INTRODUCTION

By convention seismic reflection traces are plotted vertically beneath the center of a source-receiver pair. The implication is that the waves come from directly below this central point. However this is not the reality in most situations.

Seismic waves from a source such as an explosion, a hammer blow or a vibroseis truck spread out more or less equally in all directions. In the same manner a conventional geophone will pick up waves traveling in all directions (recall how our vertical geophones detected the horizontally traveling direct wave). Thus the location of the true source of an echo can not be determined with just one shot-receiver pair. Migration combines many pairs to find the true location. We’ll only examine the two-dimensional case.

ONE DIPPING REFLECTING PLANE

Let’s first consider a simple dipping reflector and a zero-separation profile over it. Zero source-receiver separation profiles look very similar to CMP profiles. Simple trigonometry allows us to construct the seismic "record section" to compare with the actual cross-section ("earth section"). The former has horizontal location and travel-time as axes whereas the axes of the latter are horizontal location and depth.

Comparison of the two sections reveals three differences. One is that the dip appears less on the record section (time section) than it really is. A second difference is that the echo on the record section appears to extend farther down-dip than it really does. Thirdly the record section shows a diffraction hyperbola with its apex at the end of the reflection. THESE EFFECTS ARE NOT FROM VERTICAL EXAGGERATION; THEY ARE FROM THE PHYSICS OF WAVE PROPAGATION.

VELOCITY DISTORTIONS

Because our record section is plotted with travel-time as one axis we see distortions if the velocity is not constant. High velocity layers will appear relatively thinner than they really are whereas low-velocity layers will appear relatively too thick. Below a thick lens of high-velocity material echoes from deeper reflectors will be "pulled up" and appear shallower than they are. Similarly echoes below a thick low-velocity lens will be "pulled down" and look relatively deeper than they really are.

Our task now is to figure out how to "migrate" reflections back to their true horizontal and vertical locations. How can we do this? We’ll look at three computer methods in common use.

DIFFRACTION MIGRATION

Diffraction or Kirchoff migration is based on Huygen’s Principle that any point on a wavefront acts as a tiny source of waves. These waves spread out in all directions but interfere constructively to form the new position of our initial wavefront. An implication of this is that any eflecting interface can be replaced, in our mind and models, by a set of closely spaced diffracting points. Thus, if we can migrate a diffraction hyperbola, we can migrate any and all reflections.

The algorithm starts with a profile of CMP stacks ("traces") and consists of three nested loops very similar to those we used for automatic velocity analysis.

The outer loop cycles over successive sets of an odd number of traces working from one end of the profile to another (say traces 1-5, 2-6, etc.). The next loop cycles over travel-time on the central trace (say trace 3). The inner loop cycles over the five traces adding the amplitudes present at times where the diffraction hyperbola should be based on the velocity and normal moveout formula. This summation amplitude is "posted" (displayed) at the location and travel time of the apex of the diffraction hyperbola. In other words we are testing every possible vertical travel time and every possible horizontal location as apices of hyperbolae.

The final step is to "stretch" the migrated traces vertically according to suitable average velocity versus time values.

Compared to other computer methods, elementary Kirchoff migration is fast and cheap. It works well if dips are low and if velocity depends mainly on depth. The method fails for steep dips and for strong lateral velocity variations. In addition, because it is based on ray physics, the final amplitudes are not correct. A related method, wave front migration, suffers from the same limitations (Hermance, 2001). We can attempt to get better amplitudes on the migrsted section by weighting the recorded amplitudes prior to summing but it’s better to use a method based more soundly on the complete physics of wave propagation.

WAVE EQUATION MIGRATION

These methods are based on numerical solutions of the equation of wave propagation that we studied in ES658. Depending on details of the various necessary approximations, these methods preserve relative amplitudes. In addition they can deal with velocities varying both vertically and horizontally.

In "reverse time migration" the seismic traces are treated as seismic sources that are applied to the ground in reverse order (Baysal et al., 1983). In other words the last wiggles on the traces are the first to go into the ground. The waves from these "sources" propagate into the ground according to the wave equation. However we use half the estimated velocity. The amplitudes at actual zero time (when we stop propagating the wave) are the strength of the reflectors at those places.

Wave equation migration takes a lot of computer time and thus is expensive. A bonus, however, is that essentially the same finite difference equation can be used for migration or for forward modeling.

REFERENCES

Baysal et al., 1983, Geophysics, 48, 1514-.

Hermance, 2001, Geophysics, 66, 379-.