Summary Ray Research Articles

Path-averaged delay times for seismic tomography

SUMMARYThe delay times reported for 427 17 476 mantle and core phases in the ISC-EHB Bulletins for the years 1964–2018 are averaged into 120 41 671 summary rays. For each summary ray, I estimate the standard variance of the mean delay and an upper bound for use in tomographic inversion. The trade-off between model norm and the fit to mantle P-wave data shows that the regularized tomographic problem for these data is well posed, with data adequately fit between their error bounds.

Ray tracing and inhomogeneous dynamic ray tracing for anisotropy specified in curvilinear coordinates

SUMMARY Ray tracing has recently been expressed for anisotropy specified in a local Cartesian coordinate system, which may vary continuously in a model specified by elastic parameters. It takes advantage of the fact that anisotropy is often of a simpler nature locally (and is thus specified by a smaller number of elastic parameters) and that the orientation of its symmetry elements may vary. Here we extend this approach by replacing the local Cartesian coordinate system with a curvilinear coordinate system of global extent and by applying the new approach to ray tracing and inhomogeneous dynamic ray tracing. The curvilinear coordinate system is orthogonal and is constructed so that the coordinate axes are consistent with the considered anisotropy of the medium. Our formulation allows for computation of ray attributes (e.g. ray velocity vector and paraxial ray attributes) in the curvilinear coordinate system, while rays are computed in global Cartesian coordinates. Compared to the classic formulation in terms of 21 elastic moduli in global Cartesian coordinates, the main advantages are improved efficiency, lower computer-memory requirements, and conservation of anisotropic symmetry throughout the model.

Adaptive whole Earth tomography

An approach for seismic tomography is presented which allows the parameterization to be refined during the inversion. The objective is to use the data to refine the mesh and the velocity model together, and hence both are considered part of the solution. Some simple rules are used to identify the volumes of a three‐dimensional model in need of refinement. The self‐adaptive parameterization is applied to an initial mesh built from uniformly distributed spherical triangles and Delaunay tetrahedra. Application of the technique to a typical summary ray P‐wave arrival time data set shows it to be both feasible and practical for large scale whole Earth tomography. A noticeable trend in the resulting models, as the parameterization is refined, is the thinning of the Farallon and Tethys subduction features imaged in the mid mantle, together with an increase in amplitude of the velocity perturbation.

Travel times of P and S from the global digital seismic networks: Implications for the relative variation of P and S velocity in the mantle

We present new data sets of P and S arrival times which have been handpicked from long‐period vertical and transverse component recordings of the various global seismic networks. Using events which occurred from 1976 to 1994 results in ∼38,000 globally well‐distributed measurements of teleseismic P and ∼41,000 measurements of S. These data are particularly useful for looking at the relative variation of S and P velocities in the lower mantle. We describe both the measurement techniques and the gross characteristics of the data sets. The size of our data sets allows us to exploit the internal consistency of the data to identify outliers using a summary ray analysis. Since the polarity of each arrival is also known, we can construct fault plane solutions and/or compare with polarities predicted by the Harvard centroid moment tensor solutions to further diagnose phase misidentification. This analysis results in ∼5% of the data being identified as outliers. An analysis of variance indicates that the S residual travel times are dominated by the effects of three‐dimensional structure but the P data have comparable contributions from noise and source mislocation effects. The summary ray analysis reveals the basic character of lower mantle structure, and there are large‐scale patterns in both the S and P data sets that correlate quite well with each other. This analysis suggests that on average, d ln vs/d ln vp is an increasing function of depth in the mantle going from a value of ∼1.7 at the top of the lower mantle to an apparent value of 4 near the base of the mantle. This latter extreme value of R seems to result mainly from data which sample one region in the lowermost mantle under the central Pacific, where large positive S residuals are associated with very small P residuals. Such an anomaly cannot be thermal in origin.

Signature of remnant slabs in the North Pacific from P-wave tomography

SUMMARY A 3-D ray-tracing technique was used in a global tomographic inversion in order to obtaintomographicimagesoftheNorthPacific.ThedatareportedbytheGeophysical SurveyofRussia(1955‐1997)wereusedtogetherwiththecataloguesoftheInternational SeismologicalCenter(1964‐1991)andtheUSGeologicalSurveyNationalEarthquake Information Center (1991‐1998), and the recompiled catalogue was reprocessed. The final data set, used for following the inversion, contained 523430 summary ray paths. The whole of the Earth’s mantle was parametrized by cells of 2°◊2° and 19 layers. The large and sparse system of observation equations was solved using an iterative LSQR algorithm. Asubhorizontalhigh-velocityanomalyisrevealedjustabovethe660kmdiscontinuity beneaththeAleutiansubductionzone.Thishigh-velocityfeatureisobservedatlatitudes of up to~70°N and is interpreted as a remnant of the subducted Kula plate, which disappeared through ridge subduction at about 48Ma. A further positive velocity perturbation feature can be identified beneath the Chukotka peninsula and Okhotsk Sea, extending from~300 to~660km depth and then either extending further down to~800km (Chukotka) or deflecting along the 660km discontinuity (Okhotsk Sea). This high-velocity anomaly is interpreted as a remnant slab of the Okhotsk plate accreted to Siberia at~55Ma.

Inner core rotation: a test at the worldwide scale

In order to determine the angular velocity Ω of a possible inner core differential rotation with respect to the mantle, we analyse travel times of waves sampling the inner core at the worldwide scale over a period of about 30 years. For situations in which the inner core heterogeneities are dominant with respect to the mantle heterogeneities, travel times of summary rays (i.e., bunches of seismic rays with the same geometry) must have smaller mean standard deviations when computed in the inner core reference frame than when computed in the mantle reference frame. Thus, we build summary rays by applying a backward rotation of the inner core at angular velocity − Ω, in order to sum residuals which sample the same inner core region at different times. The mean standard deviation of the summary ray residuals, computed for different values of Ω, should exhibit a minimum for the effective Ω-value. This method has been applied to absolute PKP(DF) residuals, in considering only polar paths, for which inner core heterogeneities are dominant with respect to mantle heterogeneities. The data used come from a high-quality file derived by Engdahl et al. [Engdahl, E.R., van der Hilst, R., Buland, R., 1998. Global teleseismic earthquake relocation with improved travel times and procedures for depth determination. Bull. Seismol. Soc. Am. 88, 722–743.] from the catalogs of the International Seismological Centre (ISC) for the period 1964–1995. The minimum standard deviation is obtained for the null rotation rate. A similar analysis on the absolute PKP(BC) residuals, which correspond to paths that do not sample the inner core, shows that mantle heterogeneities are not at the origin of the observed minimum. The same method has also been applied to a set of more than 1200 worldwide-distributed PKP(BC)–PKP(DF) differential travel time residuals, hand-picked by ourselves or gathered by different authors for specific core studies. Here again, the minimum standard deviation is obtained for the null rotation, implying a synchronous rotation of the inner core with the mantle. However, the limited number of very old data, and the geographic distribution of the paths, prevent us from obtaining a resolution better than 1°/year. In particular, a small super-rotation with Ω=1°/year cannot be ruled out with the presently available data. The problem of the level of the inner core heterogeneities compared to that of the mantle, which is another possible limitation of the method, is also discussed.

A high‐resolution P wave model for the top 1200 km of the mantle

This work attempts to map aspherical velocities in the top 1200 km of the mantle that are illuminated by the International Seismological Centre P wave data. The processing includes summary ray sorting and hypocentral redetermination. To cope with the extreme variation of ray coverage, a multicell inversion is used to simultaneously constrain anomalies in overlapping cells. The multicell solutions are summed after inversion to form the final model, named P1200, at cell size of 1° × 1° and about 50 km in thickness. The lateral resolution of the model is about 1°–2° in major subduction zones and about 5°–15° in most places. Major features in the model withstood tests on signal coherency and different reference models. The long‐wavelength parts of the P1200 model correlate well with previous global models, especially the S12 model of shear wave. Among the large features in the transition zone and below, there are two fast bands, one from Canada to South America and another from Siberia, through East Asia and the Philippine Sea, to western Australia. There are also slow patches under the central Pacific and from India to eastern Africa. Though most subduction zones have slab‐like high‐velocity anomalies, they are sandwiched by broad slow velocities at shallow depths. Some of the slabs appear to penetrate the 660‐km discontinuity, and stagnant slabs are also seen in the transition zone. Plume‐like slow anomalies exist below some prominent hot spots like Hawaii, Iceland, Yellowstone, and French Polynesia.

Interrelationships between ultrastructure, sugar levels, and frost hardiness of ray parenchyma cells during frost acclimation and deacclimation in poplar ( Populus × canadensis Moench ‹ robusta›) Wood

Summary Ray parenchyma cells of poplar wood ( Populus × canadensis Moench ‹ robusta ›) were studied for changes in ultrastructure and in the level of various sugars in fall, winter and spring in relation to freezing resistance as measured by LT 50 (temperature at which 50% of the cells are injured). Before acclimation (mid-October), frost hardiness is low (LT 50 −3 to −5 °C), sugar levels have not yet increased while the ray cell protoplast has already become dense in structure. This is due mainly to the elimination of large vacuoles, their replacement by small storage protein vacuoles, and the vast accumulation of storage material in general, i.e. in amyloplasts, protein bodies, and oleosomes. Frost acclimation started during leaf fall (end of October) concomitant with a sudden increase of total sugar content, consisting mainly of sucrose and its galactosides. An increase in these sugars is paralleled at the ultrastructural level by a prominent vesicular and cisternal ER-system originating in the cytoplasm. Full acclimation (LT 50 lower than −50 °C) is obtained within 3 weeks (mid-November) and coincides with maximum content of sucrose and its galactosides (reaching 300 µg mg −1 DW in ray cells) and with a prominent development of vesicular and cisternal ER. The evidence for the accumulation in this elaborated ER-system of these sugars is discussed and a proposal for its functional significance both as an intrinsic device for dehydrating the protoplast during frost hardening and for delivery of membrane material to the plasma membrane is made. Transient rises of winter temperature above freezing were paralleled by a noticeable decrease in frost resistance (LT 50 −40 to −30 °C), declining sugar levels, and by gradual diminution of the particular ER-vesicles in cytoplasm. In March, at temperatures of 5 to 10 °C, sugar content rapidly lowers concomitandy with complete vanishment of vesicles from the cells. A decrease in frost resistance, in contrast, begins later, e.g. in mid-April, suggesting the involvement in frost hardiness of non-sugar compounds at this stage.

Constraints on inner core anisotropy from PKP(DF) travel times

PKP(DF) travel times reported by the International Seismological Centre (ISC) are analyzed to study the depth dependence of inner core anisotropy. Plots of PKP(DF) travel time residuals versus range for different subsets of the data clearly show large anomalies at ranges beyond about 149° for those paths with inner core ray angles within 30° of the rotation axis. On average, the times are about 3 to 5 s earlier for these paths than for rays at other angles, consistent with recent waveform measurements of anomalous PKP(DF) times by Creager (1992) and Song and Helmberger (1993), and indicative of inner core anisotropy in which the P velocity is about 3% faster for N‐S paths than E‐W paths. The size of these apparent anomalies in the ISC data are much larger than those inferred from several previous analyses of ISC PKP(DF) data which indicated 0.5 to 2 s anomalies at these ranges. The earlier studies applied binning and averaging schemes to the data using windows that were overly tight, thus excluding some of the most anomalous data points and biasing the results toward smaller anomalies. To avoid this problem, I apply a method for computing summary ray residuals that finds the maxima in smoothed histograms of the residuals and does not rely on a fixed interval to window the data points. Standard errors on each summary ray residual are estimated using a bootstrap technique that randomly resamples the data. Plots of the summary ray residuals for several range intervals reveal a small number of independent paths nearly parallel to the rotation axis for which the arrivals are 3 to 5 s early. Although the data are sparse for paths within 30° of the rotation axis, anisotropy does not appear to diminish with depth within the inner core and may cause a much larger anomaly near 180° than the 2‐s value obtained in several prior studies of ISC data. The uniform anisotropy models of Creager (1992) and Song and Helmberger (1993) fit the bulk of the data reasonably well. However, the observed residuals at ranges between 132° and 140° are smaller than those predicted by most anisotropy models, suggesting that the anisotropy may diminish within the outermost 50 km of the inner core.

A revisit to P wave travel time statics at teleseismic stations

To account for local deviations from the one‐dimensional reference velocity model, the station correction formulation of Dziewonski and Anderson (1983) is extended to include the travel time variation with respect to the ray takeoff angle. The station statics are obtained through an iterative retrieval of the lower‐degree spherical harmonics normalized over the data takeoff angle range on station residual spheres, and similar event statics are introduced to decouple the effects of near‐station and near‐event heterogeneities. Except the zero degree constant term, all higher‐degree static terms are obtained on a regional rather than an individual station/event location. The P wave static corrections up to degree and order 2 are derived for 1343 stations using nearly 3 million International Seismological Center travel time residuals with respect to the iasp91 model and with event relocation and summary ray sorting. The resulting station statics account for nearly 27% of the variance of the P wave travel time residuals. Patterns of the zero‐degree statics correlate well with surface tectonic features and with previous results. At most stations over subduction zones the slowest direction of the secondorder statics, or the second‐degree in azimuth, is perpendicular to the trench.

Confidence regions for mantle heterogeneity

Least squares tomographic models of mantle P and S velocity structure from travel times have shown large‐scale variations correlated with surface tectonic features, as well as coherent structures in the lowermost mantle. The reliability of these global features of velocity models depends on whether velocity throughout the features can be estimated well simultaneously: one needs to be able to say with confidence that a feature involving many voxels is likely to be real. We find a lower bound on how wide a 95% simultaneous confidence region for least squares estimates of mantle P or S velocity perturbations must be, as a function of position in the mantle. Suppose (perhaps optimistically) that summary ray travel time errors are independent with mean zero and standard deviation 0.25 s for direct P phases, and standard deviation 0.5 s for all other phases, regardless of distance. If summary travel time residuals are fitted by least squares to a mantle model parametrized by 4872 10° by 10° voxels, the half width of a 95% confidence region for P velocity (using 889,909 summary P rays, 84,046 summary PP rays, 67,228 summary pP rays, 23,024 summary PcP rays, 11,239 summary PKPab rays, 47,227 summary PKPbc rays, and 141,843 summary PKPdf rays) ranges from ±0.025 to ±99.8 km s−1. It is wider than 0.193 km s−1 in half of the mantle and wider than 0.593 km s−1 in a quarter of the mantle, by volume. Under the same assumptions and using the same parametrization, a 95% confidence region for S velocity based on 163,354 summary S rays, 13,781 summary SS rays, 7441 summary ScS rays, and 12,494 summary sS rays ranges from ±0.031 to ±∞ km s−1, is wider than 0.254 km s−1 in half of the mantle and is wider than 0.554 km s−1 in a quarter of the mantle. With this ray set, parametrization, and error model, the 95% confidence region around the least squares estimate of P velocity includes the radially symmetric iasp91 model for 86.6% of the mantle's volume. A 95% confidence region around the least squares S estimate includes iasp91 for 88.9% of the mantle's volume. On a global scale, the mantle's velocity structure is nearly consistent with the iasp91 radially symmetric model. Smaller voxels, more realistic assumptions about the errors, more accurate modeling of the physics, and three‐dimensional structure outside the mantle would make the confidence regions still wider.

Stochastic analysis of global traveltime data: mantle heterogeneity and random errors in the ISC data

Analysis of global traveltime data has been formulated in terms of the stochastic properties of the Earth's heterogeneity pattern and random errors in the data. The formalism relates the coherency of traveltime residuals within bundles of rays (summary rays) of varying size to the spherical harmonic power spectrum of the slowness field of the medium. It has been applied to mantle P-wave data from the ISC catalogue. The measure of coherency is the variance within summary rays. It is estimated within bins in source depth, epicentral distance and the scale size of the area defining a summary ray. The variance at infinitesimal scale length represents the incoherent component of the data (random errors). The variation of the variance with scale length contains information about the autocorrelation function or power spectrum of slowness perturbations within the Earth. The variation with epicentral distance reflects the depth variation of the spectrum. The formalism accounts for the uneven distribution (clustering) of stations and events. We find that estimates of random errors correlate well with complexities on the traveltime curve of P-waves. The variance peaks at 1.0–2.0 s^2 at Δ ≈ 20°, where triplications occur on the traveltime curve, drops to 0.15–0.8s^2 at teleseismic distances, and rises to 0.4–1.3 s^2 approaching the core shadow, where the traveltime curves of P-waves and PcP-waves merge. These estimates should be considered upper bounds for the random error variance of the data. The signal to random noise ratio in the teleseismic ISC P-wave data is about S/N ≈ 2. Inversion of the scale-dependent structural signal in the data yields models that concentrate heterogeneity strongly in the upper mantle. The product of correlation length and power drops by about two orders of magnitude from the surface of the Earth to the lower mantle. About half of this quantity in the upper mantle is due to small-scale features (<300km). The lower mantle is devoid of small-scale structure. It contains 0.1 per cent velocity variations at a characteristic scale of about 1000km. This corresponds to a spectral band-width of l ≈ 7. The D″ layer at the bottom 100–200 km of the mantle shows up as a distinct layer in our results. It has 0.3 per cent velocity variations at a characteristic scale of 350km. The top of the lower mantle contains 0.3 per cent velocity variations on a scale of 500km and also contains some small-scale power. These results are compatible with previous deterministic lower mantle studies, although some details differ. The strength of heterogeneity in the upper mantle may obscure attempts to model the Earth's deep interior.

Ray tracing in azimuthally anisotropic media-I. Results for models of aligned cracks in the upper crust

SUMMARY Ray tracing through gradients in anisotropic materials is complicated by singularities where the two quasi-shear wave slowness sheets cross or touch. Difficulties associated with such points can be removed by explicitly including polarization in the ray tracing equations. Slowness sheet and wavefront plots show the polarization and velocity behavior of various anisotropy models of aligned cracks in the upper crust. A simple scaling of the elastic tensor with depth can be shown to be approximately correct for models of aligned cracks within an isotropic host matrix with a linear velocity gradient. Ray tracing examples for models of aligned cracks within a strong vertical velocity gradient in the uppermost crust demonstrate various features of azimuthal anisotropy, including amplitude and polarization anomalies and shear-wave splitting. Quasi-shear wave polarizations typically twist along ray paths, with stronger twisting near the symmetry axis in hexagonally symmetric media. Strong anisotropy can cause unusual effects, such as ray paths which have three turning points in laterally homogeneous models.

Two-point ray tracing using Fermat's principle

SUMMARY Ray tracing through models consisting of constant-velocity layers separated by curved interfaces, may be implemented by finding ray-paths whose traveltimes are stationary with respect to changes in ray/interface intersection positions. The amplitude of the resulting event is related to the second derivative of the traveltime with respect to changes in rayhnterface intersection position. For complex multilayered models the traveltime functions can be multivalued. Stationary points are then found by decomposition of the function into a number of single valued functions.

Authoritarianism in Queensland: a reply to Ray.

Summary Ray's criticisms of a previous paper by the author are examined and refuted. Reference is made to published evidence which supports the contention that Queensland has a relatively authoritarian culture compared with liberal Northen United States communities, and that racial issues are a salient factor in some aspects of Australian life. Examination of Ray's argument concerning the alleged low validity of the Conservatism scale shows they are not supported by the evidence he refers to, and that he has misrepresented previous studies conducted by himself.