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Theoreticians are delighted by velocity increasing linearly with depth
because it happens that many equations work out in closed form.
For example, rays travel in circles.
We will need convenient expressions for velocity
as a function of traveltime depth
and RMS velocity as a function of traveltime depth.
Let us get them.
We take the **interval velocity** *v*(*z*) increasing linearly with depth:
| |
(42) |

This presumption can also be written as a differential equation:
| |
(43) |

The relationship between *z* and vertical two-way traveltime (see equation (27)) is also given by a differential equation:
| |
(44) |

Letting and applying the chain rule
gives the differential equation for :
| |
(45) |

whose solution gives us the desired expression for **interval velocity**
as a function of traveltime depth.
| |
(46) |

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Stanford Exploration Project

12/26/2000