Our theoretical analysis will abandon the geophone axis *g* in favor of a
radial-like axis characterized by a Snell parameter *p*.
(This really says nothing about the implied data processing itself,
since it would be simple enough to transform final equations back to offset).
The coordinate system being defined will be called
a retarded, moveout-corrected, Snell trace, coordinate system.
Ideal data in this coordinate system in a zero-dip earth
is unchanged as it is downward continued.
Hence the amount of work the differential equations have to do
is proportional to the departure of the data from the ideal.
Likewise the necessity for spatial sampling of the data
increases in proportion to the departure of data from the ideal.
Define

p |
(, the Snell ray parameter |

t_{p} |
any one-way time from the surface along a ray with
parameter p |

g |
the surface separation of the shot from the geophone |

t' |
one-way time, surface to reflector, along a ray |

travel-time depth of buried geophones, one-way time along a ray | |

t |
travel time seen by buried geophones |

v(p, t_{p}) |
a stratified velocity function v'(z),
in the new coordinates |

The coordinate system is based on the following simple statements:
(1) travel time from shot to geophone is twice the travel time
from shot to reflector,
less the time-depth of the geophone; and
(2) the horizontal distance traveled by a ray
is the time integral of = *pv ^{2}*;
(3) the vertical distance traveled by a ray is computed the
same way as the horizontal distance, but with a cosine instead of a sine.

(8) | ||

(9) | ||

(10) |

(11) |

(12) |

It should be noted that (12) is a linear relation involving the Fourier variables, but the coefficients involve the original time and space variables. So (12) is in both domains at once. This is useful and valid so long as it is assumed that second derivatives neglect the derivatives of the coordinate frame itself. This assumption is often benign, amounting to something like spherical divergence correction.

Here we could get bogged down in detail, were we to continue to attack the nonzero offset case. Specializing to zero offset, namely, , we get

(13) |

10/31/1997