In geodetic inversions such as estimation of coseismic slip and/or afterslip distribution on faults, the displacements on the surface calculated under an assumption of homogeneous elastic half space have been mostly used as the Green's functions (GF's). However, this seems not adequate for better estimations of such slip distribution, because the subsurface structures are more or less inhomogeneous, especially those in and around Japan where the structure must be much complicated. In this study, to examine how much the inhomogeneous subsurface structure affects on the surface displacements, we conduct some 3-D finite element calculations with a grid for the region of 1400 km (EW) × 1200 km (NS) × 200 km (depth) including the Tohoku and Hokkaido, northeastern Japan. Assuming homogeneous and inhomogeneous elastic models with various values for the Young's modulus and Poisson's ratio, we calculated the surface displacements due to a dip-slip type dislocation of 1 m on many cell-like subfaults assumed on the interface between the Pacific and land side plates. Comparing the results, we find a large discrepancy in the surface displacements between the homogeneous and inhomogeneous elastic models and less dependency of the surface displacements on the Poisson's ratio. The discrepancy is found to be more than 20% and can be as large as ∼40% in some cases. Such a large discrepancy indicates that the surface displacements calculated for inhomogeneous elastic medium with realistic subsurface structure, unlike as in usual cases, should be used as the GF's for better geodetic inversions.

We derive the 1-yr afterslip distribution following the 2003 Tokachi-oki (Hokkaido, northeastern Japan) earthquake (M W 8.0) by inverting geodetic data, i.e., horizontal and vertical displacements, at 142 land stations of the Global Positioning System (GPS) in Hokkaido and northernmost Tohoku districts, together with vertical displacements at two offshore stations of pressure gauge (PG) off the Pacific coast of Hokkaido. We use the Green's functions (GFs), calculated with a finite element method, for an inhomogeneous elastic (IE) model incorporating subsurface structure. Obtained results show a striking feature of the distribution pattern of significant afterslip, namely, a U-shaped afterslip zone encircling the co-seismic rupture zone of the 2003 event. Amounts of the 1-yr afterslip reach up to 0.9 m, and total seismic moments released from all afterslip zones are of the order of 10 21 N m, corresponding to an earthquake of M W 8.0. For comparison, we also estimate the 1-yr afterslips based on GFs for the homogeneous elastic (HE) model to find that the total seismic moment with GFs for the IE model is larger than that with GFs for the HE model by ∼33% (when the most probable values are compared) if we assume a rigidity of 40 GPa. This result implies that inhomogeneities due to subsurface structure have an important role in geodetic inversions.

S U M M A R YThe Earth's core is a contained rotating fluid. In contrast to the classical theory of rotating, incompressible fluids in rigid containers, a realistic treatment of the geophysical problem requires allowance for the compressibility of the fluid core , and for the elasticity of the shell (mantle and crust) and inner core. It is shown in this paper that when these realistic properties are included in the model, the governing sub-seismic wave equation and boundary conditions are fully Hermitean and a general functional governing the problem can be constructed. Compressibility is treated through a decompression factor which multiplies the displacement field to generate a new, solenoidal, displacement vector field, while elasticity of the boundaries is incorporated by means of internal load Love numbers. Both the decompression factor for the fluid core and the internal load Love numbers for the shell and inner core are computed for realistic Earth models.

S U M M A R Y This paper describes the numerical implementation of a variational principle for the calculation of the very low-frequency (<300 pHz) modes of oscillation of the fluid outer core using realistic models of Earth structure. The scalar, generalized displacement potential is represented by a new set of local, polynomial basis functions which are either purely even or purely odd in the equatorial plane, depending on the parity of the mode being computed. Elastic boundary conditions are matched through the use of Love numbers for the shell and inner core and by the imposition of linear constraints on the bilinear symmetric functional. The observed compressibility of the outer core is incorporated in the calculation and small, non-neutral stratification is allowed for the through the introduction of I&, the signed square of the Vaisala angular frequency, as a parameter. The resulting eigenvalue problem involves polynomial band matrices for which a special iterative procedure for the calculation of eigenvectors and corrections to eigenvalues is developed. Modes are ordered in terms of increasing spatial complexity by using an equivalent average wavenumber. A selection of numerical examples, including plots of the generalized displacement potential and the displacement fields themselves, are presented to illustrate the method of calculation. The work is expected to provide the basis for identification of anomalies in gravimeter signals and nutation series, and possibly, with the recent discovery (Beroza & Jordan 1990) of unexpectedly high levels of seismic energy at long time-scales in 'slow' and 'silent' earthquakes, the detection of core modes excited by earthquakes.

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