On a Stacked Section, Sources and Receivers Are Coincident

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).