Seismic Scattering in the Deep Earth

Peter M. ShearerInstitute of Geophysics and Planetary Physics

Scripps Institution of OceanographyUniversity of California, San Diego

La Jolla, CA 92093-0225

October 21, 2005

Abstract

Recent observations suggest that seismic scatter-ing occurs in the lower mantle and inner core,in addition to the long-established strong scat-tering in the lithosphere. A number of differ-ent scattering theories for random media con-taining small-scale heterogeneities have been de-veloped to model coda and other scattered ar-rivals in short-period seismic waves, includingboth single (Born) and multiple scattering ap-proaches. Computer advances now permit de-tailed finite difference modeling as well as MonteCarlo simulations based on radiative transfer the-ory. Scattering observations from a variety ofbody-wave phases, including S, P , Pdiff , andPKiKP coda, and PKP , PKKP and P ′P ′ pre-cursors, constrain deep-Earth scattering and sug-gest significant small-scale compositional heter-ogeneity throughout the mantle and within atleast the uppermost inner core. The increasedavailability of large global seismic data sets willenable more detailed future studies and betterseparation between intrinsic and scattering at-tenuation mechanisms.

1 INTRODUCTION

Most seismic analyses of Earth structure rely onobservations of the travel times and waveforms ofdirect seismic waves that travel along ray pathsdetermined by Earth’s large-scale velocity struc-ture. These observations permit inversions for ra-dially averaged P -wave and S-wave velocity pro-files as well as three-dimensional perturbations.However, smaller-scale velocity or density pertur-bations cause some fraction of the seismic energyto be scattered in other directions, usually arriv-

ing following the main phase as incoherent en-ergy over an extended time interval. This later-arriving wavetrain is termed the coda of the di-rect phase. Given the number of different scat-tering events and the complexity of the scatteredwavefield, it is generally impossible to resolve in-dividual scatterers. Instead, coda-wave observa-tions are modeled using random media theoriesthat predict the average energy in the scatteredwaves as a function of scattering angle, given thestatistical properties of the velocity and densityperturbations. In this way, it is possible to char-acterize Earth’s heterogeneity at much smallerscales than can be imaged using tomography orother methods.

The fact that direct seismic waves can beobserved in the Earth indicates that this scat-tering must be relatively weak so that a signif-icant fraction of the seismic energy remains inthe primary arrivals. In contrast, scattering onthe Moon is proportionally much stronger thanin the Earth, preventing the easy observation ofdirect P and S waves at global distances (at leastat the recorded frequencies of the available data)and complicating inversions for lunar structure.In addition to facilitating observations of directarrivals, weak (as opposed to strong) scatteringalso can simplify modeling by permitting use ofsingle-scattering theory (i.e., the Born approx-imation). However, it is now clear that accu-rate modeling of scattering in the lithosphere,and possibly deeper in the mantle as well, re-quires calculations based on multiple scatteringtheories. Fortunately, increased computer powermakes these calculations computationally feasi-ble.

Although both body waves and surface wavesexhibit scattering, my emphasis in this review is

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on observations and modeling of deep Earth scat-tering, for which body waves provide the primaryconstraints. In addition, I also will give more at-tention to the mantle and core than the litho-sphere, which has been the focus of the majorityof coda studies to date. Finally, I will only brieflysummarize the different scattering theories. Formore details on these topics, the reader shouldconsult the book by Sato and Fehler (1998),which provides an extensive review of scatteringtheory and analysis methods, as well as a com-prehensive summary of crustal and lithosphericstudies.

2 SCATTERING THEORY

Wave scattering from random heterogeneities isa common phenomenon in many fields of scienceand theoretical modeling approaches have beenextensively developed in physics, acoustics andseismology. Solving this problem for the full elas-tic wave equation (i.e., for both P and S waves) inthe presence of strong perturbations in the elas-tic tensor and density is quite difficult, so vari-ous simplifying approximations are often applied.These include assuming an isotropic elastic ten-sor, using first-order perturbation theory in thecase of weak scattering, using the diffusion equa-tion for very strong scattering, and assuming cor-relations among the velocity and density pertur-bations.

2.1 Single Scattering Theory and Ran-dom Media

For sufficiently weak velocity and density pertur-bations, most scattered energy will have experi-enced only one scattering event and can be ad-equately modeled using single-scattering theory.The mathematics in this case are greatly simpli-fied if we assume that the primary waves are un-changed by their passage through the scatteringregion (the Born approximation). The total en-ergy in the seismic wavefield therefore increasesby the amount contained in the scattered wavesand energy conservation is not obeyed. Thusthis approximation is only valid when the scat-tered waves are much weaker than the primarywaves, which is the case in the Earth when thevelocity and density perturbations are relativelysmall (quantifying exactly how small dependsupon the frequency of the waves and the source-

to-receiver distance). Single-scattering theory issometimes called Chernov theory after Chernov(1960). Detailed descriptions of Born scatteringtheory for elastic waves are contained in Wu andAki (1985a,b), Wu (1989) and Sato and Fehler(1998). A review of the properties (elasticity,conductivity, permeability) and statistics of ran-dom heterogeneous materials is given in the textby Torquato (2002).

Single-scattering theory provides equationsthat give the average scattered power as a func-tion of the incident and scattered wave types (i.e.,P or S), the power of the incident wave, thelocal volume of the scattering region, the bulkand statistical properties of the random medium,the scattering angle (the angle between the in-cident wave and the scattered wave), and theseismic wavenumber (k = 2π/Λ, where Λ is thewavelength). A general random medium couldhave separate perturbations in P velocity, S ve-locity and density, but in practice a commonsimplification is to assume a linear scaling re-lationship among the perturbations (e.g., Sato,1990) and/or to assume zero density perturba-tions. However, as pointed out by Hong et al.(2004) density variations can have an importantinfluence on scattering properties. Performingthe actual calculation for a specific-source re-ceiver geometry involves integrating the contri-butions of small volume elements over the scat-tering region of interest. Each volume elementwill have a specific scattering angle and geomet-rical spreading factors for the source-to-scattererand scatterer-to-receiver ray paths.

***Figure 1 near here***The nature of the scattering strongly depends

upon the relative length scales of the heterogen-eity and the seismic waves. The scale of pertur-bations in a random medium can be character-ized by the autocorrelation function (ACF), withthe correlation distance, a, providing a roughmeasure of the average size of the “blobs” inmany commonly assumed forms for the ACF(e.g., Gaussian, exponential, van Karman, etc.).Figure 1 shows examples of random realizationsof the Gaussian and exponential ACF models. Ifthe heterogeneity is large compared to the seis-mic wavelength (a � Λ, ka is large), then for-ward scattering predominates and becomes in-creasingly concentrated near the direction of theincident wave as ka increases. In the limit oflarge ka, the energy remains along the primaryray path and scattering effects do not need to be

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taken into account. Alternatively, if the blobsare small compared to the seismic wavelength(a � Λ, ka is small), then the scattering is of-ten approximated as isotropic and the scatteredpower scales as k4a3. In the limit of small ka, thescattering strength goes to zero and the mediumbehaves like a homogeneous solid. As discussedby Aki and Richards (1980, p. 749-750), scat-tering effects are strongest when a and Λ are ofcomparable size (i.e., when ka ∼ 2π).

Aki and Chouet (1975) presented an impor-tant application of single scattering theory topredict coda decay rates for local earthquakes.For a co-located source and receiver and homo-geneous body-wave scattering in 3-D media, theyobtained

AC(t) ∝ t−1e−ωt/2QC (1)

where AC is the coda amplitude at time t (fromthe earthquake origin time) and angular fre-quency ω. QC is termed the coda Q and therehas been some uncertainty regarding its physi-cal meaning, in particular whether it describesintrinsic attenuation, scattering attenuation, orsome combination of both. I will discuss thismore later in the context of more complete theo-ries. Regardless of its interpretation, this formulahas proven successful in fitting coda decay ratesin a large number of studies.

Single-scattering theory has also been im-portant for modeling deep Earth scattering interms of random heterogeneity models, includ-ing interpretation of PKP precursor observa-tions (e.g., Haddon and Cleary, 1974; Doornbos,1976), PP precursors (King et al., 1975), P ′P ′

precursors (Vinnik, 1981), Pdiff coda (Earle andShearer, 2001) and PKiKP coda (Vidale andEarle, 2000). Born theory has also been usedto model expected travel time variations in di-rect arrivals that travel through random velocityheterogeneity (e.g., Spetzler and Snieder, 2001;Baig et al., 2003; Baig and Dahlen, 2004a,b). Al-though my focus in this paper is largely on inco-herent scattering from random media, it shouldbe noted that the Born approximation can alsobe used to model the effect of specific velocitystructures, provided their perturbations are weakcompared to the background velocity field. Inthis case, true synthetic seismograms can be com-puted, not just the envelope functions. For exam-ple, Dalkolmo and Friederich (2000) recently usedthis approach to model the effect of several dif-ferent hypothesized velocity anomalies near the

CMB on long-period P waves. In addition, Borntheory forms the basis for computing sensitivitykernels in finite-frequency tomography methods(e.g., Dahlen et al., 2000; Nolet et al., 2006)

2.1.1 Q notation and definitions

Coda Q, intrinsic Q and scattering Q will betermed QC , QI and QSc, respectively. P -waveand S-wave Q are termed αQ and βQ, respec-tively. These can be combined so that, for exam-ple, βQI is intrinsic S-wave Q. This conventioneliminates any chance of confusing shear-waveQ and scattering Q (both have sometimes beentermed QS). The transmission Q, QT , describesthe total attenuation (both intrinsic and scatter-ing) suffered by the direct wave

Q−1T = Q−1

I + Q−1Sc (2)

and the amplitude reduction of the transmittedpulse for a constant QT medium is

A(t) = A0 e−ωt/2QT (3)

where A0 is the amplitude of the pulse at t = 0and we have ignored any geometrical spreading.

The scattering coefficient, g, is defined asthe scattering power per unit volume (e.g., Sato,1977) and has units of reciprocal length. Thetotal scattering coefficient, g0, is defined as theaverage of g over all directions and can also beexpressed as

g0 = `−1 = Q−1Sc k (4)

where ` is the mean free path and k is thewavenumber. One common way to estimate g0

for S waves has been to compare the energy inthe S coda to the total radiated S energy. Fi-nally, following Wu (1985) we define the seismicalbedo as the ratio of scattering attenuation tototal attenuation

B0 =Q−1

Sc

Q−1Sc + Q−1

I

=g0

g0 + Q−1I k

(5)

These definitions of Q, g0, and B0 are general andcan be applied to the multiple scattering theoriesdiscussed later in this paper.

2.2 Finite Difference Calculations andthe Energy Flux Model

Finite difference methods provide a direct, albeitcomputationally intensive, solution to the seismic

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wave equation for media of arbitrary complex-ity, and they (together with the finite elementmethod) have become one of the most widelyused techniques in seismology. Their earliest ap-plications to study scattering involved modelingsurface-wave and body-to-surface wave scatteringfrom surface topography, sediment-filled basins,and other buried interfaces (e.g., Levander andHill, 1985). Here I will discuss only their use inmodeling body-wave scattering in random media.Reviews of this topic are contained in Frankel(1990) and Sato and Fehler (1998).

As computing power has improved, finite dif-ference simulations have progressed from the 2-D parabolic approximation, to 2-D using thefull wave equation, to full 3-D synthetics. Theparabolic approximation considers only forwardscattering and is useful when the heterogeneitycorrelation length is large compared to the seis-mic wavelength. Complete finite difference sim-ulations in 2-D random media have been per-formed by Frankel and Clayton (1984, 1986),McLaughlin et al. (1985), McLaughlin and An-derson (1987), Frankel and Wennerberg (1987),Gibson and Levander (1988), Roth and Korn(1993), and Saito et al. (2003). Frenje and Juh-lin (2000) computed both 2-D and 3-D finite dif-ference simulations. Hong and Kennett (2003),Hong (2004) and Hong et al. (2005) used awavelet-based numerical approach to compute 2-D synthetics for random media and Hong and Wu(2005) computed 2-D synthetics for anisotropicmodels.

The Frankel studies provided key results informulating the influential energy-flux model ofseismic coda (Frankel and Wennerberg, 1987)so I will describe them in some detail. Frankeland Clayton (1986) modeled teleseismic P -wavetravel-time variations with a∼1-Hz plane waveletvertically incident on a layer 150 km wide by55 km thick, with a finite difference grid spac-ing of 500 m. They found that observed traveltime variations of about 0.2 s (rms) among sta-tions spaced 10 to 150 km apart could be ex-plained with 5% rms random P velocity varia-tions, provided the correlation length was 10 kmor greater. Frankel and Clayton (1986) alsomodeled high frequency coda from local earth-quakes using a ∼20-Hz explosive source at thebottom corner of a layer 8 km long by 2 km thick.They found that the amplitude of high-frequencycoda depends strongly on the presence of highwavenumber velocity perturbations. Gaussian

and exponential models with correlation lengthsof 10 km or greater (required to fit observed tele-seismic travel time variations) do not have suf-ficient small-scale structure to produce observedlevels of high-frequency coda. In contrast, a self-similar random medium model with a correlationdistance of at least 10 km and rms velocity vari-ations of 5% can account for both sets of obser-vations.

By measuring peak amplitude versus distancein their synthetics, Frankel and Clayton (1986)estimated Q for their random medium. Becausetheir finite difference calculation did not containany intrinsic attenuation, this represents a mea-sure of scattering Q (QSc). The predicted atten-uation (Q−1) peaks at ka values between 1 and 2for Gaussian random media and between about1 and 6 for exponential random media. This isconsistent with the strongest scattering occurringwhen the seismic wavelength is comparable to thesize of the scatterers. However, attenuation isconstant with frequency for self-similar randommedia, as expected since the velocity fluctuationshave equal amplitudes over a wide range of scales.Frankel and Clayton showed that these resultswere in rough agreement with those predictedby single-scattering theory in two-dimensions (tomatch the geometry of the finite-difference simu-lations).

Frankel and Clayton (1986) also measuredcoda decay rates for finite-difference syntheticscomputed for sources within a 12 km by 12 kmgrid at 20 m spacing. They found that theirobserved coda decay rates were significantly lessthan those predicted by single scattering theoryin the case of moderate to large scattering at-tenuation (QSc ≤ 200), indicating that multiplescattering is contributing a substantial portion ofthe coda energy. This implies that in these casescoda Q (QC) as determined from coda falloff andthe single-scattering model of coda (e.g., Aki andChouet, 1975) does not provide a reliable esti-mate of transmission Q.

***Figure 2 near here***Motivated by these finite-difference results

and the limitations of the single-scattering modelof coda generation, Frankel and Wennerberg(1987) introduced what they termed the en-ergy flux model of coda. This phenomenolog-ical model is based on the idea that the codaenergy behind the direct wave front can be ap-proximated as homogeneous in space. This ob-servation had previously been reported for mi-

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croearthquake coda for lapse times more thantwice the S-wave travel time (e.g., Aki, 1969;Rautian and Khalturin, 1978), and Frankel andWennerberg (1987) showed that it also could beseen in finite-difference synthetics (see Figure 2).It implies that at sufficiently long times, thecoda amplitude at all receivers is approximatelythe same (scaling only with the magnitude ofthe source), regardless of the source-receiver dis-tance. The energy flux model permits the timedecay of the coda amplitude to be modeled verysimply and to separate the effects of scatteringand intrinsic attenuation in the medium.

By considering the energy density of the codauniformly distributed in an expanding volume be-hind the direct wave front, Frankel and Wenner-berg derived an expression for the predicted timedecay of the coda amplitude

AC(t) ∝ t−3/2e−ωt/2QI

√1− e−ωt/QSc (6)

where t is time, ω is angular frequency, Q−1I is

intrinsic attenuation, and Q−1Sc is scattering at-

tenuation. For short time and/or high QSc (i.e.,weak scattering, tQ−1

Sc is very small), this equa-tion reduces to

AC(t) ∝ t−1e−ωt/2QI (7)

This is equivalent to the Aki and Chouet(1975) expression (equation 1) for the single-scattering model, assuming coda Q and intrin-sic Q are equivalent (QC = QI). This agreeswith the original interpretation of QC given byAki and Chouet (1975) and contradicts the Aki(1980) statement that in the context of single-scattering theory, QC should be considered asan effective Q that includes both absorptionand scattering effects. Frankel and Wennerberg(1987) showed that the energy flux model pre-dicts the amplitude and coda decay observedin finite-difference synthetics for random mediawith a wide range of scattering Q, and is moreaccurate than the single-scattering model for me-dia with moderate to strong scattering attenua-tion (QSc ≤ 150). Finally, they used the energyflux model to estimate QSc and QI from the codaof two M ∼ 3 earthquakes near Anza, California.

There have been numerous other studies thathave attempted to resolve QSc and QI from localearthquake coda (e.g., Wu and Aki, 1988; Toksozet al., 1988; Mayeda et al., 1991; Fehler et al.,1992). Notable is Mayeda et al. (1992), who an-alyzed S-wave coda from Hawaii, Long Valley,

and central California. They found a complicatedrelationship between theoretical predictions andobserved QC , QSc and QI and argued that mod-els with depth-dependent scattering and intrinsicattenuation are necessary to explain their results.

The energy flux model was developed to ex-plain local earthquake coda and finite differencesimulations of spherical wavefronts in media withuniform scattering. It is not directly applica-ble to modeling teleseismic coda because of thestrong concentration of scattering in the crustand lithosphere compared to much weaker scat-tering deeper in the mantle. This has motivatedthe development of extended energy flux modelsinvolving the response of one or more scatteringlayers to a wave incident from below (e.g., Korn,1988, 1990, 1997; Langston, 1989).

The resulting formulas for the coda decay rateare more complicated than the simple energyflux model because they depend upon severaladditional parameters, including the travel timethrough the layer and the amount of leakage backinto the half-space. Korn (1988) developed thetheory for a spherical wave with a cone of energyincident upon a scattering zone and used it tomodel regional earthquakes recorded by the War-ramunga array in Australia. Langston (1989) de-veloped a scattering layer-over-half-space modeland showed that it was consistent with coda de-cay in teleseismic P waves recorded at two sta-tions (PAS and SCP) in the United States. Korn(1990) tested a scattering layer over homoge-nous half-space EFM using a 2-D acoustic finite-difference code and found that it gave reliable re-sults for both weak and strong scattering regimes.Korn (1997) further extended the EFM to ex-plicitly include depth dependent scattering andshowed that it gave reliable results when com-pared to synthetics computed for a 2-D elastic(P − SV ) finite difference code.

Wagner and Langston (1992a) computed 2-D acoustic and elastic finite-difference syntheticsfor upcoming P waves incident on 150 differentmodels of heterogeneous layers over a homoge-neous half-space. These models varied in theirlayer thickness, random heterogeneity correlationlength (different vertical and horizontal correla-tion lengths were allowed), and rms velocity het-erogeneity. They found that the scattering atten-uation of the direct pulse depends upon ka and isstrongest for spatially isotropic heterogeneity, inwhich case most of the coda energy was containedin low apparent velocity S waves and surface

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waves. In contrast, anisotropic models with hor-izontally elongated heterogeneities produce codawith mostly vertically propagating layer rever-berations.

More recent finite difference calculations forrandom media include the 2-D whole-Earth pseu-dospectral calculations of Furumura et al. (1998)and Wang et al. (2001), Thomas et al. (2000)who computed 2-D whole-Earth acoustic synthet-ics to model PKP precursors, Cormier (2000)who used a 2-D elastic pseudospectral method tomodel the effects of D′′ heterogeneity on the Pand S wavefields, and Korn and Sato (2005) whocompare 2-D finite difference calculations withsynthetics based on the Markov approximation.

2.3 Multiple Scattering Theories

If the energy in the scattered wavefield is a sig-nificant fraction of the energy in the direct wave,then the Born approximation is inaccurate anda higher-order theory should be used that takesinto account the energy reduction in the primarywave and the fact that the scattered waves mayexperience more than one scattering event. Theseeffects are all naturally accounted for using thefinite-difference calculations discussed above, butthese are computationally intensive and there isa need for faster approaches that also providephysical insight into the scattering process. Inthe case of very strong scattering, the diffusionequation can be applied by assuming a randomwalk process. Although this approach preservesenergy, it violates causality by permitting someenergy to arrive before the direct P wave. Thefirst applications of the diffusion equation in seis-mology include the coda wave analyses of Wesley(1965) and Aki and Chouet (1975) and the lu-nar seismogram studies of Nakamura (1977) andDainty and Toksoz (1977, 1981).

At large times and small distances from thesource, the diffusion equation predicts that thecoda amplitude varies as

A(t) ∝ t−3/4e−ωt/2QI (8)

where QI is the intrinsic attenuation. Noticethat QSc does not appear in this equation be-cause the exact level of scattering is not impor-tant provided it is strong enough that the energyis obeying a random-walk process. The diffusionequation can also be used to model the case of astrong scattering layer over a homogeneous half-space (e.g., Dainty et al., 1974; Margerin et al.,

1998, 1999; Wegler, 2004), in which case an addi-tional decay term exists to account for the energyleakage into the half-space.

Another approach to modeling multiple scat-tering is to sum higher-order scattered energy,and the predicted time dependence of scatteredenergy was obtained in this way for double-scattering (Kopnichev, 1977) and multiple scat-tering up to seventh order (Gao et al., 1983a,b).Hoshiba (1991) used a Monte Carlo approach(see below) to correct and extend these resultsto 10th order scattering. Richards and Menke(1983) performed numerical experiments on 1-Dstructures with many fine layers to characterizethe effects of scattering on the apparent atten-uation of the transmitted pulse and the relativefrequency content of the direct pulse and its coda.

Most current approaches to synthesizing mul-tiple scattering use radiative transfer theory tomodel energy transport. Radiative transfer the-ory was first used in seismology by Wu (1985)and Wu and Aki (1988) and recent reviews of thetheory are contained in Sato and Fehler (1998)and Margerin (2004). Other results are detailedin Shang and Gao (1988), Zeng et al. (1991),Sato (1993), and Sato et al. (1997). Sato andNishino (2002) use radiative transfer theory tomodel multiple Rayleigh wave scattering. Ana-lytical solutions are possible for certain idealizedcases (e.g., Wu, 1985; Zeng, 1991; Zeng et al.,2003; Sato, 1993) but obtaining general resultsrequires extensive computer calculations.

Two analytical results are of particular inter-est (and can be used as tests of numerical simu-lations). For the case of no intrinsic attenuation,Zeng (1991) showed the coda power converges tothe diffusion solution at long lapse times

PC(t) ∝ t−3/2 (9)

For elastic waves with no intrinsic attenuation,the equilibrium ratio of P and S energy densityis given by (e.g., Sato, 1994; Ryzhik et al., 1996;Papanicolaou et al., 1996)

EP /ES = 12 (β/α)3 (10)

Assuming a Poisson solid, this predicts about 10times more S energy than P energy at equilib-rium, a result of the relatively low efficiency ofS-to-P scattering compared to P -to-S scattering(e.g., Malin and Phinney, 1985; Zeng, 1993). Formedia with intrinsic attenuation, an equilibriumratio also exists but will generally differ from

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the purely elastic case (Margerin et al., 2001).Shapiro et al. (2000) showed that this ratio canbe estimated from the divergence and curl of thedisplacement as measured with a small-aperturearray and that its stability with time provides atest of whether the coda is in the diffusive regime.

Wu (1985) used radiative transfer theory toaddress the problem of separating scattering fromintrinsic attenuation. He showed that the codaenergy density versus distance curves have differ-ent shapes depending upon the seismic albedo,B0 (see equation 5), and thus in principle it ispossible to separate scattering and intrinsic Qby measuring energy density distribution curves.Hoshiba (1991) pointed out that in practice theuse of finite window lengths for measuring codawill lead to underestimating the total energy(compared to the infinite lapse time windows inWu’s theory), likely biasing the resulting esti-mates of seismic albedo. To deal with this prob-lem, Fehler et al. (1992) introduced the multi-ple lapse-time window (MLTW) analysis, whichmeasures the energy in consecutive time windowsas a function of epicentral distance. The MLTWapproach has been widely used in studies of Scoda (see below).

***Figure 3 near here***A powerful method for computing synthetic

seismograms based on radiative transfer theory isto use a computer-based Monte Carlo approachto simulate the random walk of millions of seismicenergy “particles” which are scattered with prob-abilities derived from random media theory. Fig-ure 3 illustrates a simple example of this methodapplied to 2-D isotropic scattering. Variationson this basic technique are described by Gusevand Abubakirov (1987), Abubakirov and Gusev(1990), Hoshiba (1991, 1994, 1997), Margerinet al. (2000), Bal and Moscoso (2000), Yoshi-moto (2000), Margerin and Nolet (2003a,b), andShearer and Earle (2004). Because of the po-tential of the Monte Carlo method for modelingwhole-Earth, high-frequency scattering, these re-sults are now summarized in some detail.

Gusev and Abubakirov (1987) used the MonteCarlo method to model acoustic wave scatteringin a whole space and considered both isotropicscattering and forward scattering with a Gaus-sian angle distribution. They parameterized thescattering in terms of a uniform probability perunit volume, resulting in an exponential distri-bution of path lengths. Intrinsic attenuation wasnot included. They showed that their results

agree with the diffusion model for large lapsetimes. Abubakirov and Gusev (1990) containsa more detailed description of this method andits application to model S-coda from Kamchatkaearthquakes. They obtained an S-wave meanfree path for the Kamchatka lithosphere of 110to 150 km over a 1.5 to 6 Hz frequency range.

Hoshiba (1991) modeled the spherical radi-ation of S-wave energy in a constant veloc-ity medium using a Monte Carlo simulationthat included isotropic scattering with uniformprobability. He showed that the results agreedwith single-scattering theory for weak scattering(i.e., travel distances less than 10% of the meanfree path) and agreed with the diffusion modelfor strong scattering (i.e., travel distances morethan 10 times longer than the mean free path).Hoshiba (1991) was able to use his Monte Carloresults to correct and extend the multiple scat-tering terms of Gao et al. (1983a,b). He foundthat his simulations at long lapse times wereconsistent with the radiative transfer theory ofWu (1985) but that reliable estimates of seismicalbedo are problematic from short time windows.Finally, Hoshiba (1991) showed that for multiplescattering, coda Q is much more sensitive to in-trinsic Q than to scattering Q (as was argued byFrankel and Wennerberg, 1987, on the basis ofthe energy flux model).

Wennerberg (1993) considered the implica-tions of lapse-time dependent observations of QC

and methods for separating intrinsic and scat-tering attenuation with respect to the single-scattering model, Zeng’s (1991) multiple-scatter-ing model, Hoshiba’s (1991) model, and Abuba-kirov and Gusev’s (1990) results. Hoshiba (1994)extended his Monte Carlo method to considerdepth dependent scattering strength and intrin-sic attenuation, and Hoshiba (1997) included theeffects of a layered velocity structure to modellocal earthquake coda at distances up to 50 km.Hoshiba simulated SH-wave reflection and trans-mission coefficients at layer interfaces as proba-bilities of reflection or transmission of particlesin the Monte Carlo method but did not includeP waves and the conversions between P and Swaves. The results showed that coda amplitudesdepend upon the source depth even late into thecoda.

Margerin et al. (1998) applied the MonteCarlo approach to a layer over a half-space model,representing the crust and upper mantle, andincluded both surface- and Moho-reflected and

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transmitted phases. As in Hoshiba (1997), reflec-tion and transmission coefficients are convertedto probabilities for the individual particles. Swaves only are modeled (using the scalar waveapproximation, i.e., no P to S conversions) andno intrinsic attenuation is included. Margerinet al. (1998) compared their numerical resultsin detail with solutions based on the diffusionequation and found good agreement for suitablemean-free-path lengths. They point out the im-portance of the crustal wave guide for trappingenergy near the surface and that the possibilityof energy leakage into the mantle should be takeninto account in calculations of seismic albedo.

Bal and Moscoso (2000) explicitly includedS-wave polarization and showed that S wavesbecome depolarized under multiple scattering.Yoshimoto (2000) introduced the direct simula-tion Monte Carlo (DSMC) method, which usesa finite-difference scheme for ray tracing and canthus handle velocity models of arbitrary complex-ity, including lateral varying structures. How-ever, intrinsic attenuation and directional scat-tering were not included. Yoshimoto showed thata velocity increase with depth strongly affects theshape of the coda envelope, compared with uni-form velocity models, and that it is important toproperly model energy that may be trapped atshallow depths.

Margerin et al. (2000) extended the MonteCarlo approach to elastic waves, taking into ac-count P to S conversions and S-wave polariza-tion. They considered scattering from randomlydistributed spherical inclusions within a homoge-neous background material, using the solutionsof Wu and Aki (1985a). For both Rayleighscatterers (spheres much smaller than the seis-mic wavelength) and Rayleigh-Gans scatterers(spheres comparable to the seismic wavelength)they found good agreement with single-scatteringtheory at short times and with the diffusion equa-tion solution at long times. In addition, the P -to-S energy density ratio and the coda decay rateat long times converged to their theoretical ex-pected values.

Margerin and Nolet (2003a,b) further ex-tended the Monte Carlo approach to model whole-Earth wave propagation and scattering. Theyshowed that their Monte Carlo synthetics for thePKP AB and BC branches produced energy ver-sus distance results in good agreement with ge-ometrical ray theory. They computed scatteringproperties based on random media models char-

acterized by velocity perturbations with an ex-ponential correlation length. For whole mantlescattering, they found that the Born approxima-tion is only valid up to mean free paths of about400 s, corresponding to 0.5% RMS velocity per-turbations. They also applied their method tomodel PKP precursor observations; these resultswill be discussed later in this paper.

Shearer and Earle (2004) implemented a par-ticle-based Monte Carlo method for computingwhole-Earth scattering. They included both Pand S waves radiated from the source, mode con-versions, S-wave polarizations, and intrinsic at-tenuation. For a simple whole-space model, theyshowed that their approach agreed with theoreti-cal results for the S/P energy ratio and expectedt−1.5 falloff in power at large times. For model-ing the whole Earth, they included the effectsof reflection and transmission coefficients at thefree surface, Moho, CMB and inner core bound-ary (ICB). Scattering probabilities and scatteringangles were computed assuming random velocityand density variations characterized by an expo-nential autocorrelation function. They appliedthis method to model the time and distance de-pendence of high-frequency P coda amplitudes(see section 3.2).

All of these results suggest that body-wavescattering in the whole Earth can now be ac-curately modeled using ray theory and particle-based Monte Carlo methods. Although some-what computationally intensive, continued im-provements in computer speed make them prac-tical to run on modest machines. They can han-dle multiple scattering over a range of scatteringintensities, bridging the gap between the Bornapproximation for weak scattering and the dif-fusion equation for strong scattering. They alsocan include general depth-dependent or even 3-D variations in scattering properties, includingnon-isotropic scattering, without a significant in-crease in computation time compared to simplerproblems.

2.4 Other Theoretical Methods

Lerche and Menke (1986) presented an inversionmethod to separate intrinsic and scattering at-tenuation for a plane layered medium. Gusev(1995) and Gusev and Abubakirov (1999a,b) de-veloped a theory for reconstructing a vertical pro-file of scattering strength from pulse broaden-ing and delay of the peak amplitude. Saito et

8

al. (2002, 2003) modeled envelope broadening inS waves by applying the parabolic approxima-tion to a von Karman random medium. Sato etal. (2004) extended this approach to develop ahybrid method for synthesizing whole-wave en-velopes that uses the envelope obtained from theMarkov approximation as a propagator in theradiative transfer integral and showed that theresults agreed with finite difference calculations.The effect of anisotropic random media (wherethe scattering properties depend upon the an-gle of the incident wave) was considered numeri-cally by Wagner and Langston (1992a) and Rothand Korn (1993), and theoretically by Muller andShapiro (2003) and Hong and Wu (2005). Re-cent work on using small-aperture seismic arrayson coda to constrain the directions of individ-ual scatterers includes Schissele et al. (2004) andMatsumoto (2005).

3 SCATTERING OBSERVA-TIONS

Seismic scattering within the Earth is mainly ob-served in the incoherent energy that arrives be-tween the direct seismic phases, such as P , S, andPKP . In addition, occasionally specific scatter-ers can be imaged using seismic arrays. Scatter-ing can also influence the direct phases throughamplitude reduction and pulse broadening, ef-fects characterized by the scattering attenuationparameter, Q−1

Sc . In this way, studies of seis-mic attenuation are also resolving scattering, al-though they often do not attempt to separate in-trinsic and scattering attenuation. The incoher-ent scattered seismic wavefield usually follows adirect seismic arrival, and is termed the coda ofthat phase (e.g., P coda, S coda), but occasion-ally the ray geometry is such that scattered en-ergy can arrive before a direct phase (e.g., PKPprecursors). These precursory arrivals are partic-ularly valuable for studying deep-Earth scatter-ing because they are less sensitive to the strongscattering in the lithosphere. Scattering is usu-ally studied at relatively high frequencies (1 Hzor above) where coda is relatively strong and lo-cal earthquake records have their best signal-to-noise.

***Figure 4 near here***S-wave coda from local and regional events

has been the focus of many studies, has moti-vated much of the theoretical work, and contin-

ues to be an active field of research. Here, I willbriefly review S coda studies and their implica-tions for scattering in the crust and lithosphere,but will devote more attention to other partsof the scattered seismic wavefield, which providebetter constraints on deep Earth structure. Pre-vious review articles that discuss deep Earth scat-tering include Bataille et al. (1990) and Sheareret al. 1998). Figure 4 shows where much of thescattered energy arrives with respect to travel-time curves for the major seismic phases. In prin-ciple, any seismic phase that travels through thelower mantle or the CMB will be sensitive to deepEarth scattering. Although separating deep scat-tering effects from crust and lithospheric scatter-ing can be challenging, this is possible in somecases, either from a fortunate ray geometry or bycareful comparison of coda amplitudes at differ-ent distances or between different phases.

The picture that is emerging from these stud-ies is that seismic scattering from small-scale ve-locity perturbations is present throughout theEarth, with the exception of the fluid outer core.However, the exact strength, scale length, anddepth dependence of the scattering remains un-resolved, particularly in the vicinity of the core-mantle and inner-core boundaries, where theymay also be contributions from topographic ir-regularities or boundary layer structure.

3.1 S Coda

S coda measurements are of two main types: (1)those that simply fit the coda decay rate and esti-mate βQC without attempting to separate scat-tering and intrinsic attenuation, and (2) thosethat measure energy density and estimate thescattering coefficient g0 (or its reciprocal, themean free path). The latter studies typically ob-tain a separate estimate for intrinsic attenuationand thus can compute the seismic albedo, B0.The basic relationships among these parametersare given in equations (4) and (5). In both typesof studies, frequency dependence can also be ex-amined by filtering the data in different bands.

Coda Q studies include Rautian and Khal-turin (1978), Roecker et al. (1982), Rodriquez etal. (1983), Singh and Herrmann (1983), Biswasand Aki (1984), Pulli (1984), Rhea (1984), Jin etal. (1985), Del Pezzo et al. (1985), Jin and Aki(1988), van Eck (1988), Kvamme and Havskov(1989), Matsumoto and Hasegawa (1989), Bask-outas and Sato (1989), Oancea et al. (1991), Tsu-

9

ruga et al. (2003), Giampiccolo et al. (2004), andGoutbeek et al. (2004).

Reviews of QC measurements include Her-raiz and Espinosa (1987), Matsumoto (1995), andSato and Fehler (1998). In general, QC is fre-quency dependent and increases from about 100at 1 Hz to 1000 at 20 Hz (i.e., there is less atten-uation at higher frequencies). However there areregional variations of about a factor of ten andtypically QC is lower in tectonically active areasand higher in stable regions, such as shields. Asdiscussed above, how to interpret QC in terms ofQSc and QI has been the subject of some debateand it is becoming increasingly clear that depth-dependent calculations (which include variationsin the background velocity, the scattering char-acteristics, and the intrinsic attenuation) are re-quired in many cases to fully describe coda ob-servations.

Some studies (e.g., Rautian and Khalturin,1978; Roecker et al., 1982; Gagnepain-Beyneix,1987; Kvamme and Havskov, 1989; Akamatsu,1991; Kosuga, 1992, Gupta et al., 1998; Singh etal., 2001; Giampiccolo et al., 2004, Goutbeek etal., 2004) have observed that QC increases withincreasing lapse time (i.e., equation (1) does notfit the entire coda envelope), suggesting that thelater part of the coda contains energy that propa-gated through less attenuating material than theearly part of the coda. Gusev (1995) showed thatan increase in QC with time in the coda is pre-dicted from a single isotropic scattering model inwhich QSc increases with depth. Margerin et al.(1998) applied radiative transfer theory to showthat a model of scattering in the crust abovemuch weaker scattering in the mantle predictsQC values that depend upon the reflection coef-ficient at the Moho, implying that energy leakageinto the mantle has implications for the interpre-tation of coda Q.

***Figure 5 near here***Following Sato and Fehler (1998), we may di-

vide scattering attenuation estimates into thoseobtained using the single-scattering model andthose based on multiple lapse-time window analy-sis. Single scattering studies include Sato (1978),Aki (1980), Pulli (1984); Dainty et al. (1987),Kosuga (1992), and Baskoutas (1996). Multiplelapse-time studies are summarized in Figure 5and include Fehler et al. (1992), Mayeda et al.(1992), Hoshiba (1993), Jin et al. (1994), Akinciet al. (1995), Canas et al. (1998), Ugalde et al.(1998, 2002), Hoshiba et al. (2001), Bianco et al.

(2002, 2005), Vargas et al. (2004), and Goutbeeket al. (2004). Values of the coefficient g0 for S-to-S scattering range from about 0.002 to 0.05km−1 (mean free paths of 20 to 500 km) for fre-quencies between 1 and 30 Hz. Some papers (e.g,Mayeda et al., 1992; Hoshiba, 1993; Jin et al.,1994; Goutbeck et al., 2004; Bianco et al., 2005)have found a frequency dependence in the seismicalbedo and among different regions, but in gen-eral this dependence is not as consistent as thatseen in QC studies. Several recent papers notedthe importance of considering depth-dependentvelocity structure in computing scattering atten-uation (e.g., Hoshiba et al., 2001; Bianco et al.,2002, 2005).

Lacombe et al. (2003) modeled S coda inFrance at epicentral distances between 100 and900 km using acoustic radiative transfer theoryapplied to a two layer model. They found that amodel with scattering confined to the crust anduniform intrinsic attenuation could explain theirdata at 3 Hz, but that the tradeoff between scat-tering and intrinsic attenuation was too strongto reliably determine the relative contribution ofeach parameter. At the regional epicentral dis-tances of their model, the crustal waveguide hada dominating effect on the S coda.

In addition to the S coda decay rates at rel-atively long lapse times used to determine codaQ, there are other aspects of the coda that pro-vide additional constraints on scattering. Thecomplete seismogram envelope can be studied, in-cluding the broadening of the direct S envelopeand the delay in its peak amplitude (e.g., Sato,1989, 1991; Scherbaum and Sato, 1991; Obaraand Sato, 1995). Deviations from the averagecoda decay rate for a number of stations can beinverted to construct a 3-D model of scatteringintensity in the crust and upper mantle (e.g.,Nishigami, 1991, 1997; Obara, 1997; Chen andLong, 2000; Taira and Yomogida, 2004). Similarmethods were used by Revenaugh (1995, 1999,2000) to back-project P coda recorded in south-ern California, and by Hedlin and Shearer (2000)to invert PKP precursor amplitudes (see below).Periodic rippling of SH coda envelopes in north-eastern Japan was noted by Kosuga (1997) whosuggested this may be caused by trapped waveswithin a low-velocity layer in the top part of thesubducting slab.

Spudich and Bostwick (1987) showed howGreen’s function reciprocity can be used to ob-tain information about the ray takeoff directions

10

from the earthquake source region, using a clus-ter of earthquakes as a virtual array. Applyingthis method to aftershocks of the 1984 MorganHill earthquake in California, they found thatthe early S coda was dominated by multiplescattering within 2 km of each seismic station.Scherbaum et al. (1991) used this approach tostudy two microearthquake clusters in northernSwitzerland and found that the early S coda con-tained energy leaving the source at close to thesame angle as the direct wave, but that later (atleast 1.5 to 2 times the S travel time) coda con-tains energy from waves leaving the source in avariety of directions. Spudich and Miller (1990)and Spudich and Iida (1993) showed how aninterpolation approach using distributed earth-quake sources can be used to estimate scatteringlocations in the vicinity of the 1986 North PalmSprings earthquake in California.

3.2 P Coda

Local and many regional coda studies have mainlyfocused on the S-wave coda because of its higheramplitude and longer duration than P -wave coda(which is truncated by the S-wave arrival). How-ever, at teleseismic distances, P waves and theircoda are more prominent than S waves at highfrequencies because of the severe effect of man-tle attenuation on the shear waves. A number ofstudies both from single stations and arrays haveattempted to resolve the origin of the scatteredwaves in the P coda and to distinguish betweennear-source and near-receiver scattering.

Aki (1973) modeled scattering of P waves be-neath the Montana LASA array using Chernovsingle-scattering theory. At 0.5 Hz, he achieved agood fit to the data with a random heterogeneitymodel extending to 60 km depth beneath LASA,with RMS velocity fluctuations of 4% and a cor-relation distance of 10 km. Stronger scatteringat higher frequencies exceeded the validity limitsof the Born approximation.

Frankel and Clayton (1986) compared tele-seismic travel-time anomalies observed across theLASA and NORSAR arrays, as well as veryhigh frequency coda (f ≥ 30 Hz) observed formicroearthquakes, to synthetic seismograms forrandom media generated using a finite differencemethod. They found that a random medium withself-similar velocity fluctuations (a ≥ 10 km) of5% within a ∼100 km thick layer could explainboth types of observations.

Dainty (1990) reviewed array studies of tele-seismic P coda, such as that recorded by theLASA and NORESS arrays. He distinguishedbetween “coherent” coda, which has nearly thesame slowness and back azimuth as direct P, and“diffuse” coda, which is characterized by energyarriving from many different directions. He ar-gued that coherent coda is generated by shallow,near-source scattering in the crust, rather thandeeper in the mantle, because it is absent or weakfor deep-focus earthquakes. In contrast, the dif-fuse coda is produced by near-receiver scatteringand has power concentrated at slownesses typicalof shear, Lg or surface waves. Lay (1987) showedthat the dispersive character of the first 15 s ofthe P coda from nuclear explosions was due tonear-source effects.

Gupta and Blanford (1983) and Cessaro andButler (1987) addressed the question as to howscattering can cause transversely polarized en-ergy to appear in P and P coda, an issue rel-evant to discrimination methods between earth-quakes and explosions. Their observations atdifferent distances and frequency bands sug-gested that both near-source and near-receiverscattering must be present. Flatte and Wu(1988) performed a statistical analysis of phaseand amplitude variations of teleseismic P wavesrecorded by the NORSAR array. They fit theirresults with a two-overlapping-layer model oflithospheric and asthenospheric heterogeneity be-neath NORSAR, consisting of the summed con-tributions from a 0 to 200 km layer with a flatpower spectrum and a 15 to 250 km layer witha k−4 power spectrum (the deeper layer corre-sponds to an exponential autocorrelation modelwith scale larger than the array aperture of 110km). The RMS P velocity variations are 1% to4%. This model has relatively more small-scalescatterers in the shallow crust (which the authorsattribute to clustered cracks or intrusions) andrelatively more large-scale scatterers in the as-thenosphere (which the authors suggest may betemperature or compositional heterogeneities).

Langston (1989) studied teleseismic P wavesrecorded at station PAS in California and SCPin Pennsylvania. He showed that the coda am-plitude cannot be explained by horizontally lay-ered structures of realistic velocity contrasts andthat three-dimensional scattering is required. Hecould explain the observed coda decay by adopt-ing the energy flux model of Frankel and Wen-nerberg (1987) for the case of a scattering layer

11

above a homogenous half-space. He found thatscattering is more severe at PAS than SCP, as in-dicated by higher coda levels and a slower decayrate, obtaining a scattering Q estimate for PASof 239 (2-s period) compared to 582 for SCP.

Korn (1988) examined the P coda from In-donesian earthquakes recorded at the Warra-munga array in central Australia and found thatthe coda energy decreased with increasing sourcedepth. He computed the power in the coda atdifferent frequency bands between 0.75 and 6 Hzand fit the results with the energy flux model(EFM) of Frankel and Wennerberg (1987). Theresults indicated frequency dependence in bothintrinsic attenuation and scattering attenuation(QI = 300 at 1 Hz, increasing almost linearlywith frequency above 1 Hz; QSc = 340 at 1 Hz,increasing as f0.85). Assuming random velocityfluctuations with an exponential autocorrelationfunction, results from Sato (1984) can be used toestimate that RMS velocity variations of 5% at5.5 km scale length are consistent with the ob-served QSc below the Warramunga array. Korn(1990) extended the energy flux model to accom-modate a scattering layer over a homogeneoushalf space (a modeling approach similar to thatused by Langston, 1989) and applied the methodto the short-period Warramunga array data ofKorn (1988) for deep earthquakes. With the newmodel, he found an average scattering Q of 640at 1 Hz for the lithosphere below the array andhigher values of intrinsic Q, implying that dif-fusion rather than anelasticity is the dominantfactor controlling teleseismic coda decay rates.Korn (1993) further applied this approach to tele-seismic P coda observations from nine stationsaround the Pacific. He found significant differ-ences in scattering Q among the stations (QSc =100 to 500). The observed frequency dependenceof QSc is in approximate agreement with single-scattering theory for random heterogeneities, andfavors von Karman type autocorrelation func-tions over Gaussian or self-similar models.

Bannister et al. (1990) analyzed teleseismic Pcoda recorded at the NORESS array using botharray and three-component methods. They re-solved both near-source and near-receiver scat-tering contributions to coda, with the bulk ofreceiver-side scattering resulting from P -to-Rgconversions from two nearby areas (10 and 30 kmaway) with significant topography. Gupta et al.(1990) performed frequency wavenumber analysison high-frequency NORESS data and identified

both near-receiver P -to-Rg scattering and near-source Rg-to-P scattering. Dainty and Toksoz(1990) examined scattering in regional seismo-grams recorded by the NORESS, FINESA andARCESS arrays. P coda energy was concen-trated in the on-azimuth direction, but appearedat different phase velocities, suggesting differentcontribution mechanisms.

Wagner and Langston (1992b) applied fre-quency-wavenumber analysis to P ′P ′ (PKPPKP )waves recorded by the NORESS arrays for deepearthquakes. Most of the coda was verticallypropagating but an arrival about 15 s afterthe direct wave can be identified as body-to-Rayleigh-wave scattering from a point 30 kmwest-southwest of the array (this scattering loca-tion was previously identified in P -wave coda byBannister et al., 1990, and Gupta et al., 1990).Revenaugh (1995, 1999, 2000) applied a Kirch-hoff migration approach to back-project P codarecorded in southern California to image lat-eral variations in scattering intensity within thecrust and identified correlations between scatter-ing strength and the locations of faults and othertectonic features.

Neele and Snieder (1991) used the NARS andGRF arrays in Europe to study long-period tele-seismic P coda and found it to be coherent withenergy arriving from the source azimuth. Theyconcluded that long-period P coda does not con-tain a significant amount of scattered energyand can be explained with spherically symmet-ric models, and is particularly sensitive to struc-ture in upper-mantle low-velocity zones. Rit-ter et al. (1997, 1998) studied the frequency de-pendence of teleseismic P coda recorded in theFrench Massif Central, which they modeled asa 70-km thick layer with velocity fluctuations of3% to 7% and heterogeneity scale lengths of 1 to16 km. Rothert and Ritter (2000) studied P codain intermediate depth Hindu Kush earthquakesrecorded at the GRF array in Germany about 44◦

away. They applied a method based on the the-ory of Shapiro and Kneib (1993) and Shapiro etal. (1996) and found that the observed wavefieldfluctuations are consistent with random crustalheterogeneities of 3% to 7% and isotropic correla-tion lengths of 0.6 to 4.8 km. Ritter and Rochert(2000) used a similar approach on teleseismic Pcoda to infer differences in scattering strength be-neath two local networks in Europe. Hock et al.(2000, 2004) used teleseismic P coda to charac-terize random lithospheric heterogeneities across

12

Europe using the energy flux model. They ob-tained a range of different scale lengths and rmsvelocity fluctuations on the order to 3% to 8%.Nishimura et al. (2002) analyzed transverse com-ponents in teleseismic P coda for stations in thewestern Pacific and noted stronger scattering forstations close to plate boundaries compared tothose on stable continents.

In some cases, analysis of P coda can revealindividual scatterers and/or discontinuities atmid-mantle depths below subduction zone earth-quakes (Niu and Kawakatsu, 1994, 1997; Cas-tle and Creager, 1999; Kaneshima and Helffrich,1998, 1999, 2003; Kruger et al., 2001; Kaneshima,2003). Other individual scatterers (or regions ofstrong scattering) have been observed near thecore-mantle boundary from PcP precursors (We-ber and Davis, 1990; Weber and Kornig, 1990;Scherbaum et al., 1997; Brana and Helffrich,2004) and PKP precursors (Vidale and Hedlin,1998; Thomas et al., 1999).

Shearer and Earle (2004) attempted to sys-tematically characterize and model globally aver-aged short-period teleseismic P coda from bothshallow and deep earthquakes. This was the firstattempt to fit a global data set of P -wave am-plitudes and coda energy levels with a compre-hensive energy-preserving model that specifiedscattering and intrinsic attenuation throughoutthe Earth. Examining global seismic networkdata between 1990 and 1999 at source-receiverdistances between 10◦ and 100◦, they identi-fied high signal-to-noise records and stacked over7500 traces from shallow events (depth ≤ 50 km)and over 650 traces from deep events (depth ≥400 km). The stacking method involved sum-ming envelope functions from 0.5 to 2.5 Hz band-pass filtered traces, normalized to the maximumP amplitude. Peak P amplitudes were sepa-rately processed so that absolute P amplitudeversus source-receiver distance information waspreserved. The coda shape was markedly differ-ent between the shallow- and deep-event stacks(see Figure 6). The shallow earthquake coda wasmuch more energetic and long-lasting than thedeep-event coda, indicating that the bulk of theteleseismic P coda from shallow events is causedby near-source scattering above 600 km depth.

***Figure 6 near here***Shearer and Earle modeled these observa-

tions using a Monte Carlo, particle-based ap-proach (see above) in which millions of seismicphonons are randomly sprayed from the source

and tracked through the Earth. They found thatmost scattering occurs in the lithosphere and up-per mantle, as previous results had indicated,but that some lower-mantle scattering was alsorequired to achieve the best fits to the data.Their preferred exponential autocorrelation ran-dom heterogeneity model contained 4% RMS ve-locity heterogeneity at 4-km scale length fromthe surface to 200 km depth, 3% heterogeneityat 4-km scale between 200 and 600 km, and 0.5%heterogeneity at 8-km scale length between 600km and the CMB. They assumed equal and cor-related P and S fractional velocity perturbationsand a density/velocity scaling ratio of 0.8. Intrin-sic attenuation was αQI = 450 above 200 km andαQI = 2500 below 200 km, with βQI = (4/9) αQI

(an approximation that assumes all the attenua-tion is in shear). This model produced a reason-able overall fit, for both the shallow- and deep-event observations, of the amplitude decay withepicentral distance of the peak P amplitude andthe P coda amplitude and duration (see Figure6). These numbers imply that for both P - andS-waves the seismic albedo, β0, is about 75%to 90% in the upper mantle above 600 km, andabout 25% to 35% in the lower mantle, consis-tent with the total attenuation being dominatedby scattering in the upper mantle and by intrinsicenergy loss in the lower mantle.

The 4% velocity perturbations for the up-permost layer in the Shearer and Earle (2004)model are in rough agreement with previous Pcoda analyses by Frankel and Clayton (1986),Flatte and Wu (1988) and Korn (1988); and theS-wave mean free path of 50 km for the up-per 200 km is within the range of mean-free-paths estimated from regional measurements oflithospheric scattering. However, limitations ofthe Shearer and Earle (2004) study include therestriction to single-station, vertical-componentdata (i.e., wave polarization, slowness and az-imuth constraints from three-component and/orarray analyses are not used) and a single fre-quency band near 1 Hz. In addition the raytheoretical approach cannot properly accountfor body-to-surface-wave converted energy, whichsome array studies suggest are an important com-ponent of P coda.

Resolving possible lower-mantle scattering us-ing P coda is difficult because of the muchstronger scattering from shallow structure. How-ever, Shearer and Earle (2004) found that ∼0.5%RMS velocity heterogeneity in the lower mantle

13

was required to achieve the best fit to P coda am-plitudes at epicentral distances beyond 50◦. Thisvalue is between the estimates for lower-mantlevelocity perturbations derived from PKP pre-cursors of 1% from Hedlin et al. (1997) and 0.1%to 0.2% from Margerin and Nolet (2003b).

3.3 Pn Coda

A prominent feature of long-range records of nu-clear explosions across Eurasia is Pn and its coda,which can be observed to distances of more than3000 km (e.g., Ryberg et al., 1995, 2000; Morozovet al., 1998). As discussed by Nielson and Thybo(2003), there are two main models for upper-mantle structure that have been proposed to ex-plain these observations: (1) Ryberg et al. (1995,2000) and Tittgemeyer et al. (1996) proposedrandom velocity fluctuations between the Mohoand about 140-km depth in which the verticalcorrelation length (0.5 km) is much smaller thanthe horizontal correlation length (20 km). Thesefluctuations cause multiple scattering that forma waveguide that can propagate high-frequencyPn to long distances. (2) Morozov et al. (1998),Morozov and Smithson (2000), and Nielsen et al.(2003b) favored a model in which Pn is a whis-pering gallery phase traveling as multiple under-side reflections off the Moho, with the coda gen-erated by crustal scattering. Nielsen and Thybo’s(2003) preferred model has random crustal veloc-ity perturbations between 15 and 40 km depthwith a vertical correlation length of 0.6 km and ahorizontal correlation length of 2.4 km. However,Nielsen et al. (2003a) found that the scattered ar-rivals seen at 800 to 1400 km distance for profile“Kraton” required scattering within a layer be-tween about 100 and 185 km depth and couldbe modeled with 2-D finite difference syntheticsassuming 2% rms velocity variations.

An important aspect of all these models isthat their random velocity perturbations are hor-izontally elongated (i.e., anisotropic). At leastfor the lower crust there is also some evidence forthis from reflection seismic profiles (e.g., Wenzelet al., 1987; Holliger and Levander, 1992). Thiscontrasts with most modeling of S and P coda,which typically assumes isotropic random heter-ogeneity.

3.4 Pdiff Coda

Another phase particularly sensitive to deep Earthscattering is Pdiff and its coda. Pdiff containsP energy diffracted around the CMB and is ob-served at distances greater than 98◦. The directphase is seen most clearly at long periods, buthigh-frequency (∼1 Hz) Pdiff and its coda can bedetected to beyond 130◦ (e.g., Earle and Shearer,2001). Pdiff coda is a typically emergent wave-train that decays slowly enough that it can com-monly be observed for several minutes until it isobscured by the PP and PKP arrivals. Husebyeand Madariaga (1970) concluded that Pdiff coda(which they termed P (diff) tail) could not be ex-plained as simple P diffraction at the CMB orby reflections from the core, and suggested thatit originated from reflections or multiple pathsin the upper mantle (similar to the proposed ex-planation for PP precursors given by Bolt et al.,1968). However, later work has shown this to beunlikely, given the large differences seen betweenobservations of deep-turning direct P coda andPdiff coda. Bataille et al. (1990) reviewed pre-vious studies of Pdiff coda and suggested that itis caused by multiple scattering near the CMB,with propagation to long distances possibly en-hanced by the presence of a low velocity layerwithin D′′.

Tono and Yomogida (1996) examined Pdiff codain 15 short-period records from deep earthquakesat distances of 103◦ to 120◦ and found consid-erable variation in the appearance and durationof Pdiff coda. Comparisons between Pdiff anddirect P waves at shorter distances, as well asparticle motion analysis of Pdiff coda, indicatedthat the coda was caused by deep Earth scat-tering. Tono and Yomogida computed syntheticsusing the boundary integral method of Benites etal. (1992), applied to a simplied model of an inci-dent wave grazing an irregular CMB. They wereable to fit a subset of their observations in whichthe Pdiff coda duration was relatively short (<50 s), with bumps on the CMB with minimumheights of 5 to 40 km. Such large CMB topogra-phy is unrealistic given PcP studies, which haveindicated a relatively flat and smooth CMB (e.g.,Kampfmann and Muller, 1989; Vidale and Benz,1992), and PKKP precursor observations thatlimit CMB rms topography to less than 315 mat 10 km wavelength (Earle and Shearer, 1997,see below), but it is likely that volumetric het-erogeneity within D′′ could produce similar scat-

14

tering. Although the Tono and Yomogida (1996)model included multiple scattering, they couldnot fit the long tail (> 50 s) of some of theirPdiff observations, and they suggested that forsuch cases a low velocity zone just above theCMB is channeling the scattered energy. Strongheterogeneity and low velocity zones of varyingthicknesses have been observed above the CMB(e.g., see review by Garnero, 2000, and Lay, thisvolume) but it is not yet clear if these models canexplain Pdiff observations.

Bataille and Lund (1996) argued for a deeporigin for Pdiff coda by comparing coda shapesfor P near 90◦ range and Pdiff at 102◦ to 105◦.The Pdiff coda is more emergent and lasts muchlonger than the direct P coda. This arguesagainst a shallow source for the coda becausethis would produce roughly the same effect onboth P and Pdiff . Baitaille and Lund foundthat their observed coda decay rate for a sin-gle Pdiff observation at 116◦ could be fit with amodel of multiple scattering within a 2-D shell atthe CMB. Tono and Yomogida (1997) examinedPdiff in records of the 1994 Bolivian deep earth-quake at epicentral distances of 100◦ to 122◦.They analyzed both global broadband stationsand short-period network stations from NewZealand. They found that short-period energycontinued to arrive for over 100 s after Pdiff itself,more than twice as long as the estimated sourceduration of the mainshock. Comparisons be-tween the coda decay rate of Pdiff with otherphases, as well as polarization analysis, indicateda deep origin for Pdiff coda.

***Figure 7 near here***Earle and Shearer (2001) stacked 924 high-

quality, short-period seismograms from shallowevents at source-receiver distances between 92.5◦

and 132.5◦ to obtain average P and Pdiff codashapes. The results confirm the Bataille andLund (1996) observation that P coda changes incharacter near 100◦ . In particular, Pdiff becomesincreasingly extended and emergent at longer dis-tances. Its peak amplitude also diminishes withincreasing distance, but Pdiff can still be ob-served in the stacks at 130◦. Earle and Sheareralso performed a polarization stack (Earle, 1999),which showed that the polarization of Pdiff codais similar to Pdiff (see Figure 7). To model theseresults, Earle and Shearer (2001) applied single-scattering theory for evenly distributed scatter-ers throughout the mantle. The resulting syn-thetics included P -to-P , Pdiff -to-P , and P -to-

Pdiff scattering paths. They applied a hybridscheme that used reflectivity to compute deep-turning P and Pdiff amplitudes and ray the-ory for the shallower turning rays. Scatter-ing was computed assuming a random mediummodel with an exponential autocorrelation func-tion. Synthetics generated for a scale-length of2 km and 1% rms velocity variations achieved agood fit to the amplitude and shape of P andPdiff coda and a reasonable fit to the polariza-tion angles. Thus a fairly modest level of whole-mantle scattering appears sufficient to explainthe main features in Pdiff observations, althoughEarle and Shearer could not exclude the possibil-ity that multiple scattering models could achievesimilar fits.

3.5 PP and P ′P ′ Precursors

The decay in short-period coda amplitude withtime following direct P and Pdiff stops andamplitudes begin to increase some time beforethe surface-reflected PP phase at distances lessthan about 110◦ (at longer distances PKP in-tercedes). This energy is typically incoherentbut sometimes forms distinct arrivals; both aretermed PP precursors. Early explanations forPP precursor observations involved topside andbottomside reflections off discontinuities in theupper mantle (Nguyen-Hai, 1963; Bolt et al.,1968; Bolt, 1970; Husebye and Madariaga, 1970;Gutowski and Kanasewich, 1974). It is now rec-ognized that such arrivals do exist and form glob-ally coherent seismic phases that can readily beobserved at long periods (e.g., Shearer, 1990,1991). In particular, the 410- and 660-km discon-tinuities create discrete topside reflections thatfollow direct P by about 1.5 to 2.5 minutes andunderside reflections that proceed PP by sim-ilar time offsets. However, the high-frequencyPP precursor wavefield is much more continu-ous and it is difficult to identify discrete arrivalsfrom upper-mantle discontinuities, although Wa-jeman (1988) was able to identify underside P re-flections from discontinuities at 300 km and 670km depth by stacking broadband data from theNARS array in Europe.

Wright and Muirhead (1969) and Wright(1972) used array studies to show that PP pre-cursors often have slownesses that are signifi-cantly less than or greater than that expectedfor underside reflections beneath the PP boun-cepoint, which is consistent with asymmetric re-

15

flections at distances near 20◦ from either thesource or receiver. However, this explanationdoes not explain the generally continuous na-ture of the PP precursor wavefield. The cur-rently accepted explanation for the bulk of thePP precursor energy involves scattering from thenear surface and was first proposed by Cleary etal. (1975) and King et al. (1975). Cleary et al.(1975) proposed that PP precursors result fromscattering by heterogeneities within and near thecrust, as evidenced by travel time and slownessobservations of two Novalya Zemlya explosionsrecorded by the Warramunga array in Australia.King et al. (1975) modeled PP precursor obser-vations from the Warramunga and NORSAR ar-rays using Born single-scattering theory. Theyassumed 1% rms variations in elastic propertiesand a Gaussian autocorrelation model with a12 km scale length within the uppermost 100 kmof the crust and upper mantle. This model suc-cessfully predicted the onset times, duration, andslowness of the observed PP precursors but un-derpredicted the precursor amplitudes, suggest-ing that stronger scattering, perhaps too largefor single-scattering theory, would be required tofully explain the observations. King et al. notedthat the focusing of energy at 20◦ distance bythe mantle transition zone could explain the highand low slowness observations of their study andof Wright (1972).

A related discussion has concerned precur-sors to PKPPKP (or P ′P ′), for which the mainphase also has an underside reflection near themidpoint between source and receiver. In thiscase, however, short-period reflections from the410- and 660-km discontinuities are much eas-ier to observe and this has become one of thebest constraints on the sharpness of these fea-tures (e.g., Engdahl and Flinn, 1969; Richards,1972; Davis et al., 1989; Benz and Vidale, 1993;Xu et al., 2003). However, these reflections arrive90 to 150 s before P ′P ′ and cannot explain thelater parts of the precursor wavetrain. Whitcomb(1973) suggested they were asymmetric reflec-tions at sloping interfaces, a mechanism similarto that proposed by Wright and Muirhead (1969)and Wright (1972) for PP precursors. King andCleary (1974) proposed that near-surface scatter-ing near the P ′P ′ bouncepoint could explain theextended duration and emergent nature of P ′P ′

precursors. Vinnik (1981) used single-scatteringtheory to model globally averaged P ′P ′ precur-sor amplitudes at three different time intervals,

and obtained a good fit with a Gaussian auto-correlation function of 13 km scale length withrms velocity perturbations in the lithosphere ofabout 1.6%. Recently Tkalcic et al. (2006) ob-served P ′P ′ precursors at epicentral distances lessthan 10◦, which they interpret as backscatteringfrom small-scale heterogeneities at 150 to 220 kmdepth beneath the P ′P ′ bouncepoints because ar-ray studies show that the precursors have thesame slowness as the direct phase.

3.6 PKP Precursors

Perhaps the most direct evidence for deep-Earthscattering comes from observations of precursorsto the core phase PKP . They were first notedby Gutenberg and Richter (1934). The precur-sors are seen at source-receiver distances betweenabout 120◦ and 145◦ and precede PKP by up to∼20 s. They are observed most readily at highfrequencies and are usually emergent in characterand stronger at longer distances. Older, and nowdiscredited, hypotheses for their origin includerefraction in the inner core (Gutenberg, 1957),diffraction of PKP from the core-mantle bound-ary (Bullen and Burke-Gaffney, 1958; Doorn-bos and Husebye, 1972), and refraction or reflec-tion of PKP at transition layers between the in-ner and outer cores (Bolt, 1962; Sacks and Saa,1969). However, it is now understood that PKPprecursors are not caused by radially symmetricstructures but result from scattering from small-scale heterogeneity at the core-mantle boundary(CMB) or within the lowermost mantle (Haddon,1972; Cleary and Haddon, 1972). This scatter-ing diverts energy from the primary PKP raypaths, permitting waves from the AB and BCbranches to arrive at shorter source-receiver dis-tances than the B caustic near 145◦ and earlierthan the direct PKP (DF ) phase. It should benoted that although the PKP precursors arrivein front of PKP (DF ), they result from scatter-ing from different PKP branches. Scattered en-ergy from PKP (DF ) itself contributes only tothe coda following PKP (DF ), not to the precur-sor wavefield. In addition, the scattering regionmust be deep to create the precursors. Scatter-ing of PKP (BC) from the shallow mantle willnot produce precursors at the observed times anddistances. Thus deep-Earth scattering can beobserved uncontaminated by the stronger scat-tering that occurs in the crust and upper man-tle. This unique ray geometry, which results from

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the velocity drop between the mantle and theouter core, makes PKP precursors invaluable forcharacterizing small-scale heterogeneity near thecore-mantle boundary.

The interpretation of PKP precursors interms of scattering was first detailed by Had-don (1972) and Cleary and Haddon (1972). Theprimary evidence in favor of this model is thegood agreement between the observed and theo-retical onset times of the precursor wavetrain forscattering at the core-mantle boundary. How-ever, analyses from seismic arrays (e.g., Daviesand Husebye, 1972; Doornbos and Husebye, 1972;Doornbos and Vlaar, 1973; King et al., 1973,1974; Husebye et al., 1976; Doornbos, 1976) alsoshowed that the travel times and incidence anglesof the precursors were consistent with the scat-tering theory. Haddon and Cleary (1974) usedChernov scattering theory to show that the pre-cursor amplitudes could be explained with 1%random velocity heterogeneity with a correlationlength of 30 km in a 200-km-thick layer in thelowermost mantle just above the CMB. In con-trast, Doornbos and Vlaar (1973) and Doornbos(1976) argued that the scattering region extendsto 600 to 900 km above the core mantle bound-ary and calculated (using the Knopoff and Hud-son, 1964, single-scattering theory) that muchlarger velocity anomalies must be present. Later,however, Doornbos (1978) used perturbation the-ory to show that short wavelength CMB topog-raphy could also explain the observations (seealso Haddon, 1982), as had previously been sug-gested by Haddon and Cleary (1974). Batailleand Flatte (1988) concluded that their observa-tions of 130 PKP precursor records could be ex-plained equally well by 0.5% to 1% rms velocityperturbations in a 200 km thick layer at the baseof the mantle or by CMB topography with rmsheight of ∼300 m (see also Bataille et al., 1990).

***Figure 8 near here***One difficulty in comparing results among

these older PKP precursor studies is that itis not clear how many of their differences aredue to differences in observations (i.e., the selec-tion of precursor waveforms they examine) com-pared to differences in theory or modeling as-sumptions. PKP precursor amplitudes are quitevariable and it is likely that studies that fo-cus on the clearest observations will be biased(at least in terms of determining globally aver-aged Earth properties) by using many recordswith anomalously large amplitudes. To obtain a

clearer global picture of average PKP precursorbehavior, Hedlin et al. (1997) stacked envelopefunctions from 1600 high signal-to-noise PKPwaveforms at distances between 118◦ and 145◦

(see Figure 8). They included all records, regard-less of whether precursors could be observed, toavoid biasing their estimates of average precursoramplitudes. In this way, they obtained the firstcomprehensive image of the precursor wavefieldand found that precursor amplitude grows withboth distance and time. The growth in averageprecursor amplitude with time continues steadilyuntil the direct PKP (DF ) arrival; no maximumamplitude peak is seen prior to PKP (DF ) ashad been suggested by Doornbos and Husebye(1972). This is the fundamental observation thatled Hedlin et al. (1997) to conclude that scatter-ing is not confined to the immediate vicinity ofthe CMB but must extend for some distance intothe mantle.

Hedlin et al. (1997) and Shearer et al. (1998)modeled these observations using single-scatteringtheory for a random medium characterized withan exponential autocorrelation function. Theyfound that the best overall fit to the observationswas provided with ∼1% rms velocity heterogene-ity at 8-km scale length extending throughout thelower mantle, although fits almost as good couldbe obtained for 4 and 12 km scale lengths. Simi-lar fits could also be achieved with a Gaussian au-tocorrelation function using a slightly lower rmsvelocity heterogeneity. To explain the steady in-crease in precursor amplitude with time, thesemodels contained uniform heterogeneity through-out the lower mantle and Hedlin et al. (1997)argued that the data require the scattering toextend at least 1000 km above the core-mantleboundary and that there is no indication for asignificant concentration of the scattering nearthe core-mantle boundary. This conclusion wassupported by Cormier (1999), who tested bothisotropic and anisotropic distributions of scalelengths and found that the PKP precursor en-velope shapes are consistent with dominantlyisotropic 1% fluctuations in P velocity in the 0.05to 0.5 km−1 wavenumber band (i.e., 12 to 120 kmwavelengths).

Most modeling of PKP precursors has usedsingle scattering theory and the Born approxima-tion. The validity of this approximation for man-tle scattering was questioned by Hudson and Her-itage (1981). However, Doornbos (1988) foundsimilar results from single versus multiple scatter-

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ing theory for core-mantle boundary topographyof several hundred meters (i.e., the amount pro-posed by Doornbos, 1978, to explain PKP pre-cursor observations) and Cormier (1995) foundthe Born approximation to be valid for model-ing distributed heterogeneity in the D′′ regionwhen compared to results from the higher ordertheory of Korneev and Johnson (1993a,b). Morerecently, Margerin and Nolet (2003a,b) modeledPKP precursors using radiative transfer theoryand a Monte Carlo method (see above). Their re-sults supported Hedlin et al. (1997) and Sheareret al. (1998) in finding that whole-mantle scat-tering fit the data better than scattering re-stricted to near the CMB, but they obtainedmuch smaller P velocity perturbations of 0.1%to 0.2% versus the 1% of Hedlin et al. In addi-tion, they found that the Born approximation isaccurate for whole-mantle scattering models onlywhen the velocity heterogeneity is less than 0.5%.Finally they concluded that exponential correla-tion length models do not fit the distance depen-dence in PKP precursors amplitudes as well asmodels containing more small-scale structure.

The reasons for the discrepancy in heterogen-eity amplitude between the models of Hedlin etal. (1997), Cormier (1999) and Margerin and No-let (2003b) are not yet clear. Differences be-tween the single-scattering (Born) and multiplescattering predictions do not appear to be suf-ficient to account for the size of the discrep-ancy. There are subtleties in the data stack-ing and modeling that may have a significant ef-fect on the results. These include the weight-ing and normalization of the waveforms, the as-sumed form of the random heterogeneity powerspectral density function (PSDF), the correctionof PKP (DF ) amplitudes for inner core Q, andthe correction for realistic effective source-timefunctions. However, despite the uncertainty inthe heterogeneity amplitude, it now seems firmlyestablished that PKP precursor observations re-quire that small-scale mantle heterogeneity ex-tends above the lowermost few hundred kilome-ters of the mantle.

Determining exactly how far the scatteringmust extend above the CMB is a challengingproblem because only a small fraction of the pre-cursor wavefield is sensitive to scattering above600 km from the CMB, and this fraction contin-ues to shrink for shallower scattering depths. Forexample, at 1200 km above the CMB, precur-sors are produced only between 138◦ and 145◦,

and arrive at most 3 s before PKP (DF ) (seeFigure 8). Separating the amplitude contribu-tion from these arrivals from that produced byscattering at deeper depths in the mantle is verydifficult, particularly when the pulse broaden-ing caused by realistic source-time functions andshallow scattering and reverberations are takeninto account. An alternative approach to con-straining the strength of mid-mantle scatteringis to examine the scattered PKP wavefield af-ter the direct PKP (DF ) arrival (Hedlin andShearer, 2002). This is comprised of a sum ofPKP (DF ) coda and scattered PKP (AB) andPKP (BC) energy. In principle, if the contribu-tion from the PKP (DF ) coda could be removed,the remaining scattered wavefield would provideimproved constraints on the strength of mid-mantle scattering. However, Hedlin and Shearer(2002) found that this approach did not provideimproved depth resolution of mantle scatteringcompared to previous analyses of PKP precur-sors, owing at least in part to relatively large scat-ter in coda amplitudes as constrained by a dataresampling analysis. Their results were, however,consistent with uniform mantle scattering with1% rms random velocity perturbations at 8-kmscale length.

The preceding discussion considered only av-erage PKP precursor properties and models withradial variations in scattering. However, PKPprecursor amplitudes are quite variable, as notedby Bataille and Flatte (1988), Bataille et al.(1990), and Hedlin et al. (1995), suggesting lat-eral variations in scattering strength. Vidaleand Hedlin (1998) identified anomalously strongPKP precursors for ray paths that indicatedintense scattering at the CMB beneath Tonga.Wen and Helmberger (1998) observed broadbandPKP precursors from near the same region,which they modeled as Gaussian-shaped ultra-low velocity zones (ULVSs) of 60 to 80 km heightwith P velocity drops of 7% or more over 100to 300 km (to account for the long-period partof the precursors), superimposed on smaller-scaleanomalies to explain the high-frequency part ofthe precursors. Thomas et al. (1999) used Ger-man network and array records of PKP precur-sors to identify isolated scatterers in the lowermantle. Niu and Wen (2001) identified strongPKP precursors for south American earthquakesrecorded by the J-array in Japan, which theymodeled with 6% velocity perturbations withina 100-km-thick layer just above the CMB in a

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200 km by 300 km area west of Mexico.Hedlin and Shearer (2000) attempted to sys-

tematically map lateral variations in scatter-ing strength using a global set of high-qualityPKP precursor records. Their analysis wascomplicated by the limited volume sampled byeach source-receiver pair, the ambiguity betweensource- and receiver-side scattering, and the sparseand uneven data coverage. However, they wereable to identify some large-scale variations inscattering strength that were robust with respectto data resampling tests. These include strongerthan average scattering beneath central Africa,parts of North America, and just north of In-dia; and weaker than average scattering beneathsouth and central America, eastern Europe, andIndonesia.

Finally, it should be noted that the earliestonset time of observed PKP precursors agreesclosely with that predicted for scattering at theCMB (e.g., Cleary and Haddon, 1972; Shearer etal., 1998). If scattering existed in the outer coreat significant depths below the CMB, this wouldcause arrivals at earlier times than are seen inthe data. This suggests that no observable scat-tering originates from the outer core, althougha quantitative upper limit on small-scale outercore heterogeneity based on this constraint hasnot yet been established.

3.7 PKKP Precursors and PKKPX

PKKP is another seismic core phase that pro-vides information on deep Earth scattering. Pre-cursors to PKKP have been observed within twodifferent distance intervals. PKKP (DF ) pre-cursors at source-receiver distances beyond theB caustic near 125◦ are analogous to the PKPprecursors discussed in the previous section andresult from scattering in the mantle. Doorn-bos (1974) detected these precursors in NORSAR(Norwegian Seismic Array) records of SolomonIslands earthquakes and showed that their ob-served slownesses were consistent with scatter-ing from the deep mantle. At ranges less than125◦, PKKP (BC) precursors can result fromscattering off short-wavelength CMB topogra-phy. Chang and Cleary (1978, 1981) observedthese precursors from Novaya Zemlya explosionsrecorded by the LASA array in Montana at about60◦ range. These observations were suggestiveof CMB topography but had such large ampli-tudes that they were difficult to fit with realis-

tic models. Doornbos (1980) obtained additionalPKKP (BC) precursor observations from NOR-SAR records from a small number of events atsource-receiver distances between 80◦ and 110◦.He modeled these observations with CMB topog-raphy of 100 to 200 m at 10 to 20 km horizontalscale length. Motivated by PKKP precursor ob-servations (see below), Cleary (1981) suggestedthat some observations of P ′P ′ precursors (e.g.,Adams, 1968; Whitcomb and Anderson, 1970;Haddon et al., 1977; Husebye et al., 1977) mightbe explained as CMB scattered PKKKP pre-cursors.

A comprehensive study of PKKP (BC) pre-cursors was performed by Earle and Shearer(1997), who stacked 1856 high-quality PKKPseismograms, obtained from the Global Seis-mic Network (GSN) at distances between 80◦

and 120◦. PKKP is most readily observed athigh frequencies (to avoid interference from low-frequency S coda), so the records were bandpassfiltered to between 0.7 and 2.5 Hz. To avoid bi-asing the stacked amplitudes, no considerationwas given to the visibility or lack of visibility ofPKKP precursors on individual records. Theresulting stacked image showed that energy ar-rives up to 60 s before direct PKKP (BC) andthat average precursor amplitudes gradually in-crease with time. Earle and Shearer (1997) mod-eled these observations using Kirchhoff theory forsmall-scale CMB topography. Their best-fittingmodel had a horizontal scale length of 8 km andrms amplitude of 300 m. However, they identifieda systematic misfit between the observations andtheir synthetics in the dependence of precursoramplitude with source-receiver distance. In par-ticular, the Kirchhoff synthetics predict that pre-cursor amplitude should grow with range but thistrend is not apparent in the data stack. Thus, themodel underpredicts the precursor amplitudes atshort ranges and overpredicts the amplitudes atlong ranges.

Earle and Shearer (1997) and Shearer et al.(1998) explored possible reasons for this discrep-ancy between PKKP (BC) precursor observa-tions and predictions for CMB topography mod-els. They were not able to identify a very satis-factory explanation but speculated that scatter-ing from near the inner-core boundary might beinvolved because it could produce precursor on-sets that agreed with the observations (see Figure13 from Shearer et al., 1998). However, scatter-ing angles of 90◦ or more are required and it is

19

not clear, given the expected amplitude of the di-rect PKKP (DF ) phase, that the scattered am-plitudes would be large enough to explain thePKKP (BC) precursor observations. They con-cluded that strong inner-core scattering would berequired, which could only be properly modeledwith a multiple scattering theory. Regardless ofthe possible presence of scattering from sourcesoutside the CMB, these PKKP (BC) precursorobservations can place upper limits on the size ofany CMB topography. Earle and Shearer (1997)concluded that the RMS topography could notexceed 315 m at 10 km wavelength.

***Figure 9 near here***Earle and Shearer (1998) stacked global seis-

mograms using P ′P ′ as a reference phase andidentified an emergent, long-duration, high-fre-quency wavetrain near PKKP (see Figure 9),which they named PKKPX because it lackeda clear explanation. PKKPX extends backfrom the PKKP (C) caustic at 72◦ to a distanceof about 60◦. Its 150-s long duration, appar-ent moveout, and proximity to PKKP suggesta deep scattering origin. However, Earle andShearer were not able to match these observa-tions with predictions of single-scattering theoryfor scattering in the lower mantle, CMB or ICB.They speculated that some form of multiple-scattering model at the CMB might be able toexplain the observations, perhaps involving a lowvelocity zone just above the CMB to trap high-frequency energy, a model similar to that pro-posed to explain Pdiff coda by Bataille et al.(1990), Tono and Yomogida (1996), and Batailleand Lund (1996). Earle (2002) further exploredthe origin of PKKPX and other scattered phasesnear PKKP by performing slant stacks on LASAdata. His results suggested that near-surface P -to-PKP scattering is likely an important contrib-utor to high-frequency energy around PKKPat distances between 50◦ and at least 116◦. Inparticular, such scattering arrives at the sametime as observations of PKKP precursors andPKKPX , thus providing a possible explanationfor why PKKP precursor amplitudes are hardto fit purely with CMB scattering models. How-ever, quantitative modeling of P -to-PKP scat-tering has not yet been performed to test thishypothesis.

3.8 PKiKP and PKP Coda and InnerCore Scattering

The inner-core boundary (ICB) reflected phasePKiKP is of relatively low amplitude and ob-servations from single stations are fairly rare,particularly at source-receiver distances less than50◦. However, improved signal-to-noise and moredetails can be obtained from analysis of short-period array data. Vidale and Earle (2000) ex-amined 16 events at 58◦ to 73◦ range recordedby the Large Aperture Seismic Array (LASA)in Montana between 1969 and 1975. As shownin Figure 9, a slowness versus time stack of thedata (bandpass filtered at 1 Hz) revealed a 200 slong wavetrain with an onset time and slownessin agreement with that predicted for PKiKP .They attributed this energy to scattering fromthe inner core because it arrived at a distinctlydifferent slowness from late-arriving P coda andit was much more extended in time than LASAPcP records for the same events. The PKiKPwavetrain takes 50 s to reach its peak amplitudeand averages only about 2% of the amplitude ofPcP . Direct PKiKP is barely visible, with anamplitude close to its expected value, which issmall because the ICB reflection coefficient hasa node at distances near 72◦ (e.g., Shearer andMasters, 1990).

***Figure 10 near here***Vidale and Earle (2000) fit their observations

with synthetics computed using single-scatteringtheory applied to a model of random inner-coreheterogeneity with 1.2% rms variations in den-sity and the Lame parameters (λ and µ) anda correlation distance of 2 km, assuming an ex-ponential autocorrelation model. They assumedαQI = 360 uniformly throughout the inner core,a value obtained from a study (Bhattacharyya etal., 1993) of pulse broadening of PKP (DF ) com-pared to PKP (BC). They noted that withoutinner-core attenuation, the predicted scatteredPKiKP wavetrain would take 100 s to attainits peak value and would last 350 s. The lowvalue of αQI resulted in only the shallow pene-trating P -waves retaining sufficient amplitude tobe seen. Model predicted scattering angles werenear 90◦, making the scattering most sensitiveto variations in inner-core λ. Vidale and Earle(2000) found a tradeoff between the various freeparameters in their model and picked their 2-kmcorrelation length because it minimized the re-quired rms variation of 1.2% necessary to fit the

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observations. They computed a fractional energyloss of 10% from scattering in the top 300 km ofthe inner core, helping to justify the use of theBorn single-scattering approximation. Vidale etal. (2000) examined LASA data for two nuclearexplosions 3 years apart and separated by lessthan 1 km. They identified systematic time shiftsin PKiKP coda, which they explained as result-ing from differential inner-core rotation, as previ-ously proposed by Song and Richards (1996). Incontrast, much smaller time differences are ob-served in PKKP and PKPPKP arrivals, sup-porting the idea that the time dependence orig-inates in the inner core. Vidale et al. (2000) es-timated an inner core rotation rate of 0.15◦ peryear.

Cormier et al. (1998) measured pulse broad-ening in PKP (DF ) waveforms and showed thatthey could be fit either with intrinsic inner-coreattenuation or with scattering caused by randomlayering (1-D) with 8% P velocity perturbationsand 1.2 km scale length. Cormier and Li (2002)inverted 262 broadband PKP (DF ) waveformsfor a model of inner-core scattering attenuationbased on the dynamic composite elastic mediumtheory of Kaelin and Johnson (1998) for a ran-dom distribution of spherical inclusions. Theyobtained a mean velocity perturbation of ∼8%and a heterogeneity scale length of ∼10 km, butalso observed path-dependent differences in theseparameters, with both depth dependence andanisotropy in the size of the scattering attenu-ation. They suggested that scattering attenua-tion is the dominant mechanism of attenuationin the inner core in the 0.02 to 2-Hz frequencyband. Cormier and Li (2002) argued that thelarge discrepancy in rms velocity perturbationand scale length between their study (rms = 8%,scale length = 10 km) and the model (rms =1.2%, scale length = 2 km) of Vidale and Earle(2002) may be due to the significant intrinsic at-tenuation assumed by Vidale and Earle and dif-ferences in depth sensitivity between the studies.

Poupinet and Kennett (2004) analyzed PKiKPrecorded by Australian broadband stations andthe Warramunga array for events at short dis-tances (<∼45◦). They found that PKiKP couldfrequently be observed on single traces filteredat 1 to 5 Hz. In contrast to Vidale and Earle(2000), they found that PKiKP typically hadan an impulsive onset with a coda that decayedrapidly to a relatively small amplitude that per-sisted for more than 200 s. Although most of the

Poupinet and Kennett results were for shortersource-receiver distances than the 58◦ to 73◦

range of the Vidale and Earle study, they did an-alyze one event at 74◦ that exhibited similar be-havior. Poupinet and Kennett suggested that thedifferences in the appearance of PKiKP coda be-tween the studies may reflect different samplinglatitudes at the inner-core boundary and differ-ent frequency filtering. They pointed to simi-larities in the appearance of PKiKP coda andPdiff coda (see above) and speculated that bothresult from their interaction with a major inter-face that may involve energy channeling to pro-duce an extended wavetrain. Poupinet and Ken-nett (2004) argued the high-frequency nature ofPKiKP coda excludes an origin deep within theinner core where attenuation is high, and sug-gested that multiple scattering near the ICB is amore likely mechanism.

Koper et al. (2004) analyzed PKiKP codawaves recorded by short-period seismic arraysof the International Monitoring System (IMS)at source-receiver distances ranging from 10◦ to90◦. Stacked beam envelopes for the 10◦ to50◦ data showed impulsive onsets for PcP , ScPand PKiKP , but a markedly different coda forPKiKP , which maintained a nearly constantvalue that lasted for over 200 s. This is con-sistent with the Poupinet and Kennett (2004) re-sults at similar distances and supports the ideathat inner-core scattering is contributing to thePKiKP coda. At distances from 50◦ to 90◦,Koper et al. (2004) found one event at 56◦ witha PKiKP coda that increased in amplitude withtime, peaking nearly 50 s after the arrival of di-rect PKiKP , behavior very similar to the LASAobservations of Vidale and Earle (2000). At 4 Hz,13 out of 36 PKiKP observations had emergentcodas that peaked at least 10 s into the wavetrain.However, more commonly the peak coda ampli-tude occurred at the onset of PKiKP . Koper etal. (2004) found that the average PKiKP codadecay rate was roughly constant between stacksat short and long distance intervals, supportingthe hypothesis of inner-core scattering. They ar-gued that the best distance range to study inner-core scattering is 50◦ to 75◦, where the directPKiKP amplitude is weak (because of a verysmall ICB reflection coefficient) so that scatter-ing from the crust and mantle is unlikely to con-tribute as much to the observed coda as scatter-ing from the inner core. However, they also dis-cussed the possibility that CMB scattering could

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deflect a P -wave into a PKiKP wave that couldreflect from the ICB at an angle with a muchhigher reflection coefficient and contribute to theobserved coda.

PKP (C)diff is the P wave that diffractsaround the inner-core boundary and is seen asan extension of the PKP (BC) branch to dis-tances beyond 153◦. Nakanishi (1990) analyzedJapanese records of PKP (C)diff coda from adeep earthquake in Argentina and suggested thatscattering near the bottom of the upper man-tle could explain its times and slowness. Tanaka(2005) examined PKP (C)diff coda from 28 deepearthquakes recorded using small aperture short-period seismic arrays of the IMS at epicen-tral distances of 153◦ to 160◦. Beam form-ing at 1 to 4 Hz resolved the slownesses andback azimuths of PKP (DF ), PKP (C)diff , andPKP (AB). The PKP (C)diff coda lasted longerthan PKP (AB), but the wide slowness distribu-tion of PKP (C)diff coda is difficult to explainas originating solely from the ICB, and Tanakasuggested that scattering near the CMB is an im-portant contributor to PKP (C)diff coda.

Vidale and Earle (2005) studied PKP codafrom seven Mururoa Island nuclear explosionsover a 10 year period recorded by the NORSARarray at an epicentral distance of 136◦. Theyobserved complicated arrivals lasting ∼10 s thatwere more extended than the relatively simplepulses observed for direct P waves from explo-sions recorded in the western United States. Vi-dale and Earle suggested that these complica-tions likely arose from scattering at or near theinner-core boundary. They showed that smalltime shifts in the PKP coda were consistent withshifts predicted for point scatterers in an innercore that rotated at 0.05◦ to 0.1◦ per year, al-though they could not entirely rule out system-atic changes in source location.

Several recent studies point to complicatedstructures near the inner-core boundary. Strou-jkova and Cormier (2004) found evidence for athin low-velocity layer near the top of the in-ner core. Koper and Dombrovskaya (2005) an-alyzed a global set of PKiKP/PcP amplituderatios and found large, spatially coherent, varia-tions suggestive of significant heterogeneity at ornear the ICB. Krasnoshchekov et al. (2005) ob-served large variations in PKiKP amplitudes,which they attributed to spatial variations inICB properties, such as a thin partially liquidlayer, interspersed with patches with a sharp

transition. PKP (DF ) travel-time studies havealso indicated significant inner-core heterogene-ity, albeit at scale lengths considerably longerthan those required to scatter high-frequency Pwaves. Breger et al. (1999) noted sharp changesin PKP (BC −DF ) travel time residuals, whichthey attributed to lateral variations on lengthscales shorter than a few hundred kilometerswithin the top of the inner core or the base of themantle. Garcia and Souriau (2000) also analyzedPKP (DF ) versus PKP (BC) travel times, con-cluding that inner-core heterogeneity is no morethan 0.3% at scale lengths longer than 200 kmbut that large lateral variations in anisotropy arepresent between 100 and 400 km below the ICB.

3.9 Other Phases

Emery et al. (1999) computed the effect of dif-ferent types of D′′ heterogeneity on Sdiff , us-ing both the Langer and Born approximations.They found that their long period Sdiff obser-vations are not particularly sensitive to the typesof small-scale heterogeneities proposed to explainother data sets. Cormier (2000) used the codapower between P and PcP and S and ScS, to-gether with limits on pulse broadening in PcPand ScS waveforms, to model D′′ heterogeneityusing a 2-D pseudospectral calculation. He at-tempted to resolve the heterogeneity power spec-trum over a wide range of scale lengths, to bridgethe gaps among global tomography studies, D′′

studies, and PKP precursor analysis.Lee et al. (2003) noted an offset in S coda

observations for central Asian earthquakes (150to 250 km deep) recorded about 750 km away atstation AAK. The offset occurred near the ScSarrivals in coda envelopes at 10 and 15 s period.At shorter periods (1 to 4 s), a change in codadecay rate appeared associated with the ScS ar-rival. They simulated these observations with aMonte Carlo method based on radiative trans-fer theory and isotropic scattering. For a two-layer mantle model (separated at 670 km), theirbest-fitting synthetics at 4 s had a total scatter-ing coefficient g0 of about 1.3 × 10−3 km−1 and6.0× 10−4 km−1 for the upper and lower layers,respectively. Corresponding results at 10 s wereabout 4.7 × 10−4 km−1 and 2.6 × 10−4 km−1.Rondenay and Fischer (2003) identified a coher-ent secondary phase in the SKS+SPdKS wave-field on paths sampling the CMB below NorthAmerica. They showed that this phase could be

22

modeled with an ultra low velocity zone (ULVZ)just above the CMB on one side of the SPdKSpath, but speculated that more complicated 3-Dstructure could also explain their observations.

4 DISCUSSION

Scattering from small-scale irregularities has nowbeen detected at all depths inside the Earth withthe exception of the fluid outer core, althoughmany details of this heterogeneity (power spec-tral density, depth dependence, etc.) remainpoorly resolved, at least on a global scale. Scat-tering is strongest near the surface, but signifi-cant scattering also occurs throughout the lowermantle (e.g., Hedlin et al., 1997; Shearer et al.,1998; Cormier, 1999; Earle and Shearer, 2001;Margerin and Nolet, 2003b; Shearer and Earle,2004). Small-scale heterogeneity within the deepmantle is almost certainly compositional in originbecause thermal anomalies would diffuse awayover relatively short times (Hedlin et al., 1997;Helffrich and Wood, 2001) and supports mod-els of incomplete mantle mixing (e.g., Olson etal., 1984; Allegre, 1986; Gurnis and Davies, 1986;Morgan and Morgan, 1999). Helffrich and Wood(2001) discussed the implications of small-scalemantle structure in terms of convective mixingmodels and suggested that the scatterers aremost likely remnants of lithospheric slabs. As-suming subduction-induced heterogeneities are11–16% of the volume of the mantle, they pro-posed that most of this heterogeneity occurs atscale lengths less than 4 km, where it would havelittle effect on typically observed seismic wave-lengths. Meibom and Anderson (2003) discussedthe implications of small-scale compositional het-erogeneity in the upper mantle, where partialmelting may also be an important factor.

There is a large gap between the smallest scalelengths resolved in global mantle tomographymodels and the ∼10 km scale length of the ran-dom heterogeneity models proposed to explainscattering observations. Chevrot et al. (1998)showed that the amplitude of the heterogeneityin global and regional tomography models obeysa power law decay with wavenumber, i.e., thatmost of the power is concentrated at low spher-ical harmonic degree, a result previously notedby Su and Dziewonski (1991) for global models.For the shallow mantle, this is probably caused inpart by continent–ocean differences (G. Masters,

personal communication, 2005), but a decay oforder k−2 to k−3 is also predicted for heterogen-eity caused by temperature variations in a con-vecting fluid (Hill, 1978; Cormier, 2000). How-ever, this decay cannot be extrapolated to verysmall scales because it would predict heterogen-eity much weaker than what is required to ex-plain seismic scattering observations at ∼10 kmscale. As discussed by Cormier (2000), the mostlikely explanation is a change from thermal- tocompositional-dominated heterogeneity and thatsmall-scale (< 100 km) mantle perturbations arechemical in origin. Spherical heterogeneities ofradii 38 km or smaller can be entrained in mantleflow, assuming a mantle viscosity of 1021 Pa s, adensity contrast of 1 g/cc, and a convective veloc-ity of 1 cm/yr (Cormier, 2000). Because settlingrate scales as the radius squared, smaller blobswill be entrained even at much smaller densitycontrasts.

The role of the core-mantle and inner-coreboundaries in small-scale scattering is not yetclear. The D′′ region has stronger heterogen-eity than the mid-mantle in tomography mod-els and large velocity contrasts have been identi-fied in specific regions, including ultra-low veloc-ity zones and strong individual PKP scatterers(e.g., Vidale and Hedlin, 1998; Wen and Helm-berger, 1998; Niu and Wen, 2001). However,globally averaged PKP precursor studies do notfind evidence for stronger scattering at the CMBthan in the rest of the lower mantle (Hedlin etal., 1997; Margerin and Nolet, 2003b). CMB to-pography can also produce scattering, and hasbeen invoked by some authors to explain PKPand PKKP precursors, but fails to predict glob-ally averaged PKKP precursor amplitude versusdistance behavior (Earle and Shearer, 1997). Vi-dale and Earle modeled PKiKP coda observa-tions with bulk scattering within the inner core,while Poupinet and Kennett (2004) suggestedthat scattering from near the inner-core bound-ary, where several recent studies have found ev-idence for anomalous structures (e.g., Cormier,2004; Koper and Dombrovskaya, 2005; Kras-noshchekov et al., 2005), was more likely respon-sible for their PKiKP coda observations. Strongattenuation is observed in the inner core, but itis not yet clear how much of this is caused by in-trinsic versus scattering attenuation. Inner-corescattering might be caused by small-scale textu-ral anisotropy (Cormier et al., 1998; Vidale andEarle, 2000) or by compositionally induced vari-

23

ations in partial melt (Vidale and Earle, 2000).At shallow depths, it is clear that there is

both strong scattering and significant lateral vari-ations in scattering strength, but the numberand diversity of studies on lithospheric scatter-ing makes it difficult to draw general conclu-sions. There is a large literature on both thetheory of seismic scattering and on coda observa-tions, but there has been much less effort to inte-grate these studies into a comprehensive pictureof scattering throughout Earth’s interior. Reviewarticles (e.g., Aki, 1982; Herraiz and Espinosa,1987; Sato, 1991; Matsumoto, 1995; Fehler andSato, 2003) and the book by Sato and Fehler(1998) are certainly helpful, but their summariesoften involve comparisons among studies that dif-fer in many key respects, such as their choice ofseismic phase (P , S, etc.), their epicentral dis-tance, frequency range, and time window, andtheir modeling assumptions (e.g., single scatter-ing, multiple lapse-time window, finite difference,radiative transfer, etc.). It is not always clearwhether models proposed to explain one type ofdata are supported or contradicted by other typesof data. For example, models with horizontallyelongated crust and/or upper-mantle heterogen-eity appear necessary to explain long-range Pnpropagation across Eurasia and many models oflower crustal reflectors are anisotropic (e.g., Wen-zel et al., 1987; Holliger and Levander, 1992). Yetalmost all modeling of local earthquake coda as-sumes isotropic random heterogeneity. A promis-ing development is the increasingly open avail-ability at data centers of seismic records from lo-cal and regional networks, portable experiments,and the global seismic networks. This shouldenable future coda studies to be more compre-hensive and analyze larger numbers of stationsaround the world using a standardized approach.This would help to establish a baseline of globallyaveraged scattering properties as well as mapsof lateral variations in scattering strength overlarge regions. Ultimately more detailed infor-mation on lithospheric heterogeneity (amplitude,scale length, and anisotropy) will enable more de-tailed comparisons to geological and petrologicalconstraints on rock chemistry (e.g., Levander etal. 1994; Ritter and Rothert, 2000).

Analyses of deep Earth scattering have alsoused a variety of different phase types and model-ing approaches. Even today, there are still funda-mental features in the high-frequency wavefield,such as PKKPX (Earle, 2002), that lack defini-

tive explanations and have never been quantita-tively modeled. However, given recent improve-ments in modeling capabilities (e.g., Monte Carlocalculations based on radiative transfer theory,whole Earth finite difference calculations, etc.),there is also some hope that within the nextdecade we will see the first generally accepted 1-D models of Earth’s average scattering propertiesand a clear separation between scattering and in-trinsic attenuation mechanisms. It appears thata substantial part of seismic attenuation at highfrequencies is caused by scattering rather thanintrinsic energy loss, but fully resolving tradeoffsbetween Qsc and QI will require analysis of scat-tering observations at a wide range of frequenciesand epicentral distances. An interesting compar-ison can be made with seismologist’s efforts tomap bulk seismic velocity variations. Regionalvelocity profiling of the upper mantle gave wayin the 1980s to comprehensive velocity inversions(i.e., “tomography”) to image global 3-D mantlestructure. This required working with large datasets of body-wave travel-time and surface-wavephase-velocity measurements and developing andevaluating methods to invert large matrices. Theearliest models were crude and controversial intheir details, but rapid progress was made as dif-ferent groups began comparing their results. Wemay be poised to make similar advances in resolv-ing Earth scattering. But progress will requireimproved sharing of data from local and regionalnetworks as well as greater testing and standard-ization of numerical simulation codes. As mod-els of small-scale random heterogeneity becomemore precise, comparisons to geochemical andconvection models will become increasingly rel-evant.

5 Acknowledgments

Barbara Romanowicz and an anonymous re-viewer provided helpful comments. This researchwas supported by National Science Foundationgrant EAR02-29323.

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Figure 1: Examples of random media defined bya Gaussian autocorrelation function (top) and anexponential autocorrelation function (bottom).The correlation distance, a, is indicated in thelower left corner. The exponential medium hasmore structure at short wavelengths than theGaussian medium.

Figure 2: Envelopes of finite-difference syntheticseismograms for receivers at distances of 180 to3780 m from the source as computed by Frankeland Wennerberg (1987) for a random mediumwith a correlation distance of 40 m. Note thatthe envelopes decay to a common level followingthe initial pulse, indicating spatial homogeneityof coda energy.

Figure 3: Example of a Monte Carlo computersimulation of random scattering of seismic en-ergy particles, assuming 2-D isotropic scatteringin a uniform whole space. Particles are sprayed inall directions from the source with constant scat-tering probability defined by the indicated meanfree path length, `. As indicated in (a), blackdots show particles that have not not been scat-tered, red dots show particles that have scatteredonce, blue dots show particles that have scatteredtwice, and green dots show particles scatteredthree or more times. (b) Results for 1000 particlesafter t = 0.8`/v, where v is velocity. (c) Resultsfor 1000 particles after t = 1.25`/v. Note that theparticle density is approximately constant for thescattered energy inside the circle defining the di-rect wavefront, as predicted by the energy fluxmodel.

Figure 4: Travel-time curves for teleseismic bodywaves and some of the regions in the short-periodwavefield that contain scattered arrivals. Theseinclude: (1) P coda, (2) Pdiff coda, (3) PP pre-cursors, (4) P ′P ′ precursors, (5) PKP precur-sors, (6) PKKP precursors, (7) PKKPX , and(8) PKiKP coda. The plotted boundaries areapproximate.

Figure 5: A summary of observations of the totalscattering coefficient g0 for S-wave scattering ver-sus frequency, obtained using the multiple lapse-time window (MLTW) method. Results from var-ious studies around the world are indicated withdifferent colors.

Figure 6: Comparisons between envelope functionstacks of teleseismic P -wave arrivals (solid lines)with predictions of a Monte Carlo simulationfor a whole-Earth scattering model (thin lines)as obtained by Shearer and Earle (2004). Theleft panels show results for shallow earthquakes(≤50 km); the right panels are for deep earth-quakes (≥400 km). The top panels show peak P -wave amplitude versus epicentral distance. Thebottom panels show coda envelopes in 5◦ dis-tance bins plotted as a function of time from thedirect P arrivals, with amplitudes normalized tothe same energy in the first 30 s.

Figure 7: Global seismogram stacks and scatter-ing model predictions for high-frequency Pdiff

from Earle and Shearer (2001). (a) A stackedimage of Pdiff polarization and linearity. Timesare relative to the theoretical PP arrival times,and direct P and PKP (DF ) are also shown. (b)An amplitude stack of Pdiff envelopes (wigglylines) compared to model predictions (smoothlines) for a uniform mantle scattering model. Am-plitudes are normalized to the maximum PPamplitude. To better show the small-amplitudePdiff arrivals, the traces are magnified by 10 fortimes 20 s before PKP at distances greater than112.5◦. The width of the magnified data traces isequal to the two-sigma error in the data stack,estimated using a bootstrap method.

Figure 8: PKP precursors as imaged by stack-ing 1600 short-period seismograms by Hedlin etal. (1997). The colors indicate the strength ofthe precursors, ranging from dark blue (zero am-plitude) through green, yellow, red and brown(highest amplitudes). The onset of PKP (DF )is shown by the edge of the brown region atzero time. The precursors exhibit increasing am-plitude with both distance and time. The whitecurves show the theoretical precursor onset timesfor scattering at 400-km depth intervals abovethe core-mantle boundary (CMB).

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Figure 9: A stack of 994 seismograms from Earleand Shearer (1998), showing the PKKPX phase(red) in the high-frequency wavefield precedingP ′P ′. The stacked traces are normalized to themaximum P ′P ′ amplitude and are plotted withrespect to the origin time of a zero depth earth-quake. To better show the PKKPX arrival, am-plitudes are magnified by 8 for traces precedingP ′P ′ at epicentral distances less than 73◦ (in-dicated by the heavy lines). Labels indicate ob-served phases and the position of the P ′P ′ b andPKKP c caustics.

Figure 10: PKiKP coda as observed and mod-eled by Vidale and Earle (2000) for data from12 earthquakes and 4 nuclear explosions recordedat the LASA array in Montana. The top panel(Zhigang Peng, personal communication) showsa slowness versus time envelope function stack ofenergy arriving between 950 and 1270 s from theevent origin times. The predicted direct PKiKParrival is shown with the ×. Late P coda formsthe stripe seen at a slowness near 0.5 s km−1.PKiKP coda is the stripe seen near zero slow-ness. The bottom panel compares a PKiKP am-plitude stack (wiggly line) to the predicted scat-tering envelope for an inner-core random hetero-geneity model (dashed line).

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a

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Figure 1

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Figure 4

Fehler et al. (1992)Mayeda et al. (1992)Hoshiba (1993)Jin et al. (1994)Akinci et al. (1995)Canas et al. (1998)Hoshiba et al. (2001)Ugalde et al. (2002)Bianco et al. (2002)Bianco et al. (2005)Goutbeek et al. (1994)

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Figure 9

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Figure 10