Time and depth migration via raytracing

4.2.3 Time and depth migration via raytracing

When the first wavefield migration methods were being developed, the finite-difference approach was pioneered by Jon Claerbout (Claerbout, 1976) and his students at Stanford University. This very innovative work produced dramatic improvements in the quality of seismic images but the algorithms were a strong challenge for the available computers. As a result approximations were sought which could reduce the numerical effort required. A key approximation was Claerbout’s use of a moving coordinate system to derive an approximate wave equation that essentially substituted vertical traveltime, τ , for depth. When the finite difference machinery was implemented with this approximate equation the resulting migrated images had a vertical coordinate of time not depth. It also resulted, for reasons to be discussed when finite-difference methods are presented, in much faster run times. One way to see why this might be is to imagine a stacked section consisting only of events with ∆t/∆x = 0. This is not far from the appearance of seismic sections from many of the world’s sedimentary basins. Such basins are good v(z) environments and the normal rays for flat events are all vertical. Thus, they are already positioned correctly in vertical traveltime on the stacked section and the migration algorithm has very little to do. This is a drastic over simplification that will be more correctly stated later, but it comes close to the truth.

These early finite-difference methods were used with great success in the sedimentary basins around the world. However, when the technology was taken into thrust belts, continental margins, and other areas where the v(z) approximation is not a good model, it was gradually realized that systematic imaging errors were occurring. Today, the reason is well understood and it was a conse- quence of the approximate wave equation discussed in the previous paragraph. Figure 4.10 suggests the problem. Here an anticline is shown positioned beneath a slower-velocity near-surface that has a dipping bottom interface. The normal ray from the crest of the structure is generally not the ray of minimum traveltime even though it has the shortest path to the surface. Instead, a ray from a point to the right is the least-time ray because it spends more of its path in the fast material. Thus, on a normal incidence section, the traveltime signature of the anticline will have a crest at the emergence

90 CHAPTER 4. ELEMENTARY MIGRATION METHODS

least-time crest ray

ray v slow

v fast true

distorted image

Figure 4.10: The least-time ray does not always come from the crest of an anticline. Depth migration

Time migration

sin a

= sin b sin A sin B

Figure 4.11: (Left) When a ray encounters a dipping velocity interface, Snell’s law predicts the transmitted ray using a relation between angles with-respect-to the interface normal. (Right) In a time migration algorithm, Snell’s law is systematically violated because the vertical angles are used. Both techniques give the same result if the velocity interface is horizontal.

point of the least-time ray. Generally, a traveltime crest will have zero ∆t/∆x which means that the corresponding ray emerges vertically. It turned out that the migration schemes were producing

a distorted image in cases like this and the distortion placed the crest of the migrated anticline at the emergence point of the least time ray. Just like in the sedimentary basin case, the algorithm was leaving flat events alone even though, in this case, it should not.

The understanding and resolution of this problem resulted in the terms time migration and depth migration. The former refers to any method that has the bias towards flat events and that works correctly in v(z) settings. The latter refers to newer methods that were developed to overcome these problems and therefore that can produce correct images even in strong lateral velocity gradients. Robinson (1983) presented the raytrace migration analogs to both time and depth migration and these provide a great deal of insight. The raytrace migration algorithm presented previously in section 4.2.2 is a depth migration algorithm because it obeys Snell’s law even when ∂ x v is significant.

Figure 4.11 shows how Snell’s law must be systematically altered if a time migration is desired. Instead of using the angles that the incident and transmitted rays make with respect to the normal

91 in Snell’s law, the time migration algorithm uses the angles that the rays make with respect to the

4.2. FUNDAMENTAL MIGRATION CONCEPTS

vertical. It is as though the dipping interface is rotated locally to the horizontal at the incident point of the ray. Thus, any vertical ray (i.e. ∆t/∆x = 0) will pass through the velocity interface without deflection regardless of the magnitude of the velocity contrast.

This explains why time migration gets the wrong answer and it also explains why the technology remains popular despite this now well-understood shortcoming. The fact that flat events are unaf- fected by time migration regardless of the velocity model means that the process has a “forgiving” behavior when the velocity model has errors. Velocity models derived automatically from stacking velocities tend to be full of inaccuracies. When used with time migration, the inaccuracies have little effect but when used with depth migration they can be disastrous. Even today, the majority of seismic sections come from v(z) settings and they consist of mostly flat events. Such data can be migrated with relatively little care using time migration and well-focused sections are obtained. Of course, these are still time sections and a pleasing time migration does not mean that the migration velocities can be used for depth conversion. In general, the velocity errors that are “ignored” by time migration will cause major problems in a subsequent depth conversion. Much more care must

be taken if an accurate depth section is required. The terms time migration and depth migration arose, in part, because the early versions of both technologies tended to produce only time and depth sections respectively. However, this is no longer the case and either technology can produce both time and depth sections. Therefore, the terms should be treated simply as jargon that indicates whether or not an algorithm is capable of producing a correct image in the presence of strong lateral velocity gradients. The mere presence of

a time or depth axis on a migrated seismic display says nothing about which technology was used. In this book, time migration will be used to refer to any migration technique that is strictly valid only for v(z) while depth migration will refer to a technique that is valid for v(x, z). Thus, when employed in constant velocity or v(z) settings, both techniques are valid and there is no meaningful distinction. Under these conditions, a migrated time section can always be converted to a depth section (or vice-versa) with complete fidelity.

The first time-migration algorithms were almost always finite-difference methods. Today, a better understanding of the subject allows virtually any migration method to be recast as a time migration. Therefore, it is very important to have some knowledge of the algorithm, or to have a statement from the developer, to be sure of the nature of a method. The issue has been clouded even further by the emergence of intermediate techniques. Sometimes, it is worth the effort to build a synthetic seismic section to test a migration algorithm whose abilities are uncertain. One thing remains clear, depth migrations always require much more effort for a successful result. Not only is more computer time required; but, much more significantly, vastly greater human time may be necessary to build the velocity model.