In the upper plot the layers run vertically; there is no tendency for the wave energy to ``slip sideways'' and the flat part of the wavefront impacts the receiver transducer well centered. We can see from the figure that things are working correctly in this case. Mathematically this is because the group and phase velocities happen to be the same for this propagation direction. The same good behavior also occurs when the layers in the core run horizontally, because this also sends the waves along a symmetry direction where the group and phase velocities are identical (for all wavetypes).
The situation is not so happy in the lower plot. Here the layers are at a angle, running from the lower right to the upper left. The P wave travels faster along the layers than across them; as a result the ``flat'' part of the wavefront (containing the main focus of energy) tends to follow the layers and slips sideways to the left. In our particular example the accumulated sideways slip from bottom to top happens to be about the same as the transducer widths, so the ``flat'' part of the wavefront only just grazes the edge of the receiver. What traveltime will be measured in this case?
The answer is contained in Figure . The left part of this figure shows the result of a tiny ``seismic survey'' over the shale-core model. The source transducer is held fixed. The horizontal axis shows how the trace recorded at the receiver would vary with offset if the receiver were moved around (instead of glued in place). The first break on the earliest arrival occurs at 10.29s for an offset of -12mm. (The ``correct'' phase-velocity arrival time as defined by running the model with infinite-length transducers is very slightly later, 10.30s. We will remark further on this in the discussion.) The first break at zero offset (and thus what corresponds to the signal recorded in the actual experiment) occurs at 10.34s. If this time were used to calculate the vertical phase velocity it would cause an error of only , which is smaller than the typical errors Vernik and Nur encountered in picking first breaks in their experiment, about for P waves and for S waves. (The only significant effect of the borderline miss in our simulated experiment is a drop in trace amplitude.)
|offset (mm)||break||% delay||peak||% delay||% amplitude|
|transducer size (mm)||break||% delay||peak||% delay|
Table 1 shows how the first-break time varied with offset for the traces shown in the left part of Figure . From the table we can discern that if the miss had been smaller (say if the anisotropy had been less, the transducers had been wider, or the core had been shorter), there would have been an insignificant delay in the first break and only a very small drop in amplitude. If the miss had been by more, on the other hand (say if the anisotropy had been slightly greater, the transducers had been narrower, or the core had been taller), the first break would have been significantly delayed. Essentially, if any of the leading ``flat part'' of the wavefront manages to hit the receiver it is enough to initiate a first break at the true phase-velocity arrival time; if the flat part misses completely the error can be substantial.
The right part of Figure demonstrates this another way, by showing the results of re-running the numerical experiment for a range of transducer sizes. Zero-offset traces corresponding to what would be measured by a laboratory experiment are shown. With a point source and point receiver, the first break measures the group-velocity arrival time, 10.63s. As the transducer size is increased towards 12mm the first break moves rapidly earlier to within of the phase-velocity arrival time. As the transducer size is increased yet further the first-break time closes in on the phase-velocity arrival time more slowly, finally reaching it when the transducer width is 20mm. All wider transducers measure the phase-velocity arrival time. (Table 2 gives the numbers for this case.)