3-D traveltimes and amplitudes by gridded rays

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 277-282 ◽  
Author(s):  
Giancarlo Bernasconi ◽  
Giuseppe Drufuca
Keyword(s):  
Horizontal Plane ◽  
Eikonal Equation ◽  
Order Approximation ◽  
Finite Difference Methods ◽  
Three Dimensions ◽  
Grid Sampling ◽  
Velocity Models ◽  
Grid Step ◽  
Slowness Vector ◽  
Vertical Grid

Seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium. They can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite‐difference methods. Unfortunately, this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used. We propose a new method for computing traveltimes and amplitudes in 3-D media that is simple, fast, unconditionally stable, and robust. Defining the slowness vector as [Formula: see text] and assuming an isotropic medium, the ray velocity v is related to the slowness vector by the relation [Formula: see text]. Rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane. The components of the slowness vector are then interpolated to gridpoints on this next horizontal plane. This is termed regridding; the process of downward propagation of rays, one vertical grid step at a time, is continued until some prescribed depth is reached. Computation of amplitudes is achieved using a method similar to that for obtaining the zero‐order approximation in asymptotic ray theory. We show comparisons with a full‐wave method on readily accessible 3-D velocity models.

Kirchhoff migration using eikonal-based computation of traveltime source derivatives

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. S211-S219 ◽  
Author(s):  
Siwei Li ◽  
Sergey Fomel
Keyword(s):  
Differential Equation ◽  
Eikonal Equation ◽  
Order Partial Differential Equation ◽  
Fast Marching ◽  
First Order ◽  
Velocity Models ◽  
Kirchhoff Migration ◽  
Input Source ◽  
First Arrival ◽  
Synthetic Models

The computational efficiency of Kirchhoff-type migration can be enhanced by using accurate traveltime interpolation algorithms. We addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopted a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative did not depend on input source sampling. For smooth velocity models, the first-order traveltime source derivatives enabled a cubic Hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into Kirchhoff antialiasing schemes. We provided an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semirecursive Kirchhoff migration of the Marmousi model demonstrated the applicability of the proposed method.

Solving two-point seismic-ray tracing problems in a heterogeneous medium

1980 ◽  
Vol 70 (1) ◽  
pp. 79-99 ◽  
Author(s):  
V. Pereyra ◽  
W. H. K. Lee ◽  
H. B. Keller
Keyword(s):  
Numerical Method ◽  
Ray Tracing ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Nonlinear Boundary Conditions ◽  
Variable Order ◽  
Flexible Tool ◽  
Velocity Models ◽  
Jump Discontinuities ◽  
Variable Mesh

abstract A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

scholarly journals Interdiffusion in many dimensions: mathematical models, numerical simulations and experiment

2020 ◽  
Vol 25 (12) ◽  
pp. 2178-2198
Author(s):  
Lucjan Sapa ◽  
Bogusław Bożek ◽  
Katarzyna Tkacz–Śmiech ◽  
Marek Zajusz ◽  
Marek Danielewski
Keyword(s):  
Finite Difference Methods ◽  
Mass Conservation ◽  
Three Dimensions ◽  
Uniqueness Of Solutions ◽  
Strongly Coupled ◽  
Nonlinear Parabolic ◽  
Implicit Difference Schemes ◽  
Coupled Boundary Conditions ◽  
Difference Methods ◽  
The Difference

Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

Computer-graphic representation of mandibular movements in three dimensions. Part I. The horizontal plane

1978 ◽  
Vol 39 (4) ◽  
pp. 378-383 ◽  
Author(s):  
William H. Roedema ◽  
John G. Knapp ◽  
Judson Spencer ◽  
Michael K. Dever
Keyword(s):  
Horizontal Plane ◽  
Computer Graphic ◽  
Graphic Representation ◽  
Three Dimensions ◽  
Mandibular Movements

INDUCED PARITY-VIOLATION ANOMALY FOR A SPIN-3/2 FIELD IN THREE DIMENSIONS

1992 ◽  
Vol 07 (27) ◽  
pp. 2519-2525
Author(s):  
J. R. S. DO NASCIMENTO ◽  
E. R. BEZERRA DE MELLO
Keyword(s):  
Vector Field ◽  
Parity Violation ◽  
Effective Action ◽  
Order Approximation ◽  
Three Dimensions ◽  
Chern Simons

A diagrammatic computation of the parity-violating effective action for the massless Rarita-Schwinger field induced by its interaction with a spin 0 and 1/2 fields is presented. The lowest-order approximation is compared with the Chern-Simons term previously obtained for the spinor-vector field.

3-D traveltime computation using second‐order ENO scheme

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1867-1876 ◽  
Author(s):  
Seongjai Kim ◽  
Richard Cook
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Eikonal Equation ◽  
Second Order ◽  
Order Accuracy ◽  
Unconditionally Stable ◽  
Velocity Models ◽  
Box Scheme ◽  
Initialization Scheme ◽  
Eno Scheme

We consider a second‐order finite difference scheme to solve the eikonal equation. Upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives, whereas centered differences improve the accuracy of the computed traveltime. A second‐order upwind essentially non‐oscillatory (ENO) scheme satisfies these requirements. It is implemented with a dynamic down ’n’ out (DNO) marching, an expanding box approach. To overcome the instability of such an expanding box scheme, the algorithm incorporates an efficient post sweeping (PS), a correction‐by‐iteration method. Near the source, an efficient and accurate mesh‐refinement initialization scheme is suggested for the DNO marching. The resulting algorithm, ENO-DNO-PS, turns out to be unconditionally stable, of second‐order accuracy, and efficient; for various synthetic and real velocity models having large contrasts, two PS iterations produce traveltimes accurate enough to complete the computation.

scholarly journals Three-dimensional tuning of head direction cells in rats

2019 ◽  
Vol 121 (1) ◽  
pp. 4-37 ◽  
Author(s):  
Michael E. Shinder ◽  
Jeffrey S. Taube
Keyword(s):  
Horizontal Plane ◽  
Reference Frames ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Ventral Surface ◽  
Three Dimensions ◽  
Head Direction ◽  
Horizontal Canal ◽  
Cell Responses ◽  
The Earth

Head direction (HD) cells fire when the animal faces that cell’s preferred firing direction (PFD) in the horizontal plane. The PFD response when the animal is oriented outside the earth-horizontal plane could result from cells representing direction in the plane of locomotion or as a three-dimensional (3D), global-referenced direction anchored to gravity. To investigate these possibilities, anterodorsal thalamic HD cells were recorded from restrained rats while they were passively positioned in various 3D orientations. Cell responses were unaffected by pitch or roll up to ~90° from the horizontal plane. Firing was disrupted once the animal was oriented >90° away from the horizontal plane and during inversion. When rolling the animal around the earth-vertical axis, cells were active when the animal’s ventral surface faced the cell’s PFD. However, with the rat rolled 90° in an ear-down orientation, pitching the rat and rotating it around the vertical axis did not produce directionally tuned responses. Complex movements involving combinations of yaw-roll, but usually not yaw-pitch, resulted in reduced directional tuning even at the final upright orientation when the rat had full visual view of its environment and was pointing in the cell’s PFD. Directional firing was restored when the rat’s head was moved back-and-forth. There was limited evidence indicating that cells contained conjunctive firing with pitch or roll positions. These findings suggest that the brain’s representation of directional heading is derived primarily from horizontal canal information and that the HD signal is a 3D gravity-referenced signal anchored to a direction in the horizontal plane. NEW & NOTEWORTHY This study monitored head direction cell responses from rats in three dimensions using a series of manipulations that involved yaw, pitch, roll, or a combination of these rotations. Results showed that head direction responses are consistent with the use of two reference frames simultaneously: one defined by the surrounding environment using primarily visual landmarks and a second defined by the earth’s gravity vector.

Viscoacoustic waveform inversion of velocity structures in the time domain

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. R103-R119 ◽  
Author(s):  
Jianyong Bai ◽  
David Yingst ◽  
Robert Bloor ◽  
Jacques Leveille
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Field Data ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Waveform Inversion ◽  
Data Set ◽  
Velocity Models ◽  
Difference Methods ◽  
The Time Domain

Because of the conversion of elastic energy into heat, seismic waves are attenuated and dispersed as they propagate. The attenuation effects can reduce the resolution of velocity models obtained from waveform inversion or even cause the inversion to produce incorrect results. Using a viscoacoustic model consisting of a single standard linear solid, we discovered a theoretical framework of viscoacoustic waveform inversion in the time domain for velocity estimation. We derived and found the viscoacoustic wave equations for forward modeling and their adjoint to compensate for the attenuation effects in viscoacoustic waveform inversion. The wave equations were numerically solved by high-order finite-difference methods on centered grids to extrapolate seismic wavefields. The finite-difference methods were implemented satisfying stability conditions, which are also presented. Numerical examples proved that the forward viscoacoustic wave equation can simulate attenuative behaviors very well in amplitude attenuation and phase dispersion. We tested acoustic and viscoacoustic waveform inversions with a modified Marmousi model and a 3D field data set from the deep-water Gulf of Mexico for comparison. The tests with the modified Marmousi model illustrated that the seismic attenuation can have large effects on waveform inversion and that choosing the most suitable inversion method was important to obtain the best inversion results for a specific seismic data volume. The tests with the field data set indicated that the inverted velocity models determined from the acoustic and viscoacoustic inversions were helpful to improve images and offset gathers obtained from migration. Compared to the acoustic inversion, viscoacoustic inversion is a realistic approach for real earth materials because the attenuation effects are compensated.

An asymptotic theory of the jet flap in three dimensions

1971 ◽  
Vol 46 (4) ◽  
pp. 705-726 ◽  
Author(s):  
Naoyuki Tokuda
Keyword(s):  
Order Approximation ◽  
Three Dimensional ◽  
Outer Region ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Simple Method ◽  
Aspect Ratios ◽  
Line Theory ◽  
Lifting Line ◽  
Lifting Line Theory

A uniformly valid asymptotic solution has been constructed for three-dimensional jet-flapped wings by the method of matched asymptotic expansions for high aspect ratios. The analysis assumes that the flow is inviscid and incompressible and is formulated on the thin airfoil theory in accordance with the well-established Spence (1961) theory in two dimensions.A simple method emerges in treating the bound vortices along the jet sheet which forms behind the wing with the aid of the following physical picture. Three distinct flow regions—namely inner, outer and Trefitz—exist in the problem. Close to the wing the flow approximates to that in two dimensions. Therefore, Spence's solution in two dimensions applies. In the outer region a wing shrinks to a line of singularities from which the main disturbances of flow in this region arise. In particular, we find that the shape of the jet sheet, hence the strength of vortices, is now predetermined by the strength of the singularities there. Hence a complete flow field in the outer region can now be determined first by evaluating the flow due to various degrees of singularities along this line and then adding the effect of the jet bound vortices which is now known. Far removed from the wing, the well-known Trefftz region exists in which calculations of aerodynamic forces can be most easily done.The result has been applied to various wing planforms such as cusped, elliptic and rectangular wings. The present result breaks down for rectangular wings. However, we can apply Stewartson's (1960) solution for lifting-line theory for semi-infinite rectangular wings, because, to this second-order approximation it is established that the jet sheet in the outer region makes no contribution to lift, with the direct contribution of the deflected jet at the exit being cancelled by the reduced circulation in the jet vortices. This result for the rectangular wing gives excellent agreement with the experiments made on a rectangular wing, while the result for elliptic wings underestimates them considerably.

Robust and efficient upwind finite‐difference traveltime calculations in three dimensions

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1108-1117 ◽  
Author(s):  
William A. Schneider
Keyword(s):  
Finite Difference ◽  
Seismic Data ◽  
Velocity Model ◽  
Radial Component ◽  
Three Dimensions ◽  
Spherical Coordinate ◽  
Depth Migration ◽  
Slowness Vector ◽  
Kirchhoff Depth Migration ◽  
Upwind Finite Difference

First‐arrival traveltimes in complicated 3-D geologic media may be computed robustly and efficiently using an upwind finite‐difference solution of the 3-D eikonal equation. An important application of this technique is computing traveltimes for imaging seismic data with 3-D prestack Kirchhoff depth migration. The method performs radial extrapolation of the three components of the slowness vector in spherical coordinates. Traveltimes are computed by numerically integrating the radial component of the slowness vector. The original finite‐difference equations are recast into unitless forms that are more stable to numerical errors. A stability condition adaptively determines the radial steps that are used to extrapolate. Computations are done in a rotated spherical coordinate system that places the small arc‐length regions of the spherical grid at the earth’s surface (z = 0 plane). This improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated, thereby maximizing the radial steps. Adaptive gridding allows the angular grid spacings to vary with radius. The computation grid is also adaptively truncated so that it does not extend beyond the predefined Cartesian traveltime grid. This grid handling improves efficiency. The method cannot compute traveltimes corresponding to wavefronts that have “turned” so that they propagate in the negative radial direction. Such wavefronts usually represent headwaves and are not needed to image seismic data. An adaptive angular normalization prevents this turning, while allowing lower‐angle wavefront components to accurately propagate. This upwind finite‐difference method is optimal for vector‐parallel supercomputers, such as the CRAY Y-MP. A complicated velocity model that generates turned wavefronts is used to demonstrate the method’s accuracy by comparing with results that were generated by 3-D ray tracing and by an alternate traveltime calculation method. This upwind method has also proven successful in the 3-D prestack Kirchhoff depth migration of field data.

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