Zero-offset migration

Hand migration

Geophysicists recognized the need to correct these positioning errors on zero-offset sections long before it was practical to use computers to make the corrections. Thus a number of hand-migration techniques arose. It is instructive to see how one such scheme works. Equations (5.3) and (5.4) require knowledge of three quantities: $t$, $v$, and $\theta$. Of these, the event time $t$ is readily measured on the zero-offset section. The velocity $v$ is usually not measurable on the zero offset section and must be estimated from finite-offset data, as was shown in chapter . That leaves the dip angle $\theta$. This can be related to the reflection slope $p$ of the observed event, which is measurable on the zero-offset section:

$$p_0 \eq {\partial t \over \partial y} ,$$ (5)

where $y$ (the midpoint coordinate) is the location of the source-receiver pair. The slope $p_0$ is sometimes called the ``time-dip of the event'' or more loosely as the ``dip of the event''. It is obviously closely related to Snell's parameter, which we discussed in chapter . The relationship between the measurable time-dip $p_0$ and the dip angle $\theta$ is called ``Tuchel's law'':

$$\sin\theta \eq {v p_0 \over 2} .$$ (6)

This equation is clearly just another version of equation (), in which a factor of $2$ has been inserted to account for the two-way traveltime of the zero-offset section.

Rewriting the migration shift equations in terms of the measurable quantities $t$ and $p$ yields usable ``hand-migration'' formulas:

$$\Delta x \eq {v^2 p t \over 4}$$ (7)

$$\tau \eq t \sqrt{1 - {v^2 p^2 \over 4} } .$$ (8)

Hand migration divides each observed reflection event into a set of small segments for which $p$ has been measured. This is necessary because $p$ is generally not constant along real seismic events. But we can consider more general events to be the union of a large number of very small dipping reflectors. Each such segment is then mapped from its unmigrated $(y,t)$ location to its migrated $(y,\tau)$ location based on the equations above. Such a procedure is sometimes also known as ``map migration.''

Equations (5.7) and (5.8) are useful for giving an idea of what goes on in zero-offset migration. But using these equations directly for practical seismic migration can be tedious and error-prone because of the need to provide the time dip $p$ as a separate set of input data values as a function of $y$ and $t$. One nasty complication is that it is quite common to see crossing events on zero-offset sections. This happens whenever reflection energy coming from two different reflectors arrives at a receiver at the same time. When this happens the time dip $p$ becomes a multi-valued function of the $(y,t)$ coordinates. Furthermore, the recorded wavefield is now the sum of two different events. It is then difficult to figure out which part of summed amplitude to move in one direction and which part to move in the other direction.

For the above reasons, the seismic industry has generally turned away from hand-migration techniques in favor of more automatic methods. These methods require as inputs nothing more than

• The zero-offset section
• The velocity $v$ .

There is no need to separately estimate a $p(y,t)$ field. The automatic migration program somehow ``figures out'' which way to move the events, even if they cross one another. Such automatic methods are generally referred to as ``wave-equation migration'' techniques, and are the subject of the remainder of this chapter. But before we introduce the automatic migration methods, we need to introduce one additional concept that greatly simplifies the migration of zero-offset sections.

Zero-offset migration