Time-shift imaging condition in seismic migration

Angle transformation in wave-equation imaging

Using the definitions introduced in the preceding section, we can make the standard notations for source and receiver coordinates: ${ \bf s}= { \bf m}- { \bf h}$ 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 $t = t \left ({ \bf m},{ \bf h}\right )$. 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:

$${ \bf p}_{ \bf m} = { \bf p}_{ \bf r}+ { \bf p}_{ \bf s}\;,$$ (11)

$${ \bf p}_{ \bf h} = { \bf p}_{ \bf r}- { \bf p}_{ \bf s}\;.$$ (12)

From equations (11)-(12), we can write

$$2 { \bf p}_{ \bf s} = { \bf p}_{ \bf m}- { \bf p}_{ \bf h}\;,$$ (13)

$$2 { \bf p}_{ \bf r} = { \bf p}_{ \bf m}+ { \bf p}_{ \bf h}\;.$$ (14)

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Figure 1. Geometric relations between ray vectors at a reflection point.

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

$$\vert{ \bf p}_{ \bf h}\vert^2 = \vert{ \bf p}_{ \bf s}\vert^2 + \vert{ \bf p}_{ \bf r}\vert^2 - 2 \vert{ \bf p}_{ \bf s}\vert\vert{ \bf p}_{ \bf r}\vert\cos(2 \theta ) \;,$$ (15)

$$\vert{ \bf p}_{ \bf m}\vert^2 = \vert{ \bf p}_{ \bf s}\vert^2 + \vert{ \bf p}_{ \bf r}\vert^2 + 2 \vert{ \bf p}_{ \bf s}\vert\vert{ \bf p}_{ \bf r}\vert\cos(2 \theta ) \;.$$ (16)

Equations (15)-(16) relate wavefield quantities, ${ \bf p}_{ \bf h}$ and ${ \bf p}_{ \bf m}$, to a geometric quantity, reflection angle $\theta$. Analysis of these expressions provide sufficient information for complete decompositions of migrated images in components for different reflection angles.

Time-shift imaging condition in seismic migration