, find the
minimum phase sequence with the same amplitude spectra.
Factorize a polynomial expression
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Yilmaz, “Seismic Data Analysis”
Phase
Exercise
Q: is the sequence (2, -5, -4, 3) a minimum phase? If not, find the
minimum phase sequence with the same amplitude spectra.
Z-transform: 2-5Z-4Z2+3Z3=(Z+1)(Z-2)(3Z-1)
Is this a minimum phase? No. Because the factor (3Z-1) is not a minimum phase.
Change (-1,3) to (3,-1) to make it minimum phase, or Z-transform 3-Z.
(Z+1)(Z-2)(3-Z)=-Z3+4Z2-Z-6
The minimum phase sequence is (-6,-1, 4,-1).
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Yilmaz, “Seismic Data Analysis”
Phase
Exercise
Four, three-point wavelets, the same amplitude spectrum (Robinson, 1966).
Wavelet A: (4,0,-1)
Wavelet B: (2,3,-2)
Wavelet C: (-2,3,2)
Wavelet D: (-1,0,4)
Compute the cumulative energy of each wavelet at any one time.
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Yilmaz, “Seismic Data Analysis”
Phase
Exercise
Four, three-point wavelets (Robinson, 1966)
Wavelet A: (4,0,-1)
Wavelet B: (2,3,-2)
Wavelet C: (-2,3,2)
Wavelet D: (-1,0,4)
A: minimum phase
B&C: mixed phase
D: maximum phase
Zero-phase wavelet has a mixed phase.
Cumulative Energy at Time Sample
The wavelet with the least energy delay is called minimum delay or minimum
phase.
A minimum-phase wavelet has the least energy delay. The minimum phase
sequence has more energy concentrated at the front than any other sequence
with the same amplitude spectrum.
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Yilmaz, “Seismic Data Analysis”
Phase
A: minimum phase
D: maximum phase
B&C: mixed phase
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Yilmaz, “Seismic Data Analysis”
Phase
A time sequence (or a wavelet) is
minimum phase if its energy is
maximally concentrated at its
onset.
A wavelet is maximum phase if its
energy is maximally concentrated
at its end.
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Yilmaz, “Seismic Data Analysis”
Phase
A: least phase change; minimum phase
D: largest phase change; maximum phase
B and C: in between
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Yilmaz, “Seismic Data Analysis”
Summary
Phase
Constant phase shift
Linear phase shift
Zero-phase filter
Minimum phase, maximum phase, & mixed phase
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