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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 *t*(*x*,*z*).
On a plane of constant *z*, such as the earth's surface,
Snell's parameter *p* is measurable.
It is

| |
(24) |

In a borehole there is the constraint that measurements
must be made
at a constant *x*, where the relevant measurement from an
*upcoming*
wave would be
| |
(25) |

Recall the time-shifting partial-differential equation and its
solution *U* as some arbitrary functional form *f*:
| |
(26) |

| (27) |

The partial derivatives
in equation (26) are taken to be at constant *x*,
just as is equation (25).
After inserting (25) into (26) we have
| |
(28) |

Fourier transforming the wavefield over (*x*,*t*), we
replace by .Likewise, for the traveling wave
of the Fourier kernel ,constant phase means that .With this, (28) becomes
| |
(29) |

The solutions to (29) agree with those to the scalar wave equation
unless *v* is a function of *z*, in which case
the scalar wave equation has both upcoming and downgoing solutions,
whereas (29) has only upcoming solutions.
Chapter taught us how to go into the lateral space
domain by replacing *i k*_{x} by .The resulting equation is useful for superpositions of many local plane waves
and for lateral velocity variations *v*(*x*).

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Stanford Exploration Project

10/31/1997