DETERMINATI ON OF SEISMIC ATTENUATION
USING OBSERVED PHASE SHIFT
IN SEDIMENTARY ROCKS
by
JEAN M. BARANOWSKI
B.S.,
Universit y of Arizona
(1980 )
SUBMITTED TO THE DEPARTMENT OF
EARTH AND PLANE TARY SCIENCE
IN PARTIAL FULFI LLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN GEOPHYSICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1982
©
Massachusetts Institute of Technology 1982
Signature of Author ......
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Planetary Science
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FROM 1932
MIT LIBFRARIES
Theodore R. Madden
Chairman, Department
Graduate Committee
DETERMINATION OF SEISMIC ATTENUATION USING
OBSERVED PHASE SHIFT IN SEDIMENTARY ROCKS
by
JEAN M. BARANOWSKI
Submitted to the Department of Earth and
Planetary Science on May 24, 1982 in partial
fulfillment of the requirements for the Degree
of Master of Science in Geophysics
ABSTRACT
Seismic attenuation has been measured in dry and water
saturated samples of Berea sandstone, under varying pressure
conditions. Ultrasonic P waveforms were studied to determine
seismic quality factor, Q.
Two methods of data processing were used, both of which
assume that Q is constant over the frequency range analyzed.
The first is an amplitude spectral ratio technique, which
involves comparing the amplitude spectrum of a rock sample
waveform to the spectrum of an aluminum calibration
standard.
Viscoelastic theory predicts that seismic attenuation
must be accompanied by phase velocity dispersion. The
second method of measurement exploits the model-dependent
relationship between attenuation and dispersion to calculate
Q from observed phase spectra. As with amplitudes, the
phase spectrum of an aluminum standard is needed to cancel
source and geometric effects.
Q is found to increase with pressure, and is lower for
saturated than for dry samples. Q values determined from
both methods are not systematically in agreement,
particularly in cases where dispersion is slight over the
narrow frequency range sampled.
Thesis Supervisor:
Title:
Dr. M. Nafi Toksiz
Pof~essor of Geophysics
CONTENTS
page
I
II
III
IV
V
VI
Title
1
Abstract
2
Contents
3
Introduction
4
Definitions
6
Attenuation-Dispersion Theory
9
Data Analysis Method
17
Results
24
Conclusions
28
List of Symbols Used in
Text
30
Acknowledgements
31
References
32
Table 1
34
Figure Captions
35
Figures
38
I. INTRODUCTION
Amplitude spectral ratio methods are commonly used to
measure seismic
attenuation in laboratory experiments and
sei smi c
studies (McD ona 1 et al .,1958,
al.,
fi
197 9).
and Toks6z et
thematical model ing of seismic attenuation
requires including velocity di spersion in order to satisfy
causal it y.
The sam e models pos tulate a specific relationship
between velocity di spersion and
attenu ation.
Mor*e detailed discussions of the viscoelastic models
that wil 1 be presen ted in this study are found in Toks6z and
Johnston (1981), B1 and (1960), Aki and
Brennan and Smylie
Richards (1980) and
(1981).
The purpose of this study is to test
a compl imentary
method o)f measuring attenuation using the phase
velocity -attenuatio n relationsh ip.
Phase shift is measured
as a funIction of fr equency when the wave propagates a known
distance .
When pha se shift can
be measured accurately, this
method miay prove practical for measuring Q from seismic
refl ecti on data.
T he analysis also provides a test of the
dispersi on theory; if the mater ial
attenuat ion,
general,
has significant
dispersion is more accurately measurable.
sed imentary
I
rocks are more attenuating (lower Q)
than igneous
rocks and thus may provide a good medium for
testing this
theory.
Laboratory and field measurements of dispersion and its
5
relationship to attenuation in'body waves has also been
reported by Wuenschel(1965) and Ganley and Kanasewich(1980).
They have found that measured dispersion is in agreement with
viscoelastic models.
The form of phase velocity variation may also be used to
investigate the mechanism of attenuation.
Pore fluid
influence on the elastic propetries of sedimentary rocks and
semi -consol i dated sediments are of p rimary interest in
seismic explorati on. Frequency depe n dence of Q in these
materials may be modeled by the Bio t
effect (Biot,1956),
taking into accou nt the relative mot ion of the fluid and
solid components.
The Biot effect i s predicted to become
dominant at lower frequencies for media with low Q.
The data for this study consis t s of P waveforms reco rded
with ultrasonic 1aboratory apparatus, using Berea sandstone
under varying pressure and saturation conditions.
Vertical
seismic profiling data would also be compatible with a very
similar analysis.
Attenuation calculated using amplitude spectral
ratios
can be compared with that calculated using phase velocity, as
a check on the theory and the internal consistency of data
analysis.
II.
DEFINITIONS
Standard definitions of Q (seismic quality factor) are
usually given as
(2.1)
energy loss per deformation
cycle,
- AE
(2.1)
2 7E
where E is the total energy i-n a cycle or (2.2) in terms of
the phase lag between stress and strain,
=
where tan
tan
6 = Im(M(w) )/Re(M(w)),
friction coefficient
(2.2)
6
tan 6 is
the internal
and M(w) is a complex modulus function
relating stress and strain.
To show the equi valence of these definitions, and to
cast them in a form useful for data analysis, one first
expresses equation
than energy.
(2 .1) in terms of wave amplitude rather
Since for linear media,
E
ca
then:
AE a
and
I
2AAA
AA
=
(2.3)
TA
Now AA is the amplitude difference from one cycle to the
next;
it
can be expressed in differential
AA
=
x dA
dx
form as
(2.4)
where X is
the wavelength.
Substituting for wavelength in
terms of phase velocity, c,
1
=
dA
=
yields
X = 2rrc/w.
Hence,
-dA
2c
(2.5)
-w
dx
(2.6)
YT-T-
--
One dimensional wave propagation in the x-direction has
been assumed.
Integrating equation
A(x)
(2.6)
=
AO exp{-wx/2Qc}
=
AO exp{-ax} ,
and solving for A,
(2.7)
which defines the attenuation coefficient, a = w/2Qc.
The complex modulus M(w)
can be related to Q by
reference to the complex wavenumber K, where
K(w)
w
c-T
+
i
a(w)
.
(2.8)
From the one-dimensi onal wave equation,
pu
=
aa
ax
(2.9)
where p = density, u = di splacement in the x-direction, and
a = stress, substituting:
a(w)
=
M(w).E(w) =
M(w)au/ax.
(2.10)
Assume a solution of the form u a exp{i(Kx-wt)}:
p( 2
K 2 MM(W)
=
Then
M()
=
(7ETw)
-
(2.11)
.
+
i a(TTT
c 2 (w)p
(1 + i/2Q)
(2.12)
Taking the ratio of real and imaginary parts,
=
-Q/(Q2-.25)
-1/Q
(2.13)
Rem
Note that the approximation requires Q >> 1/2.
order of 10,
for example, a .25% error is
For Q on the
introduced,
but Q
is not considered measurable to such an accuracy in geologic
materials.
III.
ATTENUATION-DISPERSION THEORY
Attenuation
(Q) of seismi c waves has been reported by
many investigators to be const ant over a large frequency
range; i.e.,
a(w)
is
approxima tely linear with frequency.
Examples available in the geop hysical literature are
numerous; see, for example, Kn opoff(1964), Gardner et
al.,(1964), Gordon and Davis(1l968),
Mathematical
and Toksiz et al.(1979).
models of at tenuation that fit this common
observation require velocity d ispersion in order to produce
physically possible pulse prop agation; i.e.,
to satisfy
causality.
As shown by Stacey(1975), if a delta funct ion is the
input,
and it
travels a time T in a medium with both Q and c
(phase velocity) constant, the. att enuated spect rum is:
Ft(w)
(3.1)
= exp{-wT/2Q - iwT}
or, in the time domain,
ft(t)
=
I I
2r
1 .
Tr
Ft(w)exp{i t }dw
-
T/2Q
(3.2)
(T/2Q) 2 + (T-t) 2
In this equation it can be readily seen that ft(O)*O; that
is, the derivation impli es non-causality.
This
pulse peaks at t=T and i s symmetric about T;
attenuated
half the energy
has arrived before the assumed velocity predicts.
Observation predicts sharp rise times and longer fall
times.
Clearly the assumptions have violated physical reality
somewhere.
Models allowing velocity dispersion have been
formulated that do not encounter this contradiction.
Models that have been proposed are based on causality
criteria.
assuming a wave number of the form
In general,
given in equation
K(w)
(2.8),
'W
=
+ ict(w)
c w)
and requiring causality,
= 0
u(x,t)
implies that any
t < x/c(.)
for
attenuation-dispersion pair relationship
must satisfy:
c T
(Aki
=
and Richards,
transform,
+
T
1981)
H[a(w)]
where H[f(w)]
(3.3)
is
the Hilbert
defined by:
00
H[f(w)]
=
I
r -00
J')
f
(
-
dw'
(3.4)
W)
The Hilbert transform returns the function with a 7/2 phase
advance to all components.
A causality relation analagous to
equation
(3.3)
in circuit theory is known as the
Kramers-Kronig dis persion
relation.
From equation (3.3), therefore, and the definition of Q
(equation
(2.7)),
m
c--3
this requires
=
=
2Qa(w)
+
c(
H[a( ) ]
(3.5)
c ( 00)
One function set which satisfies (3.5)
Azimi
et al.(1968),
frequency,
assumes that a(w)
that was suggested by
is
almost
linear with
except at very high frequen cies:
a(w) =
(3.6a)
( alw<<l)
a
1 + al"
2
H[a(w)] =
0an
w
7(1-
aFT-a
In[1/alw]
.
(3.6b)
The corresponding dispersion has been found to agree with
many seismic observations (Brennan and Smylie,1 981),
and this
form of a(w) has become widely accepted in rece nt
seismological
literature.
The phase velocity is found from (3.5) and
c
C( )
or,
neglecting
I
c(w)
+
2a
_(7
In [1/alw]
(3.6):
(3.7)
c( )
(alw)
2
= 1
c ( co
+
2a
IT
In[1/alw]
(3.8)
Taking a ratio of phase velocities at two different
frequencies,
1 + 2acco)
c(wI)
In[W2/W1]
7r
1 +
1
Q
In[w2 /wl]
(3.9)
where terms of order greater than 1/Q are neglected.
A recent thorough discussion and comparison of many
attenuation models can be found in Brennan and Smylie(1981)
As they have pointed out,
.
although many models have been
derived from various mechanisms, most give basically
equivalent attenuation-dispersion
relations.
Models can be
considered either a)empirical; worked out to satisfy
causality and the known attenuation response of the medium,
or b) based on an assumed form of the medium's viscoelastic
behavior.
An approach to attenuation modeling based on viscoelasticity uses frequency dependent stress-strain functions
formulated in terms of creep-relaxation mechanisms.
The
creep function is defined as the strain response to
application of a constant stress.
Conversely, the relaxati
function is the stress response to a unit step of strain.
That is,
then
if
a(t)
= u(t)
E(t)
= e(t),
(3.10)
on
where:
u(t)
= 0,
t<0
1, t>0
= creep function
a(t
G(t
= stress
e(t
= strain.
S(t)
= u(t)
then
a(t)
= *(t),
where:
O(t)
= relaxation function.
Similarly, if
(3.11)
To relate th e creep-relaxation functions to the complex
modulus,
consider a time domain vers ion of equation (2.10);
a(t)
= m(t)*E(t).
(3.12)
Substituting from equation (3.11),
*(t)
or,
= m(t)*u(t)
=
=
fm(t-
m(t).
T)dT
(3.13)
(3.14)
dt
Then
M(w)
= {F(
(t)}-
(3.15)
where F denotes the Fouri er transform.
The inverse relation ship to equation
complex compliance functi on,
C(t)
(3.12) defines the
s(t):
= s(t)*a(t),
(3.16)
so that there is a corresponding relationship with creep:
d
(t)
=
(3.17)
s(t).
Once a form for the function M(w) is found, then,
analagous versions of the classical one-dimensional wave
equation are found by substituting the frequency dependent
elastic modulus in equation (2.10).
Here two proposed creep functions and their
corresponding dispersion relations will
be discussed.
The
Lomnitz(1956) logarithmic creep law is based on laboratory
data.
In the notation used above, the creep function
becomes:
*L(t)
= _
(-1 + q In(1+at)),
t>0
(3.18)
M
where:
M
= elastic modulus
a00 = initial
stress
a
= high frequency limit
q
= constant determined in laboratory
measurements.
Aki and Richards( 1980) have shown that this type of creep
function results in an approximate dispersion relation
equivalent to the Azimi(1968) relation noted earlier(3.9):
c(w )
1 + 1 ln(wl/w
2
)
where
Q = 2/wq,
Q>>1, and wl,
2
are different seismic
frequencies.
Another creep law has been proposed by Kjartansson
(1981),
formulated to result in frequency-independent Q.
This function is
basically a power law:
k(t)
Then:
where wo,
M(w)
t
IM+2
0
= MO(w
, t>O.
12
/ mO )ei7TY
W,
>O.
(3.19)
(3.20)
to are any reference frequency and time, and y is
the exponent to be fit to the power law. Note that the
argument of the modulus is independent of w.
From equation
(2.2) therefore,
=
The dispersion formula
tan(rry).
(3.21)
is found by substitution into
equations (2.8) and (2.11):
+ ic(w)
W
=
zC(Wy
Li3
= w(p/Mo)
Therefore,
c(w)
1/ 2
(/o0)
= (MO/p
= co(w/w
OY
0
(I
Y-
ei n
+
i/2Q)
/ 2
1 /2(cos(wy/2)
(3.22)
)-
1 (wlwoO)
= phase velocity
(3.23)
and
a(w)
(3.24)
= (w/c(w)) tan(ry/2)
= absorption coefficient.
One fundamental
difference between these two
formulations is that the Kjartansson relation assumes no
response; for high Q the response
instantaneous elastic
becomes
"almost" insta ntaneous.
Approximation s show the
relationship between t he two dispersion laws.
(3.23)
Expanding
in a Maclaurin seri es,
c(w) a (W/ oO)Y Y
(yln(w/w 0 )]2
where terms of order
neglected.
Since y ~ 1/7Q for
of a given observation will
and the Kjartansson
(1962) dispersion
**0
and higher have been
high Q and the frequency range
generally not be large, the Azimi
dispersion laws can be seen as
equivalent for high Q.
velocity
1 + yln(w/o 0 ) +
Similarly, expansion of Futterman's
law gives the same expression
as equation
for phase
(3.9).
Sedimentary rocks can be highly attenuating
(low Q).
Therefore in the work described here the dispersion
of equation
(3.23)
is
used.
relation
IV.
APPLICATION TO LABORATORY DATA
The initial
data set consists of P waveforms recorded
after propagation through laboratory rock
aluminum calibration sample.
generation,
as
samples or an
Assuming that the waveform
propagation, and recording system can be modeled
a linear system, output waveforms can be expressed by:
Oa(t)
= S(t)*Pa(t)*U(t-(ta+tO)
(4.1)
Or(t)
= S(t)*Pr(t)*U(t-(tr+tO)
(4.2)
where:
O(t) = output waveform (a-aluminum,r-rock)
*
convolution operator
P(t)
= effects introduced by propagation through the
medium
S(t)
= effects introduced by the system:
waveform, transducer coupling,
instrument effects.
includes source
recording
These are assumed to be the
same for the aluminum standard and rock
U(t-ti)
ta =
= 0, t
< t
1, t
> ti
samples.
i
travel time through the aluminum calibration
sampl e
tr
t
0
time through the rock
being tested
=
travel
=
travel time through the system components,
not
including the sample
(or standard) itself.
In the frequency domain, then,
Oa(w
)
= S(w)
e-i w(ta+t0)
(4.3)
Or(w
)
= S(w)
e-i w(tr+t0)
(4.4)
To get the desired express ion for the propagation effects
introduced by the rock med ium, rearrange equations
(4.3)
and
(4.4),
Pr(w)
=
Pa(w) Or(w) e-i
(tr-ta)
(4.5)
wOaT
Since aluminum has been observed to have Q values on the
order of 150,000 (Zamanek and Rudnick, 1961)
, it will
be
assumed to be nondispersive and nonabsorptive compared to
the rock.
This is equivalent to assuming that propagation
through the aluminum does not change any of the source
amplitude or phase information, i.e.,
(4.6)
Pa(f) = 1
Rewriting equation (4.5) using amplitude and phase
representations,
lapr(w)
e i pr(w) =
e-i(tr-ta)
Jaor(
Iiaoa(
ei or(m)
w
)
eI Poat-Y
where p subscripts refer to propagation effects and o
(4.7)
subscripts to observed outputs.
It is by equating the amplitudes in (4.7) that Q values are
calculated with amplitude spectra,
lapr(w)
= laor(w) I/aoa(w)l
(4.8)
Each amplitude can be described by a function of the form,
a(w)
= Am(x)exp{-am(w)x}
(4.9)
where x is the propagation distance and Am depends on the
geometry of the medium being tested.
Si nce the sample and
standard are of the same dimensions and are tested using the
same labo ratory equipment,
it
will be as sumed that the
function Am is not frequency'dependent.
Then am is
found by
taking the ratio of Fourier amplitudes:
=
a
Ar e-(ar-aa)x
(4.10)
ln
(4.11)
(aw7)
or,
In Iaoa
i
(Toksiz et al.,1979).
Ar
a
-
( ar-a)x
Since a=w/2Qc and Qa>>Qr,
aa can be
neglected and Qr can be found from the slope of a line
fitted to Inlaor(w)/aoa(w)
vs.
frequency.
By equating the phases in (4.7)
lpr(w) = 4or(w)
where n is
-
an integer.
oa(w)
- w(tr - ta)
Phase velocity is
(4.12)
+ 27n
defined by c(w)=w/
where k is the wavenumber, which is constant in this case.
is
related to the inverse of the time delay for each
frequency component, w = 2r/t, where t is the delay for the
individual
frequency.
given frequency.
Then,
pr(w)/w + tr = delay, for a
The phase velocity can then be calculated
from the measurable quantities in equation
C(w)
=
(4.12)
by:
xll
pr(w)+mtr
or()
From equation
(4.13)
-Foa(WT + ata
+
2nn
(4.13)
phase velocities are calculated using
the observed ouput phase spectra from rock samples and an
aluminum standard of the same length.
Since the phase could
equivalently be equal to the determined value plus or minus a
multiple of 2n there is a degree of nonuniqueness in the
phase spectrum.
In practice,
this has not been a problem
since the group velocity is quite well
known and only a
single value of n gives reasonable phase velocities.
Phase
unwrapping is
done using the algorithm developed by Tribolet
(1977).
Equations (4.11) and (4.13) then, are the basis for the
computational procedure.
The P and S waveforms are
digitized as 1024 point time series with a sampling rate of
100 nanoseconds.
The Fortran software used in this analysis was written by
Mark Willis
(Willis, 1982) for borehole seismolo gy
applications; it
required only minor modificatio ns to be
applied to ultrasonic data.
The waveforms are first windowed to include only the
first few cycles, thereby removing later arrival s which are
due to reflections within the medium and the sys tem.
point sine taper is applied to the edges of the
A 15
windowed
waveform.
To measure attenuation from amplitude spect ra, two time
series,
corresponding to the standard and sample waveforms,
are Fast Fourier Transformed
The two amplitude spectra are
plotted along with a plot of the log of their spectral
vs.
frequency.
The user int eractively defines a frequency
window, and Q is found from the slope of the best fit
squares line fit
slope is
is
over the wi ndow.
equal to ar;
found from a=w/2cQ.
somewhat
ratio
From equation
least
(4.11) the
since the velocity has been measured Q
The determined Q turns out to be
sensitive to the wi ndow chosen,
so
the results were standardized by using window cutoffs
defined by 25% of the maximum amplitude in the rock.
For phase velocity determination, the phase spectra of
the same pair of windowed waveforms is used to calculate
phase velocity, using equation (4.13) and the measured travel
time in the aluminum calibration.
Phase velocities are
plotted simultaneously for three different values of the
integer constant n.
The criteria used for choosing a
value of n were: minimum dispersion and velocity near the
velocity measured in the lab.
The appropriate dispersion curve is
calculate Q vs.
then used to
frequency based on the dispersion relation,
equation (3.23):
*1(tan- (1/Q))
c/co
=(0/00)'
(4.14)
Taking logarithms of both sides and solving for Q,
Q = cot[w(ln(c(w))-ln(c)/ln()/
Q(w)
)-ln(w0o))]
calculated from equation (4.15 ) is
(4.15)
plotted with the
dispersion curve and the rock amplitude spectrum,
removing th e
singularity at wO by averaging around it .
The choice of w0 i
a source of ambiguity; theoretically it is
any reference
frequency for which the phase velocity is known.
The energy
bandwidth of the amplitude spectrum is fairly narrow,
and any
s
c0 ,w0 pair must come from the data within this range.
It
might be preferable to have a reference velocity
corresponding to a known frequency outside the bandwidth. It
would therefore be useful to perform the same experiments with
varying transducer resonant frequencies.
However, the data
acquisition apparatus currently precludes this.
The calculation of equation (4.15)
dispersion is small.
can get unstable when
The reference frequency used in the
reported results was 1 MHz.
This is the resonant frequency of
the transducers and is above the frequency corresponding to
the maximum of the experimental amplitude spectra.
Since the
least variation in phase velocity occurs at high frequency,
seems reasonable to assume an wO in this range. Although the
choice of wO is presumably arbitrary, the same wO was used
consistently for all
results.
it
V.
RESULTS
The analysis described in the previous section has been
performed on waveforms recorded by Karl Coyner (Coyner,
1982) at MIT.
The rock was Berea sandstone, tested
under both oven dry and distilled water saturated
conditions.
Data recorded with differential pressures of
100, 250, and 450 bars were analyzed.
The resulting Q values calculated for P waves are
summarized in Table 1 and Figure 1.
For dry samples, Q
calculated from amplitude spectral ratios
monotonically with increasing pressure.
(Qpa) increases
Q calculated from
dispersion (Qpd) increases for the lower pressures but is
apparently low for the higher pressure.
This discrepancy
may be related to a difference in filter settings on the
recording electronics.
For the other five comparisons the high pass filter in
the recording system was set with the same band edge for
comparable aluminum and rock tests.
The data recorded for
dry Berea sandstone at 400 bars used a lower pass band edge
than was used for aluminum at the same pressure.
difference in filter settings is not large (70 vs.
compared to the signal bandwidth.
The
38 kHz)
The difference seems to
occur at a frequency that is much too low to affect the
amplitude spectra.
However, the filter has an unknown and
possibly significant effect on the phase.
For a more
conclusive comparison of the high pressure results the data
should be collected using
identical
filter
settings.
Preliminary analysis of S wave phase velocity data was
hampered by an apparently greater sensitivity to the filter
settings employed in the measurement.
No consistent results
have been obtained from limited S wave data because sample
and reference spectra at comparable pressure and filter
parameters
are not yet available.
For water saturated samples, Qpd and Qpa are in fairly
close agreement at the three pressures.
The Q values
increase with pressure but are significantly lower than
those for the dry samples, and appear to approach a constant
value with
pressure.
Plots of the amplitude and phase spectra,
amplitude spectral
ratios,
fits for
phase velocities, and dispersive
Q are shown for the six data sets
in Figures 2-28.
Figures 2-7 are plots of the fits of Q determined from
amplitude spectral
ratios, with pressure and saturation as
detailed in figure captions.
Data reference numbers
are at
the top of each figure and are listed in Table 1 for
reference.
The lower of the two graphs
in each of figures 2-7
plots the amplitude spectrum of the aluminum as a solid line
and the amplitude spectrum of the rock as
a dashed line.
The upper half shows the log of the ratio of rock to
aluminum amplitude spectra.
A straight line is fit over the
portion of this graph delineated by the vertical
calculated from the slope of the line,
x is sample length and c is
bars.
M, by M=x/2Qc,
velocity as measured
Q is
where
in the
laboratory.
Inspection of the plots of amplitude spectral
ratios
shows that the assumption of' constant Q is a reasonable one
over the frequency range sampled, as the log of the ratios
does not vary greatly from a straight line over the
frequency
range fit.
Phase velocities as calculated with equation
shown in Figures 8-13.
(4.13) are
Curves for three different values of
n are given, and the curve chosen
in Q calculations
is the
one showing the least dispersion over the sampled frequency
range.
Phase velocity goes to zero for this data at low
frequencies effectively
since no energy at these frequencies
has arrived during the sampled time.
Figures 14-19 illu
strate the results when Q is
calculated using the appropriate phase velocity data and a
reference frequency of 1MHz in equation (4.15).
The lower
half of each figure sho ws the chosen dispersion curve with
c(w)
in Kft/sec,
along with the rock amplitude spectrum.
The upper half plots Q from equation (4.15),
where rock amplitude is greater than
25%
in the region
of the maximum
amplitude.
When the phase velocity at any frequency is close to
the reference vel ocity,
extremely large.
the calculated Q values become
For most of the cases tested,
(Figures
16-19), there is a distinct region where the Q is apparently
constant.
For two cases (Figures
low pressure ther e is
14 and 15)
of dry rock at
no distinct region of cons tant Q, and
minima have been chosen as determined values.
error analysis ju stifies this choice,
No systematic
but trials with
varying reference frequencies indicate that a lo wer
reference frequen cy would result in a flat porti on of the
curve around this minimum.
The ambiguity of ref erence
frequencies does not allow a conclusive result for this
method.
Only when dispersion is expected to be significant
over the frequency range measured)
(>1%
can calculation of Q by
this method be expected to be very useful.
However, this
includes many of the crustal sedimentary rocks and sediments
being explored with various seismic techniques, such as VSP
and reflection seismology.
Figures 29-37 are the waveforms analysed.
Ideally,
both amplitude and phase operators can be developed
produce synthetic waveforms for comparison with the observed
pulse shapes, thereby using more of the information
contained in the waveform than amplitude or phase alone.
VI.
CONCLUSION
Both of the methods used in this paper to evaluate Q,
analysis of amplitude spectral ratios and of phase velocity
dispersion, give similar results, but the comparison is not
consistent for all data.
Viscoelastic theory predicts that
the two approaches are mathematically
certain assumptions,
equivalent under
so the choice of which is more useful
depends upon the application.
Where dispersion in the medium
under study predominates compared with other sources of
dispersion (for example, geometric eff ects in borehole
seismic applications), calculation of phase velocity may
reveal attenuation simply.
Comparison of the results presented here with those of
Johnston
(1978) shows that his Q values for the same
sandstone exhibit the same behavior with pressure and
saturation but are somewhat higher.
Q can vary within a
formation due to clay content, and the accuracy of
measurement decreases with increasing Q.
Since the various theoretical models predict forms of
dispersion-attenuation relations that are quite similar for
our purposes, there is no particular model that is favored
the results.
Investigation of attenuation over a wider
frequency range would be necessary to distinguish among
model s.
Models assuming constant and near constant Q have been
considered for practical
purposes, keeping in mind their
potential
pitfalls when dealing with real
measurement of Q in
earth is
a particular upper crustal
the objective,
consistent results.
earth data.
When
region of the
a constant Q approximation gives
When extrapolating Q over wide frequency
ranges or to great earth depths, the approximations break
down.
The assumption of constant Q has been made in this
analysis.
Attenuation varies significantly between different
rock types and from var ying rock saturation conditions.
Some apparent variation from a constant Q value is
but this does not limit
parameter.
observed
the usefulness of Q as a seismic
LIST OF SYMBOLS USED IN TEXT
A
=
amplitude
c
=
phase velocity
E
=
energy
F[ ] =
Fourier Transform
H[ ] =
Hilbert Transform
K(w)
=
complex wavenumber
M(w)
=
complex elastic modulus
Q
=
seismic quality factor
Qpa
=
P-wave Q calculated from amplitude spectral
ratios
Qpd
=
P-wave Q calculated from observed phase
velocity dispersion
t
=
time
u
=
displacement
x
=
sample length, or propagation direction
=
attenuation coefficient
=
exponent fit in proposed power law creep
function
=
strain
X
=
wavelength
p
=
density
a
=
stress
=
relaxation
S
=
creep function
W
=
angular frequency
y
in one dimension
function
ACKNOWLEDGMENTS
I would like to first thank my advisor,
Professor Naf i
Toksiz, for his continuing support, guidance and encourage
-
ment during my stay at MIT.
My work on this thesis was completed only with the
gracious assistance of Mark Willis and Karl Coyner.
to Mark for the use of his programs
Thank
S
and for his generosity
with time given to help and discussions.
Thanks to Karl
for
the use of ultrasonic data collected in his lab.
Others whose help is most appreciated are Darby Dyar,
for introducing me to yet another computer, and Steve
McDonald, for convincing two computers to talk to each oth er.
A special
word of thanks to Ron Marks for help in data
transmission, editing, and legwork, not to mention all
purpose cheerful moral support.
I'd like to also take this chance to thank all my
friends in the Earth and Planetary Science Department
at MIT-
staff, students, and faculty - for making the department a
friendly and supportive work environment.
During part of this work I was supported by a Chevron
Fellowship and as an employee of the Chevron Oil Field
Research Co.,
and I acknowledge their help.
REFERENCES
Aki,
K., and Richards, P.G., Quantitative Seismology: Theory
and Methods, W.H. Freeman and Co., San Francisco, 1980
Azimi, S.A., Kalinan, A.Y.,Kalinan. V.V., and P ivovarov,
B.L., Impulse and Transient Characteristics of media with
linear and quadra tic Absorbtion Laws, Izv. Acad. Sci
USSR Phys. Solid Earth, 1968, 2, p.88-93
Biot, M.A., Theory of Propagation of Elastic Waves in a
Fluid-Saturated Porous Solis. II. Higher Frequency Range,
Jour. Acoust. Soc. Am., 1956, 28, 179-191.
Bland, D.R., The Theory of Linear Viscoelasticity, Pergamon
Press, Oxford, 1960
Brennan, B.J. and Smylie, D.E., Linear Viscoelasticity and
Dispersion in Seismic Wave Propagation, Rev. Geophys. and
Space Phys., 1981, 19, 2,p.233-246
Coyner, K.B., Ph.D. Thesis, in preparation,
M.I.T., 1982
Futterman, W.I., Dispersive Body Waves, Jour. Geophys. Res.,
1962, 67, p.5279-5291
Ganley, D C., and Kanasewich, 'E.R., Measurement of Absorption
and D spersion from Check Shot Surveys, Jour. Geophys.
Res.,
1980, 85, 5219-5226.
Gardner, G.H.F., Wyllie, M.R.J., and Droschak, D.M., Effects
of Pressure and Fluid Saturation on the Attenuation of
elastic waves in sands, Jour.Petrol. Tech., 1964, 16,
p.189-198
Gordon, R.B., and Davis, L.A., Velocity and Attenuation of
Seismic Waves in Imperfectly Elastic Rock, Jour.Geophys.
Res., 1968, 7 3, p. 3917-3935
Johnston, D.H., The attenuation of Seismic Waves in Dry and
Saturated Rocks, Ph.D. Thesis, M.I.T., 1978
Kjartansson, E., Constant Q-Wave Propagation and Attenuation,
Jour. Geophy s. Res., 1979, 84, p. 4737-4748
Knopoff, L., Q, Rev. Geophys. Space Phys., 1964, 2,
p. 625-660
Lomnitz, C., Creep Measurements in Igneous Rocks,
1956, 64, p. 473
J.
Geol.,
McDonal, F.J., Angona, F.A., Mi lls,
R.L., Sengbush, R.L., Van
Nostrand, R.G., and White, J.E., Attenuation of Shear
and Compressional Waves in Pierre Shale, Geophysics,
1958, 23, p. 421-439
Stacey, F.D , Gladwin, M.T., McKavanagh , B., Linde, A.T., and
Hastie, L.M., Anela stic Damping of Acoust ic and Sei smi c
Pulses, Geophysical Surv eys, 1975, 2, p. 133-151
Toksiz, M.N., and Johnston, D .H.,
eds., Seismic Wave
Attenuation, Geophysics r eprint series, no.2, 1981
Toksiz, M.N., Johnston, D.H., and Timur, A., Attenuation of
Seismic Waves in Dry and Saturated Rocks: I.Laboratory
Measurements, Geophysics, 1979, 44, p. 681-690
Tribolet, J.M. A New Phase Unwrapping Algorithm
IEEE Trans.
on Acoustics, Speech and Signal Porcessing, 1977, ASSP25,
no.2, 170-177
White, J.E., Seismic Waves: Radiation, Transmission,
Attenuation, McGraw-Hil i, New York, 1965
Willis, M.E.,
Ph.D. Thesis, in preparation,
M.I.T.,
and
1982
Wuenschel , P.C., Dispersive Body Waves - An Experimental
Study, Geophysics, 1965 , 30, 539-551
Zamanek, J.Jr., and Rudnick, J., Attenuation and dispersion
of elastic waves in a cylindrical bar, Jour. Acoust.
Soc. Am., 1961, 33, p. 1283-1288
34
Table 1
DRY BEREA SANDSTONE
Data
pressure
(bars)
Vp (ft/sec)
Qpd
Qpa
100
22
25
11272
BR1118.2A
250
45
46
12810
BR1118.5A
400
35
74
13244
BR1118.7A
reference no.
WATER SATURATED BEREA SANDSTONE
Data
reference no.
*pressure
(bars)
Qpd
Qpa
100
15
16
12805
BR1122.5B
250
30
30
13330
BR1124.8B
400
27
32
13588
BR1210.10B
*
Vp (ft/sec)
differential pressure, fluid pressure in all
saturated cases is 100 bars.
FIGURE CAPTIONS
Figure 1.
Summary of res ults of Q calculations. Q calculated
from amplitude spectral ratios of P waves is referred to as
Q calculated from P-wave phase velocity dispersion is
Qpa.
referred to as Qpd.
Figures 2-7.
Berea Sandstone.
spectral ratios, showing Q fits
two ampl itude spectra, aluminum
line).
Amplitude spectra and
Lower half is
plot of the
olid line) and rock(dashed
hal f of each figure is a plot of the log of
Upper
ampl itu de rat ios.
graph used to fit
Verti cal bars delineate portion of upper
slope M.
V is velocity measured in lab.
Key to data reference number s at top of each figure is
given
in Tabl e 1.
Figure 2.
pressure 100 bars
dry rock
Figure 3.
pressure 250 bars
dry rock
Fi gure 4.
pressure 400 bars
dry rock
Fi gure 5.
pressure 100 bars
water satu rat
rock
Fi gure 6.
pressure 250 bars
water satu rat
rock
Fi gure 7.
pressure 400 bars
water satu rat
rock
Figures 8-13.
Phase velocity(Kft/sec) vs.
frequency(KHz).
Shown are phase velocities for three different values of the
constant n.
The curve with the least dispersi on in the
frequency range corresponding to the energy in the wave
chosen
from these to calculate Q.
Figure 8.
Figures
pressure 100 bars,
dry rock
gure
pressure 250
dry rock
gure
pressure 400
dry rock
gure
pressure 100
wat er saturated rock
gure
pressure 250
wat er saturated rock
gure
pressure 400,
wat er saturated rock
14-19.
frequency.
c(Kft/sec),
,
ampl itude, and Qd vs.
Lower half plots the dispersion curve used,
amplitude spectrum of rock.
from equation
(4.15)
Upper half plots Q calculated
for region ar ound amplitude maximum.
Figure 14. pressure 100 bars, dry rock
Figure 15.
pressure 250 bars, dry rock
Figure 16.
pressure 400 bars, dry rock
Figure 17.
pressure 100 bars, water saturated rock
Figure 18.
pressure 250 bars,
Figure 19.
pressure 400 bars, water saturated rock
Figures 20-28.
and
water saturated rock
Amplit ude and phase spectra of waveforms.
Phase is in radi ans.
Figure 20. pressu
100 bars,
aluminum
Figure 21.
pressu
250 bars,
aluminum
Figure 22. pressu
400 bars,
aluminum
Figure 23.
pressu
100 bars,
dry rock
Figure 24.
pressu
250 bars, dry rock
Figure 25.
pressure 400 bars, dry rock
Figure 26.
pressure 100 bars, water saturated rock
Figure 27.
pressure 250 bars, water saturated rock
Figure 28.
pressure 400 bars, water saturated rock
Figures 29-37.
Windowed waveforms.
Time in microseconds.
Figure 29.
pressure 100 bars, aluminum
Figure 30.
pressure 250 bars, aluminum
Figure 31.
pressure 400 bars, aluminum
Figure 32.
pressure 100 bars, dry rock
Figure 33.
pressure 250 bars, dry rock
Figure 34.
pressure 400 bars, dry rock
Figure 35.
pressure 100 bars,
Figure 36.
pressure 250 bars, water saturated rock
Figure 37.
pressure 400 bars, water saturated rock
water saturated rock
Figure 1
80
60
dry
sat
Qpd
40
*
Qpa
20
100
200
300
400
pressure (bars)
0
9
w
BR1118.2A P
L
10-
N
R
A
T
0
2
0
200
400
A
P
S
1--i
l
1
FREQUENCY, KHZ
Figure 2
BR1118.SB P
xis
5.0
0
200
400
600
800
1 00
800
1000
A
M
P
S 2.5
0.0
FREQUENCY, KHZ
Figure 3
1200
1400
1600
BR1118.7B P
L
-
.00
N
R
A
T
O
XTP
5.0
0
200
400
600
800
1000
0
200
400
600
800
1000
I
I
1200
1400
1600
1200
1400
1600
A
M
P
S 2.5
0.0
Figure 4
FREQUENCY, KHZ
Figure 4
9
9
W
BR1122.SB P
FREQUENCY, KHZ
Figure 5
SB
BR1124.8B P
L
*
1
.00
N
R
A
T
x II
0
400
200
600
800
1000
5.0-
A
M
P
S
1200
1400
QU
30.4
1600
Ma
0.129380E-05
V*
13330.000
2.5--
0.0-
-
0
200
Figure 6
"1
400
600
800
1000
1200
1400
1600
FREQUENCY, KHZ
0
0
BRi210.109
P
Do-
4.00
0
XiP
S.0
--
-
aloo
!-
I
4 d
"
bto t
-
800
IrI
1200ow
I1folo
1400
0.0
FREQUENCY, KHZ
Figure 7
0
9
10
00
BRiil8.2A P
0
200
400
600
800
1000
FREQUENCY, KHZ
Figure 8
1200
1400
1600
PR1118.SB P
0
200
400
600
800
FREQUENCY,
1000
1200
1400
160CI
KHZ
Figure 9
9
9
9
BRiiI8.7B P
10
1600
Figure 10
FREQUENCY, KHZ
BRI182.SB P
0
200
400
600
800
1000
1200
1600
1400
FREQUENCY, KHZ
Figure
9
9
BR1124.8B P
0
Figure
200
400
600
800
FREQUENCY
1000
KHZ
1200
1400
1600
BRI210.10B P
1600
FREQUENCY,
KHZ
Figure
0
9
BRl118.2A P
100
50
0
1600
20
FT
10
s
S
°
o
oo
L,
200
'
I
400
'
' .....
'
600
800
...
1000
FREQUENCY, KHZ
Figure 14
1200
1400
1600
BR1118.5B
P
100
50
0
20
0
200
400
600
800
1000
1200
1400
1600
0
200
400
600
800
1000
1200
1400
1600
FREGUENCY,
KHZ
Figure 15
9
9
BRi118.7B P
100
Q
D-
4.00
-
5SO
I
0
I
I
'
I
I .
I
I ,
0
200
400
6e
I
800
0
200
400
600
800
Figure 16
1
'
FREQUENCY,
I
1000
1000
KHZ
1.. _ !I _
'
I
1200
1200
'
I
1400
1600
1400
1600
BR1122.SB P
100
50
0
1600
20
a
T
10
-C
"
n
,--,.......
,-,,",
...
. ,**
I I
-
I/**-.
"....
...
*
ee a
***
****
0
0
200
400
600
800
1000
FREQUENCY, KHZ
Figure 17
0
0
1200
1400
--
1600
BR1124. 8B P
100-
DI
0-
20- o
0
I'
I
I
200
400
I
600
200
400
600
'
'
4.00
I
I
800
1000
800
1000
FREQUENCY, KHZ
Figure 18
'
I
'
I
1200
1400
1600
1200
1400
1600
BRi210.10B P
100
0
50
0
20
0
200
400
600
800
1000
1200
1400
1600
FREQUENCY, KHZ
Figure 19
S*
t
*
w
0
BR1ii18.A P
A40000M
P
0400
500ee
100eee
2000
1500
2500
-
P
-
H
A
S
E
200-
A
-1
S I I I II
500
Figure
20
I I I
I
I II
I
I
I I
1see500
1000
FREQUEHCY, KHZ
I
I
000
I'
2s500
BRl124.8B
P
A10000M
5000 -
0
0
400-
500
1000
1500
2000
2500
500
1000
1500
2000
2500
200 -
01
Figure 21
FREQUENCY, KHZ
BR1ii8.7B P
10000
A
P
400
0
500
2500
2000
1500
1000
200
0
-200
I
''
I
0
500
i
'
i
2000
1500
1000
FREQUENCY, KHZ
Figure 22
e
i
I
a
I
2500
BRII18.2A P
A10000
M
P
400
200
FREQUENCY, KHZ
Figure
23
PRi118.5B P
5000
M
2500
0
400
P
H
A
S
E
200.
0
FREQUENCY, KHZ
Figure
24
BR1118.7?
P
AA5000
M
P
2500
0
400
P
H
A
S
E
200
m
e
500
1000
1500
FREQUENCY, KHZ
Figure 25
2500
BR1122.SB P
A1 0 0 0 0
M
P
5000
400
D
4.00
-
0
1500
500
1000
500
1500
1000
FREQUENCY, KHZ
2000
2500
2000
2S00
P
H
A
S
E
200 --
0
Figure 26
BRI124.8B P
4000
2000
0
400
200
2500
FREQUENCY, KHZ
Figure
BRI210.OB P
D
A 4000
4.00
2000
0 -
II
0
500
1000
1500
2000
2500
0
500
1000
1500
2000
2500
400
H
S
E
200-
Figure 28
FREQUENCY, KHZ
4000
D-
2000 --
-4000
0
10
20
30
TIME (MICROSEC)
Figure 29
4.00
1500
1000
500
0
-500
-1000
0
40
10
Figure 30
TIME (MICROSEC)
60
1000
500-
co
0
-500-
-1eee
,
0
J
j
, ,- , ,,I "I,, ,I ,,I i , , ,I1,
10
20
30
TIME (MICROSEC)
Figure 31
40
50
,,,I
60
400
200--
0-
-200 --
-400 --
-600-
0
1
0
20
30
TIME (MICROSEC)
Figure 32
40
50
68
400
200-
0-A
-200 -
0
10
2T
3(
TIME (MICROSEC)
Figure 33
40
50
0R
600
400 --
200 --
0-
1
-4000
10
I
20
i
30
TIME (MICROSEC)
Figure 34
I
40
II
50
I
60
600
400
200
M
j
0
(\
V
-200 --
-400 --
-600
0
10
I
I
I
I
20
30
40
50
TIME (MICROSEC)
Figure 35
'
60
300
2000-- -
-100 - -
0
10
20
30
TIME (MICROSEC)
Figure 36
40
SO
60
400
200
0
-200
0
10
20
30
TIME (MICROSEC)
Figure 37
40
50
60