Fundamentals of signal processing exercises
Exercise 1-1. Refer to Figure 1.E-1. Using equation (2), compute the alias frequencies at 4- and 8-ms sampling intervals for the three frequency components — A, B and C from the amplitude spectrum of the 2-ms time series. Confirm the results of your computation by the amplitude spectra.
Exercise 1-2. Using the hyperbolic traveltime equation, compute the average velocity down to reflector A in Figure 1.3-8. Assume a constant velocity between A and the surface. The required information about Record 8 is in Table 1-13.
Exercise 1-3. What is the dispersive event C in Figure 1.3-8?
Exercise 1-4. Refer to the walk-away noise test record in Figure 1.3-19. Measure the phase velocity (Δx/Δt) of the ground-roll energy at location A1. Also measure the dominant frequency at the same location. Then, estimate the dominant wavelength (velocity/dominant frequency) of the ground roll. The receiver array length needed to suppress this energy in the field should be equal to or greater than the longest noise wavelength. The required information about Record 19 that you need for your computations is in Table 1-13.
Exercise 1-5. Measure the group velocity (x/t) of ground-roll energy A in Figure 1.3-25. The required information for Record 25 is in Table 1-13.
Exercise 1-6. Refer to the shot record in Figure 1.3-27. The near-offset 28 channels were recorded using a 50-m group interval. What is the group interval for the remaining 64 channels? The required information for Record 27 is in Table 1-13.
Exercise 1-7. What is event A in Figure 1.3-29? Are events C, D, and E multiples of B?
Exercise 1-8. Refer to Figure 1.3-30. Compute the water velocity using both the direct arrivals A and the water-bottom reflection B. Use one-way time for A and two-way time for B. The required information for Record 30 is in Table 1-13.
Exercise 1-9. Identify events A, B, C, D, E, and F in Figure 1.3-33.
Exercise 1-10. Identify events A, B, C, D, and E in Figure 1.3-34.
Exercise 1-11. Gain application involves multiplying the gain function with the seismic trace (Figure 1.4-8). Use the appropriate entry in Table A-1 to describe the effect of the gain application in the frequency domain.
Exercise 1-12. From Figure 1.E-2, identify the common-shot and common-midpoint gathers.
Exercise 1-13. Use equation (12) to compute fold nf for each of the following recording geometries:
|Number of Channels||Shot Spacing, m||Receiver Spacing, m|
Exercise 1-14. Consider the recording geometry in Figure 1.5-8. Sketch the traveltime curves on a common-shot gather associated with point scatterers (a) beneath the cable, (b) behind, and (c) in front of the cable. Assume all scatterers are on the plane of recording.
Exercise 1-15. Suppose that the shot associated with gather 1 in Figure 1.5-10 is missing. Identify the midpoints that are affected by this missing shot; that is, the midpoints with a lower fold of coverage. Suppose the receiver associated with gather 2 in Figure 1.5-10 is missing. Identify the midpoints that are affected by this.
Exercise 1-16. Prove the shifting, scaling, and differentiation rules applied to the Fourier transform of a function given by the entires (1), (2), and (3) in Table A-1.
Figures and tables
Figure 1.E-2 A field data set displayed in two different domains — common-shot and common-midpoint (see Exercise 1-12).
Figure 1.4-8 Gain is a time-variant scaling defined by a function, g(t). Based on some criteria, this function is defined at the time samples (shown by solid circles) that are usually at the center of specified time gates along the trace as indicated by 1, 2, 3, and 4. Gain application simply involves multiplying g(t) by the input trace amplitudes.
Figure 1.5-8 (a) Seismic data acquisition is done in shot-receiver (s, g) coordinates. The raypaths are associated with a planar horizontal reflector from a shot point (indicated by the solid circles) to several receiver locations (indicated by the triangles). The processing coordinates, midpoint-(half) offset, (y, h) are defined in terms of (s, g): y = (g + s)/2, h = (g − s)/2. The shot axis here points opposite the profiling direction, which is to the left. On a flat reflector, the subsurface is sampled by reflection points which span a length that is equal to half the cable length. (b) Seismic data processing is done in midpoint-offset (y, h) coordinates. The raypaths are associated with a single CMP gather at midpoint location M. A CMP gather is identical to a CDP gather if the depth point were on a horizontally flat reflector and if the medium above were horizontally layered.
Figure 1.5-10 (a) A hypothetical stacking chart (adapted from ). Each dot represents a single trace with the time axis perpendicular to the plane of the page. Shot-geophone (s, g), and midpoint-offset (y, h) coordinates are superimposed with the (y, h) plane rotated 45 degrees with respect to the (s, g) plane. Here, (1) is a common-shot gather, (2) is a common-receiver gather, (3) is a CMP gather, (4) is a common-offset section, and (5) is a CMP-stacked section. The remaining notation is defined in the text. (b) Raypaths associated with (1) a common-shot gather, (2) a common-receiver gather, (3) a CMP gather, and (4) a common-offset section. Solid triangles denote receiver locations and solid circles denote shot locations, x is the effective cable length — the difference between the maximum and minimum offsets, E denotes a midpoint and E’ denotes a depth point on a flat reflector.
|Record Number||Area||Number of Samples per Trace||Number of Traces||Sampling Interval, ms||Trace Interval, ft or m||Inner Offset, ft or m||Source|
|1||South Texas||1,275||48||4||330 ft||990 ft||V|
|2||West Texas||1,025||120||4||100 ft||400 ft||V|
|3*||Louisiana||1,500||24||4||340 ft||340 ft||D|
|4||Turkey||1,275||48||4||100 m||250 m||V|
|5||South America||3,000||48||2||100 m||200 m||D|
|6||Far East||1,250||48||4||100 m||150 m||D|
|7||South America||2,600||48||2||100 m||300 m||V|
|8||Central America||1,300||96||4||50 m||100 m||D|
|9||Alaska||1,000||96||4||220 ft||990 ft||V|
|10||North Africa||1,325||120||4||25 m||300 m||V|
|11||Alaska||1,000||96||4||220 ft||990 ft||V|
|12||Mississippi||1,275||48||4||330 ft||990 ft||V|
|13||Offshore Texas||2,025||48||4||220 ft||875 ft||A|
|14||Offshore Texas||1,525||48||4||220 ft||690 ft||P|
|15||Offshore Canada||2,500||48||2||25 m||360 m||A|
|16||South America||1,275||48||4||25 m||233 m||A|
|17||South America||2,000||48||4||50 m||250 m||A|
|18||Offshore Louisiana||1,500||120||4||82 ft||716 ft||A|
|19||Turkey||1,250||216||4||10 m||50 m||D|
|20||South Aleutians||2,025||120||4||82 ft||921 ft||A|
|21||Denver Basin||1,550||48||2||220 ft||220 ft||V|
|22||Williston Basin||1,550||48||2||110 ft||110 ft||V|
|23||San Juaquin Basin||1,550||48||2||220 ft||220 ft||V|
|24||Arctic||3,000||48||2||220 ft||220 ft||S|
|25||Alberta||2,000||96||2||50 m||50 m||D|
|26||Alberta||1,500||48||2||67 m||67 m||D|
|27||Canada||1,791||92||4||50 m||200 m||A|
|28||Canada||2,500||48||2||25 m||300 m||A|
|29||Offshore Spain||2,000||48||4||50 m||250 m||M|
|30||Offshore Crete||2,125||96||4||25 m||230 m||A|
|31||North Sea||1,550||96||4||25 m||228 m||A|
|32||North Sea||1,550||96||4||25 m||178 m||A|
|33||North Sea||1,625||96||4||25 m||200 m||A|
|34||Celtic Sea||1,500||60||4||50 m||253 m||A|
|35||Denmark||2,500||52||2||100 m||100 m||D|
|36||Middle East||1,024||48||4||50 m||250 m||V|
|37||Turkey||1,000||48||4||75 m||187 m||V|
|38||North Africa||2,500||60||2||100 m||100 m||V|
|39||Middle East||2,500||60||2||50 m||100 m||G|
|40||West Africa||2,600||96||2||30 m||120 m||D|
|*Analog recording. V: vibroseis, D: dynamite, A: Air gun, P: Aquapulse, M: Maxipulse, G: Geoflex, S: Aquaseis. All vibroseis records have been correlated. Aquapulse and Maxipulse are registered trademarks of Western Geophysical Company of America. Aquaseis and Geoflex are registered trademarks of Imperial Chemical Industries.|
|Operation||Time Domain||Frequency Domain|
|(1) Shifting||x(t − τ)||exp(−iωτ)X(ω)|
|(4) Addition||f(t) + x(t)||F(ω) + X(ω)|
|(5) Multiplication||f(t) x(t)||F(ω)* X(ω)|
|(6) Convolution||f(t)* x(t)||F(ω) X(ω)|
|(7) Autocorrelation||x(t)* x(−t)|
|(8) Parseval’s theorem|
|* denotes convolution.|
- Claerbout, 1976, Claerbout, J. F., 1976, Fundamentals of geophysical data processing: McGraw-Hill Book Co.
- Bracewell, 1965, Bracewell, R. N., 1965, The Fourier transform and its applications: McGraw-Hill Book Co.
- Introduction to fundamentals of signal processing
- The 1-D Fourier transform
- The 2-D Fourier transform
- Worldwide assortment of shot records
- Gain applications
- Basic data processing sequence
- A mathematical review of the Fourier transform