Time-shift imaging condition for converted waves

Angle decomposition

We can develop procedures for angle-decomposition starting from the multi-lag cross-correlations constructed in the preceding section. At this stage, we need to exploit the physical meaning of the various lags, either along space axes or along the time axis. In general, 3-D angle-decomposition should define reflectivity function of reflection and azimuth angles. In this paper, we concentrate on decomposition function of the reflection angle only.

Using the definitions introduced in the preceding section, we can make the standard notations for source and receiver coordinates, respectively: ${ \bf s}= { \bf m}- { \bf h}$ and ${ \bf r}= { \bf m}+ { \bf h}$. The traveltime from a source to a receiver is a function of all spatial coordinates of the seismic experiment $t = t \left ({ \bf m},{ \bf h}\right)$. Suppose we could identify the function $t\left ({ \bf m},{ \bf h}\right)$, then differentiating $t$ with respect to all components of the vectors ${ \bf m}$ and ${ \bf h}$, and using the standard notations ${\bf p}_\alpha = \nabla_\alpha t$, where $\alpha=\{{ \bf m},{ \bf h},{ \bf s},{ \bf r}\}$, we can write ${ \bf p}_{ \bf m}= { \bf p}_{ \bf r}+ { \bf p}_{ \bf s}$ and ${ \bf p}_{ \bf h}= { \bf p}_{ \bf r}- { \bf p}_{ \bf s}$. Therefore, we can also write equivalent relations $2{ \bf p}_{ \bf s}= { \bf p}_{ \bf m}- { \bf p}_{ \bf h}$ and $2{ \bf p}_{ \bf r}= { \bf p}_{ \bf m}+ { \bf p}_{ \bf h}$.

By analyzing the geometric relations of various vectors at an image point, Figure 1(b), we can write the trigonometric expressions:

$$4 \vert{ \bf p}_{ \bf h}\vert^2 = \vert{ \bf p}_{ \bf s}\vert^2 + \vert{ \bf p}_{ \bf r}\vert^2 - 2 \vert{ \bf p}_{ \bf s}\vert\vert{ \bf p}_{ \bf r}\vert\cos(2 \theta ) \;,$$ (9)

$$4 \vert{ \bf p}_{ \bf m}\vert^2 = \vert{ \bf p}_{ \bf s}\vert^2 + \vert{ \bf p}_{ \bf r}\vert^2 + 2 \vert{ \bf p}_{ \bf s}\vert\vert{ \bf p}_{ \bf r}\vert\cos(2 \theta ) \;.$$ (10)

Time-shift imaging condition for converted waves