We present a growth tectonic model of Earth's inner core and the resulting model of the seismic anisotropy. The inner core grows anisotropically if the convection in the outer core is of Taylor column type. The anisotropic growth produces a flow field of the poloidal zonal order 2 type as a result of the isostatic adjustment of the viscous inner core. Crystals in the inner core align themselves under the stress field produced by the flow. We model the anisotropic structure of the inner core, using the theory of Kamb [1959] and elastic constants of Stixrude and Cohen [1995b]. We consider models for both hcp iron and fcc iron, which are the probable crystal structures for the inner core iron according to Stixrude and Cohen [1995a]. We have found that the c axis for hcp iron and [111] direction for fcc iron align in the polar direction. The alignment is consistent with seismic observations, which have revealed that the P wave velocity is faster in the polar direction. Our model predicts that the degree of the alignment decreases near the inner core boundary in accord with recent body wave observations. The radial dependence of the alignment would result from the following three effects: (1) crystals near the surface have not undergone stressed state long enough to acquire anisotropy after precipitation, (2) stress near the surface is different from that in the interior of the inner core due to shear stress free boundary condition, and (3) partially molten structure results in transversely isotropic stress condition near the inner core surface due to compaction. Thus the application of Kamb's theory successfully explains the seismic anisotropy in the inner core provided that the crystals have been subjected under the same stress condition for the timescale of the order of 109 years.

Ultrasonic wave velocities in single-crystal forsteritc (F) and single-crystal olivine (0) have been measured as a function of pressure and of temperature near ambient conditions. Shear and longitudinal velocities were measured in eighteen independent modes, so that each of the nine elastic constants could be calculated by at least two independent equations. The adiabatic stiffness constants c{j (in Mb), their temperature derivatives dc,j/dT (10 -4 Mb/ deg), and their pressure derivatives dc{ffdP, are ij 11 22 33 44 55 66 23 31 12 Cii (F) 3.284 1.998 2.353 0.6515 0.8120 0.8088 0.738 0.688 0.639 (O) 3.237 1.976 2.351 0.6462 0.7805 0.7904 0.756 0.716 0.664 2Now at: by Verma [1960], who presented the secondorder elastic constants of a single crystal. A few reports have been made on the isotropic elastic constants of polycrystalline aggregates of forsteritc made by hot-pressing ([Schreiber and Anderson, 1967] and Marsh (personal communication, 1969) ). Our purpose here is to present our measured values of the elastic constants, their pressure derivatives at room temperature, and their temperature derivatives at i atm; to describe the methods of measurements on a crystal of orthorhombic symmetry; and to list a number of physical constants derived from the basic data.The technique used to measure sound velocity is known as the pulse superposition technique [McSkimin, 1961], and has been 5961 5962

S U M M A R YA detailed study of the physics of a 1-D sedimentary compaction of a viscous medium was carried out both numerically and analytically for columnar and self-gravitating spherical cases, in view of applying it to the inner-core growth process of the Earth. The effects of sedimentation rate and surface porosity upon the porosity profile were investigated. It was found that the porosity profile differs depending on whether or not the sedimentation rate is larger than the Darcy velocity (velocity of the solid matrix when the fluid flows by buoyancy alone). When the sedimentation rate is larger than the Darcy velocity, a thick, constant-porosity layer develops at the surface, and below it, the porosity decreases gradually towards the bottom. When the sedimentation rate is smaller than the Darcy velocity, the porosity profile is characterized by a mushy layer at the top, where the fluid is expelled by the deformation of the solid, underlain by a thick layer of constant porosity, termed the residual porosity. Such a porosity profile can be understood as the propagation of a half-sided solitary wave. The study was extended further for the self-gravitating spherical case. Formation of an unstable porosity structure and the appearance of solitary waves were discovered for the case of monotonically decreasing sedimentation rate. Given the size of the sphere formed by sedimentary compaction, according to the magnitude of the ratio of sedimentation rate to Darcy velocity, three types of porosity structure, which differ in force balance and the typical length scale required for porosity decrease, were discovered. One such structure is where a low-porosity layer forms at the top, accompanied by solitary waves beneath it, indicating that a crust-like region can develop at the surface of the inner core.

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S U M M A R YWe present a detailed description of a new method of spectral analysis named 'Sompi'. The basic idea of this method originates in the physical concept of the characteristic property of the linear dynamic system that is described by a linear differential equation. The time series modelling in the Sompi method consists essentially of estimating the governing differential equation of the hypothetical linear dynamic system that has yielded the given time series data. Due to the equivalence of a linear differential equation and a linear difference equation [or an autoregressive (AR) equation], this method takes the form of the familiar AR method. However, our basic concept of the AR model and the exact formulation based on the maximum likelihood principle have led to a model estimation algorithm different from previous AR methods, and further, to spectral estimation with higher resolution and reliability. By the Sompi method, a time series is deconvoluted into a linear combination of coherent oscillations with amplitudes decaying (or growing) exponentially with time, and additional noise. In other words, it yields a line-shaped spectrum in complex frequency space, unlike the traditional harmonic decomposition in real frequency space, and is powerful for the analysis of the decaying characteristics, as well as the periods, of the oscillations. Also, the variances of the spectral estimates by the Sompi method can be given in simple formulae unlike most modern parametric methods. Although some practical problems still remain unresolved, the theory presented here will provide the theoretical prototype for a new discipline of physical spectral analysis.