From the assumption that experimental data
can be fit to hyperbolas
(each with a different velocity and each with a different apex )let us next see how
we can fit an earth model of layers,
each with a constant velocity.
Consider the horizontal reflector
overlain by a stratified **interval velocity** *v*(*z*)
shown in Figure 10.

stratrms
Raypath diagram for normal moveout in a stratified earth.
Figure 10 |

The separation between the source and geophone,
also called the offset, is 2*h* and the total travel time is *t*.
Travel times are not be precisely hyperbolic,
but it is common practice to find the best fitting hyperbolas,
thus finding the function .

(24) |

An example of using equation (24)
to stretch *t* into is shown in Figure 11.
(The programs that
find the required and do the stretching are coming up in
chapter .)

Figure 11

Equation (21) shows that
is
the ``root-mean-square'' or
``RMS'' velocity defined by
an average of *v ^{2}* over the layers.
Expressing it for a small number of layers we get

(25) |

(26) |

(27) |

Next we examine an important practical calculation,
getting interval velocities from measured RMS velocities:
Define
in layer *i*,
the interval velocity *v*_{i}
and the two-way vertical travel time .Define the RMS velocity
of a reflection
from the bottom of the *i*-th layer
to be *V*_{i}.
Equation (25) tells us that for
reflections from the bottom of the first, second, and third layers we have

(28) | ||

(29) | ||

(30) |

Normally it is easy to measure the times of the three hyperbola tops,
,
and
.Using methods in chapter
we can measure the RMS velocities *V _{2}* and

(31) | ||

(32) |

(33) |

12/26/2000