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SUMMARY Non-linear interaction of ocean waves is a widely accepted mechanism of microseism excitation for frequencies approximately between 0.05 and 0.5Hz. Longuet-Higgins published the most influential paper on this subject in 1950 and his main contributions can be summarized in two points; the first is on the double-frequency mechanism and the second is on the dependency of excitation on ocean depth. The two results were derived for different media, the first point for an incompressible liquid (ocean) and the second point for a compressible liquid layer over an elastic half-space. The two features naturally come out in the normal-mode formulae. We note, however, that the use of the Longuet-Higgins formula, that showed efficiency of excitation based on ocean depth, is not suitable for interpreting land-based observations because his formula was for displacement at ocean bottom. Reevaluation of this dependency for a land-based observation shows similar results for the depth of maximum excitation, but the sharpness of this peak is reduced significantly. Some recent observational studies tried to identify the source area of excitation by using this ocean-depth dependency, but such a result needs to be revised to a larger source region according to our results. We confirm that the maximum excitation of microseisms exists and occurs when the ocean depth is about 2.7km. It is thus not surprising to find an efficient pelagic source for microseisms, although existence of the wave‐wave interaction in the source region is essential.

S U M M A R Y We present an approach for seismic noise which improves Rayleigh-wave ellipticity estimates by reducing the influences from body waves and Love waves. The method requires three-component seismographs and uses the phase-shift information between the vertical and horizontal components. We select data that show the phase-shift of about 90◦ in order to separate Rayleigh waves from body waves and Love waves. In comparison to the usual H/V approach, the estimates for Rayleigh-wave ellipticity are systematically different by about 20 per cent because the existence of S waves and Love waves systematically increases horizontal amplitudes. The differences in the inverted S-wave velocity structure reach up to about 20 per cent in the upper 20 km. The method is limited to a relatively low-frequency range, below 0.3 Hz, as the 90◦ phase-shift peaks become difficult to identify above 0.3 Hz. Despite this limitation in the frequency range, this approach can improve the S-wave velocity structure in the upper 5–10 km, which should lead to a better prediction of long-period ground motions for periods of about 5–10 s. An improved prediction of such long-period ground motion is important for hazard mitigation in the world’s metropolitan areas with high-rise buildings. This approach is applied to the structure in the broader Los Angeles basin in Southern California and improvement of the reference velocity structure in the upper 5–10 km is demonstrated.

SUMMARY An approach to detect afterslips in the source process of giant earthquakes is presented in the normal-mode frequency band (0.3–2.0 mHz). The method is designed to avoid a potential systematic bias problem in the determination of earthquake moment by a typical normal-mode approach. The source of bias is the uncertainties in Q (modal attenuation parameter) which varies by up to about ±10 per cent among published studies. A choice of Q values within this range affects amplitudes in synthetic seismograms significantly if a long time-series of about 5–7 d is used for analysis. We present an alternative time-domain approach that can reduce this problem by focusing on a shorter time span with a length of about 1 d. Application of this technique to four recent giant earthquakes is presented: (1) the Tohoku, Japan, earthquake of 2011 March 11, (2) the 2010 Maule, Chile earthquake, (3) the 2004 Sumatra-Andaman earthquake and (4) the Solomon earthquake of 2007 April 1. The Global Centroid Moment Tensor (GCMT) solution for the Tohoku earthquake explains the normal-mode frequency band quite well. The analysis for the 2010 Chile earthquake indicates that the moment is about 7–10 per cent higher than the moment determined by its GCMT solution but further analysis shows that there is little evidence of afterslip; the deviation in moment can be explained by an increase of the dip angle from 18° in the GCMT solution to 19°. This may be a simple trade-off problem between the moment and dip angle but it may also be due to a deeper centroid in the normal-mode frequency band data, as a deeper source could have steeper dip angle due to changes in geometry of the Benioff zone. For the 2004 Sumatra-Andaman earthquake, the five point-source solution by Tsai et al. explains most of the signals but a sixth point-source with long duration improves the fit to the normal-mode frequency band data. The 2007 Solomon earthquake shows that the high-frequency part of our analysis (above 1 mHz) is compatible with the GCMT solution but the low-frequency part requires afterslip to explain the increasing amplitude ratios towards lower frequency. The required slip has the moment about 19 per cent of the GCMT solution and the rise time of 260 s. The total moment of these earthquakes are 5.31 × 1022 N m (Tohoku), (1.86–1.96) × 1022 N m (Chile), 1.33 × 1023 N m (Sumatra) and 1.86 × 1021 N m (Solomon). The moment magnitudes are 9.08, 8.78–8.79, 9.35 and 8.11, respectively, using Kanamori’s original formula between the moment and the moment magnitude. However, the trade-off problem between the moment and dip angle can modify these estimates for moment up to about 40–50 per cent and the corresponding magnitude ±0.1.

SUMMARY Direct detection of Rayleigh-wave azimuthal anisotropy is reported by applying an array analysis to broad-band seismic data in Southern California, USA. Our approach has excellent resolution for frequencies between 30 and 60 mHz and good resolution between 10 and 30 mHz. Limitation from array size limits accuracy below 10 mHz and complicated wave propagation effects lead to difficulty above 60 mHz. Between 30 and 60 mHz, azimuthal anisotropy of Rayleigh-wave phase velocity is detected cleanly. Phase velocity results show that the 2θ components dominate the 4θ components. Strength of anisotropy hovers around 1 per cent for the 2θ components with standard error (1 σ) of about 0.4 per cent. Fast axes orientations show stable orientation in the direction 110° clockwise from north and its opposite direction 290°. Depth sensitivity kernels for P-wave and S-wave anisotropy indicate that anisotropy is confined to depths in the upper 100 km and may even be to shallower depths. We speculate that the fast axes alignment may be associated with a strong shearing process over the thickness of lithosphere due to obliquely collisional motions between the Pacific Plate and the North America Plate in the region.

SUMMARY Colliding ocean waves is considered to be the major source of seismic noise. In this paper, we examine the question, what would be the equivalent forces for such colliding ocean waves? In addition to the vertical pressure force term pointed out by Longuet-Higgins, we find that a lateral pressure gradient term is included in this system. This leads to a lateral dipole force and the excitation of noise that is different from vertical forcing. Numerical evaluation with PREM-like models shows that this term may become more important than vertical pressure term for frequencies below 5 millihertz (mHz). Importance of this horizontal term at low frequencies is related to the fact that horizontal motions of ocean waves become dominant in the low frequency band and its square is proportional to the excitation.

SUMMARY Rayleigh wave ellipticity has high sensitivity to shallow depth and contains complementary information to phase and group velocity data whose sensitivity extends to deeper region. The inversion of this parameter has often been conducted by using numerically derived depth sensitivity kernels. We present compact formulas based on a variational principle which includes both volumetric seismic velocity perturbations and boundary perturbations. Use of this formula can speed up the inversion process considerably.

SUMMARY The particle motion of surface waves, in addition to phase and group velocities, can provide useful information forS-wave velocity structure in the crust and upper mantle. In this study, we applied a new method to retrieve velocity structure using the ZH ratio, the ratio between vertical and horizontal surface amplitudes of Rayleigh waves. Analysing data from the GEOSCOPE network, we measured the ZH ratios for frequencies between 0.004 and 0.05 Hz (period between 20 and 250 s) and inverted them for S-wave velocity structure beneath each station. Our analysis showed that the resolving power of the ZH ratio is limited and final solutions display dependence on starting models; in particular, the depth of the Moho in the starting model is important in order to get reliable results. Thus, initial models for the inversion need to be carefully constructed. We chose PREM and CRUST2.0 in this study as a starting model for all but one station (ECH). The eigenvalue analysis of the least-squares problem that arises for each step of the iterative process shows a few dominant eigenvalues which explains the cause of the inversion’s initial-model dependence. However, the ZH ratio is unique in having high sensitivity to near-surface structure and thus provides complementary information to phase and group velocities. Application of this method to GEOSCOPE data suggest that low velocity zones may exist beneath some stations near hotspots. Our tests with different starting models show that the models with low-velocity anomalies fit better to the ZH ratio data. Such low velocity zones are seen near Hawaii (station KIP), Crozet Island (CRZF) and Djibuti (ATD) but not near Reunion Island (RER). It is also found near Echery (ECH) which is in a geothermal area. However, this method has a tendency to produce spurious low velocity zones and resolution of the low velocity zones requires further careful study. We also performed simultaneous inversions for volumetric perturbation and discontinuitydepth perturbation. While its formulation and inversion were straightforward, there seemed to be a difficult trade-off problem between volumetric perturbation and discontinuity-depth perturbation.

SUMMARY Exact normal-mode solutions for the seismic noise-correlation method are derived by assuming that excitation sources of seismic noise are distributed over the globe uniformly. While a previous derivation by Snieder was for propagating waves, using the stationary-phase approach, our results show that the method could work for normal modes. Comparison of the formula for the cross-correlated seismogram (correlogram) against the Green's function shows differences in four aspects: (1) the correlogram is proportional to the number of sources (density of sources), (2) depends on excitation-source spectra, (3) shows a larger weight on high-frequency band (inversely proportional to the square of modal mass) and (4) has time dependence which requires one extra integration over time. Waveform modelling of the correlograms seems difficult in practice, particularly because of the lack of information on the excitation sources, often the temporarily changing ocean waves. On the other hand, phase and group velocity measurements should be possible as long as the correlograms contain good surface wave signals.

SUMMARY Recent development in seismology has shown that seismic noise is a useful source of Earth structure study. Non-linear interaction of ocean waves is considered to be one of the important causes for seismic noise, although source locations and detailed mechanisms are not clear yet. In order to improve our understanding of this mechanism quantitatively, we derive a normal-mode excitation formula due to this effect. Longuet-Higgins' pressure formula naturally comes out from the vertical forcing term in the excitation coefficient. A novel aspect of our formula is the horizontal forcing term that becomes comparable to the vertical forcing term for frequencies below about 5 mHz (millihertz), especially in shallow oceans. This term is not likely to be important for microseisms because the main frequency range of microseisms is about 0.05–0.4 Hz but it may make significant contributions to the excitation of the continuous background oscillations (the hum) whose frequency range is 2–7 mHz.

SUMMARY Teleseismic surface wave data from the dense seismographic network in Southern California (CISN, California Integrated Seismic Network) are used to construct S-wave velocity maps for the region. Surface wave phase velocity measurements are achieved by way of a waveform-matching spectral-domain technique that was developed for use within the CISN array. A two-station phase velocity measurement technique was developed, which takes advantage of the fact that, for surface wave arrivals from any azimuth, there are multiple pairs of CISN stations, which are almost exactly on the same great-circle paths. To ensure the validity of this assumption, particle motions were analysed for measured pairs within the network, showing that the deviation between expected backazimuth from ray theory and observed backazimuths are small enough to be analysed by a simple distance correction. This method was applied to 114 large earthquakes for time periods between 1999 and 2002 and led to the retrieval of phase velocity maps for Rayleigh waves between 20 and 55 mHz and for Love waves between 25 and 45 mHz. Finite frequency effect for propagating surface waves was approximately taken into consideration using a geometrical technique. An ellipse, determined by wavelength and great-circle distance between the source and receiver can describe an influence zone for the finite frequency effect. Phase velocity map reconstructions, using a method which incorporates this effect, show minimal changes in phase velocity maps for our region, but do extend solvable velocity regions. Inversion for S-wave velocity distribution with depth shows that deep structural contrast across the San Andreas extends into the upper mantle. A slow velocity anomaly is evident under the Southern Sierras and the Salton Sea, the latter region indicating the northward extension of tectonic activity in the Gulf of California. Fast velocity roots that reach 60 km are found under the western part of Transverse Ranges and the Northern Peninsular Ranges. Reconstruction of past location of the Western Transverse Range microplate implies that these two ranges may have been adjacent to one another historically. Both the Western Transverse range fast velocity root and the Peninsular range fast velocity root may be a remnant of old oceanic plate.

SUMMARY Microseisms are a continuous source of seismic signal which mainly consist of Rayleigh waves but are known to contain some other types of seismic waves. We developed a simple processing procedure for single-station three-component seismic data which allows us to select Rayleigh-wave dominated portions. Application of this procedure to data from Southern California led us to confirm that excitation of the secondary-peak microseisms (the predominant energy at about 0.15 Hz) occurs in coastal regions; the source directions, viewed at each station, do not change very much throughout the year. We also discovered that the ratio of the horizontal to vertical amplitudes in Rayleigh waves, termed H/Z in this paper, is shown to have seasonal variations. The H/Z estimates typically reach their maximum in winter and their minimum in summer. Seasonal variations are observed at most stations but peak-to-peak amplitudes of seasonal variations vary greatly from station to station, ranging between 0 and 40 per cent. Two hypotheses were examined to explain this phenomenon; the first hypothesis is that it is caused by seasonal changes in seismic velocities in a layer between the surface and the groundwater level. The second hypothesis is that it is caused by seasonal changes in relative excitation of higher modes compared to dominant fundamental-mode Rayleigh waves. The first hypothesis is not likely, because predicted amplitudes of seasonal variations in H/Z are too small to explain observed variations. The second hypothesis seems quite plausible if source regions move seasonally among regions with different ocean depths. The technique developed in this paper, however, is not sufficient to answer this question conclusively.

SUMMARY S-wave velocity in the shallow crust is an important controlling parameter for ground motion amplification. It is a key parameter for prediction of ground motion and thus for earthquake hazard mitigation in general. Shallow S-wave velocity structure has often been obtained from phase velocity of microtremors and microseisms. We present a new approach which directly inverts the ratio of horizontal to vertical amplitudes of microseisms which is referred to as the HZ ratio in this paper. Our approach consists of (i) isolation of Rayleigh-wave dominant portions in seismograms by using the 90-degree phase shift between horizontal and vertical component, (ii) measurement of the HZ ratios as a function of frequency and (iii) iterative inversion for S-wave velocity structure in the crust. Depth sensitivity kernels of the HZ ratios for density, P-wave velocity and S-wave velocity are derived by a numerical method. Examples of the application of this method to data in southern California show that the HZ ratio data require modification of the standard seismic velocity model in this region (Southern California Earthquake Center Community Velocity Model 3.0 or SCEC CVM) in the upper 10–15 km. The derived models show amplification of incident SH waves by factors of 2–8, relative to the prediction by the SCEC CVM. The proposed approach has promise for applications to other regions in the world because microseisms are observed everywhere in the world. The method may become even more powerful if it is combined with phase velocity data.

Continuous oscillations of the Earth are observed for frequencies between 2 and 7 mHz at almost every seismically quiet site in the world. These oscillations ride on a broad noise peak which spans the frequency band from 3 to 15 mHz and reaches its maximum at 7-9 mHz. We propose an oceanic excitation hypothesis which explains both the modal oscillations and the broad noise peak, specifically the action of oceanic infragravity waves on the solid Earth. Using the estimated amplitudes of oceanic infragravity waves from observation, although they are limited in number at the moment, we show that there is sufficient energy in these waves to excite observed seismic signals; for a given time window, the contribution from a small area (which may be as small as 100 km x 100 km) is all that is required to explain the seismic observations. The advantage of this oceanic mechanism over the previously proposed atmospheric mechanism is in the simultaneous explanation of the above two features in seismograms, whereas the atmospheric hypothesis has only explained the modal oscillations. The oceanic mechanism naturally explains the predominant 6-month periodicity as a result of semi-hemispheric ocean-wave activities in the Northern and Southern hemispheres, showing a good match between seismic data and satellite ocean-wave data both in the amplitude and phase of seasonal variation. Our Earth seems to be filled with ubiquitous propagating Rayleigh waves, generated directly by oceanic infragravity waves, for the frequency band 3-15 mHz.

SUMMARY Compact analytical formulae are derived for particle motions of surface waves in a weakly anisotropic, flat-layered medium. Theory only incorporates coupling between fundamental mode Rayleigh and Love waves but comparison against numerical results show that a good match is achieved for anisotropy up to approximately 10 per cent. Among various types of particle motions for Rayleigh and Love waves, it is proposed that Love wave vertical polarization, defined by the ratio of the vertical to the transverse displacement, may be a good, cleanly measurable quantity. Derived formulae show that vertical polarization contains 2θ and 4θ azimuthal dependence, similar to the well-known phase velocity variations. Formulae show depth kernels explicitly and can be used for the inversion of anisotropic parameters. The measurement of polarization as a function of azimuth then provides constraints on anisotropy under a seismic station. The derived formulae are a natural extension of Smith & Dahlen's results on phase velocity to particle motions.

Summary Turbulent atmosphere can exert force on the solid Earth by the Reynolds stress, which can in principle generate motions of the solid Earth. Using a stochastic formulation and the Kolmogorov picture of isotropic turbulence, an expression for normal-mode excitation by atmospheric turbulence is derived. The final analytical formula for average acceleration contains the average convective velocity of atmosphere as the only practical parameter in the theory. Comparison with IDA gravimeter data shows that long-period noise below 3 mHz, which is ubiquitous in all good signal-to-noise ratio (S/N) records, can be explained by an average atmospheric convective velocity of 6.5 m s− 1. While this is somewhat larger than the value expected for a simplified atmospheric model (4 m s− 1), its proximity in value as well as its frequency dependence strongly support the view that the cause of long-period noise is atmospheric turbulence.

SUMMARY The state of stress within a bending spherical shell has some special features that are caused by sphericity. While most lithospheres are more like spherical shells than flat plates, our ideas of the state of stress have been dominated by flat-plate models. As a consequence, we might be missing some important aspects of the state of stress within subducting lithospheres. In order to examine this problem, we analyse spherical-shell bending problems from basic equations. We present two approaches to solve sphericalshell bending problems: one by the variational approach, which is suitable for globalscale problems, and the other by the asymptotic equation, which is valid to first order in h/R, where h is the thickness of the lithosphere and R is its curvature radius (i.e. under the assumption of small curvature). The form of the equation for displacement shows that wavelengths of deformation are determined by the spherical (elastic) eVect and the gravitational buoyancy eVect, for which only the latter eVect is included in the usual flat-plate formulations. In the case of the Earth, the buoyancy force is dominant and, consequently, spherical eVects are suppressed to a large extent; this explains why flat-plate models have been successful for Earth’s lithospheric problems. On the other hand, the state of stress shows interesting spherical eVects: while bending (fibre) stress along the subduction zone is always important, bending stress along the trench-strike direction can also be important, in particular when the subduction zone arc is small. Numerical results also indicate that compressive normal stress along the trench-strike direction is important when a subduction zone arc is large. These two stresses, the bending stress and the compressive normal stress, both along the trench-strike direction, may have important implications for intraplate earthquakes at subduction zones.

SUMMARY Effects of sphericity are commonly ignored in the lithospheric bending problem. In order to examine its effects, I solve a simple axisymmetric spherical-shell model. The full solution and the asymptotic solution are derived from the basic equations, and their relationship to the flat-plate solution is examined. For displacement, effects of sphericity are small, and use of the flat-plate solution produces results that are numerically indistinguishable from those of the spherical solution. The most significant effect of sphericity appears in the stress, in particular the normal stress along the strike direction of the trench. This stress is approximately given by Eu,/R, where E is Young’s modulus, u, is the vertical deformation of the shell and R is its radius of curvature. If the shell (lithosphere) is bent downwards and reaches 30 km, this stress can become about 5 kbar in the Earth. While plastic behaviour may set in under such high pressure conditions and analysis beyond elasticity theory may be required, sphericity may be a cause of large compressive stress in the trench strike direction. This stress may play an important role in forming the overall shape of the Earth’s subduction zones.

SUMMARY A simple modification of the waveform inversion formula, based on the normal mode perturbation theory, is shown to lead to a formula for traveltime anomalies. The kernel which is derived can be used for traveltime inversion with automatic inclusion of finite frequency effects. Inversion for Earth structure with such kernels will lead to better resolution estimates than ray-theoretical traveltime inversion. Examples of kernels for transverse component seismograms are shown for direct S waves, ScS, Love waves and diffracted S waves. A measure of finite frequency effects is also proposed by comparing our formula with the one from ray theory. A quantity which should be 1 in the case of ray theory is computed for the finite frequency kernels and is shown to have deviations up to about 30 per cent from 1. Therefore, the use of ray theory for long-period body waves applies incorrect weight along a ray path and may introduce a small bias to an earth model.

Summary Considering the coupling of normal modes of different dispersion branches, we present a formulation for synthesizing waveforms of long-period body waves in a slightly aspherical Earth model. A general asymptotic result, which is valid when angular degrees of modes are much higher than the maximum harmonic degree of lateral heterogeneity, is derived for arbitrary source mechanisms and receiver components. If we restrict the coupling to the modes along a single dispersion branch, this result reduces to the pre-existing theory for surface waves. For body waves, we demonstrate numerically that cross-branch coupling is important to set up the correspondence between normal mode and ray theories. Actually, the coupling of the modes which interfere constructively in the spherically symmetric reference model represents the largest effects of heterogeneity on the waveform of a body wave packet. This would lead to a significant reduction in computation time. Numerical tests show that the formulation is able to bring out the ray character of body waves: only the structural perturbation in the neighbourhood along the geometrical ray affects the seismogram.

SUMMARY Earth's heterogeneity near the surface is so severe that surface waves with period 20 s exhibit complicated propagation effects such as curved paths and multiple rays. In order to take account of these effects, a new method to analyse surface waves is developed. the method, which we call the membrane surface wave method, is an approximation to the Green's function method in a 3-D medium and is valid for a laterally, smoothly varying media. We also present the formulae to invert the seismograms. As an example, we apply the method to a preliminary California model. Use of the method for the forward calculation shows that heterogeneity in the model is so severe that the wavefront due to the Whittier Narrows earthquake is split in two by the time it reaches the San Francisco Bay area. the kernels in the inverse formula to the six Berkeley stations show curved paths, multipathings and finite wavelength effects and visually demonstrate how waves sample the medium. the method is attractive in that these complications cause no problems in its application.

SUMMARY Long-wavelength S-wave velocity anomalies are obtained for the entire mantle by inverting 6OOO long-period (40-100s) SH body waves and lo00 long-period (100-500 s) Love waves. In the shallow mantle, the patterns of fast and slow velocity anomalies correlate well with age and tectonics. In the depth range of 400-1000 km, we find two antipodal fast velocity anomalies, one in the western Pacific and the other in South America. They form a fairly strong I = 2 anomaly, Moreover the 1 = 2 anomaly is the largest spherical harmonic component throughout the entire mantle except for in the layer of 1000-1300 km depth. A fast-velocity anomaly near the Caribbean, previously reported by several other investigators, is found between the depths of 1000 and 1300km. With diminishing amplitudes, it can be traced to Canada and the Aleutians and may be related to old subducted slabs. In the same depth interval, similar size anomalies can be found in the middle of the Pacific and in the Indian Ocean. These could be checked by more detailed studies to judge the reliability of this study. Heterogeneity power has a peak at the surface and a secondary peak at the bottom of the mantle (D”), consistent with the existence of chemical and/or thermal boundary layers. At the base of the mantle we also find a fast-velocity anomaly which surrounds the Pacific Ocean. This feature has been found by other researchers previously, but its interpretation is difficult due to the trade-off between velocity anomaly and (core-mantle) boundary deformation.

Summary. The three-dimensional (3-D) shear wave structure of the mantle, down to the depth of about 900 km, is obtained by inverting waveforms of radial component seismograms. Radial component seismograms contain large amplitude overtone signals which circle the Earth as wave packets and are sometimes called X1, X2, X3, … We use data which contain R1, X1 and X2 and filtered between 2 and 10mHz. It is shown that, unless each seismogram is weighted, all seismograms are not fitted uniformly. Only data from large earthquakes are fitted and the final velocity anomalies are biased by the small number of large earthquake data. Resolution is good at shallow depths, becomes worse in the intermediate depth range between about 400 and 500 km and then becomes better at greater depth ranges (600–900km). Even though we use only spheroidal mode data, velocity anomalies in the shallow structure show excellent correlation with the age of the surface rocks of the Earth. In the deeper regions, between about 600 and 900km, South America shows a fast velocity anomaly which may indicate the slab penetration beyond 700 km there. Another region which shows a fast velocity anomaly is the Mariana trench, but other subduction regions do not show such features.

Summary. A simple modification of Gilbert's formula to account for slight lateral heterogeneity of the Earth by the first-order Born approximation leads to a convenient formula to calculate synthetic long-period seismograms. Partial derivatives are easily calculated, thus the formula is suitable for direct inversion of seismograms for lateral heterogeneity of the Earth. The derivation is much simpler but is equivalent to the one derived by Woodhouse, who obtained it by working in the time domain.

SUMMARY The 3-D upper-mantle structure from the surface of the Earth to a depth of about 1000 km is obtained by inverting waveforms of fundamental and higher-mode surface waves. Minor- and major-arc waves of R-waves, G-waves and X-waves (higher-mode wave packets) are used as basic data. Two features of the solution seem fairly robust; one is the shallow structure in the upper few hundred kilometres which is strongly correlated with the age of the crust. The second feature is the deep fast-velocity anomaly near the western Pacific (Indonesia, Philippines and Mariana) region and its antipodal region of South America, both of which are subduction zones, which forms a strong l= 2 pattern. Simple slab models with 100 km thicknesses of 10 per cent faster-velocity material do not explain such large-scale fast-velocity anomalies. This may indicate large-scale deformation of the 670 km discontinuity, piling up of slabs in the upper mantle or thickening lithosphere in the lower mantle or large horizontal motions of subducted material. These results are stable whether an earth model 1066A or PREM is used as a starting model.

Summary The effect of anisotropy on the free oscillations of the Earth is examined under the assumptions that anisotropy is general, i.e. there are 21 independent components, and a small perturbation from an isotropic media. It is shown that the deviation of a multiplet frequency δω from an average spherically symmetric earth depends upon 15 functions of θ (co-latitude) and φ (longitude) as opposed to three in the isotropic case shown previously by Woodhouse & Girnius. In the asymptotic limit, δω is an average of a local functional δωbar;local (θ, φ, ψ) along a great circle path which contains the source and the receiver, where δωbar;local (θ, φ, ψ) is a function of θ, φ and azimuth ψ. Also δωbar;local (θ, φ, ψ)is a function of the elastic constants beneath a location (θ, φ). The azimuth dependence of δωbar;local (θ, φ, ψ)is 1, cos 2ψ, sin2ψ, cos 4ψ and sin4ψ, which is the same as a flat-layered case. It is shown that formulae equivalent with those of Smith & Dahlen are recovered by an earth flattening transformation. In the non-asymptotic case, in which the angular order l is low, δω shows odd-order azimuthal dependence, i.e. ψ and 3ψ dependence. There exist seven radial kernels in the spherical case as opposed to four in the flat-layered case, although radial resolution remains similar because of similarity of the kernels.

Summary. Phase velocity variations obtained in the previous paper are inverted by the Backus–Gilbert method for the velocity structure of the upper mantle. Spheroidal modes and toroidal modes in the period range of 125–260 s are used in the inversion. The data cannot constrain all six parameters in a transversely isotropic medium and we chose to perturb only two parameters, SH and SV velocities. SV velocities are resolved between the depths of about 200 and 400 km and SH velocities between 0 and 200 km. Resolution kernels have half-peak widths of about 200–300 km in depth, becoming broader for deeper target depths. SV velocity kernels show secondary peaks near the surface of the Earth, with widths varying from 50 to 100 km. The deeper the target depths, the wider the secondary peaks near the surface. SH velocity kernels do not possess such secondary peaks. The trade-off between SV and SH velocities is small. SV velocity is essentially determined by spheroidal modes and SH velocity by toroidal modes. Because of the broad width of the resolution kernels, the structure in the resolved region is difficult to detect from our data set; for example the differences in SV velocity structure between 250 and 350 km or the differences in SH velocity between 100 and 200 km are difficult to distinguish. Considering the horizontal resolution of about 2000 km, obtained in the previous paper, averaging kernels for 3-D structure are quite elongated in the horizontal dimension.

SUMMARY Long-period (75-250 s) fundamental-mode Love and Rayleigh waves are inverted for the seismic velocity variations in the upper mantle under the Pacific Ocean and surrounding area. The data are inverted to map the regional variation of phase velocity, and then the velocity structure of the upper mantle. No a priori regionalization is used in this study, and the lateral heterogeneity is expanded horizontally into blocks (lo' X 10'). To estimate the reliability of the velocity distribution, we have calculated the resolving kernels. In the central, NW, NE and SW regions, the resolving kernels have fairly sharp peaks. In the SE region, however, resolution is not so good. The results show good correlation with surface tectonics. In the velocity map, shields and ridges are the major features in the depth interval 50-100km. Anomalies associated with the East Pacific Rise and the shields persist to 300 km. At 400 and 500 km, high-velocity anomalies appear under the eastern and southern Pacific ridges, and a low-velocity anomaly appears beneath Australia. The present study provides better resolution than previous studies because of the better path coverage. For example, a large low-velocity anomaly emerges under Hawaii down to the depth of 200 km for SH velocity. The horizontal extent of this anomaly is comparable to the sue of the Hawaiian swell. This feature supports that the Hawaii hotspot is an isolated, truly anomalous feature in the Pacific Ocean.

Summary. The Backus–Gilbert method is applied to obtain the phase velocity variations on a sphere from the measured phase velocity. Narrow peak kernels, with radii of about 2000 km, are obtained for almost everywhere on the sphere. The phase velocity results are thus interpreted as an average within such regions. The most trouble comes from the antipodal peak in the resolution kernel. This is evaluated as contamination and is incorporated in the error estimation. The total error, which is a root mean square of contamination from the antipodal peak and statistical error estimated from the data covariance matrix, is about 1 per cent of the phase velocity in the average earth model, which is the Preliminary Reference Earth Model (PREM). However, there is about a factor of 2 variation of errors on the sphere. Maximum variations of phase velocity are about 3–4 per cent of the phase velocity in the average earth model, and thus there still remain anomalies which exceed estimated errors. The estimated errors correspond to one standard deviation under the assumptions of uncorrelated Gaussian distribution. For high confidence interval, they show that statistically significant anomalies are scarce for the current data set. Generally, Love-wave phase velocity maps show more resolved features than Rayleigh-wave maps and we can see, in high confidence maps, fast velocities in old oceans and old continents and slow velocities in tectonically active regions like the East Pacific Rise and various back-arc regions.

The waveform inversion method described in Woodhouse & Dziewonski (1984) was modified to retrieve regional scale 3-D heterogeneities by using the minor arc part of seismograms. The lateral heterogeneities are expanded horizontally into blocks (10°× 10°) and radially into Legendre polynomials up to order 3 (0–670 km), and thus the results show much fine details of lateral variation than previous global scale studies. We assumed that the heterogeneities produce the perturbation of eigenfrequencies which are the minor arc average of local eigenfrequency shift. We applied the method to the upper mantle beneath the Atlantic Ocean and its environments. Care was taken about the weighting of the data set. We found that the fit of each seismogram became better when the weighting of each seismogram is proportional to the inverse of initial data residuals. Resolution is good in the triangular region surrounded by South America, Europe, and North America. Resolution is not good in the South Atlantic because of the poor path coverage. Depth resolution is not clear, because of the use of Legendre polynomials, though the results suggest a broad half-width of the order of 200 km or more. We found some similarities between previous global studies and our results. For example, low velocities beneath the East Pacific Rise, Chile Rise and Azores triple junction and a high velocity Canadian shield are obtained. However, there are also differences; the high-velocity zone beneath the Brazilian shield at shallow depth is not a prominent feature in this study. Instead, we found a somewhat unexpected feature near the Romanche and Vema fracture zones where shallow positive anomalies exist. Smoothed results calculated by the spherical harmonic expansion are also shown for the purpose of comparison with global studies.

Normal mode theory, extended to the slightly laterally heterogeneous Earth by the first-order Born approximation, is applied to the waveform inversion of mantle Love waves (200-500 sec) for the Earth's lateral heterogeneity at l=2 and a spherically symmetric anelasticity (Q sub mu) structure. The data are from the Global Digital Seismograph Network (GDSN). The l=2 pattern is very similar to the results of other studies that used either different methods, such as phase velocity measurements and multiplet location measurements, or a different data set, such as mantle Rayleigh waves from different instruments. The results are carefully analyzed for variance reduction and are most naturally explained by heterogeneity in the upper 420 km. Because of the poor resolution of the data set for the deep interior, however, a fairly large heterogeneity in the transition zones, of the order of up to 3.5% in shear wave velocity, is allowed. It is noteworthy that Love waves of this period range can not constrain the structure below 420 km and thus any model presented by similar studies below this depth are likely to be constrained by Rayleigh waves (spheroidal modes) only.

Summary. The effect of modal coupling is examined for torsional eigenvibrations of an anelastic earth with oceanic and continental structure. While a coupling effect is hard to observe in the eigenspectrum, some eigenfunctions show significant coupled energy into a secondary neighbouring multiplet. The deeper the heterogeneity, the greater the coupling. Also, stronger coupling is observed in an elastic earth model than in an anelastic model because the poles of the two modes in the complex s-plane are closer in the elastic case than in the anelastic case. The scatter of Q values in a multiplet is less than 5 per cent in our examples but, in extreme cases, strong coupling produces Q values that range between the two Q values for the average earth models. The result indicates that caution is needed in determining Q values for modes that interfere. The results strongly suggest that necessity of incorporating coupling effects in the formulation of inversion aimed at resolving heterogeneity in the Earth.

Summary. Ray tracing equations for surface waves in an anisotropic earth are derived in two ways: first, from the Hamilton’s canonical equations, and secondly, from Fermat’s principle. Phase velocity, including its azimuthal variation, is required to solve the equations, but group velocity is eliminated from the equations. The difference of direction between the wave vector and the ray path is one of the features of wave propagation in an anisotropic media and the equations explicitly show dependence upon such an angle. By putting that angle to be zero, ray tracing equations in a transversely isotropic (or simply isotropic) medium are obtained. In an isotropic medium, phase traveltime, which is an integration of phase slowness along the ray path, is stationary. In an anisotropic medium, phase travel-times is not stationary. Instead, phase slowness projected onto the ray path and integrated along the ray path is stationary. Ray tracing by the bending method in an anisotropic media should utilize such a stationary quantity. In a weakly anisotropic medium, however, the angle ($a) between the wave vector and the ray path is small and cos $, - 1 up to first order in $a. Thus phase traveltime is approximately stationary in a weakly anisotropic medium.