Elementary wavefront techniques

4.2.4 Elementary wavefront techniques

Though very versatile, the raytrace method described in section 4.2.2 has its limitations. For exam- ple, it is not obvious how to make the migrated amplitudes any different from the unmigrated ones and it is equally obvious that amplitudes should change under migration. Recall that bandlimited reflectivity is the goal and that the geometrical spreading of wavefronts must be accounted for to estimate this. (See section 4.1.1.) Wavefront methods provide amplitude adjustments in a very natural way though they are more difficult to adapt to complex media than the raytrace approach.

Hagedoorn (1954) provided one of the first, and still very relevant, explanations of wavefront migration. The ideas presented in this section are all essentially attributable to him. Consider the ZOS image of a dipping reflector overlain by a constant velocity medium as shown in Figure 4.12A.

92 CHAPTER 4. ELEMENTARY MIGRATION METHODS

Figure 4.12: (A) Normal-incidence raypaths are shown for a dipping reflector in a constant velocity medium. (B) The traveltime section for (A) plots the arrival times for each raypath

beneath its emergence point. (C) When (B) is plotted at apparent depth, z a = vt/2, an apparent dip can be defined.

93 The reflector dip, δ, can be related to the lengths of the raypaths and their emergence points by

4.2. FUNDAMENTAL MIGRATION CONCEPTS

The time section corresponding to this raypath diagram is shown in Figure 4.12B. Since velocity is constant, the arrival times are directly proportional to the path lengths through t k = 2l k /v and the ZOS image of the reflector is seen to have a time dip of

which is in agreement with equation (4.27) that the time dip on a ZOS gives twice the normal- incidence ray parameter. A further conceptual step can be taken by mapping the traveltimes to

apparent depth, z a , by z a = vt/2 as shown in Figure 4.12C. At this stage, the unmigrated display has both coordinate axes in distance units so that it makes sense to define an apparent dip through

Finally, comparison of equations (4.32) and (4.34) gives the relation

(4.35) Equation (4.35) is called the migrator’s equation and is of conceptual value because is illustrates

sin δ = tan α.

the geometric link between unmigrated and migrated dips. However, it should not be taken too far. It is not generally valid for non-constant velocity and cannot be applied to a time section. Though this last point may seem obvious, it is often overlooked. It is not meaningful to talk about a dip in degrees for an event measured on a time section because the axes have different units. A change of time-scale changes the apparent dip. On an apparent depth section, an apparent dip can be meaningfully defined and related to the true dip through equation (4.35). The apparent dip can never be larger than 45 ◦ because sin δ ≤ 1 which is a re-statement of the fact that the maximum time dip is ∆t/∆x = 1/v as discussed in section 2.11.1. Though intuitively useful, the apparent dip concept is not easily extended to variable velocity. In contrast, the limitation on maximum time dip is simply restated in v(z) media as ∆t/∆x = 1/v min where v min is the minimum of v(z).

Comparison of Figures 4.12A and 4.12C shows that, for a given surface location x k , the normal- incidence reflection point and the corresponding point on the ZOS image both lie on a circle of radius l k centered at x k . This relationship is shown in Figure 4.13 from which it is also apparent that

• The ZOS image lies at the nadir (lowest point) of the circle. • The reflection point is tangent to the circle. • The ZOS image and the reflector coincide on the datum (recording plane).

These observations form the basis of a simple wavefront migration technique which is kinemati- cally exact for constant velocity:

i) Stretch (1D time-depth conversion) the ZOS image from t to z a .

ii) Replace each point in the (x, z a ) picture with a wavefront circle of radius z a . Amplitudes along the wavefront circle are constant and equal to the amplitude at (x, z a ).

94 CHAPTER 4. ELEMENTARY MIGRATION METHODS

Datum

Zos k

on image ts

Figure 4.13: The normal-incidence reflection Figure 4.14: The wavefront migration of a dip- point and the corresponding point on the ZOS

ping reflector is shown. Each point on the ZOS image both lie on a circle of radius l k centered

image is replaced by a wavefront circle of radius at x k .

l k . The wavefronts interfere constructively at the location of the actual reflector.

iii) The superposition of all such wavefronts will form the desired depth section. Figure 4.14 shows the migration of a dipping reflector by wavefront superposition. The actual

mechanics of the construction are very similar to a convolution. Each point in the ZOS image is used to scale a wavefront circle which then replaces the point. A 2D convolution can be constructed by

a similar process of replacement. The difference is that the replacement curve in a 2D convolution is the same for all points while in the migration procedure it varies with depth. Thus the migration process can be viewed as a nonstationary convolution.

Wavefront migration shows that the impulse response of migration is a wavefront circle. That is, if the input ZOS contains only a single non-zero point, then the migrated section will be a wavefront circle. The circle is the locus of all points in (x, z) that have the same traveltime to (x k , 0). Two equivalent interpretations of this are either that the circle is the only earth model that could produce

a single output point or that the circle is the curve of equal probability. The first interpretation is

a deterministic statement that only one earth model could produce a ZOS image with a single live sample. The second interpretation is a stochastic one that suggests why the generalization from one live sample to many works. By replacing each point with it’s curve of equal probability, the migrated section emerges as the most likely geology for the superposition of all points. The constructive interference of a great many wavefronts forms the migrated image.

Exercise 4.2.1. Describe the appearance of: • a migrated section that contains a high amplitude noise burst on a single trace. • a migration that results from a ZOS image whose noise level increases with time.

• the edge of a migrated section.