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P waves

Figure [*] shows the situation for P waves for two different core layering orientations in Vernik and Nur's experiment.

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 $45^\circ$ 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?

Figure 4
Snapshots showing the behavior of qP waves in our $90^\circ$ (top) and $45^\circ$ (bottom) core-sample simulations. (The top snapshot shows the situation at 7.5$\mu$s, the bottom at 10.$\mu$s.) The dashed lines show the width of Vernik and Nur's cores, while the thick solid lines at the bottom and top show the size and positions of the P-wave source and receiver transducers. Note in the $45^\circ$ case how the leading part of the wavefront is aiming to miss its intended target, hitting the top of the core somewhat to the left of the receiver instead.


Figure 5
A double plot showing how the signal recorded in the $45^\circ$ P-wave case (lower plot in Figure [*]) would change as a function of two normally fixed parameters, source-receiver transducer offset (left subplot) and source and receiver transducer width (right subplot). In the ``real life'' case the transducer offset was fixed at mm and the transducer width was fixed at 12mm. In the left subplot the transducer width is held at 12mm, but the transducer offset is allowed to vary from -24mm to +12mm (with individual wiggle traces shown every 6mm). In the right subplot the transducer offset is held at mm, but the transducer width is allowed to vary by 4mm steps from mm to 16mm (and then is abruptly jumped to $\infty$). First-break times on various traces are marked by short horizontal lines. Three significant first-break times are labeled. The mm-width mm-offset trace defines the group-velocity arrival time. The 12mm-width mm-offset trace occurs in both subplots and shows the measured arrival time for the ``real life'' case. The $\infty$-width trace (offset is irrelevant) defines the phase-velocity arrival time. Except for a tiny mismatch (caused by a wavelet phase shift) the $\infty$-width first-break arrival time is the same as the -12mm-offset first-break arrival time.


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 10.29$\mu$s 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, 10.30$\mu$s. 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 10.34$\mu$s. If this time were used to calculate the vertical phase velocity it would cause an error of only $.5\%$, which is smaller than the typical errors Vernik and Nur encountered in picking first breaks in their experiment, about $1\%$ for P waves and $2\%$ for S waves. (The only significant effect of the borderline miss in our simulated experiment is a $48\%$ drop in trace amplitude.)

Table 1: Arrival times from the left part of Figure [*], showing how the picked arrival time would vary with source-receiver transducer offset. Two different sorts of arrival times are shown, the first-break time and the first-peak time. The first-break time is somewhat less sensitive to side-slipping. The rightmost column shows how the peak amplitude of the recorded trace drops off with offset.
offset (mm) break % delay peak % delay % amplitude
-12 (peak) 10.29   10.79   100
-6 10.30 .1 10.82 .3 85
0 (actual) 10.34 .5 10.94 1.5 52
6 10.48 1.8 11.17 3.7 29
12 10.78 4.8 11.52 7.1 16

Table 2: Arrival times from the right part of Figure [*], showing how the picked arrival times depart from the ideal $\infty$-transducer time for various finite transducer sizes. Two different sorts of arrival times are shown, the first-break time and the first-peak time.
transducer size (mm) break % delay peak % delay  
$\infty$ (phase) 10.30   10.84    
24 10.30 .0 10.84 .0  
20 10.30 .0 10.85 .1  
16 10.31 .1 10.88 .4  
12 (actual) 10.34 .3 10.94 .9  
8 10.42 1.2 11.02 1.7  
4 10.54 2.3 11.08 2.2  
0 (group) 10.63 3.2 11.10 2.4  

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, 10.63$\mu$s. As the transducer size is increased towards 12mm the first break moves rapidly earlier to within $.5\%$ 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.)

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