Imaging in shot-geophone space |

An equation was derived for paraxial waves.
The assumption of a
*single*
plane wave means that the arrival time
of the wave is given by a single-valued .
On a plane of constant , such as the earth's surface,
Snell's parameter is measurable.
It is

Recall the time-shifting partial-differential equation and its solution as some arbitrary functional form :

The partial derivatives in equation (9.4) are taken to be at constant , just as is equation (9.3). After inserting (9.3) into (9.4) we have

Fourier transforming the wavefield over , we replace by . Likewise, for the traveling wave of the Fourier kernel , constant phase means that . With this, (9.6) becomes

The solutions to (9.7) agree with those to the scalar wave equation unless is a function of , in which case the scalar wave equation has both upcoming and downgoing solutions, whereas (9.7) has only upcoming solutions. We go into the lateral space domain by replacing by . The resulting equation is useful for superpositions of many local plane waves and for lateral velocity variations .

Imaging in shot-geophone space |

2009-03-16