Large‐scale three‐dimensional even‐degree structure of the Earth from splitting of long‐period normal modes
Authors: Li Xiang‐Dong , Giardini Domenico , Woodhouses John H. 
Published: February 2008

Spectra of the Earth's free oscillations, which depart significantly from those predicted for spherically symmetric Earth models, contain important information on the large‐scale aspherical structure of the Earth. In this paper we present theory, techniques, and numerical results for retrieval of Earth models (including mantle heterogeneity, topography of the core‐mantle boundary (CMB), and inner core anisotropy), using such data. The inversions for Earth models in this study are based upon spectral splitting of 33 isolated multiplets, observed in long‐period accelerograms of 10 large events recorded by the International Deployment of Accelerometers network. Approximately 1000 spectra are involved. Although the data set is insufficient to yield independent results for perturbations in P velocity, S velocity, and density, it is demonstrated that it is possible to obtain large‐scale (spherical harmonic degrees s = 0, 2, 4) models of mantle heterogeneity from such a data set under the constraint that aspherical perturbations in seismic velocities and density are proportional to one another. The mantle models developed from modal data are remarkably similar to preexisting models based upon other kinds of seismic data, demonstrating that heterogeneity in seismic velocities is, at most, weakly dependent on frequency. The pattern of the inferred CMB topography is consistent with geodynamic predictions and agrees to a fair extent with results based on travel time anomalies of PcP and PKP, indicating that modal data can add independent constraints on CMB topography. The anomalous splitting of core modes is attributed to inner core anisotropy which is assumed to possess cylindrical symmetry about the Earth's rotation axis. We consider a relatively simple model which varies smoothly with radius (only with radially constant terms and terms varying with r2). Theoretically, 14 parameters are required to describe such an anisotropic tensor field if it is restricted to spherical harmonic degrees 2 and 4 and if analyticity of the field is required. The inversion for such an anisotropic inner core yields a model which can explain the splitting of anomalously split modes, without violating the constraints imposed by PKIKP travel time information. We solve the inverse problem by following two approaches: (1) using splitting function coefficients as data, we derive Earth models by solving linear inverse problems for each spherical harmonic degree and order; and (2) we directly solve the nonlinear inverse problem in which the data are observed modal spectra and the unknowns are the structural parameters. The second procedure has advantages in the case that there are insufficient data to obtain stable results for the splitting functions of some modes. The mantle models generated in these two ways are essentially identical, verifying that splitting functions can serve as a very convenient intermediate stage in modeling Earth structure using the split spectra of free oscillations.