Determination of lateral heterogeneity of the earth by means of iterative inversion.

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The results of this research are as follows:1. Computation of normal modes of rotating, elliptical, laterally heterogeneous, and attenuating earth model by variational method.2. Calculation and derivation of the partial derivatives of eigenfrequencies and eigenfunctions of the normal modes with respect to the parameters of laterally heterogeneous earth models.We have calculated normal modes of general aspherical earth model by variational method and investigated the splitting and coupling of these modes completely. This type of calculation has never done before because it requires huge numerical computation. We could solve up to 1000x1000 matrix eigenvalue problem by the supercomputer of the computer centre of the University of Tokyo.We found the coupling of adjacent multiplets and different mode type is very strong even if the wavelength of the heterogeneity is much longer than that of modes.We also derived the partial derivatives of eigenfrequencies and eigenfunctions with respect to the model paramters and tested them if they have enough accuracy when applied to the actual inversion. This type of test has also never done before. The results of calculation show that the partials have an enough accuracy and can be used in inversion. These partials never use the ray-theoretical approximation and can give a accurate models of earth's lateral heterogeneity.As an independent work, we derived the genral equation of motion for laterally heterogeneous self-gravitating elastic sphere. This equation of motion does not require the knowledge of the initial stress which cannot be obtained if the earth is laterally heterogeneous and initial stress is non-hydrostatic. We will perform inversion to obtain laterally heterogeneous structure of the earth by using the above eigenfrequencies and eigenfunctions and their partial derivatives with respect to the model parameters.