In the past two decades, seismic velocity tomography has benefited from rapidly growing data quality, coverage, and computational capability, and has provided snapshots of the present velocity variations in the Earth's mantle. On the other hand, the study of attenuation has lagged behind that of the elastic velocities because of more 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 velocity models.
Lower mantle imaging is especially difficult compared with that of the upper mantle because surface waves lose their sensitivity to such deep structures. Therefore, lower mantle tomography mostly relies on deep-turning teleseismic body waves and normal mode data. In addition to the contamination from the upper-mantle structure, body-wave datasets suffer from an uneven distribution of events and stations, which can bias the images by over-interpreting the unsampled regions in the lower mantle. Since the Earth's free oscillations involve the vibration of the whole planet, mode observations have the capacity to resolve deeper structures in the mantle at long wavelengths, and are much less likely to be biased by the uneven distribution of earthquake sources and seismic receivers.
Contamination caused by elastic effects and source complexity can lead to large uncertainties in attenuation measurements. Even if we minimize the uncertainties caused by the source term, elastic processes, especially the effects of scattering and multipathing (i.e. the focusing/defocusing phenomena due to the transverse gradients of elastic structure), are not well constrained, and yield generally very noisy attenuation datasets. Therefore, we will first start by establishing a new and higher-quality three-dimensional lower mantle elastic structure from modal constraints. This new lower mantle tomographic model can not only serve as a prerequisite for resolving attenuation structure, but can also be used to test the quality of the data set and the accuracy of our measurement and inversion methods.
The constraints on the Earth's three-dimensional structure extracted from
normal modes data mainly rely on detailed analysis of the free oscillation spectrum.
In a spherically symmetric, non-rotating, purely elastic, and isotropic idealized 1-D
earth model (the SNREI model), the spectrum of each mode is expected to be a
sharp and narrow peak. However, the spectra peaks of the real earth demonstrate
splitting, broadening, and overlapping in the observed modes, which indicates
that the fine structure of the spectra of these records carries important information
on the interior three dimensional elastic and anelastic structure of the Earth.
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, especially the 2004 great Sumatra-Andaman earthquakes
mentioned above, putting new constraints on the large-scale attenuation in
the lower mantle from normal modes is promising.
Three main stages are involved in retrieving structural information
from modal data: 1) Gather many spectra from time series of a large earthquake
(7.5); 2) Retrieve spectra of a mode or modes combination
from the spectra, and repeat the process for many other large events;
3) Repeat for other modes, build up a greatly expanded catalogue of normal
modes sensitive to the lower mantle, and invert for the 3D model of the lower mantle.
Two main approaches frequently used in deriving tomographic models from normal mode spectra will both be applied, and results will be compared with each other: (1) The ``one-step'' method: directly derive tomographic models from fitting modal spectra by solving a non-linear inverse problem in a least-square iterative way (e.g. Li et al. 1991, Hara and Geller 2000; Kuo and Romanowicz 2002); (2) The ``two-step'' method: the splitting matrix is simplified into its equivalent function on the sphere, known as the ``splitting function'' (Woodhouse and Giardini 1985). It is a similar procedure to that commonly used in surface wave tomography, in which one first determines 2D maps of phase velocity over a range of frequencies and then uses these to infer the 3D structure perturbations needed to explain the inferred phase velocity maps. Because the nonlinear stage of this approach only needs a relatively smaller number of parameters represented by the splitting function, the computation cost is smaller than the ``one-step'' method.
We start with the ``two-step'' method for the elastic structure inversion, where the Iterative Spectral Fitting (ISF) method (Ritzwoller et al. 1986, 1988) is applied in the process of splitting function inversion. Figure 2.48 shows an example of the ISF method. For mode , we can clearly see the improvement of spectra fitting within six iterations. The associated splitting function images are shown in Figure 2.49.
We hope to obtain constraints on the long wavelength attenuation structure at the base of the mantle by combining the mode data with the Berkeley waveform dataset, which provides constraints on upper mantle attenuation (following the Ph.D. work of Vedran Lekic). 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 ).
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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