Eikonal Equation Research Articles

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge–Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 4$$\\end{document}. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

Gravitational spinoptics in a curved space-time

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

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

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

Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation

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.

An existence result for accretive growth in elastic solids

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.

Physical characterization of mine goafs using high-resolution seismic cross-hole tomography via an adjoint method

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 [Formula: see text] 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.

Stable neural network-based traveltime tomography using hard-constrained measurements

Traveltime tomography, or travel time inversion, has been one of the primary seismological tools for decades and has been used to understand Earth's properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the travel time field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed those challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multi-term nature of the loss function. To improve this, we propose a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We employ the theory of functional connections to the traveltime tomography problem, which allows for the use of a single loss term for training the PINN model. Its efficiency, stability, and flexibility in tackling various cases, including topography-dependent and 3D models, are attested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography.

On the quantum Guerra–Morato action functional

Given a smooth potential W:Tn→R on the torus, the Quantum Guerra–Morato action functional is given by I(ψ)=∫(DvDv*2(x)−W(x))a(x)2dx, where ψ is described by ψ=aeiuℏ, u=v+v*2, a=ev*−v2ℏ, 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 ′ =ddτ. 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.

Fast approximation of fiber reinforced injection molding processes using eikonal equations and machine learning

Injection molding is a popular production process for short fiber reinforced components. The mechanical properties of such components depend on process-induced fiber orientations which are commonly predicted via numerical simulations. However, high computational costs prevent process simulations from being used in iterative procedures, such as topology optimization or finding optimal injection locations. We propose a fast approximation method that extracts nodal features and train a regression model to predict fill states, cooling times, volumetric shrinkage, and fiber orientations. The features are determined by solving eikonal equations with a fast iterative method and computing spatial moments to characterize node-adjacent material distributions. Subsequently, we use these features to train feed forward neural networks and gradient boosted regression trees with simulation data of a large dataset of geometries. This approach is significantly faster than conventional methods, providing 20x speed-up for single simulations and more than 200x speed-up in gate location optimization. It generalizes to arbitrary geometries and injection locations.

General Burnback Analysis for Anisotropic and Heterogeneous Solid Propellants

For several decades, researchers have successfully computed surface burnback to perform complex analyses and simulations related to the combustion of solid propellants. Many of these methods are heuristic or empirical, lacking a rigorous foundation, whereas others introduce numerical issues not related with the involved physics. This paper establishes a common theoretical and numerical core that bases the general study of burnback problems even in the case of designing nonconventional propellants. It starts with Piobert’s statement as a first principle, derives that the eikonal formulation holds for propellants with heterogeneous and anisotropic recession rates, and gives a general mathematical structure for both situations. Then, several configurations found by rocket designers are numerically solved in order to show the efficiency and the accuracy of the theory. The main conclusion is that a direct numerical integration of the eikonal equation using a time marching method is enough in terms of computational efficiency and accuracy to track the combustion surface of any anisotropic and heterogeneous solid propellant.

Smoothness: The key factor in well-log information-assisted PINNtomo

Seismic traveltime tomography is popularly used to provide a smooth background model for subsequent depth migration or full waveform inversion (FWI). The core of the traveltime tomography process requires an eikonal solver for seismic traveltime calculation. Most of the conventional eikonal solvers, suffer from first-order convergence errors and difficulties in dealing with irregular topography, which will lead to inaccurate seismic tomography results. To solve these issues, physics-informed neural networks (PINNs) have been developed and demonstrated remarkable success in simultaneous traveltime calculation and velocity construction, which is referred to as PINNtomo. It combines the physics constraint (eikonal equation) and data constraint (traveltime fitting) to form the loss function of the networks. Though PINNtomo can manage to recover a smooth background velocity, it will fail in the inversion for a complicated high-resolution velocity model. We take advantage of the well-log velocity as a prior information to provide an additional model constraint in the loss function to improve the performance of PINNtomo. We found that the well-log information will provide high-resolution components in the inverted velocity model. However, it will destroy the lateral structure of the true model. We propose to use a smoothed version of the well-log velocity as the prior model constraint in PINNtomo. As a result, the accuracy of the inverted background velocity will be improved, which will be beneficial to the subsequent FWI.

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

3D Refraction Travel Time Accuracy Study in High Contrasted Media

Among all the existing methods to solve the eikonal equation, three methods are chosen to verify accuracy, symmetry, reciprocity and error propagation along large offsets of refracted waves in seismic near surface exploration context. Performance is extremely highlighted nowadays and accuracy is being neglected, then an eikonal solver poorly explored in geoscience is used. A classical solver, the Fast Iterative and the modified Fast Sweeping Method are applied in three modeling schemes: a simple two layers model, a large four layers and a complex benchmark model. The three methods compute the first arrival of refracted waves in high contrast media and the results are compared to the analytical solution. A circular geometry is considered in all experiments to explore the method applicability using full azimuth angles. On the first scheme, the errors in travel time are computed among the three methods using different model sample spacing and we discuss accuracy, symmetry and reciprocity of first arrivals. On the second scheme, three circular receivers are placed in different offsets to check errors along refracted wave propagation. Finally, the third scheme, four shots are strategically positioned over the SEG/EAGE Overthrust model in order to compare the full acoustic wavefield with the eikonal solvers and then check the similarities. Although the focus is on methods accuracy, the algorithm run time is also considered and the comparison shows that the modified Fast Sweeping Method is the most accurate. The most computational efficient eikonal solver is the Fast Iterative Method, but its geoscience applicability needs to be cautious, because of its inaccurate results.

A shortest‐path‐aided fast‐sweeping method to improve the accuracy of traveltime calculation in vertically transverse isotropic media

AbstractThe 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.

Applications of Asymptotic Solutions of the Diffusivity Equation to Infinite Acting Pressure Transient Analysis

Summary Understanding how pressure propagates in a reservoir is fundamental to the interpretation of pressure and rate transient measurements at a well. Unconventional reservoirs provide unique technical challenges as the simple geometries and flow regimes [wellbore storage (WBS) and radial, linear, spherical, and boundary-dominated flow] applied in well test analysis are now replaced by nonideal flow patterns due to complex multistage fracture completions, nonplanar fractures, and the interaction of flow with the reservoir heterogeneity. In this paper, we introduce an asymptotic solution technique for the diffusivity equation applied to pressure transient analysis (PTA), in which the 3D depletion geometry is mapped to an equivalent 1D streamtube. This allows the potentially complex pressure depletion geometry within the reservoir to be treated as the primary unknown in an interpretation, compared with the usual method of interpretation in which the depletion geometry is assumed and parameters of the formation and well are the unknown properties. The construction is based upon the solution to the Eikonal equation, derived from the diffusivity equation in heterogeneous reservoirs. We develop a Green’s function that provides analytic solutions to the pressure transient equations for which the geometry of the flow pattern is abstracted from the transient solution. The analytic formulation provides an explicit solution for many well test pressure transient characteristics such as the well test semi-log pressure derivative (WTD), the depth of investigation (DOI), and the stabilized zone (SZ) (or dynamic drainage area), with new definitions for the limit of detectability (LOD), the transient drainage volume, and the pseudosteady-state (PSS) limit. Generalizations of the Green’s function approach to bounded reservoirs are possible (Wang et al. 2017) but are beyond the scope of the current study. We validate our approach against well-known PTA solutions solved using the Laplace transform, including pressure transients with WBS and skin. Our study concludes with a discussion of applications to unconventional reservoir performance analysis for which reference solutions do not otherwise exist.

A hierarchy of the optimal velocity model with optimal path for pedestrian evacuation: From microscopic to macroscopic models

We study the evacuation process of pedestrians by adopting an optimal velocity model. The Eikonal equation is coupled to the optimal velocity model for optimal path to guide the movement directions of pedestrians. It depends on pedestrian density. We derive the corresponding mean field equation, hydrodynamic and scalar models from the scaled microscopic optimal velocity model. Several numerical experiments are performed in a corridor with two exits. We show and compare results on the microscopic as well as on the hydrodynamic and scalar models. Results from microscopic model are closed to the hydrodynamic and scalar models when a large number of particles are considered in microscopic simulation. The computation time increases as number of particles in microscopic simulation increases. The computation times of the hydrodynamic and scalar models are less than the computation time of the microscopic model with large number of particles. Hence it is beneficial to apply the hydrodynamic and scalar models over the microscopic model when a large number of particles in microscopic system are considered.

High Order Absolutely Convergent Fast Sweeping Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations

High Order Absolutely Convergent Fast Sweeping Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations

Converted wave tomography based on inverse level set and adjoint formulation

SUMMARY 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.

Conformal diagrams for stationary and dynamical strong-field hyperboloidal slices

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.

DG-based joint transmission-reflection traveltime tomography and its application of borehole seismic

Abstract The limitations of the coverage range and density of transmission wave often result in less-than-ideal results in traveltime tomography. In 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 (VSP, RVSP and crosswell seismic) data to obtain near-wellbore structures. In the forward part, we solve the factored equation by the discontinuous Galerkin (DG) method to calculate the transmission/reflection traveltime. Due 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. And 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.