The elastic velocities and of the samples can be calculated using the simple relations:

and

where *L* is the length of the rock sample, and are the appropriate
¶ and § first-break arrival times, and *t*_{o} is the delay time
for the wave to travel through the experimental instrument setup without
the rock sample in place, as measured in procedure (1).

We can also calculate the absolute *rms* errors and associated with
and . Letting *V* stand for either or , the equations
above can be generalized to

*V* = *L* (*t*-*t*_{o})^{-1} .

The absolute error can be calculated by partial differentiation,

and the relative error is evaluated to be

This relative error estimate tends to be too conservative. A less
pessimistic error is the *rms* relative error estimate:

Finally, the absolute error estimate can be determined from the relative error estimate as

We now show a detailed example of calculating and for dry Massillon
sandstone. All other velocities will be calculated in the same manner, but
we will not show those tedious detailed calculations. For dry Massillon
sandstone, = 33.65 and = 0.10 . The ¶-wave
delay time from procedure (1) is *t*_{o} = 23.30 with a measurement
error estimate of = 0.05 . The length of the core sample
was measured as *L* = 1.189 inches with a measurement error of
= 0.001 inches. The ¶-wave velocity is

The relative *rms* error in the ¶-wave velocity is

The absolute *rms* error is

Finally, we can write the total velocity estimate as m/s.

Performing an identical set of analogous calculations for all of the lab
measurements, we calculate the following velocities and *rms* error estimates:

= 5743 75 m/s,
= 3092 22 m/s,

= 2918 32 m/s,
= 1731 99 m/s,

= 3380 78 m/s,
= 1744 201 m/s,

11/18/1997