Time-shift imaging condition in seismic migration |

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An image is formed when the Kirchoff stacking
curve (dashed line) touches the true reflection response.
Left: the case of under-migration; right: over-migration.
Figure 3. |
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off
Common-image gathers for
space-shift imaging (left column) and
time-shift imaging (right column).
Figure 4. |
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ssk
Common-image gathers after slant-stack
for space-shift imaging (left column) and
for time-shift imaging (right column).
The vertical line indicates the migration velocity.
Figure 5. |
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We can use the Kirchhoff formulation to analyze the moveout behavior of the time-shift imaging condition in the simplest case of a flat reflector in a constant-velocity medium (Figures 3-5).

The synthetic data are imaged using shot-record wavefield extrapolation migration. Figure 4 shows offset common-image gathers for three different migration slownesses , one of which is equal to the modeling slowness . The left column corresponds to the space-shift imaging condition and the right column corresponds to the time-shift imaging condition.

For the space-shift CIGs imaged with correct slowness, left column in Figure 4, the energy is focused at zero offset, but it spreads in a region of offsets when the slowness is wrong. Slant-stacking produces the images in left column of Figure 5.

For the time-shift CIGs imaged with correct slowness, right column in Figure 4, the energy is distributed along a line with a slope equal to the local velocity at the reflector position, but it spreads around this region when the slowness is wrong. Slant-stacking produces the images in the right column of Figure 5.

Let and represent
the true slowness and reflector depth, and and stand for the
corresponding quantities used in migration. An image is formed when
the Kirchoff stacking curve
touches the true reflection response
(Figure 3).
Solving for from the envelope condition
yields two solutions:

Substituting solutions 27 and 28 in the condition produces two images in the space. The first image is a straight line

and the second image is a segment of the second-order curve

Applying a slant-stack transformation with turns line (29) into a point in the space, while curve (30) turns into the curve

The curvature of the curve at is a clear indicator of the migration velocity errors.

By contrast, the moveout shape appearing in wave-equation
migration with the lateral-shift imaging condition is (Bartana et al., 2005)

which is applicable for velocity analysis. A formal connection between -parameterization in equation (31) and -parameterization in equation (33) is given by

or

where . Equation (35) is a special case of equation (23) for flat reflectors. Curves of shape (31) and (33) are plotted on top of the experimental moveouts in Figure 5.

Time-shift imaging condition in seismic migration |

2007-04-08