Finite-difference migration in practice
As described in migration principles, finite-difference migration is implemented using implicit and explicit schemes. In this section, we shall include in our discussion the 15-degree finite-difference algorithm because of its historical significance. Nevertheless, we shall primarily discuss practical aspects of the steep-dip implicit and explicit schemes in the frequency-space domain. Specifically, we shall deal with the impulse responses, depth step size and response to velocity errors in implicit and explicit schemes.
The first finite-difference migration algorithm that was introduced to the seismic industry was based on the parabolic approximation to the scalar wave equation . The algorithm was implemented in the time-space domain and designed using an implicit scheme. The parabolic approximation theoretically limits the algorithm to handling dips up to 15 degrees (Section D.3). Nevertheless, in practice, it can handle dips up to 35 degrees with sufficient accuracy due to the bandlimited nature of seismic data. Steeper dips, in principle, can be migrated by a cascaded application of the 15-degree algorithm .
Finite-difference migration of stacked data currently is performed using steep-dip algorithms based on the continued fractions expansion to the scalar wave equation. This approximation provides a theoretical dip accuracy up to 45 degrees. The basic 45-degree scheme can be improved to handle steeper dips up to 80 degrees with reasonable accuracy (Section D.4). The 45-degree finite-difference algorithm commonly is implemented using an implicit scheme in the frequency-space domain.
First, as we did for the Kirchhoff migration, we examine the impulse response of the 15-degree implicit scheme. The shape of the impulse response of a desired migration algorithm with no dip limitation is a semicircle. The shape of the impulse response of the 15-degree equation is, in theory, an ellipse  as seen in Figure 4.3-1. The nature of the dispersive noise pattern inside the ellipse is discussed in the next section on depth step size. Isolated noise spikes in field data can introduce such noise patterns on migrated sections.
The parts of the responses above the small circles in Figure 4.3-1 correspond to the evanescent energy, while the parts below the circles correspond to the propagating energy . The parts below the circles are the useful part of the response. The evanescent energy travels horizontally and is characterized by imaginary wavenumbers kz, which occur when the quantity in the square root in equation (13b) becomes negative. This means that the evanscent region corresponds to horizontal wavenumbers kx > 2ω/v. For negative kz, the exact extrapolator exp(−ikzz) is no longer a wave propagator; instead, it causes waves to decay rapidly in depth. Thus, evanescent energy is not expected to be present in recorded wavefields. The impulse response of the 15-degree finite-difference algorithm, however, suggests propagation in the region of evanescence. This is not desirable; the parts of the elliptical wavefront above the small circles should be removed. Use of a depth step size that is greater than the input sampling rate tends to suppress the response in the evanescent region. Excessively large depth steps, however, cause truncation of the wavefront further into the propagating zone below the small circles in Figure 4.3-1.
Figure 4.3-1 Desired impulse response of a 90-degree migration algorithm is a semicircle (top), while the impulse response of the 15-degree migration algorithm is an ellipse (bottom). The propagation zone is defined by the portion of the ellipse below the small circles, and the evanescent zone is defined by the portions above the small circles. For comparison, the desired response has been superimposed on the impulse response of the 15-degree equation.
Figure 4.3-2 (a) CMP stack, (b) desired migration by phase-shift method, (c) 15-degree finite-difference migration. The finite-difference migration based on the parabolic equation has the inherent property of undermigrating the steep flank of the diffraction and the steeply dipping event. See Figure 4.3-3 for a sketch of the migration results.
The impulse response (Figure 4.3-1) is used to estimate the maximum dip that the implicit finite-difference algorithm can handle without serious amplitude distortions or phase errors. This is done by super-imposing the desired semicircular response and measuring the angle between the indicated lines. Note from the measured angle in Figure 4.3-1 that the 15-degree implicit scheme can be used to migrate dips up to approximately 35 degrees with sufficient accuracy. This is primarily because errors associated with finite-difference approximations used in particular implementations of the 15-degree equation usually are adjusted to cancel some of the theoretical error associated with the 15-degree differential equation.
The dip-limited nature of the parabolic equation causes undermigration of steeply dipping events and steep flanks of diffractions. This is demonstrated by the field data example in Figure 4.3-2. The two prominent features, diffraction D and dipping event B, are located as shown in Figure 4.3-3 before and after migration.
- Claerbout and Doherty, 1972, Claerbout, J.F. and Doherty, S.M., 1972, Downward continuation of moveout-corrected seismo-grams: Geophysics, 37, 741–768.
- Larner and Beasley, 1990, Larner, K. L. and Beasley, C., 1990, Cascaded migrations: improving the accuracy of finite-difference migration: Geophysics, 52, 618–643.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Introduction to migration
- Migration principles
- Kirchhoff migration in practice
- Frequency-space migration in practice
- Frequency-wavenumber migration in practice
- Further aspects of migration in practice
- Mathematical foundation of migration