J SeismolDOI 10.1007/s10950-010-9196-5

ORIGINAL ARTICLE

Attenuation coefficients of Rayleigh and Lg waves

Igor B. Morozov

Received: 17 October 2009 / Accepted: 23 May 2010© Springer Science+Business Media B.V. 2010

Abstract Analysis of the frequency dependenceof the attenuation coefficient leads to significantchanges in interpretation of seismic attenuationdata. Here, several published surface-wave at-tenuation studies are revisited from a uniformviewpoint of the temporal attenuation coefficient,denoted by χ . Theoretically, χ( f ) is expected tobe linear in frequency, with a generally non-zerointercept γ = χ(0) related to the variations ofgeometrical spreading, and slope dχ /df = π /Qe

caused by the effective attenuation of the medium.This phenomenological model allows a simpleclassification of χ( f ) dependences as combina-tions of linear segments within several frequencybands. Such linear patterns are indeed observedfor Rayleigh waves at 500–100-s and 100–10-speriods, and also for Lg from ∼2 s to ∼1.5 Hz.The Lg χ( f ) branch overlaps with similar linearbranches of body, Pn, and coda waves, whichwere described earlier and extend to ∼100 Hz.For surface waves shorter than ∼100 s, γ valuesrecorded in areas of stable and active tectonicsare separated by the levels of γD ≈ 0.2 × 10−3 s−1

(for Rayleigh waves) and 8 × 10−3 s−1 (for Lg).The recently recognized discrepancy between the

I. B. Morozov (B)Department of Geological Sciences,University of Saskatchewan,Saskatoon, SK S7N 5E2, Canadae-mail: igor.morozov@usask.ca

values of Q measured from long-period surfacewaves and normal-mode oscillations could alsobe explained by a slight positive bias in the geo-metrical spreading of surface waves. Similarly tothe apparent χ , the corresponding linear variationwith frequency is inferred for the intrinsic atten-uation coefficient, χi, which combines the effectsof geometrical spreading and dissipation withinthe medium. Frequency-dependent rheological orscattering Q is not required for explaining anyof the attenuation observations considered in thisstudy. The often-interpreted increase of Q withfrequency may be apparent and caused by usingthe Q-based model of attenuation and followingpreferred Q( f ) dependences while ignoring thetrue χ( f ) trends within the individual frequencybands.

Keywords Attenuation · Geometrical spreading ·Lg · Mantle · Normal modes · Rayleigh waves

1 Introduction

As they travel through the Earth, seismic wavesattenuate due to three general factors: geomet-rical spreading (GS), intrinsic (anelastic) energydissipation, and scattering. At the first glance,the first of these factors can be easily separatedtheoretically, as GS is usually attributed to elasticenergy spreading within expanding wavefronts.

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The distinction between the other two factors alsoseems relatively straightforward theoretically,and it roughly relates to considering either thevisco-elastic or elastic wave mechanics in a het-erogeneous medium. However, in practical obser-vations, these effects are not so easy to separate,because they are always superimposed over eachother and complicated by experimental noise andlimited knowledge of the Earth’s structure. Inparticular, the definitions of GS and scattering-attenuation require the most attention.

Considering the concept of GS first, note that inrealistic Earth models, its simple wavefront-basedmodel breaks down, and GS is actually not easyto define. Neither wavefronts nor rays exist in re-alistic wavefields, in which refractions, reflections,and mode conversions are abundant, and “multi-pathing” is pervasive. Simplified approximationsfor GS commonly used in attenuation measure-ments often cause major uncertainties in the re-sults and misinterpretations of structural effectsas “scattering” (Morozov 2009a, b). In globalattenuation tomography studies, GS variationsare often described as “focusing” and modeled byusing the ray theory in smoothly varying media(e.g., Romanowicz and Mitchell 2007; Dalton et al.2008). From the same studies, it is also knownthat the total effects of focusing (such as the datavariance reduction in tomography) may exceedthose of Q (Dalton and Ekström 2006). Moreover,if we also wish to recognize the difference of thereal GS from its theoretical approximations, GScan apparently be only defined in a relative sense,in respect to the other two attenuation factors.

Throughout this paper, I therefore use a gen-eral definition of GS as a “measure of wavefieldamplitude in the absence of true attenuation.” Thebasis of this definition is in attributing the atten-uation to the propagating medium and assumingthat the attenuation can be hypothetically “turnedoff,” corresponding to setting the attenuationparameter Q−1 = 0. The entire remaining effectof crustal and mantle structure on the attenuationmeasurement is then attributed to GS. In a limitednumber of cases (such as uniform half-space witha one-layer lid) such GS can be modeled analyti-cally, and for more complex structures, it can besimulated by using the ray theory or numerically.On the other hand, such GS can also be treated

phenomenologically and directly measured fromthe data (Morozov 2008). This approach is takenhere.

Scattering attenuation is the most difficult toisolate in the presence of GS and intrinsic at-tenuation. It appears that scattering can only beconsidered in respect to some “theoretical” back-ground model, such as the uniform half-space usedin most studies (e.g., Wu 1985; Sato and Fehler1998). By contrast, when approaching attenua-tion measurements in realistic structures, scatter-ing can hardly be identified unambiguously. Forexample, in a perfectly known structure, the en-tire wavefield is predictable and non-random, andtherefore there is no room for scattering. If thedeterministic structure is limited to a certain scale-length, then the effects of smaller-scale randomheterogeneities could be described by randomscattering; however, in this case, the empirical GSdefined above would still be sufficient to com-pletely describe the intrinsic-Q−1 free wavefield.For these reasons, I argued that the use of scat-tering attenuation can be abandoned in practi-cal observations (Morozov 2008, 2009a), althoughit still represents a very interesting subject fortheoretical studies.

Surface waves are particularly important forstudying the seismic attenuation. They providethe most complete coverage of the Earth’s sur-face, particularly when used in multiple-stationand recent interferometery approaches. Due totheir broad frequency bands, surface waves covera great range of depths, yield the strongest con-straints on the upper-mantle structure, and pro-vide the links between traveling and standingwaves within the Earth. Because of their po-sition in the middle of the seismic spectrum,surface-wave data are also critical for testing thehypothesis of frequency-dependent attenuationwithin the Earth. Importantly, long-period surfacewaves also have the most accurate theoretical GSpredictions.

In over half a century of studying the seis-mological Q, several theoretical models and par-adigms of the expected attenuation–frequencyrelationships were developed, and unfortunately,these paradigms also influenced the very analy-sis of attenuation data. Conventions for present-ing the raw data became established and carefully

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followed. For example, body wave and coda re-sults are typically presented in Q, Q−1, or lnQvs. ln f plots, from which the power-law Q( f ) =Q0 f η dependence is usually interpreted (e.g., Aki1980). The same data are also often presented byusing the “stacked spectral ratios,” which are pro-portional to the temporal attenuation coefficientχ( f ) = π f Q−1 (Xie and Nuttli 1988). As we willsee below, this form could be the most useful;however, this quantity is invariably plotted inlog–log scales versus frequency, which again em-phasizes the power law above. In surface-wavestudies, the spatial attenuation coefficient α( f )

(often denoted γ in these studies) is typicallyshown versus period, T = f −1 (e.g., Mitchell1995). Normal-mode and long-period surfacewaves data are conventionally presented as1,000 Q−1 vs. harmonic degree (e.g., Romanowiczand Mitchell 2007). Such standard forms are use-ful for comparing the results from different stud-ies; however, in several cases, they also complicateobservations of attenuation dependences differentfrom the power-law Q0 f η.

In this paper, I employ one form of data pre-sentation that appears natural and particularlyuseful from both theoretical and practical view-points yet is almost completely overlooked inthe attenuation studies. This form is χ itself ina linear frequency scale. The reasons for under-rating this form are unclear and may be histor-ical; one potential explanation could lie in χ( f )

dependences often not supporting the expectedfrequency-dependent Q. The use of χ( f ) may castdoubts in the pervasive frequency dependenceof the in situ seismological Q, which we discusslater in this paper. Most attenuation data canusually be fit similarly well in either the ( f , χ)or (lnf , lnχ) forms, and therefore the distinctionbetween these parameterizations lies not in com-paring the data fitting errors but in the underlyingtheoretical principles and interpretational values(Morozov 2010).

Although consisting in a simple transforma-tion, presentation of attenuation data in the ( f ,χ) plane often leads to serious changes in theinterpretation. The most significant result fromswitching to this view is the recognition that χ( 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), χ( f ) shows linear dependenceson f with non-zero intercepts γ = χ(0) and con-stant (within data uncertainties) slopes corre-sponding to the “effective attenuation” Q−1

e =[χ( f ) − γ ]/π f . Morozov (2008) interpreted theintercepts γ as related to the residual GS andfound them to be variable and correlated withcrustal structures, including a clear decrease of γ

with tectonic ages.In this paper, I extend the analysis of Morozov

(2008) to Rayleigh waves within ∼10–500 s pe-riods and 0.5–1.5-Hz Lg by using the atten-uation coefficient and Q data compiled fromRaoof and Nuttli (1984), Mitchell (1995), Durekand Ekström (1996), Weeraratne et al. (2007)and also taken from IGPP Reference EarthModel web pages (http://igppweb.ucsd.edu/∼gabi/rem.html). Lg waves can be associated withhigher-mode surface waves (Knopoff et al. 1973;Panza and Calcagnile 1975), which allows us-ing them for extending the frequency band ofthe fundamental-mode Rayleigh waves. The ap-proach is strictly empirical and quantitative, andreduces to summarizing the observed χ( f ) de-pendences without using any underlying modelsbeyond the linear expression above. Analysis ofthis wide frequency band demonstrates an almostamazing commonality in the χ( f ) patterns andeven in the values of γ and Qe. In addition, recog-nition of the frequency-independent shift γ insurface-wave data leads to a potential explanationfor the discrepancy between the surface-wave andnormal-mode attenuation measurements at 200–500-s periods (Durek and Ekström 1997).

Perhaps the greatest general implication of theattenuation-coefficient approach is in its revealingno indications of frequency-dependent crustal ormantle 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 ob-served surface- and Lg wave attenuation at allfrequencies. From the observed γ and Qe, thecorresponding in situ attenuation model alsonaturally becomes frequency independent. InSection 10, the new picture is used to explain howdisregarding the χ( f ) trends while reconciling the

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Rayleigh wave and Lg attenuation levels (Congand Mitchell 1988) may lead to an apparent fre-quency dependence of Q. Finally, I also showwhy this apparent Q( f ) consistently increases,although at variable rates, across the broad fre-quency band of this study.

2 Motivation for using χ( f )

There are several reasons why plotting χ( f )

should be tried in most attenuation studies. First,χ is essentially the principal parameter directlyobtained from GS- or source-spectrum correctedamplitude measurements. The quality factor Qis derived from this parameter by a frequency-dependent transformation Q( f ) = π f/χ( f ), whichdistorts both its values and error bounds. Second,although this subject deserves a special discussion,note that Q does not actually represent a trueproperty of the propagating medium, but χ ismuch closer to being such a property. To see thisinadequacy of Q, note that in its definition:

Q−1 = δE2π Emax

(1)

where δE the energy lost per one oscillation cycle,and Emax is the maximum elastic energy in thatcycle (e.g., Aki and Richards 2002, p. 162), the“cycle” belongs to the incident wave and doesnot characterize the propagating medium. By con-trast, the temporal attenuation coefficient

χ( f ) = −∂ ln E(t, f )

2∂t, (2)

describes the relative rate of energy dissipation atany given point within the medium, irrespectivelyof the wave process. To derive a Q−1 from thisquantity, one has to divide it by the frequency,which leads to the characteristic near- f −1 depen-dence of Q−1( f ) that is often observed (e.g., Aki1980).

General theoretical arguments also suggest thatχ( f ) should be tested for linearity in f first (andnot in lnf or f −1 as above), which I denote byχ( f ) = γ + κ f . This can be seen from an exampleof a linear oscillator, which is the simplest dissipa-

tive system known in mechanics. The oscillator isdescribed by its Lagrangian (e.g., Razavy 2005)

L(x, x) = 1

2mx2 − 1

2mω2

0x2, (3)

where x is the displacement, x is the velocity, m isthe mass, and ω0 is the natural frequency. Energydissipation is described by the force of viscous fric-tion, which is proportional to the velocity: fD =−mω0ξ x, where ξ is a unitless dissipation constantrelated to the oscillator’s Q as ξ = Q−1. In theHamilton variational approach, this force arisesfrom the Rayleigh dissipation function

D = ξmω0

2x2. (4)

Note that fD is proportional to particle velocity,and D is functionally similar to the kinetic energyin Eq. 3. In a harmonic elastic wave, the timederivatives in Eq. 4 correspond to multiplicationswith frequency, and consequently all dissipationprocesses have leading linear dependences on thefrequency. This is reflected in the presence of f inthe attenuated amplitude law: u ∝ exp(−π f t/Q),and consequently in χ( f ) ∝ f . Finally, consider-ing that a zero-order term in f should gener-ally also be present in χ( f ), we see that lineardependences χ( f ) = γ + κ f should be naturallyexpected in dissipative systems.

The existence of a non-zero γ above, referredto as the “geometrical attenuation” by Morozov(2008), is the most important point of the pro-posed model. Observations of non-zero γ mayhave two key implications: (1) they indicate thefailure of the conventional Q( f ) = Q0 f η modeland (2) they provide ways for measuring the vari-ations of geometrical spreading from the data.Geometrical attenuation may arise from near-surface reflections (Morozov 2008), small-scalereflectivity and ray curvatures, mode summationswithin the coda (Morozov et al. 2008), and mostimportantly for surface waves, it is also associatedwith dispersion.

3 Effects of dispersion

Because of dispersion, wave packets spread outwith propagation distance, causing the amplitudes

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to decrease in addition to their reduction due toenergy loss. Xie and Nuttli (1988) included suchpulse broadening in the geometrical spreading andproposed a method for its estimation, which is inbroad use today (e.g., Li et al. 2009). These au-thors suggested that because of pulse broadening,energy density in a propagating wave reduces byfactor 1/U , where

U(r) = r�t0

(1

Vmin− 1

Vmax

), (5)

r is the travel distance, Vmin and Vmax are theminimum and maximum group velocities withinthe frequency band of interest, respectively. Para-meter �t0 here is a constant used for normalizingthis factor so that U(r0) = 1 at some referencedistance r0. However, Eq. 5 further simplifies toU(r) = r/r0 and actually does not depend on thevelocity dispersion parameters. This is not sur-prising, because the geometrical spreading, as wellas other wave-amplitude factors, is defined up toan arbitrary scaling which has to be removed bynormalization. The factor containing velocities in(5) is simply a part of such scaling.

The reason for missing the dispersion effectin expression (5) is in assuming the wave-pulsewidth to be zero at r = 0. To correct this problem,let us consider a pulse of finite duration �t0 atdistance r0 and linearly expanding from this point.Equation 5 then modifies to

U(r) = 1 + r − r0

�t0

(1

Vmin− 1

Vmax

). (6)

For weak dispersion, if r0 can be selected so thatthe second term in (6) is small within the rangeof observation distances, then U(r) can also berendered in the attenuation-coefficient form:

U(r) ≈ e2ad(r−r0), (7)

where the spatial attenuation coefficient due todispersive pulse broadening is

αd = 1

2�t0

(1

Vmin− 1

Vmax

)≈ δ ln V

2V�t0, (8)

and δlnV is the relative group velocity variationacross the observation frequency band. For exam-ple, for fundamental-mode Rayleigh waves at f <

0.3 Hz, (δlnV)/V ≈ 0.07 s/km (Fig. 1), and by tak-ing �t0 ≈ 500 s, we obtain αd ≈ 0.7 × 10−4 km−1,

which is comparable to the observed values ofα (Fig. 2a, b). Note that higher-frequency wave-lets (with smaller �t0) broaden stronger; how-ever, within the same wave, the effect of αd isfrequency-independent, i.e., “geometrical.”

Dispersion also affects the relation betweenthe spatial and temporal attenuation coefficients.Without taking pulse broadening into account,χ = Vα, where V is the group velocity (Aki andRichards 2002, p. 293). This relation was obtainedby equating the harmonic-wave amplitude at thedominant frequency, exp(−χ t) with the amplitudeof the wavelet maximum, exp(−αr), which is lo-cated at travel distance r = Vt. However, becauseof pulse broadening, the relation between theseamplitudes should be

1√U

exp(−χ t) = exp(−αr), (9)

leading to

χ = Vα − ln U2t

≈ V(α − αd). (10)

This formula also shows that in the presence ofdispersion, α should always exceed αd. The limit ofα = αd corresponds to a wave attenuating by purepulse broadening and without energy loss.

4 Observations

Figure 2a and b show the measured Rayleigh-wave attenuation coefficients at 10–100-s pe-riods in several tectonically active and stableareas around the world, given in the traditionalform, as functions of wave periods (Mitchell 1995;Weeraratne et al. 2007). Generally, the values ofα are within 0–1 × 10−3 s−1, comparatively high intectonic and low in oceanic areas, and also quicklyincreasing below ∼20-s periods (Mitchell 1995;Fig. 2a, b).

Note that the trends for α(T) increasing towardshorter periods T = 1/ f appear hyperbolic, whichsuggests that the dependences on f might actu-ally look simpler. This becomes clear when thesame data are plotted against frequency (Fig. 2c,d). In these plots, α( f ) was transformed into thetemporal attenuation coefficient, χ( f ), by usingEq. 10, with dispersion parameters taken from

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Fig. 1 Group velocities of Rayleigh waves, after Panzaand Calcagnile (1975): a continental structure without alow-velocity channel in the upper mantle and b with thelow-velocity channel. Labels F, 1, 2, and 3 indicate the

fundamental and three higher modes. In c and d, the samevelocity values shown as functions of frequency. Grey linesin c and d are the simplified linear trends used in Fig. 2cand d

the simplified, linear fundamental-mode group-velocity trend in Fig. 1c and d, and αd = 0.4 ×10−4 km−1. This αd value was selected empirically,close to the low bound on the observed α, so thatthe resulting intercepts of χ( f ) in Fig. 1d becamenear-zero. Note that the dispersion curves (Fig. 1c,d) show significant variations at frequencies below∼0.03–0.05 Hz, which may contribute to the “spec-tral scalloping” of χ( f ) amplitudes observed inthe data.

In the χ( f ) form, the separation betweenthe tectonic, stable, and oceanic areas becomesclearer, as well as the differences between theseveral study areas. Several separate linear trendsof χ( f ) can be recognized, as indicated by thedashed lines in Fig. 2c and d. Notably, when ex-

tended to the frequency axis, most of these linescross it at positive intercept values, particularlyif the correction for αd is not performed. Similarpositive γ values were found in recent body waveand coda observations (Morozov 2008).

Noting that when projected to zero frequencies,the observed χ( f ) trends are non-zero, we replacethe conventional attenuation parameters Q0 and η

with another two, γ and Qe, defined by

χ( f ) = γ + π fQe

, (11)

where Qe can be frequency dependent. How-ever, with data fluctuations and measurementnoise, there seems to be no reason to look for afrequency-dependent Qe (Fig. 2c, d).

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Fig. 2 Fundamental-mode Rayleigh wave attenuation-coefficient data from Mitchell (1995) and Weeraratne et al.(2007): a tectonically active and oceanic areas; b stableareas; c and d—same as a and b, respectively, but in χ( f )

form. Typical error bars from Mitchell (1995) are indicated.Dashed lines indicate the interpreted linear trends within

similarly colored data subsets. The corresponding Qe val-ues are given in labels. Yellow bars highlight the character-istic χ( f ) intercept levels for each group of tectonic areas.Black dashed line indicates the level of χ into which valuesα = 0 are mapped by the dispersion correction

From the interpreted linear trends in χ( f )

(Fig. 2c, d) several important observations canbe made:

1. Although lower values of Qe ≈ 230 arepresent in the data from regions of active-tectonics compared to the stable areas, theseranges of Qe overlap almost completely.Therefore, although Qe may generally in-

crease with tectonic age (Morozov 2008), itdoes not significantly discriminate betweenthe stable and active tectonic types.

2. Nevertheless, the intercept value of γD ≈ 2 ×10−4 s−1 (compare the yellow bars in Fig. 2cand d) separates most of the tectonic andactive areas. If the dispersion correction is notapplied, this threshold should be measuredrelative to the dashed black line in Fig. 2c

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and d, and equals γD ≈ 3.2 × 10−4 s−1. A simi-lar relationship was found for crustal body andcoda waves, for which the stable and activeareas were separated by the level of γD ≈0.8 × 10−2 s−1 (Morozov 2008).

3. In relation to this γD discriminant, theoceanic-area data from Canas and Mitchell(1978) (solid lines in Fig. 2c) generally alignwith the continental stable-tectonic group(Fig. 2d), although recent data by Weeraratneet al. (2007; green triangles in Fig. 2c) areclose to the edge of the active-tectonic group.Values of γ for oceanic recordings also ap-pear to increase with age (Fig. 2c), which isan opposite trend compared to the continen-tal lithosphere. In addition, the oceanic datashow consistently higher Qe.

Note that the above observations are entirelyempirical and independent of the traditionalgeometrical-spreading and Q( f ) = Q0 f h assump-tions. However, they reveal several important re-lationships in the data that have not been noticedin the original χ(T) and Q( f ) interpretations(Mitchell 1995). This shows that raw data repre-sentation and classification is very important inthe analysis of attenuation.

In the above analysis (Fig. 2c, d), we did not at-tempt rigorous estimations of statistical parametererrors and confidence intervals. Unfortunately,the published data do not allow a complete erroranalysis in the spirit of the proposed approach.The individual measurement errors are significant(error bars in Fig. 2c, d); however, the ampli-tude deviations from the interpreted linear trendsare non-random and should be mostly related towave-mode interferences within the specific struc-tures, known as “tuning” in reflection seismology.A proper inversion for χ( f ) would require revisit-ing the full raw-amplitude datasets, which are notavailable to us at present. At the same time, in thispaper, we focus only on the fact of distinct linearχ( f ) dependences and their characteristic para-meters, and consequently can rely on interpretive“visual” analysis and line-fitting. It is quite clearfrom Fig. 2c and d that: (1) multiple χ( f ) trendsexist and (2) these trends may be considered aslinear at best.

From the arguments above, “turning off” theattenuation in the interpreted linear χ( f ) trendswould correspond to setting Q−1

e , or alternatively,f equal to zero. This suggests an interpretation ofγ as a measure of the residual GS or dispersionremaining in the surface-wave amplitudes aftertheir correction (Morozov 2008). For example, forthe cut-off value of γ ≈ 2 × 10−4 s−1, this residualGS correction amounts in only γ t ≈ 8% for a 400-sRayleigh wave propagation time. This relativelevel of this residual GS is similar to that estimatedfor body waves (Morozov 2008).

Notably, all values of γ are above the minimallevel (−adV| f=0; dashed line in Fig. 2c and d),showing that Rayleigh waves are systematically“under-corrected” by the theoretical GS correc-tion. This is again similar to the observationsof lithospheric body and coda waves (Morozov2008). The systematic character of this GS termshows that it is caused not only by focusing and de-focusing on lateral variations of velocity (Daltonand Ekstrom 2006), but also generally deviatesfrom the theoretical �−ν(sin�)1/2 dependence(Nuttli 1973). The variability is also significantbetween different regions, and particularly withinthe tectonically active lithosphere (Fig. 2c).

The relative significance of the residual GSin attenuation measurements can be character-ized by the “cross-over” frequency fc = |γ | Qe/π

(Morozov 2008). Below this frequency, the effectsof the residual GS exceed those of attenuation.For the characteristic values of γ = 4 × 10−4 s−1

and Qe = 500, we have fc ≈ 0.05 Hz, with somevariations for the different regions. This frequencycorresponds to the ∼20-s period below which theapparent attenuation factor χ(T) starts quicklyincreasing (Fig. 2a, b; Mitchell 1995).

Let us now consider the ∼100 to ∼300–400-sRayleigh waves. Interestingly, the global-averageQ−1 curve in this range is also hyperbolic (Fig. 3a),and the corresponding χ( f ) again shows a well-defined linear dependence (Fig. 3b). By con-trast to the shorter-wave case, its Qe ≈ 84 issignificantly lower, and the intercept γ ≈ −8 ×10−6 s−1 is negative, showing that these wavesare “over-corrected” by the background GS cor-rection. Taking the same characteristic traveltime (400 s), the relative amount of this over-correction is only |γ |t ≈ 3%. This small value is not

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Fig. 3 Spheroidal-mode (black dots) and Rayleigh wave(other symbols) attenuation data from IGPP referencemodel web site: a in the original 1,000 Q−1 form; b trans-formed to χ( f ). Dashed line shows the interpreted linear

trend χ( f ) = −8 · 10−6 + π f/Qe[s−1], with Qe = 84. fc isthe “cross-over” frequency at which the change from sur-face wave to normal mode regime occurs

surprising, because at such wavelengths, thespherical-Earth model used in accounting for theGS effects is quite accurate. The cross-over fre-quency for the long-period band equals fc ≈2 mHz. Although the interpretation of this quan-tity is not as straightforward as in the case ofunder-corrected GS, note that this frequency isclose to the transition from the surface wave tonormal mode regime (Fig. 3b).

As shown in the following sections, Qe stillrepresents an apparent quantity characterizing theobservations on the surface. For relatively shortwaves localized within comparatively uniformlayers, Qe should be somewhat greater than thelowest intrinsic Qi sampled by the correspondingwave. However, it appears that for long surfacewaves, the above Qe ≈ 84 may actually be belowthe lowest Qi within the upper mantle and repre-sent the redistribution of wave energy density withchanging frequency.

Below ∼2.0–2.5 mHz, the attenuation coeffi-cient flattens out with the transition into thelow-order fundamental spheroidal modes. This isthe only studied frequency range in which the be-havior of χ( f ) strongly deviates from piecewise-linear. Such change could be related to thecharacteristic wavelengths reaching the thicknessof the entire upper mantle, and consequently the

transition from predominantly traveling to stand-ing waves.

5 Summary of χ( f ) observations

Combining the above observations with short-period results by Morozov (2008, 2010), we cansummarize the available attenuation-coefficientdata as consisting of three nearly linear branchesof χ( f ) within 100–400 s, 10–100 s, and ∼0.5–100 Hz period/frequency bands (Fig. 4). A broadgap from ∼1–2 to ∼10 s still remains, in whichno attenuation measurements are available. Thedifficulty of measurements and interpretation inthis frequency band are well known and caused bythe complexity of the lithospheric structure and bythe complex character of the wavefield changingfrom predominantly surface- to body-wave typeacross this band.

Whereas the first of these χ( f ) branches (100–400 s) appears to be well-defined and “global”in character, the two higher-frequency branchesare sensitive to the tectonic types and geologicstructures (Fig. 4). Both γ and Qe values varyregionally for both Rayleigh and body waves, withγ being systematically lower within stable regions.In oceanic areas, γ has lower, continental-type

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Fig. 4 Schematic summary of the observed χ( f ) depen-dences for the two bands of Rayleigh waves of this studyand short-period body, Lg, and coda waves from Morozov(2008), and Pn from Morozov (2010). “Frequency-reduced” χ values are shown, so that the linear depen-

dences corresponding to Qe = 1000 appear horizontal.Typical ranges of Qe, and γ levels discriminating betweenthe stable and active tectonic regimes are indicated. Valuesof γD are labeled in grey boxes; the value not corrected fordispersion given in parentheses

values, and Qe is high (∼1,000; Fig. 2c). Theseobservations generally agree with the recent nu-merical modeling by Morozov et al. (2008), whofound that γ is principally controlled by the upper-crustal structure, which is more heterogeneous inactive continental environments (Christensen andMooney 1995; Mitchell 1995) and virtually ab-sent in the oceanic crust. Note that the transitionbetween the two Rayleigh-wave branches occursnearly continuously at χ( f ) ≈ 3 × 10−4 s−1 at f =0.01 Hz (Figs. 2c, d, and 3b), whereas χ( f ) jumpsupward by over ∼(3–6) × 10−3 s−1 when a changeto crustal modes occurs (Fig. 4). This once againsuggests that the upper crust should be the causeof the increased γ values.

Assuming that the attenuation-coefficient datacan be summarized by a collection of piecewise-linear χ( f ) branches (Fig. 4), it appears thatsuch empirical χ( f ) practically excludes the needfor a frequency-dependent Q within the man-tle. Originally, the dependence of the attenuationcoefficient on the period (Fig. 2a, b) was viewedas the primary indication of the frequency depen-dent Q (Mitchell 1995). However, in the presentinterpretation, this argument is reversed, and theattenuation-coefficient observations only indicate

spatially variable but frequency-independent γ

and Qe (Fig. 2c, d).

6 Intrinsic attenuation coefficient

The observed near-linear χ( f ) trends can be ex-plained by using a very general model with afrequency-independent in situ attenuation. Con-sider the expression for the observed path factorP(t, f ),

P(t, f ) = G0(t, f )δP(t, f ), (12)

where G0(t, f ) is the theoretical GS factor (forexample, �−ν(sin�)−1/2 with additional disper-sion or focusing factors for surface waves), andδP(t, f ) includes the remaining effects of theimperfectly predicted GS and attenuation. Thesource and site effects are assumed to have beenremoved from P(t, f ). Conventionally, the entireδP(t, f ) is attributed to Q( f ) along the wave path(e.g., Der and Lees 1985):

δP(t, f ) = e−π f t∗ , (13)

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where

t∗ =∫

path

Q−1( f )dτ , (14)

and τ is the time within the travel path, measuredfrom 0 to t. The exponential form of path correc-tion (13) reflects the fact that δP(t, f ) = 1 whent = 0, with ln[δP(t, f )] increasing with time ap-proximately linearly. This is the typical approxi-mation used in the perturbation (such as weakscattering) theory. However, note that with inac-curate G0(t, f ), the exponent in path correction(13) is not guaranteed to be proportional to f , andtherefore we need to generalize this equation to

δP (t, f ) = e−χ t, (15)

where the path-average attenuation coefficient inEq. 11 is

χ = 1

t

∫path

χidτ , (16)

and χi is the “intrinsic” differential attenuationcoefficient. Thus, in its use and meaning, χ( f ) isquite similar to the conventional π f Q−1( f ), in thesense that it can be averaged over the wave pathsto predict corrections to the logarithms of seismicamplitudes. Its only yet critical difference is therecognition of χ( f ) being generally non-zero atf → 0.

For surface waves, the meanings of the “wavepath” integrals in Eqs. 14 and 16 are of courseheuristic, because such waves do not follow anyparticular paths between the source and receiver.In such cases, these integrals can be rigorouslyrepresented by Feynman path integrals, summa-tions over all normal modes of the field, or bythe full treatment of the perturbation theoryproblem. For example, for layered elastic mediawith weak lateral variations, Woodhouse (1974)and Babich et al. (1976) developed perturba-tion theories in which such effective “rays” wererigorously defined. However, regardless of theirsymbolic forms, expressions (14) and (16) cor-rectly illustrate the essential conclusion that ismost important for us. This conclusion is thatthe observed quantity χ = γ + π f/Qe representsa weighed average of the corresponding intrin-sic quantities of the medium, χi = γi + π f Q−1

i ,

with weights (known as Fréchet kernels in thenormal-mode attenuation theory) determined bythe wave-amplitude distribution.

7 Frequency-independent attenuationwithin the Earth?

From Eqs. 15 and 16, it is apparent that χ shouldgenerally include both frequency-independentand dependent parts, and consequently a lineardependence χ on f should be expected as thefirst-order possibility. The same argument appliesto the intrinsic attenuation coefficient χi. Be-cause a constant Qe appears to be the case inall seismological data we considered in this paperand elsewhere (Morozov 2008, 2009a, 2010, andunpublished), it is therefore natural to considera medium with frequency-independent intrinsicQi first. In such a medium, Eq. 16 predicts afrequency-independent Qe

Q−1e = 1

t

∫path

Q−1i dτ , (17)

and a similar equation relating γi to γ . Note thatQ−1

e is therefore dominated by the zone of high-est attenuation along the wave path (Morozov2009a), which corresponds to the asthenosphere inthe case of most mantle waves. By using the stan-dard inversion techniques (e.g., mode summationsor tomography), χ( f ) measured on the surfacecan be inverted for the in situ differential χi.

Equation 17 is valid when the integration“paths” do not significantly change with frequencyor at least stay within the zones of similar levelsof Q−1

i . This should be the case for crustal bodywaves and shorter-period surface waves (Fig. 2c),but for long-period surface waves and normalmodes (Fig. 3), wave energy is progressively re-moved from the attenuative upper mantle whenfrequency decreases. This causes the apparentχ( f ) to also reflect the shapes of the Fréchetkernels in depth, and the resulting very low Qe ≈84 (Fig. 3b)—to overestimate the Qi in the uppermantle. In detail, these effects are studied else-where.

Note that in the presence of significant struc-tural contrasts, the GS is frequency dependent

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(e.g., Yang et al. 2007), and therefore the cor-rection for it included in ln[δP(t, f )] may befrequency dependent as well. In such cases, sep-arating the effects of γi and Qi in the frequency-dependent part of χi( f ) becomes ambiguous. Thisambiguity stems from the general uncertainty ofthe concept of medium Q and can hardly beresolved unequivocally. The interpretation usedabove and in Morozov (2008) assumed that theresidual GS is frequency independent, which ap-peared reasonable in the most common casesof frequency-independent background G0(t, f ).However, by treating the entire attenuation of themedium as a single χi( f ) quantity, this ambiguitycan be avoided.

Note that the trade-off between the fre-quency dependence and depth layering of Q (e.g.,Mitchell 1991) can still be utilized to introducea frequency-dependent Q in the Earth models.Frequency dependence would increase the num-ber of model variables and therefore allow fittingthe data even better. Frequency dependence ofQ may also be sufficiently small to be unnotice-able within the individual frequency bands, butswitching between the branches (i.e., between sig-nificantly different wave modes and penetrationdepths) may require different Q values. However,also considering the existing successful frequency-independent global 3D Q models (e.g., Daltonet al. 2008), it still appears unlikely that frequency-dependent material Q should be necessary forfitting seismological data.

Finally, in this paper, I prefer staying withinstrictly seismological, quantitative, and empiri-cal arguments. Evidence from laboratory studies(e.g., Faul et al. 2004; Romanowicz and Mitchell2007) often serves as the principal motivation forlooking for a frequency-dependent Q within themantle (e.g., Lekic et al. 2009). However, correla-tion of Q values arising from such different typesof observations may be thwarted with difficultiesof reconciling the assumptions and models used,extrapolating the results to mantle conditions, andeven with the differences in the types of quantitiesmeasured. Bourbié et al. (1987) summarized anumber of Q-measurement types and noted thatalthough most of them can be described by visco-elastic models, there is little agreement betweenthe resulting values of Q. It can also be shown

that “geometric” factors similar to those discussedabove could be found in lab measurements, yetthis would take us far from the subject of thepresent study.

8 Crustal model

As shown above, in all cases where a frequency-dependent Qint( f ) is interpreted, an alternatequantity, which is the intrinsic attenuation co-efficient, χi = π f/Qint( f ) = γi + π f Q−1

i , can beused to describe the attenuative property of themedium. In this expression, Qi shall be firsttried as frequency independent, and frequencydependence further considered if required bythe data.

To illustrate how the traditional attenuationmodels look in the (γi, Qi) form, Fig. 5 showsmodels for tectonically extended crust in the Basinand Range province (BR; Mitchell and Xie 1994)and for the stable eastern United States (EUS;Cong and Mitchell 1988). Both Q models arefrequency dependent; however, the power-lawQ( f ) = Q0 f ζ dependences in them were deriveddifferently. In the BR model, the upper 15 km ofthe crust was taken as frequency independent, andζ was set equal 0.5 everywhere else in the twomodels.

Fig. 5 Frequency-dependent crustal S-wave Q model forthe Basin and Range province (Mitchell and Xie 1994) andfor the eastern USA (Cong and Mitchell 1988), sampled at0.1, 0.3, and 1.0 Hz

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By approximating the local Q( f ) values bydepth distributions of γi and Qi, the models be-come somewhat easier to compare (Fig. 6). Qi

values in the EUS model are high (over 1,000)even in the upper crust, and their variation withdepth is weak. By contrast, the BR model showsan over 10-time stronger attenuation within theupper crust (Qi ≈ 100), which quickly drops to∼1,000 within the lower crust. Values of γi wereforced to equal 0 in the upper crust of the BRmodel, which was probably not a very good ap-proximation, particularly in its extensional tec-tonic setting. Apart from this contradiction, the γi

curves are in agreement with the differentiationproposed above (dashed line in Fig. 6b): γi is sig-nificantly higher than γD ≈ 3.2 × 10−4 s−1 withinthe tectonically active zone (BR), and for stablecrust (EUS), values of γ are at near or belowthis level. Note that we use the value of γD notcorrected for dispersion, because the model byMitchell and Xie (1994) also included no suchcorrections.

The above comparison was based on an ad hoctransformation of the crustal models built withinthe Q( f ) = Q0 f ζ paradigm. This transformationresults in approximately the same Qint( f ) valueswithin the crust, and consequently these modelsshould reproduce the attenuation data fit used byCong and Mitchell (1988). In view of this mod-eling, an interesting question arises: how can weinterpret the γ values in Fig. 6b, considering thatthe forward modeling approach used by Cong andMitchell (1988) did not include variable GS? Theanswer is that deviations of the actual structure

from their layered crustal models (i.e., γ ) can beinterpreted as the “scattering attenuation,” Q−1

s .Combined with the intrinsic attenuation Q−1

i , itproduces the resulting apparent Q−1

int ( f ):

Q−1int ( f ) = γ

π f+ Q−1

i ≡ Q−1s + Q−1

i . (18)

Therefore, Qs = π f/γ ∝ f , which is typical forscattering attenuation (Dainty 1981; Padhy 2005;Morozov 2008).

Thus, when limited-accuracy modeling is used,γ can be inverted from the apparent intrinsicQ−1

int ( f ) results and interpreted as caused by elas-tic scattering. Note that even with such inter-pretation, the upper-crust of the BR model withQ−1

s = 0 (i.e., non-scattering) but very high Q−1i

(Fig. 6) appears contradictory. In a full and ac-curate modeling, one would need to start fromQ−1

i = 0 and adjust the structure until a correctγ is achieved. After this, the need for Qs woulddisappear, and Q−1

i could be inverted for from thevalue of Qe. Such modeling and inversion needsto be addressed from the raw χ( f ) data and isbeyond the scope of this paper.

9 Discrepancy between normal-modeand surface-wave Q−1

The phenomenological argument above also sug-gests a potential explanation of the discrep-ancy among the measurements of traveling andstanding-wave attenuation noted by Durek andEkström (1997). The discrepancy consists in

Fig. 6 Crustal modelsfrom Fig. 5 transformedinto (γi, Qi) form. Notethe low attenuation(Q > 1,000) in the easternUSA and strongdifference between thetwo upper-crustal models.Dashed line indicates theproposed γD thresholdseparating the active andtectonic structures

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systematic, ∼15% differences in the attenuationlevels measured by the surface wave compared tothe normal-mode techniques (Fig. 3a). In the χ( f )

form, this difference amounts in a near-constant,∼10−5 s−1 upward shift of the surface-wave χ

(Fig. 3b). Note that the amount of this shift is closeto the surface-wave γ ≈ −8 × 10−6 s−1 and repre-sents only ∼3% of the measured crustal GS effectfor surface waves at 100–10-s periods (γ ≈ 3.2 ×10−4 s−1 before the correction for dispersion;Fig. 4). Thus, such shift could be expected froma slightly inaccurate surface-wave GS correction.

As Durek and Ekström (1997), Masters andLaske (1997), and Roult and Clévédé (2000)argued, long-period surface-wave measurementsmay be affected by noise and difficulties indefining the time windows for separating thefundamental modes from the various overlap-ping wave trains. They concluded that normal-mode estimates can generally be carried out moreaccurately and may be more reliable. At thesame time, normal-mode χ estimates also tend tobe decreased by noise, particularly in the pres-ence of heterogeneity, which may account fora part of their gap with surface-wave estimates(Romanowicz and Mitchell 2007). Although theorigin of this discrepancy has still not been estab-lished, our empirical observations above show thatlong-period surface-wave measurements allow alot of room for adjustments by recognizing theirGS component. Note that according to Eq. 16,the GS factor is effectively accumulated alongthe paths from the source to receiver. There-fore, for example, predominance of continentalsurface-wave recordings (which are mostly con-ducted in tectonically active areas with higher γi)from deep-focus earthquakes (also likely withhigher γi with respect to the overlaying layeredmantle and crust) could cause increased γ valueswhen globally averaged for the correspondingwave modes in Fig. 3b. By contrast, normal-modemeasurements are dominated by the oceanic areaswith presumably lower γi.

10 Discussion

This paper focuses on classification of attenuationmeasurements irrespective of the models for at-

tenuation or Q. When practically raw values of χ

are presented in a linear frequency scale, they re-veal piecewise-linear frequency dependences andsuggest many modifications of the existing inter-pretations. These observations also show newdirections for research, some of which were out-lined above. For example, the causes of GS under-and over-compensation of Rayleigh waves withinthe shorter- and longer-period bands, respec-tively, need to be established by detailed analysisand modeling. The amount of potential bias inglobal-average χ for long periods needs to beevaluated and compared to the discrepancy withthe normal-mode estimates. Modeling and inver-sion techniques for χi( f ) need to be developed,and potentially many datasets revisited at theraw-data level. Values of Qe, and consequentlyof Qi, are often strongly increased (up to ∼20–30 times, Morozov 2008) compared to Q0, andQs is removed, leading to dramatic changes ininterpreting the nature of attenuation. The sepa-ration of the geometrical parameter γi from Q−1

icasts serious doubts on the validity of interpretingthe entire in situ Q−1 as the complex argumentof the medium’s elastic modulus (e.g., Andersonand Archambeau 1964; Aki and Richards 2002).Note that most modern inversions for global at-tenuation (e.g., Dahlen and Tromp 1998; Daltonet al. 2008) are based on this assumption, whichallows using the velocity sensitivity kernels forderiving Q−1.

Apparently, the most significant implication ofthe proposed χ( f ) view relates to the problemof the frequency dependence of Q within themantle. This problem can obviously never besolved in favor of the frequency-independentmodel, merely because it is far more restrictive,and new data conflicting with it may arise. Bycontrast, the frequency-dependent Q model isextremely permissive, and its inherent trade-offsallow easy reconciling different datasets. Dueto its rich theoretical implications, this modelis also favored by most seismologists sinceearly 1960s. Modern visco-elastic models rou-tinely start by postulating rheological relaxationmechanisms and complex-valued elastic moduliwithin the Earth (e.g., Dahlen and Tromp 1998;Borcherdt 2009), which automatically lead to afrequency-dependent Q. However, also because

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the frequency-independent model is more restric-tive, ascertaining its validity would have advancedus much further in understanding the Earth’sstructure, properties of its materials, and the me-chanics of seismic wave propagation. Therefore,I suggest that this avenue should be explored tothe end, and the frequency-independent modelis not ruled out until conclusive and unbiasedexperimental evidence against it is found.

Unfortunately, reviewing the continuouslymounting evidence for frequency-dependent Qwithin the crust and mantle shows that in manycases, the initial presentation of experimental datais done with a definitely frequency-dependent Qin mind. In a vast majority of papers, attenuationdata are presented only by apparent Q( f ) depen-dences (e.g., Aki 1980), and if the attenuationcoefficient is used, it is shown as a function ofperiod, f −1 (e.g., Mitchell 1995). However, asshown above and in Morozov (2008, 2009a), pre-senting the raw χ as in linear f scales revealslinear dependences that should be indicativeof some fundamental properties of attenuationprocesses. Such two key observations from theabove examples are: (1) χ( f ) usually containsa non-zero frequency-independent contribution,which can be measured by the intercept χ(0)

and interpreted as caused by the residual GSand dispersion; (2) frequency-dependent incre-ments χ( f ) − χ(0) are linear for all wave typesand datasets considered (Fig. 4); and (3) non-linearities of the observed Q( f ) and χ( f −1) de-pendences are usually spurious and caused bythe corresponding transformations from χ( f ).The second of these observations suggests thata frequency-independent Q model is viable andnatural from such observations.

Seismic measurements rarely span a continu-ous range of frequencies wide enough to allowdetection of a frequency-dependent Qe. However,the apparent Q or t∗ values vary even within therelatively narrow observation bands (e.g., Fig. 2a),which makes them problematic for comparisons.By contrast, parameters γ and Qe characterizethe entire sets of near-linear χ( f ) observations(Fig. 2a), and consequently they should provide amore consistent basis for analysis.

The most reliable experimental indicationsof frequency-dependent attenuation come from

comparing different frequency bands (e.g., Sipkinand Jordan 1979), and often from combiningdifferent wave types (Der et al. 1986; Cong andMitchell 1988). To re-examine the frequency de-pendence of crustal attenuation across about twodecades in frequency, let us correlate the 3–70 sRayleigh wave results for South America fromHwang and Mitchell (1987) and from 0.4–1.4 HzLg Q0 and η measurements by Raoof and Nuttli(1984; Fig. 7). These data were interpreted byCong and Mitchell (1988), who concluded that thefrequency dependence of crustal Qβ in its tectonic(western) part is weak (with exponent ζ < 0.3 inthe Qβ = Q0 f ζ law) and within the stable (east-ern) part—much stronger (ζ ≈ 0.7). According toMitchell (1995), strong frequency dependence istypical for stable areas. A strong contrast in at-tenuation levels was also found (from Q0 ≈ 900in eastern part to Q0 ≈ 200 in the western partof this area). However, by looking at the data inFig. 7 without tailoring them to the (Q0, η) model,we arrive at quite different conclusions.

Cong and Mitchell (1988) derived their ζ valuesby using a procedure schematically illustrated bythe dotted arrows in Fig. 7. They first constructedcrustal models consistent with the Rayleigh-waveattenuation (left arrow in each plot), then scaledtheir Qβ values by using trial ζ parameters and nu-merically modeled the Lg-phase attenuation. Theresulting values of ζ were established by match-ing the Lg attenuation with the measurementsby Raoof and Nuttli (1984) at 1-s periods (rightarrows and dark-grey bars in Fig. 7a, b). However,in respect to this procedure, note that: (1) in eachcase, it used only a single point, namely that at1 Hz, and ignored the rest of the measured Lgattenuation-coefficient trends and (2) this choiceof 1-Hz reference, although well-established byconvention, is completely arbitrary, and differentchoices for this frequency would have changed thevalues of ζ .

In χ( f ) diagrams (Fig. 7a, b), Q−1( f ) valuesof the Rayleigh and Lg waves correspond to theslopes of the corresponding radius-vectors shownby dotted arrows: Q−1( f ) = χ( f )/π f . Conse-quently, larger values of ζ required to reconcilethese Q’s in the stable-area case corresponds tothe wider angle between these arrows (Fig. 7b).Thus, the increased interpreted ζ in the stable

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Fig. 7 Comparison between surface wave and Lg atten-uation data converted to χ( f ) from: a western part ofSouth America (tectonically active); and b its eastern part(stable). “Frequency-reduction” was applied to χ( f ), sothat linear trends with Qe = 800 appear horizontal. Errorbars show fundamental-mode Rayleigh wave data fromHwang and Mitchell (1987; labeled H&M). Grey-shaded

areas labeled R&N show Lg χ( f ) derived from Q0 and η

values reported by Raoof and Nuttli (1984). Black dashedlines are the linear χ( f ) interpretations of Lg waves asin Fig. 2c,d. Dotted arrows labeled C&M and grey barsat 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

area (Cong and Mitchell 1988) is actually causedby larger Qe and lower γ in this area (i.e., morehorizontal and lower-placed grey-shaded ( f , χ)distribution for Lg waves in Fig. 7b).

Plotting the raw attenuation-coefficient data(for Lg, here reconstructed from Q0 and η mapsby Raoof and Nuttli 1984) allows seeing the ba-sic relationships between them without the useof assumption-prone Q( f ) models and numericalmodeling. In Fig. 7, we see that Lg χ( f ) distrib-utions have the same patterns as shown in Fig. 4,and representative linear Lg trends (black dashedlines) can be identified. As above, we only usean interpretive approach by drawing “the mostlikely” linear χ( f ) trends through the positionsof the dark-grey observation bars at 1 Hz inFig. 7a and b. Several observations can be madeby directly comparing these trends to those forRayleigh waves:

1. In the stable area (Fig. 7b), there is no sig-nificant difference between the Rayleigh waveand Lg χ( f ) across the entire frequency band.For both types of waves, γ ≈ (0.2–0.3) ×10−3 s−1 and frequency-independent Qe ≈ 800.

2. In the active area (Fig. 7a), both the Rayleighwave and Lg data are again consistent withthe same values of Qe ≈ 330, possibly increas-ing to ∼400 for Lg. These values are sig-nificantly higher than Q0 ≈ 170–220 by Raoofand Nuttli (1984), and there is no frequencydependence.

3. However, Lg γ in the active area is muchhigher, ∼6 × 10−3 s−1, than the Rayleighwave γ , which is ∼6 × 10−4 s−1 (Fig. 7a,see also Fig. 2c). This is just below thelower threshold for body and coda wave γD ≈8 × 10−3 s−1 proposed for tectonic areas byMorozov (2008). However, note that valuesabove γD are still within the uncertainty ofthe reconstructed Lg χ( f ) data (dash-dottedline with γ ≈ 9 × 10−3 s−1), in which case Qe

would likely increase to ∼400.

The similarity of Rayleigh-wave Qe values in thetwo areas and the difference of their γ ‘s suggestthat the principal difference between them con-sists in the structure of the upper crust. At 3–70-speriods, Rayleigh-wave Qe should be princi-pally controlled by the mid- and lower crust,

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whereas the lower velocity, lower Q uppercrust modifies the distribution of wave ampli-tudes, which is described by the geometric fac-tor. By contrast, the high-frequency Lg-waveQe likely closely corresponds to the Qi of theupper crust. The corresponding high γ could beexplained by the upper-crustal velocity gradientsand reflectivity, which for surface waves leadto reduced amplitudes recorded at the surface.Detailed theoretical treatment and numericalmodeling of these effects will be given elsewhere;at this time, it is important to ascertain that aphenomenological physical model can correctlyaccount for all observations (Figs. 2c, d, 3b, 4,and 7) without postulating a frequency-dependentmaterial Q.

If a quality-factor picture is still desired, Fig. 8shows the distribution of χ( f ) trends (Fig. 4)transformed into Q( f ) = π f/χ( f ) and plotted inlogarithmic frequency scale. The apparent Q( f )

consistently increases with frequency, but this in-crease mostly occurs within two bands (0.01–0.1and 1–100 Hz), below and between which Q isnearly constant. A log(Q) plot (Fig. 8b) shows thatwithin the different sub-bands, Q( f ) dependencescan also be approximately described by the Q0 f η

power law with exponent varying around η ≈0.5–1. However, this appearance should mostly bedue to the notorious universality of log–log plots.

The above reworking of several publisheddatasets shows that changes in the frequency de-

pendence of attenuation indeed occur at ∼300-s,∼100-s, and 10–1-s periods (Fig. 4), but they cor-respond to the transitions between different wavetypes dominating these frequency bands. Withineach of these wave types, the attenuation qualityQe and geometrical spreading γ are regionallyvariable and correlate with tectonics (for periodsshorter than ∼100 s) yet are frequency indepen-dent within the available observational uncertain-ties. Moreover, the values of Qe are generallyclose for all wave types below ∼100 s periods(Fig. 4). The widespread notion of Q pervasivelyincreasing with frequency may thus be due tothe fact that structural effects (γ ) are positiveand consistently increase during the transitionsto higher-frequency wave modes. Note that asmentioned above, such transitions can be formallyattributed to “scattering Q.” However, such as-sociation works only within the limited tasks ofattenuation measurements and may incorrectlydescribe the effects of the first-order Earth’s struc-ture as mere random scattering. The concept ofgeneralized GS, in the sense defined in Section 1and measured by parameter γ , is much more suit-able for this purpose.

Although a vast volume of other observationsstill remains to be reviewed in a similar manner,the examples presented here already indicatethat this general picture of surface-wave,Lg, and body-wave attenuation will likelyremain correct. It appears that no microscopic

Fig. 8 a Apparent Q( f ) corresponding to characteristic (γ , Qe) ranges in Fig. 4. Areas of active and stable tectonic typesare indicated. b The same plot in log10 Q( f ) form. Slopes corresponding to η = 0, 0.5, and 1 are indicated by arrows

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frequency-dependent attenuation mechanismsare required to explain the key observations.Although frequency-dependent elastic scatteringcertainly occurs within the Earth, as well asseismic-wave induced relaxation and creep insome structures, these effects are not nearly asdominant in attenuating seismic waves as it iscommonly thought. In the models discussed here,these effects appear indistinguishable from thefrequency-independent γi and Qe.

Finally, note that several specific and quanti-tative observations and correlations with geologybecome possible simply by presenting the attenu-ation observations in their generic form, as the at-tenuation coefficient χ plotted against frequency.This form is so naturally suggested by the scat-tering theory and by the character of attenuationmeasurements that it quite surprising that it isso rarely used. However, when utilized, this formreveals similarities even among such disparatewave types as normal modes, Lg, and body waves,and suggests a general classification of attenua-tion patterns (Fig. 4). Due to its generality, theapproach applies to most wave types used in atten-uation studies, including surface, Lg (this paper),crustal body waves and coda (Morozov 2008), Pnand synthetic surface waves, long-period P, andScS waves. Analysis of attenuation coefficientsis also free from underlying model assumptionsabout the geometrical spreading, which causegreat uncertainties in conventional (Q0, η) inter-pretations. The available data compilations showthat parameter Qe, and especially γ clearly cor-relate with tectonic ages (Morozov 2008). Fi-nally, values of γ also appear to be predictablefrom structural data, by a completely independentwaveform modeling (Morozov et al. 2008).

11 Conclusions

Analysis of the frequency dependence of the at-tenuation coefficient leads to significant changesin the interpretation of seismic attenuation data.Continuing the study of crustal body and codawaves by Morozov (2008), several publishedRayleigh wave and Lg attenuation studies are re-visited from a uniform standpoint of the temporalattenuation coefficient, χ . In all cases considered,

the dependence of χ on frequency is found tofollow the linear χ( f ) ≈ γ + π f/Qe law expectedfrom a phenomenological theory.

The observed piecewise-linear pattern of χ( f )

allows a simple classification of attenuation-coefficient dependences within a broad range offrequencies from ∼500 s to ∼1.5 Hz. Three groupsof linear patterns are revealed: Rayleigh waves at500–100 and 100–10 s, and Lg waves from ∼2 sto ∼1.5 Hz. The last of these segments overlapswith similar linear χ( f ) patterns of body, Pn,and coda waves (Morozov 2008), which extend to∼100 Hz. Within each of these frequency bands,parameters γ and Qe can be considered as con-stant, and they rapidly increase between thesebands. Such increases are related to changingwave types, particularly as a result of crustaleffects.

For both Rayleigh waves at 100–10 s and Lg,the levels of γ are lower within stable areas andhigher in the areas of active tectonics, which areseparated by the levels of γD ≈ 0.2 × 10−3 s−1

and 8 × 10−3 s−1, respectively. The threshold forLg also coincides with the corresponding γD forhigh-frequency body and coda waves described byMorozov (2008).

The proposed χ( f ) phenomenology forRayleigh waves suggests an explanation for therecently recognized discrepancy between thevalues of Q measured from long-period surfacewaves and from normal-mode oscillations. Thediscrepancy amounts in �χ ≈ 10−5 s−1, whichcan be explained by a small uncertainty in themeasured geometrical effect for surface waves.Such uncertainty could be related, for example, topredominantly continental observations.

To model the observed χ , the “intrinsic atten-uation coefficient” of the propagating medium isdefined and denoted χi. This parameter general-izes the intrinsic attenuation by incorporating thevariations of geometrical spreading and dispersionwithin the medium. Two frequency-dependentcrustal Q models are recast in this form and showthat the difference between the tectonically activeand stable crust should primarily be related to thedifferences in γi and Qi within the upper crust.

Frequency-dependent rheological or scatteringQ is not required for explaining the observationsconsidered in this study. The often-interpreted

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increase of Q with frequency from ∼2 mHz to∼100 Hz is shown to be apparent. Three generalcauses for interpreting a frequency-dependent insitu Q are identified: (1) presenting the attenu-ation data in various forms obscuring the linearχ( f ) dependences, such as Q( f ) or α(period),(2) ignoring the fact of γ being non-zero andvariable in realistic structures, and (3) follow-ing preferred theoretical Q( f ) dependences whilecutting across the χ( f ) trends observed within theindividual frequency bands.

Acknowledgements Two anonymous reviewers havegreatly helped in improving the presentation and suggestedseveral references. This research was supported by NSERCDiscovery Grant RGPIN261610-03. GNU Octave software(http://www.gnu.org/software/octave/) and GMT programs(Wessel and Smith 1995) were used in preparation ofseveral illustrations.

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