Transcript Seismic Methods Geoph 465/565 ERB 2104 Lecture 2 – Sept 6

Lee M. Liberty
Associate Research Professor
Boise State University
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Dates  Mar 17, 19, 21
15-20 minute oral presentation
Metrics: presentation, style (professionalism),
organization, accuracy, references
Include: history of topic, theory, approach to
addressing/solving topic, relevance to
industry/society
Topic examples:
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Site response method comparisons (Gribler)
Episodic tremor & slip (Terbush)
AVO methods (Lindsay)
Seismo-electric/electroseismic effects (Hetrick)
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Process dataset (e.g. reflection, surface wave,
microseismicity, refraction, modeling)
Report – SEG style: Summary, methods,
acquisition, processing, interpretation,
discussion/conclusions, references
Topic examples:
◦ Process marine/land reflection/surface wave,
refraction dataset, microseismicity dataset
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Seismic refraction analysis
◦ First break picks, 2-layer model
◦ surface wave dispersion curves
◦ reflection data analysis
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meet in the computer lab
T, msec
400
200
Refracted
Reflected
Direct
0
Distance, km
x
t
Dispersion
Ewing, Jardetzky and Press (1957)
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Bandwidth is the frequency range contained
in a source wavelet or seismic trace.
Mono-frequency waves extend forever (e.g.
sine wave).
Waves with an infinite bandwidth (all
frequencies) can be infinitely short
◦ Dirac Delta function = spike
side lobes
The same
bandwidth at
higher
frequencies has
the same
number of side
lobes. For
distinguishing
thin layers, it is
better to have
more
bandwidth,
even if freqs are
lower.
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We want to minimize the sample rate (maximize
the digitizing interval) so that we minimize the
computer storage requirements and the processing
time.
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What is the largest digitizing interval (minimum
sample rate) that we can use?
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For a given sample rate, what is the highest
frequency wave that is correctly sampled?
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Fourier transforms – we must be able to accurately
take FTs of the time function for filtering and other
processes.
trace
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digitizing interval = dt (sample
rate=1/dt)
duration (period) = T
T
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
dt
0.1
0.2
0.3
time (sec)
0.4
0.5
1.00
-0.31
-0.81
0.81
0.31
-1.00
0.31
0.81
-0.81
-0.31
1.00
-0.31
-0.81
0.81
0.31
-1.00
0.31
0.81
-0.81
-0.31
1.00
-0.31
-0.81
0.81
0.31
-1.00
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If not sampled frequently enough, the
time series does not provide an accurate
representation of the wave.
Other frequencies also fit the time series.
1.5
1.5
Amplitude
Amplitude
1 1
0.5
0.5
0 0
-0.5
-0.5
-1
-1
-1.5
-1.5
0 0
0.10.1
0.2
0.2
0.3
0.3
time(sec)
(sec)
time
0.4
0.4
0.5
0.5
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Time series must be sampled so that the
highest frequency is sampled at least twice.
dt = 1/2fmax
fmax = 1/(2dt)
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
0.1
0.2
0.3
time (sec)
0.4
0.5
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fmax is called the aliasing frequency, the
folding frequency, or the Nyquist
frequency fN.
Frequencies above the Nyquist frequency
are “aliased” or “folded” to lower
frequencies.
Amp
frequency
fN
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fN = 12.5 Hz
dt = 0.04
10 Hz
15 Hz
1.5
Amplitude
1
0.5
0
-0.5
-1
-1.5
0
0.1
0.2
0.3
time (sec)
0.4
0.5
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Before resampling a seismic trace to a larger
digitizing interval, or when you collect
seismic data, you MUST use an anti-alias
filter first to prevent aliasing!
>>>It’s not just signal – you also MUST
sample the noise properly, or filter it out
before sampling.
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The time sampling interval and the
maximum frequency are related.
dt=1/2fmax
fmax=1/(2dt)
Similarly, the frequency sampling interval
(df) and the maximum time (T) are related.
df = 1/T
T=1/df
Lowest frequency wave = 1 cycle/period
Fundamental mode
df = 1/T
3df
.
.
.
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Time domain
=convolution
G(t)
F(t)
frequency domain
=complex multiplication
G(t)
F(t)
Fourier transform
G(w)
convolution
F(w)
G(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function
h(t) = f(t) * g(t) = ∫f(t)g(t-t)dt
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Given two arrays:
◦ 1. Reverse moving array
◦ 2. Multiply and add
◦ 3. Shift, repeat step 2
Multiply two functions together….
time
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2
*
-1
=
Source wavelet: 2, 4, 3
Reflectivity sequence: 0, 0, 1, 0, 0, 0, 0
0 0 1 0 0 0 0
3 4 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 2
0
0
2
4
3
0
0
0
0
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Two arrays (wavelets):
◦ Source wavelet: 2, 4, 3
◦ Reflectivity sequence: 1, 4, 3, 2
(output length = sum of two wavelets – 1)
1 4 3 2
3 4 2
2
3 4 2
12
3 4 2
25
3 4 2
28
3 4 2
17
3 4 2
6
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When convolving a wavelet (source signature) with
a spike train of reflection coefficients, we are just
filtering. We can reverse this process to extract the
spike train.
Spike = infinite frequency reflectivity function
Filtered = band-limited reflectivity function
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Change spike to band-limited wavelet.
2
*
-1
=
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S(t)*R(t)*G(t)
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S = source
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R=reflections = earth model
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G=recording instrument (geophone)
response
input
filter
output
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the Z-transform converts a discrete time-domain
signal, which is a sequence of real or complex
numbers, into a complex frequency-domain
representation.
A function can be represented as a polynomial with a
“dummy” variable z that is equal to the unit time
interval:
Function: 2, 5, 3, -2, 1
z-transform: 2 + 5z + 3z2 –2z3 + z4
dt
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The convolution of two functions can be
accomplished by multiplying their ztransforms:
1+4z+3z2+2z3 * 2+4z+3z2
= 2+12z+25z2+28z3+17z4+6z5
(multiplication of polynomials)
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Two arrays (wavelets):
◦ Source wavelet: 2, 4, 3
◦ Reflectivity sequence: 1, 4, 3, 2
(output length = sum of two wavelets – 1)
1 4 3 2
3 4 2
2
3 4 2
12
3 4 2
25
3 4 2
28
3 4 2
17
3 4 2
6
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z-transforms allow us to define “minimum
phase” and “maximum phase” wavelets.
Break a wavelet’s z-transform polynomial
into its roots:
Polynomial: 30+17z-11z2-6z3
Roots: (3+2z), (2+1z), (5-3z)
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The roots all have the form:
a0 + a1z
If a0 > a1, it is a minimum-phase wavelet
If a0 < a1, it is a maximum-phase wavelet
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A minimum-phase wavelet has only minimumphase roots (2+1z)(3+2z)
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A maximum-phase wavelet has only
maximum-phase roots (1+2z)(3+5z)
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A mixed-phase wavelet has both minimum and
maximum-phase roots (2+1z)(3+5z)
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Time domain
=convolution
T(t)
F(t)
frequency domain
=complex multiplication
T(t)
F(t)
Fourier transform
T(w)
convolution
F(w)
T(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function
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The FFT is a faster version of the Discrete
Fourier Transform (DFT). The FFT utilizes
some clever algorithms to do the same thing
as the DFT, but in much less time.
Function must have 2n numbers
◦ For shorter functions, add zeros to the end to make
a function with length 2n samples.
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FFTs make it faster to filter in the frequency
domain than in the time domain.
Fourier transforms (and FFTs)
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Result in the frequency domain
Imaginary (sin)
is a complex number
For filtering, multiply each
frequency component by the
corresponding frequency
component of the filter
(complex multiplication).
Ac + iAs
As
A
f
Ac
Real (cos)
Phase f = tan-1(As/Ac)
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(a+bi)(c+di) = ac + (ad+bc)i – bd
=(ac – bd) + (ad + bc)i
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Time domain
=convolution
T(t)
F(t)
frequency domain
=complex multiplication
T(t)
F(t)
Fourier transform
T(w)
convolution
F(w)
T(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function
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An operation closely related to convolution
is correlation.
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Convolution:
h(t) = ∫f(t)g(t-t)dt
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Correlation:
(t) = ∫f(t)g(t-t)dt
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In practice, correlation is done the same
way as convolution in the time domain,
except that neither wavelet is reversed
before the multiplication process.
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In the frequency domain, correlation is
carried out as a complex multiplication, but
first the complex conjugate is taken of one of
the functions.
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Convolution: H(w) = F(w) * G(w)
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Correlation:
(w) = F(w) * G(w)
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Two arrays (wavelets):
◦ Source wavelet: 2, 4, 3
◦ Reflectivity sequence: 3, 2, 1, 4, 3, 2, 4, 1
(often, output length = wavelet1 – wavelet2 + 1)
3 2 1 4 3 2 4 1
2 4 3
17
2 4 3
20
2 4 3
27
2 4 3
26
2 4 3
26
2 4 3
23
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Shows how well two functions match. Largest
value shows how well the functions match.
Location of largest value shows time of best
match.
Wavelet 1:
Wavelet 2:
3 2 1 4 3
4 3 2
4 3 2
4 3 2
4 3
4
4, 3, 2
3, 2, 1, 4, 3, 2, 1, 4
2 1 4
2
3 2
4 3 2
20
19
22
29
20
19
best match
Vibroseis
sources use
cross-correlation
to synthesize a
short, zerophase wavelet
from a long
source sweep.
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Correlation between a function and itself is
called an autocorrelation.
The autocorrelation of a function gives the
zero-phase wavelet, scaled to the total
energy in the trace.
Autocorrelation is used to retrieve the zerophase wavelet from a seismic trace.
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Correlation of function with itself
◦ wavelet: 1, 4, 3, 2
1
1 4 3 2
1 4 3
1 4
1
Symmetric,
zero-phase wavelet
4 3 2
2
3 2
4 3 2
1 4 3 2
1 4 3 2
1 4 3 2
2
11
22
30
22
11
2
Total power in
wavelet (amplitude2)
sin2x
x2
b
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Rayleigh chose to keep the mathematical relationships
involved simple.
When applied to wavelets other than sin2x / x2, the
“dimple-to-dimple” amplitude ratios may vary.
0.81A
A
b
Seismic Resolution of Zero-Phase Wavelets,
R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977
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The ability to
distinguish two
separate reflectors
(top and bottom of
a layer).
(Kallweit demo)
Zero-phase wavelets
33 Hz dominant frequency
7
Dirac delta function
(spike)
all frequencies
5
3 octaves; 7-56 Hz
3
Amplitude
Side lobes make it
difficult to
distinguish closelyspaced reflectors.
More bandwidth
reduces side lobes.
1
0.5 octaves; 56-80 Hz
0.5 octaves; 27-38 Hz
-1
mono-frequency; 33 Hz
-3
-5
0.2
0.5
Time (seconds)
0.8
Wavelets from the
top and bottom of
the layer interfere
(tuning). Minimum
bed thickness that
can be distinguished
is l/4 (one quarter
wavelength). This
causes a time delay
of l/2 (two-way
traveltime through
layer. (Widess, 1973)
“quarter wavelength rule”
Lateral Resolution (pre-migration)
Reflections represent the
contribution of a reflector
area, not a point. This area is
determined by ¼ of the
dominant wavelength of the
source wavelet.
Points within the Fresnel zone
cannot be distinguished from
each other without additional
information.
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The width of the Fresnel zone can be
computed from:
F = v t/fdom
This width is generally much greater
than the vertical resolution.
The Pythagorean theorem allows one to
calculate the radius of the Fresnel
Zone.
Cannot determine location
of reflector within box
Vertical
resolution
Horizontal
resolution
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The width of the Fresnel zone is the limit of
lateral (horizontal) resolution on stacked
sections.
Migration, however, can increase the lateral
resolution so that we can distinguish (image)
objects much smaller than a Fresnel zone.
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Migration increases the lateral resolution by
making use of the non-symmetric (nonspecular) reflections contained in the
diffraction curve.
non-specular
*
specular
(strongest)
Diffractions are waves
incident from the side, and
thus can have a horizontal
resolution of l/4 under
ideal conditions
(horizontally-traveling
wave).
Thus, horizontal and
vertical resolution can be
the same.
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Vertical (time) resolution
>>improves with greater bandwidth
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Horizontal resolution
>>improves with greater range of incidence angles
(more complete diffraction curves)
>>MUST migrate data to get this resolution