Consider the dipping reflector CD of the simple geologic section in Figure 4.1-1a. We want to obtain a zero-offset section along the profile Ox. As we move the source-receiver pair (s, g) along Ox, the first normal-incidence arrival from the dipping reflector is recorded at location A. In this discussion, we assume a normalized constant-velocity medium v = 1 so that time and depth coordinates become interchangeable. The reflection arrival at location A is indicated by point C′ on the zero-offset time section in Figure 4.1-1b. As we move from location A to the right, normal-incidence arrivals are recorded from the dipping reflector CD. The last arrival is recorded at location B, which is indicated by point D′ in Figure 4.1-1b. In this experiment, diffractions off the edges of reflector CD are excluded to simplify the discussion.
Compare the geologic section in Figure 4.1-1a, which is in depth, with the zero-offset seismic section in Figure 4.1-1b, which is in time. The true subsurface position of reflector CD is superimposed onto the time section for comparison. Clearly the true geologic position of reflector CD is not the same as the reflection event position C′D′.
From this simple geometric construction, note that the reflection in the time section C′D′ must be migrated to its true subsurface position CD in the depth section. The following observations can be made from the geometric description of migration in Figure 4.1-1:
- The dip angle of the reflector in the geologic section is greater than in the time section; thus, migration steepens reflectors.
- The length of the reflector, as seen in the geologic section, is shorter than in the time section; thus, migration shortens reflectors.
- Migration moves reflectors in the updip direction.
The example in Figure 4.0-1 demonstrates the above observations. In particular, the dipping event (B) has moved in the updip direction, become shorter, and steepened after migration (A).
As mentioned in the previous section, conventional migration output is displayed in time, as is the input stacked section. To distinguish the two time axes, we will denote the time axis on the stacked section as t — event time in the unmigrated position, and the time axis on the migrated section as τ — event time in the migrated position.
We shall now examine the horizontal and vertical displacements as seen on the migrated time section. From Figure 4.1-2, consider a reflector segment CD. Assume that CD migrates to C′D′ and that point E′ on C′D′ migrates to point E on CD. The horizontal and vertical (time) displacements — dx and dt, and the dip Δτ/Δx, all measured on the migrated time section (Figure 4.1-2), can be expressed in terms of medium velocity v, traveltime t, and apparent dip Δt/Δx as measured on the unmigrated time section (Figure 4.1-2). Chun and Jacewitz  derived the following formulas:
To gain a quantitative insight into these expressions, we consider a numerical example. For a realistic velocity function that increases with depth, consider five reflecting segments at various depths. For simplicity, assume that quantity Δt/Δx is the same for all (10 ms per 25-m trace spacing). From equations (1), (2), and (3), compute the horizontal and vertical displacements dx and dt and the dips (in ms/trace) after migration. The results are summarized in Table 4-2.
- The time dip Δτ/Δx on the migrated section is always greater than the time dip Δt/Δx on the unmigrated section.
- The horizontal displacement dx increases with event time t in the unmigrated position. At 4 s, the horizontal displacement is more than 6 km.
- The horizontal displacement dx is a function of the velocity squared. If there is a 20 percent error in the velocity used in migration, then the event is misplaced by an error of 44 percent.
- The vertical displacement dt also increases with time and velocity.
- The steeper the event dip, the more the horizontal and vertical displacements after migration.
|t (s)||v (m/s)||dx (m)||dt (s)||Δt/Δx (ms/trace)||Δτ/Δx (ms/trace)|
In Figure 4.1-1a, assume that the zero-offset section was recorded only between surface locations A and B. The time section would include the event C′D′, but when migrated, the event would migrate out of the section, resulting in a blank migrated section (Figure 4.1-1b). Therefore, the data on a stacked section are not necessarily confined to the subsurface below the seismic line. The converse is even more important; the structure below the seismic line may not be recorded on the seismic section. Suppose that the data were recorded only between surface locations O and A. This time, the resulting time section would be blank. So, we should not record between A and B, and neither should we record between O and A. Instead, we should record between O and B in order to record the reflector of interest properly and also to migrate it properly.
In areas with a structural dip, line length must be chosen by considering the horizontal displacements of dipping events from the structures causing the events. This is an important consideration, especially in 3-D seismic work. The areal surface coverage of a survey usually is larger than the areal subsurface coverage of interest.
To achieve a complete image of a dipping reflector, also, the recording time must be long enough. For example, if only OE seconds were recorded (Figure 4.1-1), then the recorded segment C′D″ would yield only part of the complete image CD. An excellent example of recording deeper in time and with longer line length for steeper dips is shown in Figure 4.0-1. Proper imaging of the salt dome boundary required that data be recorded for more than 6 s.
Figure 4.0-1 A CMP stack (a) before, (b) after migration; (c) sketch of a prominent diffraction D and a dipping event before (B) and after (A) migration. Migration moves the dipping event B to its assumed true subsurface position A and collapses the diffraction D to its apex P. The dotted line indicates the boundary of a salt dome.
Figure 4.1-2 Quantitative analysis of the migration process. Dipping event AB on the unmigrated section (left) is moved to A′B′ on the migrated section (right). The event after migration also is superimposed on the unmigrated section to compare the position of the event before and after migration. Point C on dipping reflector AB is moved to C′ after migration. The amount of horizontal displacement dx, vertical displacement dt, and the dip Δτ/Δx after migration is calculated from equations (1), (2) and (3).
The migration concepts described above are demonstrated further by the dipping events model in Figure 4.1-3. The edge diffractions are included here. The dipping reflectors on the zero-offset section are steepened, shortened, and moved in the updip direction as a result of migration. A field data example of a series of dipping events on a stacked section before and after migration is shown in Figure 4.1-4. Note that the steeper the dip, the more the event moves after migration.
So far, only linear reflectors were considered. We now consider a more realistic geologic situation that involves curved reflecting interfaces. Figure 4.1-5 shows three synclines and a small anticlinal feature. The synclines appear as bowties on the zero-offset section. By using the principles deduced from the geometry of Figure 4.1-1, note that as a result of migration, segment A of the bow tie moves in the updip direction to the left. Similarly, segment B moves to the right, while flat-topped segment C does not move much at all. Consequently, after migration the flanks of bow ties associated with synclines are opened up. On the other hand, the small anticline seems to be broader on the zero-offset section than it is on the migrated section. Again note that segment D moves updip to the right, while segment E moves updip to the left as a result of migration. Thus, synclines broaden and anticlines compress as a result of migration. Migration velocities also affect the apparent size of the structure; higher velocities mean more migration and, hence, smaller anticlinal structure.
Figure 4.1-3 (a) A zero-offset section modeled from the dipping reflectors shown in (b). The medium velocity is constant 3500 m/s and the trace spacing is 25 m. The true dip angles of the reflectors vary from 0 to 45 degrees at 5-degree increment. Migration of the zero-offset section (a) yields the model of the dipping reflectors (b).
Figure 4.1-5 Curved reflecting interfaces (synclines and anticlines) (a) before and (b) after migration. See text for details. (Modeling courtesy Union Oil Company.)
Why does a syncline look like a bowtie on the stacked section? The answer is in Figure 4.1-6, where a symmetric syncline is modeled. Given the subsurface model in Figure 4.1-6a, the normal-incidence rays can be computed to derive the zero-offset traveltime section in Figure 4.1-6b. Only five CMP locations are shown for clarity. At locations 2 and 4, there are two distinct arrivals, while at location 3, there are three distinct arrivals. By filling in the intermediate raypaths, the bow tie character of the syncline can be constructed on the time section. Complete the procedure by tracing the traveltime trajectory in Figure 4.1-6b.
Two field data examples containing synclinal and anticlinal structures are shown in Figures 4.1-7 and 4.1-8. In Figure 4.1-7, note that the synclinal feature broadens and the anticlinal feature narrows as a result of migration. In Figure 4.1-8, the bow ties associated with two small synclinal basins A and B grow larger in depth. After migration, the bowties are untied and the synclines are delineated.
Figure 4.1-7 A portion of a CMP stack (a) before and (b) after migration. Anticlines appear bigger, while synclines appear smaller than their actual sizes on the unmigrated section (a).
- Chun and Jacewitz, 1981, Chun, J.H. and Jacewitz, C., 1981, Fundamentals of frequency-domain migration: Geophysics, 46, 717–732.
- Introduction to migration
- Kirchhoff migration in practice
- Finite-difference migration in practice
- Frequency-space migration in practice
- Frequency-wavenumber migration in practice
- Further aspects of migration in practice
- Mathematical foundation of migration