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Chapter 7
Migration
Introduction
- The goal of migration is to make the stacked section as close to the geologic cross section as possible. Migration is also called seismic imaging.
- Migrating a stacked section is called poststack migration, while migrating the pre-stacked data is called prestack migration.
- On a stacked section, sources and receivers are coincident.
- A reflector is assumed to be directly below its associated receiver on the stacked section.
- This is true for horizontal reflectors; however, dipping reflectors reflect energy at non-vertical direction. Therefore, the above assumption is not true and should be corrected.
- In general, a reflection can come from any point on a semicircle whose center is the receiver and radius equal to the traveltime (or depth) to the reflection.
- Migration in this case is done by spreading the energy at every point over such a semicircle. Constructive interference of these semicircles will produce the migrated section.
- Alternatively, we can think of every point in the subsurface as a point scatterer that produces a diffraction hyperbola whose apex is at the position of the scatterer. Constructive interference of these hyperbolae produces the unmigrated section.
- Migration in this case is done by summing the energy over every possible hyperbola and collapsing it at its apex. The result of this process is the migrated section.
- Migration moves dipping reflectors to their true subsurface positions and collapses diffractions to their apexes. Therefore, it enhances the horizontal (spatial) resolution.
Time migration versus depth migration
- Diffractions are hyperbolic only if there are no lateral heterogeneities because they can distort the shape of diffractions.
- Time migration assumes hyperbolic diffractions and collapses them to their apexes.
- Depth migration assumes a known velocity model and estimates the correct shape of diffractions by ray tracing or wavefront modeling.
- Time migration produces a migrated time section while depth migration produces a migrated depth section.
- Evidently, using time migration followed by time-to-depth conversion does not produce a depth-migrated section.
- Time migration is valid only when lateral velocity variations are mild to moderate. When this assumption fails, we use depth migration.
- Migrated sections are commonly displayed in time rather than depth for the following reasons:
- To avoid errors introduced by inaccurate time-to-depth conversion.
- To facilitate the comparison of migrated sections with unmigrated sections, which are usually displayed in time.
Prestack migration
- Prestack migration takes into account the location of the source and receiver for each trace when determining the reflector position.
- Before stack, a reflection can come from any point on an ellipse whose foci are the source and receiver.
- Prestack migration is done by spreading the energy at every point over an ellipse. Constructive interference of these ellipses will produce the migrated section.
2-D migration versus 3-D migration
- In 2-D migration, we migrate the data once along the profile. This might generate misties on intersecting profiles.
- In addition, 2-D migration is prone to sideswipe effects. Sideswipes are reflections from out of the plane of the profile.
- In 3-D migration, we first migrate the data in the inline direction then take that migrated data and migrate it again in the crossline direction. This is the two-pass 3-D migration.
- One-pass 3-D migration can also be done using a downward continuation approach.
- Therefore, considering 2-D versus 3-D, prestack versus poststack, and time versus depth, we can have the following types of migrations (ordered from fastest but least accurate to slowest but best accurate):
- 2-D poststack time migration.
- 2-D poststack depth migration.
- 2-D prestack time migration.
- 2-D prestack depth migration.
- 3-D poststack time migration.
- 3-D poststack depth migration.
- 3-D prestack time migration.
- 3-D prestack depth migration.
Geometrical aspects of migration
- Before migration, synclines look like bowties because of the interference among diffraction hyperbolae. These bowties are untied into synclines after migration.
- Graphically, a linear reflector can be migrated by drawing two semicircles from two points on the reflector. The tangent to these semicircles is the migrated position.
- When a dipping reflector is migrated, it is moved updip, steepened, and shortened.
- The amount of horizontal displacement (dx), vertical displacement (dt) and dip (/x) introduced by the migration are given by the following relations:
, (8.1)
, (8.2)
, (8.3)
where v: medium velocity,t: TWTT, and t/x: dip onunmigrated section.
- dx, dt, and x increase with time, velocity, and dip of reflector on the unmigrated section.
- Another important relation is the migrator’s equation given by:
sin = tan, (8.4)
where is dip angle on unmigrated section and is dip angle on migrated section.
- Migration of noise usually produces “migration smiles” due to the non-interference of the semicircles.
Migration algorithms
- The main migration algorithms are:
- Kirchhoff migration.
- F-K (Stolt) migration.
- Finite-difference (downward continuation, phase-shift) migration.
Kirchhoff migration
- This method employs Huygen’s principle, which states that every point on the wavefront can be regarded as a secondary source that generates seismic waves in the forward direction.
- Huygen’s secondary sources are point apertures, which produce waves that depend on propagation angle; unlike point sources, which are isotropic.
- A Huygen’s secondary source generates a diffraction hyperbola in the (x,t) plane.
- The purpose of Kirchhoff migration is to sum up the energy produced by every Huygen’s secondary source and map it into its point of generation.
F-K migration
- This method uses the 2-D Fourier transform to convert the input t-x section into an f-k section.
- F denotes frequency (the Fourier transform of time) while K denotes wavenumber (the Fourier transform of space or distance).
- While in the f-k domain, the data is migrated using a simple algorithm.
- The inverse 2-D Fourier transform provides the migrated (x,) section.
Finite-difference migration
- This method works on a conceptual volume of information (x,z,t) rather than two time (x,t) and depth (x,z) planes.
- The method can be summarized in the following steps:
- Input the top surface seismic section (x,z=0,t).
- Compute the entire volume (x,z,t) using a finite-difference extrapolation approach.
- Extract the structure (x,z,t=0).