Huygens’ principle and point diffractors

4.2.5 Huygens’ principle and point diffractors

Christian Huygens was an early physicist and astronomer (a contemporary of Newton) who made

a number of advances in the understanding of waves. Most notable was his principle that, if the

4.2. FUNDAMENTAL MIGRATION CONCEPTS

Datum

point sources

Point diffractors (scatterers)

Reef Figure 4.15: Huygens’ principle reconstructs the

t + ∆t

Figure 4.16: The seismic response of a contin- wavefront at t + ∆t by the superposition of the

uous geological structure is synthesized as the small wavefronts from secondary point sources

superposition of the responses of many point placed on the wavefront at t.

diffractors.

position of a wavefront at time t is known, its position at t + ∆t can be computed by a simple strategy. Each point on the wavefront at time t is considered to be a secondary source of a small spherical (3D) or circular (2D) wavefront called a Huygens’ wavelet as shown in Figure 4.15. For the seismic problem, Huygens’ principle can be adapted to consider the response of a continuous reflector as the superposition of the responses of a great many point diffractors or scatterpoints(Figure 4.16).

Figure 4.17 shows the concept of a point diffractor that scatters any incident wave in all directions. Thus the complete imaging of any reflector is only possible if all of this scattered energy is captured and focused at the scatterpoint. Each scattered raypath is characterized by its scattering angle θ. If the scatterpoint lies on a dipping reflector then the scattered ray that coincides with the normal ray will be the strongest after superposition of many scatterpoints. For this ray the scattering angle is the same as the reflector dip which suggests why the scattering angle is commonly called the “migration dip” in a migration program. However, migration dip still conveys a misleading impression to many untutored users who wish to equate it with geologic dip. They draw the mistaken conclusion that, for data with low geologic dips, it suffices to correctly handle only low migration dips. The correct interpretation of migration dip as scattering angle shows that even for low geologic dips it is desirable to handle large scattering angles. In fact, a high-resolution migration requires that scattering angle

be as large as possible. In the context of section 4.1.3 scattering angle is directly related to spectral components through k x /f = .5 sin θ/v or equivalently k x = .5f sin θ. This says that high k x values require large values of θ. It is a fundamental component of resolution theory that the k x spectrum must be as large as possible for high resolution.

The traveltime curve of a point diffractor, for zero-offset recording and constant velocity, has a raypath geometry that is essentially equivalent to that for a CMP gather and a zero-dip reflector (Figure 4.18). Recall that the NMO traveltime curve is given by t 2 H =t 2 0 +H 2 /v 2 where H is the full source-receiver offset. This means that the traveltime curve for a zero-offset point diffractor is given by

t 2 x =t 2 4(x − x 0 ) 0 2 +

where the diffractor is at (x 0 , z), t 0 = 2z/v, and the velocity, v, is constant. The factor of 4 in the numerator arises because x − x 0 corresponds to the half-offset. For v(z), the conclusion is immediate

96 CHAPTER 4. ELEMENTARY MIGRATION METHODS

cmp 1 cmp 3 datum .

cmp n

sr / sr / sr / sr / s s s sr / r r r

θ is the”migration dip”

reflector

point diffractors

(scatterers)

Figure 4.17: Each point on a reflector is consid- Figure 4.18: (A) The ray paths for the left side of ered to be a point diffractor that scatters energy

a zero-offset point diffractor. (B) The raypaths to all surface locations.

for a CMP gather assuming a zero-dip reflector. These raypaths are identical for v(z) leading to the conclusion that the traveltime curves for the point diffractor are the same as for the NMO experiment.

seconds 1 1

1.5 -3000 -2000 -1000

meters 3000 meters

Figure 4.19: (A) This diffraction chart shows the traveltime curves for five point diffractors in a constant velocity medium. All five curves are asymptotic to the line x=vt/2. (B) The second hyperbola has been migrated by wavefront superposition.

4.2. FUNDAMENTAL MIGRATION CONCEPTS

met er s 1000

Figure 4.20: The ZOS image of the dipping reflector is formed by replacing each point on the reflector with the ZOS image of a point diffractor. The superposition of these many hyperbolae forms the dipping reflector’s ZOS image.

that the Dix equation proof that the NMO hyperbola is approximately characterized by v rms means that the diffraction traveltimes are approximately

where t 0 = 2z/v ave . Figure 4.14 shows how wavefront superposition migrates the ZOS image of a dipping reflector.

The ZOS image of a single point diffractor is the hyperbola given by equation (4.36) so it is instructive to see how wavefront migration can collapse such hyperbolae. Figure 4.19A shows a diffraction chart that gives the traveltime curves for five point diffractors in a constant velocity medium. The wavefront migration of this chart should convert it into five impulses, one at the apex of each hyperbola. Figure 4.19B shows the wavefront migration of the second hyperbola on the chart. Each point on the second hyperbola has been replaced by a wavefront circle whose radius is the vertical time of the point. Since the chart is drawn with a vertical coordinate of time rather than depth, these wavefront curves may not appear to be exactly circular. The geometry of hyperbola and circles is such that the wavefront circles all pass through the apex of the second hyperbola. Thus the amplitude of the wavefront superposition will be large at the apex and small elsewhere. The same process will also focus all other hyperbolae on the chart simultaneously.

The diffraction curves can be used to perform the inverse of migration or modelling. Figure 4.20 shows the construction of the ZOS image of a dipping reflector using Huygens’ principle. Each point on the dipping reflector is replaced by the ZOS image of a point diffractor. The superposition of these many hyperbolae constructs the ZOS image of the dipping reflector. For a given hyperbola, its apex lies on the geology and it is tangent to the ZOS image. This directly illustrates that the wavefront migration of dipping reflectors is completely equivalent to the migration of diffraction responses. In fact, the collapse of diffraction responses is a complete statement of the migration problem and a given algorithm can be tested by how well it achieves this goal.

98 CHAPTER 4. ELEMENTARY MIGRATION METHODS Receivers

Receivers x f.ψ 0 (x ,t)= 1 2 V ∞ φ (k π ,0 ,f)e2 x i(k x x − f t)d k x f. d d k φ ψ t)d (x ,t)= 1 2 V ∞ 0 (k x ,0 ,f)e2 π i(k x x − f

Wavefield snapshot Wavefield at time 0

Reflector Figure 4.21: The exploding reflector model at

Figure 4.22: A snapshot of the ERM wavefield the instant of explosion establishes a wavefield

at some instant of time as it approaches the re- that is isomorphic with the reflector.

ceivers