Search

A two-dimensional argument by Bagnold for the flux over the brink line of a shape-invariantly moving dune is generalized to three dimensions. This is achieved by describing the slip face as the solution to an eikonal equation with an unusual Dirichlet boundary condition where part of the boundary is to be determined. With the assumption of potential flow the flux over a heap is obtained based on kinematics, by solving a Poisson equation and without making reference to the wind profile or sand flux laws. Matching it with the brink line flux can be used in the results of field observations by Sauermann et al. (Geomorphology 36:47–62, 2000) to explain one of the five measured shape parameters of a barchan, the brink line curvature, from the other four. More generally the brink line flux formula proposed here could serve as an evolution equation for the brink line position in a given height and flux profile, in the limit that the avalanching processes are much faster than the rest of the surface evolution. [ABSTRACT FROM AUTHOR]

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained. [ABSTRACT FROM AUTHOR]

SUMMARY: Seismic traveltime tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linearized inverse problem, it maps differences between observed and expected wave traveltimes to interior regions where waves propagate faster or slower than the expected average. The Earth's interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions.Within the past decade the (Learning) Inverse Problem Matching Pursuits [(L)IPMPs] have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov–Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven their applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for traveltime tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial traveltime delays obtained from a contrived model for velocity differences. The corresponding code is available online. [ABSTRACT FROM AUTHOR]

The high accuracy and efficiency of traveltime calculation are critical in seismic tomography, migration, static corrections, source locations and anisotropic parameter estimation. The fast‐sweeping method is an efficient upwind finite‐difference approach for solving the eikonal equation. However, the fast‐sweeping method is accurate only along the axis directions. In two‐dimensional or higher dimensional cases, the accuracy is severely decreased in the diagonal directions due to the numerical errors in these directions. These similar numerical errors also arose in higher order fast‐sweeping method and anisotropic fast‐sweeping method. To improve the accuracy of traveltime calculation in two‐dimensional or higher dimensional space, a shortest‐path‐aided fast‐sweeping method is proposed. The shortest‐path‐aided solution is embedded into the sweeping process of the standard fast‐sweeping method to improve the traveltime accuracy in the diagonal directions. Shortest‐path‐aided fast‐sweeping method is very easy to implement nearly without additional computational cost and memory consumption. Furthermore, this method is easy to extend from two‐dimensional to higher dimensional, from low‐order to higher‐order and from isotropic to anisotropic cases. [ABSTRACT FROM AUTHOR]

There are two main approaches to the numerical solution of the eikonal equation: reducing it to asystemofODES(methodofcharacteristics)andconstructingspecializedmethodsforthenumericalsolutionof this equation in the form of a partial differential equation. The latter approach includes the FSM (Fast sweeping method) method. It is reasonable to assume that a specialized method should have greater versatility. The purpose of this work is to evaluate the applicability of the FSM method for constructing beams and fronts. The implementation of the FSM method in the Eikonal library of the Julia programming language was used. The method was used for numerical simulation of spherical lenses by Maxwell, Luneburg and Eaton. These lenses were chosen because their optical properties have been well studied. A special case of flat lenses was chosen as the easiest to visualize and interpret the results. The results of the calculations are presented in the form of images of fronts and rays for each of the lenses. From the analysis of the obtained images, it is concluded that the FSM method is well suited for constructing electromagnetic wave fronts. An attempt to visualize ray trajectories based on the results of his work encounters a number of difficulties and in some cases gives an incorrect visual picture.

Mine goafs often cause severe ground settlement and collapse, posing significant endangers to the safety of buildings and engineering constructions. Quantitatively detecting the spatial distribution and physical characteristics of goafs has always been challenging. To evaluate the potential threat of a fluorite mine goaf on a proposed expressway, we inverted cross-hole dataset to image the P-wave velocities between boreholes using a newly developed Eikonal equation based adjoint-state traveltime tomography method (ATTOMO). Independent comprehensive geophysical prospecting, including core analysis, borehole optical televiewer imaging and ultrasonic wave logging, indicates that our tomographic results at the central section could discern anomalies on a scale of 1 m or even smaller and detects velocity variations at a scale of 100–200 m/s. Our results show that the mine goaf is distributed in a northeastward trend with a width of 20–30 m, consistent with the exploiting features of the fluorite mine. The goaf is located approximately 20 m away from the bridge site, where only a small-scale goaf tunnel with dimensions of 1.5×1.5 m was found at depths of 18–19.5 m. Our results suggest that the safety of the proposed expressway would not be significantly affected by the goaf, and thus no design changes are required. [ABSTRACT FROM AUTHOR]

Given a smooth potential W : T n → R on the torus, the Quantum Guerra–Morato action functional is given by I (ψ) = ∫ ( D v D v * 2 (x) − W (x)) a (x) 2 d x , where ψ is described by ψ = a e i u ℏ , u = v + v * 2 , a = e v * − v 2 ℏ , v, v* are real functions, ∫a2(x)dx = 1, and D is the derivative on x ∈ Tn. It is natural to consider the constraint div(a2Du) = 0, which means flux zero. The a and u obtained from a critical solution (under variations τ) for such action functional, fulfilling such constraints, satisfy the Hamilton-Jacobi equation with a quantum potential. Denote ′ = d d τ . We show that the expression for the second variation of a critical solution is given by ∫a2D[v′] D[(v*)′] dx. Introducing the constraint ∫a2Du dx = V, we also consider later an associated dual eigenvalue problem. From this follows a transport and a kind of eikonal equation. [ABSTRACT FROM AUTHOR]

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semi-continuous sub and super solutions, obtaining a unique characterization of the value function. [ABSTRACT FROM AUTHOR]

We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless, they share several of their fine properties with BV functions: we show in particular that the set of non-Lebesgue points has at least one co-dimension. [ABSTRACT FROM AUTHOR]

The effects of low-lying core(30S) and projectile(31Cl) excited states on the full-width half maxima(FWHM) of longitudinal momentum distribution(LMD) of core fragment and single proton breakup cross-section of 12C(31Cl,30S)X breakup reaction at 44 MeV/nucleon beam energy have been analyzed quantitatively for the stripping and diffraction dissociation reaction mechanisms. The calculations are carried out using standard MOMDIS code based on Glauber eikonal approximation. Our analysis shows that the inclusion of core and projectile excitation enhances the LMD width and suppresses the magnitude of the single proton breakup cross-section significantly. Additionally, in comparison with the experimental LMD width, the contribution of s and d-state in admixed sd-state have also been explored. The obtained results are informative for the lucid structural study of the 31Cl nucleus. [ABSTRACT FROM AUTHOR]

Shear wave velocity (Vs) is a fundamental property of elastic media whose estimation from PS converted waves is challenging and requires modelling the boundary where P to S conversion occurs. This paper presents a PS tomography where seismic wave conversion/reflection points correspond to reflectors modelled with the level-set function set to zero [ ϕ (x, z) = 0]. The proposed method aims for stable Vs inversion in a seismic acquisition setting using multicomponent receivers. Synthetic models simulating true Vs , Vp and the location of the geological reflector are used in the study. The inversion starts by locating a flat reflector, ϕ (x, z) = 0, which defines the zone Ω1 between the surface and the reflector, where the initial Vs and Vp fields are also set. To calculate the traveltimes of incident PT (P wave that propagates in Ω1 from source to the reflector), converted PS and reflected PP waves, for both observed and modelled data (forward problem), the methodology proposed by Rawlinson and Sambridge is adopted. This method uses the arrival times of the P waves, Tpt , from the seismic source at each reflector point as secondary sources generating the times Tps and Tpp. These times are calculated as a solution to the eikonal equation by using the Fast Marching method. The PS and PP residual times are minimized by updating Vs , Vp and ϕ (x, z) = 0 through adjoint variables designed from a formulation using Lagrange Multipliers in a variational context. The performance of the algorithm is evaluated for models with synclinal, sinusoidal and monoclinal reflector geometries using numerical tests considering the inversion of: (1) ϕ , given the true values of Vs and Vp ; (2) ϕ and Vs , given the true value of Vp ; (3) ϕ and Vp , given the true value of Vs and (4) the three parameters ϕ, Vs and Vp , simultaneously. Good results are obtained by inverting Vs and ϕ , given the true value of Vp. The simultaneous inversion of the three parameters exhibits promising results, despite the illumination problems caused by the different distribution of the PS, PP and PT time gradients due to the geometry of the reflectors and the acquisition setting (sources–receivers in the same plane). The proposed tomography estimates Vs and reflector positions which could help in statics corrections and improve the lithological characterization of near surface. [ABSTRACT FROM AUTHOR]

Recent advancements in empirical observations of human psychological responses have revealed that the governing rules of human behavioral aspects can not only be predicted with adequate deterministic precision but also forged into mathematical formulations. These modeling studies enable crisis managers with reliable simulation tools to gain insights into critical facets of crowd disasters and enhance their decision-making abilities to facilitate safe egress in emergency situations. The macroscopic scale of representation in this context is often invoked to comprehend large-scale collective pattern formation in vast built environments, owing to its computational viability. The associated governing equations are typically constructed in a system of hyperbolic conservation law form, and conducting simulations in real-life complex geometric configurations can pose computational challenges. This article puts forward a high-resolution, shock-capturing meshfree particle method in an Eulerian framework for the numerical approximation of several widely adopted macroscopic pedestrian flow models. It serves as a superior alternative to traditional mesh-based methods, eliminating the need for specific mesh structures and connectivity between them. A classical Generalized Finite Difference Method (GFDM) in conjunction with several consistency conditions on spatial derivative coefficients is adopted to obtain a non-oscillatory and fairly conservative Godunov-type discretization of the governing system. The intercell flux is evaluated using a suitable Riemann solver, and a slope-limited reconstruction procedure of the corresponding Riemann states, adapted to arbitrary point distribution, is employed for improved accuracy. Moreover, the preferred direction of human motion is determined from a local density-dependent eikonal-type equation, which is solved using a generalized version of the Fast Marching Method. A thorough numerical investigation of three well-known second-order macroscopic models, namely Payne-Whitham (PW), Aw-Rascle (AR), and Aw-Rascle-Zhang (ARZ) is carried out through two hypothetical scenarios of crowd evacuation, not only to demonstrate the accuracy and applicability of the proposed approach but also to deepen understanding of their distinctive characteristics. A comparison of the flow flux with the fundamental diagram suggests that the density profile obtained with the ARZ model is equivalent to Hughes' first-order model. In contrast, the PW and AR models have proven to effectively replicate complex, unstable pedestrian phenomena such as stop-and-go waves. Finally, Helbing's counter-intuitive idea that strategically positioned barriers upstream of a bottleneck can essentially expedite evacuation is numerically investigated across various parameter choices. • A high-resolution, shock-capturing mesh-free particle method for three widely adopted pedestrian flow models. • Classical Generalized Finite Difference Method in conjunction with geometric conservation. • Complete characterization of complex, unstable crowd phenomena such as stop-and-go waves. • Investigation of bottleneck evacuation in the presence of strategically placed obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]

The precise determination of earthquake location is the fundamental basis in seismological community, and is crucial for analyzing seismic activity and performing seismic tomography. First arrivals are generally used to practically determine earthquake locations. However, first-arrival traveltimes are not sensitive to focal depths. Moreover, they cannot accurately constrain focal depths. To improve the accuracy, researchers have analyzed the depth phases of earthquake locations. The traveltimes of depth phases are sensitive to focal depths, and the joint inversion of depth phases and direct phases can be implemented to potentially obtain accurate earthquake locations. Generally, researchers can determine earthquake locations in layered models. Because layered models can only represent the first-order feature of subsurface structures, the advantages of joint inversion are not fully explored if layered models are used. To resolve the issue of current joint inversions, we use the traveltimes of three seismic phases to determine earthquake locations in heterogeneous models. The three seismic phases used in this study are the first P-, sPg- and PmP-waves. We calculate the traveltimes of the three seismic phases by solving an eikonal equation with an upwind difference scheme and use the traveltimes to determine earthquake locations. To verify the accuracy of the earthquake location method by the inversion of three seismic phases, we take the 2021 MS6.4 Yangbi, Yunnan earthquake as an example and locate this earthquake using synthetic and real seismic data. Numerical tests demonstrate that the eikonal equation-based earthquake location method, which involves the inversion of multiple phase arrivals, can effectively improve earthquake location accuracy. [ABSTRACT FROM AUTHOR]

The picking efficiency of seismic first breaks (FBs) has been greatly accelerated by deep learning (DL) technology. However, the picking accuracy and efficiency of DL methods still face huge challenges in low signal-to-noise ratio (SNR) situations. To address this issue, we propose a regression approach to pick FBs based on bidirectional long short-term memory (BiLSTM) neural network by learning the implicit Eikonal equation of 3D inhomogeneous media with rugged topography in the target region. We employ a regressive model that represents the relationships among the elevation of shots, offset and the elevation of receivers with their seismic traveltime to predict the unknown FBs, from common-shot gathers with sparsely distributed traces. Different from image segmentation methods which automatically extract image features and classify FBs from seismic data, the proposed method can learn the inner relationship between field geometry and FBs. In addition, the predicted results by the regressive model are continuous values of FBs rather than the discrete ones of the binary distribution. The picking results of synthetic data shows that the proposed method has low dependence on label data, and can obtain reliable and similar predicted results using two types of label data with large differences. The picking results of 9380 shots for 3D seismic data generated by vibroseis indicate that the proposed method can still accurately predict FBs in low SNR data. The subsequent stacked profiles further illustrate the reliability and effectiveness of the proposed method. The results of model data and field seismic data demonstrate that the proposed regression method is a robust first-break picker with high potential for field application. [ABSTRACT FROM AUTHOR]

The limitations of the coverage range and density of transmission wave often result in less-than-ideal results in traveltime tomography. By contrast, joint transmission-reflection traveltime tomography can not only recover deep structures that transmission tomography cannot detect, but also optimize its inversion results. In this article, we perform joint tomography on borehole seismic (vertical seismic profile, reverse vertical seismic profile, and crosswell seismic) data to obtain near-wellbore structures. In the first part, we solve the factored eikonal equation by the discontinuous Galerkin (DG) method to calculate the transmission/reflection traveltime. Owing to the large wavefront curvature near the source point, the traveltime errors generated by the numerical simulation will propagate from the source to all the calculation domains. According to the factorization principle, the equation solution is decomposed into two parts to solve the point-source singularity. To further improve the accuracy of solving traveltime, we use the DG method to solve the factored eikonal equation with additive factors (the factored DG method), obtaining second-order accuracy solution. The adjoint-state method is employed in the inversion section to calculate the gradient of the misfit function. We use the traveltime difference observed inside the model to define the misfit function, which is more suitable for borehole seismic and avoids the influence of surface normal vectors on gradients. Numerical tests applied on models indicate that the joint tomography method has the potential to accurately inverse the seismic structure information near the well and recover the deep underground structure. [ABSTRACT FROM AUTHOR]

We present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with Nv discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all Nv cell centers to each other. The overall CPU time and memory requirement scale as O(Nv log2 Nv) for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an O(Nv4/3) CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.

Conformal Carter–Penrose diagrams are used for the visualization of hyperboloidal slices, which are smooth spacelike slices reaching null infinity. The focus is on the Schwarzschild black hole geometry in spherical symmetry, whose Penrose diagrams are introduced in a pedagogical way. The stationary regime involves time-independent slices. In this case, different options are given for integrating the height function—the main ingredient for constructing hyperboloidal foliations. The dynamical regime considers slices changing in time, which are evolved together with the spacetime using the eikonal equation. It includes the relaxation of hyperboloidal Schwarzschild trumpet slices and the collapse of a massless scalar field into a black hole, for which Penrose diagrams are presented. [ABSTRACT FROM AUTHOR]

Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the space-time dependent source function from the measurement of the wave, collected at a single point $ x $ for a large enough interval of time, generated by a small scaled droplets, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel-time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small droplets) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an Eulerian type of numerical finite difference approximation on structured grids for an advection dominated level set equation enhanced with a small curvature term. The proposed numerical scheme is based on upwind discretization in both the advection and the curvature term. This class of approximations is used in algebraic solvers, such as fast sweeping or fast marching methods, which respect the causality of solutions for which any information is propagating along characteristics, in order to solve the algebraic system in a finite number of computationally simple iterations or updates. An implicit numerical scheme for a time relaxed level set equation with non-zero right-hand side is used in this paper and solved by a nonlinear fast sweeping method. The results of the von Neumann stability analysis of the proposed scheme are presented, and numerical examples are used to verify the analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a direct probing method to solve the inverse problem of the Eikonal equation. This problem involves the determination of the inhomogeneous wave-speed distribution from first-arrival time data at measurement surfaces corresponding to distributed point sources. The viscosity solution of the point-source Eikonal equation represents the least traveltime of wave fields from the source to the point at the high-frequency limit. We show that this inverse problem is highly ill-posed. To address this issue, we develop a direct probing method that incorporates solution analysis of the Eikonal equation and several aspects of the velocity models. Specifically, we use the filtered back-projection method to reconstruct the inhomogeneous wave-speed distribution when it has a small variation from the homogeneous medium. For the scenarios involving highcontrast media, we assume a background medium and develop an adjoint-based back-projection method to identify the variations of the medium from the assumed background. [ABSTRACT FROM AUTHOR]

We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation ∥ ∇ φ ∥ = f , with a positive continuous function f connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock's primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Riemannian warped product map as a generalization of warped product isometric immersion and Riemannian warped product submersion with examples and obtain some characterizations. First, we define Riemannian warped product map and find conditions for a Riemannian map to be Riemannian warped product map. We show that Riemannian warped product map is a composition of Riemannian warped product submersion followed by warped product isometric immersion locally. In addition, we show that Riemannian warped product map satisfies the generalized eikonal equation which is a well known bridge between geometrical and physical optics. We also find necessary and sufficient conditions for the fibers, range space of the derivative map of Riemannian warped product map and horizontal distributions to be totally geodesic and minimal. Further, we give some fundamental geometric properties for the study of such smooth maps. Precisely, we construct Gauss formula (second fundamental form), Weingarten formula and tension field. We obtain necessary and sufficient conditions for a Riemannian warped product map to be totally geodesic, harmonic and umbilical. Comparatively, we analyse the obtained results with the existing results for a Riemannian map between Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical algorithm based on a cell-centered finite volume method to compute a distance from given objects on a three-dimensional computational domain discretized by polyhedral cells. Inspired by the vanishing viscosity method, a Laplacian regularized eikonal equation is solved and the Soner boundary condition is applied to the boundary of the domain to avoid a non-viscosity solution. As the regularization parameter depending on a characteristic length of the discretized domain is reduced, a corresponding numerical solution is calculated. A convergence to the viscosity solution is verified numerically as the characteristic length becomes smaller and the regularization parameter accordingly becomes smaller. From the numerical experiments, the second experimental order of convergence in the L 1 norm error is confirmed for smooth solutions. Compared to solve a time-dependent form of the eikonal equation, the Laplacian regularized eikonal equation has the advantage of reducing computational cost dramatically when a more significant number of cells is used or a region of interest is far away from the given objects. The implementation of parallel computing using domain decomposition with 1-ring face neighborhood structure can be done straightforwardly by a standard cell-centered finite volume code. [ABSTRACT FROM AUTHOR]

We provide a numerical realization of an optimal control problem for pedestrian motion with agents that was analyzed in Herzog et al. (Appl. Math. Optim. 88(3):87, 2023). The model consists of a regularized variant of Hughes’ model for pedestrian dynamics coupled to ordinary differential equations that describe the motion of agents which are able to influence the crowd via attractive forces. We devise a finite volume scheme that preserves the box constraints that are inherent in the model and discuss some of its properties. We apply our scheme to an objective functional tailored to the case of an evacuation scenario. Finally, numerical simulations for several practically relevant geometries are performed. [ABSTRACT FROM AUTHOR]

In this study we recast surface wave traveltime tomography as an inverse problem constrained by an eikonal equation and solve it using the efficient adjoint‐state method. Specifically, recognizing that large topographic variations and high surface wave frequencies can make the topographic effect too significant to ignore, we employ an elliptically anisotropic eikonal equation to describe the traveltime fields of surface waves on undulated topography. The sensitivity kernel of the traveltime objective function with respect to shear wave velocity is derived using the adjoint‐state method. As a result, the newly developed method is inherently applicable to any study regions, whether with or without significant topographic variations. Hawaii is one of the most seismically and magmatically active regions. However, its significant topographic variations have made it less accurate to investigate using conventional surface wave traveltime tomography methods. To tackle this problem, we applied our new method to invert ambient noise Rayleigh wave phase traveltimes and construct a 3D shear wave velocity model. Our results reveal features that are consistent with geological structures and previous tomography results, including high velocities below Mauna Loa Volcano and Kilauea Volcano, and low velocities beneath the Hilina Fault Zone. Additionally, our model reveals a high‐velocity anomaly to the South of Hualalai's summit, which may be related to a buried rift zone. Our findings further demonstrate that including topography can lead to a correction of up to 0.8% in the shear wave velocity model of Hawaii, an island spanning approximately 100 km with volcanoes reaching elevations exceeding 4 km. Plain Language Summary: Surface wave traveltime tomography is a commonly used technique for constructing subsurface shear wave velocity models based on surface wave traveltime measurements, which has been widely used to investigate the structures of the crust and upper mantle. The accuracy of tomography relies on effective forward modeling, that is, computing traveltime fields in a given velocity model. In this study, we employ an elliptically anisotropic eikonal equation to model the traveltime fields of surface waves on undulated topography, ensuring precise forward modeling in regions with significant topographic variations. The inversion of surface wave traveltimes is formulated as an optimization problem constrained by this elliptically anisotropic eikonal equation, which is solved using the efficient adjoint‐state method. We have applied the new tomography method to the Island of Hawaii. High shear wave velocities are revealed beneath the summits and rift zones of Mauna Loa Volcano and Kilauea Volcano, outlining the subsurface structures associated with magma storage and transportation. Key Points: An elliptically anisotropic eikonal equation is used to model the traveltime fields of surface waves on an undulated topographyAn adjoint‐state surface wave tomography method based on the anisotropic eikonal equation is developed to determine shear wave velocityThe case study in Hawaii suggests limited topography influence on velocity anomaly amplitudes in regions comparable or larger than Hawaii [ABSTRACT FROM AUTHOR]

The recently developed adjoint-state traveltime tomography (ATT) method offers an alternative approach to conduct traveltime tomography without the need for ray tracing or waveform modelling. Instead, it utilizes the eikonal equation to depict the minimal traveltime field from an earthquake location to any position in the computational domain. The process of tomographic inversion is formulated as an optimization problem with the goal of minimizing the difference between observed and theoretical first arrival times, which is subsequently solved using the efficient adjoint method. One advantage of differential arrival time data is that it cancels or reduces the influence of common factors, making it more sensitive to a specific subset of model parameters compared to first arrival times. To take advantage of this property, two variants of the ATT method are derived to determine velocity structure and earthquake locations in this study. The first variant, adjoint-state common-source differential arrival time tomography (ATT-CS), uses common-source differential arrival times; while the second variant, adjoint-state common-receiver differential arrival time tomography (ATT-CR), inverts common-receiver differential arrival times. Numerical examples demonstrate that the ATT-CS method is a valuable tool for imaging receiver-side fine-scale velocity structures. Conversely, the ATT-CR method is well suited for resolving source-side velocity structures. Differential arrival times also place constraints on earthquake locations. Compared to common-source differential arrival times, common-receiver differential arrival times are less sensitive to velocity errors and suitable for earthquake location determination. Both common-source and common-receiver differential arrival times are considered first-order differential arrival times. To demonstrate the ease with which the ATT method can be extended to higher-order differential arrival times, we also derive the adjoint-state second-order differential arrival time tomography method. Finally, we discuss how the adjoint-state tomography methods address multipathing. [ABSTRACT FROM AUTHOR]

We propose a novel framework for teleseismic traveltime tomography that requires no ray tracing. The tomographic inverse problem is formulated as an Eikonal equation‐constrained optimization problem, aiming at the determination of a slowness model that minimizes the difference between observational and predicted differential traveltimes. Two improvements have been made over previous ray‐based methods. First, the traveltimes from the source outside the study region to any positions within the study region are computed using a hybrid approach. This involves solving a 2D Eikonal equation to obtain the traveltimes from the source to the boundary of the study region and solving a 3D Eikonal equation to compute the traveltimes from the boundary to any positions within the study region. Second, we compute the sensitivity kernel using the adjoint‐state method. This method avoids the computation of ray paths and makes the computational cost nearly independent of the number of receivers. We apply our new method in Thailand and adjacent regions. The final velocity model reveals a thick lithosphere beneath the Khorat Plateau and two mantle upwelling branches beneath its southern and western margins. The mantle upwelling may result from the mantle convection triggered by surrounding subduction systems and/or a slab window of the Indian Plate. The presence of the mantle upwelling corresponds to the source zone of the erupted Cenozoic basalts in the Khorat Plateau, indicating lithospheric modification beneath the plateau. The insightful tomographic result verifies our method and provides new perspectives on the structural heterogeneities and dynamics of the Indochina Block. Plain Language Summary: We propose a novel teleseismic traveltime tomography method based on the adjoint‐state method. The critical procedure of computing sensitivity kernels is realized by solving the Eikonal and adjoint equations. This approach eliminates the need for calculating ray paths, which is required in conventional ray‐based tomography methods. As a result, our approach avoids the potential failures of conventional shooting and bending methods for ray tracing and makes the computational cost independent of the number of receivers. We test our new method in Thailand, using the new data from the Thai Seismic Array. Our model reveals a thick lithosphere beneath the Khorat Plateau as a high‐velocity anomaly. In addition, two mantle upwelling branches are imaged beneath the southern and western margins of the plateau as low‐velocity anomalies. These anomalies are likely driven by surrounding subduction systems and may also be associated with the slab window of the Indian Plate. The presence of mantle upwelling aligns with the source zone of the exposed Cenozoic basalts and may suggest the lithospheric modification beneath the Khorat Plateau. Key Points: A novel framework based on the adjoint‐state method is developed for teleseismic traveltime tomographyThe low‐velocity anomalies beneath the southern and western margins of Khorat Plateau suggest mantle upwelling due to mantle convectionThe existence of mantle upwelling indicates lithospheric modification beneath the Khorat Plateau [ABSTRACT FROM AUTHOR]

We discuss the control of a human crowd whose dynamics is governed by a regularized version of Hughes’ model, cf. Hughes (Transp Res Part B: Methodol 36(6):507–535, 2002. ). We assume that a finite number of agents act on the crowd and try to optimize their paths in a given time interval. The objective functional can be general and it can correspond, for instance, to the desire for fast evacuation or to maintain a single group of individuals. We provide an existence and regularity result for the coupled PDE-ODE forward model via an approximation argument, study differentiability properties of the control-to-state map, establish the existence of a globally optimal control and formulate optimality conditions. [ABSTRACT FROM AUTHOR]

In modern seismic exploration, the migration procedure plays an important role for further interpretation of the observed data. It is the migration that makes it possible to display the deep structure of the geological section based on the dynamic characteristics of the registered wave fields. When processing WARRP (wide angle reflection/refraction profiling) seismic data, standard migration methods are ineffective, which is due to the peculiarities of observation systems. All existing migration methods are mainly based on reflected waves, which have a limited registration interval. The wave field on WARRP is observed at distances from sources that reach several hundred kilometers, and the uneven step between receivers is on average 1—3 km. Under such conditions, it is difficult and sometimes impossible to separate the reflected waves. The finite-difference reflection/refraction migration developed at the Subbotin Institute of Geophysics of the National Academy of Sciences of Ukraine has proven its effectiveness in constructing migration images of the depth structure of the section based on WARRP data observed in various regions of the world. The main difference of this migration method is the use of refracted waves as reference waves registered in the large offsets. At the same time, there is a question of the correctness of the reproduction of the depth structure of the section on the migration image, which depends on the correctness of the calculation methods. The algorithm of finite-difference reflection/refraction migration contains a continuation of the time and wave fields. The article presents a proof of the mathematical correctness of the solution of the eikonal differential equations and the scalar wave equation by the finite-difference method, on which are based the continuation of the time and wave fields, respectively. The given comparison of the formed migration images and the velocity models, calculated by the ray-tracing method, of the upper crust along the PANCAKE and TTZ-South profiles allows us to assert that the combination of the results of kinematic and dynamic processing of WARRP data allows to increase the informativeness of their further interpretation

Many natural landscapes maintain steep planar hillslopes bounded at a typical angle, beyond which shallow landslides or slope failures remove the excess sediment volume by mass wasting. Here we show that the celebrated eikonal equation, derived from a landscape evolution model in conditions of negligible soil diffusion and fluvial erosion, accurately portrays the organization of these topographies. Referred to as "eikonal landscapes," such solutions feature constant‐slope hillslopes originating from downstream boundary conditions and culminating in sharp upstream ridges. We demonstrate that the eikonal landscapes reproduce well a variety of natural landforms, including small islands, a volcano, and an extended mountain ridge. The boundary condition for the eikonal representation is specified through the natural landscape's slope‐area relation. Going beyond merely representing landscape statistical features, the present results provide a first‐of‐kind direct match of mathematical and natural landscapes. Plain Language Summary: Natural terrains often feature steep and nearly flat hillslopes, which remain stable up to a threshold angle or the angle of repose. Beyond this angle, mass failure ensues, resulting in the removal of excess sediment volume and preserving the topography. Here we demonstrate the effective use of the eikonal model, originally developed to represent the propagation of light rays two centuries back, in describing the organization of threshold hillslopes in natural landscapes. We directly compare eikonal model‐generated topographies with various natural landscapes, including islands, volcanoes, and inland mountains, differing in size, underlying geology, and climatic conditions. The presented model achieves these results with minimal computational cost compared to traditional landscape evolution simulations. Key Points: The simple eikonal model describes the landscape morphology with threshold hillslopesThe natural landscape's slope‐area relation establishes the boundary condition for the modelThe model results faithfully reproduce diverse landforms, such as islands, an inland volcano, and a mountain range [ABSTRACT FROM AUTHOR]

We present a theoretical framework that links Fermat's principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency. [ABSTRACT FROM AUTHOR]

Recently, a fluid volume enrichment strategy based on the asymptotic solutions near the crack tip was proposed for fluid‐driven fracture propagation problems. Despite its successes in various benchmark and field‐scale problems of hydraulic fracturing simulations, the aforementioned enrichment strategy has the following limitations. First, the tightly coupled solid‐fluid nonlinear system cannot be consistently linearized due to the fast marching method applied to solve the Eikonal equation for fracture tracking. As a result, an approximated Jacobian had to be deployed for the Newton‐Raphson iterations. This is particularly troublesome when the fracture front propagates into newly fractured cells, since a large number of nonlinear iterations are required for convergence because of the inconsistent linearization. Second, the existing method only focused on the viscosity‐dominated fracture propagation regime. Even though the extension of the method to the toughness‐dominated regime could be relatively straightforward, it is not immediately clear how to apply the enrichment technique to the transition regime. This work is dedicated to address the above two limitations. Specifically, a unified fracture propagation criterion is proposed, which not only works for the viscosity‐dominated regime, but also for the toughness‐dominated regime and the transition regime in between. The techniques to consistently linearize the coupled solid‐fluid system and properly initialize the primary unknowns are demonstrated, which result in the significant reduction of required number of nonlinear iterations for convergence. The proposed technique is demonstrated in the context of the Khristianovic‐Geertsma‐de Klerk (KGD) problems due to its relative simplicity. [ABSTRACT FROM AUTHOR]

Contemporary cardiac imaging allows for detailed non-invasive assessment of structural alterations of the atria implicated in the initiation and maintenance of atrial arrhythmias. Left atrial wall thickness (LAWT) may be an important parameter reflecting atrial remodelling as it is thought that LAWT increases in the presence of AF and may further promote its perpetuation. Tissue thickness may also be important to success and safety of AF ablation given the importance of creating durable, transmural lesions without causing collateral damage. Different calculation methods for quantifying local LAWT exist. An understanding of quantitative differences between estimation methods is relevant if LAWT-guided ablation strategies are pursued. We aimed to compare the results of LAWT measurement within the same cohort using a semi-automated eikonal-equation based workflow against a proprietary automated nearest-point LAWT estimation approach. In a prospective case-control study, patients with AF and controls without AF but with clinical indication for cardiac computed tomographic angiography (CCTA) were recruited. Prospective, ECG-gated CCTAs were acquired and the best diastolic sequence identified. 42 AF patients (35.7% female, mean age 64.6 +/- 10.2, CHA2DS2-VASc 2.48 +/- 1.5, 69.0% paroxysmal AF, 31% persistent AF, mean LVEF 57.9 +/- 10.5%) and 37 controls (35.1% female, mean age 56.6 +/- 7.2, CHA2DS2-VASc 1.54 +/- 1, mean LVEF 60.4 +/- 4.9%) were recruited. A total of 76 contrast enhanced CTs (37 Control, 39 AF) were acquired. Mean wall thickness between the two methods showed moderate, statistically significant correlation (r 0.499 p < 0.001). Mean LAWT was higher using ADAS LAWT estimation in all groups (total, AF and control) with a mean difference of 0.39mm (p 0.001). Both methods identified a statistically significant increase in LAWT in AF patients when compared to controls. There was no significant difference in standard derivation between estimation methods. In conclusion, fully automated nearest point LAWT estimation and semi-automated eikonal based LAWT estimation method yield a moderate, statistically significant correlation with mean difference in the sub-millimetre range. Even though the latter is statistically significant, for clinical ablation procedures these differences are likely to be of negligible clinical relevance. Available local expertise and software for LAWT estimation may determine the preferred employed approach. [ABSTRACT FROM AUTHOR]

We develop a fast method for approximating the solution to the generalized eikonal equation in a moving medium. Our approach consists of the following two steps. First, we convert the generalized eikonal equation in a moving medium into a Hamilton–Jacobi–Bellman equation of anisotropic eikonal type for an anisotropic minimum-time control problem. Second, we modify the Neighbor–Gradient Single-pass method (NGSPM developed by Ho et al.), so that it not only suits the converted Hamilton–Jacobi–Bellman equation but also can be faster than original NGSPM. In the case of that Mach number is not comparable than 1, we compare our method and Characteristic Fast Marching Method (CFMM developed by Dahiya) via several numerical examples to show that our method is faster and more accurate than CFMM. We also compare the numerical solutions obtained from our method with the solutions obtained using the ray theory to show that our method captures the viscosity solution accurately even when the Mach number is comparable to 1. We also apply our method to 3D example to show that our method captures the viscosity solution accurately in 3D cases. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the nonlocal problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and nonlocal problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the nonlocal problem and that of the local one, both in continuous time and forward Euler time discretization. We then turn to studying continuum limits of nonlocal problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows then, almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large. [ABSTRACT FROM AUTHOR]

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes. [ABSTRACT FROM AUTHOR]

We study the possibilities of using the eikonal equation to construct theoretical models of objects that become invisible under certain illumination conditions. The characteristics of the cloaking coating, which performs its functions for a given direction of wave incidence on the cloaked volume of space, are calculated. A method for constructing models of cylindrical objects that are invisible at arbitrary azimuthal and given zenith angles of illumination is proposed. It is shown that by employing the eikonal equation, it is possible to design invisible objects from materials with refractive indices of different signs. [ABSTRACT FROM AUTHOR]

We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant–Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered. [ABSTRACT FROM AUTHOR]

Accurate temporal resolution of the eikonal equation forms the cornerstone of seismological studies, including microseismic source localization, and travel-time tomography. Physics-informed neural networks (PINNs) have gained significant attention as an efficient approximation technique for numerical computations. In this study, we put forth a novel model named Eiko-PIGCNet, a graph convolutional neural network that incorporates physical constraints. We demonstrate the effectiveness of our proposed model in solving the 3D eikonal equation for travel-time estimation. In our approach, the discretized grid points are converted into a graph data structure, where every grid point is regarded as a node, and the neighboring nodes are interconnected via edges. The node characteristics are defined by incorporating the velocity and spatial coordinates of the respective grid points. Ultimately, the efficacy of the Eiko-PIGCNet and PINNs is evaluated and compared under various velocity models. The results reveal that Eiko-PIGCNet outshines PINNs in terms of solution accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

Surface wave tomography is a widely used geophysical method to measuring seismic velocity and anisotropy in the crust and upper mantle. This paper briefly reviewed the history of surface wave tomography, and summarized the principles and advantages of multiple surface wave tomography methods (i.e., two-station method, two-plane-wave method, Eikonal and Helmholtz surface wave tomography, ambient noise tomography, and direct surface wave tomography). The theory and application of the two-station method are straightforward, but this method requires the earthquake and seismic stations to be located in the same great circle. This restriction results in a low lateral resolution phase velocity map for regions where stations are sparse and deployment times short. The two-plane-wave method can overcome the effect of scattering and multipathing of seismic wave propagation on phase velocity dispersions, but this method requires high-quality surface wave waveforms and is usually suitable for regional seismic networks. The Eikonal and Helmholtz surface wave tomography can directly compute the phase velocities and azimuthal anisotropy straightforwardly without any processes of forward modeling and inversion, but this method is limitedly suitable for high-density and orderly seismic arrays. In comparison to the Eikonal surface wave tomography method, the Helmholtz surface wave tomography approach analyzes both the phase and amplitude of the waveform and can produce better results. Ambient noise tomography can obtain high-resolution crustal structure without seismic events. However, this method is difficult to obtain long-period phase velocities, resulting in a lack of constraints on the mantle and lithosphere structure. Direct surface wave tomography can obtain the shear-wave velocity and azimuthal anisotropy from dispersive curves without the inversion process of phase velocity maps. We conducted detailed comparisons of the phase velocity maps at the short-intermediate periods (20～40 s) previously obtained from these surface wave tomography methods in the central-northern, northeastern and southern Tibetan Plateau. The comparisons demonstrated that the Rayleigh and Love wave phase velocity maps from different tomography methods show highly consistency in velocity variation patterns at same periods. The results show that the plateau's interior by characterized with low-velocities in phase velocity maps at intermediate and long periods, whereas relatively high velocities dominate adjacent regions, such as the Qaidam Basin and Sichuan Basin. These features indicate that the Tibetan Plateau's mid-lower crust and upper mantle tend to be weaker and easily deform during the northward movement of the Indian plateau and the barrier of the strong blocks (e.g., the Qaidam Basin and the Alashan Platform). The short periods phase velocity maps (i.e., 20～40 s) reveal that the mid-lower crustal flow of the Tibetan Plateau escapes southward surrounding the weakening zones (i.e., the Red River Fault and the Xianshui River Fault) due to the barrier of the strong Chuandian tectonic block. The Qilian Orogen also shows low velocities in the phase velocity maps at the short-intermediate periods, which is likely due to thermal anomaly with localized mantle upwelling. It is noteworthy that the Rayleigh-wave and Love-wave phase velocities at different period ranges (e.g., 4～200 s) can be obtained via incorporating earthquake surface wave tomography and ambient noise tomography. Those phase velocities can be used as input to inverting three-dimensional shear wave velocities and radial anisotropy models simultaneously. This paper envisions that integrating higher-mode earthquake surface wave tomography, adjoint tomography, and joint inversion could obtain higher resolution and more reliable crustal and upper mantle structures.

Abstract Many natural landscapes maintain steep planar hillslopes bounded at a typical angle, beyond which shallow landslides or slope failures remove the excess sediment volume by mass wasting. Here we show that the celebrated eikonal equation, derived from a landscape evolution model in conditions of negligible soil diffusion and fluvial erosion, accurately portrays the organization of these topographies. Referred to as “eikonal landscapes,” such solutions feature constant‐slope hillslopes originating from downstream boundary conditions and culminating in sharp upstream ridges. We demonstrate that the eikonal landscapes reproduce well a variety of natural landforms, including small islands, a volcano, and an extended mountain ridge. The boundary condition for the eikonal representation is specified through the natural landscape's slope‐area relation. Going beyond merely representing landscape statistical features, the present results provide a first‐of‐kind direct match of mathematical and natural landscapes.

We study the existence of target patterns in oscillatory media with weak local coupling and in the presence of an impurity, or defect. We model these systems using a viscous eikonal equation posed on the plane and represent the defect as a perturbation. In contrast to previous results, we consider large defects, which we describe using a function with slow algebraic decay, i.e., g ∼ O (1 / | x | m) for m ∈ (1 , 2 ] . We prove that these defects are able to generate target patterns and that, just as in the case of strongly localized impurities, their frequency is small beyond all orders of the small parameter describing their strength. Our analysis consists of finding two approximations to target pattern solutions, one which is valid at intermediate scales and a second one which is valid in the far field. This is done using weighted Sobolev spaces, which allow us to recover Fredholm properties of the relevant linear operators, as well as the implicit function theorem, which is then used to prove existence. By matching the intermediate and far-field approximations, we then determine the frequency of the pattern that is selected by the system. [ABSTRACT FROM AUTHOR]

Tong has proposed an adjoint-state traveltime tomography method to determine velocity heterogeneity and azimuthal anisotropy. This method, however, ignores the Earth's curvature when deriving the eikonal equation for azimuthally anisotropic media. Thus, further coordinate transformation or approximation is required to ensure the accuracy of traveltime prediction in large-scale tomography. To address this problem, we derive the eikonal equation for azimuthally anisotropic media in spherical coordinates, which naturally considers the Earth's curvature. Another key ingredient is the forward modelling algorithm, whose accuracy and efficiency dominate the numerical error and computational cost of the inversion. In this study, we apply a modified fast sweeping method to solve the eikonal equation in spherical coordinates. Two approaches, including the third-order weighted essentially non-oscillatory approximation and multiplicative factorization technique, are applied to improve the accuracy. According to the numerical experiments, this new eikonal solver achieves a second-order accuracy and is about two orders of magnitude more accurate than the commonly used first-order fast sweeping method with similar runtime. Taking advantage of the two improvements, we develop a novel eikonal equation-based adjoint-state traveltime tomography method for azimuthally anisotropic media in spherical coordinates. This method is applicable for large-scale tomography, and its performance is verified by a synthetic checkerboard test and a practical seismic tomographic inversion in central California near Parkfield. [ABSTRACT FROM AUTHOR]