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SUMMARYInversion for seismological structures of the oceanic lithosphere–asthenosphere system is important to understand the mechanisms of plate tectonics. Previous models of the oceanic upper mantle have been primarily obtained via global tomography using surface waveforms. However, besides scarcity of waveform data in the oceanic regions, difficulties in fitting phases for shorter-period components in the previous global tomography have yielded resultant models that possess poor resolutions above $\sim \, 50$ km depth. Recent developments of broad-band ocean-bottom seismometer (BBOBS) arrays provide larger amount of seismic data with epicentral distances of

Koki Idehara,1 Satoru Tanaka2 and Nozomu Takeuchi3 1Department of Earth and Planetary Science, University of Tokyo, 7–3-1 Hongo, Bunkyo-ku, Tokyo 113–0033, Japan. E-mail: idehara@eps.s.u-tokyo.ac.jp 2Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, 2–15, Natsushima-cho, Yokosuka 237–0061, Japan 3Earthquake Research Institute, University of Tokyo, 1–1-1 Yayoi, Bunkyo-ku, Tokyo 113–0032, Japan

SUMMARY A whole mantle SH velocity model is obtained by using a unique data set and techniques. Body and surface waveforms including major and multi-orbit phases are used as a data set and are inverted by using 3-D Born kernels. The resultant model, SH18CE, reveals the different natures of the two major upwelling systems: the strong low velocity anomalies beneath Africa extend for more than 1000 km from the core–mantle boundary (CMB), whereas those beneath the Pacific are restricted to 300–400 km from the CMB. The results also show the variable natures of stagnant slabs on the 670 discontinuity around Japan: the depths of the strongest high velocity anomalies within the stagnant slabs are different region by region, which is consistent with the detailed delay time tomography model in this area.

SUMMARY In the waveform inversion for global 3-D heterogeneous mantle structure, it is critical to accurately compute the perturbation of synthetic seismograms caused by the crustal heterogeneities. In actual applications, to reduce the required CPU time, a weak form equation of motion in terms of global trial functions is widely used. For those trial functions, the principal difficulty is how to compute the effect of the perturbation in the location of boundaries (such as Moho and the surface). In previous studies, the formulation has been based on the first-order perturbation theory of the free oscillation. However, this method holds only for weakly heterogeneous media and for lower frequencies. In this study, we derive the exact weak form elastic equation of motion for finite boundary perturbations. Our method can be applied to arbitrary strongly heterogeneous media and to arbitrary frequencies, and should facilitates accurate crustal correction.

SUMMARY We present a new method for data weighting in waveform inversion to improve the obtained seismological earth model. We define ‘sensitivity’ as an index to show the homogeneity of the waveform inversion data set. The sensitivity is directly evaluated from the partial derivatives of the synthetic seismograms and can be evaluated regardless of the method used to compute the synthetics. In this study we use the Direct Solution Method, which is a highly suitable method because of its accuracy and efficiency. If we weight the data so that the sensitivity is more or less homogeneous throughout the whole Earth, we can achieve homogeneous resolution for the entire region. The obtained model is an improvement over models obtained by previous inversion schemes, in the sense that the new model efficiently retrieves the robust information in the data. The inversion of a large data set using our methods can be expected to lead to further improvement of seismological earth models.

SUMMARY In our previous studies we derived a general criterion for optimally accurate numerical operators for computing synthetic seismograms. In this paper we extend our results by deriving source representations that are ‘tuned’ to match the errors of the numerical operators, so that optimally accurate synthetic seismograms are obtained. The same accuracy can be obtained whether the sources are at nodes of the numerical grid or between nodes. The accuracy of the synthetic seismograms is confirmed by numerical experiments. These results should facilitate inversion of seismic waveform data for 3-D Earth structure and earthquake source parameters.

SUMMARY We previously derived a general criterion for optimally accurate numerical operators for the calculation of synthetic seismograms in the frequency domain (Geller & Takeuchi 1995). We then derived modi¢ed operators for the Direct Solution Method (DSM) (Geller & Ohminato 1994) which satisfy this general criterion, thereby yielding signi¢cantly more accurate synthetics (for any given numerical grid spacing) without increasing the computational requirements (Cummins et al .1 994; Takeuchi, Geller & Cummins 1996; Cummins, Takeuchi & Geller 1997). In this paper, we derive optimally accurate time-domain ¢nite diierence (FD) operators which are second order in space and time using a similar approach. As our FD operators are local, our algorithm is well suited to massively parallel computers. Our approach can be extended to other methods (e.g. pseudo-spectral) for solving the elastic equation of motion. It might also be possible to extend this approach to equations other than the elastic equation of motion, including non-linear equations.

SUMMARY We derive modified matrix operators that minimize the numerical error of solutions of the discretized elastic equation of motion. The criterion for obtaining the modified matrix operators is that the net error of the discretized equation of motion must be approximately equal to zero whenever the operand is an eigenfunction and the frequency is equal to the corresponding eigenfrequency. As it is not necessary to know the explicit values of the eigensolutions, our approach can be applied to arbitrarily heterogeneous media. In this paper we primarily consider frequency domain solutions calculated using the direct solution method (DSM) (Geller et al. 1990; Hara, Tsuboi & Geller 1991; Geller & Ohminato 1994). We present explicit formulations of the modified operators and numerical examples for P-SV and SH wave propagation in laterally homogeneous, isotropic media. The numerical solutions obtained using the modified operators are about 30 times more accurate than those obtained using the unmodified operators for the same CPU time. Our methods are readily applicable to problems in spherical coordinates or involving laterally heterogeneous media, as well as to time-domain solutions. It should also be possible to apply the methods of this paper to numerical methods other than the DSM.