The exploding reflector model

4.2.6 The exploding reflector model

In the preceding sections, two methods have been described to migrate stacked data. In section

4.2.2 the stacked section was treated as a normal-incidence seismogram and raytrace techniques were developed to migrate it. In section 4.2.4 the wavefront migration method was presented for constant-velocity zero-offset sections. Further progress is greatly facilitated by a more abstract model of the stacked section. Ultimately, the goal is to formulate the post-stack migration problem as a solution to the wave equation where the measured data plays the role of a boundaryvalue. However, this is complicated by the fundamental fact that the CMP stack is not a single wavefield but the composite of many individual wavefields. There is no single physical experiment that could record a ZOS and so a ZOS cannot be a single physical wavefield.

There is a useful thought experiment, called the exploding reflector model (ERM), that does yield something very similar to a ZOS and serves as the basis for most post-stack migration methods. (The following discussion is presented in 2D and the generalization to 3D is elementary.) As motivation, note that equation (4.36) can be rewritten as

(x − x 0 ) 2

t 2 x =t 2 0 +

(4.38) where ˆ v = v/2 is called the exploding reflector velocity and t 0 = z/ˆ v. This trivial recasting allows

the interpretation that the point diffractor is a seismic source of waves that travel at one-half of the physical velocity. As shown in Figure 4.21 the exploding reflector model adopts this view and postulates a model identical to the real earth in all respects except that the reflectors are primed with explosives and the seismic wave speeds are all halved. Receivers are placed at the CMP locations and at t = 0 the explosives are detonated. This creates a wavefield that is morphologically identical to the geology at the instant of explosion.

If ψ(x, z, t) denotes the exploding reflector wavefield, then the mathematical goal of the migration problem is to calculate ψ(x, z, t = 0). Thus ψ(x, z, t = 0) is identified with the geology and represents the migrated depth section. The ERM wavefield is allowed to propagate only upward (the −z direction) without reflections, mode conversions, or multiples. It refracts according to Snell’s law and the half-velocities mean that the traveltimes measured at the receivers are identical to those of the ZOS. In the constant velocity simulation of Figure 4.22, a snapshot of the ERM wavefield, ψ(x, z, t =

4.2. FUNDAMENTAL MIGRATION CONCEPTS

receivers

zosWimage Geology

Figure 4.23: The ERM seismogram is shown superimposed on the migrated depth section.

const), is shown as the wavefield approaches the receivers. The raypaths are normal-incidence raypaths (because of the isomorphism between the geology and ψ(x, z, t = 0)) and spreading and buried foci are manifest. The figure emphasizes three Huygen’s wavelets that emanate from the three corners of the geologic model. The migrated depth section is also a snapshot, but a very particular one, so the term snapshot will be used to denote all such constant-time views.

In Figure 4.23, the ERM seismogram, ψ(x, z = 0, t), is shown in apparent depth superimposed on top of the depth section. Wavefront circles are shown connecting points on the geology with points on the seismogram. The ERM seismogram is kinematically identical (i.e. the traveltimes are the same) with the normal-incidence seismogram and is thus a model of the CMP stack. It is also kinematically identical with all normal-incidence, primary, events on a ZOS image. This allows the migration problem to be formulated as a solution to the wave equation. Given ψ(x, z = 0, t) as a boundary condition, a migration algorithm solves the wave equation for the entire ERM wavefield, ψ(x, z, t), and then sets t = 0 to obtain the migrated depth section. This last step of setting t = 0 is sometimes called imaging though this term has also come to refer to the broader context of migration in general. Precisely how the wavefield is evaluated to extract the geology is called an imaging condition. In the post-stack case the imaging condition is a simple evaluation at t = 0. Later, it will be seen that there are other imaging conditions for pre-stack migration.

Thus far, the ERM has given simple mathematical definitions to the migrated depth section, the recorded seismogram, and the wavefield snapshot. In addition, the extrapolated seismogram can be defined as ψ(x, z = ∆z, t). Both the ERM seismogram and the extrapolated seismogram are time sections while the snapshot and the migrated section are in depth. The extrapolated seismogram is a mathematical simulation of what would have been recorded had the receivers been at z = ∆z rather than at z = 0. The construction of ψ(x, z = ∆z, t) from ψ(x, z = 0, t) is called downward continuation and, synonymously, wavefield extrapolation.

As a summary, the ERM has defined the following quantities: ψ(x, z, t) . . . the ERM wavefield. ψ(x, z, t = 0) . . . the migrated depth section. ψ(x, z, t = const) . . . a wavefield snapshot. ψ(x, z = 0, t) . . . the ERM seismogram. ψ(z, z = ∆z, t) . . . an extrapolated section.

100 CHAPTER 4. ELEMENTARY MIGRATION METHODS

cmp stack

extrapolated

∆z

section migrated depth

section

Figure 4.24: This prism shows the portion of (x, z, t) space that can be constructed from

a finite-extent measurement of ψ(x, z = 0, t) (upper face). Also shown are the migrated depth section, ψ(x, z, t = 0), (vertical slice) and an extrapolated section, ψ(x, z = ∆z, t), (horizontal slice).

These quantities are depicted in Figure 4.24. It is significant that any extrapolated seismogram can

be evaluated at t = 0 to give a single depth sample of the migrated depth section. The process of deducing a succession of extrapolated seismograms, and evaluating each at t = 0, is called a recursive migration. It is recursive because the extrapolated seismogram ψ(x, z = z k , t) is computed from the previous seismogram ψ(x, z = z k−1 , t). Examples of recursive migrations are the finite- difference methods and the phase-shift techniques. An alternative to the recursive approach is a direct migration that computes ψ(x, z, t = 0) directly from ψ(x, z = 0, t) without the construction of any intermediate products. Examples of direct migration are (k x , f ) migration and Kirchhoff migration.

A direct migration tends to be computationally more efficient than a recursive approach both in terms of memory usage and computation speed. However, the direct methods must deal with the entire complexity of the velocity structure all at once. In contrast, the recursive approach forms a natural partitioning of the migration process into a series of wavefield extrapolations. The extrapolation from z = z k−1 to z = z k need only deal with the velocity complexities between these two depths. If the interval z k−1 →z k is taken sufficiently small, then the vertical variation of velocity can be ignored and v can be considered to depend only upon the lateral coordinates. In this way the migration for a complex v(x, z) variation can be built from the solution of many v(x) problems.