Published in: J. Seismol., 2010, doi 10.1007/s10950-010-9196-5
Attenuation coefficients of Rayleigh and Lg waves
Igor B. Morozov
Department of Geological Sciences, University of Saskatchewan, Saskatoon, SK S7N
5E2, Canada; tel. 1-306-966-2761, Fax 1-306-966-8593
igor.morozov@usask.ca
Abstract
Analysis of the frequency dependence of the attenuation coefficient leads to significant changes in
interpretation of seismic attenuation data. Here, several published surface-wave attenuation studies
are revisited from a uniform viewpoint of the temporal attenuation coefficient, denoted by .
Theoretically, (f) is expected to be linear in frequency, with a generally non-zero intercept  =
(0) related to the variations of geometrical spreading, and slope d/df = /Qe caused by the
effective attenuation of the medium. This phenomenological model allows a simple classification
of (f) dependences as combinations of linear segments within several frequency bands. Such
linear patterns are indeed observed for Rayleigh waves at 500– 100-s and 100–10-s periods, and
also for Lg from ~2 s to ~1.5 Hz. The Lg (f) branch overlaps with similar linear branches of body,
Pn, and coda waves, which were described earlier and extend to ~100 Hz. For surface waves
shorter than ~100 s,  values recorded in areas of stable and active tectonics are separated by the
levels of D ≈ 0.2·10-3 s-1 (for Rayleigh waves) and 8·10-3 s-1 (for Lg). The recently recognised
discrepancy between the values of Q measured from long-period surface waves and normal-mode
oscillations could also be explained by a slight positive bias in the geometrical spreading of
surface waves. Similarly to the apparent , the corresponding linear variation with frequency is
inferred for the intrinsic attenuation coefficient, i, which combines the effects of geometrical
spreading and dissipation within the medium. Frequency-dependent rheological or scattering Q is
not required for explaining any of the attenuation observations considered in this study. The ofteninterpreted increase of Q with frequency may be apparent and caused by using the Q-based model
of attenuation and following preferred Q(f) dependences while ignoring the true (f) trends within
the individual frequency bands.
Key words:
Attenuation; Geometrical spreading; Lg; Mantle; Normal modes; Rayleigh waves
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Published in: J. Seismol., 2010, doi 10.1007/s10950-010-9196-5
1 Introduction
As they travel through the Earth, seismic waves attenuate due to three general factors:
geometrical spreading (GS), intrinsic (anelastic) energy dissipation, and scattering. At the
first glance, the first of these factors can be easily separated theoretically, as GS is usually
attributed to elastic energy spreading within expanding wavefronts. The distinction
between the other two factors also seems relatively straightforward theoretically, and it
roughly relates to considering either the visco-elastic or elastic wave mechanics in a
heterogeneous medium. However, in practical observations, these effects are not so easy
to separate, because they are always superimposed over each other and complicated by
experimental noise and limited knowledge of the Earth’s structure. In particular, the
definitions of GS and scattering-attenuation require the most attention.
Considering the concept of GS first, note that in realistic Earth models, its simple
wavefront-based model breaks down, and GS is actually not easy to define. Neither
wavefronts nor rays exist in realistic wavefields, in which refractions, reflections, and
mode conversions are abundant, and “multi-pathing” is pervasive. Simplified
approximations for GS commonly used in attenuation measurements often cause major
uncertainties in the results and misinterpretations of structural effects as “scattering”
(Morozov, 2009a,b). In global attenuation tomography studies, GS variations are often
described as “focusing” and modeled by using the ray theory in smoothly-varying media
(e.g., Romanowicz and Mitchell, 2007; Dalton et al., 2008). From the same studies, it is
also known that the total effects of focusing (such as the data variance reduction in
tomography) may exceed those of Q (Dalton and Ekstrőm, 2006). Moreover, if we also
wish to recognize the difference of the real GS from its theoretical approximations, GS
can apparently be only defined in a relative sense, in respect to the other two attenuation
factors.
Throughout this paper, I therefore use a general definition of GS as a “measure of
wavefield amplitude in the absence of true attenuation.” The basis of this definition is in
attributing the attenuation to the propagating medium and assuming that the attenuation
can be hypothetically “turned off,” corresponding to setting the attenuation parameter Q-1
= 0. The entire remaining effect of crustal and mantle structure on the attenuation
measurement is then attributed to GS. In a limited number of cases (such as uniform halfspace with a 1-layer lid) such GS can be modeled analytically, and for more complex
structures, it can be simulated by using the ray theory or numerically. On the other hand,
such GS can also be treated phenomenologically and directly measured from the data
(Morozov, 2008). This approach is taken here.
Scattering attenuation is the most difficult to isolate in the presence of GS and intrinsic
attenuation. It appears that scattering can only be considered in respect to some
“theoretical” background model, such as the uniform half-space used in most studies
(e.g., Wu, 1985; Sato and Fehler, 1998). By contrast, when approaching attenuation
measurements in realistic structures, scattering can hardly be identified unambiguously.
For example, in a perfectly-known structure, the entire wavefield is predictable and nonrandom, and therefore there is no room for scattering. If the deterministic structure is
limited to a certain scale-length, then the effects of smaller-scale random heterogeneities
could be described by random scattering; however, in this case, the empirical GS defined
above would still be sufficient to completely describe the intrinsic-Q-1 free wavefield. For
these reasons, I argued that the use of scattering attenuation can be abandoned in practical
observations (Morozov, 2008, 2009a), although it still represents a very interesting
subject for theoretical studies.
Surface waves are particularly important for studying the seismic attenuation. They
provide the most complete coverage of the Earth’s surface, particularly when used in
multiple-station and recent interferometery approaches. Due to their broad frequency
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bands, surface waves cover a great range of depths, yield the strongest constraints on the
upper-mantle structure, and provide the links between travelling and standing waves
within the Earth. Because of their position in the middle of the seismic spectrum, surfacewave data are also critical for testing the hypothesis of frequency-dependent attenuation
within the Earth. Importantly, long-period surface waves also have the most accurate
theoretical GS predictions.
In over half a century of studying the seismological Q, several theoretical models and
paradigms of the expected attenuation-frequency relationships were developed, and
unfortunately, these paradigms also influenced the very analysis of attenuation data.
Conventions for presenting the raw data became established and carefully followed. For
example, body-wave and coda results are typically presented in Q, Q-1, or lnQ vs. lnf
plots, from which the power-law Q(f) = Q0f dependence is usually interpreted (e.g., Aki,
1980). The same data are also often presented by using the “stacked spectral ratios,”
which are proportional to the temporal attenuation coefficient (f) = fQ-1 (Xie and Nuttli,
1988). As we will see below, this form could be the most useful; however, this quantity is
invariably plotted in log-log scales versus frequency, which again emphases the power
law above. In surface-wave studies, the spatial attenuation coefficient (f) (often denoted
in these studies) is typically shown versus period, T = f-1 (e.g., Mitchell, 1995). Normalmode and long-period surface waves data are conventionally presented as 1000Q-1 vs.
harmonic degree (e.g., Romanowicz and Mitchell, 2007). Such standard forms are useful
for comparing the results from different studies; however, in several cases, they also
complicate observations of attenuation dependences different from the power-law Q0f
In this paper, I employ one form of data presentation that appears natural and particularly
useful from both theoretical and practical viewpoints yet is almost completely overlooked
in the attenuation studies. This form is  itself in a linear frequency scale. The reasons for
underrating this form are unclear and may be historical; one potential explanation could
lie in (f) dependences often not supporting the expected frequency-dependent Q. The use
of (f) may cast doubts in the pervasive frequency dependence of the in situ
seismological Q, which we discuss later in this paper. Most attenuation data can usually
be fit similarly well in either the (f, ) or (lnf, ln) forms, and therefore the distinction
between these parameterizations lies not in comparing the data fitting errors but in the
underlying theoretical principles and interpretational values (Morozov, 2010).
Although consisting in a simple transformation, presentation of attenuation data in the (f,
) plane often leads to serious changes in the interpretation. The most significant result
from switching to this view is the recognition that (f) may be non-zero when
extrapolated to f  0, which is excluded by the power-law Q(f) model. For example, in
all cases of body-wave, Lg, Pn, and coda waves considered so far (Morozov, 2008, 2010,
and below), (f) shows linear dependences on f with non-zero intercepts  = (0) and
constant (within data uncertainties) slopes corresponding to the “effective attenuation”
Qe-1 = [(f) - ]/f. Morozov (2008) interpreted the intercepts  as related to the residual
GS and found them to be variable and correlated with crustal structures, including a clear
decrease of  with tectonic ages.
In this paper, I extend the analysis of Morozov (2008) to Rayleigh waves within ~10–500
s periods and 0.5–1.5-Hz Lg by using the attenuation-coefficient and Q data compiled
from Raoof and Nuttli (1984), Mitchell (1995), Durek and Ekstrőm (1996), Weeraratne et
al. (2007), and also taken from IGPP Reference Earth Model web pages
(http://igppweb.ucsd.edu/~gabi/rem.html). Lg waves can be associated with higher-mode
surface waves (Knopoff et al., 1973; Panza and Calcagnile, 1975), which allows using
them for extending the frequency band of the fundamental-mode Rayleigh waves. The
approach is strictly empirical and quantitative, and reduces to summarizing the observed
(f) dependences without using any underlying models beyond the linear expression
above. Analysis of this wide frequency band demonstrates an almost amazing
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commonality in the (f) patterns and even in the values of  and Qe. In addition,
recognition of the frequency-independent shift  in surface-wave data leads to a potential
explanation for the discrepancy between the surface-wave and normal-mode attenuation
measurements at 200–500-s periods (Durek and Ekström, 1997).
Perhaps the greatest general implication of the attenuation-coefficient approach is in its
revealing no indications of frequency-dependent crustal or mantle attenuation. Similarly
to the recent body-, Pn-, and coda-wave observations by Morozov (2008 and 2010), a
frequency-independent Qe is found to be sufficient for describing the observed surfaceand Lg wave attenuation at all frequencies. From the observed  and Qe, the
corresponding in-situ attenuation model also naturally becomes frequency-independent.
In Discussion, the new picture is used to explain how disregarding the (f) trends while
reconciling the Rayleigh-wave and Lg attenuation levels (Cong and Mitchell, 1988) may
lead to an apparent frequency dependence of Q. Finally, I also show why this apparent
Q(f) consistently increases, although at variable rates, across the broad frequency band of
this study.
2 Motivation for using (f)
There are several reasons why plotting (f) should be tried in most attenuation studies.
First,  is essentially the principal parameter directly obtained from GS- or sourcespectrum corrected amplitude measurements. The quality factor Q is derived from this
parameter by a frequency-dependent transformation Q(f) = f/(f), which distorts both its
values and error bounds. Second, although this subject deserves a special discussion
(Morozov, 2009 and in review I), note that Q does not actually represent a true property
of the propagating medium, but  is much closer to being such a property. To see this
inadequacy of Q, note that in its definition:
Q 1 
E
2Emax
(1)
where E the energy lost per one oscillation cycle, and Emax is the maximum elastic
energy in that cycle (e.g., Aki and Richards, 2002, p. 162), the “cycle” belongs to the
incident wave and does not characterize the propagating medium. By contrast, the
temporal attenuation coefficient
 f   
 ln E t , f 
,
2t
(2)
describes the relative rate of energy dissipation at any given point within the medium,
irrespectively of the wave process. To derive a Q-1 from this quantity, one has to divide it
by the frequency, which leads to the characteristic near-f-1 dependence of Q-1(f) that is
often observed (e.g., Aki, 1980).
General theoretical arguments also suggest that (f) should be tested for linearity in f first
(and not in lnf or f-1 as above), which I denote by (f) =  +  f . This can be seen from an
example of a linear oscillator, which is the simplest dissipative system known in
mechanics. The oscillator is described by its Lagrangian (e.g., Razavy, 2005)
L( x, x ) 
1 2 1
mx  m02 x 2 ,
2
2
(3)
where x is the displacement, x is the velocity, m is the mass, and 0 is the natural
frequency. Energy dissipation is described by the force of viscous friction, which is
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proportional to the velocity: f D   m0x , where  is a unitless dissipation constant
related to the oscillator’s Q as  = Q-1. In the Hamilton variational approach, this force
arises from the Rayleigh dissipation function
D 
m0 2
x .
2
(4)
Note that fD is proportional to particle velocity, and D is functionally similar to the kinetic
energy in Eq. (3). In a harmonic elastic wave, the time derivatives in Eq. (4) correspond
to multiplications with frequency, and consequently all dissipation processes have leading
linear dependences on the frequency. This is reflected in the presence of f in the
attenuated amplitude law: u  exp(-ft/Q), and consequently in (f)  f. Finally,
considering that a zero-order term in f should generally also be present in (f), we see that
linear dependences (f) =  +  f should be naturally expected in dissipative systems.
The existence of a non-zero above, referred to as the "geometrical attenuation" by
Morozov (2008), is the most important point of the proposed model. Observations of nonzero may have two key implications: 1) they indicate the failure of the conventional Q(f)
= Q0f model, and 2) they provide ways for measuring the variations of geometrical
spreading from the data. Geometrical attenuation may arise from near-surface reflections
(Morozov, 2008), small-scale reflectivity and ray curvatures (Morozov, in review II),
mode summations within the coda (Morozov et al., 2008), and most importantly for
surface waves, it is also associated with dispersion.
3 Effects of dispersion
Because of dispersion, wave packets spread out with propagation distance, causing the
amplitudes to decrease in addition to their reduction due to energy loss. Xie and Nuttli
(1988) included such pulse broadening in the geometrical spreading and proposed a
method for its estimation, which is in broad use today (e.g., Li et al., 2009). These authors
suggested that because of pulse broadening, energy density in a propagating wave reduces
by factor 1/U, where
U r  
r  1
1 


,
t0  Vmin Vmax 
(5)
r is the travel distance, Vmin and Vmax are the minimum and maximum group velocities
within the frequency band of interest, respectively. Parameter t0 here is a constant used
for normalizing this factor so that U(r0) = 1 at some reference distance r0. However, Eq.
(5) further simplifies to U(r) = r/r0 and actually does not depend on the velocity
dispersion parameters. This is not surprising, because the geometrical spreading, as well
as other wave-amplitude factors, is defined up to an arbitrary scaling which has to be
removed by normalization. The factor containing velocities in (5) is simply a part of such
scaling.
The reason for missing the dispersion effect in expression (5) is in assuming the wavepulse width to be zero at r = 0. To correct this problem, let us consider a pulse of finite
duration t0 at distance r0 and linearly expanding from this point. Equation (5) then
modifies to
U r   1
r  r0  1
1 


.
t0  Vmin Vmax 
(6)
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For weak dispersion, if r0 can be selected so that the second term in (6) is small within the
range of observation distances, then U(r) can also be rendered in the attenuationcoefficient form:
U r   e
2 ad  r  r0 
,
(7)
where the spatial attenuation coefficient due to dispersive pulse broadening is
d 
1  1
1


2t0  Vmin Vmax
  ln V

 2V t0
,
(8)
and lnV is the relative group velocity variation across the observation frequency band.
For example, for fundamental-mode Rayleigh waves at f < 0.3 Hz, (lnV)/V ≈ 0.07 s/km
(Fig. 1), and by taking t0 ≈ 500 s, we obtain d ≈ 0.7·10-4 km-1, which is comparable to
the observed values of  (Fig. 2a,b). Note that higher-frequency wavelets (with smaller
t0) broaden stronger; however, within the same wave, the effect of d is frequencyindependent. i.e. “geometrical.”
Dispersion also affects the relation between the spatial and temporal attenuation
coefficients. Without taking pulse broadening into account,  = V, where V is the group
velocity (Aki and Richards, 2002, p. 293). This relation was obtained by equating the
harmonic-wave amplitude at the dominant frequency, exp(-t) with the amplitude of the
wavelet maximum, exp(-r), which is located at travel distance r = Vt. However, because
of pulse broadening, the relation between these amplitudes should be
1
exp    t   exp   r  ,
U
(9)
ln U
 V    d  .
2t
(10)
leading to
  V 
This formula also shows that in the presence of dispersion,  should always exceed d.
The limit of  = d corresponds to a wave attenuating by pure pulse broadening and
without energy loss.
4Observations
Figs. 2a and b show the measured Rayleigh-wave attenuation coefficients at 10-100-s
periods in several tectonically-active and stable areas around the world, given in the
traditional form, as functions of wave periods (Mitchell, 1995; Weeraratne, 2007).
Generally, the values of  are within 0–1·10-3 s-1, comparatively high in tectonic and low
in oceanic areas, and also quickly increasing below ~20-s periods (Mitchell, 1995; Figs.
2a and b).
Note that the trends for (T) increasing toward shorter periods T=1/f appear hyperbolic,
which suggests that the dependences on f might actually look simpler. This becomes clear
when the same data are plotted against frequency (Figs. 2c and d). In these plots, (f) was
transformed into the temporal attenuation coefficient, (f), by using Eq. (10), with
dispersion parameters taken from the simplified, linear fundamental-mode group-velocity
trend in Figs. 1c, d, and d = 0.4·10-4 km-1. This d value was selected empirically, close
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to the low bound on the observed , so that the resulting intercepts of (f) in Fig. 1d
became near-zero. Note that the dispersion curves (Figs. 1c, d) show significant variations
at frequencies below ~0.03–0.05 Hz, which may contribute to the “spectral scalloping” of
(f) amplitudes observed in the data.
In the (f) form, the separation between the tectonic, stable, and oceanic areas becomes
clearer, as well as the differences between the several study areas. Several separate linear
trends of (f) can be recognized, as indicated by the dashed lines in Figs. 2c,d. Notably,
when extended to the frequency axis, most of these lines cross it at positive intercept
values, particularly if the correction for d is not performed. Similar positive  values
were found in recent body-wave and coda observations (Morozov, 2008).
Noting that when projected to zero frequencies, the observed (f) trends are non-zero, we
replace the conventional attenuation parameters Q0 and  with another two,  and Qe,
defined by
( f )   
f
Qe
,
(11)
where Qe can be frequency-dependent. However, with data fluctuations and measurement
noise, there seems to be no reason to look for a frequency-dependent Qe (Figs. 2c and d).
From the interpreted linear trends in (f) (Figs. 2c and d) several important observations
can be made:
1) Although lower values of Qe ≈ 230 are present in the data from regions of
active-tectonics compared to the stable areas, these ranges of Qe overlap
almost completely. Therefore, although Qe may generally increase with
tectonic age (Morozov, 2008), it does not significantly discriminate
between the stable and active tectonic types.
2) Nevertheless, the intercept value of D ≈ 2·10-4 s-1 (compare the yellow
bars in Figs. 2c and d) separates most of the tectonic and active areas. If
the dispersion correction is not applied, this threshold should be
measured relative to the dashed black line in Figs. 2c and d, and equals D
≈ 3.2·10-4 s-1. A similar relationship was found for crustal body and coda
waves, for which the stable and active areas were separated by the level
of D ≈ 0.8·10-2 s-1 (Morozov, 2008).
3) In relation to this D discriminant, the oceanic-area data from Canas and
Mitchell (1978) (solid lines in Fig. 2c) generally align with the
continental stable-tectonic group (Fig. 2d), although recent data by
Weeraratne et al. (2007) (green triangles in Fig. 2c) are close to the edge
of the active-tectonic group. Values of  for oceanic recordings also
appear to increase with age (Fig. 2c), which is an opposite trend
compared to the continental lithosphere. In addition, the oceanic data
show consistently higher Qe.
Note that the above observations are entirely empirical and independent of the traditional
geometrical-spreading and Q(f) = Q0fh assumptions. However, they reveal several
important relationships in the data that have not been noticed in the original (T) and Q(f)
interpretations (Mitchell, 1995). This shows that raw data representation and
classification is very important in the analysis of attenuation.
In the above analysis (Fig. 2c,d), we did not attempt rigorous estimations of statistical
parameter errors and confidence intervals. Unfortunately, the published data do not allow
a complete error analysis in the spirit of the proposed approach. The individual
measurement errors are significant (error bars in Figs. 2c,d); however, the amplitude
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deviations from the interpreted linear trends are non-random and should be mostly related
to wave-mode interferences within the specific structures, known as “tuning” in reflection
seismology. A proper inversion for (f) would require revisiting the full raw-amplitude
datasets, which are not available to us at present. At the same time, in this paper, we only
focus only on the fact of distinct linear (f) dependences and their characteristic
parameters, and consequently can rely on interpretive “visual” analysis and line-fitting. It
is quite clear from Figs. 2c,d that: 1) multiple (f) trends exist and 2) these trends may be
considered as linear at best.
From the arguments above, “turning off” the attenuation in the interpreted linear (f)
trends would correspond to setting Qe-1, or alternatively, f equal to zero. This suggests an
interpretation of  as a measure of the residual GS or dispersion remaining in the surfacewave amplitudes after their correction (Morozov, 2008). For example, for the cut-off
value of  ≈ 2·10-4 s-1, this residual GS correction amounts in only t ≈ 8% for a 400-s
Rayleigh wave propagation time. This relative level of this residual GS is similar to that
estimated for body waves (Morozov, 2008).
Notably, all values of  are above the minimal level (–adV|f=0) (dashed line in Figs. 2c and
d), showing that Rayleigh waves are systematically “under-corrected” by the theoretical
GS correction. This is again similar to the observations of lithospheric body and coda
waves (Morozov, 2008). The systematic character of this GS term shows that it is caused
not only by focusing and defocusing on lateral variations of velocity (Dalton and
Ekstrőm, 2006), but also generally deviates from the theoretical (sin)1/2 dependence
(Nuttli, 1973). The variability is also significant between different regions, and
particularly within the tectonically-active lithosphere (Fig. 2c).
The relative significance of the residual GS in attenuation measurements can be
characterized by the “cross-over” frequency fc = ||Qe/ (Morozov, 2008). Below this
frequency, the effects of the residual GS exceed those of attenuation. For the
characteristic values of  = 4·10-4 s-1 and Qe = 500, we have fc ≈ 0.05 Hz, with some
variations for the different regions. This frequency corresponds to the ~20-s period below
which the apparent attenuation factor (T) starts quickly increasing (Fig. 2a, b; Mitchell,
1995).
Let us now consider the ~100 to ~300– 400-s Rayleigh waves. Interestingly, the globalaverage Q-1curve in this range is also hyperbolic (Fig. 3a), and the corresponding (f)
again shows a well-defined linear dependence (Fig. 3b). By contrast to the shorter-wave
case, its Qe ≈ 84 is significantly lower, and the intercept  ≈ -8·10-6 s-1 is negative,
showing that these waves are “over-corrected” by the background GS correction. Taking
the same characteristic travel time (400 s), the relative amount of this over-correction is
only ||t ≈ 3%. This small value is not surprising, because at such wavelengths, the
spherical-Earth model used in accounting for the GS effects is quite accurate. The crossover frequency for the long-period band equals fc ≈ 2 mHz. Although the interpretation of
this quantity is not as straightforward as in the case of under-corrected GS, note that this
frequency is close to the transition from the surface-wave to normal-mode regime (Fig.
3b).
As shown in the following sections, Qe still represents an apparent quantity characterizing
the observations on the surface. For relatively short waves localized within comparatively
uniform layers, Qe should be somewhat greater than the lowest intrinsic Qi sampled by
the corresponding wave. However, it appears that for long surface waves, the above Qe ≈
84 may actually be below the lowest Qi within the upper mantle and represent the
redistribution of wave energy density with changing frequency.
Below ~2.0–2.5 mHz, the attenuation coefficient flattens out with the transition into the
low-order fundamental spheroidal modes. This is the only studied frequency range in
which the behaviour of (f) strongly deviates from piecewise-linear. Such change could
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be related to the characteristic wavelengths reaching the thickness of the entire upper
mantle, and consequently the transition from predominantly traveling to standing waves.
5 Summary of (f) observations
Combining the above observations with short-period results by Morozov (2008, 2010),
we can summarize the available attenuation coefficient data as consisting of three nearlylinear branches of (f) within 100– 400 s, 10–100 s, and ~0.5–100 Hz period/frequency
bands (Fig. 4). A broad gap from ~1–2 to ~10 s still remains, in which no attenuation
measurements are available. The difficulty of measurements and interpretation in this
frequency band are well known and caused by the complexity of the lithospheric structure
and by the complex character of the wavefield changing from predominantly surface- to
body-wave type across this band.
Whereas the first of these (f) branches (100–400-s) appears to be well-defined and
“global” in character, the two higher-frequency branches are sensitive to the tectonic
types and geologic structures (Fig. 4). Both  and Qe values vary regionally for both
Rayleigh and body waves, with  being systematically lower within stable regions. In
oceanic areas,  has lower, continental-type values, and Qe is high (~1000; Fig. 2c). These
observations generally agree with the recent numerical modeling by Morozov et al.
(2008), who found that  is principally controlled by the upper-crustal structure, which is
more heterogeneous in active continental environments (Christensen and Mooney, 1995;
Mitchell, 1995) and virtually absent in the oceanic crust. Note that the transition between
the two Rayleigh-wave branches occurs nearly continuously at (f) ≈ 3·10-4 s-1 at f = 0.01
Hz (Figs. 2c,d, and 3b), whereas (f) jumps upward by over ~(3– 6)·10-3 s-1 when a
change to crustal modes occurs (Fig. 4). This once again suggests that the upper crust
should be the cause of the increased  values.
Assuming that the attenuation-coefficient data can be summarized by a collection of
piecewise-linear (f) branches (Fig. 4), it appears that such empirical (f) practically
excludes the need for a frequency-dependent Q within the mantle. Originally, the
dependence of the attenuation coefficient on the period (Figs. 2a,b) was viewed as the
primary indication of the frequency dependent Q (Mitchell, 1995). However, in the
present interpretation, this argument is reversed, and the attenuation-coefficient
observations only indicate spatially-variable but frequency-independent  and Qe (Figs.
2c,d).
6 Intrinsic attenuation coefficient
The observed near-linear (f) trends can be explained by using a very general model with
a frequency-independent in situ attenuation. Consider the expression for the observed
path factor P(t, f),
Pt , f   G0 (t , f )P(t , f ) ,
(12)
where G0(t,f) is the theoretical GS factor (for example, -(sin)-1/2 with additional
dispersion or focusing factors for surface waves), and P(t,f) includes the remaining
effects of the imperfectly-predicted GS and attenuation. The source and site effects are
assumed to have been removed from P(t, f). Conventionally, the entire P(t,f) is attributed
to Q(f) along the wave path (e.g., Der and Lees, 1985):
P(t , f )  eft
*
,
(13)
where
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t* 
 Q  f d ,
1
(14)
path
and  is the time within the travel path, measured from 0 to t. The exponential form of
path correction (13) reflects the fact that P(t,f) = 1 when t = 0, with ln[P(t,f)] increasing
with time approximately linearly. This is the typical approximation used in the
perturbation (such as weak scattering) theory. However, note that with inaccurate G0(t,f),
the exponent in path correction (13) is not guaranteed to be proportional to f, and
therefore we need to generalize this equation to
 P(t , f )  e  t ,
(15)
where the path-average attenuation coefficient in Eq. 11 is

1
 i d ,
t path
(16)
and i is the “intrinsic” differential attenuation coefficient. Thus, in its use and meaning,
(f) is quite similar to the conventional fQ-1(f), in the sense that it can be averaged over
the wave paths to predict corrections to the logarithms of seismic amplitudes. Its only yet
critical difference is the recognition of (f) being generally non-zero at f  0.
For surface waves, the meanings of the “wave path” integrals in eqs. (14) and (16) are of
course heuristic, because such waves do not follow any particular paths between the
source and receiver. In such cases, these integrals can be rigorously represented by
Feynman path integrals, summations over all normal modes of the field, or by the full
treatment of the perturbation-theory problem. For example, for layered elastic media with
weak lateral variations, Woodhouse (1974) and Babich et al (1976) developed
perturbation theories in which such effective “rays” were rigorously defined. However,
regardless of their symbolic forms, expressions (14) and (16) correctly illustrate the
essential conclusion that is most important for us. This conclusion is that the observed
quantity  =  + f/Qe represents a weighed average of the corresponding intrinsic
quantities of the medium, i = i + fQi-1, with weights (known as Fréchet kernels in the
normal-mode attenuation theory) determined by the wave-amplitude distribution.
7 Frequency-independent attenuation within the Earth?
From Eqs. 15 and 16, it is apparent that  should generally include both frequencyindependent and dependent parts, and consequently a linear dependence  on f should be
expected as the first-order possibility. The same argument applies to the intrinsic
attenuation coefficient i. Because a constant Qe appears to be the case in all
seismological data we considered in this paper and elsewhere (Morozov, 2008, 2009a,
2010, in review III, and unpublished), it is therefore natural to consider a medium with
frequency-independent intrinsic Qi first. In such a medium, Eq. (16) predicts a frequencyindependent Qe
Qe1 
1
1
 Qi d ,
t path
(17)
and a similar equation relating i to . Note that Qe-1 is therefore dominated by the zone of
highest attenuation along the wave path (Morozov, 2009a), which corresponds to the
asthenosphere in the case of most mantle waves. By using the standard inversion
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techniques (e.g., mode summations or tomography), (f) measured on the surface can be
inverted for the in-situ differential i.
Equation (17) is valid when the integration “paths” do not significantly change with
frequency or at least stay within the zones of similar levels of Qi-1. This should be the
case for crustal body waves and shorter-period surface waves (Fig. 2c), but for longperiod surface waves and normal modes (Fig. 3), wave energy is progressively removed
from the attenuative upper mantle when frequency decreases. This causes the apparent
(f) to also reflect the shapes of the Fréchet kernels in depth, and the resulting very low
Qe ≈ 84 (Fig. 3b) – to overestimate the Qi in the upper mantle. In detail, these effects are
studied elsewhere (Morozov, in review I).
Note that in the presence of significant structural contrasts, the GS is frequencydependent (e.g., Yang et al., 2007), and therefore the correction for it included in
ln[P(t,f)] may be frequency-dependent as well. In such cases, separating the effects of i
and Qi in the frequency-dependent part of i(f) becomes ambiguous. This ambiguity
stems from the general uncertainty of the concept of medium Q and can hardly be
resolved unequivocally. The interpretation used above and in Morozov (2008) assumed
that the residual GS is frequency independent, which appeared reasonable in the most
common cases of frequency-independent background G0(t,f). However, by treating the
entire attenuation of the medium as a single i(f) quantity, this ambiguity can be avoided.
Note that the trade-off between the frequency dependence and depth layering of Q (e.g.,
Mitchell, 1991) can still be utilized to introduce a frequency-dependent Q in the Earth
models. Frequency dependence would increase the number of model variables and
therefore allow fitting the data even better. Frequency-dependence of Q may also be
sufficiently small to be unnoticeable within the individual frequency bands, but switching
between the branches (i.e., between significantly different wave modes and penetration
depths) may require different Q values. However, also considering the existing successful
frequency-independent global 3D Q models (e.g., Dalton et al., 2008), it still appears
unlikely that frequency-dependent material Q should be necessary for fitting
seismological data.
Finally, in this paper, I prefer staying within strictly seismological, quantitative, and
empirical arguments. Evidence from laboratory studies (e.g., Faul et al., 2004;
Romanowicz and Mitchell, 2007) often serves as the principal motivation for looking for
a frequency-dependent Q within the mantle (e.g., Lekić et al., 2009). However,
correlation of Q values arising from such different types of observations may be thwarted
with difficulties of reconciling the assumptions and models used, extrapolating the results
to mantle conditions, and even with the differences in the types of quantities measured.
Bourbié et al. (1987) summarized a number of Q-measurement types and noted that
although most of them can be described by visco-elastic models, there is little agreement
between the resulting values of Q. It can also be shown that “geometric” factors similar to
those discussed above could be found in lab measurements, yet this would take us far
from the subject of the present study.
8 Crustal model
As shown above, in all cases where a frequency-dependent Qint(f) is interpreted, an
alternate quantity , which is the intrinsic attenuation coefficient, i = f/Qint(f) = i + fQi1
, can be used to describe the attenuative property of the medium. In this expression, Qi
shall be first tried as frequency-independent, and frequency dependence further
considered if required by the data.
To illustrate how the traditional attenuation models look in the (i, Qi) form, Fig. 5 shows
models for tectonically-extended crust in the Basin and Range province (BR; Mitchell
and Xie, 1994) and for the stable eastern United States (EUS; Cong and Mitchell, 1988).
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Both Q models are frequency dependent; however, the power-law Q(f) = Q0f
dependences in them were derived differently. In the BR model, the upper 15 km of the
crust was taken as frequency-independent, and  was set equal 0.5 everywhere else in the
two models.
By approximating the local Q(f) values by depth-distributions of i and Qi, the models
become somewhat easier to compare (Fig. 6). Qi values in the EUS model are high (over
1000) even in the upper crust, and their variation with depth is weak. By contrast, the BR
model shows an over 10-time stronger attenuation within the upper crust (Qi ≈ 100),
which quickly drops to ~1000 within the lower crust. Values of i were forced to equal 0
in the upper crust of the BR model, which was probably not a very good approximation,
particularly in its extensional tectonic setting. Apart from this contradiction, the i curves
are in agreement with the differentiation proposed above (dashed line in Fig. 6b): i is
significantly higher than D ≈ 3.2·10-4 s-1 within the tectonically-active zone (BR), and for
stable crust (EUS), values of  are at near or below this level. Note that we use the value
of D not corrected for dispersion, because the model by Mitchell and Xie (1994) also
included no such corrections.
The above comparison was based on an ad hoc transformation of the crustal models built
within the Q(f) = Q0f paradigm. This transformation results in approximately the same
Qint(f) values within the crust, and consequently these models should reproduce the
attenuation-data fit used by Cong and Mitchell (1988). In view of this modeling, an
interesting question arises: how can we interpret the  values in Fig. 6b, considering that
the forward-modeling approach used by Cong and Mitchell (1988) did not include
variable GS? The answer is that deviations of the actual structure from their layered
crustal models (i.e., ) can be interpreted as the “scattering attenuation,” Qs-1. Combined
with the intrinsic attenuation Qi-1, it produces the resulting apparent Qint-1(f):
1
f 
Qint

 Qi1  Qs1  Qi1 .
f
(18)
Therefore, Qs = f/  f, which is typical for scattering attenuation (Dainty, 1981; Padhy,
2005; Morozov, 2008).
Thus, when limited-accuracy modeling is used,  can be inverted from the apparent
intrinsic Qint-1(f) results and interpreted as caused by elastic scattering. Note that even
with such interpretation, the upper-crust of the BR model with Qs-1 = 0 (i.e., nonscattering) but very high Qi-1 (Fig. 6) appears contradictory. In a full and accurate
modeling, one would need to start from Qi-1 = 0 and adjust the structure until a correct  is
achieved. After this, the need for Qs would disappear, and Qi-1could be inverted for from
the value of Qe. Such modeling and inversion needs to be addressed from the raw (f)
data and is beyond the scope of this paper.
9 Discrepancy between normal-mode and surface-wave Q-1
The phenomenological argument above also suggests a potential explanation of the
discrepancy among the measurements of traveling and standing-wave attenuation noted
by Durek and Ekström (1997). The discrepancy consists in systematic, ~15% differences
in the attenuation levels measured by the surface-wave compared to the normal-mode
techniques (Fig. 3a). In the (f) form, this difference amounts in a near-constant, ~ 10-5 s-1
upward shift of the surface-wave  (Fig. 3b). Note that the amount of this shift is close to
the surface-wave  ≈ -8·10-6 s-1 and represents only ~3% of the measured crustal GS effect
for surface waves at 100–10-s periods (≈ 3.2·10-4 s-1 before the correction for dispersion;
Fig. 4). Thus, such shift could be expected from a slightly inaccurate surface-wave GS
correction.
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As Durek and Ekström (1997), Masters and Laske (1997), and Roult and Clévédé (2000)
argued, long-period surface-wave measurements may be affected by noise and difficulties
in defining the time windows for separating the fundamental modes from the various
overlapping wave trains. They concluded that normal-mode estimates can generally be
carried out more accurately and may be more reliable. At the same time, normal-mode 
estimates also tend to be decreased by noise, particularly in the presence of heterogeneity,
which may account for a part of their gap with surface-wave estimates (Romanowicz and
Mitchell, 2007). Although the origin of this discrepancy has still not been established, our
empirical observations above show that long-period surface-wave measurements allow a
lot of room for adjustments by recognizing their GS component. Note that according to
Eq. (16), the GS factor is effectively accumulated along the paths from the source to
receiver. Therefore, for example, predominance of continental surface-wave recordings
(which are mostly conducted in tectonically-active areas with higher i) from deep-focus
earthquakes (also likely with higher i with respect to the overlaying layered mantle and
crust) could cause increased  values when globally-averaged for the corresponding wave
modes in Fig. 3b. By contrast, normal-mode measurements are dominated by the oceanic
areas with presumably lower i.
10 Discussion
This paper focuses on classification of attenuation measurements irrespectively of the
models for attenuation or Q. When practically raw values of  are presented in a linear
frequency scale, they reveal piecewise-linear frequency dependences and suggest many
modifications of the existing interpretations. These observations also show new directions
for research, some of which were outlined above. For example, the causes of GS underand over-compensation of Rayleigh waves within the shorter- and longer-period bands,
respectively, need to be established by detailed analysis and modeling. The amount of
potential bias in global-average  for long periods needs to be evaluated and compared to
the discrepancy with the normal-mode estimates. Modeling and inversion techniques for
i(f) need to be developed, and potentially many datasets revisited at the raw-data level.
Values of Qe, and consequently of Qi, are often strongly increased (up to ~20–30 times,
Morozov, 2008) compared to Q0, and Qs is removed, leading to dramatic changes in
interpreting the nature of attenuation. The separation of the geometrical parameter i from
Qi-1 casts serious doubts on the validity of interpreting the entire in-situ Q-1 as the
complex argument of the medium’s elastic modulus (e.g., Anderson and Archambeau,
1964; Aki and Richards, 2002). Note that most modern inversions for global attenuation
(e.g., Dahlen and Tromp1998; Dalton et al., 2008) are based on this assumption, which
allows using the velocity sensitivity kernels for deriving Q-1.
Apparently the most significant implication of the proposed (f) view relates to the
problem of the frequency dependence of Q within the mantle. This problem can
obviously never be solved in favour of the frequency-independent model, merely because
it is far more restrictive, and new data conflicting with it may arise. By contrast, the
frequency-dependent Q model is extremely permissive, and its inherent trade-offs allow
easy reconciling different datasets. Due to its rich theoretical implications, this model is
also favoured by most seismologists since early 1960’s. Modern visco-elastic models
routinely start by postulating rheological relaxation mechanisms and complex-valued
elastic moduli within the Earth (e.g., Dahlen and Tromp, 1998; Borcherdt, 2009), which
automatically lead to a frequency-dependent Q. However, also because the frequencyindependent model is more restrictive, ascertaining its validity would have advanced us
much further in understanding the Earth’s structure, properties of its materials, and the
mechanics of seismic wave propagation. Therefore, I suggest that this avenue should be
explored to the end, and the frequency-independent model is not ruled out until
conclusive and unbiased experimental evidence against it is found.
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Unfortunately, reviewing the continuously mounting evidence for frequency-dependent Q
within the crust and mantle shows that in many cases, the initial presentation of
experimental data is done with a definitely frequency-dependent Q in mind. In a vast
majority of papers, attenuation data are presented only by apparent Q(f) dependences
(e.g., Aki, 1980), and if the attenuation coefficient is used, it is shown as a function of
period, f-1 (e.g., Mitchell, 1995). However, as shown above and in Morozov (2008,
2009a), presenting the raw  as in linear f scales reveals linear dependences that should
be indicative of some fundamental properties of attenuation processes. Such two key
observations from the above examples are: 1) (f) usually contains a non-zero frequencyindependent contribution, which can be measured by the intercept (0) and interpreted as
caused by the residual GS and dispersion; 2) frequency-dependent increments (f) - (0)
are linear for all wave types and datasets considered (Fig. 4); and 3) non-linearities of the
observed Q(f) and (f-1) dependences are usually spurious and caused by the
corresponding transformations from (f). The second of these observations suggests that a
frequency-independent Q model is viable and natural from such observations.
Seismic measurements rarely span a continuous range of frequencies wide enough to
allow detection of a frequency-dependent Qe. However, the apparent Q or t* values vary
even within the relatively narrow observation bands (e.g., Fig. 2a), which makes them
problematic for comparisons. By contrast, parameters  and Qe characterize the entire sets
of near-linear (f) observations (Fig. 2a), and consequently they should provide a more
consistent basis for analysis.
The most reliable experimental indications of frequency-dependent attenuation come
from comparing different frequency bands (e.g., Sipkin and Jordan, 1979), and often from
combining different wave types (Der et al., 1986; Cong and Mitchell, 1988). To reexamine the frequency dependence of crustal attenuation across about two decades in
frequency, let us correlate the 3–70 s Rayleigh wave results for South America from
Hwang and Mitchell (1987) and from 0.4–1.4 Hz Lg Q0 and  measurements by Raoof
and Nuttli (1984) (Fig. 7). These data were interpreted by Cong and Mitchell (1988), who
concluded that the frequency dependence of crustal Q in its tectonic (western) part is
weak (with exponent in the Q = Q0f law) and within the stable (eastern) part –
much stronger (≈ 0.7). According to Mitchell (1995), strong frequency dependence is
typical for stable areas. A strong contrast in attenuation levels was also found (from Q0 ≈
900 in eastern part to Q0 ≈ 200 in the western part of this area). However, by looking at
the data in Fig. 7 without tailoring them to the (Q0, ) model, we arrive at quite different
conclusions.
Cong and Mitchell (1988) derived their values by using a procedure schematically
illustrated by the dotted arrows in Fig. 7. They first constructed crustal models consistent
with the Rayleigh-wave attenuation (left arrow in each plot), then scaled their Q values
by using trial  parameters and numerically modelled the Lg-phase attenuation. The
resulting values of  were established by matching the Lg attenuation with the
measurements by Raoof and Nuttli (1984) at 1-s periods (right arrows and dark-grey bars
in Figs. 7a,b). However, in respect to this procedure, note that: 1) in each case, it used
only a single point, namely that at 1 Hz, and ignored the rest of the measured Lg
attenuation-coefficient trends, and 2) this choice of 1-Hz reference, although wellestablished by convention, is completely arbitrary, and different choices for this
frequency would have changed the value s of .
In (f) diagrams (Figs. 7a,b), Q-1(f) values of he Rayleigh and Lg waves correspond to the
slopes of the corresponding radius-vectors shown by dotted arrows: Q-1(f) = (f)/f.
Consequently, larger values of  required to reconcile these Q’s in the stable-area case
corresponds to the wider angle between these arrows (Fig. 7b). Thus, the increased
interpreted  in the stable area (Cong and Mitchell, 1988) is actually caused by larger Qe
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and lower  in this area (i.e., more horizontal and lower-placed grey-shaded (f,)
distribution for Lg waves in Fig. 7b).
Plotting the raw attenuation-coefficient data (for Lg, here reconstructed from Q0 and 
maps by Raoof and Nuttli, 1984) allows seeing the basic relationships between them
without the use of assumption-prone Q(f) models and numerical modeling. In Fig. 7, we
see that Lg (f) distributions have the same patterns as shown in Fig. 4, and representative
linear Lg trends (black dashed lines) can be identified. As above, we only use an
interpretive approach by drawing “the most likely” linear (f) trends through the positions
of the dark-grey observation bars at 1 Hz in Figs. 7a and b. Several observations can be
made by directly comparing these trends to those for Rayleigh waves:
1) In the stable area (Fig. 7b), there is no significant difference between the
Rayleigh-wave and Lg (f) across the entire frequency band. For both types of
waves,  ≈ (0.2– 0.3)·10-3 s-1 and frequency-independent Qe ≈ 800.
2) In the active area (Fig. 7a), both the Rayleigh-wave and Lg data are again
consistent with the same values of Qe ≈ 330, possibly increasing to ~400 for Lg.
These values are significantly higher than Q0 ≈ 170–220 by Raoof and Nuttli
(1984), and there is no frequency-dependence.
3) However, Lg  in the active area is much higher, ~6·10-3 s-1, than the Rayleighwave ., which is ~6·10-4 s-1 (Fig. 7a, see also Fig. 2c). This is just below the
lower threshold for body- and coda-wave D ≈ 8·10-3 s-1 proposed for tectonic
areas by Morozov (2008). However, note that values above D are still within the
uncertainty of the reconstructed Lg (f) data (dash-dotted line with  ≈ 9·10-3 s-1),
in which case Qe would likely increase to ~400.
The similarity of Rayleigh-wave Qe values in the two areas and the difference of their 's
suggest that the principal difference between them consists in the structure of the upper
crust. At 3–70-s periods, Rayleigh-wave Qe should be principally controlled by the midand lower crust, whereas the lower-velocity, lower-Q upper crust modifies the
distribution of wave amplitudes, which is described by the geometric factor (Morozov, in
review I). By contrast, the high-frequency Lg-wave Qe likely closely corresponds to the
Qi of the upper crust. The corresponding high  could be explained by the upper-crustal
velocity gradients and reflectivity, which for surface waves lead to reduced amplitudes
recorded at the surface. Detailed theoretical treatment and numerical modeling of these
effects will be given elsewhere; at this time, it important to ascertain that a
phenomenological physical model can correctly account for all observations (Figs. 2c,d,
3b, 4, and 7) without postulating a frequency-dependent material Q.
If a quality-factor picture is still desired, Fig. 8 shows the distribution of (f) trends (Fig.
4) transformed into Q(f) = f/(f) and plotted in logarithmic frequency scale. The
apparent Q(f) consistently increases with frequency, but this increase mostly occurs
within two bands (0.01–0.1 Hz and 1–100 Hz), below and between which Q is nearly
constant. A log(Q) plot (Fig. 8b) shows that within the different sub-bands, Q(f)
dependences can also be approximately described by the Q0f power law with exponent
varying around ≈ 0.5–1. However, this appearance should mostly be due to the
notorious universality of log-log plots.
The above reworking of several published datasets shows that changes in the frequency
dependence of attenuation indeed occur at ~ 300-s, ~100-s, and 10–1-s periods (Fig. 4),
but they correspond to the transitions between different wave types dominating these
frequency bands. Within each of these wave types, the attenuation quality Qe and
geometrical spreading  are regionally-variable and correlate with tectonics (for periods
shorter than ~100 s) yet are frequency-independent within the available observational
uncertainties. Moreover, the values of Qe are generally close for all wave types below
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~100 s periods (Fig. 4). The widespread notion of Q pervasively increasing with
frequency may thus be due to the fact that structural effects () are positive and
consistently increase during the transitions to higher-frequency wave modes. Note that as
mentioned above, such transitions can be formally attributed to “scattering Q.” However,
such association works only within the limited tasks of attenuation measurements and
may incorrectly describe the effects of the first-order Earth’s structure as mere random
scattering. The concept of generalized GS, in the sense defined in Introduction and
measured by parameter , is much more suitable for this purpose.
Although a vast volume of other observations still remains to be reviewed in a similar
manner, the examples presented here already indicate that this general picture of surfacewave, Lg, and body-wave attenuation will likely remain correct. It appears that no
microscopic frequency-dependent attenuation mechanisms are required to explain the key
observations. Although frequency-dependent elastic scattering certainly occurs within the
Earth, as well as seismic-wave induced relaxation and creep in some structures, these
effects are not nearly as dominant in attenuating seismic waves as it is commonly
thought. In the models discussed here, these effects appear indistinguishable from the
frequency-independent i and Qe.
Finally, note that several specific and quantitative observations and correlations with
geology become possible simply by presenting the attenuation observations in their
generic form, as the attenuation coefficient  plotted against frequency. This form is so
naturally suggested by the scattering theory and by the character of attenuation
measurements that it quite surprising that it is so rarely used. However, when utilized, this
form reveals similarities even among such disparate wave types as normal modes, Lg, and
body waves, and suggests a general classification of attenuation patterns (Fig. 4). Due to
its generality, the approach applies to most wave types used in attenuation studies,
including surface, Lg (this paper), crustal body waves and coda (Morozov, 2008), Pn and
synthetic surface waves (Morozov, in review I), long-period P, and ScS waves (Morozov,
in review III). Analysis of attenuation coefficients is also free from underlying model
assumptions about the geometrical spreading, which cause great uncertainties in
conventional (Q0,) interpretations. The available data compilations show that parameter
Qe, and especially  clearly correlate with tectonic ages (Morozov, 2008). Finally, values
of  also appear to be predictable from structural data, by a completely independent
waveform modeling (Morozov et al., 2008).
11 Conclusions
Analysis of the frequency-dependence of the attenuation coefficient leads to significant
changes in the interpretation of seismic attenuation data. Continuing the study of crustal
body and coda waves by Morozov (2008), several published Rayleigh-wave and Lg
attenuation studies are revisited from a uniform standpoint of the temporal attenuationcoefficient, . In all cases considered, the dependence of  on frequency is found to
follow the linear (f) ≈  + f/Qe law expected from a phenomenological theory.
The observed piecewise-linear pattern of (f) allows a simple classification of
attenuation-coefficient dependences within a broad range of frequencies from ~500 s to
~1.5 Hz. Three groups of linear patterns are revealed: Rayleigh waves at 500–100 s and
100–10 s, and Lg waves from ~2 s to ~1.5 Hz. The last of these segments overlaps with
similar linear (f) patterns of body, Pn, and coda waves (Morozov, 2008), which extend
to ~100 Hz. Within each of these frequency bands, parameters  and Qe can be considered
as constant, and they rapidly increase between these bands. Such increases are related to
changing wave types, particularly as a result of crustal effects.
For both Rayleigh waves at 100–10 s and Lg, the levels of  are lower within stable areas
and higher in the areas of active tectonics, which are separated by the levels of D ≈
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0.2·10-3 s-1 and 8·10-3 s-1, respectively. The threshold for Lg also coincides with the
corresponding D for high-frequency body and coda waves described by Morozov (2008).
The proposed (f) phenomenology for Rayleigh waves suggests an explanation for the
recently recognised discrepancy between the values of Q measured from long-period
surface waves and from normal-mode oscillations. The discrepancy amounts in  ≈ 10-5
s-1, which can be explained by a small uncertainty in the measured geometrical effect for
surface waves. Such uncertainty could be related, for example, to predominantly
continental observations.
To model the observed , the “intrinsic attenuation coefficient” of the propagating
medium is defined and denoted i. This parameter generalizes the intrinsic attenuation by
incorporating the variations of geometrical spreading and dispersion within the medium.
Two frequency-dependent crustal Q models are recast in this form and show that the
difference between the tectonically-active and stable crust should primarily be related to
the differences in i and Qi within the upper crust.
Frequency-dependent rheological or scattering Q is not required for explaining the
observations considered in this study. The often-interpreted increase of Q with frequency
from ~2 mHz to ~100 Hz is shown to be apparent. Three general causes for interpreting a
frequency-dependent in-situ Q are identified: 1) presenting the attenuation data in various
forms obscuring the linear (f) dependences, such as Q(f) or (period), 2) ignoring the
fact of  being non-zero and variable in realistic structures, and 3) following preferred
theoretical Q(f) dependences while cutting across the (f) trends observed within the
individual frequency bands.
Acknowledgments
Two anonymous reviewers have greatly helped in improving the presentation and suggested
several references. This research was supported by NSERC Discovery Grant RGPIN261610-03.
GNU Octave software (http://www.gnu.org/software/octave/) and GMT programs (Wessel and
Smith, 1995) were used in preparation of several illustrations.
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Figures
Fig. 1. Group velocities of Rayleigh waves, after Panza and Calcagnile (1975): a) continental
structure without a low-velocity channel in the upper mantle, and b) with the low-velocity channel.
Labels F, 1, 2, and 3 indicate the fundamental and three higher modes. In c) and d), the same
velocity values shown as functions of frequency. Grey lines in c) and d) are the simplified linear
trends used in Fig. 2c and d.
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Fig. 2. Fundamental-mode Rayleigh wave attenuation-coefficient data from Mitchell (1995) and
Weeraratne (2007): a) tectonically-active and oceanic areas; b) stable areas; c) and d) – same as a)
and b), respectively, but in (f) form. Typical error bars from Mitchell (1995) are indicated.
Dashed lines indicate the interpreted linear trends within similarly-coloured data subsets. The
corresponding Qe values are given in labels. Yellow bars highlight the characteristic (f) intercept
levels for each group of tectonic areas. Black dashed line indicates the level of  into which values
 = 0 are mapped by the dispersion correction.
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Fig. 3. Spheroidal-mode (black dots) and Rayleigh-wave (other symbols) attenuation data from
IGPP reference model web site: a in the original 1000Q-1 form; b) transformed to(f). Dashed line
shows the interpreted linear trend (f) = -8·10-5 +f/Qe [s-1], with Qe = 84. Frequency fc is the
“cross-over” frequency at which the change from surface-wave to normal-mode regime occurs.
Fig. 4. Schematic summary of the observed (f) dependences for the two bands of Rayleigh waves
of this study and short-period body, Lg, and coda waves from Morozov (2008), and Pn from
Morozov (in review I). “Frequency-reduced”  values are shown, so that the linear dependences
corresponding to Qe = 1000 appear horizontal. Typical ranges of Qe, and  levels discriminating
between the stable and active tectonic regimes are indicated. Values of D are labelled in grey
boxes; the value not corrected for dispersion given in parentheses.
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Fig. 5. Frequency-dependent crustal S-wave Q model for the Basin and Range province (Mitchell
and Xie, 1994) and for the eastern United States (Cong and Mitchell, 1988), sampled at 0.1, 0.3,
and 1.0 Hz.
Fig. 6. Crustal models from Fig. 5 transformed into (i, Qi) form. Note the low attenuation (Q >
1000) in the eastern U.S. and strong difference between the two upper-crustal models. Dashed line
indicates the proposed D threshold separating the active and tectonic structures.
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Fig. 7. Comparison between surface-wave and Lg attenuation data converted to (f) from: a)
western part of South America (tectonically-active); and b) its eastern part (stable). “Frequencyreduction” was applied to (f), so that linear trends with Qe = 800 appear horizontal. Error bars
show fundamental-mode Rayleigh-wave data from Hwang and Mitchell (1987; labelled H&M).
Grey-shaded areas labelled R&N show Lg (f) derived from Q0 and  values reported by Raoof
and Nuttli (1984). Black dashed lines are the linear (f) interpretations of Lg waves as in Figs.
2c,d. Dotted arrows labelled C&M and grey bars at 1.0 Hz illustrate the procedure for correlating
Q(f) between the Rayleigh and Lg-waves by Cong and Mitchell (1988). See text for discussion.
Fig. 8. a) Apparent Q(f) corresponding to characteristic (, Qe) ranges in Fig. 4. Areas of active and
stable tectonic types are indicated. b) The same plot in log10Q(f) form. Slopes corresponding to
 = 0, 0.5, and 1 are indicated by arrows.
24