# Fundamentals of signal processing exercises

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

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.

 ${\displaystyle {{f}_{a}}=\left|2m\,{{f}_{N}}-{{f}_{s}}\right|,}$ (2)

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 (Δxt) 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 240 25 25 240 50 25 240 100 25 240 25 50

 ${\displaystyle {n_{f}}={\frac {{n_{g}}\Delta g}{2\Delta s}},}$ (12)

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

 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) (2) Scaling x(at) ${\displaystyle {{\left|a\right|}^{-1}}X\left({\omega }/{a}\;\right)}$ (3) Differentiation dx(t)/dt iωX(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) ${\displaystyle {{\left|X\left(\omega \right)\right|}^{2}}}$ (8) Parseval’s theorem ${\displaystyle \int {{\left|x\left(t\right)\right|}^{2}}\ dt}$ ${\displaystyle \int {{\left|X\left(\omega \right)\right|}^{2}}\ d\omega }$
 * denotes convolution.

## References

1. Claerbout, 1976, Claerbout, J. F., 1976, Fundamentals of geophysical data processing: McGraw-Hill Book Co.
2. Bracewell, 1965, Bracewell, R. N., 1965, The Fourier transform and its applications: McGraw-Hill Book Co.