Towards Constraining Lateral Variation of Attenuation Structure With Low-Degree Normal Modes Splitting Coefficients

Shan Dou and Barbara Romanowicz


The study of attenuation is very challenging because of the complexity in its measurements and interpretation. However, attenuation is important for at least two reasons:

1) Attenuation is considerably more sensitive to temperature variations than elastic velocities. While elastic velocities have a quasi-linear dependence upon temperature variations, seismic attenuation depends exponentially on temperature (e.g., Jackson, 1993; Karato, 1993). Therefore, attenuation tomography is important for studying temperature variations within the Earth, and combining elastic and anelastic studies has the potential to separate different effects of chemical composition, water content, partial melting, etc.

2) Attenuation causes physical dispersion of seismic velocities, and this effect needs to be corrected for in velocity models.

Motivation and Theory

The two large regions of low shear velocity (Very Low Shear Velocity Provinces, denoted as LLSVPs in the following content) in the lowermost mantle located beneath the south-central Pacific Ocean and southern Africa/southern Atlantic/Southern Indian ocean regions are unusual deep structures that could offer important indications about the dynamic structures. Because these two LLSVPs each have a lateral extent that is much greater than what might be expected for a hot upwelling plume rising from a thermal boundary layer, they are usually referred to as ``superplumes.'' However, resolving the dynamical significance of these large scale features is nontrivial as well as complicated: Low velocities may be caused by high temperatures or by chemical differences or the competing effects of the two. A variety of body wave studies have reported large velocity anomalies (varying from 1% to 10%) and strong lateral gradients (e.g. Ritsema et al., 1998; Ni and Helmberger, 2003a, 2003b; Wen et al., 2001; Wang and Wen, 2004, 2006; Tanaka, 2002; Toh et al., 2005; Ford et al., 2006; He et al., 2006). Shear wave velocities tend to reduce at deep-mantle pressures, and thus lateral variations of $500 \sim 1000_{\circ}$ over $\sim100$ km are needed to contribute to the observed velocity anomalies. Thermal anomalies to this extent could lead to the onset of partial melting that can generate strong velocity reductions. On the other hand, of chemical variations appear to be important in LLSVPs, the temperature contrasts may be far lower.

Comparing P-wave and S-wave velocities can offer important indications that LLSVPs involve chemical heterogeneities. One of the most important results from previous studies is that LLSVPs have a bulk-sound-velocity anomaly that is anti-correlated with the shear wave velocity anomalies (e.g. Resovsky and Trampert, 2003; Trampert et al., 2004). Several normal modes studies Ishii and Tromp, 1999; Trampert et al., 2004) also indicate that density heterogeneity exists at the base of the mantle, which is dominated by the two LLSVPs on a large scale. The anti-correlation between density and shear velocity anomalies that are proposed in these studies favors chemical heterogeneity. This remains a topic of controversy (e.g. Romanowicz, 2001; Kuo and Romanowicz, 2002), but at the same time, it is equally critical to better resolve the density and anelastic structure to assess the effect of thermal buoyancy and chemical negative buoyance. Nonetheless, resolving the attenuation signature of LLSVPs is quite challenging due to the contamination from the elasticity effect and strong lateral variations existing in the upper mantle. Because surface waves lose their sensitivity to such deep structures, lower mantle tomography mostly relies on deep-turning teleseismic body waves and normal mode data. Different from body wave datasets that could be degraded by uneven distribution of events and stations, the Earth's free oscillations involve the vibration of the whole planet and thus are much less likely to be biased by source-receiver geometry. In this way, information carried by normal modes signals can serve quite well for the purpose of exploring the physics properties of the large scale lateral variations in the lowermost mantle.

For normal mode multiplets well isolated in complex frequency, the effect of even-order aspherical structure on the splitting behavior of the spectra can be quantitatively represented by a discrete set of ``splitting coefficients.'' These coefficients determine the coupling of the singlets within a multiplet. The splitting coefficients describe a radial average of three-dimensional heterogeneity, and can be related to internal properties by:

\begin{displaymath}c_{st} = e_{st} + ia_{st}\end{displaymath}

\begin{displaymath}e_{st} = \int^{a}_{0}\delta m_{s}^{t}(r) K_{s}(r)r^2dr + \sum_{d}h^{t}_{sd}B_{sd}r_{d}^{2}\end{displaymath}

\begin{displaymath}a_{st}=\int^{a}_{0}(\delta q_{\kappa s}^{t}K_{q\kappa}+\delta q_{\mu s}^{t}K_{q_{\mu}})r^{2}dr\end{displaymath}

Owing to the high quality digital data set assembled in the last 20 years on the global broadband seismic network, and owing to the occurrence of several very large earthquakes, putting new constraints on the large-scale attenuation in the lower mantle from normal modes is promising.

Preliminary Results and Prospective Work

We applied the Iterative Spectral Fitting (ISF) method (Ritzwoller et al. 1986, 1988) in the study. In the ISF approach, the technique breaks down naturally into two parts: A discrete regression for the interaction coefficients for a number of lower mantle sensitive modes followed by a continuous inverse problem to solve for the three-dimensional structure from the splitting coefficients. Figure 2.31 shows examples of elastic splitting functions obtained from the ISF approach, and we can clearly see the dominant degree-2 pattern in all of the mantle modes shown in the figure.

We applied the same technique to resolve anelastic splitting coefficients, but due to the data noise and very limited size of the dataset used in the study, the retrieved anelastic coefficients are generally below the error level, and appear to be quite unstable and strongly rely on the elastic starting model. With more data involved in the inversion, and more optimum regularization design, we hope to improve the stability of the anelastic splitting coefficients and then go further to invert for a three-dimensional anelastic model of the lower mantle from normal modes. Even if we can only resolve the longest wavelengths (degrees 2 or possibly up to 4), this will be important for the understanding of the nature of the two low velocity regions at the base of the mantle, commonly referred to as ``superplumes,'' whose thermo-chemical nature is still under debate (e.g. Masters et al., 1982; Romanowicz, 1998; Bijwaard and Spakman, 1999; Ishii and Tromp, 1999; Romanowicz, 2001; Trampert et al., 2004; Gung and Romanowicz, 2004; Anderson, 2005 ).

Figure 2.31: Examples of splitting function for mantle-sensitive modes ${}_{0}S_{6}$, ${}_{4}S_{4}$, ${}_{1}S_{7}$, and ${}_{5}S_{3}$
\epsfig{, width=8cm}\epsfig{, width=8cm}\end{center}\end{figure}


Ritzwoller M, Masters G, and Gilbert F, Observations of anomalous splitting and their interpretation in terms of aspherical structure with low frequency interaction coefficients: Application to uncoupled multiplets, Journal of Geophysical Research, 91, 10203-10228, 1986.

Ritzwoller M, Masters G, and Gilbert F, Constraining aspherical structure with low frequency interaction coefficients: Application to uncoupled multiplets, Journal of Geophysical Research, 93, 6369-6396, 1988.

Romanowicz B and Mitchell B, Deep Earth Structure -- Q of the Earth from Crust to Core, Treatise on Geophysics, Volume 1, 775-803, 2007.

Widmer-Schnidrig R and Laske G, Theory and Observations -- Normal Modes and Surface Wave Measurements, Treatise on Geophysics, Volume 1, 67-125, 2007.

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