Finite-frequency Sensitivity Kernels Research Articles

SItomo – A toolbox for splitting intensity tomography and application in the Eastern Alps

The tomographic inversion of shear wave splitting data for upper mantle anisotropy has been a longstanding challenge. This is due to the ray-based approximation of classical approaches and the near-vertical incidence of the core-mantle converted phases such as SKS that are often used. Recent developments include the calculation of finite-frequency sensitivity kernels for SKS splitting intensity observations, which allows us to accurately take into account the sensitivity to anisotropic structure with depth. A requirement of this tomographic technique is a dense station spacing, which results in overlapping sensitivity kernels at depth and allows for the localization of anisotropic structure. This is satisfied by a growing number of temporary seismic deployments, which motivates the desire to image anisotropic complexities with depth. Here, we introduce and make available a toolbox for the MATLAB environment that facilitates the application of finite-frequency splitting intensity tomography to dense seismic arrays. Our implementation includes several key features, including: 1) A forward calculation of splitting intensities and sensitivity kernels for a complex anisotropic model space. 2) Consideration of the dominant period of the wave, allowing for multiple-frequency analysis, as well as the incoming wave’s non-vertical incidence. 3) The inversion can be based on a classical gradient descent, on a form of the conjugate gradient method known as the BFGS algorithm, or on a gradient-informed stochastic reversible jump algorithm, allowing for a data-driven parametrization of the model space. 4) Importing splitting intensity measurements from waveforms processed in SplitRacer allows for fast pre-processing of large data sets due to its fully automatic design. To illustrate our method, we present both synthetic tests and an application to real data. We apply our inversion procedure to data from the Swath-D network, which densely covers the transition of the Central to the Eastern Alps. Previous studies showed evidence for an abrupt lateral change of layered seismic anisotropy that had been attributed to an opening for channeled asthenospheric flow. Using an SKS splitting intensity tomography approach, we can confirm previous inferences while providing additional constraints on the distribution of anisotropy laterally and with depth.

Reproducing complex anisotropy patterns at subduction zones from splitting intensity analysis and anisotropy tomography

SUMMARYMeasurements of seismic anisotropy provide a lot of information on the deformation and structure as well as flows of the Earth's interior, in particular of the upper mantle. Even though the strong and heterogeneous seismic anisotropic nature of the upper mantle has been demonstrated by a wealth of theoretical and observational approaches , most of standard teleseismic body-wave tomography studies overlook P- and S-wave anisotropy, thus producing artefacts in tomographic models in terms of amplitude and localization of heterogeneities. Conventional methods of seismic anisotropy measurement have their limitations regarding lateral and mainly depth resolution. To overcome this problem much effort has been done to develop tomographic methods to invert shear wave splitting data for anisotropic structures, based on finite-frequency sensitivity kernels that relate model perturbations to splitting observations. A promising approach to image the upper mantle anisotropy is the inversion of splitting intensity (SI). This seismic observable is a measure of the amount of energy on the transverse component waveform and, to a first order, it is linearly related to the elastic perturbations of the medium through the 3-D sensitivity kernels, that can be therefore inverted, allowing a high-resolution image of the upper mantle anisotropy. Here we present an application of the SI tomography to a synthetic subduction setting. Starting from synthetic SKS waveforms, we first derived high-quality SKS SI measurements; then we used the SI data as input into tomographic inversion. This approach enables high-resolution tomographic images of upper-mantle anisotropy through recovering vertical and lateral changes in anisotropy and represents a propaedeutic step to the real cases of subduction settings. Additionally this study was able to detect regions of strong dipping anisotropy by allowing a 360° periodic dependence of the splitting vector.

Finite-frequency sensitivity kernels and hierarchical traveltime and waveform inversion of direct and reflected waves from vertical seismic profile data

Wave-equation-based traveltime or waveform inversion updates the velocity model along finite-frequency sensitivity kernels and exhibits higher accuracy than ray-based tomography. Owing to special geometry, the kernels for vertical seismic profile (VSP) data are different from those for surface seismic data. We have used the Born approximation to compute the traveltime and waveform kernels for direct and reflected waves from VSP data. Based on the property of sensitivity kernels for different information and arrivals, we develop a hierarchical inversion scheme: joint wave-equation traveltime inversion (JWETI) of direct and reflected waves, joint waveform inversion (JWI) of direct and reflected waves, and full-waveform inversion (FWI) of the reflected wave. The traveltime kernels, the waveform kernels, and the migration isochrones are used to retrieve the long-, intermediate-, and short-wavelength components of the velocity model, respectively. Numerical tests on synthetic and real walkway VSP data reveal that the JWETI stage obtains a plausible velocity macromodel, which can be treated as a starting model for subsequent JWI and FWI stages. The JWI stage is a transition between JWETI and FWI and helps to improve the structures of the deep layers of the model. The FWI stage offers some high-wavenumber contents to the inverted model. For the VSP survey, the transmitted direct waves can be used to reduce the dependence of FWI on the initial model. However, the conventional reflection-based wave-equation traveltime inversion and FWI still suffer from cycle skipping when the initial model is far away from the real model. In contrast, the hierarchical inversion scheme can yield reasonable inversion results around the borehole, even when large velocity errors are present in the starting model.

Finite-Frequency Delay Times of Phase Segments for Body Waves

ABSTRACT We propose a new way of measuring the delay times of body waves, based on the time differences between short segments of a phase. Using this proposed methodology, which we call the delay time of phase segments (DTPSs) method, we believe it is possible to (1) optimize the reduction mode that reduces finite-frequency kernels to ray-theoretical kernels, (2) reduce computation and memory storage costs by reducing the volume of finite-frequency sensitivity kernels, and (3) achieve greater linearity between delay times and velocity variations for larger velocity perturbations up to ±30%. The DTPS kernel can also be used in adjoint methods. Theory and our calculations indicate that the width of the DTPS kernel decreases as the length of the phase segment decreases from the length of the entire phase. The scattering caused by inhomogeneity is more likely to complicate the latter parts of a phase more than its beginning. For this reason, the DTPS method using a phase segment in the first quarter of a phase is robust for velocity perturbations up to ±30% from the initial model, whereas traditional methods using the entire phase are only robust for velocity perturbations up to ±10%. The DTPS method may reduce computation times by up to 70% because the size of the DTPS kernels is smaller than that of other methods by up to 70%. Synthetic tests indicate that the DTPS method produces inverse models nearly as accurate as generalized seismological data functionals.

An analysis of core–mantle boundary related seismic waves using full-waveform modelling and adjoint methods

SUMMARYThe core–mantle boundary (CMB) is the most abrupt internal discontinuity in the Earth, marking the solid–fluid boundary between mantle and outer core that strongly affects the dynamics of the Earth’s interior. However, good agreement between models of CMB topographic variations is still lacking. This is probably due to difficulties relating to observations on seismograms and to the lack of good models of lowermost mantle velocity structure. Using spectral-element synthetic seismograms and adjoint methods, we perform traveltime analyses of seismic waves interacting with the CMB. We focus on reflected and refracted P and S waves. We select some of the most important and routinely used seismic phases: ScS, SKS, SKKS, PcP, PKP, PKKP and PcS, given their path through mantle and core and their interaction with the CMB. These seismic waves have been widely deployed by seismologists trying to image CMB topography and lowermost mantle structure. To analyse the reliability of measuring their traveltimes to infer CMB topography, we perform experiments in two ways. First, we compute synthetic seismograms with a dominant period of T ≈ 11s, for computational efficiency, using existing models of CMB topography. We compare traveltime perturbations measured by cross-correlation on the synthetics to those predicted using ray theory. We find deviations from a perfect agreement between ray theoretical predictions of time shifts and those measured on synthetics with and without CMB topography. Second, we calculate Fréchet sensitivity kernels of traveltimes with respect to shear and compressional wave speeds. We also explicitly compute boundary sensitivities with respect to the CMB interface. We observe that the overall sensitivity of the traveltimes is mostly due to volumetric velocity structure and that imprints of CMB on traveltimes are less pronounced. Our study explains the observed difficulties relating to inferring CMB topography using traveltimes and provides a suite of finite frequency sensitivity kernels computed with the adjoint method. The kernels allow us to qualitatively explain the behaviour of measured traveltimes and understand the trade-off between velocity and CMB topography. They can also serve as reference of finite frequency effects on traveltimes of observed seismic phases. From our analyses we conclude that: i) traveltime anomalies measured on Swaves are more in accord with ray theoretical predictions, ii) PcP, PKP, ScS and SKS phases have more pronounced sensitivity to the boundary and iii) separating the greater effects of velocity from those due to the boundary structure is difficult, as they intricately affect the traveltime. We propose that jointly inverting for CMB topography and lowermost mantle velocity structure using full-waveform synthetics and adjoint sensitivity kernels can progress our understanding of deep Earth structure and finite frequency effects on observed waveforms.

Fast calculation of spatial sensitivity kernels for scattered waves in arbitrary heterogeneous media using graph theory

SUMMARYP-to-S and S-to-P receiver functions are widely used to constrain the seismic discontinuity structures of the Earth. Typically, receiver functions are projected to the depth and location of conversion assuming a 1-D layered Earth structure. Receiver function finite frequency sensitivity kernels have the potential to increase resolution. Here we present a method for rapidly calculating the P- and S-wave receiver function sensitivity kernels, based upon the shortest path method and Dijkstra's algorithm to calculate the traveltime fields, and accounting for geometrical spreading in heterogeneous media. The validity of the approach is evaluated by comparing with amplitudes derived from a finite difference elastic full waveform simulation in a complex subduction zone geometry. We show P-to-S and S-to-P kernels calculated using our method for three examples cases: a half space, a layer with topography, and a sinusoidal discontinuity. We also demonstrate the kernel recovery of discontinuities with these topographies by inverting synthetic data from SPECFEM2D. We find that P-to-S kernels recover the structure of strong topography better than S-to-P kernels, although S-to-P kernels may be useful in some situations. P-to-S kernels also show better recovery of the amplitude of the discontinuities in comparison to S-to-P, although both typically achieve values within a few percent of the input model. The computational cost of our approach for improved kernel calculation in heterogeneous media is up to a few tens of seconds per station for typical regional scale models on the scale of several 100s of kilometres.

Imaging upper mantle anisotropy with teleseismicP-wave delays: insights from tomographic reconstructions of subduction simulations

SUMMARYDespite the well-established anisotropic nature of Earth’s upper mantle, the influence of elastic anisotropy on teleseismic P-wave imaging remains largely ignored. Unmodelled anisotropic heterogeneity can lead to substantial isotropic velocity artefacts that may be misinterpreted as compositional heterogeneities. Recent studies have demonstrated the possibility of inverting P-wave delay times for the strength and orientation of seismic anisotropy. However, the ability of P-wave delay times to constrain complex anisotropic patterns, such as those expected in subduction settings, remains unclear as synthetic testing has been restricted to the recovery of simplified block-like structures using ideal self-consistent data (i.e. data produced using the assumptions built into the tomography algorithm). Here, we present a modified parametrization for imaging arbitrarily oriented hexagonal anisotropy and test the method by reconstructing geodynamic simulations of subduction. Our inversion approach allows for isotropic starting models and includes approximate analytic finite-frequency sensitivity kernels for the simplified anisotropic parameters. Synthetic seismic data are created by propagating teleseismic waves through an elastically anisotropic subduction zone model created via petrologic-thermomechanical modelling. Delay times across a synthetic seismic array are measured using conventional cross-correlation techniques. We find that our imaging algorithm is capable of resolving large-scale features in subduction zone anisotropic structure (e.g. toroidal flow pattern and dipping fabrics associated with the descending slab). Allowing for arbitrarily oriented anisotropy also results in a more accurate reconstruction of isotropic slab structure. In comparison, models created assuming isotropy or only azimuthal anisotropy contain significant isotropic and anisotropic imaging artefacts that may lead to spurious interpretations. We conclude that teleseismic P-wave traveltimes are a useful observable for probing the 3-D distribution of upper mantle anisotropy and that anisotropic inversions should be explored to better understand the nature of isotropic velocity anomalies particularly in subduction settings.

Helioseismic Finite-frequency Sensitivity Kernels for Flows in Spherical Geometry including Systematic Effects

Abstract Helioseismic inferences of large-scale flows in the solar interior necessitate accounting for the curvature of the Sun, both in interpreting systematic trends introduced in measurements as well as the sensitivity kernel that relates photospheric measurements to subsurface flow velocities. Additionally, the inverse problem that relates measurements to model parameters needs to be well posed to obtain accurate inferences, which necessitates a sparse set of parameters. Further, the sensitivity functions need to be computationally easy to evaluate. In this work, we address these issues by demonstrating that the sensitivity kernels for flow velocities may be computed efficiently on the basis of vector spherical harmonics. We are also able to account for line-of-sight projections in Doppler measurements, as well as center-to-limb differences in line-formation heights. We show that given the assumed spherical symmetry of the background model, it is often cheap to simultaneously compute the kernels for pairs of observation points that are related by a rotation. Such an approach is therefore particularly well suited to inverse problems for large-scale flows in the Sun, such as meridional circulation.

Comparing ray-theoretical and finite-frequency teleseismic traveltimes: implications for constraining the ratio of S-wave to P-wave velocity variations in the lower mantle

SUMMARY A number of seismological studies have indicated that the ratio R of S-wave and P-wave velocity perturbations increases to 3–4 in the lower mantle with the highest values in the large low-velocity provinces (LLVPs) beneath Africa and the central Pacific. Traveltime constraints on R are based primarily on ray-theoretical modelling of delay times of P waves (ΔTP) and S waves (ΔTS), even for measurements derived from long-period waveforms and core-diffracted waves for which ray theory (RT) is deemed inaccurate. Along with a published set of traveltime delays, we compare predicted values of ΔTP, ΔTS, and the ΔTS/ΔTP ratio for RT and finite-frequency (FF) theory to determine the resolvability of R in the lower mantle. We determine the FF predictions of ΔTP and ΔTS using cross-correlation methods applied to spectral-element method waveforms, analogous to the analysis of recorded waveforms, and by integration using FF sensitivity kernels. Our calculations indicate that RT and FF predict a similar variation of the ΔTS/ΔTP ratio when R increases linearly with depth in the mantle. However, variations of R in relatively thin layers (< 400 km) are poorly resolved using long-period data (T > 20 s). This is because FF predicts that ΔTP and ΔTS vary smoothly with epicentral distance even when vertical P-wave and S-wave gradients change abruptly. Our waveform simulations also show that the estimate of R for the Pacific LLVP is strongly affected by velocity structure shallower in the mantle. If R increases with depth in the mantle, which appears to be a robust inference, the acceleration of P waves in the lithosphere beneath eastern North America and the high-velocity Farallon anomaly negates the P-wave deceleration in the LLVP. This results in a ΔTP of about 0, whereas ΔTS is positive. Consequently, the recorded high ΔTS/ΔTP for events in the southwest Pacific and stations in North America may be misinterpreted as an anomalously high R for the Pacific LLVP.

Strong seismic anisotropy in the deep upper mantle beneath the Cascadia backarc: Constraints from probabilistic finite-frequency SKS splitting intensity tomography

The volcanically active High Lava Plains (HLP) region is a striking tectonomagmatic feature of eastern Oregon, in the backarc of the Cascadia subduction zone. It features young (<12 Ma), bimodal volcanic activity; the rhyolitic volcanism has a spatiotemporal trend that is oblique to that expected from the absolute motion of the North American plate. Several models have been proposed to explain the tectonic evolution of Cascadia backarc and the relationships between upper mantle processes and volcanic activity; however, consensus remains elusive. Because seismic anisotropy in the upper mantle reflects processes such as mantle flow and partial melting, constraints on anisotropic structure can shed light on the connections between mantle dynamics and tectonomagmatic activity in the Cascadia backarc. Anisotropy is often constrained via SKS splitting measurements; however, their interpretation is typically ambiguous because they lack depth resolution. Here we present new constraints on upper mantle anisotropy beneath the HLP region from probabilistic finite-frequency SKS splitting intensity tomography, which provides both lateral and depth constraints on anisotropic structure. Our technique is based on a Markov chain Monte Carlo approach to searching parameter space, and we use finite-frequency sensitivity kernels to relate model perturbations to splitting intensity observations. We use data from broadband stations of the dense High Lava Plains experiment, which provide good resolution of upper mantle anisotropic structure, as demonstrated via resolution tests. We find evidence for particularly strong seismic anisotropy in the deep upper mantle (200-400 km depth) beneath the Cascadia backarc, suggesting that flow in the deep upper mantle, rather than alignment of partial melt in the shallow mantle, provides the first-order control on shear wave splitting delay times. Our model provides additional support for the idea that mantle flow beneath the Cascadia backarc is controlled by rollback subduction. Our results suggest that anisotropy in the deep upper mantle may be more important to the interpretation of SKS splitting measurements in some settings than commonly appreciated, and provides an avenue for reconciling apparently contradictory constraints on anisotropic structure from surface waves and SKS splitting.

Finite-frequency imaging of the global 410- and 660-km discontinuities using SS precursors

SUMMARY We report finite-frequency imaging of the global 410- and 660-km discontinuities using boundary sensitivity kernels for traveltime measurements made on SS precursors. The application of finite-frequency sensitivity kernels overcomes resolution limits in previous studies associated with large Fresnel zones of SS precursors and their interferences with other seismic phases. In this study, we calculate the finite-frequency sensitivities of SS waves and their precursors based on a single-scattering (Born) approximation in the framework of travelling-wave mode summation. The global discontinuity surface is parametrized using a set of triangular gridpoints with a lateral spacing of about 4°, and we solve the linear finite-frequency inverse problem (2-D tomography) based on singular value decomposition (SVD). The new global models start to show a number of features that were absent (or weak) in ray-theoretical back-projection models at spherical harmonic degree l &amp;gt; 6. The thickness of the mantle transition zone correlates well with wave speed perturbations at a global scale, suggesting dominantly thermal origins for the lateral variations in the mantle transition zone. However, an anticorrelation between the topography of the 410-km discontinuity and wave speed variations is not observed at a global scale. Overall, the mantle transition zone is about 2–3 km thicker beneath the continents than in oceanic regions. The new models of the 410- and 660-km discontinuities show better agreement with the finite-frequency study by Lawrence &amp; Shearer than other global models obtained using SS precursors. However, significant discrepancies between the two models exist in the Pacific Ocean and major subduction zones at spherical harmonic degree &amp;gt;6. This indicates the importance of accounting for wave interactions in the calculations of sensitivity kernels as well as the use of finite-frequency sensitivities in data quality control.

Tomographic Imaging of Slab Segmentation and Deformation in the Greater Antilles

AbstractWe present a new tomographic P wave model of the upper mantle in the east Caribbean region. The model was built using 3‐D finite frequency sensitivity kernels and ~20,000 teleseismic P and PP traveltime residuals from 535 events recorded across 130 broadband seismometers. We observe high‐velocity features corresponding to a Caribbean beneath northern South America and an arcuate slab beneath the Lesser and Greater Antilles island arcs. The latter exhibits an along strike gradient in dip with steep edges and a reclined middle, consistent with ongoing slab rollback and collision. We divide the arcuate slab into three sections from two lateral discontinuities. The southern and northern Lesser Antilles sections are separated by a gap ~15°N down to ~200 km. Between Puerto Rico and Hispaniola, another gap down to ~300 km separates the northern Lesser Antilles slab from a narrow slab fragment further east. We relate these discontinuities to the subducted North American‐South American plate boundary and a slab segmentation tear, respectively. The northern and southern ends of the Lesser Antilles trench are actively deforming from collision and differential rollback. However, these areas exhibit different styles of lithospheric tearing, as manifest in the morphology of the slab. We infer the contrast in tearing relates to the presence of microplates at the northern boundary of the Caribbean plate. Microplates facilitate block divergence and differential trench retreat/rollback, which drive slab segmentation. These results offer new insight into the tectonics of the Caribbean region and the factors driving lithospheric tearing in slabs generally.

Finite-frequency Sensitivity Kernels in Spherical Geometry for Time–Distance Helioseismology

Abstract The inference of internal properties of the Sun from surface measurements of wave travel times is the goal of time–distance helioseismology. A critical step toward the accurate interpretation of shifts in travel time is the computation of sensitivity functions linking seismic measurements to internal structure. Here we calculate finite-frequency sensitivity kernels in spherical geometry for two-point measurements of travel time. We numerically build Green’s function by solving for it at each frequency and each degree of spherical harmonic and summing over all these pieces. These computations are performed in parallel (“embarrassingly”), thereby achieving significant speedup in wall-clock time. Kernels are calculated by invoking the first-order Born approximation connecting deviations in the wave field to perturbations in the operator. Validated flow kernels are shown to produce travel times within 0.47% of the true value for uniform flows up to . We find that travel time can be obtained with errors of 1 ms or less for flows having magnitudes similar to meridional circulation. Alongside flows, we also compute and validate a sensitivity kernel for perturbations in sound speed. These accurate sensitivity kernels might improve the current inferences of subsurface flows significantly.

Fast Computation of Global Sensitivity Kernel Database Based on Spectral-Element Simulations

Finite-frequency sensitivity kernels, a theoretical improvement from simple infinitely thin ray paths, have been used extensively in recent global and regional tomographic inversions. These sensitivity kernels provide more consistent and accurate interpretation of a growing number of broadband measurements, and are critical in mapping 3D heterogeneous structures of the mantle. Based on Born approximation, the calculation of sensitivity kernels requires the interaction of the forward wavefield and an adjoint wavefield generated by placing adjoint sources at stations. Both fields can be obtained accurately through numerical simulations of seismic wave propagation, particularly important for kernels of phases that cannot be sufficiently described by ray theory (such as core-diffracted waves). However, the total number of forward and adjoint numerical simulations required to build kernels for individual source–receiver pairs and to form the design matrix for classical tomography is computationally unaffordable. In this paper, we take advantage of the symmetry of 1D reference models, perform moment tensor forward and point force adjoint spectral-element simulations, and save six-component strain fields only on the equatorial plane based on the open-source spectral-element simulation package, SPECFEM3D_GLOBE. Sensitivity kernels for seismic phases at any epicentral distance can be efficiently computed by combining forward and adjoint strain wavefields from the saved strain field database, which significantly reduces both the number of simulations and the amount of storage required for global tomographic problems. Based on this technique, we compute traveltime, amplitude and/or boundary kernels of isotropic and radially anisotropic elastic parameters for various ( $$P$$ , $$S$$ , $$P_{\mathrm{diff}}$$ , $$S_{\mathrm{diff}}$$ , depth, surface-reflected, surface wave, S 660 S boundary, etc.) phases for 1D ak135 model, in preparation for future global tomographic inversions.

Finite-frequency sensitivity kernels of seismic waves to fault zone structures

Finite-frequency sensitivity kernels of seismic waves to fault zone structures

Upper mantle structure beneath southern African cratons from seismic finite-frequency P- and S-body wave tomography

We present a 3D high-resolution seismic model of the southern African cratonic region from teleseismic tomographic inversion of the P- and S-body wave dataset recorded by the Southern African Seismic Experiment (SASE). Utilizing 3D sensitivity kernels, we invert traveltime residuals of teleseismic body waves to calculate velocity anomalies in the upper mantle down to a 700 km depth with respect to the ak135 reference model. Various resolution tests allow evaluation of the extent of smearing effects and help defining the optimum inversion parameters (i.e., damping and smoothness) for regularizing the inversion calculations.The fast lithospheric keels of the Kaapvaal and Zimbabwe cratons reach depths of 300–350 km and 200–250 km, respectively. The paleo-orogenic Limpopo Belt is represented by negative velocity perturbations down to a depth of ∼250 km, implying the presence of chemically fertile material with anomalously low wave speeds. The Bushveld Complex has low velocity down to ∼150 km, which is attributed to chemical modification of the cratonic mantle. In the present model, the finite-frequency sensitivity kernels allow to resolve relatively small-scale anomalies, such as the Colesberg Magnetic Lineament in the suture zone between the eastern and western blocks of the Kaapvaal Craton, and a small northern block of the Kaapvaal Craton, located between the Limpopo Belt and the Bushveld Complex.

Validity of the Rytov Approximation in the Form of Finite-Frequency Sensitivity Kernels

The first-order (or linear) Rytov or Born approximation is the foundation for formulation of wave-equation tomography and waveform inversion, so the validity of the Rytov/Born approximation can substantially affect the applicability of these theories. However, discussions and research reported in literature on this topic are insufficient or limited. In this paper we introduce five variables in scattering theory to help us discuss conditions under which the Rytov approximation, in the form of the finite frequency sensitivity kernels (RFFSK), the basis of waveform inversion and tomography, is valid. The five variables are propagation length L, heterogeneity scale a, wavenumber k, anisotropy ratio ξ, and perturbation strength ɛ. Combined with theoretical analysis and numerical experiments, we conclude that varying the conditions used to establish the Rytov approximation can lead to uninterpretable or undesired results. This conclusion has two consequences. First, one cannot rigorously apply the linear Rytov approximation to all theoretical or practical cases without discussing its validity. Second, the nonlinear Rytov approximation is essential if the linear Rytov approximation is not valid. Different from previous literature, only phase (or travel time) terms for the whole wavefield are discussed. The time shifts of two specific events between the background and observed wavefields measured by cross-correlation will serve as a reference for evaluation of whether the time shifts predicted by the FFSKs are reasonably acceptable. Significantly, the reference “cross-correlation” should be regarded as reliable only if the condition “two specific similar signals” is satisfied. We cannot expect it to provide a reasonable result if this condition is not met. This paper reports its reliability and experimental limitations. Using cross-correlation (CC) samples as the X axis and sensitivity kernel (SK) or ray tracing (RT) samples as the Y axis, a chart of cross validation among the three is established. According to experimental analysis for a random and deterministic medium, the RFFSK works well for diffraction and geometric optics enclosed by the boundaries ka > 1 and (Λ 1, Φ < 2π), where Λ is a diffraction variable and Φ is a variable used to measure scattering strength. Both Λ and Φ depend on the five variables mentioned above. However, a large perturbation will limit the applicability of the Rytov approximation and narrow the range in which it is valid, because larger ɛ results in higher-amplitude fluctuations and larger back-scattering. Therefore, large ɛ is important in making SK/RT predictions discrepant from correct CC measurements. We also confirm that the RFFSK theory has wider range of application than ray theory. All the experiments point out the importance, both practical and theoretical, of respecting the conditions used to establish the Rytov approximation.

Seismic velocity structure and anisotropy of the Alaska subduction zone based on surface wave tomography

AbstractSouthcentral Alaska is a complex tectonic region that transitions from subduction of Pacific crust to flat slab subduction—and collision—of overthickened Yakutat crust. Because much of the Yakutat crust has been subducted, seismic imaging is needed in order to understand the crustal and upper mantle structural framework for this active tectonic setting. Here we use teleseismic Rayleigh waves to image large‐scale variations in shear wave structure. Our imaging technique employs a two‐plane wave representation with finite frequency sensitivity kernels. Our 3‐D isotropic model reveals several features: the subducting Pacific/Yakutat slab, slow wave speeds characterizing the onshore Yakutat collision zone, slow wave speeds of the Wrangell subduction zone, and a deep tomographic contrast at the eastern edge of the Pacific/Yakutat slab. We produce anisotropic phase velocity maps that exhibit variations in the fast direction of azimuthal anisotropy. These maps show the dominance of the Yakutat slab on the observed pattern of anisotropy. West of the Yakutat slab the fast directions are approximately aligned with the plate convergence direction. In the region of the Yakutat slab the pattern is more complicated. Along the margins of the slab the fast directions are roughly parallel to the margins. We identify notable differences and similarities with published SKS splitting measurements. Integrative modeling using 3‐D anisotropy models and different seismic measurements will be needed in order to establish a detailed 3‐D anisotropic velocity model for Alaska. This study provides a large‐scale starting point for such an effort.

Gaussian beam based finite-frequency turning wave tomography

Abstract We introduce a Gaussian beam based method to calculate finite-frequency sensitivity kernels and use these kernels in seismic turning-wave tomography. The Gaussian beam summation method has the advantage of high computation efficiency and without any angle limitation. Thus, it is suitable in generating sensitivity kernels for high-frequency, long-distance and wide-angle waves, particularly for turning waves in models with vertical velocity gradient. We first validate the Gaussian beam calculated sensitivity kernels by comparing them with those calculated using analytical and finite-difference solutions. Then, we build an inversion system based on these kernels for turning wave travel-time tomography. The proposed method incorporates the wave phenomena into the inversion, avoiding drawbacks usually encountered by the high-frequency asymptotic method. On the other hand, this method still keeps the simplicity of the ray-based travel-time tomography. The numerical results are used to demonstrate the potential applications of this method in building velocity models using turning waves.

Solving large tomographic linear systems: size reduction and error estimation

We present a new approach to reduce a sparse, linear system of equations associated with tomographic inverse problems. We begin by making a modification to the commonly used compressed sparse-row format, whereby our format is tailored to the sparse structure of finite-frequency (volume) sensitivity kernels in seismic tomography. Next, we cluster the sparse matrix rows to divide a large matrix into smaller subsets representing ray paths that are geographically close. Singular value decomposition of each subset allows us to project the data onto a subspace associated with the largest eigenvalues of the subset. After projection we reject those data that have a signal-to-noise ratio (SNR) below a chosen threshold. Clustering in this way assures that the sparse nature of the system is minimally affected by the projection. Moreover, our approach allows for a precise estimation of the noise affecting the data while also giving us the ability to identify outliers. We illustrate the method by reducing large matrices computed for global tomographic systems with cross-correlation body wave delays, as well as with surface wave phase velocity anomalies. For a massive matrix computed for 3.7 million Rayleigh wave phase velocity measurements, imposing a threshold of 1 for the SNR, we condensed the matrix size from 1103 to 63 Gbyte. For a global data set of multiple-frequency P wave delays from 60 well-distributed deep earthquakes we obtain a reduction to 5.9 per cent. This type of reduction allows one to avoid loss of information due to underparametrizing models. Alternatively, if data have to be rejected to fit the system into computer memory, it assures that the most important data are preserved.