How can you tell if you're in trouble?

We have shown that unless the anisotropy is too strong, the transducers are too narrow, or the core sample is too tall, core-sample first-break measurements should measure the phase velocity. But what precisely do we mean by ``too'' strong, narrow, or tall? It depends on the elastic properties of the rock being measured. Of course if we knew the elastic constants we wouldn't have to do the experiment in the first place! Is there some warning sign to be found among the measurements in hand?

If the amplitude for the $45^\circ$ case is anomalously low compared to the untroublesome $0^\circ$ and $90^\circ$ cases, this could indicate the energy is missing the receiver and thus the $45^\circ$ traveltime measurement should be regarded with suspicion. Unfortunately amplitude variations in such experiments can be caused in many ways, so this is not a reliable indicator.

Another cross check is possible because some of the elastic constants are overdetermined. In particular, (as we already saw) the key elastic constant C₁₃ can be found from either the $45^\circ$ qP or qSV traveltime measurements; assuming the velocity measurements at $0^\circ$ and $90^\circ$ can be trusted, we can compare the values of C₁₃ calculated from each of the two $45^\circ$ measurements independently as a check. There should not be a problem with both values of C₁₃ erring in the same direction due to side-slip effects. Since the phase-velocity arrival time is the earliest possible, a ``partial miss'' must always result in a delay, resulting in a velocity measurement that is too low. The effect of such a mismeasurement on the calculated value of C₁₃ depends on the wavetype; a too-slow $45^\circ$ qP phase-velocity results in finding a C₁₃ (and $\delta$) that is too low, while a too-slow $45^\circ$ qSV phase-velocity measurement results in a C₁₃ (and $\delta$) that is too high.

Probably the best way to estimate whether there might be significant side-slip problems, though, is to use the (possibly inaccurate) measured elastic constants to model the problem. Analytically, the side-slip velocity is simply $- d V_{\hbox{\rm phase}}(\theta) / d \theta$.In the case of qP and qSV waves in transverse isotropy the total side-slip at $45^\circ$can be expressed directly in terms of the core height H and the measured phase velocities:  

$$- H { \bigl(V_P(90^\circ)^2 - V_P(0^\circ)^2\bigr) \over 2 V... ...}(90^\circ)^2 + V_{SV}(0^\circ)^2\bigr) - 4 V_X(45^\circ)^2 } ,$$ (7)

where V_X is either V_P or V_SV, depending on which wavetype the sideslip is being calculated for. The corresponding equation for SH waves is quite simple:  

$$- H { \bigl(V_{SH}(90^\circ)^2 - V_{SH}(0^\circ)^2\bigr) \over 2 V_{SH}(45^\circ)^2 } .$$ (8)

Conveniently, these equations appear to be rather insensitive to the precise value used for the $45^\circ$ phase velocity; for our example $V_P(45^\circ)$ could be varied by $10\%$ with only a 1mm resulting change in the calculated qP sideslip. (The denominator does start to blow up if $V_P(45^\circ) \approx V_{SV}(45^\circ)$, but hopefully this fact alone would already have suggested there just might be a problem!) If equations (7) or (8) indicate the total sideslip is greater than about half the transducer width for a given wavetype, extra care should be taken when interpreting the corresponding $45^\circ$ measurement as a phase velocity.