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

本研究的主要结果如下：1.用变分法计算旋转、椭圆、横向非均匀和衰减地球模型的简正波.计算和推导了横向非均匀地球模型的简正模本征频率和本征函数对参数的偏导数，用变分法计算了一般非球面地球模型的简正模，并对简正模的分裂和耦合进行了全面的研究。这种类型的计算以前从未做过，因为它需要巨大的数值计算。我们用东京大学计算机中心的超级计算机求解了高达1000 × 1000的矩阵本征值问题。我们发现，即使非均匀性的波长远大于模的波长，相邻多重态和不同模型之间的耦合也很强。我们还导出了本征频率和本征函数对模型参数的偏导数，并检验了它们是否具有应用于实际反演时具有足够的精度。这种类型的测试也从来没有做过。计算结果表明，这些分波具有足够的精度，可用于反演。这些偏微分方程不使用射线理论近似，可以给出地球横向非均匀性的精确模型，作为独立的工作，我们导出了横向非均匀自引力弹性球的一般运动方程。这个运动方程不需要初始应力的知识，如果地球是横向非均匀的和初始应力是非流体静力学的，则不能获得初始应力。我们将利用上述本征频率和本征函数及其对模型参数的偏导数进行反演，以获得地球的横向非均匀结构。

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.