Eikonal Equation Research Articles

A neural network based global traveltime function (GlobeNN)

Global traveltime modeling is an essential component of modern seismological studies with a whole gamut of applications ranging from earthquake source localization to seismic velocity inversion. Emerging acquisition technologies like distributed acoustic sensing (DAS) promise a new era of seismological discovery by allowing a high-density of seismic observations. Conventional traveltime computation algorithms are unable to handle virtually millions of receivers made available by DAS arrays. Therefore, we develop GlobeNN—a neural network based traveltime function that can provide seismic traveltimes obtained from the cached realistic 3-D Earth model. We train a neural network to estimate the traveltime between any two points in the global mantle Earth model by imposing the validity of the eikonal equation through the loss function. The traveltime gradients in the loss function are computed efficiently using automatic differentiation, while the P-wave velocity is obtained from the vertically polarized P-wave velocity of the GLAD-M25 model. The network is trained using a random selection of source and receiver pairs from within the computational domain. Once trained, the neural network produces traveltimes rapidly at the global scale through a single evaluation of the network. As a byproduct of the training process, we obtain a neural network that learns the underlying velocity model and, therefore, can be used as an efficient storage mechanism for the huge 3-D Earth velocity model. These exciting features make our proposed neural network based global traveltime computation method an indispensable tool for the next generation of seismological advances.

Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen–Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.

A Pedestrian Evacuation Model with Leaders during the Smoke Dispersion Based on a Social Force Model

The pedestrian evacuation process during the propagation of smoke in case with and without guider is investigated. The effects of leaders on the evacuation are considered. The model is based on the social force model for pedestrians’ motions. The advection–diffusion equation is applied for the dispersion of smoke. The movement direction of a guider is guided by the solution of the Eikonal equation. It relies on the desired speed and the smoke density. A pedestrian who is not a guider follows the rule termed ‘flow with the stream’ and ‘following the wall’. We perform different numerical experiments in a room with one and two exits. The results show that the guiders effect on the evacuation time when they are large number of individuals in simulation. It can help to increase the number of evacuees. With small number of individuals in the experiment, the effect of guiders on evacuation time is not obvious. Further, simulation results are found that the domain with two exits provides higher number of outside pedestrians than the domain with a single door. Longer evacuation time period can increase the number of evacuees. The visibility range of a pedestrian is reduced when an additional smoke source is added to the system. It decreases the number of evacuees. The results of the proposed model are discussed and compared with the existing models.

Tug of War games and PDEs on graphs with applications in image and high dimensional data processing

The aim of this note is to revisit the connections between some stochastic games, namely Tug-of-War games, and a class of nonlocal PDEs on graphs. We consider a general formulation of Tug-of-War games which is shown to be related to many classical PDEs in the continuous setting. We transcribe these equations on graphs using ad hoc differential operators and we show that it covers several nonlocal PDEs on graphs such as infty -Laplacian, game p-Laplacian and the eikonal equation. This unifying mathematical framework allows us to easily design simple algorithms to solve several inverse problems in imaging and data science, with a particular focus on cultural heritage and medical imaging.

Numerical geometric acoustics: An eikonal-based approach for modeling sound propagation in 3D environments

We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved recursively to generate boundary conditions for a tree of eikonal/transport equation pairs, describing the phase and amplitude of a geometric optic wave propagating in a complicated domain, including reflection and diffraction. Edge diffraction is modeled using the uniform theory of diffraction. For simplicity, we limit our attention to domains with piecewise linear boundaries and a constant speed of sound. The domain is discretized into a conforming tetrahedron mesh. For the eikonal equation, we extend the jet marching method to tetrahedron meshes. Hermite interpolation enables second order accuracy for the eikonal and its gradient and first order accuracy for its Hessian, computed using cell averaging. To march the eikonal on an unstructured mesh, we introduce a new method of rejecting unphysical updates by considering Lagrange multipliers and local visibility. To handle accuracy degradation near caustics, we introduce several fast Lagrangian initialization algorithms. We store the dynamic programming plan uncovered by the marcher in order to propagate auxiliary quantities along characteristics. We introduce an approximate origin function which is computed using the dynamic programming plan, and whose 1/2-level set approximates the geometric optic shadow and reflection boundaries. We also use it to propagate geometric spreading factors and unit tangent vector fields needed to compute the amplitude and evaluate the high-frequency edge diffraction coefficient. We conduct numerical tests on a semi-infinite planar wedge to evaluate the accuracy of our method. We also show an example with a more realistic building model with challenging architectural features. Finally, we demonstrate a simple approach to extending the method to handle nonconstant speeds of sound by modifying the semi-Lagrangian updates to account for a varying speed.

Accessibility constraints in structural optimization via distance functions

This paper is concerned with a geometric constraint, the so-called accessibility constraint, for shape and topology optimization of structures built by additive manufacturing. The motivation comes from the use of sacrificial supports to maintain a structure, submitted to intense thermal residual stresses during its building process. Once the building stage is finished, the supports are no longer useful and should be removed. However, such a removal can be very difficult or even impossible if the supports are hidden deep inside the complex geometry of the structure. A rule of thumb for evaluating the ease of support removal is to ask that the contact zone between the structure and its supports can be accessed from the exterior by a straight line which does not cross another part of the structure. It mimicks the possibility to cut the head of the supports attached to the structure with some cutting tool. The present work gives a new mathematical way to evaluate such an accessibility constraint, which is based on distance functions, solutions of eikonal equations. The main advantage is the possibility of computing shape derivatives of such a criterion with respect to both the structure and the support. We numerically demonstrate in 2D and 3D that, in the context of the level-set method for topology optimization, such an approach allows us to optimize simultaneously the mechanical performance of a structure and the accessibility of its building supports, guaranteeing its manufacturability.

An Eikonal Equation with Vanishing Lagrangian Arising in Global Optimization

We show a connection between global unconstrained optimization of a continuous function f and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution v of the critical Hamilton–Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then v is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of v. Such trajectories are proved to converge to the set of minima of f, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.

A topography-dependent eikonal solver for accurate and efficient computation of traveltimes and their derivatives in 3D heterogeneous media

Accurate and efficient computation of traveltimes and their spatial derivatives for 3D heterogeneous media is essential in many areas of seismology, such as traveltime/slope tomography and quantitative migration. A popular method of obtaining this information is to solve the eikonal equation using a finite-difference (FD) scheme. However, the complex surface of the earth causes numerical complications when applying FD methods based on rectangular grids. To overcome the challenge of taking into account topographic effects, we make use of the 3D topography-dependent eikonal equation (TDEE) which is a 3D surface-flattening scheme that uses curvilinear grids to transform irregular surfaces in the physical domain into regular surfaces in the computational domain. Unfortunately, solving the TDEE in three dimensions starting from the initial point-source condition needs to get around the point-source singularity. Therefore, we formulate the 3D factored TDEE (FTDEE) according to a factorization principle. Furthermore, to numerically solve the 3D original TDEE, as well as the FTDEE, we apply a lock sweeping method, which avoids the problem of repetitive computation when using the fast sweeping method. Several numerical examples demonstrate the efficiency of the new lock scheme in terms of computation time and accuracy of the 3D FTDEE solver for the calculation of traveltimes and their derivatives. This scheme provides a good choice for the calculation of traveltimes and their spatial derivatives using models with irregular surfaces, which means greater potential for seismic data processing under conditions of complex topography.

A deep learning workflow for weak reflection extraction in pitch-catch measurements in the cased hole

As a key technology to evaluate cement bonds in the cased hole, an advanced ultrasonic logging tool combines pulse-echo and pitch-catch measurements in which the latter one provides reflections from the cement-formation interface (called third-interface-echo [TIE]) to evaluate the bond condition and determine casing eccentering as well as cement velocity. However, the TIE would be weak and not easy to pick due to the eccentered tool and casing and it would overlap with the strong multiple reflections between the casing inner surface and the transducer-housing tool. We have developed a deep learning workflow to extract weak TIE from noisy data and to preserve its amplitude at the same time. First, we use synthetic waveforms from thousands of finite-difference simulations as initial training data sets to train a deep learning network, which is modified from a network in speech separation. Then, the trained model is used to predict the field data through an active-learning strategy. The improved network is further used to extract the weak TIEs, which are not easy to pick in the initial deep learning model. Finally, the TIE waves image is converted to a pseudovelocity image to obtain the minimum traveltime path by solving the eikonal equation. The shortest traveltime path is used as the TIE arrival time. In addition, a 3D visualization is used to display the borehole shape from the picked arrival time. The applications in synthetic data and data set from a calibration well illustrate a good performance of our workflow in which the weakest TIE extracted from the network can reach 50 dB compared to the maximum amplitude in the full waveform. The picked arrival times can be used to reconstruct a borehole shape.

A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources

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 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 cell centers to each other. The overall CPU time and memory requirement scale as for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an 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.

Solving Traveltime Tomography with Deep Learning

This paper introduces a neural network approach for solving two-dimensional traveltime tomography (TT) problems based on the eikonal equation. The mathematical problem of TT is to recover the slowness field of a medium based on the boundary measurement of the traveltimes of waves going through the medium. This inverse map is high-dimensional and nonlinear. For the circular tomography geometry, a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction. Motivated by this and filtered back-projection, we propose an effective neural network architecture for the inverse map using the recently proposed BCR-Net, with weights learned from training datasets. Numerical results demonstrate the efficiency of the proposed neural networks.

Adjoint-state traveltime tomography for azimuthally anisotropic media in spherical coordinates

SUMMARYTong 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.

Black and gray box learning of amplitude equations: Application to phase field systems.

We present a data-driven approach to learning surrogate models for amplitude equationsand illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equationsdescribing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equationsfor the dynamics of the phase field interface (a higher-order eikonal equationand its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equationsthat accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equationsoutperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.

An aberration correction approach for single and dual aperture ultrasound imaging of the abdomen

Abdominal ultrasound image quality is hampered by phase aberration, that is mainly caused by the large speed-of-sound (SoS) differences between fat and muscle tissue in the abdominal wall. The mismatch between the assumed and actual SoS distribution introduces general blurring of the ultrasound images, and acoustic refraction can lead to geometric distortion of the imaged features. Large aperture imaging or dual-transducer imaging can improve abdominal imaging at deep locations by providing increased contrast and resolution. However, aberration effects for large aperture imaging can be even more severe, which limits its full potential. In this study, a model-based aberration correction method for arbitrary acquisition schemes is introduced for delay-and-sum (DAS) beamforming and its performance was analyzed for both single- and dual-transducer ultrasound imaging. The method employs aberration corrected wavefront arrival times, using manually assigned local SoS values. Two wavefront models were compared. The first model is based on a straight ray approximation, and the second model on the Eikonal equation, which is solved by a multi-stencils fast marching method. Their accuracy for abdominal imaging was evaluated in acoustic simulations and phantom experiments involving tissue-mimicking and porcine material with large SoS contrast (∼100 m/s). The lateral resolution was improved by up to 90% in simulations and up to 65% in experiments compared to standard DAS, in which the use of Eikonal beamforming generally outperformed straight ray beamforming. Moreover, geometric distortions were mitigated in multi-aperture imaging, leading to a reduction in position error of around 80%. A study on the sensitivity of the aberration correction to shape and SoS of aberrating layers was performed, showing that even with imperfect segmentations or SoS values, aberration correction still outperforms standard DAS.

Eikonal equation-based earthquake location with irregular surfaces

SUMMARYEarthquake location is a basic seismological problem and has a key role in many quantitative seismic analyses. The significant anomaly of traveltimes observed in mountainous areas with irregular surfaces has made earthquake location a challenge. To cope with this problem, we develop an effective eikonal equation-based earthquake location method based on unstructured mesh for 2-D/3-D isotropic and anisotropic media with irregular surfaces. First, the location misfit function is established by the reciprocity principle. Then, we use a global search algorithm to find the optimal origin times and hypocentres. Afterward, we apply eikonal equation-based master-event relocation method to relocate earthquakes on lateral boundaries of the region where the location results may be biased. To accurately compute the traveltime in 3-D vertically transversely isotropic and tilted transversely isotropic models with irregular surfaces or interfaces, we also propose a 3-D iterative fast sweeping method for eikonal equation on the unstructured tetrahedral mesh. Finally, we verify the proposed method by performing numerical experiments in 2-D/3-D irregular isotropic and anisotropic models. The numerical tests indicate that the proposed eikonal equation-based earthquake location method provides an effective way to find accurate hypocentre and origin time in 2-D/3-D isotropic and anisotropic media with irregular surfaces, even for inhomogeneous complex structures. In addition, the eikonal equation-based master-event location method has also yielded promising relocation results.

The inhomogeneous p-Laplacian equation with Neumann boundary conditions in the limit pto infty

We investigate the limiting behavior of solutions to the inhomogeneous p-Laplacian equation -Delta _{p} u = mu _{p} subject to Neumann boundary conditions. For right-hand sides, which are arbitrary signed measures, we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-1 distance. In the regular case with continuous right-hand sides, we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.

Limits and consistency of nonlocal and graph approximations to the Eikonal equation

Abstract 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.

Bayesian Physics-Informed Neural Networks for the Subsurface Tomography Based on the Eikonal Equation

The high cost of acquiring a sufficient amount of seismic data for training has limited the use of machine learning in seismic tomography. In addition, the inversion uncertainty due to the noisy data and data scarcity is less discussed in conventional seismic tomography literature. To mitigate the uncertainty effects and quantify their impacts in the prediction, the so-called Bayesian Physics-Informed Neural Networks (BPINNs) based on the eikonal equation are adopted to infer the velocity field and reconstruct the travel-time field. In BPINNs, two inference algorithms including Stein Variational Gradient Descent (SVGD) and Gaussian variational inference (VI) are investigated for the inference task. The numerical results of several benchmark problems demonstrate that the velocity field can be estimated accurately and the travel-time can be well approximated with reasonable uncertainty estimates by BPINNs. This suggests that the inferred velocity model provided by BPINNs may serve as a valid initial model for seismic inversion and migration.

ON THE NONCHARACTERISTICITY OF A CAUCHY SURFACE AND APPLICATIONS

In mathematical physics, Cauchy surfaces play a crucial role in defining initial data for the evolution of physical systems. The characteristic nature of Cauchy surface means that information about the future evolution of the system can be obtained from the data on the surface itself. However, not all Cauchy surfaces are characteristic. This article aims to investigate the noncharacteristic nature of Cauchy surfaces and their applications in various physical scenarios. We demonstrate that the noncharacteristicity of a Cauchy surface has important implications for the behavior of physical systems, including the existence of singularities and the formulation of initial value problems. We explore the applications of noncharacteristic Cauchy surfaces in classical and quantum field theory, general relativity, and quantum gravity. Our analysis provides a deeper understanding of the mathematical and physical properties of Cauchy surfaces and their significance in the study of fundamental physical phenomena. In this work we show that under the noncharacteristic condition one can prove a local existence result for the Cauchy problem similar to the semi-linear and quasilinear cases. For this, we consider the Eikonal equation and Hamilton-Jacobi equations. Therefore, we solve simultaneously for the characteristics and the solution value, resulting in curves that live in the solution graph.

Riemannian geometric modeling of underwater acoustic ray propagation—Basic theory

Underwater sound propagation models are generally established from the extrinsic perspective, that is, embedding acoustic channels in Euclidean space with a fixed coordinate system. Riemannian geometry is intrinsic for curved space, which can describe the essential properties of background manifolds. The underwater acoustic Gaussian beam is originally adopted from seismology. Till now it has been the most important method used in acoustic ray based modeling and applications. Owing to the advantages of Gaussian beam method over the traditional ray counterpart, it is the mainstream technology of ray propagation computational software such as the famous Bellhop. With the assumption of Euclidean space, it is hard to grasp the naturally curved characteristics of the Gaussian beam. In this work, we propose the Riemannian geometry theory of underwater acoustic ray propagation, and obtain the following results. 1) The Riemannian geometric intrinsic forms of the eikonal equation, paraxial ray equation and the Gaussian beam under radially symmetric acoustic propagation environments are established, which provide a Riemannian geometric interpretation of the Gaussian beam. In fact, the underwater acoustic eikonal equation is equivalent to the geodesic equation in Riemannian manifolds, and the intrinsic geometric spreading of the Gaussian beam corresponds to the lateral deviation of geodesic curve along the Jacobian field. 2) Some geometric and topological properties of acoustic ray about conjugate points and section curvature are acquired by the Jacobi field theory, indicating that the convergence of ray beam corresponds to the intersection of geodesics at the conjugate point with positive section curvature. 3) The specific modeling method under horizontal stratified and distance-related environment is presented by using the above theory. And we point out that the method proposed here is also applicable to other radially symmetric acoustic propagation environments. 4) Simulations and comparative analyses of three typical underwater acoustic propagation examples, confirm the feasibility of the Riemannian geometric model for underwater acoustic propagation, and show that the Riemannian geometric model has exact mathematical physics meaning over the Euclidean space method adopted by the Bellhop model. The basic theory given in this paper can be extended to the curved surface, three-dimensional and other complex propagation cases. And especially it lays a theoretical foundation for the further research of long-range acoustic propagation considering curvature of the earth.