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

本文的研究结果如下：1。用变分法计算旋转、椭圆、横向非均质和衰减地球模型的正态模态。横向非均质地球模型的正态模态的特征频率和特征函数的偏导数的计算和推导。本文用变分法计算了一般非球面地球模型的正态模态，并对这些模态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，因为它需要大量的数值计算。利用东京大学计算机中心的超级计算机，我们可以求解高达1000x1000的矩阵特征值问题。我们发现相邻多态和不同模型的耦合非常强，即使非均质波长比模型的波长长得多。我们还推导了特征频率和特征函数对模型参数的偏导数，并测试了它们在应用于实际反演时是否具有足够的精度。这种类型的测试以前也从未做过。计算结果表明，偏导数具有足够的精度，可用于反演。这些偏导数从不使用射线理论近似，可以给出地球横向非均质性的精确模型。作为一个独立的工作，我们导出了横向非均质自重力弹性球的一般运动方程。该运动方程不需要初始应力的知识，如果地球是横向非均质且初始应力是非流体静力的，则无法获得初始应力的知识。我们将利用上述特征频率和特征函数及其对模型参数的偏导数进行反演，得到地球的横向非均质结构。

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.