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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

| |
(2) |

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
| |
(3) |

Recall the time-shifting partial-differential equation and its
solution *U* as some arbitrary functional form *f*:
| |
(4) |

| (5) |

The partial derivatives
in equation (4) are taken to be at constant *x*,
just as is equation (3).
After inserting (3) into (4) we have
| |
(6) |

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, (6) becomes
| |
(7) |

The solutions to (7) 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 (7) has only upcoming solutions.
We
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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** Up:** SURVEY SINKING WITH THE
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Stanford Exploration Project

12/26/2000