Time-shift imaging condition for converted waves

Time-shift angle decomposition

If we construct a seismic image using only time-shift ${ \bf R}({ \bf m},{\tau})$, we use equation  $\left( \ref{eqn:C-t-m} \right)$ for angle decomposition of converted-mode waves. The computation and storage requirements are small, since we need to store images for 1-D cross-correlation lags. However, although the cost is smaller, we lose the option of computing the azimuth of the reflection and we are limited only to the reflection angle.

A decomposition algorithm is a follows:

$${ \bf R}({ \bf m},{\tau}) \rightarrow { \bf R}({ \bf k}_{ \b... ...}_{ \bf m}\vert/\omega ) \rightarrow { \bf R}({ \bf m},\theta)$$

where ${ \bf k}_{ \bf m}$ and $\omega$ are the Fourier duals of position ${ \bf m}$ and time-shift ${\tau}$. The decomposition from the slant-stack parameter $\vert{ \bf k}_{ \bf m}\vert/\omega$ requires a space-domain correction based on the slowness $s$ and the $v_p/v_s$ ratio $\gamma$.