Angle-gather time migration

Amplitude considerations

One simple approach to amplitude weighting for angle-gather migration is based again on Cheops' pyramid considerations. Stacking along the pyramid in the data space is a double integration in midpoint and offset coordinates. Angle-gather migration implies the change of coordinates from $\{x,h\}$ to $\{\alpha,\gamma\}$. The change of coordinates leads to weighting the integrand by the following Jacobian transformation:

$$dx dh = \left\vert \det \left( \begin{array}{cc} \frac... ...ial \gamma} \end{array} \right) \right\vert d\alpha d\gamma$$ (11)

Substituting formulas (5) and (6) into equation (11) gives us the following analytical expression for the Jacobian weighting:

$$W_{\mbox{J}} = \left\vert \det \left( \begin{array}{cc} ... ...= \frac{z^2}{\left(\cos{\alpha}^2 - \sin{\gamma}^2\right)^2}$$ (12)

Weighting (12) should be applied in addition to the weighting used in common-offset migration. By analyzing formula (12), we can see that the weight increases with the reflector depth and peaks where the angles $\alpha$ and $\gamma$ approach condition (10).

The Jacobian weighting approach, however, does not provide physically meaningful amplitudes, when migrated angle gathers are considered individually. In order to obtain a physically meaningful amplitude, we can turn to the asymptotic theory of true-amplitude migration (Goldin, 1992; Tygel et al., 1994; Schleicher et al., 1993). The true-amplitude weighting provides an asymptotic high-frequency amplitude proportional to the reflection coefficient, with the wave propagation (geometric spreading) effects removed. The generic true-amplitude weighting formula (Fomel, 1996b) transforms in the case of 2-D angle-gather time migration to the form:

$$W_{\mbox{TA}} = \frac{1}{\sqrt{2 \pi}} \frac{\sqrt{L_... ...partial^2 L_r}{\partial \xi \partial \gamma} \right\vert\;,$$ (13)

where $L_s$ and $L_r$ are the ray lengths from the reflector point to the source and the receiver respectively. After some heavy algebra, the true-amplitude expression takes the form

$$W_{\mbox{TA}} = \frac{2 z \sin{\alpha}}{\sqrt{2 \pi} v... ...mma}} {\left(\cos^2{\alpha} - \sin^2{\gamma}\right)^{5/2}}\;.$$ (14)

Under the constant-velocity assumption and in high-frequency asymptotic, this weighting produces an output, proportional to the reflection coefficient, when applied for creating an angle gather with the reflection angle $\gamma$. Despite the strong assumptions behind this approach, it might be useful in practice for post-migration amplitude-versus-angle studies. Unlike the conventional common-offset migration, the angle-gather approach produces the output directly in reflection angle coordinates. One can use the generic true-amplitude theory (Fomel, 1996b) for extending formula (14) to the 3-D and 2.5-D cases.

Angle-gather time migration