Slowness Vector Research Articles

Ray and energy-flux velocities at a contact of two viscoelastic anisotropic materials

Summary A g*-Hamiltonian method for tracing real rays was developed that can handle cusps and triplication of quasi-shear wave in a general viscoelastic anisotropic medium. We demonstrate that the g*-Hamiltonian method can produce homogeneous ray-velocity vectors (with parallel real and imaginary parts) and the slowness vectors of reflected and transmitted waves at the interface based on the real Snell's law (RSL), which constrains only the continuity of the real parts of the slowness vectors, or the real slowness direction (RSD) method, which ignores the inhomogeneous component of the slowness vector. These methods are based on the characteristic lines with different Hamiltonians. Our research indicates that these methods are limited to pre-critical incidence ranges. Moreover, we derived a complex energy velocity vector (energy flux velocity) expression, which is always homogenous. We found that directions of corresponding energy velocity calculated with complex Snell's law (CSL) at a contact of two viscoelastic anisotropic materials well match the solutions of the RSL and RSD methods for all R/T waves except post-critical incidence, whereas the RSL and RSD methods fail to obtain homogenous ray velocities. The RSL and RSD methods result in discrepancies in the ray quality factor, R/T coefficients, and energy ratios, especially for post-critical incidence. However, the exact critical angle determined by the RSD method approximates the ‘critical’ angle for anelastic/inhomogeneous waves, which was a previous challenge. Our calculations suggest that the energy velocity and energy quality factor obtained with the CSL method can be used for real ray tracing at the interface of viscoelastic anisotropic media, and the complex energy flux velocity vector is always exactly homogeneous. For the post-critical incidence, the RSL and RSD methods fail because the ray quality factor drastically changes from the infinite down to near 2, which contradicts homogeneous ray velocity in elastic anisotropic materials. However, the energy quality factor for the elastic-anisotropic material is all infinite, even for post-critical incidence, which is correct. We also show that the CSL method has the same efficiency as the RSD method.

Calculating traveltimes in 2D general tilted transversely isotropic media using fast sweeping method

Traveltime calculations play an important role in the field of exploration seismology, such as traveltime tomography and seismic imaging and so on. Seismic anisotropy poses a challenge for traveltime calculation, because anisotropic eikonal solvers are more complex than the isotropic counter part. To solve the eikonal equations in 2D tilted transversely isotropic (TTI) media, we have developed a fast algorithm combine with fast sweeping method to compute the first arrival traveltimes of quasi-P (qP)-, quasi-SV (qSV)-, and quasi-SH(qSH)-waves. For the qP- and qSV-waves, we analyzed the quartic coupled slowness surface equation derived from the Christoffel equation. Then, we constructed a local solver to relate traveltime and slowness. We found that in the local solver, one component of the slowness vector is known and the corresponding slowness equation is monotonic. This provides a strong basis for the fast iterative algorithm we proposed, where we use the Newton method to solve the qP- and qSV-wave slowness equation to determine the related traveltimes. For the qSH wave, the slowness equation is quadratic and simple to solve. Numerical experiments demonstrate that the proposed method can obtain accurate traveltimes for simple and complicated 2D TTI models.

The propagation of inhomogeneous waves in an orthotropic viscoelastic medium

Abstract The present article deals with the propagation of inhomogeneous waves in an orthotropic viscoelastic medium. For chosen directions of propagation and a real finite inhomogeneity parameter, a complex slowness vector is specified to define the propagation of an inhomogeneous incident wave. Then, the reflection and transmission of plane waves at a plane interface between two orthotropic viscoelastic half-spaces are discussed. In this incidence, horizontal slowness determines the slowness vectors for all reflected and transmitted waves. For each reflected and transmitted wave, the corresponding slowness vector is resolved to define its phase direction, phase velocity and attenuation angle. Appropriate boundary conditions on this wave field determine the amplitude ratios for reflected and transmitted waves relative to the incident wave. The numerical examples are provided to show the effect of the inhomogeneity of the incident wave on the propagation characteristic as well as the reflection and transmission coefficients. The existence of homogeneous, inhomogeneous incident waves also is investigated.

Introducing ViDA3, An Earthquake Early Warning Algorithm for Offshore Hypocenter Determination Using Onshore Seismic Networks

ABSTRACT The objective functions adopted by earthquake early warning (EEW) location algorithms are inadequate for out-of-network earthquakes. As a result, the real-time locations of these earthquakes are often erroneous. The consequences of mislocating out-of-network earthquakes are that their magnitudes are miscalculated, and the loci of their shaking predictions map are shifted. Given that the largest earthquakes occur in subduction settings, improving real-time out-of-network earthquake location is of great importance. In this study, the Virtual Dynamically Assembled Array Algorithm (ViDA3) is introduced, which addresses the location issue of offshore and off-network earthquakes. The guiding principle underlying the new EEW location algorithm is that standard seismic networks may be viewed as a collection of medium-sized seismic arrays, with each array consisting of three or more network stations. The potential of array seismology for EEW against out-of-network earthquakes stems primarily from the slowness vector, which points at the direction of the epicentral region. Thus, this region may be constrained merely by intersecting two or more such vectors. In addition, the length of the slowness vector depends on the hypocentral distance and depth and is thus vital for addressing an acute problem in a subduction setting—discriminating between upper crust and deep slab earthquakes. Furthermore, when the slowness of the P phase is known, the slowness of the S phase is deduced, and the S-phase arrival is searched for using the shift-and-sum practice. What makes ViDA3 so attractive is that, in locations where a real-time network is already in place, these added values may be achieved without extra hardware or substantial budget requirements. We present the result of ViDA3 real-time operation on a shallow earthquake offshore Vancouver Island and the result of its replay on a deep slab earthquake in northern Chile. ViDA3 performance is further assessed using a dataset of seismograms from the Mendocino Triple Junction area. It is concluded that ViDA3 location scheme outperforms currently available EEW location algorithms for out-of-network earthquakes.

Inversion of fracture weakness parameters based on 3C vertical seismic profile data — Part I: Horizontal transversely isotropic media composed of a single fracture set in the isotropic background media

Natural fractures in oil and gas reservoirs are a crucial factor that cannot be ignored because they significantly influence the reservoir’s petrophysical properties and hydrocarbon development. A horizontal transversely isotropic (HTI) medium composed of a single fracture set in an isotropic background is a typical anisotropic medium. Meanwhile, the S-wave splitting is a sensitive response of such anisotropic media, resulting in the generation of fast and slow S waves. The normal and tangential fracture weaknesses are crucial parameters that characterize the anisotropy of fractured media. We develop an inversion method for fracture weakness based on 3C vertical seismic profiling (3CVSP) data. First, assuming weak anisotropy and an HTI medium containing a single fracture set, we derive a first-order linear approximation of the traveltimes of the converted fast and slow S waves (PS1 and PS2 waves) with respect to fracture weakness parameters in the phase velocity domain. By solving for the horizontal projection of the slowness vector, approximate equations of the traveltimes of the PS1 and PS2 waves are converted from the phase velocity domain to the group velocity domain. Furthermore, we devise an inversion workflow consisting of three primary steps: (1) preprocessing the VSP data to derive the traveltimes and azimuth of the HTI medium, (2) constructing a forward model with undetermined fracture weakness parameters, and (3) following the establishment of the objective function, conducting the inversion for the fracture weakness parameters. We demonstrate the reliability of the method through numerical examples and synthetic 3CVSP data. The inversion errors are primarily influenced by the azimuth angle, with minimal influence from the receiver depth. Furthermore, a collective set of inverted results derived from all geophones is more stable and accurate than individual geophones. The application to actual 3CVSP data further confirms the effectiveness of our approach.

Slowness vectors in viscoelastic anisotropic media and their physical interpretation and significance

Under the high-frequency assumption, the slowness vector in a viscoelastic anisotropic medium is often defined by a complex-valued vector, whose direction is given by a complex unit vector that can hardly ever be explained by the intuitive physical reasoning related to the actual seismic wave propagation direction. We extend the existing conjugate real ray-tracing (C-RRT) method to compute the ray velocity vectors and then apply it to determine the slowness vectors for three body waves (qP, qSV, and qSH) in a viscoelastic anisotropic medium. Moreover, we dissect the slowness vector with two physical specifications — traveltime gradient and inhomogeneity components, to reveal the physical significance of the slowness vector and give illustrative examples of these components and the phase propagation (real traveltime) and wave-energy attenuation (imaginary traveltime) wavefronts of seismic waves. We also display the homogeneous and inhomogeneous components, as well as the inhomogeneity angles (or deviation angles between these two wavefronts) for the three body waves (qP, qSV, and qSH) in shale with different quality factors ( Q-factors). These results reveal the elusive link between the homogeneous slowness vector and the Q-factors of the medium. The theoretical and numerical dissections of the slowness vector and the extended C-RRT method are extremely helpful in understanding seismic wave propagation and offer a ray-tracing method for wave-energy compensation in seismic migration and reconstruction of subsurface images through seismic ray tomography in viscoelastic anisotropic media.

Vectorgastrogram: dynamic trajectory and recurrence quantification analysis to assess slow wave vector movement in healthy subjects

Functional gastric disorders entail chronic or recurrent symptoms, high prevalence and a significant financial burden. These disorders do not always involve structural abnormalities and since they cannot be diagnosed by routine procedures, electrogastrography (EGG) has been proposed as a diagnostic alternative. However, the method still has not been transferred to clinical practice due to the difficulty of identifying gastric activity because of the low-frequency interference caused by skin–electrode contact potential in obtaining spatiotemporal information by simple procedures. This work attempted to robustly identify the gastric slow wave (SW) main components by applying multivariate variational mode decomposition (MVMD) to the multichannel EGG. Another aim was to obtain the 2D SW vectorgastrogram VGGSW from 4 electrodes perpendicularly arranged in a T-shape and analyse its dynamic trajectory and recurrence quantification (RQA) to assess slow wave vector movement in healthy subjects. The results revealed that MVMD can reliably identify the gastric SW, with detection rates over 91% in fasting postprandial subjects and a frequency instability of less than 5.3%, statistically increasing its amplitude and frequency after ingestion. The VGGSW dynamic trajectory showed a statistically higher predominance of vertical displacement after ingestion. RQA metrics (recurrence ratio, average length, entropy, and trapping time) showed a postprandial statistical increase, suggesting that gastric SW became more intense and coordinated with a less complex VGGSW and higher periodicity. The results support the VGGSW as a simple technique that can provide relevant information on the “global” spatial pattern of gastric slow wave propagation that could help diagnose gastric pathologies.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Complex slowness vector for generalised propagation of harmonic plane waves at the boundaries of real materials

Novel methods to determine the slowness and ray-velocity vectors in viscoelastic anisotropic media

SUMMARY Determination of the slowness vector and the homogeneous ray-velocity vector is critical for seismic ray tracing in a viscoelastic anisotropic medium. Three formulae, the traditional g-Hamiltonian, newly developed conjugate real ray tracing (C-RRT) and innovative g*-Hamiltonian, are employed to calculate the ray-velocity vectors with the determined slowness vectors in a viscoelastic anisotropic medium. We demonstrate the forward and reverse searching procedures to determine the ray-velocity vectors' slowness vectors. The former implements either a linear search or an optimization method to find the slowness vectors that lead to homogeneous complex ray-velocity vectors (its real and imaginary parts are parallel). The latter is based on a new generalized cost function and applies an optimization method to find the slowness vector for a known ray direction. Using sandstone as an example material, we compare the accuracies and efficiencies of the three formulae and the two searching procedures. Our examples show that the forward searching procedure with the traditional g-Hamiltonian formula and the linearly searching method may generate unphysical solutions for qSV wave due to its cusps or triplication, but using the optimization method may not only mitigate the influence of the cusps and triplication but also significantly improve the accuracies and efficiencies almost two orders higher. For the reverse searching procedure, we propose a general form of the cost function valid for all the formulae of the ray-velocity vector and easily solved by an optimization method. The examples demonstrate that the solutions yielded by the forward and reverse searching procedures coincide well for all three body waves (qP, qSV and qSH), except for the triplication of the qSV wave. In particular, the optimization method combined with the novel g*-Hamiltonian formula may completely overcome the issues of spurious solutions and the qSV-wave cusp and triplication.

2D Analysis and simulation of quartz crystal etch penetration by revisiting a previous geometric method

This paper presents a quartz wet etching simulation procedure tailored for microelectromechanical systems (MEMS) fabrication. It utilizes geometric methods and is primarily founded on modifying some aspects of the traditional slowness vector approach. Through a theoretical analysis of the crystal etching trajectory, we provide a macroscopic elucidation for the formation of 2D etch profiles and optimize wafer etch penetration during double-sided wet etching convergence of upper and lower surfaces. Based on the proposed theoretical model, we develop a CAD-like quartz crystal etch simulator that allows for simulations of various quartz cut types and etch penetration. To validate the accuracy of the simulator, we quantitatively compare the simulation with experimental results using the angle between adjacent etched sidewall profiles as a metric. The comparison reveals a minimum error of only 2.2° and a maximum error of 11.7° between the simulation and experimental results. The developed simulator is able to predict the profiles of different etching profiles well, and it has certain engineering application capability.

Influence of orientation of fibres on surface wave in fibre‐induced anisotropic elastic medium

AbstractAn isotropic elastic solid reinforced with a family of parallel fibres behaves anisotropic to any deformation. With orientation of all fibres along a coordinate axis in rectangular Cartesian system, this anisotropy reduces to transverse isotropy. Else with arbitrary orientation of fibre‐family, induced anisotropy lacks any symmetry. In this composite medium with arbitrary anisotropy, three‐dimensional wave motion decaying with depth propagates as a surface wave along all directions on plane boundary. Real phase velocity for this surface wave lies implicit in a system of three equations with complex coefficients. An appropriate transformation of this system yields a real characteristic equation, which is solved to get the anisotropic phase velocity of generalised surface wave. This phase velocity defines a complex slowness vector, which is used to calculate the particle motion at any point in the medium. Effects of fibre‐orientation are analysed, numerically, on the phase velocity and polarisation of surface wave. For example, in the reinforced fibres alters the contours of particle motion through increased ellipticity. However, in the presence of fibres, the decay of particle motion with depth slows down.

Propagation phenomena of a vector-host disease model

This paper is devoted to the study of spreading properties and traveling wave solutions for a vector-host disease system, which models the invasion of vectors and hosts to a new habitat. Combining the uniform persistence idea from dynamical systems with the properties of the corresponding entire solutions, we investigate the propagation phenomena in two different cases: (1) fast susceptible vector; (2) slow susceptible vector when the disease spreads. It turns out that in the former case, the susceptible vector may spread faster than the infected vector and host under appropriate conditions, which leads to multi-front spreading with different speeds; while in the latter case, the infected vector and host always catch up with the susceptible vector, and they spread at the same speed. We further obtain the existence and nonexistence of traveling wave solutions connecting zero to the endemic equilibrium. We also conduct numerical simulations to illustrate our analytic results.

Machine Learning-Based Model Predictive Control of Two-Time-Scale Systems

In this study, we present a general form of nonlinear two-time-scale systems, where singular perturbation analysis is used to separate the dynamics of the slow and fast subsystems. Machine learning techniques are utilized to approximate the dynamics of both subsystems. Specifically, a recurrent neural network (RNN) and a feedforward neural network (FNN) are used to predict the slow and fast state vectors, respectively. Moreover, we investigate the generalization error bounds for these machine learning models approximating the dynamics of two-time-scale systems. Next, under the assumption that the fast states are asymptotically stable, our focus shifts toward designing a Lyapunov-based model predictive control (LMPC) scheme that exclusively employs the RNN to predict the dynamics of the slow states. Additionally, we derive sufficient conditions to guarantee the closed-loop stability of the system under the sample-and-hold implementation of the controller. A nonlinear chemical process example is used to demonstrate the theory. In particular, two RNN models are constructed: one to model the full two-time-scale system and the other to predict solely the slow state vector. Both models are integrated within the LMPC scheme, and we compare their closed-loop performance while assessing the computational time required to execute the LMPC optimization problem.

Early Results of Surface Tomography Using Automated Surface-Wave Phase-Velocity Measuring System (ASWMS) in Hawaii

The Hawaiian Islands are a prominent place to conduct seismicity research. Geologically, occurring earthquakes in this region are unique since it is caused by the presence of a hot spot rather than the common fault activity. The research was conducted using a tomography method called Surface Wave Tomography. The goals of this study are to produce tomographic images of the Hawaiian Islands and to test the accuracy of the Automated Surface-Wave Phase-Velocity Measuring System (ASWMS). ASWMS employs a cross-correlation process to calculate the phase delay between stations, then Eikonal equation to invert the slowness vector to obtain apparent phase velocities, and lastly Helmholtz equation to correct the amplitude to obtain structural phase velocities. The teleseismic data used in this study were collected from stations located throughout the Hawaiian Islands between 2004 and 2009. This study discovered a low-velocity anomaly beneath the Hawaiian Islands with a value of 3.8 – 3.9 km/s, indicating the presence of a low-velocity body. The decrease in velocity with increasing depth suggests an increase in temperature associated with the path that magma takes from the Earth’s mantle to the Earth’s crust beneath the Hawaiian Islands.

Tracking aftershock sequences using empirical matched field processing

SUMMARY Extensive aftershock sequences present a significant problem to seismological data centres attempting to produce near real-time comprehensive seismic event bulletins. An elevated number of events to process and poorer performance of automatic phase association algorithms can lead to large delays in processing and a greatly increased human workload. Global monitoring is often performed using seismic array stations at considerable distances from the events involved. Empirical matched field processing (EMFP) is a narrow-frequency band array signal processing technique that recognizes the inter-sensor phase and amplitude relations associated with wavefronts approaching a sensor array from a given direction. We demonstrate that EMFP, using a template obtained from the first P arrival from the main shock alone, can efficiently detect and identify P arrivals on that array from subsequent events in the aftershock zone with exceptionally few false alarms (signals from other sources). The empirical wavefield template encodes all the narrow-band phase and amplitude relations observed for the main shock signal. These relations are also often robust and repeatable characteristics of signals from nearby events. The EMFP detection statistic compares the phase and amplitude relations at a given time in the incoming data stream with those for the template and is sensitive to very short-duration signals with the required characteristics. Significant deviations from the plane wavefront model that typically degrade the performance of standard beamforming techniques can enhance signal characterization using EMFP. Waveform correlation techniques typically perform poorly for aftershocks from large earthquakes due to the distances between hypocentres and the wide range of event magnitudes and source mechanisms. EMFP on remote seismic arrays mitigates these difficulties; the narrow-band nature of the procedure makes arrival identification less sensitive to the signals’ temporal form and spectral content. The empirical steering vectors derived for the main shock P arrival can reduce the frequency dependency of the slowness vector estimates. This property helps us to automatically screen out arrivals from outside of the aftershock zone. Standard array processing pipelines could be enhanced by including both plane-wave and empirical matched field steering vectors. This would maintain present capability for the plane-wave steering vectors and provide increased sensitivity and resolution for those sources for which we have empirical calibrations.

Automatized localization of induced geothermal seismicity using robust time-domain array processing

The surveillance of geothermal seismicity is typically conducted using seismic networks, deployed around the power plants and subject to noise conditions in often highly urbanized areas. In contrast, seismic arrays can be situated at greater distances and allow monitoring of different power plants from one central location, less affected by noise interference. However, the effectiveness of arrays to monitor geothermal reservoirs is not well investigated and the increased distance to the source coincides with a decreased accuracy of the earthquake localizations. It is therefore essential to establish robust data processing and to obtain precise estimates of the location uncertainties. Here, we use time-domain array data processing and solve for the full 3-D slowness vector using robust linear regression. The approach implements a Biweight M-estimator, which yields stable parameter estimates and is well suited for real-time applications. We compare its performance to conventional least squares regression and frequency wavenumber analysis. Additionally, we implement a statistical approach based on changepoint analysis to automatically identify P- and S-wave arrivals within the recorded waveforms. The method can be seen as a simplification of autoregressive prediction. The estimated onsets facilitate reliable calculations of epicentral distances. We assess the performance of our methodology by comparison to network localizations for 77 induced earthquakes from the Landau and Insheim deep-geothermal reservoirs, situated in Rhineland-Palatinate, Germany. Our results demonstrate that we can differentiate earthquakes originating from both reservoirs and successfully localize the majority of events within the magnitude range of ML -0.2 to ML 1.3. The discrepancy between the two localization methods is mostly less than 1 km, which falls within the statistical errors. However, a few localizations deviate significantly, which can be attributed to poor observations during the winter of 2021/2022.

High-precision and high-efficiency first-arrival slope tomography via eikonal solvers and the adjoint-state method

Abstract First-arrival slope tomography (FAST) introduces first-arrival slopes, corresponding to the horizontal components of the slowness vectors at the receiver and source positions to supplement first-arrival traveltime for better guiding ray propagation in the media until the best match is achieved with the observed data. FAST can recover the velocity model with higher resolution and precision than first-arrival traveltime tomography (FATT) but is computationally intensive. In this context, we propose an improved approach, referred to as high-precision and high-efficiency first-arrival slope tomography (HFAST). HFAST redefines one of the slopes using the reciprocity principle and simultaneously employs the first-arrival traveltime and slopes to ensure high-quality model building. On the other hand, HFAST extracts calculated data and derives the gradient of the misfit function from the solutions of relatively limited forward and inverse problems, resulting in a low computational cost. The cost of HFAST is proportional to the minimum between the receivers and sources, whereas the cost of FAST is scaled to the sum of the receivers and sources. Numerical experiments involving the checkerboard and SEAM II Foothill models demonstrate that HFAST can achieve a higher inversion precision than FATT, especially in the recovery of small-scale anomalies and the presence of velocity reversal. Moreover, HFAST is more computationally efficient than FAST and suitable for managing large data sets. Therefore, HFAST can be regarded as a valuable supplement to current first-arrival-based model building methods and has the potential to be applied in static corrections, prestack depth migration and waveform inversion in the future.

Upper Mantle Structure Beneath the Contiguous US Resolved With Array Observations of SKS Multipathing and Slowness Vector Perturbations

AbstractContinent‐scale observations of seismic phenomena have provided multi‐scale constraints of the Earth's interior. Of those analyzed, array‐based observations of slowness vector properties (backazimuth and horizontal slowness) and multipathing have yet to be made on a continental scale. Slowness vector measurements give inferences on mantle heterogeneity properties such as velocity perturbation and velocity gradient strength and quantify their effect on the wavefield. Multipathing is a consequence of waves interacting with strong velocity gradients resulting in two arrivals with different slowness vector properties and times. The mantle structure beneath the contiguous Unites States has been thoroughly analyzed by previous seismic studies and is data‐rich, making it an excellent testing ground to both analyze mantle structure with our approach and compare with other imaging techniques. We apply an automated array‐analysis technique to an SKS data set to create the first continent‐scale data set of multipathing and slowness vector measurements. We analyze the divergence of the slowness vector deviation field to highlight seismically slow and fast regions. Our results resolve several slow mantle anomalies beneath Yellowstone, the Appalachian mountains and fast anomalies throughout the mantle. Many of the anomalies cause multipathing in frequency bands 0.15–0.30 and 0.20–0.40 Hz which suggests velocity transitions over at most 500 km exist. Comparing our observations to synthetics created from tomography models, we find model NA13 (Bedle et al., 2021, https://doi.org/10.1029/2021GC009674) fits our data best but differences still remain. We therefore suggest slowness vector measurements should be used as an additional constraint in tomographic inversions and will lead to better resolved models of the mantle.

Properties of singular points in a special case of orthorhombic media

The position of singular lines for orthorhombic (ORT) media with fixed diagonal elements of the elasticity matrix cij, i=1…6 is studied under the condition that c11, c22, c33>c66>c44>c55. In this case, the off-diagonal coefficients of the elasticity matrix c12, c13, c23 are chosen so that some of the values of d12=c12+c66, d13=c13+c55, d23=c23+c44 are zero. For orthorhombic medium, where the only one of d12, d13, d23 is zero, contains only singular points in the planes of symmetry. If two or all three dij are zero, then the ORT medium contains singular lines and discrete singular points. We call such media pathological. A degenerate ORT medium with positive d12, d13, d23 has at most two singular lines, which are the intersection of a quadratic cone with a sphere. The pathological media may have up to 6 singular lines on the surface of the slowness. Singular lines for pathological media are described by more complex equations than conventional degenerate ORT models. The article proposes to using squares x, y, z of the components of the slowness vector in the equations. In a new coordinate system, equations defining singular lines for pathological media become linear or quadratic. Intersecting with the plane x+y+z =1, they define the straight lines, ellipses, or hyperbolas. If non-zero values d12, d13, d23 increase, the singular lines pass through four fixed points on the plane x+y+z =1, which makes it possible to describe the evolution of their change. Conditions are derived under which the singular curves of pathological ORT models are limiting the singular curves for degenerate ORT models with positive values of d12, d13, d23. Formulas are derived for transforming surfaces of slowness and singular lines of pathological media into the region of group velocities. The results are demonstrated with examples of pathological models obtained from the standard model of the ORT medium by changing the elasticity coefficients c12, c13, c23 so that some of the values d12, d13, d23 are zero