Dynamic Ray Tracing Research Articles

Asymptotic Ray Method for the Double Square Root Equation

The parabolic wave equation describes wave propagation in a preferable direction, which is usually horizontal in underwater acoustics and vertical in seismic applications. For dense receiver arrays (receiver spacing is less than signal wavelength), this equation can be used for propagating the recorded wavefield back into the medium for imaging sources and scattering objects. Similarly, for multiple source and receiver array acquisition, typical for seismic exploration and potentially beneficial for ocean acoustics, one can model data in one run using an extension of the parabolic equation—the pseudo-differential Double Square Root (DSR) equation. This extended equation allows for the modeling and imaging of multi-source data but operates in higher-dimensional space (source, receiver coordinates, and time), which makes its numerical computation time-consuming. In this paper, we apply a faster ray method for solving the DSR equation. We develop algorithms for both kinematic and dynamic ray tracing applicable to either data modeling or true-amplitude recovery. Our results can be used per se or as a basis for the future development of more elaborated asymptotic techniques that provide accurate and computationally feasible results.

Two dimensional elastic dynamically focused beam migration in transversely isotropic (TI) media

The anisotropic ray tracing equations of elastic waves are derived for calculating the central ray's travel time, trajectory, and parameters P and Q. Based on the anisotropic ray tracing algorithm, we proposed a transversely isotropic elastic dynamically focused beam migration for two dimensional media. Based on the two dimensional Kirchhoff-Helmholtz integral of elastic waves, the elastic imaging weight coefficients are used to reduce the crosstalk noise. Then, based on the previous research, we introduced a sign function to eliminate the effect of polarity reversal in qPqSV wave migration. Through model tests, it can be found that the proposed method can process the multicomponent seismic data and image the complex structures for anisotropic media. The optimized dynamic ray tracing algorithm is superior to the conventional one in computational efficiency. Furthermore, the proposed method eliminates the limitations of the Gaussian beam imaging. This method not only does not affect the accuracy of shallow imaging, but also enhances the energy focusing and the amplitude of deep layers. The model tests have shown the accuracy and effectiveness of our proposed method.

Point cloud-based environment reconstruction and ray tracing simulations for railway tunnel channels

Radio propagation environment plays a critical role in the performance of wireless communication systems, and understanding channel characteristics is vital for ensuring reliable communication links and optimizing system performance. Ray tracing is an effective method to investigate propagation characteristics in a complex environment, and how to quickly and accurately obtain environmental information needs to be solved. This paper presents dynamic environment reconstruction and ray tracing simulation in railway tunnel environment based on Simultaneous Localization and Mapping (SLAM) algorithm and Poisson reconstruction algorithm. Accurate channel parameters are obtained and analyzed based on ray tracing simulation. Both straight and curved tunnels are considered and investigated, and the results show the channel characteristics in complex railway tunnel environments.

On Dynamic Ray Tracing and Anticipative Channel Prediction for Dynamic Environments

Ray tracing algorithms, that can simulate multipath radio propagation in presence of geometric obstacles such as buildings, objects or vehicles, are becoming quite popular, due to the increasing availability of digital environment databases and high-performance computation platforms, such as multi-core computers and cloud computing services. When objects or vehicles are moving, which is the case of industrial or vehicular environments, multiple successive representations of the environment ("snapshots") and multiple ray tracing runs are often necessary, which require a great human effort and a great deal of computation resources. Recently, the Dynamic Ray Tracing (DRT) approach has been proposed to predict the multipath evolution within a given time lapse on the base of the current multipath geometry, assuming constant speeds and/or accelerations for moving objects, using analytical extrapolation formulas. This is done without re-running a full ray tracing for every "snapshot" of the environment, therefore with a great reduction in both human labour and computation time. When DRT is embedded in a mobile radio system and used in real-time, ahead-of-time (or anticipative) field prediction is possible that opens the way to interesting applications. In the present work, a full-3D DRT algorithm is presented that allows to account for multiple reflections, edge diffraction and diffuse scattering for the general case where moving objects can translate and rotate. For the purpose of validation, the model is first applied to some ideal cases and then to realistic cases where results are compared with conventional ray tracing simulation and measurements available in the literature.

Decomposition and Properties of the Traveltime Sensitivity Kernels in Transversely Isotropic Elastic Media

This study discusses the properties and decompositions of the traveltime sensitivity kernels in elastic isotopic and transversely isotropic media. Analytical solutions in homogeneous isotropic elastic media are derived and help better understand their properties. One key component is the kernel phase function which determines the frequency components of the sensitivity kernels, the geometrical setting of the Fresnel zones, and the resolving ability for velocity inversion. For the purpose of exploration seismology, the sensitivity kernels for finite-frequency traveltime inversion in transversely isotropic elastic media are expressed in terms of Thomsen’s anisotropic parameters. The sensitivity kernels in anisotropic media can be decomposed into the summation of the isotropic terms constructed by the elastic properties in the symmetry direction and the correction terms due to the presence of anisotropy. Sensitivity kernels computed from different types of acquisitions contain information for different parameters. We compare the kernels computed using finite-difference with those by dynamic ray tracing. We conclude that the full wave numerical method is able to provide high-resolution kernels, which form the basis for 3D finite-frequency traveltime inversion for velocity and anisotropic parameters.

Virtual Simulation of the Effect of FMCW Laser Fuse Detector's Component Performance Variability on Target Echo Characteristics under Smoke Interference.

The laser transmitter and photoelectric receiver are the core modules of the detector in a laser proximity fuse, whose performance variability can affect the accuracy of target detection and identification. In particular, there is no study on the effect of detector’s component performance variability on frequency-modulated continuous-wave (FMCW) laser fuse under smoke interference. Therefore, based on the principles of particle dynamic collision, ray tracing, and laser detection, this paper builds a virtual simulation model of FMCW laser transmission with the professional particle system of Unity3D, and studies the effect of performance variability of laser fuse detector components on the target characteristics under smoke interference. Simulation results show that the difference in the performance of the fuse detector components causes the amplitude variation and peak migration of the beat signal spectrum, and the change in the visibility of the smoke can also affect the results, which indicates that the factors affecting the signal-to-noise ratio (SNR) of the echo signal are related to the smoke interference and performance variability of the detector. The proposed simulation model is supported by experimental results, which reflect the reliability of the proposed findings. Therefore, this study can be used for the optimization of the parameters in the laser fuse antismoke interference to avoid false alarms.

Two dimensional dynamically focused beam migration in weakly anisotropic media

Anisotropy exists widely in the earth medium. For anisotropic medium, the traditional isotropic migration will lead to the inaccurate images and unfocused energy. Especially for a deep target exploration, ignoring anisotropy will further affect the subsequent seismic data interpretation. As an improved ray method, Gaussian beam migration overcomes the imaging problems of caustics and shadow zones in Kirchhoff migration, and avoids the time-consuming problems of wave equation migrations. However, the imaging quality of the Gaussian beam migration is controlled by the initial beam width at the surface. In this paper, we extend Nowack (2011)’s algorithm to anisotropic media and present an anisotropic dynamically focused beam migration by modifying the propagator of the Gaussian beams. We use anisotropic kinematic and dynamic ray tracing with optimized coefficients to obtain travel times, trajectories and dynamic information. We then use dynamically focused beams for weakly anisotropic media to calculate the Green's function and the Claerbout imaging condition to obtain images. This strategy enables us to improve the imaging quality in the middle and deep layers without reducing the imaging quality in the shallow layers, which will help to overcome the limitation of the initial beam width in Gaussian beam migration. The results for a VTI fault model and the Shengli complex structure model show the accuracy and validity of the proposed method in this paper.

3D Dynamic Ray-tracing Propagation Model with Moving Scatterer Effects

Ray-tracing propagation model (RTPM) has been widely used for predicting channel characteristics, whereas the scenarios considered are generally static. The complexity of RTPM is significantly increased due to the rapidly time-varying scenario resulted from moving scatterers. A three-dimensional (3D) dynamic RTPM considering moving scatterer effects is advanced in this paper. First, a simplified dynamic scenario preprocessing method based on the predefined active region and face transformation is proposed. The random movement of multiple scatterers can be enabled without repeated scenario modeling. Second, an efficient dynamic ray-tracing method based on self-adaptive ray-launching technique is advanced. The computational efficiency of the dynamic RTPM can be significantly improved due to the exclusion of repeated ray-tracing process over time. Finally, the feasibility and accuracy of the RTPM is verified by comparing the simulation results with the measurements performed in an indoor scenario with pedestrians.

Reverse time migration with Gaussian beams using optimized ray tracing systems in transversely isotropic media

ABSTRACTPrestack depth migration is a key technology for imaging complex reservoirs in media with strong lateral velocity variations. Prestack migrations are broadly separated into ray‐based and wave‐equation‐based methods. Because of its efficiency and flexibility, ray‐based Kirchhoff migration is popular in the industry. However, it has difficulties in dealing with the multi‐arrivals, caustics and shadow zones. On the other hand, wave‐equation‐based methods produce images superior to that of the ray‐based methods, but they are expensive numerically, especially methods based on two‐way propagators in imaging large regions. Therefore, reverse time migration algorithms with Gaussian beams have recently been proposed to reduce the cost, as they combine the high computational efficiency of Gaussian beam migration and the high accuracy of reverse time migration. However, this method was based on the assumption that the subsurface is isotropic. As the acquired azimuth and maximum offsets increase, taking into account the influence of anisotropy on seismic migration is becoming more and more crucial. Using anisotropic ray tracing systems in terms of phase velocity, we proposed an anisotropic reverse time migration using the Gaussian beams method. We consider the influence of anisotropy on the propagation direction and calculate the amplitude of Gaussian beams with optimized correlation coefficients in dynamic ray tracing, which simplifies the calculations and improves the applicability of the proposed method. Numerical tests on anisotropic models demonstrate the efficiency and accuracy of the proposed method, which can be used to image complex structures in the presence of anisotropy in the overburden.

Weak-anisotropy approximation of P-wave geometric spreading in horizontally layered anisotropic media of arbitrary symmetry: Tilted transversely isotropic specification

Understanding the role of geometric spreading and estimating its effects on seismic wave propagation play an important role in several techniques used in seismic exploration. The spreading can be estimated through dynamic ray tracing or determined from reflection traveltime derivatives. In the latter case, derivatives of nonhyperbolic moveout approximations are often used. We offer an alternative approach based on the weak-anisotropy approximation. The resulting formula is applicable to P-waves reflected from the bottom of a stack of horizontal layers, in which each layer can be of arbitrary anisotropy. At an arbitrary surface point, the formula depends, in each layer, on the thickness of the layer, on the P-wave reference velocity used for the construction of reference rays, and on nine P-wave weak-anisotropy (WA) parameters specifying the layer anisotropy. Along an arbitrary surface profile, the number of WA parameters reduces to five parameters related to the profile. WA parameters represent an alternative to the elastic moduli and, as such, can be used for the description of any anisotropy. The relative error of the approximate formula for a multilayered structure consisting of layers of anisotropy between 8% and 20% is, at most, 10%. For models including layers of anisotropy stronger than 20%, the relative errors may reach, locally, even 30%. For any offset, relative errors remain under a finite limit, which varies with the anisotropy strength.

Tracking of Interaction Points for Improved Dynamic Ray Tracing

Ray tracing is a powerful tool to obtain deterministic and dynamic descriptions of communication channels. However, performing ray tracing simulations at each discrete time instant is computationally expensive. Instead, a new approach is proposed to extrapolate results obtained from a single ray tracing simulation. It relies on the geometric tracking of interaction points (i.e. reflection or diffraction points), enabling analytical or numerical predictions of the evolution of any ray identified during an initial ray tracing simulation. The performance of this new approach is studied on several canonical vehicle-to-vehicle configurations, as part of a statistical study. Focus is also given to the time horizon during which dynamic ray tracing is possible, related to the lifetime of the main rays. This time horizon can be directly estimated based on the knowledge of the geometry and its evolution. It is found to be the main parameter influencing the accuracy and the computational gain of the presented approach.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in ray-centred coordinates

SUMMARY Dynamic ray tracing is a robust and efficient method for computation of amplitude and phase attributes of the high-frequency Green’s function. A formulation of dynamic ray tracing in Cartesian coordinates was recently extended to higher orders. Extrapolation of traveltime and geometrical spreading was demonstrated to yield significantly higher accuracy—for isotropic as well as anisotropic heterogeneous 3-D models of an elastic medium. This is of value in mapping, modelling and imaging, where kernel operations are based on extrapolation or interpolation of Green’s function attributes to densely sampled 3-D grids. We introduce higher-order dynamic ray tracing in ray-centred coordinates, which has certain advantages: (1) such coordinates fit naturally with wave propagation; (2) they lead to a reduction of the number of ordinary differential equations; (3) the initial conditions are simple and intuitive and (4) numerical errors due to redundancies are less likely to influence the computation of the Green’s function attributes. In a 3-D numerical example, we demonstrate that paraxial extrapolation based on higher-order dynamic ray tracing in ray-centred coordinates yields results highly consistent with those obtained using Cartesian coordinates. Furthermore, in a 2-D example we show that interpolation of dynamic ray tracing quantities along a wavefront can be done with much better consistency in ray-centred coordinates than in Cartesian coordinates. In both examples we measure consistency by means of constraints on the dynamic ray tracing quantities in the 3-D position space and in the 6-D phase space.

Computational aspects of finite-frequency traveltime inversion kernels

Finite-frequency traveltime inversion offers higher accuracy for velocity model building than ray-based traveltime inversion. The adjoint force is the key for the computation of inversion kernels. Starting at the definition of inversion kernels for the acoustic wave equation, we have derived the explicit formula for the spectral distribution density function used in the adjoint force computation. Two formulations are developed for the computation of adjoint forces for receiver-side extrapolation, frequency-domain representation, and time-domain representation. The accuracy of finite-frequency traveltime inversion kernels is benchmarked with the analytical solutions for homogeneous isotropic media. We use wavefront construction to compute the first Fresnel zones for kernel conditioning. Based on dynamic ray tracing, we design a processing procedure guided by synthetic data tests to extract the desired events for wavefield backward extrapolation from the data. Unlike ray-based velocity tomography, finite-frequency inversion can resolve the velocity structures comparable with the size of Fresnel zones as we demonstrate on a marine salt model using ocean-bottom node acquisition geometry. Despite the fact that the inversion kernels are based on Born approximation, velocities with errors up to 20% can be well-resolved. For practical purposes, a simple formulation is given for the determination of the shot spacing. Our workflow for finite-frequency inversion is efficient and converges in only very few iterations.

Eigenrays in 3D heterogeneous anisotropic media, Part I: Kinematics

ABSTRACTWe present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler–Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler–Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples.

Eigenrays in 3D heterogeneous anisotropic media, Part II: Dynamics

ABSTRACTThis paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.

An efficient rectilinear Gaussian beam tracing method for sound propagation modelling in a non-turbulent medium

Large-distance sound propagation with high-frequency noise sources, multiple obstacles/geometry with varying acoustic impedance is common in real-life applications. To resolve the acoustic governing equations directly is often computationally costly, especially in three-dimensional space. Methods based on geometric acoustics can be more rapid. However, efforts are still being made to improve the efficiency, robustness, and the capability for complex configurations of such methods. In this paper, an efficient implementation of the rectilinear Gaussian beam tracing method is conducted, which combines rectilinear ray tracing with a proposed efficiency-matched dynamic ray tracing algorithm. A continuous medium stratification method is employed to improve the robustness. Also, a ray compression algorithm is proposed to save computation time. Numerical tests show that computation acceleration up to tenfold is achieved, benefiting rapid estimation of large-distance sound propagation. A standard octree data structure is employed in the code, which accelerates ray tracing in the testing cases with complex geometries. The efficiency and capability of the solver are demonstrated by studying several benchmark problems with varying complexity.

Ray tracing for modeling underwater acoustic communications channel between moving platforms, with moving ocean surface and changing bathymetry

Underwater acoustic communication systems must contend with a time-varying channel due to motion of source, receiver, point of reflection from ocean surface waves, and point of reflection from the seabed over changing bathymetry. These motions produce Doppler effects, which impact acomms systems. The signal processing to equalize such channel fading typically consists of a number of tracking loops, whose testing requires time varying realizations of such channels. A realistic channel model would enable improved modem designs and hardware-in-the-loop testing. We will focus on how to handle the ocean surface and the changing bathymetry. We propose an architecture which pre-computes the ray tracing for the part of the ocean that is not changing, then updates only small subsets of the ocean where dynamics occur. This consists of identifying the point of specular reflection, then interpolating the source-to-reflector and receiver-to-reflector rays to produce the union of these two paths. Such acoustic paths may be shadowed by the shape of the ocean surface. We will present examples of a time-varying channel impulse response, using the combined travel times, and discuss how dynamic ray tracing can be used to produce the amplitude of the joined paths.

Hybrid frequency-domain full-waveform inversion using ray+Born sensitivity kernels

Full-waveform inversion (FWI) using the scattering integral (SI) approach is an explicit formulation of the inversion optimization problem. The inversion procedure is straightforward, and the dependence of the data residuals on the model parameters is clear. However, the biggest limitation associated with this approach is the huge computational cost in conventional exploration seismology applications. Modeling from each of the source and receiver locations is required to compute the update at every iteration, and that is prohibitively expensive, especially for 3D problems. To deal with this issue, we have developed a hybrid implementation of frequency-domain FWI, in which forward modeling is combined with ray tracing to compute the update. We use the sensitivity kernels computed from dynamic ray tracing to build the gradient. The data residual is still computed using finite-difference wavefield modeling. With ray theory, the Green’s function can be approximated using a coarser grid compared to wave-equation modeling. Therefore, the memory requirements, as well as the computational cost, are reduced significantly. Considering that in transmission FWI long-to-intermediate wavelengths are updated during the early iterations, we obtain accurate inverted models. The inversion scheme captured the anomaly embedded in the homogeneous background medium. For more complex models, the hybrid inversion method helps in improving the initial model with little cost compared with conventional SI inversion approaches. The accuracy of the inversion results shows the effectiveness of the hybrid approach for 3D realistic problems.

Characterization of internal waves in SAS imagery via ray tracing

Synthetic aperture sonar imagery often captures features that appear similar to sand waves but are actually pockets of denser water traveling as isolated waves along the seafloor. These pockets of cold water refract acoustic waves like a lens, causing intensity peaks and shadows that resemble medium to large scale sand waves. This work uses dynamic ray tracing to predict the intensity return as affected by refraction. First, we explore the nature of the intensity pattern created by internal waves of various shapes and sizes. Then, we use an optimization-based approach to solve the inverse problem: given an intensity pattern, determine the size, shape, and location of the internal wave that created it.