Vertical Symmetry Axis Research Articles

Exact solutions of the theory of elasticity for a clamped rectangle

We present the formulas that describe the exact solutions of the boundary value problems in the theory of elasticity for a half-strip and a rectangle in which the horizontal sides are firmly clamped, while normal and tangential stresses are specified on the vertical ones. We consider only an even-symmetric deformation of the half-strip and the rectangle relative to the horizontal axis of symmetry as well as even-symmetric and odd-symmetric deformations relative to the vertical axis of symmetry for the rectangle. This paper is based on the previously obtained solutions for a free half-strip and a free rectangle.

Hydrodynamics of Moonpool-Type Floaters: A Theoretical and a CFD Formulation

Moonpool-type floaters were initially proposed for applications such as artificial islands or as protecting barriers around a small area enabling work at the inner surface to be carried out in relatively calm water. In recent years, a growing interest on such structures has been noted, especially in relation to their use as heaving wave energy converters or as oscillating water column (OWC) devices for the extraction of energy from waves. Furthermore, in the offshore marine industry, several types of vessels are frequently constructed with moonpools. The present paper deals with the hydrodynamics of bottomless cylindrical bodies having vertical symmetry axis and floating in a water of finite depth. Two computation methods were implemented and compared: a theoretical approach solving analytically the corresponding diffraction problem around the moonpool floater and a computational fluid dynamics (CFD) solver, which considers the viscous effects near the sharp edges of the body (vortex shedding) as non-negligible. Two different moonpool-type configurations were examined, and some interesting phenomena were discussed concerning the viscous effects and irregularities caused by the resonance of the confined fluid.

Bayesian Deterministic Inversion Based on the Exact Reflection Coefficients Equations of Transversely Isotropic Media With a Vertical Symmetry Axis

Unconventional reservoirs usually have strong anisotropy. Generally, they can be regarded as transversely isotropic media with a vertical symmetry axis [(VTI) media] in the absence of fractures. Therefore, it is of great significance to develop high-accuracy inversion method for VTI media. The three elastic parameters and two Thomsen anisotropy parameters of VTI media are usually obtained by indirect calculation or inversion methods based on the approximate formulas. However, the cumulative errors caused by indirect calculation and low calculation accuracy of the approximate formulas limit the estimation accuracy of these parameters. In this article, we propose a new method based on the exact reflection coefficients equations (ERCEs) of VTI media to improve the inversion accuracy of these elastic and anisotropy parameters. The new method adopts the Bayesian deterministic inversion (BDI) strategy to solve the inversion problem, which becomes highly nonlinear when using the ERCEs. We analyze the feasibility of the new method using the residual function maps (RFMs) and eigenvalue analysis of Hessian matrix. The analysis results show that the BDI strategy can well solve a series of problems brought by the ERCEs for nonlinear inversion, such as strong nonlinearity and more target parameters need to invert. Both synthetic and field data examples show that the proposed method can accurately estimate the elastic and anisotropy parameters, which verifies the feasibility and effectiveness of the method.

A Hierarchical Prestack Seismic Inversion Scheme for VTI Media Based on the Exact Reflection Coefficient

Seismic exploration of unconventional hydrocarbon reservoirs (e.g., shale rocks) must take into account the transverse isotropy with a vertical axis of symmetry (VTI) characteristics. Prestack inversion for VTI media is more complex than the isotropic case, since the forward engine is highly nonlinear and more unknowns (five instead of three) are involved, which aggravate the ill-posedness of the inverse problem. Here, we propose a hierarchical inversion scheme to improve the estimation of the five parameters, where the exact reflection coefficient is used as forward engine (instead of the commonly-used approximations which lack accuracy). To handle the highly nonlinear inverse problem, we perform the prestack anisotropic inversion in two steps, namely, a preliminary linear result is used to reduce the search window and to formulate the constrain term and initial models of the subsequent nonlinear step. Specifically, for a reasonable preliminary estimation, we introduce a data-driven model building algorithm to provide reliable initial models, and employ the limited-memory Broyden–Fletcher–Goldfarb–Shanno combined with the momentum technique (LBFGS-MT) optimization to increase the convergence speed. We derive the Fréchet derivatives of the exact forward operator with respect to the parameters, i.e., the key factors of the linear part. Besides, we introduce a hybrid global optimization, the particle swarm optimization aided by very fast simulated annealing (PSO-VFSA) to enhance the accuracy and computational efficiency of the nonlinear stage. Synthetic tests demonstrate the effectiveness and accuracy of the proposed scheme. The field application shows that the method is capable to obtain reliable elastic information of shale reservoirs.

Characterization of two types of shale rocks from Guizhou China through micro-indentation, statistical and machine-learning tools

Micro-indentation tests were performed adopting the grid-type of analysis on two types of shale rocks, which are porous natural materials characterized by strong geological structural and microfabric features. Emphasis was placed on understanding the influence and linkage between these microstructural-microfabric characteristics and the mechanical properties obtained from indentation, including hardness, modulus, elastic and plastic deformation fractions, and their anisotropic and heterogeneous distribution. The indentation loading-unloading curve patterns also functioned as a way to explore this linkage besides the global variation in the mechanical properties. Statistical tools and machine learning techniques were further employed in the indentation data analysis, providing a means of quantifying the intricacies involved with the mechanical properties of the shales. The creep behavior of these porous materials was quantified and linked to the anisotropic structure and mechanical properties, providing insights into one of the most complex natural materials, as many researchers have characterized. Opposite to previously published data on transversely isotropic shale rocks with vertical axis of symmetry, especially shales with high clay and organic content, the results in the present study showed higher modulus normal to the bedding planes compared to the parallel direction, which observation was attributed to the low clay (and organic) content of the shales and the possible low depth the shales were obtained. • Hardness and modulus of shale rocks were examined based on microindentation tests. • The results were interpreted and linked to the microstructure and composition of the shales. • Statistical analysis was applied to the data along with machine learning tools. • Anisotropy and the roles of creep in different directions and fissility were highlighted. • Contour maps with spatial variation of properties were developed.

Multi-Parameter True-Amplitude Generalized Radon Transform Inversion for Acoustic Transversely Isotropic Media With a Vertical Symmetry Axis

In anisotropic media, the compressional wave scattering phenomena is governed by several parameters and can be simulated using a good kinematic approximation. How to retrieve anisotropic properties from recorded seismic data free of shear waves for exploration geophysics is interesting and challenging. We present an approach to infer the material properties in acoustic transversely isotropic (TI) media with a vertical axis of symmetry (VTI) described by a combination of the normal-moveout velocity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{n}$ </tex-math></inline-formula> and anisotropic parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> . The method we consider is based on the true-amplitude generalized Radon transform (GRT) inversion strategy. Because the scattering integral is at the kernel of the inversion engine, we start our investigation from a fourth-order pseudo-acoustic VTI equation instead of the conventionally coupled system of second-order wave equations. With the single-scattering approximation and high-frequency asymptotic analysis, the integral representation of P-wave scattered wavefield is incorporated into a weighted GRT operator that contains VTI scattering patterns of each parameter perturbation, which leads the way in constructing an acoustic VTI amplitude-preserving GRT pseudo-inverse operator. We describe an appropriate survey design by shooting a fan of rays from the target area toward the acquisition system, which is necessary when calculating the pseudo-inverse operator. Numerical test results from 2-D synthetic data verify the effectiveness of our method.

Born Scattering Integral, Scattering Radiation Pattern, and Generalized Radon Transform Inversion in Acoustic Tilted Transversely Isotropic Media

Although the pseudo-acoustic wave equation has good accuracy in characterizing anisotropic wave propagation and obtaining interpretable seismic images, a high-precision multiparameter inversion accounting for anisotropy from compressional wavefields is still confronted with challenges, even in the simple transversely isotropic (TI) case, due to the complicated relationship between anisotropic properties and pressure data. To reduce difficulty in correctly inverting surface compressional data in anisotropic media, an appropriate parameterization for inversion is necessary. For acoustic TI media with a tilted symmetry axis (TTI), we describe the pseudo-acoustic TTI equations with the P-wave normal moveout velocity vn and anisotropic parameters η and δ, and aim to invert for this parameterization by the scattering integral method. Using perturbation theory in formulating the integral solution of the singly scattered pressure wavefield allows to acquire a scattering radiation pattern that explicitly illustrates the angular effect (including migration dip and scattering angles) of the TTI perturbation parameters, in which perturbations whether in the wavefield or anisotropic parameters are from the elliptical anisotropic background medium. Taking advantage of a ray-theoretical approximation to the background Green’s function, we can establish a relationship between the scattering integral and a form of TTI generalized Radon transform (GRT). As a result, we develop an acoustic TTI pseudoinverse GRT operator for estimating the corresponding perturbation parameters. Numerical tests on two simple models and a part of BP 2007 anisotropic benchmark model verify the effectiveness of the presented acoustic TTI GRT inverison/migration method and show its evident advantages over the conventional acoustic isotropic and TI with a vertical symmetry axis (VTI) approaches.

Elastic properties of fractures in transversely isotropic media

A fracture may be characterized by its normal and shear compliances that relate the discontinuity in displacement of the fracture faces to an applied traction. To assess the feasibility of using elastic waves for monitoring natural and hydraulic fractures in anisotropic shale reservoirs, an understanding of how these compliances relate to the properties of the background rock and any fracture infill material is required. Three models of vertical fractures in a VTI (transversely isotropic medium with vertical axis of rotational symmetry) background suitable for feasibility studies are discussed. The first two models are suitable for open vertical fractures and represent the fractures as aligned low aspect ratio elliptic cylindrical inclusions with long axis of the ellipse either horizontal or vertical. In these models the horizontal and vertical shear compliance differ in a way that depends on the anisotropy of the background medium and the orientation of the fractures. The third model is suitable for estimating the properties of hydraulic fractures containing proppant. In this model, proppant is represented as a random packing of identical elastic spheres subject to a uniaxial compression due to the normal stress acting on the fracture. The results are expected to help improve the reliability of fracture models used for assessing the feasibility of elastic waves for characterizing natural and hydraulic fractures.

Source-independent time-lapse full-waveform inversion for anisotropic media

Time-lapse full-waveform inversion can provide high-resolution information about changes in the reservoir properties during hydrocarbon production and CO2 injection. However, the accuracy of the estimated source wavelet, which is critically important for time-lapse FWI, is often insufficient for field-data applications. The so-called “source-independent” FWI is designed to reduce the influence of the source wavelet on the inversion results. We incorporate the convolution-based source-independent technique into a time-lapse FWI algorithm for VTI (transversely isotropic with a vertical symmetry axis) media. The gradient of the modified FWI objective function is obtained from the adjoint-state method. The algorithm is tested on a model with a graben structure and the modified VTI Marmousi model using three time-lapse strategies (the parallel-difference, sequential-difference, and double-difference methods). The results confirm the ability of the developed methodology to reconstruct the localized time-lapse parameter variations even for a strongly distorted source wavelet. The algorithm remains robust in the presence of moderate noise in the input data but the accuracy of the estimated time-lapse changes depends on the model complexity.

Linearized Frequency-Dependent Reflection Coefficient and Attenuated Anisotropic Characteristics of Q-VTI Model

Seismic wave exhibits the characteristics of anisotropy and attenuation while propagating through the fluid-bearing fractured or layered reservoirs, such as fractured carbonate and shale bearing oil or gas. We derive a linearized reflection coefficient that simultaneously considers the effects of anisotropy and attenuation caused by fractures and fluids. Focusing on the low attenuated transversely isotropic medium with a vertical symmetry axis (Q-VTI) medium, we first express the complex stiffness tensors based on the perturbation theory and the linear constant Q model at an arbitrary reference frequency, and then we derive the linearized approximate reflection coefficient of P to P wave. It decouples the P- and S-wave inverse quality factors, and Thomsen-style attenuation-anisotropic parameters from complex P- and S-wave velocity and complex Thomsen anisotropic parameters. By evaluating the reflection coefficients around the solution point of the interface of two models, we analyze the characteristics of reflection coefficient vary with the incident angle and frequency and the effects of different Thomsen anisotropic parameters and attenuation factors. Moreover, we realize the simultaneous inversion of all parameters in the equation using an actual well log as a model. We conclude that the derived reflection coefficient may provide a theoretical tool for the seismic wave forward modeling, and again it can be implemented to predict the reservoir properties of fractures and fluids based on diverse inversion methods of seismic data.

Orthotropic anisotropy analysis and parameter estimation from 3-D vertical seismic profile data

SUMMARY Orthotropic velocity models are commonly considered as to have the lowest complexity (or highest symmetry) that can realistically describe the combined effect of Earth’s velocity heterogeneity in the vertical and in the horizontal (radial) directions. Velocity heterogeneity in the vertical direction is commonly caused by horizontal stratification of subsurface rock layers with different elastic properties or caused intrinsically by microlayered rocks such as shales. In the horizontal direction, azimuthal variation of the Earth’s stress field can initially lead to preferential deformation of pore space in the rocks and eventual development of aligned (vertical) fractures that are commonly regarded as the main cause for seismic azimuthal anisotropy. Building an appropriate orthotropic velocity model that can explain seismic velocity variations in vertical planes and along any azimuthal direction can therefore enhance the quality of seismic data processing and imaging. This, however, requires knowledge of several anisotropy parameters defined in the three symmetry planes of the orthotropic model and also the orientation of these planes. A combination of P- and S-wave slowness and polarization data derived from vertical seismic profile (VSP) measurements has been routinely used to estimate parameters of less complex anisotropic media such as transverse isotropy with vertical axes of symmetry (VTI). The accuracy of anisotropy parameter estimation by these methods depends primarily on the availability of data that are used in the inversion and the size of anisotropy parameters. In this study, we develop a benchmark for the accuracy of orthotropic parameters that are estimated from 3-D VSP measurements using P-wave slowness only method. Through numerical analysis, we model a wide range of fracture-induced orthotropic symmetries by adding vertical fractures to a background VTI media and show that the accuracy of our orthotropic parameter estimation method is dependent on anisotropy in the host media, intensity of fracturing, noise level in slowness data and slowness data coverage in both the dip and azimuthal planes. We then expand our accuracy analysis work to multilayer media by examining the accuracy of P-wave slowness method for orthotropic anisotropy parameter estimation by inverting finite-difference data generated in a pseudo 3-D VSP experiment. We show that, for a typical 3-D VSP experiment, reasonable estimates of orthotropic parameters in the model can only be obtained with slowness data taken from Xmax/Z ≥ 1 and line azimuth interval ϕ ≤ 10° where Xmax is the maximum source to receiver offset and Z is the receiver depth. We finally describe how field VSP measurements can be inverted for orthotropic anisotropy parameters.

Elastic anisotropy of the Marcellus Shale

ABSTRACTThe Marcellus Shale is one of the largest shale gas resources in the world and is anisotropic due to fine layering and the partial alignment of anisotropic clay minerals and organic matter with the bedding. This anisotropy can be approximated as Vertically Transversely Isotropic (transversely isotropic with a vertical axis of rotational symmetry) with five independent density‐normalized elastic stiffnesses A11, A13, A33, A55 and A66 in the two‐index notation and with axis of rotational symmetry along x3. Compressional and dipole shear‐wave data acquired in a horizontal well allows estimation of A11, A55 and A66, while the same data in a vertical pilot well allows estimation of A33 and A55. The ratio of vertically propagating P‐ to S‐velocity is and has a quadratic dependence on in the Upper Marcellus with minimum occurring for the largest volume fraction of kerogen. This relation allows estimation of A33 along a lateral well using measured values of A55 in the well. Comparison of Thomsen's anisotropy parameter ε is found to be mostly greater than anisotropy parameter γ. Estimating A13 as the average of A33 – 2A55 and A11 – 2A66 as proposed recently by Yan and Vernik allows Thomsen's anisotropy parameter δ and a parameter K0 that relates the horizontal and vertical effective stresses to be estimated. The results are expected to help in estimating horizontal stress needed for design of hydraulic fractures, and in interpreting seismic amplitude variation with offset required for reservoir characterization.

The Characteristics of Seismic Rotations in VTI Medium

Under the assumptions of linear elasticity and small deformation in traditional elastodynamics, the anisotropy of the medium has a significant effect on rotations observed during earthquakes. Based on the basic theory of the first-order velocity-stress elastic wave equation, this paper simulates the seismic wave propagation of the translational and rotational motions in two-dimensional isotropic and VTI (transverse isotropic media with a vertical axis of symmetry) media under different source mechanisms with the staggered-grid finite-difference method with respect to nine different seismological models. Through comparing the similarities and differences between the translational and rotational components of the wave fields, this paper focuses on the influence of anisotropic parameters on the amplitude and phase characteristics of the rotations. We verify that the energy of S waves in the rotational components is significantly stronger than that of P waves, and the response of rotations to the anisotropic parameters is more sensitive. There is more abundant information in the high-frequency band of the rotational components. With the increase of Thomsen anisotropic parameters ε and δ, the energy of the rotations increases gradually, which means that the rotational component observation may be helpful to the study of anisotropic parameters.

Approximations to a generalized real ray-tracing method for heterogeneous anisotropic viscoelastic media

The real ray-tracing approach leads to an effective solution in the real space domain using a homogeneous ray velocity vector. However, it fails to yield solutions for quasi-S-waves, which suffer from triplications of the wavefronts. To address this challenging problem, a new real ray-tracing method and its two approximations are developed to solve the complex ray equation. The numerical results indicate that the new real ray-tracing method is superior to the common real ray-tracing method in the presence of triplications of the quasi-S-waves in the computation of ray velocity, ray attenuation, and ray quality factors, as well as the reflection and transmission coefficients in viscoelastic anisotropic media. Based on the assumptions of the real slowness direction and real polarization vectors, two new approximations of the new real ray-tracing method are developed for directly computing the homogeneous complex ray velocity vectors of three wave modes (qP, qS1, and qS2). These approximations significantly improve computational efficiency by avoiding the iterative process required by the new real ray-tracing method that is inherited from the common real ray-tracing method. The computational accuracies are verified through transversely isotropic and orthorhombic models with different strengths of attenuation and anisotropy. Incorporation of the new approximation into the shortest-path method turned out to be efficient and accurate for seismic ray tracing in heterogeneous viscoelastic and transversely isotropic media with a vertical axis of symmetry, even in the presence of strong attenuation and anisotropy.

Approximations for traveltime, slope, curvature, and geometric spreading of elastic waves in layered transversely isotropic media

Each seismic body wave, including quasi-, P-, S-, and converted wave modes, carries useful subsurface information. For the processing, imaging, amplitude analysis, and forward modeling of each wave mode, we need approximate equations of the traveltime, slope (ray parameter), and curvature as a function of offset. Considering the large offset coverage of modern seismic acquisitions, we have developed new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis. Our approximation for traveltime is a modified version of the extended generalized moveout approximation that comprises six parameters. Our direct approximations for the ray parameter and curvature use new algebraically simple equations with three parameters. We define these parameters for each wave mode without ray tracing so that we have similar approximate equations for all wave modes that only change based on the parameter definitions. However, our approximations are unable to reproduce S-wave triplications that may occur in some strongly anisotropic models. Using our direct approximation of traveltime derivatives, we also obtain a new expression for the relative geometric spreading. We determine the high accuracy of our approximations using numerical tests on a set of randomly generated multilayer models. Using synthetic data, we show simple applications of our approximations for the normal moveout correction and relative geometric spreading compensation of different wave modes.

Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis

Many applications in seismology involve the modeling of seismic-wave traveltimes in anisotropic media. We have developed homotopy solutions of the acoustic eikonal equation for P-wave traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over powers of an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results of our investigation with approximate formulas demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximate numerical solutions in strongly attenuating anisotropic media.

Artistic and compositional features of G. Narbut’s book graphics and possibilities of their interpretation in modern graphic design

The purpose of the article is to generalize and systematize the graphic and compositional parameters of G. Narbut's book graphics and to find ways of their interpretation in modern graphic design. The methodology of the research is represented by general scientific and special methods. Analysis, synthesis, comparative, systemic, and empirical methods have become basic in the study of literature and empirical material. Comparative, compositional, visual-graphic, and descriptive methods were used to identify the figurative and semantic characteristics of G. Narbut's graphic works and their influence on modern design. The scientific novelty of the work lies in the complex analysis of G. Narbut's graphic works, revealing the peculiarities of their graphic-compositional solutions and determining the principles of their interpretation in modern design practice. Conclusions. It is revealed that in the context of the formation of the national model of Ukrainian graphic design today, the work of G. Narbut needs not copying, but careful rethinking. The study, systematization, and generalization of the artist's experience open the prospect of development and interpretation in the works of designers of the XXI century. The algorithm of such interpretation is a holistic approach to design and composition; rethinking form from the point of view of traditional art; synthesis of geometric calculation and complex artistic image; the contrast between primary and secondary elements; rhythmic repetitions; emphasis on the vertical axis of symmetry and the author's presentation, corresponding to the demands of today.

Perturbation‐based traveltime approximation for two acoustic assumptions in the transversely isotropic media with a vertical symmetry axis

ABSTRACTThe accuracy of an explicit traveltime‐offset approximation affects the results of the velocity analysis that plays a crucial role in the seismic data processing. Seismic anisotropy is important to account for large offset and azimuth since it can provide detailed information comparing with the isotropic assumption. For the perturbation‐based method, different traveltime approximation forms result in different accuracy. It is necessary to find the optimal traveltime approximation for specific signs and magnitudes of the anellipticity and different offset ranges. In this paper, a series of perturbation‐based traveltime approximations from different acoustic assumptions and parameterizations are specified. The perturbation coefficients are derived from the corresponding acoustic eikonal equations. We test the accuracy of the defined perturbation approximations in four homogeneous transversely isotropic medium with vertical symmetry axis (VTI) medium with a different set of anisotropy parameters and one multilayered transversely isotropic medium with vertical symmetry axis medium. The sensitivity in anellipticity for different approximations is also analysed. We find that different approximation achieves different accuracy under the circumstance of specific offset range and the value of anellipticity. Therefore, the optimal approximation form with higher accuracy can be selected based on different offset ranges and the magnitudes of anellipticity.

Number of Distinct Homomorphic Images in Coset Diagrams

The representation of the action of PGL 2 , Z on F t ∪ ∞ in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by p q Δ 1 p q − 1 Δ 2 p q Δ 3 … p q − 1 Δ m ∈ PSL 2 , Z , then this circuit is titled to be a length- m circuit, denoted by Δ 1 , Δ 2 , Δ 3 , … , Δ m . In this manuscript, we consider a circuit Δ of length 6 as Δ 1 , Δ 2 , Δ 3 , Δ 4 , Δ 5 , Δ 6 with vertical axis of symmetry, that is, Δ 2 = Δ 6 , Δ 3 = Δ 5 . Let Γ 1 and Γ 2 be the homomorphic images of Δ acquired by contracting the vertices a , u and b , v , respectively, then it is not necessary that Γ 1 and Γ 2 are different. In this study, we will find the total number of distinct homomorphic images of Δ by contracting its all pairs of vertices with the condition Δ 1 &gt; Δ 2 &gt; Δ 3 &gt; Δ 4 . The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.

SPiRaL: a multiresolution global tomography model of seismic wave speeds and radial anisotropy variations in the crust and mantle

SUMMARYSPiRaL is a joint global-scale model of wave speeds (P and S) and anisotropy (vertical transverse isotropy, VTI) variations in the crust and mantle. The model is comprised of &amp;gt;2.1 million nodes with five parameters at each node that capture velocity variations for P- and S-waves travelling at arbitrary directions in transversely isotropic media with a vertical symmetry axis (VTI). The crust (including ice, water, sediments and crystalline layers) is directly incorporated into the model. The default node spacing is approximately 2° in the lower mantle and 1° in the crust and upper mantle. The grid is refined with ∼0.25° minimum node spacing in highly sampled regions of the crust and upper mantle throughout North America and Eurasia. The data considered in the construction of SPiRaL includes millions of body wave traveltimes (crustal, regional and teleseismic phases with multiples) and surface wave (Rayleigh and Love) dispersion. A multiresolution inversion approach is employed to capture long-wavelength heterogeneities commonly depicted in global-scale tomography images as well as more localized details that are typically resolved in more focused regional-scale studies. Our previous work has demonstrated that such global-scale models with regional-scale detail can accurately predict both teleseismic and regional body wave traveltimes, which is necessary for more accurate location of small seismic events that may have limited signal at teleseismic distances. SPiRaL was constructed to predict traveltimes for event location and long-period waveform dispersion for seismic source inversion applications in regions without sufficiently tuned models. SPiRaL may also serve as a starting model for full-waveform inversion (FWI) with the goal of fitting waves with periods 10–50 s over multiple broad regions (thousands of kilometres) and potentially the globe. To gain insight to this possibility, we simulated waveforms for a small set of events using SPiRaL and independent waveform-based models for comparison. For the events tested, the performance of the traveltime-based SPiRaL model is shown to be generally on par with regional 3-D waveform-based models in three regions (western United States, Middle East, Korean Peninsula) suggesting SPiRaL may serve as a starting model for FWI over broad regions.