Thomsen's anisotropy parameters and errors

These antagonistic inequalities do suggest that phase and group directions are important and should not be confused. This was shown more rigorously by Thomsen (1986), who found that for weak anisotropy associated phase and group velocities could be taken as equal to first order, but associated phase and group directions could not. Unfortunately his unnumbered equation on page 1962 giving $\delta$in terms of $0^\circ$, $45^\circ$, and $90^\circ$ P-wave rock-core velocity measurements makes just this error, using group directions which must be phase directions for the equation to be valid. Happily, from the results in this paper we know that rock-core measurements measure phase velocities, not group, so Thomsen's equation can be corrected by simply using phase velocities and directions throughout:  

$$\delta = 4 \biggl[ V_P(45^\circ) / V_P(0^\circ) - 1 \biggr] - \biggl[ V_P(90^\circ) / V_P(0^\circ) - 1 \biggr] ,$$ (4)

where $V_P(\theta)$ is the P-wave phase velocity for the phase-direction $\theta$.

Using this equation the $\delta$ 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 $\delta = .165$.Unfortunately, each of the three P-wave velocity measurements has a standard deviation of about $1\%$; the implied absolute standard deviation in the calculated $\delta$ is about .05, of nearly the same order as $\delta$ itself.

This is not the whole story; Equation (4) is a weak-anisotropy approximation to the equation  

$$\delta = { (C_{13} + C_{55})^2 - (C_{33} - C_{55})^2 \over 2 C_{33} (C_{33} - C_{55}) }$$ (5)

given in Thomsen's paper as equation (17*) (Thomsen, 1986). (Thomsen's $\delta$ is itself a weak-anisotropy approximation to a more fundamental small-offset normal-moveout parameter. See for example Lyakhovitsky and Nevsky (1971).) Our anisotropy is not weak, so it is not safe to use equation (4) as an approximation for equation (5).

The ``new, improved'' equation does have some drawbacks. Equation (4) requires only $V_P(0^\circ)$, $V_P(45^\circ)$, and $V_P(90^\circ)$;equation (5) contains the term C₅₅, and so requires in addition $V_{SV}(0^\circ)$ (and/or $V_{SV}(90^\circ)$). Working with equation (5), assuming a standard deviation of $2\%$ for SV velocity measurements, we find $\delta = .075 \pm .077$.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 $0^\circ$ and $90^\circ$ SV measurements, why not the $45^\circ$ measurement as well? There may even be compelling reasons to include it; if we assume the $45^\circ$ measurement has a standard deviation of $2\%$in Vernik and Nur's example (like we did for the $0^\circ$ and $90^\circ$ SV ones) we find that the $45^\circ$ SV measurement actually constrains C₁₃ (and thus $\delta$)almost twice as well as the $45^\circ$ 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 $45^\circ$ SV phase velocities than it does on $45^\circ$ 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 $45^\circ$ measurements will coincide. For our example we find from the P measurement $C_{13} = 4.79 \pm .78$and from the SV measurement $C_{13} = 6.20 \pm .46$.If the two values wildly disagree it probably indicates something is wrong with one of the measurements; here from the modeling we suspect the $45^\circ$ P measurement could be $.5\%$ too slow. If we adjust the $V_P(45^\circ)$ value to correct for that we get $C_{13} = 5.13 \pm .78$ 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 $45^\circ$ P measurement) we find $C_{13} = 5.94 \pm .37$, and $\delta = .189 \pm .045$.

Clearly the value of $\delta$ we find is rather disturbingly sensitive to which equation and subset of available measurements we use. (For this example we could have found $\delta$ 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 $45^\circ$ SV measurement because it implied such an (improbably?) large value for $\delta$.

The situation is not so grim for Thomsen's $\epsilon$ anisotropy parameter; the exact equation is  

$$\epsilon = V_P(90^\circ) / V_P(0^\circ) - 1 .$$ (6)

For Vernik and Nur's example we find $\epsilon = .298 \pm .019$, a value indicating a rather extreme level of anisotropy.