Exercise 4-1. Consider the special case of a 90-degree dipping reflector in Figure 4.1-1a. Sketch the corresponding zero-offset time section.
Exercise 4-2. Consider Huygens’ secondary sources along a dipping reflector. Sketch the zero-offset section by superimposing the individual responses from these sources. Remember, to do zero-offset modeling, you must map a point in the x − z plane to a hyperbola in the x − t plane with its apex as the input point.
Exercise 4-3. For which case is spatial aliasing a more serious problem, the low-velocity or high-velocity medium?
Exercise 4-4. Locate the dipping event AA′ on the migrated section in Figure 4.E-1.
Exercise 4-5. A point in the x − t plane is mapped onto a semicircle in the x − z plane. Where does it map on the x − τ plane, where τ = 2z/v.
Exercise 4-6. Refer to Figure 4.1-14. It suggests that if the subsurface consisted of a semicircular reflector (b), then the zero-offset response would be as in (a). What would the subsurface be like if you obtained (a) using a source-receiver pair with a finite separation between them? (See Figures D-5 and D-6.)
Exercise 4-7. Suppose you specified the wrong trace spacing in your migration. What effect does it have, overmigration or undermigration? Assume that you supplied the wrong sampling rate in time. What effect does it have on migration output?
Exercise 4-8. Suppose you want to do zero-offset recording of the steep flank of a salt dome. Which case would require a longer line length when the medium velocity along the raypath is (a) constant or (b) vertically increasing?
Exercise 4-9. How would Figure 4.3-6 look if you used a 15-degree phase-shift migration algorithm?
Figures and equations
Figure D-5 The response characteristics of the DSR operator (equation D-23) .]]. (a) Real part of the y − z plane at 16 Hz and h = 400 m. Note the semielliptical wavefronts. (b) Real part of the y − z plane at t = 1024 ms, h = 400 m. Because of the wraparound in h, we observe two wavefronts, one for h = 400 m and one for h = 0. (c) Real part of the y − t plane at z = 200, 400, 600, and 800 m superimposed. These are the table-top trajectories for h = 400 m. The loci of the arrival times are determined by a stationary-phase approximation to DSR (see Section D.2) . Periodicity in y and t result from approximating Fourier integrals by sums.
Figure D-6 The response characteristics of the exploding reflectors operator ER(Y) (equation D-29) . (a) Real part of the y − z plane at 16 Hz. Note the circular wavefronts. (b) Real part of the y − z plane at t = 1024 ms. (c) Real part of the y − t plane at z = 200, 400, 600, and 800 m superimposed. These are the hyperbolic trajectories. The loci of the arrival times are determined via the stationary-phase approximation to ER(Y) (see section D.2) . Periodicity in y and t result from approximating Fourier integrals by sums.
- Chun and Jacewitz, 1981, Chun, J.H. and Jacewitz, C., 1981, Fundamentals of frequency-domain migration: Geophysics, 46, 717–732.
- Yilmaz, 1979, Yilmaz, O., 1979, Prestack partial migration: Ph.D. thesis, Stanford University.
- Clayton, 1978, Clayton, R., 1978, Common midpoint migration: Stanford Expl. Proj., Rep. No. 14, Stanford University.
- Introduction to migration
- Migration principles
- Kirchhoff migration in practice
- Finite-difference migration in practice
- Frequency-space migration in practice
- Frequency-wavenumber migration in practice
- Further aspects of migration in practice
- Mathematical foundation of migration