Introduction to earth imaging in depth
Strong lateral velocity variations associated with complex overburden structures require earth imaging in depth. Examples of complex overburden include diapiric structures formed by salt tectonics, imbricate structures formed by overthrust tectonics and irregular water-bottom topography. All three are characterized as structure-dependent lateral velocity variations. There also exist structure-independent lateral velocity variations, often associated with facies changes; for instance, changes in lithology from shale to sandstone to carbonate induce lateral changes in acoustic impedance.
Figure 8.0-1a shows a field data example of a diapiric structure associated with salt tectonics. Note how the base-salt (event B) and subsalt reflections at about 2 s are pulled up in the middle of the section. Time migration yields an inaccurate image of the base-salt (Figure 8.0-1b).
Figure 8.0-2a is a field data example of an imbricate structure associated with overthrust tectonics. Time migration is adequate for imaging a target within the imbricate structure itself, but would not yield a correct image of a target below it (Figure 8.0-2b).
Refer to the field data example for a case of irregular water-bottom topography shown in Figure 8.0-3a. Note the false structures along the unconformity T because of irregular water-bottom topography. These are especially noticed below midpoint locations A, B, and C. Sagging of the reflection that is associated with the unconformity at these locations is attributed to the low-velocity overburden, here, the water layer. Once again, time migration does not provide the solution for such a complex overburden as shown in Figure 8.0-3b. Specifically, distortions along the unconformity still exist.
- Complex overburden structures often give rise to strong lateral velocity variations. In the presence of strong lateral velocity variations, an earth image in time derived from time migration is not accurate; and thus, it is imperative to obtain an earth image in depth by depth migration.
- Strong lateral velocity variations cause significant ray bending at layer boundaries.
- This then gives rise to nonhyperbolic behavior of reflection times on CMP gathers that correspond to layer boundaries below a complex overburden structure.
- As a result, amplitudes and traveltimes associated with the reflection events with nonhyperbolic moveout are distorted during conventional CMP stacking which is based on the hypberbolic move-out assumption.
- This causes CMP stack to depart from an ideal zero-offset wavefield. Therefore, when depth migration is needed, in principle, it must be done before stack and not after.
- Finally, complex overburden structures often exhibit three-dimensional (3-D) behavior. Therefore, when depth migration is needed, again in principle, it must be done not only before stack but also in three dimensions.
Figure 8.0-1 (a) A CMP stacked section over a salt-dome structure, (b) time migration. Note the similarity of the overmigration at the base-salt event in this section with that seen in the time-migrated section of the synthetic data in Figure 8.2-7.
Figure 8.0-2 (a) A CMP stacked section from an area with overthrust tectonics, (b) time migration.
Figure 8.0-3 (a) Conventional CMP stack from an area with irregular water-bottom topography, (b) time migration. (Data courtesy Hispanoil.)
Figure 7.0-16 (a) A DMO-stacked section derived from common-cell gathers along an inline from a 3-D marine survey (the same section as in Figure 7.0-13a); (b) the same inline section as in (a) after 3-D poststack time migration.
Compare the results of depth imaging in 2-D and 3-D using post- and prestack data shown in Figures 8.0-4 through 8.0-7. Note the improved imaging of the fault planes and the reflectors within the faulted zone attained by combining the advantages of prestack migration and 3-D migration. Figure 8.0-4 shows 2-D poststack depth migration of the 3-D DMO stack in Figure 7.0-16a derived from common-cell gathers along an inline from a 3-D marine survey. Velocities vary in the lateral direction across the steep faults in the middle portion of the image. The imaging within the faulted zone can be improved by prestack depth migration as shown in Figure 8.0-5. Nevertheless, the 3-D behavior of the fault planes requires 3-D imaging (Figure 8.0-6). The complete solution to the imaging problem in the presence of lateral velocity variations and the 3-D behavior of reflector geometries, of course, is 3-D prestack depth migration (Figure 8.0-7). Since lateral velocity variations can change the polarity of fault-plane reflections, the image sections in Figures 8.0-4 through 8.0-7 are displayed both with normal and reverse polarities.
Figure 8.0-4 (a) 2-D poststack depth migration of the 3-D DMO stack shown in Figure 7.0-13a derived from common-cell gathers along an inline from a 3-D marine survey; (b) the same section as in (a) displayed with reverse polarity.
Figure 8.0-5 (a) 2-D prestack depth migration of the data as in Figure 7.0-13a; (b) the same section as in (a) displayed with reverse polarity.
Figure 8.0-6 (a) An inline section from the image volume derived from 3-D poststack depth migration of the data as in Figure 7.0-16a; (b) the same inline section as in (a) displayed with reverse polarity.
Figure 8.0-7 (a) An inline section from the image volume derived from 3-D prestack depth migration of the data as in Figure 7.0-16a; (b) the same inline section as in (a) displayed with reverse polarity.
Although the migration methods discussed in migration are based on a layered media assumption, simple modifications of the basic algorithms make them accurate for situations with mild lateral velocity variations. For example, rms velocities can be varied laterally in Kirchhoff migration. In the finite-difference method, as long as lateral velocity variations are mild, the thin-lens term can be dropped (Section D.3), and the velocity function used in the diffraction term can be varied laterally. In the frequency-wavenumber methods such as Stolt migration, lateral velocity variations are accommodated by varying the stretch factor between 0 and 1. Even when velocity varies, the output from these three methods is still a time section, thus, the term time migration.
The situation is different when strong lateral velocity variations are encountered. In that case, simple algorithmic modifications no longer provide adequate accuracy, and depth migration  , rather than time migration, must be done. While both migration types use a diffraction term for collapsing energy along a diffraction hyperbola to its apex, only depth migration algorithms implement the additional thin-lens term that explicitly accounts for lateral velocity variations (Section D.4). Unlike time migration, the output from the migration algorithms that include the thin-lens term is a depth section, thus the term depth migration.
Sometimes, although not so often, the complex overburden can be defined by a single layer and the boundary between the overburden and the substratum can be determined in the form of an irregular interface with a significant velocity contrast. In that case, the layer replacement method described in layer replacement, in lieu of depth migration, can be used to remove the deleterious effect of the overburden on the geometry of the underlying reflections.
Time migration requires an rms velocity field, whereas depth migration requires an interval velocity-depth model. A velocity-depth model usually is defined by two sets of parameters — layer velocities and reflector geometries. While an rms velocity field does not contain discontinuities, an interval velocity-depth model can include discontinuities associated with layer boundaries.
A velocity-depth model is the seismic representation of an earth model in depth. An earth model and the earth image created from it are an inseparable pair of products of seismic inversion. To obtain an earth image in depth, one has to first estimate an accurate earth model in depth. We shall defer estimation of earth models in depth to the next chapter where we shall demonstrate the use of various inversion methods and depth migration itself to estimate the earth model parameters. Aside from earth modeling and imaging in depth, depth migration also is used to verify and update velocity-depth models. In this chapter, we shall be concerned only with depth migration as a tool for earth imaging in depth.
Historically, depth migration was used in an iterative fashion. Starting with an initial velocity-depth model, depth migration is performed and results are interpreted for updating layer velocities and reflector geometries. Using the updated velocity-depth model, depth migration is repeated until such time as what is inferred from depth migration as the velocity-depth model matches with the velocity-depth model input to depth migration. To achieve rapid convergence to a final velocity-depth model, only one set of parameters — usually, reflector geometries, are altered from one iteration to the next. We shall review the iterative depth migration procedure for zero-offset, CMP-stacked and prestack data, and draw conclusions as to when such a procedure yields an acceptable image in depth (2-D poststack depth migration).
In this chapter, we shall review two methods of two-dimensional (2-D) prestack depth migration (2-D prestack depth migration). In 2-D migration, we assume that the seismic line is along the dip direction and that the recorded wavefield is two dimensional. Shot-geophone migration, which is based on downward extrapolation of common-shot and common-receiver gathers using the double-square root equation (Section D.1), focuses primary reflection energy to zero-offset. The migrated section is obtained by retaining the zero-offset traces and abandoning the nonzero-offset traces. Shot-profile migration is based on migrating each common-shot gather, individually. In this method, the migrated section is obtained by sorting the migrated common-shot gathers into common-receiver gathers and summing the traces in each receiver gather.
We shall review 3-D post- and prestack depth migrations in 3-D poststack depth migration and 3-D prestack depth migration, respectively. While 3-D poststack depth migration can be performed using a wide variety of algorithms based on finite-difference (Sections G.1 and G.2), frequency-wavenumber (Sections G.3 and G.4), and Kirchhoff integral solutions (Section H.1) to the 3-D scalar wave equation, 3-D prestack depth migration often is done using the Kirchhoff integral (Section H.1) or the eikonal equation (Section H.2). This is because Kirchhoff migration is conveniently adaptable to source-receiver geometry irregularities in 3-D seismic surveys. Topographic irregularities also are conveniently accommodated in Kirchhoff migration.
The output of prestack depth migration consists of image gathers, which may be likened to moveout-corrected CMP gathers with [[the vertical axis in depth. Image gathers, however, consist of traces in their migrated positions. A stack of image gathers represents the earth image in depth obtained from prestack depth migration. If the velocity-depth model used in prestack depth migration is correct, then, events on an image gather would exhibit a flat character with no moveout. An erroneously too low or too high velocity would cause a residual moveout on the image gather. The initial velocity-depth model can be updated by analyzing this residual moveout and correcting for it (model updating).
- Lateral velocity variations
- Layer replacement
- 2-D poststack depth migration
- 2-D prestack depth migration
- 3-D poststack depth migration
- 3-D prestack depth migration
- Diffraction and ray theory for wave propagation