Geophys. J . Int. (1991) 106, 703-707
RESEARCH NOTE
Seismic-wave attenuation operators for arbitrary Q
Y. Fang and G. Muller
Institute of Meteorology and Geophysics, Feldbergstr. 47, 6OOO Frankfurt1M . , Germany
Accepted 1991 March 18. Received 1991 March 15; in original form 1990 September 5
SUMMARY
Under the minimum-phase condition attenuation operators are derived from
arbitrary intrinsic or scattering Q as a function of frequency, and a simple method is
described to calculate these operators with the aid of Fourier transformation. In the
case of scattering attenuation, the medium may contain both 1-D, 2-D or 3-D
heterogeneities. For illustration, the method is applied to a 1-D model of statistically
layered media with 12 per cent standard deviation of relative velocity fluctuation.
The theoretical scattering Q, deduced from single-scattering theory, is determined
for a numerical autocorrelation function and an exponential autocorrelation
function, as well as for a modified exponential autocorrelation function, which is
introduced here. The corresponding attenuation operators are calculated. The
theoretical Q and simulated seismograms, calculated with the attenuation operators,
agree very well with the exact results of synthetic-seismogram calculations.
Key words: attenuation operators, scattering Q, single-scattering theory.
INTRODUCTION
The attenuation of seismic waves is caused by anelasticity
and scattering, and called intrinsic and scattering attenuation, respectively. These are physically different mechanisms, but the phenomenological description by a quality
factor Q is the same. The construction of attenuation
operators from Q, which make it possible to incorporate
attenuation into the calculation of synthetic seismograms,
should also be the same. We consider the delayed
attenuation operator in the frequency domain with the form
where z is the propagation distance, Q(w) the frequencydependent intrinsic or scattering quality factor and c ( w ) the
frequency-dependent phase velocity which describes the
attenuation-related dispersion. c(m) is the high-frequency
limit of the phase velocity, i.e. the highest wave-propagation
velocity through the anelastic or scattering medium. The key
to equation (1) is to determine c ( w ) , or in other words, to
derive the dispersion law from Q(w). Usually authors
require that this law be determined from the causality
condition, as expressed by the Kramers-Kronig relations,
e.g. Weaver & Pao (1981) and Beltzer (1988), but actually
this i s not enough, since many dispersion laws are
compatible with causality. It seems to be indispensable to
use from the beginning the stricter minimum-phase
condition (under causality), which selects the dispersion law
with minimum deviation from the propagation in a medium
without anelasticity and/or scatterers. This minimumdeviation assumption is physically plausible, and in the case
of weak anelasticity it is consistent with pulse-shape
measurements in a variety of solid materials (Gladwin &
Stacey 1974). Purely mathematical arguments for minimum
phase (Aki & Richards 1980, pp. 173-175) are not
sufficient. In the case of scattering by a 1-D medium
(stratigraphic attenuation) the minimum-phase property of
the transmission filter has been proved (Sherwood & Trorey
1965). Therefore, the minimum-phase condition appears
reasonable for weak attenuation processes in general,
including scattering attenuation in 2-D or 3-D media.
Equation (1) can then be rewritten in the form
D ( 0 , Z ) = ID(&),z)l e''p(o'z)
(2)
with the magnitude
(3)
703
704
Y . Fang and G . Miiller
and the phase
obtain
44 = q(t ) + A t ) = W ) W )
(9)
with the unit step function H ( t ) . Finally, Fourier
transformation of equation (9) yields 3 ( w ) from which
follows the attenuation operator (5):
CT- 1.
D(w, T ) = e x p --s(w)
which simplifies to
(4)
because Q-' is an odd function of frequency.
Equation (4) implies that the phase of a minimum-phase
attenuation operator is related to the Hilbert transform of
Q-'(w). The complete minimum-phase attenuation operator (2) follows from (3) and (4):
where T = z / c ( m ) is the traveltime through the medium,
and the symbol X denotes Hilbert transformation. In
essence, the attenuation-operator form (5) has already been
used by Dubendorff & Menke (1986) and Gorich & Muller
(1987). The phase velocity c ( w ) , appearing in equation (I),
follows from comparison with equation (5).
In the next section we describe a simple method by which
the attenuation operator (5) can be calculated numerically
for arbitrary intrinsic or scattering Q ( w ) laws. Then the
method is illustrated in the case of stratigraphic attenuation,
i.e. the case of a 1-D scattering medium in continuation of
investigations by Gorich & Muller (1987).
NUMERICAL METHOD F O R ATTENUATION
OPERATORS
According to equation ( 5 ) the essential step in the
determination of the attenuation operator is the calculation
of the analytical signal of Q - ' ( w ) , i.e. of
Thus, equations (7), (9) and (10) form the three steps of the
calculation of the minimum-phase attenuation operator for
arbitrary intrinsic or scattering Q(w).
Eisner (1984) and Mitchel & Stokes (1986) have discussed
the principal incompatibility of bandlimitation, causality and
minimum-phase property of a signal and have pointed out
that sampling may destroy the minimum-phase property.
Both statements are correct, but for practical calculations ail
problems are avoided by adhering to the sampling theorem.
Minimum-phase signals need no special treatment, compared to other causal signals.
APPLICATION
Examples of attenuation operators for intrinsic Q are the
analytic forms, given by Muller (1983) for power-law
dependence of Q on frequency, including the constant-Q
case. Here we want to concentrate on attenuation operators
for scattering Q and (in this note) on the 1-D case, in which
the attenuation operators can be checked by exact
synthetic-seismogram calculations. In this case, scattering
(or stratigraphic) Q has the form (Sato 1982; Wenzel 1982;
Gorich & Miiller 1987)
where a0 is the mean P-wave velocity and R ( z ) the
autocorrelation function of the relative impedence fluctuation ~ ( z ) :
I
rD
a ( @ )= Q-'(w) + j x [ Q - ' ( ~ ) ] .
A simple method is to calculate
S ( w ) = j a ( w ) = q(w) + P ( w ) ,
(6)
where
q(w) =jQ-'(w),
P ( w ) = --X[Q-'(w)l,
are considered as Fourier transforms of time functions q ( t )
and p ( t ) . q ( t ) is calculated by inverse Fourier
transformation,
using standard FFT techniques which require knowledge of
Q - ' ( w ) from w = 0 to a sufficiently high Nyquist frequency;
q ( t ) is an odd function. From the properties of Hilbert
transformation
p ( t ) = q ( t ) sign t
(8)
follows; hence, p ( t ) is even. Then, from (6), (7) and (8) we
where D is the thickness of the stratified medium.
The autocorrelation function (12) for a particular
stratigraphy q (z) will be called numerical autocorrelation
function. An analytical alternative is the exponential
autocorrelation function
~ ( 2 =
) y2 e-lzl'a,
113)
where y2 is the variance of the relative impedance
fluctuation and a the correlation distance. A further
alternative is the modified exponential autocorrelation
function
R ( Z ) = y2 cos ( p n z / a ) e-lrl'a
(14)
with the additional parameter p. Equation (14) turns out to
be better suited than equation (13) to describe scattering
attenuation, since negative R ( z ) values are also possible.
Using in equation (11) the two-way traveltime t = 22/a0
instead of the space coordinate z , one obtains the
Attenuation operators
705
function, expressed as a function o f t ,
0.025
N=200
0.020
s= 1 2 %
0.015
0.010
0.005
-
0.000
We have calculated synthetic seismograms, both with
stratigraphic attenuation operators and with an exact matrix
method, for a stack of layers, embedded between two
homogeneous half-spaces with velocity cro = 4000 m sC1 and
density po = 2.39 g cmP3. The total thickness of the layers is
D = 1600 m, and the layer number N varies from 100 to
1600. Layer thickness, velocity and density in the layers
obey uniform distributions about mean values and were
determined with the aid of pseudo-random numbers. The
relative velocity fluctuation about the mean value a,)has the
standard deviation s = 12 per cent. The relative density
fluctuation about p o is assumed to be proportional to the
relative velocity fluctuation with a factor of 0.29. The
P-wave incident on the layer stack has frequencies from 0 to
300 Hz. Some more details of the layer model can be found
in Gorich & Miiller (1987).
Figure 1 shows the three autocorrelation functions of the
relative impedance fluctuation that we have used; the
-
-0.005
Lag
z (rn)
Figure 1. The numerical autocorrelation function (solid line), the
exponential autocorrelation function (long-dashed line) and the
modified exponential autocorrelation function ( p = 0.2, shortdashed line) for N = 200 layers and standard deviation s = 12 per
cent of the relative velocity fluctuations.
well-known relation
W
Q-'( W ) = - F( w ) ,
2
where T ( w ) is the Fourier transform of the autocorrelation
10.
.
.
.
N=600
.
1000
0
100
10.
.
-
.
N= 1000
.
+
10.
1000
I000
0 100
0 100
,
-
N-; I 6 0 0
.
'
'
II..;;.
Q.,
-_- -
__--
N=600
10
10
i0
10
60
110 160 210 260
Frequency
(Hz)
10
60
110 160 210 260
Frequency
(Hz)
10
60
110 160 210 i 80
Frequency
(Hz)
Figure 3. Scattering Q for the modified exponential autocorrelation function ( p = 0.2, short-dashed line) and for the exponential
autocorrelation function (long-dashed line), and the exact scattering Q (solid line), for s = 12 per cent, N = 200, 300 and 600.
Y . Fang and G. Muller
706
WAX
MAX
N
I .0
207. I
6oo
176.7
I 000
141.8
6oo
I
.o
A
1.0 .
number of layers is N=200. The correlation distance
a = 8 m agrees with the average layer thickness. Note that
the numerical autocorrelation function does not follow a
simple law, because of the way in which the stratigraphy was
constructed; it looks more like a realistic autocorrelation
function, e.g. of a sonic log. The modified exponential
autocorrelation function (14) for p = 0.2 approximates the
numerical autocorrelation function (12) considerably better
than the exponential autocorrelation function (13) for lags z
less than about 20 m, which is the most important part.
Figure 2 shows scattering Q as a function of frequency
from 10 to 260Hz for variable N. Exact Q was determined
from exact synthetic seismograms with the spectral ratio
method. Scattering Q, calculated from the numerical
autocorrehtion function by equations (15) and (161,
reproduces not only the general trend of the exact results,
but also many of their details. Figure 3 shows the same
300
200
1
.o
I
0
l
l
100 200 300 400
F-requency
623.6
100
-0.01
(H2)
0.01
0.03
T
(3)
irne
0.05
Figure 4. Attenuation operators (left: frequency domain, right:
time domain) for s = 12 per cent and variable N. The numerical
autocorrelation function was used.
(a)
Amplitude
0.340
0.333
0.289
0.274
0.218
0.213
0.189
0.201
0.343
0.350
0.619
0.562
N
1000
-u
600
-
100
1.000
5=12%
0.00
Amp I
I
0.02
0.04
0.06
0.08
'Time ( 3 1
0.10
0.?12
(b)
tude -
N
0.341
0.333
0.270
0.274
1000
0.207
0.213
600
0.220
0.201
300
0.341
0.350
0.562
0.562
1600
-
200
100
Input
1.000
5=12%
0.00
0:02
0.04
0.08
0.06
'I
ime
0.10
0.12
(3)
(C)
tude
N
0.308
0.333
1600
0.245
0.274
1000
0.204
0.213
Y 600
0.252
0.201
300
0.368
0.350
0.575
0.562
Amp
I
I
1
-
200
100
.ooo
InDut
S=12%
.Time
Figure 5. Comparison of exact synthetic seismograms (solid lines) and simulated seismograms, calculated with attenuation operators (dashed
lines), for variable N: (a) numerical autocorrelation function, (b) modified exponential autocorrelation function ( p = 0.2), (c) exponential
autocorrelation function.
Attenuation operators
comparison for scattering Q, as derived from the other two
autocorrelation functions. The modified exponential autocorrelation function provides a somewhat better fit than the
exponential autocorrelation function.
In Fig. 4 attenuation operators, calculated by equation
(10) for the numerical autocorrelation function, are shown.
Both in the frequency and the time domain the operators
have considerable fine structure, including a coda in the
time domain.
In Fig. 5 a comparison is presented of exact synthetic
seismograms and simulated seismograms which were
obtained by convolution of the input signal shown with
attenuation operators. The latter were calculated for the
numerical autocorrelation function (12), the modified
exponential autocorrelation function (14) and the exponential autocorrelation function (13), respectively. The
simulated seismograms fit the exact seismograms very well
to well, a fact which illustrates the usefulness of the
attenuation operator (10). The agreement is particularly
good for the numerical autocorrelation function (Fig. Sa),
in which case even parts of the coda of the main arrival are
reproduced; this is amazing, since stratigraphic Q is derived
solely from single-scattering theory and multiple scattering is
disregarded completely. As in Fig. 3, the modified
exponential autocorrelation function is somewhat superior
to the exponential autocorrelation function.
CONCLUSIONS
The numerical method described in this note is a convenient
way to calculate attenuation operators for arbitrary intrinsic
or scattering Q ( w ) and for 1-D, 2-D or 3-D media, provided
that Q-' is bandlimited, i.e. that a Nyquist frequency can
be chosen. In cases where this is not so this property has to
be enforced by tapering at high frequencies beyond the
seismic frequency band.
In the case of stratigraphic attenuation by a 1-D medium
the modified exponential autocorrelation function of the
relative impedance fluctuations gives an improved description, compared to the well-known exponential autocorrela-
707
tion function.
ACKNOWLEDGMENTS
This work was supported by a grant from the Deutsche
Forschungsgemeinschaft. We are grateful to Michael Korn
for discussions and comments on this note and to Ingrid
Hornchen for typing the manuscript.
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