Inversion of normal mode frequencies for estimation of lateral heterogeneity of the earth's elasticity atructure: A numerical test

An investigation was made on effect of lateral heterogenity of the earth's elasticity structure on the normal mode frequencies. The lateral heterogeneity is expressed by an expansion of spherical harmonic functions, Pιm(cosθ)sinmΦ and Pιm(cosθ)cosmΦ, up to the second order harmonics (ι≤2). Free oscillation freguencies of the heterogeneous earth were computed by the xyz algorithm. Further we derived an analytic expression of partial derivatives of eigenfrequency with respect to the expansion coefficients, and performed a numerical test to verify whether or not it is possible to estimate the heterogeneity of the earth's structure by the inversion of noemal mode frequencies. When the earth is assumed to be elastically isotropic spheroid with short polar and long equatorial radii, frequency spectra affected by longitudinal heterogeneity terms, Pιm(cosθ)(sinmΦ, cosmΦ) (m≠ 0), for given values of ι and m coincide with each other, because these two terms describe the same heterogeneity when the earth is rotated by π/2m around rotation axis. In such a case, we cannot determine accurately the expansion coefficients of tha two heterogeneity terms by inversion of normal mode frequencies, whereas the coefficients of the latitudinal heterogeneity Pι0(cosθ) can be precisely determined. Therefore it is difficult to estimate of lateral heterogeneity of the earth's elasticity structure by the inversion of normal mode frequencies.