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\lefthead{Dellinger and Vernik}
\righthead{pulse-transmission traveltimes}
\footer{\thepage}
\thispagestyle{sep}
\email{jdellinger@trc.amoco.com, lev@pangea.stanford.edu, respectively}
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\title{Do traveltimes in pulse-transmission experiments yield
anisotropic group or phase velocities?}
\author{Joe Dellinger and Lev Vernik}

%\noindent
%(running title: pulse-transmission traveltimes)

\noindent
(presented at the 62nd annual SEG meeting
under the title ``Do core-sample measurements record group
or phase velocity?'', pages 662--665.
Appeared in the November 1994 GEOPHYSICS, pages 1774--1779.)

\mhead{INTRODUCTION}
The elastic properties of layered rocks are often measured
using the pulse through-transmission technique
on sets of cylindrical cores cut at angles
of $0$, $90$, and $45$~degrees
to the layering normal
(e.g., Vernik and Nur, 1992; Lo et al{}., 1986; Jones and Wang, 1981).
In this method
transducers are attached to the flat ends of the three cores
(see Figure~\origlatexref{Cores}),
the first-break traveltimes of P, SV, and SH-waves down the axes are measured,
and a set of transversely isotropic
elastic constants are fit to the results.
The usual assumption is that frequency
dispersion, boundary reflections, and near-field effects can all be
safely ignored, and that the traveltimes measure
either vertical anisotropic group velocity
(if the transducers are very small compared to their separation)
or phase velocity
(if the transducers are relatively wide compared to their separation)
(Auld, 1973).

To discover whether typical experiments of this kind are more
likely to measure group velocity, phase velocity, or something in between,
we numerically model a laboratory
pulse-transmission experiment of Vernik and Nur (1992).
In their experiment
the separation between source and receiver
was more than three times greater than the transducer width.
Although this configuration might seem closer to the group-velocity case
than the phase-velocity one, Vernik and Nur assumed their recorded
first-breaks represented phase velocities.

Our numerical results show Vernik and Nur were (almost) correct.
Except for a slight underestimate in the P-wave
velocity measurements at 45~degrees to the layering,
they did record phase-velocity traveltimes in their experiment.
Most experiments are clearly closer to the phase-velocity case
than Vernik and Nur's; we therefore conclude that almost all
pulse-transmission experiments of this
kind should measure anisotropic phase, not group, velocities.

\mhead{METHODOLOGY}
Figure~\origlatexref{Cores} shows the core size and P-wave transducer
width for one of Vernik and Nur's (1992) early
experimental configurations to scale.
The cores were 40~mm tall and 26~mm wide;
the P-wave transducers were 12~mm wide.
The SV and SH transducers (not shown) were 20~mm wide,
nearly as wide as the core itself.
The central frequencies of the transducers were $1.0$~MHz for P-waves
and $0.7$~MHz for S.
For each core Vernik and Nur picked first-break P, SV, and SH traveltimes.

A complete numerical simulation of this experiment
would take into account the
complex three-dimensional boundary conditions
of tilted-axis transverse isotropy interacting
with a truncated cylindrical surface.
We do not need to model the entire waveform, however;
Vernik and Nur did not use the entire seismogram to make their
traveltime measurements, but only the first breaks.
Assuming the direct waves arrive ahead of waves that have interacted
with the core boundaries, a homogeneous model should suffice.

Our numerical model uses a two-dimensional finite-difference algorithm,
Fourier spectral in space and Chebychev spectral in time
(Etgen and Dellinger, 1989; Tal-Ezer, 1987).
This algorithm is accurate as long as the wavelet is not
spatially aliased; the time step can be arbitrarily large.
We use a scale of $.5$~mm per spatial gridpoint (well within the Nyquist
criteria for the source wavelet) and $.05$~$\mu$s per time step.
The transversely isotropic (TI) medium is homogeneous,
so the density has been normalized out;
in units of velocity squared, the normalized elastic constants
($W_{ij} = C_{ij} / \rho$) are
$W_{11} = 20.16$, $W_{33} = 11.97$, $W_{55} = 4.00$, $W_{66} = 6.86$, and
$W_{13} = 5.51$ $(\mbox{\rm km/s})^2  \equiv (\mbox{mm/$\mu$s})^2$.
These constants are for a core cut perpendicular to the layering.
Figure~\origlatexref{Waves} shows the shape
of the highly anisotropic {\sl q}P, {\sl q}SV, and SH wavefronts
in this medium, a sample of Bakken Shale (Vernik and Nur, 1992).

The numerical model source time function
is a Ricker wavelet with central frequency $0.9$~MHz
for both wavetypes.
The wavefield excited by a finite-width transducer is constructed
by convolving the wavefield produced by a single-gridpoint source with
a horizontal bar of impulses, 25 adjacent gridpoints for the P transducer,
41 for the S.
We defined the first-break time as the point when
the recorded wavelet first attains an amplitude of $1$~percent
of its maximum;
this seems to pick about the same point on the trace
that we would choose by eye.

\mhead{PHASE VERSUS GROUP}
We begin by examining model results for two ``canonical'' experiments.
The lower plot in Figure~\origlatexref{phasegroup} shows how
vertical group velocity (the velocity of vertically traveling energy)
can be measured.
The anisotropic wavefront radiates out from a point source at the bottom, and
a point receiver at the top detects the part of the wavefront with
vertically traveling energy.
The distance between the two transducers
is divided by the measured traveltime to find the vertical group velocity.

The upper plot in Figure~\origlatexref{phasegroup} shows
how vertical phase velocity (the velocity of a vertically traveling plane wave)
can be measured.
The source must be wide enough
to launch a reasonable facsimile of a plane wave.
If the source transducer were truly infinite,
the horizontal position of the receiver would be irrelevant;
in practice, the position of the receiver does not matter
if it samples the flat central part of the wavefront away
from the diffracting truncated edge.
The {\em vertical\/} distance between source and receiver
divided by the measured traveltime then gives the vertical phase velocity
(Mignogna, 1990).

The upper plot in Figure~\origlatexref{phasegroup} was constructed
from the lower plot in the same figure
by summing copies of the latter shifted over the range $-20$ to $+20$~mm;
the flat part of the wavefront in the upper plot thus corresponds
to the highest point on the wavefront in the lower plot.
This implies that
we could have measured the vertical phase velocity directly
from the lower plot by positioning the receiver
where the wavefront {\em first encountered the upper surface\/}
(around the $-11$~mm position) and dividing by
the {\em vertical\/} distance from source to receiver ($40$~mm).
This fundamental relationship is used by Sahay et al{}. (1992) to
design experiments that use movable transducers
to unambiguously measure phase velocities.

Note that if we instead divided by the the true distance between
the source and the offset receiver, $\sqrt{40^2 + 11^2}$~mm, we would
be finding the (nonvertical) group velocity
associated with the vertical phase velocity.
The three kinds of velocities in Figure~\origlatexref{phasegroup}
satisfy an inequality:
the associated group velocity is greater than or equal to
the vertical phase velocity, which in turn is greater than or equal to
the vertical group velocity.

\mhead{MODEL RESULTS}
\shead{P-waves}
Figure~\origlatexref{Pexam} shows snapshots of {\sl q}P-waves propagating
in our numerical model of Vernik and Nur's (1992) experiment.
In the upper plot the layers are vertical,
propagation is along a symmetry axis of the medium, and so
the wave energy is not deflected by the layering
and the flat part of the wavefront
impacts the receiver transducer centrally.

In the lower plot the layers extend from the lower right to the upper left
at a 45-degree angle to the core axis.
The {\sl q}P-wave travels faster along the layers than across them, so
the flat part of the wavefront containing the main focus of energy
is deflected to the left.
[See Green (1973) for photographs of wavefronts doing this.]
In this example the accumulated lateral displacement from
bottom to top happens to be almost the same as the transducer width.
Not surprisingly, Vernik and Nur noticed the direct {\sl q}P-waves were
distinctly weaker in their 45-degree cores than in the others.

Figure~\origlatexref{Palt} shows how this grazing miss affects
first-break traveltimes.
The left plot shows the model output for a multioffset survey
over the shale-core model; the horizontal axis shows source-receiver offset.
The earliest first break occurs at $10.29$~$\mu$s at an offset of $-12$~mm.
The first break at zero offset, corresponding to the actual
geometry in Vernik and Nur's (1992) experiment, occurs $.5$~percent
later at $10.34$~$\mu$s.
This error is smaller than the $1$~percent P-wave measurement uncertainty
Vernik and Nur found in their experiment (Vernik and Nur, 1993) but
is still detectable.

The right part of Figure~\origlatexref{Palt} shows
the results of re-running the numerical experiment
with zero offset but a range of transducer sizes.
With a point source and point receiver, the first break defines
the vertical group-velocity arrival time, here $10.63$~$\mu$s.
As the transducer size is
increased towards $12$~mm the first break moves rapidly earlier to within
$.5$~percent of the true phase-velocity arrival time; for further increases
the first-break time continues to converge on the exact phase-velocity arrival
time, but much more slowly.

As we saw in Figure~\origlatexref{phasegroup}, when {\em any\/} of the leading
flat part of the wavefront hits the receiver it theoretically initiates
a first break {\em at the phase-velocity arrival time\/}.
In this example the lateral offset of the {\sl q}P-wave energy is slightly
less than $12$~mm. Wider transducers than $12$~mm continue to record slightly
earlier times because we pick the first break when the amplitude
exceeds $1$~percent of its peak value, not at the ``theoretical'' first break.
For the same reason, the ``infinite-transducer, zero-offset''
and ``$12$~mm transducer, $-12$~mm offset'' first breaks (at either end
of the upper dotted line in Figure~\origlatexref{Palt}) are
also not quite at the same time.

\shead{S-waves}
The upper plot in Figure~\origlatexref{Sexam} shows
numerical model results for the 45-degree SH-wave case.
Although the lateral drift is nearly the same
as for the 45-degree {\sl q}P case shown in Figure~\origlatexref{Pexam},
the receiver transducer solidly encounters the key
flat part of the wavefront because of the larger S-wave transducer size.

{\sl q}SV-waves in TI media can display complex anisotropic effects.
Wavefront cusps are always associated with precursive energy,
making the definition of ``theoretical first break'' problematical.
These are point-source, not plane-source phenomena, however;
in practice they should only be significant near the edges
of the ``flat part'' of the wavefront (if they occur at all).
{\sl q}SV-wavefronts in TI media do almost always display
near-symmetry about 45~degrees, so we expect little lateral displacement
to occur; the lower plot in Figure~\origlatexref{Sexam} shows
that the displacement in our example is only $2$~mm (to the right).

\mhead{DISCUSSION}
\shead{Detecting measurement errors}
Vestrum (1994) notes that the distinction
between anisotropic group and phase velocity is not always significant;
even in a strongly anisotropic medium there may happen to be
near-symmetry about $45$~degrees.
If lateral displacement of the direct arrivals may be a problem,
a consistency check can be made
by determining the elastic constant $C_{13}$ from the
45-degree P-wave and {\sl q}SV-wave measurements independently.
Because the phase-velocity arrival time is the earliest possible,
a ``partial miss'' must always result in a delay and thus an underestimate
of the phase velocity. (Similarly, insufficiently narrow transducers
cause group velocities to be overestimated.)
An erroneously slow 45-degree {\sl q}P phase velocity results in
finding a $C_{13}$ that is too low,
while an erroneously slow 45-degree {\sl q}SV phase velocity results
in a $C_{13}$ that is too high;
thus any errors in interpreting the traveltimes as group or phase
must cause the two values of $C_{13}$ to diverge.

Eaton (1992) examines
Vernik and Nur's (1992) results using a variation of this method:
the measured {\sl q}P 45-degree velocity is used to predict
a {\sl q}SV 45-degree velocity that is then compared to the
actual measurement. Eaton finds that the predicted {\sl q}SV
velocities tend to be a few percent too high, exactly
as we should expect if the input {\sl q}P phase-velocity
estimates were around half a percent too low.

In the case of {\sl q}P and {\sl q}SV-waves in
transverse isotropy the total lateral displacement at 45~degrees
can be expressed directly in
terms of the core height $H$ and the (possibly inaccurately)
measured phase velocities:
\begin{equation}
D =
H
{
\bigl(V_P(90)^2 - V_P(0)^2\bigr)
\over
2 V_X(45)^2
}
\;
{
\bigl(V_{SV}(90)^2 + V_{SV}(0)^2\bigr)
- 2 V_X(45)^2
\over
\bigl(V_P(90)^2 + V_P(0)^2\bigr) +
\bigl(V_{SV}(90)^2 + V_{SV}(0)^2\bigr)
- 4 V_X(45)^2
}
,
\label{Sideslip1}
\end{equation}
where D is the total lateral displacement and
$V_X$ is either $V_P$ or $V_{SV}$, depending on the wavetype being considered.
$D$ is positive if the offset is in the direction of the layers,
the expected case for {\sl q}P-waves.
The corresponding equation for SH-waves is:
\begin{equation}
D =
H
{
\bigl(V_{SH}(90)^2 - V_{SH}(0)^2\bigr)
\over
2 V_{SH}(45)^2
}
.
\label{Sideslip2}
\end{equation}

Note these equations always {\em overestimate\/} the
total offset of the {\sl q}P and SH-wavefronts
if lateral displacement has caused the measured 45-degree phase velocities
to be erroneously slow.
Equations \origlatexref{Sideslip1} and~\origlatexref{Sideslip2} are
also valid for orthorhombic anisotropy measured in symmetry planes,
as in Cheadle et al{}. (1991).

\shead{Physical versus numerical models}
If a first break could be detected as soon as any of the leading flat part
of the wavefront met the receiver
our conclusions would be equally valid in either two or three dimensions.
In practice the shape of the following wavelet must matter to some extent,
and this will be different in a two-dimensional model than in a
three-dimensional one.
Our numerical model also does not attempt to include reflections
at the boundary between the transducer and the core sample.
As a check against reality we reversed Eaton's (1992) method and used
Vernik and Nur's (1992) 45-degree {\sl q}SV measurements
(presumably unaffected by lateral-displacement errors)
to predict the corresponding true 45-degree {\sl q}P phase velocities.
Although there was a large scatter in the discrepancies,
reflecting the difficulty Vernik and Nur had measuring
45-degree {\sl q}SV velocities,
the average error for
the 9~core-sample sets was found to be $.61$~percent,
in good agreement with our simple model's predictions.

\CON
We expect that the Bakken Shale example investigated here represents
about the most anisotropic geological specimen
(with the largest P and SH
lateral displacements) likely to be encountered in the laboratory.
If this is true then
the traveltimes measured in similar laboratory core-sample experiments
should also represent phase velocities,
assuming the critical ratio of core-sample height to transducer width is
no greater than it was in Vernik and Nur's (1992) experiment (i.e., $\leq 3$).
Our results also suggest that {\em group\/} velocities are very difficult
to measure directly; the ratio of core-sample height to transducer width
needs to be at least $20$, preferably even larger.

The usefulness of 45-degree {\sl q}SV measurements as an
independent source of information should not be overlooked, but
even if they are unavailable
equations (\origlatexref{Sideslip1}) and~(\origlatexref{Sideslip2})
can be used to estimate the total lateral displacement.
If the offset is predicted to be more than about half the
transducer width caution should be exercised when interpreting
the measured traveltimes.
In any case, published elastic constants should always include information
about what measurements were used and what magnitudes of error to expect
in them (Seriff and Sriram, 1991).

\ACK
The authors wish to thank Leon Thomsen and Jim Brown for helpful comments.
John Etgen provided the finite-difference modeling programs.
The Stanford Exploration Project provided computer resources.
Lev Vernik wishes to thank Amos Nur for support.
This is University of Hawai`i
School of Ocean and Earth Science and Technology contribution
number~3592.

\clearpage
\REF
\reference{Auld, B. A., 1973,
Acoustic fields and waves in solids, vol 1.: John Wiley and Sons.}

\reference{Cheadle, S. P., Brown, R. J., and Lawton, D. C., 1991,
Orthorhombic anisotropy: a physical seismic modeling study:
Geophysics, {\bf 56}, 1603--1613.}

\reference{Eaton, D. W., 1992,
Discussion on:
``Ultrasonic velocity and anisotropy of hydrocarbon source rocks'':
Geophysics, {\bf 58}, 758--759.}

\reference{Etgen, J. T., and Dellinger, J., 1989,
Accurate wave-equation modeling:
59th Ann. Internat. Mtg., Soc. Expl. Geophys.,
Expanded Abstracts, 494--497.}

\reference{Green, R. E., 1973,
Ultrasonic investigation of mechanical properties:
Academic Press Inc.}

\reference{Jones, E. A., and Wang, H. F., 1981,
Ultrasonic velocities in Cretaceous shales from the Williston basin:
Geophysics, {\bf 21}, 905--938.}

\reference{Lo, T. W., Coyner, K. B., and Toks\"oz, M. N., 1986,
Experimental determination of elastic anisotropy of Berea sandstone,
Chicopee shale, and Chelmsford granite:
Geophysics, {\bf 51}, 164--171.}

\reference{Mignogna, R., 1990,
Ultrasonic determination of elastic constants from oblique angles
of incidence in non-symmetry planes,
{\it in\/} Thompson, D. O., and Chimenti, D. E., Eds.,
Rev. Prog. in QNDE, vol. 9:
Plenum Press,
1565--1572.}

\reference{Sahay, S. K., Kline, R. A., and Mignogna, R., 1992,
Phase and group velocity considerations for dynamic modulus measurement
in anisotropic media:
Ultrasonics, {\bf 30}, 373--382.}

\reference{Seriff, A.J., and Sriram, K.P., 1991,
P-SV reflection moveouts for transversely isotropic media with a vertical
symmetry axis:
Geophysics, {\bf 56}, v. 8, 1271--1274.}

\reference{Tal-Ezer, H., 1986,
Spectral methods in time for hyperbolic problems:
SIAM J. Numer. Anal., {\bf 23}, 11--26.}

\reference{Vernik, L., and Nur, A., 1992,
Ultrasonic velocity and anisotropy of hydrocarbon source-rocks:
Geophysics, {\bf 57}, 727--735.}

\reference{Vernik, L., and Nur, A., 1993,
Reply by the authors to David W. Eaton:
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\reference{Vestrum, R.W., 1994,
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\clearpage
\mhead{Figure Captions}
\begin{figure}[h]
\caption
{
Shale cores cut at 0, 90, and 45~degrees (perpendicular, parallel,
and at a 45-degree angle to the layering respectively).
The disks at the top and bottom of each core
show the relative width of the P-wave transducers.
\label{Cores}
}
\end{figure}
\begin{figure}[h]
\caption
{
The shapes of
{\sl q}P (outer curve), {\sl q}SV (inner solid curve), and SH (dotted)
wavefronts for the medium used in our numerical model.
The 90, 45, and 0-degree labels show the
direction of vertical for the corresponding shale-core orientations.
\label{Waves}
}
\end{figure}
\begin{figure}[h]
\caption
{
Snapshots demonstrating two ``ideal'' core-sample experiments.
The positions and sizes of the source and receiver transducers are
indicated respectively
by thick horizontal lines at the bottom and top of the model.
Top: a wide-source experiment for measuring
vertical phase velocity.
Bottom: a point-source to point-receiver experiment
for measuring vertical group velocity.
\label{phasegroup}
}
\end{figure}
\begin{figure}[h]
\caption
{
Snapshots showing the behavior of {\sl q}P-waves in our
90 (top) and 45 (bottom) degree core-sample simulations.
The top snapshot is taken at $7.5$~$\mu$s,
the bottom $10.$~$\mu$s.
The dashed lines show the width of Vernik and Nur's cores; the thick
solid lines at the bottom and top
show the size and positions of the P-wave source and receiver transducers.
\label{Pexam}
}
\end{figure}
\begin{figure}[h]
\caption
{
How the recorded signal in the 45-degree P-wave case
depends on experimental geometry.
In the left subplot the transducer width is fixed at $12$~mm while
the transducer offset is varied;
in the right subplot the transducer offset is fixed at $0$~mm while
the transducer width is varied.
Short horizontal lines mark first-break times.
\label{Palt}
}
\end{figure}
\begin{figure}[h]
\caption
{
Snapshots showing the behavior of SH (top) and {\sl q}SV (bottom) waves
in our 45-degree simulation.
Both snapshots are taken at $17.5$~$\mu$s.
\label{Sexam}
}
\end{figure}
\clearpage
\end{document}
