Time-shift imaging condition for converted waves

Space-shift angle decomposition

If we construct a seismic image using only space-shift ${ \bf R}({ \bf m},{ \bf h})$, we use equation  $\left( \ref{eqn:C-h} \right)$ for angle decomposition of converted-mode waves. The computation and storage requirements are high, since we need to store images for 3-D cross-correlation lags. However, if only the reflection angle is of interest, we can reduce cost by storing only the absolute value of the space-shift vector (Sava and Fomel, 2005a).

A decomposition algorithm is a follows:

$${ \bf R}({ \bf m},{ \bf h}) \rightarrow { \bf R}({ \bf k}_{ ... ...t{ \bf k}_{ \bf m}\vert) \rightarrow { \bf R}({ \bf m},\theta)$$

where each arrow indicates a transformation from one domain to another, and ${ \bf k}_{ \bf m}$ and ${ \bf k}_{ \bf h}$ are the Fourier duals of position ${ \bf m}$ and space-shift ${ \bf h}$. The transform from $({ \bf m},{ \bf h})$ to $({ \bf k}_{ \bf m},{ \bf k}_{ \bf h})$ and back represent Fourier transforms, and the transformation from $({ \bf k}_{ \bf m},{ \bf k}_{ \bf h})$ to $({ \bf k}_{ \bf m},\vert{ \bf k}_{ \bf h}\vert/\vert{ \bf k}_{ \bf m}\vert)$ represents slant-stacking. The decomposition from the slant-stack parameter $\vert{ \bf k}_{ \bf h}\vert/\vert{ \bf k}_{ \bf m}\vert$ to the reflection angle $\theta$ requires a space-domain correction based on the $v_p/v_s$ ratio $\gamma$.