FOURIER METHODS 115

4.4. FOURIER METHODS 115

0.08 wavelike k z =.06

0.06 f/v=.06 f/v 0.06

k z =.04

0.04 f/v=.04

k z =.02

0.02 f/v=.02

Figure 4.47: The space (f /v, k x ) is shown con- Figure 4.48: The space (k x ,k z ) is shown with toured with values of k z from equation (4.54).

contours of f /v as given by equation (4.51). The dashed lines are the boundary between the wavelike and evanescent regions.

f/v

70 k x

Figure 4.49: The mapping from (k x , f ) (top) to (k x ,k z ) (bottom) is shown. A line of constant f is mapped, at constant k x , to a semi-circle in (k x ,k z ). The compressed apparent dip spectrum of (f /v, k x ) space unfolds into the uniform fan in (k x ,k z ) space. The numbers on each space indicate dips in degrees.

116 CHAPTER 4. ELEMENTARY MIGRATION METHODS is real-valued when |f/k x | ≥ ˆv and is otherwise imaginary. Only for real k z will equation (4.45)

correspond to traveling waves in the positive and negative z directions. For complex k z e ±2πik z z becomes a real exponential that either grows or decays; however, on physical grounds, growing exponentials must be excluded. Given a value for ˆ v this dual behavior of k z divides (k x , f ) space into two regions as was discussed from another perspective in section 2.11.1. The region where |f/k x | ≥ ˆv is called the wavelike or bodywave regions and where |f/k x | < ˆv it is called the evanescent region. Equation (4.54) is sometimes called a dispersion relation for one-way waves since the choice of the plus sign in front of the square root generates upward traveling waves in e ±2πik z while the minus sign generates downward traveling waves.

The geometric relationships between the spaces of (f /v, k x ) and (k x ,k z ) are shown from two different perspectives in Figures 4.47 and 4.48. In (f /v, k x ) space, the lines of constant k z are hyperbolae that are asymptotic to the dashed boundary between the wavelike and evanescent regions. In (k x ,k z ) space, the curves of constant f /v are semi-circles. At k x = 0, k z = f /v so these hyperbolae and semi-circles intersect when the plots are superimposed.

The spectral mapping required in equation (4.53) is shown in Figure 4.49. The mapping takes a constant f slice of (k x , f ) space to a semi-circle in (k x ,k z ) space. Each point on the f slice maps at constant k x which is directly down in the figure. It is completely equivalent to view the mapping as

a flattening of the k z hyperbolae of Figure 4.47. In this sense, it is conceptually similar to the NMO removal in the time domain though here the samples being mapped are complex valued. That the spectral mapping happens at constant k x is a mathematical consequence of the fact that k x is held constant while the f integral in equation (4.50) is evaluated. Conceptually, it can also be viewed as

a consequence of the fact that the ERM seismogram and the migrated section must agree at z = 0 and t = 0. On a numerical dataset, this spectral mapping is the major complexity of the Stolt algorithm. Generally, it requires an interpolation in the (k x , f ) domain since the spectral values that map to grid nodes in (k x ,k z ) space cannot be expected to come from grid nodes in (k x , f ) space. In order to achieve significant computation speed that is considered the strength of the Stolt algorithm, it turns out that the interpolation must always be approximate. This causes artifacts in the final result. This issue will be discussed in more detail in section 4.4.2.

The creation of the migrated spectrum also requires that the spectrum be scaled by ˆ vk z / k 2 x +k z 2 as it is mapped (Equation (4.53)). In the constant velocity medium of this theory, sin δ = ˆ vk x /f (δ is the scattering angle) from which it follows that cos δ = ˆ vk z /f = k z / k 2 x +k z 2 . Thus the scaling factor is proportional to cos δ and therefore ranges from unity to zero as δ goes from zero to 90 ◦ . This scaling factor compensates for the “spectral compression” that is a theoretical expectation of the ERM seismogram. Recall the migrator’s formulae (equation (4.35)) that relates apparent angles in (f /v, k x ) space to real angles in (k x ,k z ) space. If there is uniform power at all angles in (k x ,k z ) space, then the migrator’s formula predicts a spectral compression in (f /v, k x ) (with consequent increasing power) near 45 ◦ of apparent dip. As shown in Figure 4.49 it is as though (k x ,k z ) space is an oriental fan that has been folded to create (f /v, k x ) space. Migration must then unfold this fan.

f -k migration is called a steep-dip method because it works correctly for all scattering angles from

0 ◦ to 90 ◦ . The f -k migration algorithm just described is limited to constant velocity though it is exact in this case. Its use of Fourier transforms for all of the numerical integrations means that it is computationally very efficient. Though it has been used for many years in practical applications its restriction to constant velocity is often unacceptable so that it has gradually being replaced by more flexible, though usually slower, methods. Today, one major virtue still remains and that is the conceptual understanding it provides. The description of the construction of the migrated spectrum

will provide the basis for a realistic theory of seismic resolution in section 4.7.