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Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Table of contents
1 Motivation
2 Method
3 Analysis
4 Synthetic Experiment
5 Conclusion
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Motivation of this study
Conventional method:
R(t) = P(t) ∗ h(t) + n(t) (1)
Contradiction: Resolution ↔ Signal to Noise Ratio (SNR)
Improve the resolution in reflection seismology.
Improve the SNR in the seismic record.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Method - Time Coded Impulsive Seismic Technique
The principle of this method (TCIST) may be loosely described asfollows, instead of using a big energy source, a large number of smallenergy are released in a sophisticated manner according to a timecoded scheme. Finally the result wavelet achieved by decodingprocess has nearly the same shape as single wavelet sharing the samemain frequencies, but its amplitude will be enlarged by n.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Basic Principle
TCIST method:The small sources is excited at instants determined by the time codedscheme whose mathematical expression is
y(t) =
i=n∑
i=1
δ(t − ti), (2)
Single pulse p(t) is released at each time ti in coded sequence y(t)to create coded pulses Sc(t) as follows,
Sc(t) = y(t) ∗ p(t) (3)
which will be transmitted with the coded sequence y(t) as a newseismic signal.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Basic Principle
Corresponding to Sc(t), the coded seismic record Rc(t) can beexpressed as follows,
Rc(t) = Sc(t) ∗ h(t) + n(t)
= y(t) ∗ p(t) ∗ h(t) + n(t). (4)
The decode process can be expressed as:
Rd(t) = Rc(t) ⊗ y(t)
= y(t) ∗ p(t) ∗ h(t) ⊗ y(t) + n(t) ⊗ y(t)
= ACF{y(t)} ∗ p(t) ∗ h(t) + n(t) ⊗ y(t), (5)
where ⊗ represents cross-correlation operation and ACF{} theauto-correlation function.
Jianmin Lin Time Coded Impulsive Seismic Technique
If {ACF{y(t)} = kδ(t), k >> 1}, every frequency of signal isamplified with power spectrum |Y (jω)|2, but the ambient noise isamplified with square-root of the power spectrum.⇒A SNR improvement by square-root of the power spectrum.Since the total energy of the power spectrum is proportional to n, sowe can improve the SNR by
√n in TCISTmethod.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Ricker Wavelet
−0.05 0 0.05−0.5
0
0.5
1
fm
=150Hz
Time (s)
A
Ricker Wavelet
0 200 400 600 800 1000−300
−250
−200
−150
−100
−50
0
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
Ricker Wavelet
Figure: Ricker Wavelet with peak frequency equal to 150Hz and itspower spectrum density
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Number of Excitations n
0 200 400 600 800 1000 1200 1400 16000
10
20
30
40
Number of ExcitationsS
NR
Impr
ovem
ent
0 200 400 600 800 1000 1200 1400 16000
0.05
0.1
0.15
0.2
0.25
Number of Excitations
SN
R Im
prov
emen
t Inc
rem
ent
Figure: The square-root relationship between the SNR improvementand total number of impulsive excitations in TCIST method.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Random Coded Scheme
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
Time (s)
Pseudo−random Coded Sequence
0 2 4 6 8 100
100
200
300
400
500
Time (s)
Pul
se R
ate
(Hz)
Pulse Rate
0 200 400 600 800 1000−75
−70
−65
−60
−55
−50
−45
−40
−35
−30
−25
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
Pseudo−random Coded Sequence
Figure: Random coded sequence with 300 impulsive excitationsrandomly distributed in 10s and its power spectrum density
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Random Coded Scheme
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
A
Auto−correlation Function of Pseudo−random Coded Sequence
0 200 400 600 800 1000−100
−90
−80
−70
−60
−50
−40
−30
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
Auto−correlation function of Pseudo−random Coded Sequence
Figure: Auto-correlation function of random coded sequence and itspower spectrum density
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Random Coded Scheme
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (s)
A
Pseudo−random Coded Signals
0 200 400 600 800 1000−300
−250
−200
−150
−100
−50
0
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
PSD Compare
Single Signal p(t)Coded Signals p(t)*y(t)
Figure: Random coded signals and its power spectrum density
Jianmin Lin Time Coded Impulsive Seismic Technique
Decoded Signal ACF{y(t)}*P(t) ( Linear Coded Scheme )
0 200 400 600 800 1000−70
−60
−50
−40
−30
−20
−10
0
10
20
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
Decoded Signal (Linear Coded Scheme)
Figure: Decoded signal of linear coded scheme and its powerspectrum density.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
More Sophisticated Coded Schemes
To develop the TCISTmethod further, more sophisticated codedscheme has been tried borrowing ideas from the Vibrator technique.Assume monotonical sweep or coded sequence (though discrete)could be represented as follow,
s(t) = Im
{
exp
[
i2π
∫ t
0
fi (t)dt
]}
= Im{exp[iΦ(t)]} (8)
Its spectrum could be expressed with following integral,
S(fa) =
∫
T
0
exp
{
i2π
[∫
t
0
fi (t)dt − fat
]}
dt
=
∫
T
0
exp
{
i2π
[∫
t
0
(fi (t) − fa)dt
]}
dt, (9)
where fa is the frequency one wants to analysis on.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Principle of Stationary Phase
An integral tends to be nil when its integrand oscillates rapidly aboutzero value. However, if its integrand contains some part withstationary phase where the integrand keeps nearly constant, theintegral will monotonically accumulate the contribution of thisinterval to the overall value of the integral. How much it contributesdepends on the duration of this interval which in turn is directlycontrolled by the rate of change of the sweep.⇒S(fa) depends on the length of the duration when the difference(fi (t) − fa) vanishes, thus controlled by the rate of the change offi (t). ( So linear coded sequence has flat spectrum.)
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Principle of Stationary Phase
For a nonlinear coded scheme, the frequencies whose rate of changeis the lowest will contribute the most to the spectrum, while, theamplitude spectrum of the frequencies with fastest rate of change willbe depressed.
This property could be used to design seismic signals in the means offrequency modulation according to our needs.
Jianmin Lin Time Coded Impulsive Seismic Technique
Decoded Signal ACF{y(t)}*P(t) ( Exponential Coded Scheme fi(t)=100*(2−exp(0.25t)))
0 200 400 600 800 1000−300
−250
−200
−150
−100
−50
0
50
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
PSD Compare
Decoded Signal ( ECS fi(t)=100*(2−exp(0.25t)))
Original Signal
Figure: Decoded signal of exponential coded scheme(fi (t) = 100 ∗ (2 − exp{0.25t})) and its power spectrum density.
Jianmin Lin Time Coded Impulsive Seismic Technique
Time Coded
Impulsive
Seismic
Technique
Jianmin Lin
Motivation
Method
Analysis
Synthetic
Experiment
Conclusion
Exponential Coded Scheme
0 50 100 150−85
−80
−75
−70
−65
−60
−55
−50
−45
−40
−35
Frequency (Hz)
Pow
er S
pect
ral D
ensi
ty (
dB/H
z)
Auto−correlation function of ECS fi(t)=100*(2−exp(0.25t))
Figure: Power spectrum density of auto-correlation function of theexponential coded sequence (10Hz <= fi <= 100Hz), the biggerfrequencies near 100Hz are more emphasized than smaller frequenciesbecause the rate of change of the bigger frequency around 100Hz issmaller (Figure. 11).
Jianmin Lin Time Coded Impulsive Seismic Technique