Time-shift imaging condition in seismic migration

Moveout analysis

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Figure 3. 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.

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Figure 4. Common-image gathers for space-shift imaging (left column) and time-shift imaging (right column).

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Figure 5. 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.

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 $s$, one of which is equal to the modeling slowness $s_0$. 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 $s_0$ and $z_0$ represent the true slowness and reflector depth, and $s$ and $z$ stand for the corresponding quantities used in migration. An image is formed when the Kirchoff stacking curve $t(\hat{h}) = 2 s \sqrt{z^2+\hat{h}^2} + 2 \tau$ touches the true reflection response $t_0(\hat{h}) = 2 s_0 \sqrt{z_0^2+\hat{h}^2}$ (Figure 3). Solving for $\hat{h}$ from the envelope condition $t'(\hat{h})=t_0'(\hat{h})$ yields two solutions:

$$\hat{h} = 0$$ (27)

and

$$\hat{h} = \sqrt{\frac{s_0^2 z^2 - s^2 z_0^2}{s^2-s_0^2}} \;.$$ (28)

Substituting solutions 27 and 28 in the condition $t(\hat{h})=t_0(\hat{h})$ produces two images in the $\{z,\tau\}$ space. The first image is a straight line

$$z(\tau) = \frac{z_0 s_0 - \tau}{s}\;,$$ (29)

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

$$z(\tau) = \sqrt{z_0^2 + \frac{\tau^2}{s^2-s_0^2}}\;.$$ (30)

Applying a slant-stack transformation with $z = z_1 - \nu \tau$ turns line (29) into a point $\{z_0 s_0/s,1/s\}$ in the $\{z_1,\nu\}$ space, while curve (30) turns into the curve

$$z_1(\nu) = z_0 \sqrt{1 + \nu^2 \left(s_0^2-s^2\right)}\;.$$ (31)

The curvature of the $z_1(\nu)$ curve at $\nu=0$ is a clear indicator of the migration velocity errors.

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

$$z(h) = s_0 \sqrt{\frac{z_0^2}{s^2} + \frac{h^2}{s^2-s_0^2}}\;.$$ (32)

After the slant transformation $z = z_1 + h \tan{\theta}$, the moveout curve (32) turns into the curve

$$z_1(\theta) = \frac{z_0}{s} \sqrt{s_0^2 + \tan^2{\theta} \left(s_0^2-s^2\right)}\;,$$ (33)

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

$$\tan^2{\theta} = s^2 \nu^2 - 1\;,$$ (34)

or

$$\cos \theta = \frac{1}{\nu s} = \frac{{ \tau}_z}{s} \;,$$ (35)

where ${ \tau}_z = \frac{\partial { \tau}}{\partial z}$. 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