(4) |

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

(5) |

The ``new, improved'' equation does have some drawbacks.
Equation (4) requires only
, , and ;equation (5) contains the term *C _{55}*,
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 *C _{13}* (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

It is unlikely the values of *C _{13}* 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 *C _{13}*; knowledge of each measurement's error bars
along with its contribution to the answer can then be used to
put error bars on

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

(6) |

12/18/1997