Splitting Functions of Long‐Period Normal Modes of the Earth
Authors: Giardini Domenico , Li Xiang‐Dong , Woodhouse John H. 
Published: February 2008

The spectral splitting of the Earth's free oscillation eigenfrequencies departs from that predicted on the basis of ellipticity and rotation because of lateral heterogeneity in the structure of the Earth's mantle and core and because of deviations from hydrostatic ellipticity of the major internal discontinuities. For an isolated multiplet of angular degree l, singlet eigenfrequencies and eigenfunctions are fully determined by the coefficients cst of the splitting function (s = 0,2,4,…, 2l, −s ≤ t ≤ s). These coefficients constitute linear contraints on the Earth's heterogeneous structure of even spherical harmonic degree s and order t; they are analogous, in this and other respects, to the spherical harmonic expansion coefficients of the phase velocity distribution of a surface wave of given frequency. For an earthquake having known source parameters, knowledge of the splitting coefficients is also sufficient to predict, through a nonlinear relationship, the spectra of the multiplet at all stations. This paper addresses the nonlinear inverse problem of estimating the splitting coefficients, cst, using, as data, observed spectra of the multiplet for an array of sources and receivers. The problem is solved by an iterative, least squares procedure in which the individual data are the complex spectral values obtained by Fourier transformation of windowed, long‐period accelerograms from the International Deployment of Accelerometers (IDA) network. Using recordings of large events since 1977, the data set consists of approximately 1000 narrow spectral windows, relating to 27 different modes, each spectral window containing one or two multiplets. The splitting function possesses a real part, which is related to heterogeneity in density and elastic properties, and an imaginary part corresponding to asphericity of anelastic parameters. Inversion is limited to the real and imaginary parts of the splitting function for degree s = 0 and to the real part only for degrees s = 2, 4. The complex c00 term provides a very accurate determination of the degenerate eigenfrequency and attenuation of the multiplet, which constitute additional constraints on the spherically averaged Earth. We find that splitting is systematically larger than that predicted due to rotation and ellipticity. This phenomenon is particularly evident for the PKIKP‐equivalent modes (e.g., 13S2, 11S4) and for some very long period modes (3S2, 2S3) with significant sensitivity in the inner core; it is also present, however, for modes sensitive primarily to mantle structure. Comparison between splitting functions and the kernels characterizing their sensitivity with depth enables us to identify large‐scale patterns of heterogeneity in the Earth's mantle and core. Forward modeling using existing heterogeneous mantle models produces splitting functions in substantial agreement with those obtained from the modal observations if it is assumed that vp and vs heterogeneity are proportional; it is necessary, however, that d In vs/d In vp has a value in the range 2 to 2.5 in the lower mantle, a value much larger than is often supposed and roughly corresponding to the case that lateral variations in shear modulus dominate those in bulk modulus. Because the modal data are principally sensitive to vS and the prior models of the lower mantle are based on short‐period P residuals, a possible explanation is that the P heterogeneity at modal periods is greater than that inferred at ∼1s period. However, some mantle modes have nonnegligible sensitivity to vp and provide evidence that the level of vp heterogeneity is the same at modal periods as it is at 1s, thus supporting, using an entirely different kind of data, both the size and the pattern of lower mantle vp deduced from travel times.