Using this equation the anisotropy parameter can be calculated from the three experimentally measured P-wave phase velocities. Table 3 lists the velocities found by Vernik and Nur for their Bakken Shale sample; plugging those numbers into equation (4) we find .Unfortunately, each of the three P-wave velocity measurements has a standard deviation of about ; the implied absolute standard deviation in the calculated is about .05, of nearly the same order as itself.
This is not the whole story; Equation (4) is a weak-anisotropy approximation to the equation
The ``new, improved'' equation does have some drawbacks. Equation (4) requires only , , and ;equation (5) contains the term C55, and so requires in addition (and/or ). Working with equation (5), assuming a standard deviation of for SV velocity measurements, we find .This is quite a different answer from the first one and suggests we should be wary of equation (4) if the anisotropy is not weak.
Of course, this is still not the whole story; if we make use of the and SV measurements, why not the measurement as well? There may even be compelling reasons to include it; if we assume the measurement has a standard deviation of in Vernik and Nur's example (like we did for the and SV ones) we find that the SV measurement actually constrains C13 (and thus )almost twice as well as the P measurement. How is this possible, given that the P velocity measurements were assumed to be twice as exact? It happens because a small change in C13 has more effect on SV phase velocities than it does on P phase velocities. (This is not some fluke of our particular example; this should almost always be the case.)
It is unlikely the values of C13 calculated using the two different measurements will coincide. For our example we find from the P measurement and from the SV measurement .If the two values wildly disagree it probably indicates something is wrong with one of the measurements; here from the modeling we suspect the P measurement could be too slow. If we adjust the value to correct for that we get for our P-only value, which narrows the mismatch to a more acceptable level.
Ideally all available information should be used to find the most likely value for C13; knowledge of each measurement's error bars along with its contribution to the answer can then be used to put error bars on C13. Doing this for Vernik and Nur's model (using the adjusted P measurement) we find , and .
Clearly the value of we find is rather disturbingly sensitive to which equation and subset of available measurements we use. (For this example we could have found to be .075, .108, .165, .180, .189, or .217!) Of course not all values are equally justifiable; great care must be taken. Vernik and Nur decided to discard their SV measurement because it implied such an (improbably?) large value for .
The situation is not so grim for Thomsen's anisotropy parameter; the exact equation is