RESIDUAL CONSTANT-OFFSET MIGRATION

Define residual constant-offset migration as the process that converts a constant-offset section migrated with slowness w to a constant-offset section migrated with slowness w_n. Building a kinematic residual constant-offset migration operator simply requires finding where the image of a point on a dipping reflector in a migrated constant-offset section moves as the migration velocity changes. Doing this for all points and all dips in the image defines a kinematic residual migration operator. A wave-theoretic residual migration operator can be found by cascading constant-offset modeling at the original migration slowness and constant-offset migration with a new slowness. A kinematic development will give a stationary phase approximation to this wave theoretic residual migration operator. Since my main application of this operator will be velocity analysis, the errors of the stationary phase approximation should not be important. I further approximate, by building residual constant-offset migration operators using constant velocity prestack migration. Although the kinematic arguments will be strictly valid only in the constant slowness case, the residual migration operator should still be useful, even when velocity varies. Constant-offset depth migration followed by constant-velocity constant-offset residual migration will still give an approximation to constant-offset depth migration with a new interval velocity model. The approximation gets better as the amount residual migration gets smaller.

The impulse response of constant-velocity constant-offset migration is an ellipse with foci at source and receiver. Each point on the impulse response ellipse represents a different time dip in the data. A small segment of a dipping reflection event assumed to contain only a single traveltime dip, after migration becomes the tangent to the impulse response ellipse for the appropriate traveltime. The traveltime determines the size of the ellipse, and the traveltime dip determines the position of the image along the impulse response ellipse. Seen in reverse, given an image point and its depth dip, we can solve for the position of the source and receiver that have an impulse response that goes through our point and is tangent to the dipping reflector segment.

Starting at some point in depth with an assumed dip and solving for the source and receiver points that contain a specular reflection from the point, we can write an equation in polar coordinates that describes the migration ellipse that goes through the point.

$$r(\theta , w)= \sqrt{ {{\displaystyle t^2 \over \displaystyl... ...e w^2 \over \displaystyle t^2} h^2 \cos \theta}}\ \ . \eqno (1)$$

The origin is shifted to be the center of the ellipse. Equation (1) describes all image points that have specular reflections recorded at this shot receiver pair. Specify a point of interest as $r(\theta_0)$.Write the traveltime of the reflection from the dipping reflector as

$$t=w\biggl[ \sqrt{(x-h)^2+z^2}+\sqrt{(x+h)^2+z^2} \biggr] \ \ , \eqno (2)$$

where $x=r(\theta) \cos \theta$ and $z=r(\theta) \sin \theta$.Also write the time dip of any reflection from a tangent to the ellipse as

$${\partial t \over \partial y}=w \biggl[{r(\theta) \cos \thet... ...^2(\theta)+h^2+2r(\theta)h \cos \theta}}\biggr] \ \ . \eqno (3)$$

Figure 1 shows a selected point ``P" with its tangent impulse response ellipse and specular reflection rays for a specified dip.

After choosing an initial depth and depth dip and solving for the shot receiver pair, equations (1)-(3) give the traveltime and traveltime dip of the specular reflection from that point. When the migration slowness changes, the migration ellipse will move to a new position. The key to finding the position of the reflector as the slowness changes is to remember that the traveltime, midpoint, and traveltime dip of the reflector segment in the constant-offset section is fixed. Only its image in depth changes as the migration slowness changes. First, find a new impulse response after the migration slowness changes for the original shot-receiver pair that has the same traveltime as the previous impulse response at the old migration velocity. Then find a point on the new impulse response that has the same traveltime dip as the original point using the original migration slowness. This procedure gives the new location and dip of the event. Keeping t fixed as the traveltime of a reflection from our original point, the equation of the new impulse response ellipse at slowness w_n can be written as

$$r(\theta , w_n)= \sqrt{ {{\displaystyle t^2 \over \displayst... ..._n^2 \over \displaystyle t^2} h^2 \cos \theta}} \ \ . \eqno (4)$$

Identify d=t/w as the shot to image point to geophone distance of the reflector at the original migration slowness. Also substitute $\gamma=w/w_n$ (Al-Yahya, 1987). $\gamma$ can be called the residual slowness or slowness scale factor. When $\gamma=1$, residual migration leaves the image unchanged. With these changes, rewrite equation 4 as:

$$r(\theta , \gamma)= \sqrt{ {{\displaystyle \gamma^2 \display... ...e \gamma^2 \displaystyle d^2} h^2 \cos \theta}} \ \ . \eqno (5)$$

The time dip imaged at any point on the new impulse response ellipse is likewise given by:

$${\partial t \over \partial y}=w_n \biggl[{r \cos \theta -h \... ...h \over \sqrt {r^2+h^2+2rh \cos \theta}} \biggr]\ \ . \eqno (6)$$

Equating traveltime dip of the reflector at the new and original slownesses and plugging in the new relation for $r(\theta,\gamma)$ which equates the traveltimes gives a single equation to solve for $\theta_n$ as a function of $\gamma$ and the position of the original point.

$$\gamma \biggl[{r(\theta_0,d) \cos \theta_0 -h \over \sqrt {r... ...ta_0, d)+h^2+2r(\theta_0, d)h \cos \theta_0}} \biggr] \eqno (7)$$

$$=\biggl[{r(\theta_n , \gamma) \cos \theta_n -h \over \sqrt {... ..., \gamma)+h^2+2r(\theta_n , \gamma)h \cos \theta_n}} \biggr]\ .$$

Unfortunately, equation (7) is very difficult to solve analytically except for the trivial cases of zero dip or zero offset. To compute the operator I solve equation (7) numerically using Newton's method for finding roots.

Converting the solution of equation (7) back to Cartesian coordinates gives $x(r,\theta_n)$ and $z(r,\theta_n)$,the new position of the dipping reflector segment as shown in Figure 2.

Solving equation (7) for an initial depth and all initial dips will trace out the ``spraying" operator for residual constant-offset migration. The operator traces out the new positions of events for a range of dips after the migration slowness changes for a fixed original point in the image. This curve is also the summation path if we change the role of starting and final points by redefining $\gamma=w_n/w$.It is often more convenient to write the computer code in terms of a summation operator. The equations are symmetric so the summation operator for $\gamma$ is the ``spraying" operator for $1/\gamma$.

Figure 3 shows an example of the summation operator when the slowness increases. Depending on the change in migration slowness and the initial depth of the reflector, the operator can triplicate.

This happens for image points with large offset/depth ratio. Figure 4 shows an examples of the residual migration operators for $\gamma$ less than one, when slowness decreases.

To get the correct amplitudes along the summation operator, compute the points on the summation operators in equal dip angle increments in the original image. To prevent operator aliasing I resample the summation operators in equal arclength along the summation trajectory. To recover equal dip weighting of the image, the amplitude along the summation path can be taken as the jacobian of the change of variables from arclength to dip angle.