# Migration - book

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 2 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Let us back up a bit and define some terms. The term imaging refers to the formation of a computer image. For the geophysicist, the required image is one that faithfully depicts the subsurface structure of the earth. Seismic imaging can be divided roughly into two parts: signal enhancement and event movement. Signal enhancement, which we discussed in the previous section, entails separation of primary reflections (the desired signals) from signal-generated noise (the undesired signals). In this section, we discuss event movement.

The more common terminology for event movement is seismic migration. Each primary reflection on an enhanced trace must be moved (or migrated) to the spatial position of the reflecting point at depth. In other words, each amplitude value on the enhanced trace is moved to its proper spatial location (the depth point). Migration can be accomplished by superposition of the processed traces. Such superposition is similar to that used in Huygens’ construction (Figures 4 and 5).

Figure 4.  Use of Huygens’ construction to reconstruct an interface. The two-way traveltime of a primary reflection (from a ground point to the interface and back again) can be found from seismic data. The one-way travel distance is one-half the product of two-way traveltime and velocity. Draw a circle with the ground point as center and the oneway travel distance as radius. Repeat this operation for many ground points. The interface is the envelope of circles.
Figure 5.  In 1921, at Belle Isle, Oklahoma, J. C. Karcher was the first person to record a seismic reflection line. He used Huygens’ method to migrate the record. As seen in his diagram shown here, the Viola interface is the envelope of circular arcs. Adapted from an illustration by J. C. Karcher.

Superposition has an added benefit: It is one of the most effective ways to accentuate signals and suppress noise. Superposition is designed to return primary reflections to their spatial locations. The signal-generated noise still remaining on the traces is out of step with the primaries. Consequently, superposition of a huge number of processed traces results in destruction of much of the remaining signal-generated noise. Superposition provides the desired digital image of the underground structure of the earth

An anticline is a ridge-shaped fold of stratified rock in which the strata slope downward from the crest. Some of the great oil fields of the world are located in very gentle anticlines, so to a good degree of approximation, the rock layers can be considered essentially flat and horizontal. The configuration of flat horizontal layers is known as the layer-cake model of stratigraphy, the simplest type of stratigraphy encountered in oil exploration. Because the reflecting interfaces are horizontal (or nearly so), waves going straight down will be reflected nearly straight up. Thus, wave motion will be essentially vertical. If the time axes on the records are placed in the vertical position, time will appear in the same direction as the raypaths. By using the correct wave velocity, the time axis can be converted into the depth axis. The result is that the primary reflections show the locations of the reflecting interfaces. Thus, in areas that have nearly level reflecting horizons, the primary reflections as recorded essentially show the correct depth positions of the subsurface interfaces.

On the other hand, in areas that have a subsurface structure that is more complicated than the layer-cake model, the primary reflections as recorded in time do not show the correct depth positions of the subsurface interfaces. In such cases, the wave equation can be used to move (or migrate) the primary reflections to their proper spatial positions in depth. If one is satisfied with using ray theory instead of wave theory, then the eikonal equation can be used instead of the wave equation. Historically, movement (i.e., migration) of reflected events to their proper locations in space was carried out manually, sometimes using elaborate drawing instruments. These early analog implementations of seismic migration were based either on ray theory or on a graphic application of the wave equation (as in Huygens’ principle).

Digital implementation of migration involves massive data handling. All the traces must be amalgamated either by wave-equation methods (Claerbout, 1971[1]) or by the associated ray-theory methods (Hagedoorn, 1954[2]; Gray, 1986[3]). Such methods involve significant data manipulation, which in the last century generally overloaded the limited capacities of the available computers. To reduce the extent of computations, migration usually was limited to two dimensions (namely, a horizontal dimension and the depth dimension). In addition, it was expedient to break the migration problem down into smaller parts. Thus, migration was done by a sequence of partial operations, such as stacking, followed by normal move-out, followed by dip moveout, and then followed by migration after stack. The process of time migration often was used, which improved the records in time but stopped short of putting the events in their proper spatial positions. All sorts of modifications and adjustments were made to improve such piecemeal operations. This approach made seismic migration a complicated discipline and an art as much as a science. The use of this art required much insight. Seismic migration in three dimensions (namely, two horizontal dimensions and the depth dimension) rarely was used because of the prohibitive costs involved.

In the 1990s, great improvements in instrumentation and computers resulted in light, compact geophysical field equipment and affordable computers with great speed and massive capacity. Geophysicists rapidly took advantage of that new capacity. Instead of being confined to using the modest number of sources and receivers common to 2D seismic processing, geophysicists began to use the tremendous number required for 3D processing, on a regular basis, in field operations.

Computers today are large enough to handle 3D imaging. As a result, 3D methods commonly are used, and the resulting subsurface images are of extraordinary quality. Three-dimensional event movement (or migration) can be carried out by such time-honored methods as those used in Huygens’ construction. Such migration methods generally go under the name of prestack migration. Three-dimensional prestack migration significantly improves seismic interpretation because it gives the locations of geologic structures, especially faults, much more accurately. In addition, migration collapses diffractions from secondary sources, such as reflector terminations against faults, and corrects the so-called bow ties to show the synclinal structure.

Let us now outline how 3D imaging is done. The 3D volume (x,y,z) represents the earth, where (x,y) represents the surface coordinates and z represents the depth coordinate. On the surface plane (x,y), the sources and receivers are arranged in a 2D grid. The (two-way) traveltime t is the duration of the passage of a primary reflection down from the source ${\displaystyle \left(x_{s},y_{s}\right)}$ to a depth point (x,y,z) and then up to the receiver ${\displaystyle \left(x_{r},y_{r}\right)}$. A digital trace is a discrete sequence of amplitudes as a function of discrete traveltime. Thus, the amplitude ${\displaystyle f\left(x_{s},y_{s},x_{r},y_{r},t\right)}$ represents the sum of all primary reflections with traveltime t originating from source ${\displaystyle x_{s},y_{s}}$ and recorded at receiver ${\displaystyle x_{r},y_{r}}$.

Many admissible depth points (x,y,z) can contribute to this amplitude (Figure 6a). All of those admissible depth points must be taken into account (Figure 6b). The digital process called migration takes the given amplitude ${\displaystyle f\left(x_{s},y_{s},x_{r},y_{r},t\right)}$ and moves (i.e., migrates) that amplitude to each and every admissible depth point (x,y,z). This process is done for all the amplitudes on each primaries-only trace. The results then are summed, and behold, an image of the geologic structure g(x,y,z) emerges. The principle used is that of Huygens, in which the summation of all the individual responses yields the correct overall response.

Figure 6.  (a) Source point, depth point, and receiver point. (b) In a constant-velocity medium, the admissible depth points lie on the ellipse that has the source point and the receiver point as the two foci.

In summary, migration is a digital operation in which the reflections on the seismic traces are moved to their correct locations in space. Migration requires the primary reflected events. What else is required? The answer is the seismic velocity function v(x,y,z). The velocity function is required to determine the raypaths of the admissible reflections. The velocity function gives the wave velocity at each point in the given volume of the earth that is under exploration.

The word velocity generally connotes a vector, with magnitude equal to the rate (a scalar) at which a wave travels through a medium and with direction equal to the direction of movement. When velocity is a scalar, the term speed often is used instead. In isotropic media, wave velocity at any position (x,y,z) has the same magnitude when measured in different directions. Because direction does not matter, wave velocity traditionally is considered as a scalar in isotropic media. In anisotropic media, wave velocity at any position (x,y,z) varies in magnitude according to the direction of measurement. Because direction does matter, wave velocity traditionally is considered as a vector in anisotropic media. In this book, we consider only isotropic media, so we treat wave velocity as a scalar quantity v(x,y,z). Wave velocity can be determined from laboratory measurements, acoustic logs, and vertical seismic profiles or from velocity analysis of seismic data. Wave velocity tends to increase with depth in the earth because of compaction of the medium. From now on, we merely will say velocity when we mean wave velocity.

Over the years, various methods have been devised to obtain a sampling of the velocity distribution within the earth. The velocity functions so determined vary from method to method. For example, the velocity measured vertically from a check shot or vertical seismic profile (VSP) differs from the stacking velocity derived from normal-moveout measurements of common-depth-point (CDP) gathers. Ideally, we would want to know the velocity at each and every point in the volume of earth in which we are interested. The velocity is needed especially in regions where significant and intensive lateral or vertical differences in velocity occur. Migration requires an accurate knowledge of vertical and horizontal seismic velocity variations. Because velocity depends on the types of rocks through which a wave travels, a complete knowledge of the velocity v(x,y,z) is essentially equivalent to a description of the geologic structure g(x,y,z), as far as that can be obtained by conventional seismic methods. However, as we have stated above, the velocity function is required to get the geologic structure. In other words, to get the answer g(x,y,z), we must know the answer v(x,y,z).

## References

1. Claerbout, J. F., 1971, Toward a unified theory of reflector mapping: Geophysics, 36, 467–481.
2. Hagedoorn, B. B., 1954, A process of seismic reflection interpretation: Geophysical Prospecting, 2, 85–127.
3. Gray, S. H., 1986, Efficient traveltime calculations for Kirchhoff migration: Geophysics, 51, 1685–1688.