1. Seismic Methods Geoph 465/565 ERB 5104 Lecture 4
Lee M. Liberty
Research Professor
Boise State University

2. Shots from last Friday

3. Sept 4 Lab – ERB 3110 Seismic Unix primer Linux scripting primer
Seismic analysis
refraction
surface wave dispersion
reflection data analysis

4. COMPUTER LAB: Linux/scripting overview
Linux tutorials Seismic Unix overview/basics ftp://ftp.cwp.mines.edu/pub/cwpcodes/sumanual_300dpi_a4.pdf

5. 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

6. Final Project seismic data analysis
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

7. 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.

8. 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:

9. Geometrics Geode seismograph
Configurations: 3, 6, 8, 12, 16 or 24 channels
A/D Conversion: 24 bit result using Crystal Semiconductor sigma-delta converters
Dynamic Range: 144 dB (system), 110 dB (instantaneous, measured) at 2 ms, 24 dB.
Distortion: 2 ms, 1.75 to 208 Hz.
Bandwidth: 1.75 Hz to 20 kHz. 0.6 and DC low frequency option available.
Crosstalk: -125 dB at 23.5 Hz, 24 dB, 2 ms.
Noise Floor: 0.20 uV, RFI at 2 ms, 36 dB, 1.75 to 208 Hz.
Stacking Trigger Accuracy: 1/32 of sample interval.
Maximum Input Signal: 2.8V PP, 0 dB.
Input Impedance: 20 kOhm, 0.02 uf.
Preamplifier Gains: 24 and 36 db, jumpered for software selectable 12 and 24 dB or can be jumpered in four channel blocks as a single fixed gain of 0 dB for high-voltage devices.
Anti-alias Filters: -3 dB at 83% of Nyquist frequency, down 90 dB.
Acquisition and Display Filters:
Low Cut: OUT, 10, 15, 25, 35, 50, 70, 100, 140, 200, 280, 400 Hz, 24 or 48 dB/octave, Butterworth. Notch: 50, 60, 150, 180 Hz and OUT, with the 50 dB rejection bandwidth 2% of center frequency. High Cut: OUT, 32, 64, 125, 250, 500 or 1000 Hz, 24 or 48 dB/ octave.
Sample Interval: 0.02, , , 0.125, 0.25, 0.5, 1.0, 2.0, 4.0, 8.0,16.0 ms.
Record Length: 16,384 samples standard, 65,536 samples optional
Pre-trigger Data: Up to full record length.
Delay: 0 to 100 sec in steps of 1 sample interval.

10. Delta sigma A/D
The modulator is a chip circuit that operates on a 256,000 Hz clock (0.004 ms) – or – 1 bit per ms
For a 2 ms sample rate (or 512 clock cycles), 512 samples goes into the final measurement
(for a 1 ms sample rate, 256 samples goes into the final measurement)
but, we need to remove signals above the Nyquist and near DC
For a 24-bit system (144 dB) with system noise and at 2 ms SR, we can get about 110 dB (18 bits) of usable signal

11. Geophone specs
geophone equation is a second- order linear ODE characteristic of damped harmonic oscillation
V(t) is the seismograph voltage input, and vB(t) is the vertical particle velocity.
The constants h, C, and w0 are the total damping constant, total transduction constant, and natural frequency
natural frequency w0 can be adjusted by changing
the stiffness K of the spring, or the mass m of the coil

12. Geophone response (4.5 Hz)

13. A/D measured displacements
110 dB:
1.5 mm max geophone displacement
~4 nm detection threshold
But, a typical geophone only has ~70 dB of instantaneous dynamic range or ~500 nm
But, we can gain our instruments to record greater sensitivity
70 dB:

14. Digital Filtering
Convolution: a mathematical way of combining two signals to achieve a third, modified signal
Deconvolution: the reversal of the convolution process
Cross-correlation: a statistical measure used to compare two signals as a function of the time shift (lag) between them.
Autocorrelation is a special case where the signal is compared with itself

15. Digital Filtering Time domain frequency domain
=convolution =complex multiplication
G(t)
F(t)
G(t) F(t)
Fourier transform
G(w)
F(w)
convolution
G(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function

16. Filtering in the time domain - convolution
h(t) = f(t) * g(t) = ∫f(t)g(t-t)dt
Given two arrays:
1. Reverse moving array
2. Multiply and add
3. Shift, repeat step 2

17. Filtering
Multiply two functions together….
time 2
* = -1

18. Convolution Source wavelet: 2, 4, 3
Reflectivity sequence: 0, 0, 1, 0, 0, 0, 0

19. Convolution Two arrays (wavelets): Source wavelet: 2, 4, 3
Reflectivity sequence: 1, 4, 3, 2
(output length = sum of two wavelets – 1)

20. Filtering = convolution
When convolving a wavelet (source signature) with a spike train of reflection coefficients, we are just filtering. We can reverse this process (deconvolve) to extract the spike train.
Spike = infinite frequency reflectivity function
Filtered = band-limited reflectivity function

21. Filtering
Change spike to band-limited wavelet. 2
* = -1

22. The Earth as a filter S(t)*R(t)*G(t) S = source
R=reflections = earth model
G=recording instrument (geophone) response
filter
input
output

23. z-transforms
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:
a discrete-time equivalent of the Laplace transform
Function: 2, 5, 3, -2, 1
z-transform: 2 + 5z + 3z2 –2z3 + z4 dt

24. z-transforms 1+4z+3z2+2z3 * 2+4z+3z2 = 2+12z+25z2+28z3+17z4+6z5
The convolution of two functions can be accomplished by multiplying their z- transforms:
1+4z+3z2+2z3 * 2+4z+3z2
= 2+12z+25z2+28z3+17z4+6z5
(multiplication of polynomials)

25. Convolution Two arrays (wavelets): Source wavelet: 2, 4, 3
Reflectivity sequence: 1, 4, 3, 2
(output length = sum of two wavelets – 1)

26. z-transforms
z-transforms allow us to define “minimum phase” and “maximum phase” wavelets.
Break a wavelet’s z-transform polynomial into its roots:
Polynomial: z-11z2-6z3
Roots: (3+2z), (2+1z), (5-3z)

27. z-transforms: min and max phase wavelets
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
A minimum-phase wavelet has only minimum- phase roots (2+1z)(3+2z)
A maximum-phase wavelet has only maximum-phase roots (1+2z)(3+5z)
A mixed-phase wavelet has both minimum and maximum-phase roots (2+1z)(3+5z)

28. Filtering – frequency domain
Time domain frequency domain
=convolution =complex multiplication
T(t)
F(t)
T(t) F(t)
Fourier transform
T(w)
F(w)
convolution
T(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function

29. Fast Fourier Transforms - FFTs
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.
FFTs make it faster to filter in the frequency domain than in the time domain.

30. Fourier transforms (and FFTs)
Result in the frequency domain is a complex number
For filtering, multiply each frequency component by the corresponding frequency component of the filter (complex multiplication).
Imaginary (sin)
Ac + iAs As A f
Real (cos) Ac
Phase f = tan-1(As/Ac)

31. Complex multiplication
(a+bi)(c+di) = ac + (ad+bc)i – bd
=(ac – bd) + (ad + bc)i

32. Filtering Time domain frequency domain
=convolution =complex multiplication
T(t)
F(t)
T(t) F(t)
Fourier transform
T(w)
F(w)
convolution
T(w)*F(w)
Complex mult.
Inverse
Fourier transform
Filtered time function
Filtered time function

33. Correlation
An operation closely related to convolution is correlation.
Convolution: h(t) = ∫f(t)g(t-t)dt
Correlation: (t) = ∫f(t)g(t-t)dt
In practice, correlation is done the same way as convolution in the time domain, except that neither wavelet is reversed before the multiplication process.

34. Correlation
In the frequency domain, correlation is carried out as a complex multiplication, but first the complex conjugate is taken of one of the functions.
Convolution: H(w) = F(w) * G(w)
Correlation: (w) = F(w) * G(w)

35. Cross-correlation Two arrays (wavelets): Source wavelet: 2, 4, 3
Reflectivity sequence: 3, 2, 1, 4, 3, 2, 4, 1
(often, output length = wavelet1 – wavelet2 + 1)

36. Cross-correlation
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: 4, 3, 2
Wavelet 2: 3, 2, 1, 4, 3, 2, 1, 4
best match

37. Vibroseis
Vibroseis sources use cross-correlation to synthesize a short, zero- phase wavelet from a long source sweep.

38. Autocorrelation
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 zero- phase wavelet from a seismic trace.

39. Autocorrelation Correlation of function with itself
wavelet: 1, 4, 3, 2
Symmetric,
zero-phase wavelet
Total power in
wavelet (amplitude2)

40. Resolution

41. Rayleigh’s Criterion
sin2x x2 b
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. A
0.81A b
Seismic Resolution of Zero-Phase Wavelets,
R. S. Kallweit and L. C. Wood, Amoco Houston Division DGTS, January 12, 1977

42. Resolution: “How thin is a ‘thin’ bed?”
The ability to distinguish two separate reflectors (top and bottom of a layer).

43. Time (vertical) resolution
Side lobes make it difficult to distinguish closely- spaced reflectors. More bandwidth reduces side lobes.
(Kallweit demo)

44. Tuning/resolution
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”

45. Lateral Resolution (pre-migration) Fresnel zones
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.

46. Fresnel zone width 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.

47. Vertical resolution Horizontal resolution l/4 F = v t/fdom
Cannot determine location
of reflector within box
Vertical
resolution
l/4
Horizontal
resolution
F = v t/fdom

48. Lateral resolution
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.

49. Migration
Migration increases the lateral resolution by making use of the non-symmetric (non- specular) reflections contained in the diffraction curve.
specular
(strongest)
non-specular *

51. Horizontal resolution
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.

52. Resolution >>improves with greater bandwidth
Vertical (time) resolution
>>improves with greater bandwidth
Horizontal resolution
>>improves with greater range of incidence angles (more complete diffraction curves)
>>MUST migrate data to get this resolution