The elastic velocities and of the samples can be calculated using the simple relations:
where L is the length of the rock sample, and are the appropriate ¶ and § first-break arrival times, and to 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-to)-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 to = 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:
= 2918 32 m/s,
= 1731 99 m/s,
= 3380 78 m/s,
= 1744 201 m/s,