In an extensional basin, such as the Gulf of Mexico, salt tectonism in its advanced stage can cause the formation of salt diapirs with overhang structures. Consider such a salt structure as sketched in Figure 5.1-23. If velocities in the surrounding sedimentary sequence increase rapidly in depth, downward traveling waves change their direction at some depth and travel upward. When these turning waves, in their upward travel path, encounter a salt overhang, they reflect from the underside of the structure and follow a downgoing path before turning back upward to the surface. The turning-wave reflection then gives rise to a dipping event on the stacked section which conflict with the reflections associated with the surrounding gently dipping strata. The concept of imaging turning waves was introduced by Claerbout  and first demonstrated on field data by Hale . Many examples of turning-wave imaging of salt overhang structures from the Gulf of Mexico are given by Ratcliff .
Dip-moveout correction alone, even if it accounts for vertical velocity variations, does not preserve the turning-wave energy on stacked data. This is because the normal moveout associated with the turning raypaths exhibit an abnormal behavior as sketched in Figure 5.1-23. Consider a reflection point E that represents all dips on the salt flank. Sketched in Figure 5.1-23 are the zero-offset raypaths for the reflector dip of less than 90 degrees (raypath that emerges at location C), exactly 90 degrees (raypath that emerges at location B), and greater than 90 degrees (raypath that emerges at location A).
Note that when the CMP raypaths do not have an upward component (location C), the moveout trajectory in the CMP gather exhibits the usual behavior described by the hyperbolic traveltime equation:
where t is the two-way traveltime associated with a source-receiver separation 2h, t0 is the two-way zero-offset time, and vNMO is the velocity that best flattens the event after normal-moveout correction.
When, however, the CMP raypaths have an upward component associated with turning waves (location A), the moveout trajectory in the CMP gather exhibits an abnormal behavior which may be described by the traveltime equation 
Note that the abnormal moveout equation (25b) differs from the normal moveout equation (25a) by a change in the sign of the moveout term There can also exist a circumstance where the reflection off the salt flank does not exhibit any moveout (location B in Figure 5.1-23).
To make use of the turning-wave energy in imaging salt overhang structures, we must
- preserve the reflections with abnormal moveout associated with the turning waves during stacking, and
- migrate the stacked data using an algorithm that can handle dips beyond 90 degrees.
To derive a stacked section that preserves reflections associated with both normal (nonturning) and turning waves, consider the stacking process in two parts. First, use equation (25a) with positive to handle events with normal moveout and derive a stack associated with the normal waves. Second, use equation (25a) with negative to handle events with abnormal moveout and derive a stack associated with the turning waves. Finally, add the two sections to obtain the composite-wave stack.
Reflections off the flank of a salt dome, whether they are associated with normal waves or turning waves, and reflections associated with the surrounding sedimentary sequence constitute dipping events with different moveout velocities on a stacked section. As such, DMO correction needs to be applied to data prior to stacking both the normal and turning waves. The DMO operator preferably should account for the strong vertical velocity variations that give rise to turning waves .
Figure 5.1-24 (left column) shows a normal-wave stack, a turning-wave stack, and the composite-wave stack. Note that the normal-wave stack actually contains a significant portion of the turning wave energy represented by the steeply dipping event in the turning-wave stack. This is because the turning-wave energy is predominantly of low frequency. Therefore, despite its abnormal moveout behavior, much of the turning-wave energy is preserved during conventional stacking based on normal moveout.
Note from Figure 5.1-23 that turning waves propagate at angles greater than 90 degrees. This means that turning waves are evanescent waves and need to be imaged using a migration algorithm that can handle dips beyond 90 degrees. Reverse-time migration  described in migration principles is based on such an algorithm. The field data example shown in Figure 4.3-22 suggests that the salt dome at its root at about 3 s has a gentle overhang. This character is not identifiable in the image obtained by an algorithm that only handles waves propagating at less than 90 degrees (Figure 4.4-24).
Kirchhoff migration  and the phase-shift method  also can be adapted to image turning waves. To develop the conceptual basis of turning-wave migration, consider the raypath segments DA and ED as denoted in Figure 5.1-23. The raypath segment DA is associated with the upcoming wave energy contained in the normal-wave stack and the raypath segment ED is associated with the downgoing wave energy contained in the turning-wave stack (Figure 5.1-24).
Just as the stacking process was treated in two parts, the imaging process also can be treated in two parts. First, perform phase-shift migration of the normal-wave stack; this involves downward extrapolation of the upcoming wave energy along the raypath CE from the surface z = 0 to a depth zE where the wave originates (location E in Figure 5.1-23). The equation for the wave extrapolation is (Section D.1):
where ky, kz, and ω are the Fourier transform variables associated with the coordinate variables for midpoint y, depth z, and two-way zero-offset traveltime t, respectively, and P(ky, z = 0, ω) is the 2-D Fourier transform of the upcoming wavefield at the surface z = 0 represented by the stacked data P(y, z = 0, t). The vertical wavenumber kz is defined in terms of the horizontal wavenumber ky and frequency ω by the dispersion relation (Section D.1):
where v is the medium velocity. In equation (27), the region of propagation at angles less than 90 degrees is associated with ω > (v/2)|ky|, and the region of evanescense at angles greater than 90 degrees is associated with ω < (v/2)|ky|.
Figure 5.1-23 Geometry of turning rays in a medium with velocities increasing in depth, and the CMP traveltime trajectories depicted above at three locations, A, B, and C. At location A, the turning raypaths such as ADE causes an abnormal moveout. At location C, the raypaths yield a normal moveout, and at location B no moveout is observed on the CMP gather. Adapted from Hale .
Figure 4.3-22 (a) A CMP-stacked section, (b) reverse time migration.
Figure 4.4-24 Migration of the CMP stack shown in Figure 4.3-22a using the phase-shift method.
Next, perform phase-shift migration of the turning-wave stack; this involves downward extrapolation of the upcoming wave energy along the raypath AD from the surface z = 0 to a depth zD where the wave turns (location D in Figure 5.1-23). The equation for the wave extrapolation is
Since the journey of the turning waves does not end at the turning point D, we must continue with the wave extrapolation until we reach the point where the waves originate on the salt flank (location E in Figure 5.1-23). Hence, perform upward extrapolation of the downgoing wave energy along the raypath DE from the depth level zD to location E on the salt flank:
Just as the stacking of normal waves and turning waves is combined to obtain the composite-wave stack (Figure 5.1-24), the migration processes based on the extrapolation equations (26) and (28a, 28b) can also be combined . First, substitute equation (28a) into equation (28b) to obtain
Based on the concepts described above, we now outline the steps involved in a turning-wave migration algorithm that makes use of the phase-shift method  . First, consider imaging the normal waves. The process involves downward continuation of the upcoming waves from the surface z = 0 to the maximum specified depth zmax at discrete depth steps Δz.
- Start with the composite-wave stack — an approximation to the zero-offset section P(y, z = 0, t) and apply 2-D Fourier transform to get the transformed wavefield P(ky, z = 0, ω).
- For each frequency ω > (v/2)|ky| and ω ≤ (v/2)|ky| in the transform domain, extrapolate the wavefield P(ky, z, ω) at depth z with a phase-shift operator exp (−ikz Δz) to get the wavefield P(ky, z + Δz, ω) at depth z+Δz. At each depth, a new extrapolation operator with the velocity v(z) defined for that z value is computed.
- Split the wavefield P(ky, z + Δz, ω) into its propagating Pu(ky, z + Δz, ω) and evanescent Pd(ky, z + Δz, ω) components, corresponding to the normal and turning waves, respectively.
- Save the wavefield Pd(ky, z + Δz, ω) for use later in turning-wave imaging.
- As for any other migration algorithm, invoke the imaging principle t = 0 upon the wavefield Pu(ky, z + Δz, ω) at each extrapolation step to obtain the migrated section from the normal waves Pu(ky, z, t = 0) in the transform domain. The imaging condition t = 0 is met by summing over all frequency components of the extrapolated wave-field at each depth step (equation D-84).
- Repeat steps (b) through (d) for all depth steps down to a specified z = zmax to obtain the normal-wave image in the transform domain Pu(ky, z, t = 0).
- Apply inverse Fourier transform in the midpoint y direction to obtain the image Pu(y, z, t = 0) from the normal waves.
Now consider imaging the turning waves. The process involves upward continuation of the downgoing waves from the maximum specified depth zmax to the surface z = 0 at discrete depth steps Δz. At each depth, the downgoing waves are updated by adding the evansecent wave Pd(ky, z, w) saved in step (d) during the downward extrapolation of the composite-wave stack. We shall assume that the downgoing wave at z = zmax is null.
- Add the evanescent wave saved at depth z to the downgoing wave at the same depth, and for each frequency ω ≤ (v/2)|ky| in the evansecent region of the transform domain, extrapolate the new downgoing wave with a phase-shift operator exp(−ikzΔz) to get the wavefield Pd(ky, z − Δz, ω) at depth z − Δz.
- Invoke the imaging principle t = 0 upon the wavefield Pd(ky, z − Δz, ω) at each extrapolation step to obtain the migrated section from the turning waves Pd(ky, z, t = 0) in the transform domain.
- Repeat steps (a) and (b) for all depth steps to obtain the turning-wave image in the transform domain Pd(ky, z, t = 0).
- Apply inverse Fourier transform in the midpoint y direction to obtain the image Pd(y, z, t = 0) from the turning waves.
- Add the normal-wave image and turning-wave image to obtain the composite-wave image.
As described above, it is important to emphasize that the turning-wave energy can be preserved only by making use of its abnormal moveout behavior during stacking and imaged properly by making use of the evanescent energy during migration.
Figure 5.1-24 (right column) shows migration of a normal-wave stack using the phase-shift equation (28), migration of a turning-wave stack using the phase-shift equation (5-31a), and migration of the composite-wave stack using the phase-shift equation (5-31b). Note the distinctively defined salt boundary obtained from imaging the compsite-wave stack.
- Claerbout (1985), Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Hale and Artley (1992), Hale, D. and Artley, C., 1992, Squeezing dip-moveout for depth-variable velocity: Geophysics, 58, 257–264.
- Ratcliff et al. (1992), Ratcliff, D. W., Gray, S. H., and Whitmore, N. D., 1992, Seismic imaging of salt structures in the Gulf of Mexico: The Leading Edge, 11, no. 4, 15–31.
- Artley and Hale (1994), Artley, C. and Hale, D., 1994, Dip-moveout processing for depth-variable velocity: Geophysics, 59, 610–622.
- Baysal et al., 1984, Baysal, E., Kosloff, D., and Sherwood, J. W. C., 1984, A two-way nonreflecting wave equation: Geophysics, 49, 132–141.
- Prestack partial migration
- Frequency-wavenumber DMO correction
- Log-stretch DMO correction
- Integral DMO correction
- Velocity errors
- Variable velocity