1. Seismic Methods Geoph 465/565 ERB 5104 Lecture 3
Lee M. Liberty
Research Professor
Boise State University

2. Mid-term Presentations (Oct 19) special topics in seismology
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:
Refraction approaches– ray tracing vs tomography
Earthquake relocations via HYPODD or others
Volcanic seismology
Topics in hydrogeophysics

3. Earliest known seismoscope
To detect an earth quake in mouth of each dragon a bal was present if a quake was present (system disturbed) a mechanism was making that the bal was falling into the frog’s mouth.
Used to indicate the occurrence of an earthquake in the year 138
(Reynolds)

5. Shot records distance V=slope Refracted/direct arrival time
Reflections
Direct Shear
Surface waves

6. Simple harmonic motion 3 parameters describe a wave
A = amplitude
f = frequency = 1/T
f = phase
F(t) = A cos (2pft - f)
f= =zero phase wave (cosine wave)
If f=90o (p/2 radians) =sine wave
F(t) = A cos (2pft – p/2) = A sin(2pft)

7. Waves in space Degrees: (radians) 360 (2p) 720 (4p) 1080 (6p)
360 (2p)
720 (4p)
1080 (6p)
k = angular wavenumber (cycles/distance)

8. Waves in time Degrees: (radians) 360 (2p) 720 (4p) 1080 (6p)
360 (2p)
720 (4p)
1080 (6p)
f = frequency (cycles/time) (Hz)

9. Space and time version of waves are related by the wave speed (v)
v (m/s) = l/(t1-t0) = l/T = f l
l (m/cy) = v/f
f (cy/s – or Hertz) = v/l

10. For example: 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 m/sec, we need a wave of frequency
f = v/l = 1500/10 = 150 Hz

11. Phase shifts f
Degrees:
(radians)
360 (2p)
720 (4p)
1080 (6p)

12. If f=90o (p/2 radians) >> sine wave F(t) = A cos (2pft – p/2) = A sin(2pft)

13. Waves and Music 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
Infrasonic <20 Hz
Ultrasonic >20,000 Hz
Wavelengths: Low C: l = v/f
l = (330m/sec)/(128 cycles/sec)
l = 2.6 m/cycle

14. Fourier Transform
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.

15. Suppose we have several waves summed together? > interference

16. Any continuous function can be described as a series of cosine and sine waves
cosine = symmetric sine =antisymmetric

17. Fourier transforms: Multiplication by sine and cosine waves to determine a coefficient whose amplitude is a measure of how well the wave matches the function.
add
multiply
multiply
add

18. Real and imaginary axes
Imaginary (sin)
Amplitude = sqrt(Ac2 + As2) As A f
Real (cos) Ac
Phase f = tan-1(As/Ac)

19. Fourier Transforms 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

20. But cos(w) - isin(w) = e-iw and cos(w) + isin(w) = eiw
This allows us to rewrite the Fourier transforms as:
f(t) = (1/2p)∫F(ω) eiωtdω
F(ω) = ∫ f(t) e-iωtdt

21. Fourier transform results in Amplitude and phase spectra F(w)

22. Wavelet types: Zero phase Minimum phase Maximum phase Mixed phase
All phases = 0
Minimum phase
Energy is front-loaded
Maximum phase
Energy is back-loaded
Mixed phase
Most wavelets are mixed phase

23. Wavelets – seismic sources
Short burst of energy
Dynamite blast, hammer hit, airgun firing

24. Zero-phase
Symmetric
vibroseis

25. Why is phase important? 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

26. Why is phase important? 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

27. Bandwidth
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

28. Zero-phase
1 Hz
What happens to wavelet as we increase or decrease the range of frequencies?
5.5 Hz

29. side lobes

30. Bandwidth 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

31. Bandwidth 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; Hz; Hz; etc.)

32. How thin is a thin bed?
Widess (1973) showed the effects of bandwidth on resolution
¼ wavelength rule (not thin bed)
If you want better resolution, you need higher frequencies

33. Bandwidth = higher frequencies
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.

34. Bandwidth
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

35. side lobes

36. Bandwidth = higher frequencies
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.

37. Sampling frequency
We want to minimize the sample rate (maximize the digitizing interval) so that we minimize the computer storage requirements and the processing time.
What is the largest digitizing interval (minimum sample rate) that we can use?
For a given sample rate, what is the highest frequency wave that is correctly sampled?
Fourier transforms – we must be able to accurately take FTs of the time function for filtering and other processes.

38. Sampling a time function
trace
digitizing interval = dt (sample rate=1/dt)
duration (period) = T

39. Aliasing
If not sampled frequently enough, the time series does not provide an accurate representation of the wave.
Other frequencies also fit the time series.

40. Aliasing
Time series must be sampled so that the highest frequency is sampled at least twice.
dt = 1/2fmax fmax = 1/(2dt)

41. Aliasing – frequency domain
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 fN
frequency

42. Example
dt = fN = 12.5 Hz
15 Hz
10 Hz

43. Resampling
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.

44. Sampling constraints (for a harmonic Fourier Series)
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

45. df = 1/T
2df
3df .

46. A/D converter
An analog-to-digital converter (A/D) is a device that converts a continuous quantity to a discrete time digital representation.
converts an input analog voltage or current to a digital number proportional to the magnitude of the voltage or current.
A/D converter is defined by bandwidth (the range of frequencies it can measure) and its signal to noise ratio (how accurately it can measure a signal relative to the noise it introduces).
The actual bandwidth is characterized primarily by its sampling rate.

47. Dynamic range
Range which can be measured using different number of bits:
8-bit : 1 mV mV
24-bit: 1 mV - 16 V
Dynamic range is expressed in dB:
Examples: