The relative contributions of scattering and viscoelastic attenuation to the apparent attenuation of seismic body waves are estimated from synthetic and observed S waves multiply reflected from Earth's surface and the core–mantle boundary. The synthetic seismograms include the effects of viscoelasticity and scattering from small-scale heterogeneity predicted from both global tomography and from thermodynamic models of mantle heterogeneity that have been verified from amplitude coherence measurements of body waves observed at dense arrays. Assuming thermodynamic models provide an estimate of the maximum plausible power of heterogeneity measured by elastic velocity and density fluctuations, we predict a maximum scattering contribution of 43 % to the total measured attenuation of mantle S waves having a dominant frequency of 0.05 Hz. The contributions of scattering in the upper and lower mantle to the total apparent attenuation are estimated to be roughly equal. The relative strength of the coda surrounding observed ScSn waves from deep focus earthquakes is not consistent with a mantle having zero intrinsic attenuation.

Seismic tomography reveals a laterally heterogeneous velocity structure in the mantle. Constraining the locations and dimensions of such elastic heterogeneities is critical to understanding the intricate details of the dynamic mixing process of the mantle, which is closely tied to the plate tectonic evolution of the Earth. Large-scale (

The apparent attenuation of multiple ScS waves is an excellent observable to untangle the relative contributions of scattering and intrinsic attenuation. Many previous studies have used ScS and its reverberations within the mantle to obtain path-averaged values for the mantle attenuation. These attenuation measurements are usually represented in terms of a quality factor (

Our effort employs an estimate for a ScSn attenuation operator to evaluate the relative percentages of scattering and intrinsic attenuation contributing to the apparent attenuation observed from simulated mantle heterogeneity models. Observations of scattered body waves together with geodynamic modeling have established that heterogeneities of scale lengths as small as 4 to 10 km with rms (root mean square) velocity perturbations of 1 % to 8 % can persist throughout the mantle, even in the presence of constant convective stirring (Hedlin et al., 1997; Shearer and Earle, 2008; Kaneshima and Helffrich, 2010). Our investigation considers the effects of similar dimensions and perturbation strengths for heterogeneity models. We also consider the effects of a model of mantle heterogeneity power obtained by applying stochastic tomography (Zheng and Wu, 2008) to invert for the heterogeneity spectrum of the upper 1000 km of the mantle from observations of amplitude and phase fluctuations of teleseismic P waves observed by the Earthscope USArray (Cormier et al., 2019). We assumed fluctuations of S velocity and density to be correlated with those of P velocity such that

Apparent attenuations are measured from ScSn waveforms observed in synthetic seismograms for four different models of mantle heterogeneity. All of these assume PREM as the one-dimensional background velocity and density model, with the PREM shear wave attenuations providing the purely intrinsic component of attenuation. MODEL 1 does not perturb PREM with any lateral heterogeneities. Therefore, the apparent attenuation measured for this case will be purely intrinsic. MODEL 2 (Fig. 1) applies a depth-dependent shear velocity perturbation to the PREM mantle similar to those determined from many seismic tomographic studies (Megnin and Romanowicz, 2000; Ritsema et al., 2004). MODEL 3 (Fig. 2) applies scaled shear velocity and density perturbations to the PREM mantle based on the stochastic P tomography model of Cormier et al. (2019) for the upper 1000 km of the mantle. MODEL 4 (Fig. 3) is the same as MODEL 3 in the upper 1000 km of the mantle but includes an additional peak in heterogeneity power in the lowermost mantle predicted by Stixrude and Lithgow-Bertelloni (2012) from the effect of the post-perovskite phase transition. In MODEL 5, the intrinsic attenuations are turned off while still applying the thermodynamic model of mantle heterogeneity to shear velocity perturbations. Hence, the synthetic seismograms for this model will exhibit purely scattering effects in any attenuation measurement. In all models, heterogeneities are represented as stochastic random media with an exponential autocorrelation having a corner scale equal to 10 km. In MODELS 2, 3, 4 and 5, we assume a relation between P velocity and density and shear velocity perturbations such that

All simulations are performed by a numerical pseudospectral method in 2-D (Cormier, 2000), assuming an SH line source at 500 km depth with a Gaussian-shaped source–time function having a half-width of 1.2 s. Wave propagation uses a 2-D staggered grid with radial step size of 3.0 km and lateral step size of 5.427 km, with time sampling set to 0.025 s, ensuring stability and negligible grid dispersion. Intrinsic attenuation, taken to be approximately constant across a broad frequency band, is introduced by three memory functions using the methods described by Robertson et al. (1994). Waveforms are computed at a great circle distance of 18

In the inversion procedure, the predicted ScSScS velocity waveform is generated by convolving the ScS waveform with an attenuation operator corresponding to a peak attenuation

Observed and predicted ScSScS velocity waveform aligned by the arrival time of first extremum and normalized by the peak-to-trough amplitude. The least squares norm difference between these two waveforms is obtained using a summation of amplitude differences over time.

An operator to convert an ScS waveform into an ScSScS waveform is defined in the frequency domain by

The least squares norm difference between observed and predicted waveforms is calculated from

Our goal was to simply estimate an apparent attenuation parameter

We found MODEL 1, which has pure intrinsic attenuation and no small-scale heterogeneity, to have an apparent attenuation value of 0.004167 corresponding to a

Apparent attenuation parameters and their errors estimated for the five simulated models using probability density functions shown in Fig. 5.

Gaussian probability density function constructed with the least squares norm difference between predictions and simulated observations for

With MODEL 2, which has a conventional tomographic estimate of mantle heterogeneity, we find that the apparent attenuation is increased to 0.005 (

Finally, in MODEL 5, the intrinsic attenuation in the mantle is turned off while applying the mantle heterogeneity of MODEL 4. The apparent attenuation (now purely due to scattering) is measured to be 0.0029 (

Figure 6 compares the levels of scattered coda energy arriving in the vicinity (

Upper–lower bounds of coda envelopes (shaded area) calculated from five random heterogeneity realizations of MODEL 2

To obtain recordings of clear ScS and ScSScS without interference by depth phases and other arrivals (S, SS, sS), we searched for waveforms of deep focus events in the 10 to 30

Estimated relative contributions to apparent

Regional variations measured for

Contour plots of probability density functions obtained with multiple ScS observations in two regions. Event (circles) and station IU (triangles) locations for the two regions described below are shown in panel

Upper–lower bounds of coda envelopes (shaded area) calculated from five random heterogeneity realizations of MODEL 2 (left), MODEL 4 (middle) and MODEL 5 (right), compared to the squared velocity envelopes of data traces (solid blue lines) from

Strong depth dependence of mantle attenuation, both intrinsic and scattering, has long been documented. Intrinsic attenuation has been found to be relatively low in the middle and deep mantle compared to the upper mantle. Evidence of some scattering in the middle and deep mantle has been confirmed in studies of PKIKP precursors in the 120 to 140

In suggesting that scattering attenuation may dominate intrinsic attenuation throughout the mantle, Ricard et al. (2014) considered the effects of heterogeneity distributed primarily in the form of horizontal layers based on geodynamic numerical experiments that predict folding and horizontal stretching of chemical heterogeneity (e.g., Manga, 1996), whose origin primarily originates from the convective cycling of oceanic crust. The attenuative effects of horizontally layered structure have been well known since the classic paper by O'Doherty and Anstey (1971) and are simply calculated. In this paper, we have instead considered the effects of scale lengths predicted by thermodynamic models in which variations in temperature and chemistry dictate the stability of silicate mineral phases. These variations in temperature and chemistry can also be connected to the convective cycling of oceanic crust but instead predict that peaks in heterogeneity power will be concentrated near phase transitions. Such models have not yet fully considered the effects of mechanical mixing on the anisotropy of scale lengths within these relatively narrow regions of depth. Nonetheless, thermodynamic models, when verified by observations of scattering effects that supplement tomographic imaging, may at least provide a more reliable estimate of the upper bound to velocity and density fluctuations in the mantle. Experiments similar to ours may be extended to include the effects of anisotropy of scale lengths. Our results indicate that some intrinsic attenuation will always be required to explain the attenuation of body waves, regardless of the state of isotropy of scale lengths.

An inversion algorithm for apparent mantle attenuation based on L2 norm differences between observed and predicted ScSScS velocity waveforms has been verified by inversion of synthetic seismograms and applied to estimate the relative contributions of intrinsic and scattering attenuation to the total apparent attenuation. Thermodynamic models of mantle heterogeneity predict significantly higher heterogeneity power than the predictions from global tomography and a correspondingly higher relative contribution to apparent attenuation measured from body waves. Taking the depth-dependent heterogeneity power of thermodynamic models of mantle heterogeneity as the maximum plausible heterogeneity, we estimate that scattering may explain up to 41.3 % of apparent mantle attenuation with up to 3 % rms shear velocity perturbations concentrated near mantle phase transitions and 1 % everywhere else. We estimate the scattering contribution to the apparent attenuation from heterogeneity in the upper and lower mantle to be roughly equal in global averages, but regional variations between upper and lower mantle scattering contributions are likely. These estimates agree well with the excitation of coda surrounding ScSn waves observed from deep focus earthquakes. These codas can only be matched by the existence of both intrinsic and scattering attenuation.

The data set of SH component synthetic seismograms can be found at

The supplement related to this article is available online at:

SdS and VFC designed the experiments and SdS carried them out. VFC developed the simulation code. SdS developed the modeling codes and performed the simulations. SdS prepared the manuscript with contributions from VFC.

The authors declare that they have no conflict of interest.

This work was supported by grant EAR 14-46509 from the National Science Foundation.

This research has been supported by the National Science Foundation, Division of Earth Sciences (grant no. EAR 14-46509).

This paper was edited by Caroline Beghein and reviewed by Ian Jackson and one anonymous referee.