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Introduction to Geophysics
Ali [email protected].
saDepartment of Earth SciencesKFUPM
Seismic Reflection 4: Acquisiton, Processing, and Waveform Analysis
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Example: Disk Formula •Seismic Waveform: Input Seismic Signal
•Homogenous Medium •Strong Interface
•Amplitude •Minimum Phase Pulse •Zero Phase Pulse
•Acoustic Impedance •Coefficients of Reflection and Transmission
•Example: Air-water reflection •Example: Water-Air Reflection
•Reflection from Single Interface •Reflections from Several Interfaces
Previous LectureIn
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Convolution
• Earth as a Filter of Seismic Energy
G(t) * F(t) = H(t)
Source Earth = Seismogram Wavelet Ref. Coeff. In
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Source and Earth Response
• Mathematical Description of Filter
• Convolution
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See pp. 353 of Reynolds, 2002
Convolutional Model for the Earth
SOURCE * Reflection Coefficient = DATA(input) (earth)
(output)
where * stands for convolution
Reflections in the earth are viewed as equivalent to a convolution process between the earth and the input seismic wavelet.
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(MORE REALISTIC)
Convolutional Model for the Earth
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SOURCE * Reflection Coeffi cient = DATA
(input) (earth) (output)
where * stands for convolution
Example 1: ConvolutionIn
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1) Folding
2) Shifting
3) Multiplication
4) Summation
The process of convolution consists of 4 simple mathematical
operations
Seismic Wavelet (1, -1/2, 1/2)Reflectivity Sequence
(1, 1/2, 1/2)
Solution
Reflectivity Output Sequence Response
1 1/2 1/2
1/2 -1/2 1 1x1= 1
1/2 -1/2 1 -1/2x1+1/2x1= 0
1/2 -1/2 1 1/2x1+1/2x-1/2+1/2x1= 3/4
1/2 -1/2 1 …………………= 0
1/2 -1/2 1 …………= 1/4
1/2 -1/2 1 …………= 0
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Convolution
Output Response
Plot amplitudes as a seismogram using travel times
1 , 0 , 3/4 , 1/4, 0
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From Kearey, Brooks, and Hill, 2002
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For example, a double spike function, 2, 0, 1 convolved with an impulse response function 4, 3, 2, 1 .
Example 2: Convolution
Recall: Elastic waves
• Amplitude: the peak to trough height divided by two.
• Wavelength: the distance over which the wave goes through one complete cycle.
• Period: the time over which the wave is observed to complete a single cycle. V =
Velocity = wavelength x frequency
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Seismic Resolution
•Vertical – /4 to /2 wavelength
V =
V = 2000 m/sec = 25 Hz
=V/ = 80 metersResolution 20 to 40 m
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– /4 /2
What is their period? 12 sec
What is the amplitude of these waves? 1mm
What is their frequency?
1/12 Hz = 0.083 Hz
Example 1In
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V =
What is the amplitude of these waves? 1.2 mmWhat is their wavelength? 1.75 mIf the frequency of these waves is 5 Hz, what is their speed?
v = 1.75 m • 5Hz= 8.75 m/sec = 0.00875 km/sec
Example 2In
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V =
Homework, Due to April 26
What is the amplitude of these waves?
What is their period?
These waves were recorded in rock with a seismic velocity of 5km/sec. What is their wavelength?
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V =
What is the amplitude of these waves?What is their period?
What is their frequency?
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V =
Homework, Due to April 26
1. In one typed page, discuss applications of a) the seismic reflection methodb) the seismic refraction method
2. Calculating by hand Convolve the spike function:2 0 0 3 3 1 3 -1 1 -2 -1with the impulse response function:1 2 1 -0.5 0 0.5 0Show your working for the hand calculated
version.
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