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Efficient and Accurate Calculation of Ray Theory Seismic Travel Time through Variable Resolution 3D Earth Models

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... Hikida & Wieczorek (2007) obtain lunar crustal thickness based on an inversion method of gravity data using a triangular grid. Ballard et al. (2009) used a triangular grid with variable resolution for the computation of seismic traveltimes using ray tracing. Zhang & Chen (2018) proposed a method of forward calculating the gravitational effect of topography with dynamic local refinements based on the STT initialized from an icosahedron. ...
... Zhang & Chen (2018) proposed a method of forward calculating the gravitational effect of topography with dynamic local refinements based on the STT initialized from an icosahedron. Zhang et al. (2018) expanded the modelling structure of Ballard et al. (2009) to include the forward modelling of gravitational data, with the advantage of 3-D density models with relatively constant resolutions across the globe and the option for enhanced model details over structurally complex portions of the model. ...
... Fig. 1 shows some examples of the construction of the STT. As a practical matter, one can obtain such structures using the Delaunay triangulation defined on the spherical surface (Renka 1997), recursive subdividing routines (Ballard et al. 2009;Zhang & Chen 2018;Zhang et al. 2018) or simply by connecting diagonal nodes of a rectangular grid under the spherical coordinates. Subsequently, the topography of the interface is represented by geocentric radii of the STT's vertices. ...
Article
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The inversion of density interfaces using spherical coordinates is usually achieved with techniques developed in the spherical harmonic domain or by applying a rectangular partition of the spherical surface with model cells of a tesseroid. For irregular data coverage and areas at high latitude, methods developed under such a framework have difficulties utilizing multiresolution models or global models that can achieve constant resolution. We introduce the implementation of the spherical triangular tessellation as the fundamental representation for constructing 3-D density models using spherical coordinates for the inversion of the density interfaces. Methods for model construction and algorithms for forward and inverse calculations are presented with synthetic and real-world examples, including a regional and a global synthetic example and an inversion for the crustal thickness of the Moon. The proposed method can be applied to numerous regional and global research problems.
... STT formulation consists of only triangular facets, it is topological inclined for undulant surface modeling. Moreover, for the same spatial resolution, the number of the STT's vertices is about 25 per cent less as compared to grid nodes of the rectangular grid under the spherical coordinates, (Ballard et al. 2009). This fact makes the SST formulation more computational efficient for large scale forward and inversion problems. ...
... In this paper, we demonstrate processes and algorithms used to divide a spherical surface into a set of triangular regions, namely the spherical triangular tessellation (STT) initialized from an icosahedron. In addition, constraints are introduced in the process of model construction to enable varying resolution (Ballard et al. 2009) at different locations. The resulting grid could be used to construct density interface models under the spherical coordinates (Zhang & Chen 2018). ...
... Considering the need for both variable model resolution and an undulated topographic interface, the modeling method we introduced here is developed from approaches proposed by Ballard et al. (2009) and Zhang & Chen (2018). The latter paper demonstrated a method of constructing global density interface models using the polyhedral representation on the basis of the STT formulation and evaluated forward modeling for spherical or ellipsoidal topographic gravitational effects. ...
Article
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We propose a spatial approach for constructing 3D density models and carrying out the corresponding forward modeling of gravitational fields using spherical coordinates. The density model may have variable resolution and undulated interfaces. We introduce a modeling method that uses constraining conditions that provide refinements of the model at selected locations, trajectories or areas. Subsequently, an adaptive strategy is presented to forward calculate the model's exterior gravitational field. The method provides almost constant calculation accuracy from the Earth's surface up to height of satellites. Errors of the forward calculated gravitational fields are minimal and are adequate for regional and global applications given that the 3D density model has sufficient resolution. Two applications are demonstrated as examples: (1) the forward modeling the topographic effect on the gravity field; and (2) the calculation of residual mantle gravity anomalies.
... By contrast, the intrinsic method is more suitable to perform the natural representation on large-scale objects. However, the current intrinsic methods (Bassin et al. 2000, Kageyama and Sato 2004, Zhao 2004, Kageyama and Yoshida 2005, Stemmer et al. 2006, Kageyama et al. 2008, Ballard et al. 2009, 2010 have some defects on its grids, such as shrinking, overlapping, non-latitude-longitude consistent, triangular prismshaped or non-uniformly subdivided and lack of a unified method to represent geometric, attributive and topological information integrally. ...
... For example, Bassin et al. (2000) employed a 2 o × 2 o × 250 m (or 1 km) 3DLLG to develop a global crust model -Crust 2.0, while Zhao (2004) employed a 5 o × 5 o × ∼50 km 3DLLG to investigate the distribution of P-wave velocity in the mantle. Nevertheless, the 3DLLG will gradually shrink to the poles and the core of the Earth, which leads to a large range of grid granularities and result in a great deal of data redundancy and extra calculation in spatial analysing and simulation (Kageyama and Sato 2004, Stemmer et al. 2006, Chen and Xiao 2008, Ballard et al. 2009, Ohno and Kageyama 2009). Yin-Yang grid, developed for geodynamic and for mantle modelling and simulating by Kageyama and Sato (2004), Kageyama and Yoshida (2005) and Kageyama et al. (2008), is a non-shrinking and quasi-uniform 3D spheroid grid. ...
... However, the edges of the 3D spheroid grids are not composed of latitudinal and longitudinal arcs, i.e. it is non-latitude-longitude consistent, which results in a complicated algorithm for conversion between grid code and spheroid coordinates and leads to the less acceptance by spatial information community (Kimerling et al. 1999, Wu andYu 2012). Ballard et al. (2009Ballard et al. ( , 2010) developed a nonshrinking, non-overlapping 3D spheroid grid to explore the global 3D seismic wave velocity model for the planet Earth. However, the 3D spheroid grid is non-latitude-longitude consistent and is not subdivided uniformly along the radial dimension, both of which lead to a loss of universality in data representation and in spatial modelling. ...
Article
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Three-dimensional (3D) modelling is a powerful tool for spatial representation and data analysis, and large scale is the common feature for spatial objects in Global Spatial Information System/Science (GSIS), especially in Earth System Science (ESS). It is important to develop a new 3D modelling method for large-scale spatial objects to meet the demands of global change and ESS researches. The projection-based methods, which have been applied for hundreds of years, are inadequate to perform large-scale spatial modelling, while the embedding methods are unnatural to represent the gravitational features of geo-objects, making the spatial modelling complex and the global data analysis hard. Although the current intrinsic methods are capable of dealing with large-scale spatial modelling, they have some defects such as shrinking, overlapping, non-latitude–longitude consistent, triangular prism-shaped or non-uniformly subdivided and lack of a unified representation model on geometric, topologic and attributive information integrally. Spheroid Degenerated-Octree Grid (SDOG), which takes the advantages of non-shrinking, quasi-uniform, non-overlapping, latitude–longitude consistent, hexahedron-shaped, uniformly subdivided, multi-resolution, is a preferable grid for developing an intrinsic method for the 3D modelling of large-scale spatial objects. This article employed SDOG to develop a new intrinsic method for large-scale 3D modelling. A triple representation model, T(OID, S, A), was proposed to conduct a unified representation on geometric, topologic and attributive information integrally. An algorithm of triples construction, as well as a two-table data structure, was developed to make the intrinsic method operable. A large-scale 3D modelling case, with SDOG-based intrinsic method, on the lithosphere of planet Earth in intrinsic space was illustrated. It shows that the SDOG-based intrinsic method is feasible to perform the 3D modelling of large-scale spatial objects, so as to support global visualization and ESS studies.
... Model properties between nodes are determined through linear interpolation, in contrast to the constant property volume or "voxel" approach. Spherical tessellation parameterizations have been employed for multiple geophysical applications including mantle convection simulations [Baumgardner and Frederickson, 1985], magnetic field modeling [Constable et al., 1993;Stockmann et al., 2009], and the basis for representation of Earth's seismic velocity structure [e.g., Wang and Dahlen, 1995;Chiao and Kuo, 2001;Ishii and Dziewonski, 2002;Antolik et al., 2003;Sambridge and Faletič, 2003;Peter et al., 2007;Ballard et al., 2009;Gung et al., 2009;Myers et al., 2010]. The primary purpose for employing spherical tessellations is to generate a mesh of nearly evenly spaced nodes (or knots), therefore preventing the polar distortion effects of a latitudelongitude grid. ...
... Indexing the parent-daughter triangle relationships during tessellation mesh design allows for efficient communication with the model through a hierarchical version of the triangle search method described by Lawson [1984]. The interpolation approach described herein is described by Ballard et al. [2009], but reiterated for clarity of further discussion. ...
... We have therefore adapted the procedure to find such global minima by considering multiple starting raypath configurations. A similar adaptation has recently been developed by Ballard et al. [2009] where the computational process was designed within a distributed computing environment. Ballard et al. [2009] demonstrated the ability to dramatically improve the efficiency of the technique thus providing support for the implementation of the general approach for routine seismic event location. ...
Article
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We construct a model of three-dimensional P wave velocity structure in the crust and mantle that is global in extent, but with detailed upper mantle heterogeneities throughout the greater Middle East region. Fully three-dimensional ray tracing is employed to achieve accurate travel time predictions of P and Pn arrivals, requiring the characterization of irregular and discontinuous boundaries. Therefore, we explicitly represent undulating seismic discontinuities in the crust and upper mantle within a spherical tessellation modeling framework. The tessellation-based model architecture is hierarchical in that fine node sampling is achieved by recursively subdividing a base level tessellation. Determining the required node spacing to effectively model a given set of data is problematic, given the uneven sampling of seismic data and the differing wavelengths of actual seismic heterogeneity. To address this problem, we have developed an inversion process called Progressive Multilevel Tessellation Inversion (PMTI) that exploits the hierarchical nature of the tessellation-based design and allows the data to determine the level of model complexity. PMTI serves as an alternative to existing multiresolution approaches and robustly images regional trends while allowing localized details to emerge where resolution is sufficient. To demonstrate our complete modeling concept, we construct a velocity model based on teleseismic P travel time data for global events and regional Pn travel time data for events occurring throughout the Middle East. Input data are a product of the statistical procedure called Bayesloc that simultaneously models all components of a multievent system including event locations, origin times, and arrival times (described in the Myers et al. (2011) companion paper). The initial tomographic image provides a new glimpse of the complex upper mantle velocity anomalies associated with the convergence of the Arabian and Indian plates with Eurasia. More important for event monitoring, the model accurately predicts both teleseismic and regional travel times for events occurring within the Middle East region.
... Our model is derived from the latest version of the GT catalog of travel-time picks assembled by Los Alamos National Laboratory. The model is represented using the triangular tessellation system described by Ballard et al. (2009), which incorporates variable resolution in both the geographic and radial dimensions. For our starting model, we use a simpli ed layer crustal model derived from the Crust 2.0 model, overlying a mantle based on a Vp/Vs ratio from our P-velocity model. ...
... • Basic starting point is a 2D, multi-level triangular tessellation of a unit sphere (Ballard et al., 2009;Wang and Dahlen, 1995). ...
Poster
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The SALSA3D (SAndia LoS Alamos 3D) global three-dimensional P-velocity model has been shown to significantly improve seismic event location accuracy and precision using various test data sets. Recently, this improvement was demonstrated using the International Monitoring System (IMS), a sparse, pre-defined seismic network, which is par- ticularly limited by the velocity model used. Typically, 1D models varying with depth are used for location, some- times combined with empirical travel-time corrections. We developed a global 3D P wave velocity model of the Earth’s crust and mantle using seismic tomography to assess improvement to seismic event locations obtained using high quality 3D Earth models in lieu of 1D and 2/2.5D models. For relocation tests using the IMS network and a set of seismic events with ground truth (GT) levels of 5 km or better, over 80% of the defining arrivals are regional Pn (15%) or teleseismic P phases (65.1%). However, there is a small percentage of phases where a secondary phase model would be required (e.g., S, Sn, etc.) or a model that uses phases that interact with the core (e.g., PKP, PcP). We have currently updated the SALSA3D model to include an S-velocity component and plan to update the P-velocity model to use other P-type phases in the future. Our model is derived from the latest version of the GT catalog of travel-time picks assembled by Los Alamos National Laboratory. The model is represented using the triangular tessellation system described by Ballard et al. (2009), which incorporates variable resolution in both the geographic and radial dimensions. For our starting model, we use a simplified layer crustal model derived from the Crust 2.0 model, overlying a mantle based on a Vp/Vs ratio from our P-velocity model. Sufficient damping is used to reduce velocity adjustments so that ray path changes between itera- tions are small. We obtain proper model smoothness by using progressive grid refinement, refining the grid only in areas where the data warrant such a refinement. We utilize the diagonal of the model resolution matrix to control where grid refine- ment occurs, resulting in more consistent and continuous areas of refinement. Our approach produces a smooth, multi-resolution model with node density appropriate to both ray coverage and the velocity gradients required by the data. This scheme is computationally expensive, so we use a distributed computing framework based on the Java Parallel Processing Framework, providing us with ~400 processors. We compare the travel-time prediction and location capabilities of the updated S-velocity SALSA3D model to stan- dard 1D and 2/2.5D models via location tests on a global event set with GT of 5 km or better. We will compare loca- tion results using a subset of these events that have a significant number of S/Sn phases in which to produce random realizations of actual arrival data.
... For these reasons, eikonal solvers are the method of choice when (i) the domain being modeled is limited in spatial extent, (ii) the domain is characterized by strong velocity gradients, and (iii) the ratio of sources to receivers (or vice versa) is high. Ballard et al. (2009) compare the performance of their implementation of the pseudo-bending scheme to the Fast-Marching Method (FMM, a finite difference eikonal solver) and concluded that their bender yields satisfactory results with substantially fewer computer resources. Nonetheless, the monitoring community has successfully employed finite difference schemes for tomographic purposes at small spatial scales (e.g., Zhang et al., 2014;Syracuse et al., 2015). ...
... Sambridge and Rawlinson (2005) and Nolet (2008) provide excellent reviews on this topic. The approach that has become more popular inside the monitoring community to address the uneven data illumination is a multi-scale hierarchical tessellation (Ballard et al., 2009(Ballard et al., , 2016a. Under this approach, a global tessellation with a regular polyhedron is initiated and then division of the faces into smaller-size cells progresses based on some criteria prior to the inversion. ...
Book
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This document entitled “Trends in Nuclear Explosion Monitoring Research and Development – A Physics Perspective” reviews the accessible literature, as it relates to nuclear explosion monitoring and the Comprehensive Nuclear-Test-Ban Treaty (CTBT, 1996), for four research areas: source physics (understanding signal generation), signal propagation (accounting for changes through physical media), sensors (recording the signals), and signal analysis (processing the signal). Over 40 trends are addressed, such as moving from 1D to 3D earth models, from pick-based seismic event processing to full waveform processing, and from separate treatment of mechanical waves in different media to combined analyses. Highlighted in the document for each trend are the value and bene t to the monitoring mission, key papers that advanced the science, and promising research and development for the future.
... This ray-tracing scheme can deal with a general velocity model including several velocity discontinuities of complex geometry and having 3-D velocity variations everywhere in the model. It is adaptable to tracing the first arrivals as well as reflected and converted waves (for details, see Zhao et al. 1992;Ballard et al. 2009;and Chap. 2 of this book). The modified version of the 3-D ray-tracing algorithm (Zhao 2001(Zhao , 2004 yields accurate results with computational error in travel times smaller than 0.05 s, which is considered to be accurate enough, because the uncertainty of arrival-time pickings is generally 0.1 s or greater for the mantle rays at the global scale. ...
... The existence of complex structural heterogeneities in the Earth's interior requires the use of effective 3-D ray tracing in the global tomographic studies. Although there have been new progresses in the studies of 3-D ray tracing recently (e.g., Ballard et al. 2009;Huang et al. 2013a), continuing efforts are still needed to devise better ray tracing algorithms so that more seismic data can be used to image the 3-D Earth's structure in greater details and in higher resolution by taking advantage of advances in computer technology. ...
Chapter
High-resolution global tomography models shed new light on the deep structure and fate of subducting slabs and the origin of hotspots and mantle plumes, as well as deep Earth dynamics. Ray paths of direct and reflected P-waves in a 3-D global velocity model deviate up to 100 km from those in a 1-D Earth model, and the differences in their travel times in the 1-D and 3-D velocity models amount to ~ 4 s, indicating the necessity of using a 3-D ray tracing technique to calculate ray paths and travel times precisely in global tomographic studies. Ten kinds of later phases transmitted and reflected in the mantle and core are used to conduct global tomographic inversions, and it is found that the later phase data can greatly improve the ray path coverage in the mantle and hence the resolution of mantle tomography. Whole-mantle heterogeneities outside the target volume of a regional tomography can cause significant changes (~ 0.2–0.4 s) to the observed relative travel-time residuals of a teleseismic event. The pattern of regional tomography remains the same even after correcting for the whole-mantle heterogeneity, but there are some changes in the amplitude of velocity anomalies in regional tomography. Hence, it is necessary to correct for mantle heterogeneity outside the target volume in order to obtain a better regional tomography.
... [19] The LLNL-G3Dv3 model is parameterized with nodes defined by triangular tessellations of spherical surface ( Figure 4). Spherical tessellation grids have been employed in numerous global geophysical studies primarily for generating evenly spaced points and avoiding polar distortions created by latitude-longitude grids [e.g., Baumgardner and Frederickson, 1985;Constable et al., 1993; Wang and Dahlen, 1995; Chiao and Kuo, 2001;Ishii and Dziewonski, 2002;Antolik et al., 2003;Sambridge and Faletič, 2003;Peter et al., 2007;Ballard et al., 2009;Gung et al., 2009;Stockmann and Jackson, 2009;Myers et al., 2010;Simmons et al., 2011]. Spherical tessellation grids are designed through a process known as dyadic refinement [see Baumgardner and Frederickson, 1985] and are conveniently extensible to any resolution level. ...
... [20] Recently, studies by Ballard et al. [2009] and Simmons et al. [2011] demonstrate the ability to construct complex Earth models within a tessellation-based framework while preserving efficient means of communication with the models. Specifically, designing a spherical tessellation mesh is a recursive process and each subdivision step produces a new level in the grid hierarchy. ...
Article
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We develop a global-scale P wave velocity model (LLNL-G3Dv3) designed to accurately predict seismic travel times at regional and teleseismic distances simultaneously. The model provides a new image of Earth's interior, but the underlying practical purpose of the model is to provide enhanced seismic event location capabilities. The LLNL-G3Dv3 model is based on ˜2.8 millionP and Pnarrivals that are re-processed using our global multiple-event locator called Bayesloc. We construct LLNL-G3Dv3 within a spherical tessellation based framework, allowing for explicit representation of undulating and discontinuous layers including the crust and transition zone layers. Using a multiscale inversion technique, regional trends as well as fine details are captured where the data allow. LLNL-G3Dv3 exhibits large-scale structures including cratons and superplumes as well numerous complex details in the upper mantle including within the transition zone. Particularly, the model reveals new details of a vast network of subducted slabs trapped within the transition beneath much of Eurasia, including beneath the Tibetan Plateau. We demonstrate the impact of Bayesloc multiple-event location on the resulting tomographic images through comparison with images produced without the benefit of multiple-event constraints (single-event locations). We find that the multiple-event locations allow for better reconciliation of the large set of direct P phases recorded at 0-97° distance and yield a smoother and more continuous image relative to the single-event locations. Travel times predicted from a 3-D model are also found to be strongly influenced by the initial locations of the input data, even when an iterative inversion/relocation technique is employed.
... SALSA3D is a three-dimensional global seismic velocity model of the Earth's mantle that captures spatial variations of the solid earth's properties and was developed using 3D tomographic inversion of an extensive seismic travel-time dataset (Ballard et al., 2016b;Begnaud et al., 2015). This velocity model is stored in GeoTess format (Ballard et al., 2016a) and has been employed by PCalc (Ballard et al., 2009;Conley et al., 2021), a 3D ray tracer based on the pseudo-bending algorithm of Um and Thurber (1987) and Zhao and Lei (2004). The lateral heterogeneity of the Earth, which mostly affects propagation at local and regional seismic distances (less than about 2000 km) where waves spend most of their time in the highly heterogeneous shallow parts of the Earth, is then taken into account by the 3D velocity model and ray tracer. ...
Article
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The conversion from seismic to ocean-acoustic waves occurs in different places on the bottom of the ocean, often hundreds to thousands of kilometers away from the epicenter. Here, we investigate this conversion process by studying 15 large-magnitude earthquakes that occurred between 2014 and 2022 along the Kermadec Arc in the southwestern Pacific Ocean. To pinpoint the location where seismic-to-acoustic conversion takes places, we analyze hydroacoustic signals recorded by a hydrophone triplet station of the International Monitoring System in the Juan Fernández archipelago. Results from direction-of-arrival and travel-time calculations indicate that the location of the conversion zone largely matches segments of the Louisville Seamount Chain, its lateral extent ranging from approximately 300 to 1800 km, and its location depending on the geometry between earthquake epicenter and the seamounts.
... In order to compare the use of 3DTTLS to full 3D ray tracing, we used the SAL-SA3D model (Ballard et al., 2016a) [https://www. sandia.gov/salsa3d] to build the 3DTTLS as well as for 3D ray tracing (i.e., ''Bender'', Ballard et al., 2008;Ballard et al., 2009;Um & Thurber, 1987) using first-P and first-S phases. SALSA3D is a global model for P-and S-velocities (the S-model is unpublished but publicly available), defined on a 1°G eoTess tessellation, with crust, mantle, and core layers. ...
Article
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Location algorithms have historically relied on simple, one-dimensional (1D) velocity models for fast seismic event locations. 1D models are generally used as travel-time lookup tables, one for each seismic phase, with travel-times pre-calculated for event distance and depth. These travel-time lookup tables are extremely fast to use and this fast computational speed makes them the preferred type of velocity model for operational needs. Higher-dimensional (i.e., three-dimensional—3D) seismic velocity models are becoming readily available and provide more accurate event locations over 1D models. The computational requirements of these 3D models tend to make their operational use prohibitive. Additionally, comparing location accuracy for 3D seismic velocity models tends to be problematic, as each model is determined using different ray-tracing algorithms. Attempting to use a different algorithm than the one used to develop a model usually results in poor travel-time prediction. We demonstrate and test a framework to create first-P and first-S 3D travel-time correction surfaces using an open-source framework (PCalc + GeoTess, https://www.sandia.gov/salsa3d/software/geotess) that easily stores 3D travel-time and uncertainty data. This framework produces fast travel-time and uncertainty predictions and overcomes the ray-tracing algorithm hurdle because the lookup tables can be generated using the exact ray-tracing algorithm that is preferred for a model.
... Attributed to increased amounts and resolutions of the geophysical observations, as well as the rapid growth of computing power, geophysical inversions that aim to recover detailed spatial structures of the subsurface physical properties have been developing actively in the recent decades. Techniques like the travel-time tomography (Ballard et al. 2009;Bianco and Gerstoft 2018;Lelièvre et al. 2011Lelièvre et al. , 2012, surface wave tomography (Barmin et al. 2001;Ritzwoller and Levshin 1998;Sabra et al. 2005;Shapiro et al. 2005), receiver functions (Ammon et al. 1990;Zhu and Kanamori 2000) and 2D/3D inversions of the MT data (Manassero et al. 2020;Siripunvaraporn et al. 2005;Usui 2015) or potential field data Oldenburg 1996 1998;Liang et al. 2014;Zhang et al. 2019) are developed to obtain underground velocity, conductivity, density or geometric structures from regional to global studies. For most of the aforementioned methods, depending on the nature of the inversion problem, the subsurface space is firstly divided into discrete meshes with specific physical properties. ...
Article
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A new efficient random search algorithm is introduced for solving inversion problems in geophysical studies. The proposed algorithm is inherently a stochastic optimization method which is built on the concept of gradient descending and Lévy flights. Therefore, the algorithm is referred to as the Lévy gradient descent (L-GD). In which, the Lévy flights is a special class of random walk which consists of many short steps along with a few large steps. Such movements are observed in a varying range of fields, including animals’ foraging patterns, fluid dynamics, transport of light and so on. Meanwhile, the Lévy flights typically shows much higher speed in searching for sparsely located targets compared to the well-known Brownian walks, which make them preferable to drive random search algorithms. As shown in the paper, besides optimal solutions of the inverse problems, the L-GD algorithm could also produce estimations on the error distributions of the resultant model parameters. Following a detailed introduction of the methodology, parameter settings of the algorithm are discussed in length through statistical experiments. Subsequently, the proposed algorithm is evaluated using numeric tests and shows attracting properties of the global convergence and significant higher searching efficiency compared to commonly adopted stochastic optimization techniques in geophysical inversions. Moreover, the L-GD algorithm is applied to the inversions of gravity and seismic travel time data and has achieved the same accuracy as gradient-based optimization methods. Meanwhile, though error estimations generated by L-GD algorithm are essentially qualitative, they could still provide valuable information to help evaluating the resultant model parameters, which is of great importance for practical geophysical inversions.
... salsa3d/Software.html] (Ballard et al. 2008(Ballard et al. , 2009 which permits using standard 1D travel-time tables (e.g., iasp91 and ak135), 3D source-specific station corrections (SSSCs), ray-bending through 3D models, and full RSTT models for travel-time prediction. All depths were fixed at the Bayesloc result to reduce the trade-off between depth and origin time. ...
Article
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The regional seismic travel time (RSTT) model and software were developed to improve travel-time prediction accuracy by accounting for three-dimensional crust and upper mantle structure. Travel-time uncertainty estimates are used in the process of associating seismic phases to events and to accurately calculate location uncertainty bounds (i.e. event location error ellipses). We improve on the current distance-dependent uncertainty parameterization for RSTT using a random effects model to estimate slowness (inverse velocity) uncertainty as a mean squared error for each model parameter. The random effects model separates the error between observed slowness and model predicted slowness into bias and random components. The path-specific travel-time uncertainty is calculated by integrating these mean squared errors along a seismic-phase ray path. We demonstrate that event location error ellipses computed for a 90% coverage ellipse metric (used by the Comprehensive Nuclear-Test-Ban Treaty Organization International Data Centre (IDC)), and using the path-specific travel-time uncertainty approach, are more representative (median 82.5% ellipse percentage) of true location error than error ellipses computed using distance-dependent travel-time uncertainties (median 70.1%). We also demonstrate measurable improvement in location uncertainties using the RSTT method compared to the current station correction approach used at the IDC (median 74.3% coverage ellipse).
... salsa3d/Software.html] (Ballard et al. 2008(Ballard et al. , 2009 which permits using standard 1D travel-time tables (e.g., iasp91 and ak135), 3D source-specific station corrections (SSSCs), ray-bending through 3D models, and full RSTT models for travel-time prediction. All depths were fixed at the Bayesloc result to reduce the trade-off between depth and origin time. ...
Article
Full-text available
The regional seismic travel time (RSTT) model and software were developed to improve travel-time prediction accuracy by accounting for three-dimensional crust and upper mantle structure. Travel-time uncertainty estimates are used in the process of associating seismic phases to events and to accurately calculate location uncertainty bounds (i.e. event location error ellipses). We improve on the current distance-dependent uncertainty parameter-ization for RSTT using a random effects model to estimate slowness (inverse velocity) uncertainty as a mean squared error for each model parameter. The random effects model separates the error between observed slowness and model predicted slowness into bias and random components. The path-specific travel-time uncertainty is calculated by integrating these mean squared errors along a seismic-phase ray path. We demonstrate that event location error ellipses computed for a 90% coverage ellipse metric (used by the Comprehensive Nuclear-Test-Ban Treaty Organization International Data Centre (IDC)), and using the path-specific travel-time uncertainty approach, are more representative (median 82.5% ellipse percentage) of true location error than error ellipses computed using distance-dependent travel-time uncertainties (median 70.1%). We also demonstrate measurable improvement in location uncertainties using the RSTT method compared to the current station correction approach used at the IDC (median 74.3% coverage ellipse).
... salsa3d/Software.html] (Ballard et al. 2008(Ballard et al. , 2009 which permits using standard 1D travel-time tables (e.g., iasp91, ak135), source-specific correction surfaces (SSSCs) for each IMS station and phase (Firbas et al. 1998) (converted to GeoTess format), ray-bending through 3D models, and full RSTT models for travel-time prediction. All depths are fixed at the Bayesloc result to reduce the trade-off between depth and origin time. ...
Article
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A function of global monitoring of nuclear explosions is the development of Earth models for predicting seismic travel times for more accurate calculation of event locations. Most monitoring agencies rely on fast, distance-dependent one-dimensional (1D) Earth models to calculate seismic event locations quickly and in near real-time. RSTT (Regional Seismic Travel Time) is a seismic velocity model and computer software package that captures the major effects of three-dimensional crust and upper mantle structure on regional seismic travel times, while still allowing for fast prediction speed (milliseconds). We describe updates to the RSTT model using a refined data set of regional phases (i.e., Pn, Pg, Sn, Lg) using the Bayesloc relative relocation algorithm. The tomographic inversion shown here acts to refine the previous RSTT public model (rstt201404um) and displays significant features related to areas of global tectonic complexity as well as further reduction in arrival residual values. Validation of the updated RSTT model demonstrates significant reduction in median epicenter mislocation (15.3 km) using all regional phases compared to the iasp91 1D model (22.1 km) as well as to the current station correction approach used at the Comprehensive Nuclear-Test-Ban Treaty Organization International Data Centre (18.9 km).
... Yin-Yang 格网继 承了经纬格网正交的优点, 也解决了两极及球心的 收缩问题, 但其 Yin 和 Yang 两区的边界存在重叠, 物理空间与格网空间映射不唯一, 导致 Yin-Yang 边 界区空间及属性信息表达含糊, 不利于空间统一表 达、建模与分析. Ballard 等 [28] 则将球面三角网沿径向 扩展, 设计了一种非重叠的三维格网, 构建了全球地 震波三维速度场模型; 但其格网在径向上采用自然 不连续面划分, 导致径向划分非一致、非均匀; 且其 格网在球面上采用了三角划分, 丧失了格网的正交 性及经纬一致性(即边界全为经纬径线的格网). 格网 的非正交性增加了数据预处理难度和数值模拟过程 的复杂度 [25] , 而非经纬一致性导致了现有大量基于 经纬坐标的数据与算法难以直接应用, 以及所建立 的格网模型与计算结果难以被其他应用模型与研究 领 域 共 享 复 用 . ...
... To model the traveltime from one grid to all grids at once, we select the Eikonal solvers for seismic ray tracing, which are more accurate than ray shooters and ray benders, especially in the presence of strong velocity gradients [30]. The Eikonal equation is a non-linear partial differential equation encountered in problems of wave propagation. ...
Article
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The accurate location of induced seismicity is a problem of major interest in the safety monitoring of underground mines. Complexities in the seismic velocity structure, particularly changes in velocity caused by the progression of mining excavations, can cause systematic event mislocations. To address this problem, we present a novel construction method for an arbitrary 3D velocity model and a targeted hypocenter determination method based on this velocity model in underground mining. The method constructs a velocity model from 3D geological objects that can accurately express the interfaces of geologic units. Based on this model, the block corresponding to the minimum difference between the observed arrival times and the theoretical arrival times computed by the Fast Marching Method is located. Finally, a relocation procedure is carried out within the targeted block by heuristic algorithms to improve the performance. The accuracy and efficiency of the proposed method are demonstrated by the source localization results of both synthetic data and on-site data from Dongguashan Copper Mine. The results show that our proposed method significantly improves the location accuracy compared with the widely used Simplex and Particle Swarm Optimization methods.
... We applied the tomographic method of Zhao et al. 20 to invert the local earthquake arrival times for a 3-D P-wave velocity (Vp) model of the study area. The tomographic method uses a 3-D ray tracing technique combining the pseudo-bending algorithm 54 and Snell's Law to compute theoretical travel times and ray paths, which has been confirmed to be very robust and efficient by a number of studies (e.g., 10,55,56 ). This tomographic method was originally developed for a study of the Japan subduction zone and adopted the spherical coordinate system. ...
Article
In this study, we determined P-wave tomographic images of the Yongshaba deposit, an underground mining zone in Guizhou Province, China by inverting arrival-time data of micro-seismic events and blasts recorded by a passive seismic array consisting of 28 sensors during January to April 2014 using an event location technique and a travel-time tomography method that can handle complex seismic discontinuities. To reveal internal P-wave velocity changes of the study area under severe mining activities, a time-lapse data partition scheme is used. To assess the human influence on the underground structure, we introduce a parameter to measure the relative velocity changes in different periods. The resolution of the tomographic images and the robustness of the obtained features are examined by conducting a series of checkerboard resolution tests and a restoring resolution test. Our tomographic results obtained from the entire data set reveal a prominent low-velocity (low-V) zone and many obvious high-velocity (high-V) zones. These features match well with the geological setting and the excavation plan carried out in the mine, according to in-site surveys. The low-V zone may reflect empty volumes, stress releases and rock breaking and cracking caused by mining activities, whereas the high-V zones are probably the consequences of local stress concentrations caused by regional stress redistribution. The time-lapse tomographic images may reveal the process of stress concentrations caused by rock breaking due to the continuing excavations in the entire observation period, indicating that the mining activities influenced the rock property and caused complex changes of the underground structure.
... The source and station datasets were decimated to reduce ray redundancy, to include no more than one source or station within a 1°× 1°m odel cell, resulting in 1668 sources and 4221 stations ( Fig. 1). Our raytracing is performed using an infinite-frequency pseudobending method (Um and Thurber, 1987;Ballard et al., 2009Ballard et al., , 2016. The algorithm computes theoretical first-arriving P-wave ray paths through the crust and mantle from a source to a sensor within 90°distance, avoiding the P-wave shadow zone. ...
Article
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We present a comparison between global seismic ray coverage for observations at today's largely on-land seismic stations and coverage possible with the addition of proposed Science Monitoring and Reliable Telecommunication (SMART) cables equipped with closely spaced seismic sensors. Coverage is estimated by generating P-wave rays through the ak135 reference model from 1668 earthquakes havingM >6:0, traced to 4421 permanent and temporary seismic stations in existence to date. An additional 1382 anticipated sensors populating notional first-generation SMART cables provide new rays originating from the same 1668 earthquakes. The greatest improvement in the ray-path coverage is observed beneath the Indian, Atlantic, and northern Pacific Oceans, where at some depths, the number of well-sampled model cells is increased by a factor of more than three times, compared to the coverage possible using only existing stations. The improved coverage may offer clear benefits for earthquake detection and location, global seismological modeling, and the.
... 三维空间, Stemmer 等[9] 、Ballard 等[10] 以及 Stadler等 [11] 面向多种应用对三维格网进行了探索。 ...
Article
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To supplement the deficiency of the latitude and longitude existed as location code, such as complex description, non-regional characteristics and complex computation, a globe spatial grid reference system is constructed based on GeoSOT from Peking University. The grid system, built from a perfect quadtree with one degree, one minute and one second grid, could be fit for air-earth joint action. It designs a simple and practical location coding method, which also supports distance simple calculation. It could realize multi-source spatial data integrated retrieval, and develop methods of efficient code operation, framework of spatial computing, and 3D-earth grid system. Globe Spatial Grid Reference System will definitely play an important role in the future of big spatial data applications.
... (2) Trace all the rays in the tomography data set using the pseudobending algorithm (Um and Thurber, 1987;Zhao and Lei, 2004;Ballard et al., 2009). Populate the matrix of data kernels A with the model parameter weights derived by interpolating the data ray paths through the model, and the vector of data residuals Δd with the difference between observed and computed travel times, weighted by the data uncertainties. ...
Article
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The task of monitoring the Earth for nuclear explosions relies heavily on seismic data to detect, locate, and characterize suspected nuclear tests. Motivated by the need to locate suspected explosions as accurately and precisely as possible, we developed a tomographic model of the compressional wave slowness in the Earth’s mantle with primary focus on the accuracy and precision of travel-time predictions for P and Pn ray paths through the model. Path-dependent travel-time prediction uncertainties are obtained by computing the full 3D model covariance matrix and then integrating slowness variance and covariance along ray paths from source to receiver. Path-dependent travel-time prediction uncertainties reflect the amount of seismic data that was used in tomography with very low values for paths represented by abundant data in the tomographic data set and very high values for paths through portions of the model that were poorly sampled by the tomography data set. The pattern of travel-time prediction uncertainty is a direct result of the off-diagonal terms of the model covariance matrix and underscores the importance of incorporating the full model covariance matrix in the determination of travel-time prediction uncertainty. The computed pattern of uncertainty differs significantly from that of 1D distance-dependent traveltime uncertainties computed using traditional methods, which are only appropriate for use with travel times computed through 1D velocity models.
... Software that supports triangular tessellations, on the other hand, is somewhat more complicated to develop but results in grids with much more uniform cell size and ∼25% fewer vertices. Triangular tessellations have been used in a number of Earth science applications (Wang and Dahlen, 1995;Wang et al., 1998;Simmons et al., 2011;Ballard et al., 2009;Myers et al., 2010;Pasyanos et al., 2014). ...
Article
GeoTess is a model parameterization and software support library that manages the construction, population, storage, and interrogation of data stored in 2D and 3D Earth models. The software is available in Java and C++, with a C interface to the C++ library. The software has been tested on Linux, Mac, Sun, and PC platforms. It is open source and is available online (see Data and Resources).
... Among all the schemes, the gird based on regular octahedron has the characters of clear positioning, relatively simple structure and moderate deformation, and it is the most researched and applied discrete gird data model(Zhao, X. S., et al, 2002). Types of spherical solid grid are relatively few, only eight kinds, including latitude and longitude grid, the radially extending spherical triangular mesh grid [Baumgardner, J R., et al, 1985], the Cubed-sphere grid [Tsuboi, S., et al, 2008], the "Yin Yang grid" [Kageyama, A., et al, 2004], the based on triangulation and radial spherical discontinuity divided grid [Ballard, S., et al, 2009], the adaptive mesh refinement (Adaptive Mesh Refinement, AMR) [Stadler, G., et al, 2010], the degradated octree grid, and the great arc QTM octree grid [Jinxin, W., et al, 2012;. From the following discussion shows that the Grids which fit for GIS spatial data model, or the universal Earth system (Lixin, W., et al(2012) called it as Earth System Spatial Grid, ESSG) only are latter degenerated octree and octree . ...
Article
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With the development of computers, network communications, scientific computing, mapping remote sensing and geographic information technologies, Discrete Global Grids (DGGs) and Earth System Spatial Grid(ESSG)have become the integrated spatial data model facing the large-scale and global-scale problems and the complex geo-computation. This paper discusses the property and character of the global spatial data at first. Then it introduces the grid division system based on large arc QTM octree and compares this scheme with degradation octree scheme. At last, it introduces the application of the scheme in land surface, underground and aerial geographic entity modeling. The study suggests that: the grid division system based on large arc QTM octree has the potential to integrate the whole spatial data of different layers of the geospatial. And it will have a broad application prospect in complex large-scale geographic computing.
... For later arrival tracing in two point ray tracing problem, good algorithms are available, for example, a wavefront construction method (VINJE et al., 1993 and, a perturbation theory (BIJWAARD and SPAKMAN, 1999), a revised pseudo-bending method (ZHAO, 2001), a generalized ray theory method (BALLARD et al., 2009), and a dynamic ray tracing method (DAHLEN et al., 2000). Among the grid/cellbased wavefront propagation method, there is an alternative choice for tracing the first arrivals of multiply transmitted, reflected (or refracted), and converted seismic phases. ...
Article
Full-text available
Traditionally, traveltime tomography entails inversion of either the velocity field and the reflector geometry sequentially, or the velocity field and the hypocenter locations simultaneously or in a cascaded fashion, but seldom are all three types (velocities, geometry of reflectors, and source locations) updated simultaneously because of the compromise between the different classes of model variable and the lack of different seismic phases to constrain these variables. By using a state-of-the-art ray-tracing algorithm for the first and later arrivals combined with a popular linearized inversion solver, it is possible to simultaneously recover the three classes of model variables. In the work discussed in this paper we combined the multistage irregular shortest-path ray-tracing algorithm with a subspace inversion solver to achieve simultaneous inversion of multi-class variables, using arrival times for different phases to concurrently obtain the velocity field, the reflector shapes, and the hypocenter locations. Simulation and comparison tests for two sets of source–receiver arrangements (one the ideal case and the other an approximated real case) indicate that the combined triple-class inversion algorithm is capable of obtaining nearly the same results as the double-class affect inversion scheme (velocity and reflector geometry, or velocity and source locations) even if a lower ray density and irregular source-receiver geometry are used to simulate the real situation. In addition, the new simultaneous inversion method is not sensitive to a modest amount of picking error in the traveltime data and reasonable uncertainty in earthquake hypocenter locations, which shows it to be a feasible and promising approach in real applications.
... The lateral positioning of nodes is determined by a triangular tessellation of a spherical surface (i.e., spherical tessellation grids) with triangle vertices defining a grid of latitude/ longitude points. Spherical tessellation grids have been used for global-scale geophysical problems for decades, because node spacing is approximately equal at all latitudes (e.g., Baumgardner and Frederickson, 1985;Constable et al., 1993;Wang and Dahlen, 1995;Ballard et al., 2009). The LLNL-G3Dv3 spherical tessellation grid is developed by recursively subdividing the triangular facets comprising an icosahedron, thus generating what we refer to as a "hierarchical spherical tessellation grid." ...
Article
A global validation dataset of 116 seismic events and 20,977 associated Pn and P arrivals is used to assess travel-time prediction and event location accuracy for the global-scale, 3D, P-wave velocity model called LLNL-G3Dv3 (Simmons et al., 2012). Strong regional trends that are observed for ak135 travel-time residuals are largely removed when LLNL-G3Dv3 is used for prediction. The 25th-75th quantile spread of travel-time residuals is reduced by 30%-40% at teleseismic distances, and the spread is reduced by ∼60% at regional distances (<16°). Epicenter error decreases when more data are used to constrain event locations until more than ∼40 arrivals times are used. At which point, epicenter error reduction tends to plateau. Median epicenter errors for the ak135 and LLNL-G3Dv3 models plateau at ∼8:0 and ∼5:5 km, respectively, for teleseismic P datasets. Median epicenter errors for the ak135 and LLNL-G3Dv3 models plateau at ∼12:0 and ∼4:0 km, respectively, for regional Pn datasets. We demonstrate that spatially correlated travel-time residual errors for the ak135 model lead to increasing epicenter error when ∼40 to ∼100 Pn arrivals are used to constrain the location. The effect of correlated error is mitigated by LLNL-G3Dv3, for which epicenter error steadily decreases to ∼4 km when 100 Pn arrivals are used. The median area of 0.95 epicenter probability bounds for ak135 and LLNL-G3Dv3 are 1811 and 758 km2, respectively. The ak135 ellipses are inflated to achieve the desired rate of true events occurring inside the probability region, whereas LLNLG3Dv3 error ellipses based on empirical residual distributions cover the true location at the expected rate because location bias is minimal.
... For purposes of global travel-time tomography, there exist a variety of schemes to predict the travel-times for a couple of the important global phases. They include perturbation theory (Bijwaard and Spakman, 1999), dynamic ray tracing (Dahlen et al., 2000), the pseudo-bending method (Zhao, 2001) nd generalized ray theory (Ballard et al., 2009), in which most of the global phases are not predicted and accommodated, for example, later phases with the minimax time-path (see Ray Tracing for Phases with a Stationary Minimax Time Path section for detailed explanation). ...
Article
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The pioneering work of Jeffreys and Bullen (1940; JB Tables) established the first truly global seismological travel‐time tables, which have fulfilled a vital and important role in performing earthquake locations and seismic tomography. Such information has enriched our knowledge about the inner structure of the Earth and its dynamic processes. The JB travel‐time tables, which are based on 1D velocity model (radial dependence only), have gradually improved over the years through a series of modifications, for example, the PREM model (Dziewonski and Anderson, 1981), the IASP91 model (Kennett and Engdahl, 1991) and the AK135 model (Kennett et al. , 1995). The major source of error in travel‐time prediction with these tables is due to from unincorporated lateral heterogeneities in the true velocity structure of the Earth. For traditional purposes of earthquake location, the global travel‐time tables (i.e., ISAP91 or AK135) have been generalized by a ray‐parameter method, such as a tau‐p function integral method of Buland and Chapman (1983). Some modified versions of the tau‐p function integral method are available, for example, the TauP Toolkit software (Crotwell et al. , 1999) and TTBox software (Knapmeyer, 2004), which offer improvements in computational efficiency and flexibility of usage. For purposes of global travel‐time tomography, there exist a variety of schemes to predict the travel‐times for a couple of the important global phases. They include perturbation theory (Bijwaard and Spakman, 1999), dynamic ray tracing (Dahlen et al. , 2000), the pseudo‐bending method (Zhao, 2001) nd generalized ray theory (Ballard et al. , 2009), in which most of the global phases are not predicted and accommodated, for example, later phases with the minimax time‐path (see Ray Tracing for Phases with a Stationary Minimax Time Path section for detailed …
... For later arrival tracing in two point ray tracing problem, good algorithms are available, for example, a wavefront construction method (VINJE et al., 1993 and, a perturbation theory (BIJWAARD and SPAKMAN, 1999), a revised pseudo-bending method (ZHAO, 2001), a generalized ray theory method (BALLARD et al., 2009), and a dynamic ray tracing method (DAHLEN et al., 2000). Among the grid/cellbased wavefront propagation method, there is an alternative choice for tracing the first arrivals of multiply transmitted, reflected (or refracted), and converted seismic phases. ...
Conference Paper
Full-text available
Traditionally, traveltime tomography entails inversion of either the velocity field and the reflector geometry sequentially, or the velocity field and the hypocenter locations simultaneously or in a cascaded fashion, but seldom are all three types (velocities, geometry of reflectors, and source locations) updated simultaneously because of the compromise between the different classes of model variable and the lack of different seismic phases to constrain these variables. By using a state-of-the-art ray-tracing algorithm for the first and later arrivals combined with a popular linearized inversion solver, it is possible to simultaneously recover the three classes of model variables. In the work discussed in this paper we combined the multistage irregular shortest-path ray-tracing algorithm with a subspace inversion solver to achieve simultaneous inversion of multi-class variables, using arrival times for different phases to concurrently obtain the velocity field, the reflector shapes, and the hypocenter locations. Simulation and comparison tests for two sets of source–receiver arrangements (one the ideal case and the other an approximated real case) indicate that the combined triple-class inversion algorithm is capable of obtaining nearly the same results as the double-class affect inversion scheme (velocity and reflector geometry, or velocity and source locations) even if a lower ray density and irregular source-receiver geometry are used to simulate the real situation. In addition, the new simultaneous inversion method is not sensitive to a modest amount of picking error in the traveltime data and reasonable uncertainty in earthquake hypocenter locations, which shows it to be a feasible and promising approach in real applications.
... The nodes form a triangular tessellation that seamlessly covers the globe, and node spacing may be adjusted as needed. Nominal node spacing is approximately 1° for current RSTT models, and we have used 0.5° node spacing in North America, which is warranted by outstanding data coverage in the western U.S. Velocity interfaces are defined by the radial distance from the center of the Earth, which allows us to explicitly build the GRS80 ellipsoid (Moritz, 1980) into the model and obviate travel time corrections for ellipticity (e.g., Ballard et al., 2009). ...
Article
Full-text available
Lowering the global seismic detection threshold can be accomplished by introducing data recorded at regional distances (<2000 km from the event) into global monitoring systems. Unfortunately, introduction of regional data degrades average epicenter accuracy compared to locations constrained solely by data recorded at greater distances. Location accuracy degrades because regional seismic travel time (RSTT) prediction error is generally greater than prediction error for waves that travel to greater distances. In previous work we developed a computationally efficient method to capture the first-order effects of three-dimensional (3-D) crust and upper mantle structure on RSTTs. Previous results demonstrate that RSTT prediction accuracy is greatly improved by seismic tomography, in which model velocities are adjusted so that predicted travel times are in agreement with travel times that are based on a dataset of accurate event locations and arrival-time measurements. We have conducted RSTT tomography for the regional phases Pn, Pg, Sn, and Lg across Eurasia and for the Pn phase across North America. After tomography across Eurasia and North America, rigorous tests find that the standard deviation of Pn travel time residuals is reduced from approximately 1.75 seconds (ak135 model) to approximately 1.25 seconds. Further, the median location error is reduced from approximately 15 km to 9 km for network configurations that effectively average out measurement error. Reduction of epicenter error, millisecond travel time computation, and the flexibility to compute travel times between arbitrary points on/in the globe all make the RSTT method ideal for routine location work and for use in seismic monitoring systems. Extension of RSTT tomography to additional regions is a necessary step towards making the RSTT approach truly global and therefore attractive to monitoring agencies, such as the International Data Center at the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO), the National Earthquake Information Center (NEIC), and the International Seismic Centre (ISC). Geographic extension of RSTT tomography through regional cooperation is currently being pursued. In the regional cooperation approach, scientists from study regions participate in the development of high-quality data sets, tomography, and model validation. A regional cooperation approach works towards producing the most complete tomographic data set, and it provides a better understanding of the RSTT approach to the international scientific community.
... We decided to parameterize the model laterally as a tessellated surface, rather than on a fixed grid of latitude and longitude, because it more evenly samples the globe. The icosahedron tessellation parameterization used here is well established in other applications such as mantle convection simulations [Baumgardner and Frederickson, 1985], magnetic field modeling [Constable et al., 1993], and seismic modeling [Wang and Dahlen, 1995;Sambridge and Faletic, 2003;Ballard et al., 2009]. We use the same grid as in Simmons et al. [2012]. ...
Article
[1] We present the LITHO1.0 model, which is a 1° tessellated model of the crust and uppermost mantle of the earth, extending into the upper mantle to include the lithospheric lid and underlying asthenosphere. The model is parameterized laterally by tessellated nodes and vertically as a series of geophysically identified layers, such as water, ice, sediments, crystalline crust, lithospheric lid, and asthenosphere. LITHO1.0 is created by constructing an appropriate starting model and perturbing it to fit high-resolution surface wave dispersion maps (Love, Rayleigh, group, phase) over a wide frequency band (5-40 mHz). We examine and discuss the model with respect to key lithospheric parameters, such as average crustal velocity, crustal thickness, upper mantle velocity, and lithospheric thickness. We then compare the constructed model to those from a number of select studies at regional and global scales and find general consistency. It appears that LITHO1.0 represents a reasonable starting model of the earth's shallow structure (crust and uppermost mantle) for the purposes in which these models are used, such as travel time tomography or in efforts to create a 3D reference earth model. The model matches surface wave dispersion over a frequency band wider than the band used in the inversion. There are several avenues for improving the model in the future by including attenuation and anisotropy, as well as making use of surface waves at higher frequency.
... For purposes of global travel-time tomography, there exist a variety of schemes to predict the travel-times for a couple of the important global phases. They include perturbation theory (Bijwaard and Spakman, 1999), dynamic ray tracing (Dahlen et al., 2000), the pseudo-bending method (Zhao, 2001) nd generalized ray theory (Ballard et al., 2009), in which most of the global phases are not predicted and accommodated, for example, later phases with the minimax time-path (see Ray Tracing for Phases with a Stationary Minimax Time Path section for detailed explanation). ...
... In the two polar regions, a hexahedral cell ( Fig. 1d) is used to link two adjoining cells across the poles. Note that other cell types are also available for the parameterization, for example, polyhedral or tetrahedral elements (see, Ballard et al. 2009;Bai et al. 2011). To improve the ray angular coverage, secondary nodes are inserted along the surfaces of the trapezoidal prism or trapezoidal cone or hexahedral cell. ...
Article
Full-text available
The two key requirements in conducting 3-D simultaneous traveltime tomography on real data at the regional and global scale with multiple classes of arrival time information are (1) it needs an efficient and accurate arrival tracking algorithm for multiply transmitted, reflected (or refracted) and converted waves in a 3-D variable velocity model with embedded velocity discontinuities (or subsurface interfaces), and (2) a subdimensional inversion solver is required which can easily search for different types of model parameters to balance the trade-off between the different types of model parameter updated in the simultaneous inversion process. For these purposes, we first extend a popular grid/cell-based wavefront expanding ray tracing algorithm (the multistage irregular shortest-path ray tracing method), which previously worked only in Cartesian coordinate at the local scale, to spherical coordinates appropriate to the regional or global scale. We then incorporated a fashionable inversion solver (the subspace method) to formulate a simultaneous inversion algorithm, in which the multiple classes of arrivals (including direct and reflected arrivals from different velocity discontinuities) can be used to simultaneously update both the velocity fields and the reflector geometries. Numerical tests indicate that the new inversion method is both applicable and flexible in terms of computational efficiency and solution accuracy, and is not sensitive to a modest level of noise in the traveltime data. It offers several potential benefits over existing schemes for real data seismic imaging.
... In 3-D spherical coordinates we extend the above 2-D model parametrization to the 3-D case (see Fig. 3a). Note that other cells are also available, for example, polyhedron or tetrahedron (see, Ballard et al. 2009;Bai et al. 2012b). We use a trapezoidal prism (Fig. 3b) to divide the 3-D spherical earth model, except for the global and other irregular subsurface interfaces, where a trapezoidal cone or a hexahedral cell is used (Figs 3c and d). ...
Article
Full-text available
The traditional grid/cell-based wavefront expansion algorithms, such as the shortest path algorithm, can only find the first arrivals or multiply reflected (or mode converted) waves transmitted from subsurface interfaces, but cannot calculate the other later reflections/conversions having a minimax time path. In order to overcome the above limitations, we introduce the concept of a stationary minimax time path of Fermat's Principle into the multistage irregular shortest path method. Here we extend it from Cartesian coordinates for a flat earth model to global ray tracing of multiple phases in a 3-D complex spherical earth model. The ray tracing results for 49 different kinds of crustal, mantle and core phases show that the maximum absolute traveltime error is less than 0.12 s and the average absolute traveltime error is within 0.09 s when compared with the AK135 theoretical traveltime tables for a 1-D reference model. Numerical tests in terms of computational accuracy and CPU time consumption indicate that the new scheme is an accurate, efficient and a practical way to perform 3-D multiphase arrival tracking in regional or global traveltime tomography.
... The boundary-grid approach has been extended to conduct global tomography (Zhao, 2001c(Zhao, , 2004Zhao and Lei, 2004) by considering the lateral depth variations of the Moho, 410-and 670-km discontinuities. Re-cently, Ballard et al. (2009) andSimmons et al. (2011) applied the 3-D ray tracing approach of Zhao et al. (1992b) for global-scale travel-time calculations and tomographic inversions. ...
Article
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We review the significant recent results of multiscale seismic tomography of the Western-Pacific subduction zones and discuss their implications for seismotectonics, magmatism, and subduction dynamics, with an emphasis on the Japan Islands. Many important new findings are obtained due to technical advances in tomography, such as the handling of complex-shaped velocity discontinuities, the use of various later phases, the joint inversion of local and teleseismic data, tomographic imaging outside a seismic network, and P-wave anisotropy tomography. Prominent low-velocity (low-V) and high-attenuation (low-Q) zones are revealed in the crust and uppermost mantle beneath active arc and back-arc volcanoes and they extend to the deeper portion of the mantle wedge, indicating that the low-V/low-Q zones form the sources of arc magmatism and volcanism, and the arc magmatic system is related to deep processes such as convective circulation in the mantle wedge and dehydration reactions in the subducting slab. Seismic anisotropy seems to exist in all portions of the Northeast Japan subduction zone, including the upper and lower crust, the mantle wedge and the subducting Pacific slab. Multilayer anisotropies with different orientations may have caused the apparently weak shear-wave splitting observed so far, whereas recent results show a greater effect of crustal anisotropy than previously thought. Deep subduction of the Philippine Sea slab and deep dehydration of the Pacific slab are revealed beneath Southwest Japan. Significant structural heterogeneities are imaged in the source areas of large earthquakes in the crust, subducting slab and interplate megathrust zone, which may reflect fluids and/or magma originating from slab dehydration that affected the rupture nucleation of large earthquakes. These results suggest that large earthquakes do not strike anywhere, but in only anomalous areas that may be detected with geophysical methods. The occurrence of deep earthquakes under the Japan Sea and the East Asia margin may be related to a metastable olivine wedge in the subducting Pacific slab. The Pacific slab becomes stagnant in the mantle transition zone under East Asia, and a big mantle wedge (BMW) has formed above the stagnant slab. Convective circulations and fluid and magmatic processes in the BMW may have caused intraplate volcanism (e.g., Changbai and Wudalianchi), reactivation of the North China craton, large earthquakes, and other active tectonics in East Asia. Deep subduction and dehydration of continental plates (such as the Eurasian plate, Indian plate and Burma microplate) are also found, which have caused intraplate magmatism (e.g., Tengchong) and geothermal anomalies above the subducted continental plates. Under Kamchatka, the subducting Pacific slab shortens toward the north and terminates near the Aleutian-Kamchatka junction. The slab loss was induced by friction with the surrounding asthenosphere, as the Pacific plate rotated clockwise 30 Ma ago, and then it was enlarged by the slab-edge pinch-off by the asthenospheric flow. The stagnant slab finally collapses down to the bottom of the mantle, which may trigger upwelling of hot mantle materials from the lower mantle to the shallow mantle. Suggestions are also made for future directions of the seismological research of subduction zones.
... The problem of overlapping will lead to a non-unique mapping between the physical space and the grid space, and make the spatial representation confusing on the border of Yin and Yang parts, which may be harmful to the uniform spatial representation, modeling and visualization of global lithosphere. Ballard et al. [28] designed a non-overlapping global 3D grid and applied it in the 3D modeling of global seismic velocity. Its radial subdivision adopts a natural division in discontinuity, such as Moho, which may result in a non-uniform subdivision in the radial direction. ...
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The structure of global lithosphere is very important to the scientific researches of tectonic movement, geodynamic process, mantle convection, resource exploration, and disaster prevention and reduction. Three-dimensional (3D) spatial modelling and visualization is an effective tool for lithosphere researches. However, both the isoline/profile methods and the Euclidean-based 3D modelling methods cannot meet the requirement of real 3D modeling of global lithosphere, whereas the recently developed global 3D grid methods have some defects on grid design, such as grid shrinkage, overlapping, non-orthogonality, and non-latitude-longitude consistency. In this paper, Spheroid Degenerated-Octree Grid (SDOG), a non-overlapping, non-shrinking, orthogonal, latitude-longitude consistent grid in the spheroidal manifold space, was chosen as the basic grid for global lithosphere 3D modeling and visualization. The SDOG-based methods of spatial representation and modelling of lithosphere were proposed. A multi-scale model of lithosphere was designed, and the multi-scale modeling and multi-mode visualization were realized at the full advantages of SDOG in multi-hierarchical and multi-resolution and the properties of lithosphere in multi-semantic. It shows that (1) the SDOG-based method has not only overcome the defects of the current global 3D grid, but also reflected the spherical features of lithosphere more realistically and naturally than the traditional methods, providing a novel solution for global modeling, numeric simulating, and data sharing of lithosphere; and (2) more detailed plates division, more precise geo-layer structure, plates boarder and surface concave-convex, and more rich lithosphere properties are revealed as the scale-model moves on.
... Reduction in the total number of ray paths is ~50%. The model is represented using the triangular tessellation system described by Ballard et al. (2009), which incorporates variable resolution in both the geographic and radial dimensions. For our starting model, we use a simplified layer crustal model derived from the NNSA Unified model in Eurasia and Crust 2.0 model everywhere else, over a uniform ak135 mantle. ...
Conference Paper
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To test the hypothesis that high quality 3D Earth models will produce seismic event locations which are more accurate and more precise, we are developing a global 3D P wave velocity model of the Earth's crust and mantle using seismic tomography. In this paper, we present the most recent version of our model, SALSA3D version 1.5, and demonstrate its ability to reduce mislocations for a large set of realizations derived from a carefully chosen set of globally-distributed ground truth events. Our model is derived from the latest version of the Ground Truth (GT) catalog of P and Pn travel time picks assembled by Los Alamos National Laboratory. To prevent over-weighting due to ray path redundancy and to reduce the computational burden, we cluster rays to produce representative rays. Reduction in the total number of ray paths is {approx}50%. The model is represented using the triangular tessellation system described by Ballard et al. (2009), which incorporates variable resolution in both the geographic and radial dimensions. For our starting model, we use a simplified two layer crustal model derived from the Crust 2.0 model over a uniform AK135 mantle. Sufficient damping is used to reduce velocity adjustments so that ray path changes between iterations are small. We obtain proper model smoothness by using progressive grid refinement, refining the grid only around areas with significant velocity changes from the starting model. At each grid refinement level except the last one we limit the number of iterations to prevent convergence thereby preserving aspects of broad features resolved at coarser resolutions. Our approach produces a smooth, multi-resolution model with node density appropriate to both ray coverage and the velocity gradients required by the data. This scheme is computationally expensive, so we use a distributed computing framework based on the Java Parallel Processing Framework, providing us with {approx}400 processors. Resolution of our model is assessed using a variation of the standard checkerboard method. We compare the travel-time prediction and location capabilities of SALSA3D to standard 1D models via location tests on a global event set with GT of 5 km or better. These events generally possess hundreds of Pn and P picks from which we generate different realizations of station distributions, yielding a range of azimuthal coverage and ratios of teleseismic to regional arrivals, with which we test the robustness and quality of relocation. The SALSA3D model reduces mislocation over standard 1D ak135 regardless of Pn to P ratio, with the improvement being most pronounced at higher azimuthal gaps.
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The physical properties of the lithospheric and upper mantle's rock are determined by its composition and the in situ temperature and pressure conditions. Together, they have been referred to as the thermochemical structure. Information about the upper mantle's thermochemical structure could be obtained using methods from different disciplines of the earth sciences, in which the geophysical approaches show potential to map the 3D variations on both the regional and global scales. Thus, techniques for investigating the thermochemical structure in the spherical coordinates are needed, including forward modeling of the geophysical observables, calculating schemes of the thermophysical properties for the lithologies, and effective inversion algorithm, which is particularly important for large‐scale applications. This paper first demonstrates an adaptive meshing architecture based on the tetrahedral mesh by the sophisticated constructions in a spherical shell. Techniques that enable rapid calculations of the thermophysical properties of the upper mantle's rocks are introduced in length. Methodologies for constructing 3D thermochemical models and forward modeling geophysical observations, including an inversion sub‐routine that couples the lithostatic pressure and density variations to forward modeling, are introduced and examined in detail using synthetic data sets. We then introduce methods for determining 3D thermochemical structures of the upper mantle. The inverse problem is treated as a multi‐task evaluation process and solved using advanced stochastic optimizing techniques. Estimated uncertainties of the resultant thermochemical models are obtained simultaneously for error analysis. The proposed forward modeling and inversion techniques are validated using synthetic data sets in both forward and inversion circumstances. Limitations and further developments are discussed in the subsequent concluding remarks.
Thesis
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В работе представлен новый подход к расчету радиотрасс в ионосфере Земли с зафиксированными точками передатчика и приемника. Для коротких радиоволн применимо приближение геометрической оптики, основанное на вариационном принципе Ферма, согласно которому траектории радиотрасс должны удовлетворять условию стационарности фазового пути. Этот факт позволяет свести решение исходной краевой задачи к поиску стационарных точек функционала фазового пути радиолуча на основе прямой оптимизации траектории без необходимости решения вариационного уравнения и уравнения эйконала.
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It is now common practice to perform simultaneous traveltime inversion for the velocity field and the reflector geometry in reflection/refraction tomography, or the velocity field and the hypocenter locations in regional earthquake tomography, but seldom are all three classes of model parameters updated simultaneously. This is mainly due to the trade-off between the different types of model parameters and the lack of different seismic phases to constrain the model parameters. Using a spherical-coordinate ray tracing algorithm for first and later (primary reflected) arrival tracing algorithm in combination with a popular linearized inversion solver, it is possible to simultaneously recover the three classes of model parameters in regional or global tomographic studies. In this paper we incorporate the multistage irregular shortest-path ray tracing algorithm (in a spherical coordinate system) with a subspace inversion solver to formulate a simultaneous inversion algorithm for triple model parameters updating using direct and later arrival time information. Comparison tests for two sets of data (noise free and added noise) indicate that the new triple-class parameter inversion algorithm is capable of obtaining nearly the same results as the double-class parameter inversion scheme. Furthermore, the proposed multi-parameter type inversion method is not sensitive to a modest level of picking error in the traveltime data, and also performs well with a relatively large uncertainty in earthquake hypocentral locations. This shows it to be a feasible and promising approach in regional or global tomo-graphic applications.
Conference Paper
With the development of earth observation technology, especially the rapid progress of LiDAR technology in recent years, the kinds and quantity of three-dimensional spatial information is increasing. How to effectively model and visualize the 3D spatial data has become a key issue for academic and applied science community. The global discrete voxel divides the earth into a series of different scale voxels according to specific rules. However, it is usually bound to the surface, the data model below the surface belongs to 3D geological modeling while data model above is geographic 3D modeling. The earth itself is a complete whole. Splitting it to two parts or using different model frameworks will bring errors caused by the boundary. In this paper, we propose a new global discrete voxel called GeoSOT3D, which is the abbreviation of Geographical coordinates Subdivision grid with One dimension integral coding on 2n-Tree for space. GeoSOT3D framework is an illimitably divisible, multi-level, multi-scale space division pattern, which can seamlessly fit the global space. Based on the GeoSOT3D, a global multi-scale 3D spatial data representation model can be established. It is a model that is trying to represent the spatial object using subdivision unit according to accuracy of spatial data and the size of the entity. At last, in order to verify the theory presented in this issue, we construct a 3D spatial information subdivision visualization prototype system using the GeoSOT3D representation method. We then use the earth's magnetic field data to demonstrate the 3D representation model.
Chapter
In this chapter, we first introduce the basic principles of seismic tomography and discuss the common features and differences between seismic tomography and the medical CT-scan. Considering the fact that many different kinds of tomographic studies have been made and a large number of tomography-related technical terms are used in the literature, we present a classification of seismic tomography. Then we explain the meaning of multiscale seismic tomography, and discuss how to interpret the obtained tomographic images. Finally, the scope and contents of this book are outlined.
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With the development of earth observation technology, three-dimensional geo-spatial information acquisition and update capabilities have achieved rapid development. How to effectively organize and manage massive spatial data and related information has become an urgent problem to be solved. In this paper, according to the idea of global subdivision, we propose a design and implementation of the global spatial data organization. The method firstly process spatial data in multi-scale subdivision based on GeoSOT-3D, and then three-dimensional volume data and field data is organized. Compared with the existing local three-dimensional grid or global three-dimensional grid for certain purpose, our method is more suitable for worldwide large scale data organization and provide a new idea for the effective organization of the global spatial data.
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There are hundreds of spatial reference frame (SRF) being applied, and the great difference among SRFs has blocked the share of global data on planet Earth. A conceptual spheroid of radius 12,800 km and a spheroid degenerated octree grid method are applied to produce an earth system spatial grid (ESSG), which is of natural characteristics to be applied as a new common SRF. A triple CTA is designed as ESSG-based data structure to organize the big data of planet Earth, and a 2D table of a unique label and limitless records for time slices and attribute values is present to record the data of each grid. The big data on planet Earth can hence be gridded and interrelated without discipline gaps and SRF obstacles. An integral data organization mode is designed, and three potential routes are presented for users to access shareable global data in cloud environment. Furthermore, with global crust, atmosphere, DEM, and satellite image being examples, the integrated visualization of global large objects is demonstrated.
Article
The SDOG-based Earth System Spatial Grid (SDOG-ESSG) is an important tool and method in the Earth System Science and spatial information domains. This paper focuses on the key problem of transformation between the grid code of SDOG-ESSG and the existing spatial reference, the transformation between Spheroid Coordinates System (SCS) and the grid code of SDOG-ESSG. Six column-row-layer number systems were brought in and related formulas were derived. Based on this and the bit-interleaving method of Morton code, forward and backward transformation algorithms were developed. The time efficiency of both algorithms was analyzed theoretically and experimentally. The results show that: a) both algorithms are very high efficient, and the time consumption is linear to the principle subdivision level and the further subdivision level of SDOG-ESSG, where the time complexity is O(n); and (b) approximately 106~107 times of transformation operations can be done in one second under current personal computer. Each transformation operation is identical to 101~102 division operation in time.
Article
Given a set of observations within a specified time window, a fitness value is calculated at each grid node by summing station-specific conditional fitness values. Assuming each observation was generated by a refracted P wave, these values are proportional to the conditional probabilities that each observation was generated by a seismic event at the grid node. The node with highest fitness value is accepted as a hypothetical event location, subject to some minimal fitness value, and all arrivals within a longer time window consistent with that event are associated with it. During the association step, a variety of different phases are considered. Once associated with an event, an arrival is removed from further consideration. While unassociated arrivals remain, the search for other events is repeated until none are identified. Results are presented in comparison with analyst-reviewed bulletins for three datasets: a two-week ground-truth period, the Tohoku aftershock sequence, and the entire year of 2010. The probabilistic event detection, association, and location algorithm missed fewer events and generated fewer false events on all datasets compared to the associator used at the International Data Center (51% fewer missed and 52% fewer false events on the ground-truth dataset when using the same predictions).
Article
Discrete global grids, a modeling framework for big geo-spatial data, is always used to build the Digital Earth platform. Based on the sphere split bricks (Earth system spatial grids), it can not only build the true three-dimensional digital Earth model, but also can achieve integration, fusion, expression and application of the spatial data which locates on, under or above the Earth subsurface. The theoretical system of spheroid geodesic QTM octree grid is discussed, including the partition principle, analysis of grid geometry features and coding/ decoding method etc, and a prototype system of true-3D digital Earth platform with the sphere split bricks is developed. The functions of the system mainly include the arbitrary sphere segmentation and the visualization of physical models of underground, surface and aerial entities. Results show that the sphere geodesic QTM octree grid has many application advantages, such as simple subdivision rules, the grid system neat, clear geometric features, strong applicability etc. In particular, it can be extended to the ellipsoid, so it can be used for organization, management, integration and application of the global spatial big data.
Article
This paper is to deduce and conclude the methods of neighbors finding for the SDOG-based Earth System Spatial Grid (SDOG-ESSG) based on the finding of the geometrical rules of it. Firstly, the concepts of bed-region and latitude-region were brought to get an analysis on the qualitative and quantitative rules of geometric distribution, and the calculation methods between the amount of bed-regions and latitude-regions of SDOG-ESSG, and the times of equivalent bisection were deduced; Then, based on the analyses of several possible cases of face-neighbors and a temporal coordinate system named bed-column-row coordinate system, a method of face-neighbor finding for any given grid code was proposed by transforming the codes into bed-column-row coordinates, finding the neighbors in the coordinates system, and then transforming the coordinates back into the code system. ©, 2015, China University of Mining and Technology. All right reserved.
Book
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This book on multiscale seismic tomography, written by one of the leaders in the field, is suitable for undergraduate and graduate students, researchers, and professionals in Earth and planetary sciences who need to broaden their horizons about seismotectonics, volcanism, and interior structure and dynamics of the Earth and Moon. It describes the state-of-the-art in seismic tomography, with emphasis on the new findings obtained by applying tomographic methods in local, regional, and global scales for understanding the generating mechanism of large and great earthquakes such as the 2011 Tohoku-oki earthquake (Mw 9.0), crustal and upper mantle structure, origin of active arc volcanoes and intraplate volcanoes including hotspots, heterogeneous structure of subduction zones, fate of subducting slabs, origin of mantle plumes, mantle convection, and deep Earth dynamics. The first lunar tomography and its implications for the mechanism of deep moonquakes and lunar evolution are also introduced.
Article
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A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.
Article
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The current state of seismic tomography using irregular parameterizations is described. A review is given of previous work in the area covering local, regional and global seismic applications. The potential advantage of an irregular parameterization over a regular (uniform) one is that, by introducing higher resolution grids in particular parts of the model, one can maximize the information extracted from the data. Although irregular parameterizations have only recently been used in 3-D mantle tomography, their origins can be traced back more than twenty years. The use of irregular meshes in two and three dimensional tomographic imaging creates a number of `implementation' issues, not seen with uniform grids. An outline of algorithms for solving all of these bookkeeping problems arising in cubic, tetrahedral or polygonal meshes is included, and references given to more detailed descriptions. Some recent implementations of tomography are discussed where the parameterization is refined during the course of the inversion process. Two examples of adaptive tomography in non-seismic problems suggest that the gradual refinement of the parameterization may be an effective way to regularize the inverse problem. The intention here is to point out the potential advantages and pitfalls of using an irregular parameterization in tomography. At present experience with seismic problems is rather limited (especially in 3-D) and an optimal approach is yet to emerge.
Article
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Traditional grid‐based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid‐based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite‐difference eikonal solver known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single‐stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.
Article
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Increasingly, nuclear explosion monitoring is focusing on detection, location, and identification of small events recorded at regional distances. Because Earth structure is highly variable on regional scales, locating events accurately at these distances requires the use of region-specific models to provide accurate travel times. Improved results have been achieved with composites of 1D models and with approximate 3D models with simplified upper mantle structures, but both approaches introduce nonphysical boundaries that are problematic for operational monitoring use. Ultimately, what is needed is a true, seamless 3D model of the Earth. Towards that goal, we have developed a 3D tomographic model of the P velocity of the crust and upper mantle for the region of southcentral Asia centered around the Tibetan Plateau. Our model is derived from almost 140,000 Pn picks for more than 5300 events recorded at 563 stations from a Ground Truth (GT) dataset assembled by Los Alamos National Laboratory (LANL). Our starting model is the a priori model of East Asia developed by LANL, which is based on various global and regional studies. The topmost layers come from the Laske and Masters global sedimentary model from 1997. As our dataset lacks the resolution to improve this sedimentary portion of our model, we fix the velocity and depth of these layers, as well as the depths of the major mantle discontinuities (Moho, 410 km, 660 km). We invert for P velocities from the crust down through the upper mantle, along with source and receiver terms to account for the effects of event mislocation and fine-scale structure near the receiver not accounted for in the crustal model. Forward calculations are made using our own implementation of the Um and Thurber ray pseudo-bending approach (Ballard, 2008, these Proceedings) with full enforcement of Snell's Law in 3D at the major discontinuities.
Article
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We demonstrate our ability to improve regional travel-time prediction and seismic event location accuracy using ana priori, three-dimensional velocity model of Western Eurasia and North Africa (WENA1.0). Travel- time residuals are assessed relative to the iasp91 model for approximately 6,000 Pg, Pn, and P arrivals, from seismic events having 2sigma epicenter accuracy between 1 km and 25 km (GT1 and GT25, respectively), recorded at 39 stations throughout the model region. Ray paths range in length between 0 and 40 degrees epicentral distance (local, regional, and near teleseismic) providing depth sounding that spans the crust and upper mantle. The dataset also provides representative geographic sampling across Eurasia and North Africa including aseismic areas. The WENA1.0 model markedly improves travel-time predictions for most stations with an average variance reduction of 29% for all ray paths from the GT25 events; when we consider GT5 and better events alone the variance reduction is 49%. For location tests we use 196 geographically distributed GT5 and better events. In 134 cases (68% of the events), locations are improved, and average mislocation is reduced from 24.9 km to 17.7 km. We develop a travel time uncertainty model that is used to calculate location coverage ellipses. The coverage ellipses for WENA1.0 are validated to be representative of epicenter error and are smaller than those for iasp91 by 37%. We conclude that a priori, models are directly applicable where data coverage limits tomographic and empirical approaches, and the development of the uncertainty model enables merging of a priori, and data-driven approaches using Bayesian techniques. This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, Contribution UCRL- JRNL-220179.
Article
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The fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.
Article
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The traveltimes of first arriving seismic rays through most velocity structures can be computed rapidly on a three-dimensional numerical grid by finite-difference extrapolation. Head waves are properly treated and shadow zones are filled by the appropriate diffractions. Differences of less than 0.11 percent are found between the results of this technique and ray tracing for a complex but smooth model. This scheme has proven useful for earthquake location and shows promise as an inexpensive, well-behaved substitute for ray tracing in forward-modeling and Kirchhoff inversion applications.
Article
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We construct and evaluate a new three-dimensional model of crust and upper mantle structure in Western Eurasia and North Africa (WENA) extending to 700 km depth and having 1 deg parameterization. The model is compiled in an a priori fashion entirely from existing geophysical literature, specifically, combining two regionalized crustal models with a high-resolution global sediment model and a global upper mantle model. The resulting WENA1.0 model consists of 24 layers: water, three sediment layers, upper, middle, and lower crust, uppermost mantle, and 16 additional upper mantle layers. Each of the layers is specified by its depth, compressional and shear velocity, density, and attenuation (quality factors, QP and QS ). The model is tested by comparing the model predictions with geophysical observations including: crustal thickness, surface wave group and phase velocities, upper mantle n velocities, receiver functions, P-wave travel times, waveform characteristics, regional 1-D velocities, and Bouguer gr
Chapter
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We present three case studies of traveltime tomography applied to regional imaging of the earth's crust and upper mantle. The first case study investigates the imaging of Poisson's ratio using measurements of compressional and shear wave traveltimes. Inversions schemes that jointly fit both types of traveltime data and that selectively damp the compressional to shear wave velocity ratio are shown to be very effective. The second case study focuses on the ability of joint inversions of crustal and Moho- reflected wave traveltimes to image both crustal velocity structure and Moho topography. The ability of this kind of tomography to distinguish Moho topography and lower crustal heterogeneity is shown to be poor. The third case study examines the ability of teleseismic tomography to determine the shape of a mantle plume. A new test for departures from an axial shape is proposed and tested. We also present a case study of earthquake location, that compares locations made with traditional P and S wave traveltimes to those based on differential P wave traveltimes (i.e., the double- difference method). The underlying tomography software, "raytrace3d" that is used in this study and which is publicly and freely available is described, and the technical elements of traveltime tomography are fully documented in an appendix.
Article
This chapter discusses algorithm and underlying theory of software for C¹ surface interpolation. There has been practically no theory to guide the development of algorithms for triangulation and there has been no practical static global criterion to characterize a preferred triangulation. An interpolation algorithm is interfaced with algorithms for contour plotting or surface perspective plotting. If the subroutines are available for doing contour or surface perspective plotting for data given on a rectangular grid, then the surface interpolation algorithm can be used to produce the values needed at the lattice points of a rectangular grid. The computerized representation of surfaces is a central issue in the field of computer-aided geometric design and plays an important role in the field of finite element methods. Some methods of building a triangular grid with a given set of nodes start by locating the boundary points of the convex hull of the point set. In some reasonable triangulation algorithms, the number of boundary points of the convex hulls of a sequence of sub­sets has an effect on the operation count.
Article
An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finite-element schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.
Article
A class of conservative finite-difference approximations of the primitive equations is given for quasi-uniform spherical grids derived from regular polyhedrons. The earth is split into several contiguous regions. Within each region, a coordinate system derived from central projections is used, instead of the spherical coordinate system, to avoid the use of inconsistent boundary conditions at the poles. The presence of artificial internal boundaries has no effect on the conservation properties of the approximations. Examples of conservative schemes, up to the second order in the case of a cube, are given. A selective damping operator is needed to remove the two-grid interval waves generated by the existence of internal boundaries.
Article
We have developed a three-dimensional (3D) a priori geophysical model for the China/East Asia (CEA) region and are using it to construct 3D travel-time tables for seismic event relocation. Validation of this D model involves comparison of travel times to other 3D models, both a priori and tomograhic, as well as to observed arrival times from ground truth events. The CEA a priori model is constructed by patching one-dimensional velocity models from previous studies together and adjusting them to conform to known basin and crust thickness values. This type of model aids in the view of geophysical properties over a larger region than typical tomographic models can provide. Using several ray-tracing algorithms, we will produce fine-scale travel-time tables for various stations within the CEA model and compare the travel times to both tomographic models and other a priori models. The 3D travel-time tables will be used to relocate ground truth events for models and determine how these tables affect the regional locations. 3D models do display significant variation in velocities when compared to global one-dimensional models, especially for crust and upper-mantle structure. Our goal is to utilize these 3D variations and apply the 3D travel times on large portions of a seismic catalog to determine their effect. The use of 3D travel-time tables is still being investigated as a viable, operational alternative to one-dimensional models with corrections. As model resolution and accuracy improve, their use can continue to benefit regional event relocation results, but the trade-off with computational load and efficiency must be weighed for the more detailed parameterizations.
Book
This is the first textbook to cover the essential aspects of the topic at a level accessible to students. While focusing on applications in solid earth geophysics, the book also uniquely includes excursions into helioseismology, thereby highlighting the strong affinity between the two fields. The book provides a comprehensive introduction to seismic tomography, including the basic theory of wave propagation, the ray and Born approximations required for interpretation of amplitudes, and travel times and phases. It considers observational features while also providing practical recommendations for implementing numerical models. Written by one of the leaders in the field, and containing numerous student exercises, this textbook is appropriate for advanced undergraduate and graduate courses. It is also an invaluable guide for seismology research practitioners in geophysics and astronomy. Solutions to the exercises and accompanying tomographic software and documentation can be accessed online from www.cambridge.org/9780521882446.
Article
A three-dimensional (3D) ray tracing technique is used to investigate ray path variations of P, PcP, pP and PP phases in a global tomographic model with P wave velocity changing in three dimensions and with lateral depth variations of the Moho, 410 and 660 km discontinuities. The results show that ray paths in the 3D velocity model deviate considerably from those in the average 1D model. For a PcP wave in Western Pacific to East Asia where the high-velocity (1–2%) Pacific slab is subducting beneath the Eurasian continent, the ray path change amounts to 27 km. For a PcP ray in South Pacific where very slow (−2%) velocity anomalies (the Pacific superplume) exist in the whole mantle, the maximum ray path deviation amounts to 77 km. Ray paths of other phases (P, pP, PP) are also displaced by tens of kilometers. Changes in travel time are as large as 3.9 s. These results suggest that although the maximal velocity anomalies of the global tomographic model are only 1–2%, rays passing through regions with strong lateral heterogeneity (in velocity and/or discontinuity topography) can have significant deviations from those in a 1D model because rays have very long trajectories in the global case. If the blocks or grid nodes adopted for inversion are relatively large (3–5 •) and only a low-resolution 3D model is estimated, 1D ray tracing may be feasible. But if fine blocks or grid nodes are used to determine a high-resolution model, 3D ray tracing becomes necessary and important for the global tomography.
Article
We present a ‘pseudo-bending’ approach to 3-D ray tracing in a spherical earth with discontinuities. This method is based on a three-point perturbation associated with a first-order approximation, while Snell’s law in curvilinear coordinates is applied at the discontinuities. We demonstrate the computational accuracy and efficiency of the pseudo-bending method in tracing rays for various velocity models by comparing results with analytical solutions and with results from the bending method. The improvement of efficiency is significant, but is reduced as the number of discontinuities increases. Since the bending approach may be computationally unstable in some situations, even though it is exact, the pseudo-bending approach is preferable for automatic calculation of rays.
Article
We present an a priori three-dimensional ‘tomographic’ model of the upper mantle. We construct this model (called 3SMAC — three-dimensional seismological model a priori constrained) in four steps: we compile information on the thickness of ‘chemical’ layers in the Earth (water, sediments, upper and lower crust, etc); we get a 3D temperature distribution from thermal plate models applied to the oceans and continents; we deduce the mineralogy in the mantle from pressure and temperature and we finally get a three-dimensional model of density, seismic velocities, and attenuation by introducing laboratory measurements of these quantities as a function of pressure and temperature. The model is thus consistent with various geophysical data, such as ocean bathymetry, and surface heat flux. We use this model to compute synthetic travel-times of body waves, and we compare them with observations. A similar exercise is performed for surface waves and normal modes in a companion paper (Ricard et al., 1996, J. Geophys. Res., in press). We find that our model predicts the bulk of the observed travel-time variations. Both the amplitude and general pattern are well recovered. The discrepancies suggest that tomography can provide useful regional information on the thermal state of the continents. In the oceans, the flattening of the sea-floor beond 70 Ma seems difficult to reconcile with the seismic observations. Overall, our 3SMAC model is both a realistic model, which can be used to test various tomographic methods, and a model of the minimum heterogeneities to be expected from geodynamical modeling. Therefore, it should be a useful a priori model to be used in tomographic inversions, in order to retrieve reliable images of heterogeneities in the transition zone, which should, in turn, greatly improve our understanding of geodynamical processes in the deep Earth. 3SMAC and accompanying software can be retrieved by anonymous ftp at geoscope.ipgp.jussieu.fr.
Article
We present a new massively parallel method for computation of first arrival times in arbitrary velocity models. An implementation on conventional sequential computers is also proposed. This method relies on a systematic application of Huygens’ principle in the finite difference approximation. Such an approach explicitly takes into account the existence of different propagation modes (transmitted and diffracted body waves, head waves). Local discontinuities of the time gradient in the first arrival time field (e.g., caustics) are built as intersections of locally independent wavefronts. As a consequence, the proposed method provides accurate first traveltimes in the presence of extremely severe, arbitrarily shaped velocity contrasts. Associated with a simple procedure which accurately traces rays in the obtained time field, this method provides a very fast tool for a large spectrum of seismic and seismological problems. We show moreover that this method may also be used to obtain several arrivals at a given receiver, when the model contains reflectors. This possibility significantly extends the domain of potential geophysical applications.
Article
A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n + 1) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.
Seismic tomography with irregular meshes Geophysical Monograph Series 157 Research on tri-angle subdivision and cell search based on equilateral octahe-dron
  • M Sambridge
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Sambridge, M., and N. Rawlinson (2005). Seismic tomography with irregular meshes. In Seismic Earth: Array Analysis of Broadband Seismograms, ed. A. Levander and G. Nolet, 45–65. Geophysical Monograph Series 157. Washington, DC: American Geophysical Union. Shengmao, Z., W. Jianping, and G. Jiayuan (2008). Research on tri-angle subdivision and cell search based on equilateral octahe-dron. Geoinformatics 2008 and Joint Conference on GIS and Built Environment: Advanced Spatial Data Models and Analyses, ed. L.
A brief description of the natural neighbor interpola-tion A fast algorithm for two-point seis-mic ray tracing
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Sibson, R. (1981). A brief description of the natural neighbor interpola-tion. In Interpreting Multivariate Data, ed. V. Barnett, 21–36. New York: John Wiley & Sons. Um, J., and C. H. Thurber (1987). A fast algorithm for two-point seis-mic ray tracing. Bulletin of the Seismological Society of America 77, 972–986.
Tomographic imaging of P and S wave velocity structure beneath northwestern Japan Seismic ray path variations in a 3D global velocity model
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Zhao, D., A. Hasegawa, and S. Horiuchi (1992). Tomographic imaging of P and S wave velocity structure beneath northwestern Japan. Journal of Geophysical Research 97 (B13), 19,909–19,928. Zhao, D., and J. Lei (2004). Seismic ray path variations in a 3D global velocity model. Physics of the Earth and Planetary Interiors 141, 153–166.