A gridding method commonly called minimum curvature is widely used in the earth sciences. The method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature. The minimum-curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points. Minimum-curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used. These extraneous inflection points can be eliminated by adding tension to the elastic-plate flexure equation. It is straightforward to generalize minimum-curvature gridding algorithms to include a tension parameter; the same system of equations must be solved in either case and only the relative weights of the coefficients change. Therefore, solutions under tension require no more computational effort than minimum-curvature solutions, and any algorithm which can solve the minimum-curvature equations can solve the more general system. We give common geologic examples where minimum-curvature gridding produces erroneous results but gridding with tension yields a good solution. We also outline how to improve the convergence of an iterative method of solution for the gridding equations.

/pdf/gridding-with-continuous-curvature-splines-in-tension-40d6kiojd4.pdf

Gridding with continuous curvature splines in tension

Seismic tomography based on P-wave travel times and improved earthquake locations provides further evidence for mantle-wide convective flow. The use of body waves makes it possible to resolve long, narrow structures in the lower mantle some of which can be followed to sites of present-day plate convergence at the Earth's surface. The transition from subduction-related linear structures in the mid-mantle to long-wavelength heterogeneity near the core-mantle boundary remains enigmatic, but at least some slab segments seem to sink to the bottom of the mantle.

/pdf/evidence-for-deep-mantle-circulation-from-global-tomography-3unqcjsa05.pdf

Evidence for deep mantle circulation from global tomography

Late Mesozoic and Cenozoic rifting and its dynamic setting in Eastern China and adjacent areas

Recent global travel time tomography studies by Zhou [1996] and van der Hilst et al. [1997] have been performed with cell parameterizations of the order of those frequently used in regional tomography studies (i.e., with cell sizes of 1°–2°). These new global models constitute a considerable improvement over previous results that were obtained with rather coarse parameterizations (5° cells). The inferred structures are, however, of larger scale than is usually obtained in regional models, and it is not clear where and if individual cells are actually resolved. This study aims at resolving lateral heterogeneity on scales as small as 0.6° in the upper mantle and 1.2°–3° in the lower mantle. This allows for the adequate mapping of expected small-scale structures induced by, for example, lithosphere subduction, deep mantle upwellings, and mid-ocean ridges. There are three major contributions that allow for this advancement. First, we employ an irregular grid of nonoverlapping cells adapted to the heterogeneous sampling of the Earth's mantle by seismic waves [Spakman and Bijwaard, 1998]. Second, we exploit the global data set of Engdahl et al. [1998], which is a reprocessed version of the global data set of the International Seismological Centre. Their reprocessing included hypocenter redetermination and phase reidentification. Finally, we combine all data used (P, pP, and pwP phases) into nearly 5 million ray bundles with a limited spatial extent such that averaging over large mantle volumes is prevented while the signal-to-noise ratio is improved. In the approximate solution of the huge inverse problem we obtain a variance reduction of 57.1%. Synthetic sensitivity tests indicate horizontal resolution on the scale of the smallest cells (0.6° or 1.2°) in the shallow parts of subduction zones decreasing to approximately 2°–3° resolution in well-sampled regions in the lower mantle. Vertical resolution can be worse (up to several hundreds of kilometers) in subduction zones with rays predominantly pointing along dip. Important features of the solution are as follows: 100–200 km thick high-velocity slabs beneath all major subduction zones, sometimes flattening in the transition zone and sometimes directly penetrating into the lower mantle; large high-velocity anomalies in the lower mantle that have been attributed to subduction of the Tethys ocean and the Farallon plate; and low-velocity anomalies continuing across the 660 km discontinuity to hotspots at the surface under Iceland, east Africa, the Canary Islands, Yellowstone, and the Society Islands. Our findings corroborate that the 660 km boundary may resist but not prevent (present day) large-scale mass transfer from upper to lower mantle or vice versa. This observation confirms the results of previous, global mantle studies that employed coarser parameterizations.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98JB02467

Closing the gap between regional and global travel time tomography

We have developed model S40RTS of shear-velocity variation in Earth's mantle using a new collection of Rayleigh wave phase velocity, teleseismic body-wave traveltime and normal-mode splitting function measurements. This data set is an order of magnitude larger than used for S20RTS and includes new data types. The data are related to shear-velocity perturbations from the (anisotropic) PREM model via kernel functions and ray paths that are computed using PREM. Contributions to phase delays and traveltimes from the heterogeneous crust are estimated using model CRUST2.0. We calculate crustal traveltimes from long-period synthetic waveforms rather than using ray theory. Shear-velocity perturbations are parametrized by spherical harmonics up to degree 40 and by 21 vertical spline functions for a total of 35 301 degrees of freedom. S40RTS is characterised by 8000 resolved unknowns. Since we compute the exact inverse, it is straightforward to determine models associated with fewer or more unknowns by adjusting the model damping. S40RTS shares many characteristics with S20RTS because it is based on the same data types and similar modelling procedures. However, S40RTS shows more clearly than S20RTS the abrupt change in the pattern of shear-velocity heterogeneity across the 660-km phase transition and it presents a more complex patern of shear-velocity heterogeneity in the lower mantle. Utilities to visualise S40RTS and software to analyse the resolution of S40RTS (or models for different damping parameters) are made available.

/pdf/s40rts-a-degree-40-shear-velocity-model-for-the-mantle-from-4i5sq7pv6c.pdf

S40RTS: A degree-40 shear-velocity model for the mantle from new Rayleigh wave dispersion, teleseismic traveltime and normal-mode splitting function measurements

The P wave velocity structure beneath the Western Pacific is found from the International Seismological Center first arrival data. Special attention was paid to the deep structure beneath the Wadati-Benioff zone. We discretized the whole mantle into blocks with finer blocks in the region of interest to obtain the velocities of all the blocks. This way of discretization minimizes a problem with tomographic studies of regional scale: difficulty in making corrections for the effects outside the region of interest. Our solution is iterative with the alternate step of the relocation of earthquakes, using the whole mantle model of Inoue et al. (1990) as a starting model. A first-order smoothness constraint was imposed to suppress the possible fluctuation of the solution around the initial model. The essential result depends little on whether the reference spherical model is smooth or discontinuous near 400- and 670-km depths. We examined the resolution by calculating the resolving kernels for selected blocks and by reconstructing the checkerboard test patterns of velocity perturbation and the test structures of subducting lithosphere. The resolution is depth dependent but in general good enough to see the slab configuration beneath the Southern Kurile-Japan-Izu-Bonin arcs and the Java arc. It is relatively poor beneath the Northern Kurile and Mariana arcs. The seismic image of subducting slab beneath the Southern Kurile to Bonin arcs bends to subhorizontal near the leading edge of the Wadati-Benioff zone and extends continentward over a distance of more than 1000 km. The subhorizontal portion of the slab connects a high-velocity blob to the bottom that reaches a depth of at least 800 km across the 670-km discontinuity under the Japan arc. Although the image of the Java slab directly penetrates the 670-km discontinuity, it then bends to a shallow dip with a considerable spread, reaching a depth of about 1200 km. These results suggest that descending slabs of lithosphere in the Western Pacific tend to be stagnant in the transition zone under a subtle control of the 670-km discontinuity. Although stagnant slab materials eventually descend into the lower mantle, they no longer maintain their original configuration below the 670-km discontinuity.

Subducting slabs stagnant in the mantle transition zone

A new method of multivariate smooth fitting of scattered, noisy data using cubic B-splines was developed. An optimum smoothing function was defined to minimize the &#x1d4c1;2 norm composed of the data residuals and the first and the second derivatives, which represent the total misfit, fluctuation, and roughness of the function, respectively. The function is approximated by a cubic B‐spline expansion with equispaced knots. The solution can be interpreted in three ways. From the stochastic viewpoint, it is the maximum‐likelihood estimate among the admissible functions under the a priori information that the first and second derivatives are zero everywhere due to random errors, i.e., white noise. From the physical viewpoint, it is the finite‐element approximation for a lateral displacement of a bar or a plate under tension which is pulled to the data points by springs. From a technical viewpoint, it is an improved spline‐fitting algorithm. The additional condition of minimizing the derivative norms stabilizes the l...

A least-squares smooth fitting for irregularly spaced data; finite-element approach using the cubic B-spline basis

Summary. Subcrustal earthquakes occur in a slab-like zone beneath SW Japan in association with the subduction of the Philippine Sea plate. They generate a distinct pair of P and S later phases with anomalously low apparent velocities of about 7 and 4 km s−1, which cannot be explained by a laterally homogeneous velocity structure. These later phases are observed for particular source–receiver geometries: the sources at depths shallower than 50–60 km and the receivers beneath which the subcrustal seismic zone comes in contact with the bottom of the continental crust. We examined the nature of these later phases using two methods. One is to determine two types of apparent velocities, an ‘event-common’ apparent velocity and a ‘station-common’ apparent velocity, which may be different from one another for a laterally inhomogeneous velocity structure. The other is to trace seismic rays numerically for a laterally heterogeneous velocity model. An extensive analysis using these two methods leads to a conclusion that the slab-like seismic zone constitutes a low-velocity channel in the uppermost mantle with P and S velocities comparable to those of the lower continental crust beneath SW Japan. The waves trapped within the low-velocity channel escape from it through its contact with the continental Moho and are observed as a distinct pair of later phases. The head waves guided by the bottom boundary of the low-velocity channel are observed as weak arrivals of initial phases. The thickness (<10 km) and the velocities (VP=7 km s−1, VS=4 km s−1) of the subcrustal seismic zone indicates that it is made up of the ‘untransformed basaltic oceanic crust’ subducted with the underlying lithospheric mantle and that the transformation of basalt to eclogite does not take place in the subducting slab at least down to depths of 50–60 km. We present laterally heterogeneous structural models of the crust and the uppermost mantle along two profiles in SW Japan. The models are characterized by an apparent thickening of the lower continental crust in particular regions due to a direct contact of the subducting oceanic crust to the continental Moho. Such features are qualitatively consistent with the results of explosion seismology and with the Bouguer anomaly pattern in the relevant regions. The oceanic lithosphere with the untransformed ‘light’ oceanic crust at its top might still be buoyant after subduction, a factor which must be taken into account in considering the driving mechanisms of plate tectonics, the slab geometry and the earthquake-related tectonic process.

/pdf/seismic-detection-of-the-untransformed-basaltic-oceanic-oclfpn6qhj.pdf

Hiroshi Inoue

Papers

Subducting slabs stagnant in the mantle transition zone

A least-squares smooth fitting for irregularly spaced data; finite-element approach using the cubic B-spline basis

Seismic detection of the untransformed ‘basaltic’ oceanic crust subducting into the mantle

Systematic variation in anisotropy beneath the mantle wedge in the Java–Sumatra subduction system from shear-wave splitting

Deformation and mantle flow beneath the Sangihe subduction zone from seismic anisotropy