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

Lee M. Liberty
Associate Research Professor
Boise State University
Homework: Due Jan 3,1 2014
SU/Linux primer
Jan 31, 2014 - Seismic acquisition lab
Jougnot/Gonzales lectures
This week: Neumann/Arrowsmith lectures




15-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:
◦
◦
◦
◦
Site response method comparisons (Gribler)
Episodic tremor & slip (Terbush)
AVO methods (Lindsay)
Seismo-electric/electroseismic effects (Hetrick)


Acoustic (fluid)
-Defined by incompressibility (k) and density (r)
or by wave speed (v) and density (r)
-Support only compressional (acoustic) waves
Elastic (solids)
-Defined by 3 elastic parameters: Lame parameter
(l) [not to be confused with wavelength], rigidity
(m) and density (r)
-Support both compressional and shear waves
Acoustic media are much simpler because they
support only one type of wave (acoustic, or sound
wave)


One type of body wave: compressional, P,
acoustic
Wave speed v = sqrt(k/r)
k = bulk modulus (incompressibility), r = density

Two types of body waves:
◦ Compressional, P
v = a = sqrt((l+2m)/r)
◦ Shear, S
v = b = sqrt(m/r)
l = Lame parameter (incompressibility) – bulk modulus
m = rigidity – shear modulus
l >> Do not confuse Lame parameter with wavelength!

Surface waves:
◦ Rayleigh
◦ Love

Note inverse relationship to density!

a = sqrt((l+2m)/r)
>>denser = slower
b = sqrt(m/r)
BUT: l and m tend to increase with density
Influences on Rock Velocities
• Confining pressure
• Porosity
• Lithology
• Fluids – dry, wet
• Degree of compaction
• Microcracks
•In situ versus lab measurements
• Frequency differences

Porosity has large
effect on P
velocity, but little
effect on S velocity
Compressional, P
v = a = sqrt((l+2m)/r)
Shear, S
v = b = sqrt(m/r)
m = rigidity
VP 
4
K m
3
r
m
Vs 
r
Sediments and
sedimentary rock
Igneous and
metamorphic rock
Figure 3.10 of Lillie, 1999, modified from Birch, 1960



localized near the free surface of an isotropic
elastic half-space
c1 and c2 are (dimensional) speeds of
longitudinal and shear body waves, respectively.
They may be written in terms of the material
density (r), shear modulus (m), and the Poisson
ratio (u)
distance
Refracted/direct arrival
Reflections
Direct Shear
Surface waves
time
V=slope

A = amplitude
f = frequency = 1/T
f = phase

F(t) = A cos (2pft - f)

f=0

If f=90o (p/2 radians)


=zero phase wave (cosine wave)
=sine wave
◦ F(t) = A cos (2pft – p/2) = A sin(2pft)
Degrees:
(radians)
0
360 (2p)
k = angular wavenumber (cycles/distance)
720 (4p)
1080 (6p)
Degrees:
(radians)
0
360 (2p)
720 (4p)
f = frequency (cycles/time) (Hz)
1080 (6p)
t0
t1



v = l/(t1-t0) = l/T  f l
l = v/f
f = v/l


If a wave with a frequency of 10 Hz is
traveling at a speed of 1000 m/sec, the
wavelength l = v/f = 1000/10 = 100 m
If we want a wave to have a wavelength of 10
m in a material with a wave speed of 1500
m/sec, we need a wave of frequency
f = v/l = 1500/10 = 150 Hz
Degrees:
(radians)
0
f
360 (2p)
720 (4p)
1080 (6p)
p/2






1 octave = double frequency
◦ 50Hz – 100Hz = 1 octave
Low C = 128 Hz (cycles/sec)
Middle C = 256 Hz
High C = 512 Hz
Human hearing ~ 20 Hz to 20,000 Hz
Wavelengths: Low C:
l = v/f
l = (330m/sec)/(128
cycles/sec)
l = 2.6 m/cycle
A mathematical operation that decomposes a signal
into its constituent frequencies
Transform signals between time (or spatial) domain
and frequency domain
The Fourier transform
relates the function's time
domain, shown in red, to
the function's frequency
domain, shown in blue. The
component frequencies,
spread across the frequency
spectrum, are represented
as peaks in the frequency
domain.
cosine = symmetric
sine =antisymmetric
add
multiply
multiply
add
Imaginary (sin)
Amplitude = sqrt(Ac2 + As2)
As
A
f
Ac
Real (cos)
Phase f = tan-1(As/Ac)
f(t) = (1/2p)∫F(ω) cos(ωt)dω + i∫F(ω) sin(ωt)dω
F(ω) = ∫ f(t) cos(ωt)dt - i∫ f(t) sin(ωt)dt
ω = 2pf = angular frequency
(radians/sec)
Angular frequency is a measure of how fast an
object is rotating or the magnitude of the vector
quantity angular velocity
This allows us to rewrite the Fourier
transforms as:
f(t) = (1/2p)∫F(ω) eiωtdω
F(ω) = ∫ f(t) e-iωtdt

Zero phase
◦ All phases = 0

Minimum phase
◦ Energy is front-loaded

Maximum phase
◦ Energy is back-loaded

Mixed phase
◦ Most wavelets are mixed phase


Short burst of energy
Dynamite blast, hammer hit, airgun firing


Symmetric
vibroseis

Interpretation
◦ Zero-phase (Vibroseis)
 reflection is at the peak
◦ Minimum/mixed phase (dynamite, airguns)
 reflection is first deflection
◦ Maximum/mixed phase (water guns)
 Reflection is later, but we usually correct this

Deconvolution (predictive error filtering)
◦ We will independently alter the phase spectrum
and the amplitude spectrum to change the source
wavelet to a more desirable shape.
◦ i.e., we can change minimum phase to zero
phase, etc.
◦ But: maximum-phase wavelets can create
unstable inverse filters – you can delay the
energy, but moving the energy forward in time is
more difficult



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
1 Hz

What happens to
wavelet as we
increase or
decrease the
range of
frequencies?
5.5 Hz
side lobes


Mono-frequency waves extend forever.
Waves with an infinite bandwidth (all
frequencies) can be infinitely short
◦ Dirac Delta function = spike
Frequency spike = infinite time function
infinite frequencies = time spike



Broader bandwidth = sharper wavelet
>>For seismic reflection data, our resolving
power (ability to separate two thin layers)
increases as the bandwidth increases.
Therefore, we would like to get a source
that produces the largest bandwidth
possible.
Minimum of 2 octaves is suggested, 3 is
better!!
(5-40 Hz; 10-80 Hz; 20-160 Hz; etc.)
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.