Time-shift imaging condition for converted waves

Space-shift angle decomposition for converted waves

We can transform the expressions $\left( \ref{eqn:PmPh-a} \right)$- $\left( \ref{eqn:PmPh-b} \right)$ using the notations $\vert{ \bf p}_{ \bf s}\vert=s$ and $\vert{ \bf p}_{ \bf r}\vert=\gamma s$, where $\gamma ({ \bf m})$ is the $v_p/v_s$ ratio, and $s({ \bf m})$ is the slowness associated with the incoming ray at every image point:

$$\vert{ \bf p}_{ \bf h}\vert^2 = s^2 \left (1+\gamma ^2 -2\gamma \cos 2\theta \right)\;,$$ (11)

$$\vert{ \bf p}_{ \bf m}\vert^2 = s^2 \left (1+\gamma ^2 +2\gamma \cos 2\theta \right)\;.$$ (12)

If we eliminate $\omega$ and make the notations $\vert{ \bf k}_{ \bf h}\vert=\vert{ \bf p}_{ \bf h}\vert/\omega$ and $\vert{ \bf k}_{ \bf m}\vert=\vert{ \bf p}_{ \bf m}\vert/\omega$, we obtain the expression (Sava and Fomel, 2005b)

$$\tan^2 \theta = \frac {(1+\gamma )^2 \vert{ \bf k}_{ \bf h}\... ... m}\vert^2 - (1-\gamma )^2 \vert{ \bf k}_{ \bf h}\vert^2} \;,$$ (13)

that can be used for angle decomposition for converted waves after space-shift imaging condition. For PP reflections ($\gamma =1$), this expression reduces to

$$\tan^2 \theta = \frac{\vert{ \bf k}_{ \bf h}\vert^2}{\vert{ \bf k}_{ \bf m}\vert^2} \;.$$ (14)

Time-shift imaging condition for converted waves