A fast algorithm for two-point seismic ray tracing

1987 ◽  
Vol 77 (3) ◽  
pp. 972-986
Author(s):  
Junho Um ◽  
Clifford Thurber
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Three Dimensional ◽  
Linear Interpolation ◽  
Geometric Interpretation ◽  
Approximate Algorithm ◽  
Velocity Models ◽  
Limit Test ◽  
Ray Path ◽  
Lateral Variations

Abstract A new approximate algorithm for two-point ray tracing is proposed and tested in a variety of laterally heterogeneous velocity models. An initial path estimate is perturbed using a geometric interpretation of the ray equations, and the travel time along the path is minimized in a piecewise fashion. This perturbation is iteratively performed until the travel time converges within a specified limit. Test results show that this algorithm successfully finds the correct travel time within typical observational error much faster than existing three-dimensional ray tracing programs. The method finds an accurate ray path in a fully three-dimensional form even where lateral variations in velocity are severe. Because our algorithm utilizes direct minimization of the travel time instead of solving the ray equations, a simple linear interpolation scheme can be employed to compute velocity as a function of position, providing an added computational advantage.

Comparing Higher-dimensional Velocity Models for Seismic Location Accuracy using a Consistent Travel Time Framework

2021 ◽  
Author(s):  
Michael Begnaud ◽  
Sanford Ballard ◽  
Andrea Conley ◽  
Patrick Hammond ◽  
Christopher Young
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Seismic Velocity ◽  
3D Seismic ◽  
Location Accuracy ◽  
Velocity Models ◽  
Higher Dimensional ◽  
1D Model ◽  
Tracing Algorithm ◽  
2D And 3D

<p>Historically, location algorithms have relied on simple, one-dimensional (1D, with depth) velocity models for fast, seismic event locations. The speed of these 1D models made them the preferred type of velocity model for operational needs, mainly due to computational requirements. Higher-dimensional (2D-3D) seismic velocity models are becoming more readily available from the scientific community and can provide significantly more accurate event locations over 1D models. The computational requirements of these higher-dimensional models tend to make their operational use prohibitive. The benefit of a 1D model is that it is generally used as travel-time lookup tables, one for each seismic phase, with travel-time predictions pre-calculated for event distance and depth. This simple, lookup structure makes the travel-time computation extremely fast.</p><p>Comparing location accuracy for 2D and 3D seismic velocity models tends to be problematic because each model is usually determined using different inversion parameters and ray-tracing algorithms. Attempting to use a different ray-tracing algorithm than used to develop a model almost always results in poor travel-time prediction compared to the algorithm used when developing the model.</p><p>We will demonstrate that using an open-source framework (GeoTess, www.sandia.gov/geotess) that can easily store 3D travel-time data can overcome the ray-tracing algorithm hurdle. Travel-time lookup tables (one for each station and phase) can be generated using the exact ray-tracing algorithm that is preferred for a specified model. The lookup surfaces are generally applied as corrections to a simple 1D model and also include variations in event depth, as opposed to legacy source-specific station corrections (SSSCs), as well as estimates of path-specific travel-time uncertainty. Having a common travel-time framework used for a location algorithm allows individual 2D and 3D velocity models to be compared in a fair, consistent manner.</p>

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.

Two‐point ray tracing in a three‐dimensional medium consisting of homogeneous nonisotropic layers separated by plane interfaces

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 767-770 ◽  
Author(s):  
R. F. Stöckli
Keyword(s):  
Point Source ◽  
Travel Time ◽  
Ray Tracing ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Point Of View ◽  
Wave Surface ◽  
Computational Point ◽  
Time Problem

The ray‐tracing problem is considered the solution to a minimum travel time problem for media where each layer may be regarded as a transversely isotropic homogeneous solid. The wave surface‐wavefront at t = 1 s, corresponding to a wave generated at the point source, associated with each layer’s anisotropy is approximated by surfaces which are not more difficult to handle, from a computational point of view, than ellipsoidal surfaces. These approximating surfaces are those used in ray‐tracing computation; a ray being a true ray approximation is thus obtained.

Rapid solution of ray tracing problems in heterogeneous media

1980 ◽  
Vol 70 (4) ◽  
pp. 1137-1148 ◽  
Author(s):  
C. H. Thurber ◽  
W. L. Ellsworth
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Heterogeneous Media ◽  
Velocity Structure ◽  
Minimum Time ◽  
Local Minima ◽  
Complex Velocity ◽  
Efficient Manner ◽  
Tracing Techniques ◽  
Ray Path

abstract The determination of local earthquake hypocenters and orgin times from first-P-arrival times by Geiger's method requires a technique for finding the minimum travel time (and derivatives) between the source and the station. Sophisticated ray tracing techniques have been developed for this purpose for use in complex velocity structures. Unfortunately, the two common techniques, shooting and bending, are generally prohibitively expensive for routine use in data analysis. The bending method is also particularly vulnerable to the problem of local minima in travel time. A method has been developed known as the ray initializer, which can be used to circumvent these problems in many cases. First, the technique can find a reasonable estimate of the minimum-time ray path in a quick and efficient manner. The velocity in a region local to the source and receiver is laterally averaged to yield an approximate layered velocity model. One-dimensional ray tracing techniques are used to find the minimum-time path for this layered structure. The ray path estimate can then be used as the starting path in a bending routine, a procedure resulting in more rapid convergence and the avoidance of local minima. Second, the travel time found by numerical integration along the estimated ray path is an excellent approximation to the actual travel time. Thus, in many cases, the ray initializer can be substituted for a three-dimensional ray tracing routine with a tremendous increase in efficiency and only a small loss in accuracy. It is found that the location of an explosion, derived using the ray initializer, is nearly identical to a complete ray tracing solution, even for a highly complex velocity structure.

Earthquake locations by 3-D finite-difference travel times

1990 ◽  
Vol 80 (2) ◽  
pp. 395-410 ◽  
Author(s):  
Glenn D. Nelson ◽  
John E. Vidale
Keyword(s):  
Travel Time ◽  
Velocity Structure ◽  
Three Dimensional ◽  
Computation Time ◽  
Velocity Model ◽  
Model Complexity ◽  
Grid Spacing ◽  
Origin Time ◽  
Velocity Models ◽  
Fault Trace

Abstract We present a new method for locating earthquakes in a region with arbitrarily complex three-dimensional velocity structure, called QUAKE3D. Our method searches a gridded volume and finds the global minimum travel-time residual location within the volume. Any minimization criterion may be employed. The L1 criterion, which minimizes the sum of the absolute values of travel-time residuals, is especially useful when the station coverage is sparse and is more robust than the L2 criterion (which minimizes the RMS sum) employed by most earthquake location programs. On a UNIX workstation with 8 Mbytes memory, travel-time grids of size 150 by 150 by 50 are reasonably employed, with the actual geographic coverage dependent on the grid spacing. Location precision is finer than the grid spacing. Earthquake recordings at six stations in Bear Valley are located as an example, using various layered and laterally varying velocity models. Locations with QUAKE3D are nearly identical to HYPOINVERSE locations when the same flat-layered velocity model is used. For the examples presented, the computation time per event is approximately 4 times slower than HYPOINVERSE, but the computation time for QUAKE3D is dependent only on the grid size and number of stations, and independent of the velocity model complexity. Using QUAKE3D with a laterally varying velocity model results in locations that are physically more plausible and statistically more precise. Compared to flat-layered solutions, the earthquakes are more closely aligned with the surface fault trace, are more uniform in depth distribution, and the event and station travel-time residuals are much smaller. Hypocentral error bars computed by QUAKE3D are more realistic in that the trade-off of depth versus origin time is implicit in our error estimation, but ignored by HYPOINVERSE.

MODELING AND ANALYSIS OF AN ACOUSTIC CHANNEL WITH A MOVING SURFACE

2013 ◽  
Vol 21 (04) ◽  
pp. 1350015 ◽  
Author(s):  
YOUNGMIN CHOO ◽  
WOOJAE SEONG
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Path Dependence ◽  
Moving Surface ◽  
Modeling And Analysis ◽  
Surface Movement ◽  
Delay Times ◽  
Tracing Algorithm ◽  
The Difference ◽  
Ray Path

A ray tracing algorithm for moving surfaces is derived to enable the analysis of surface movement effects. For this, a ray tracing algorithm for frozen surface is modified. By comparing the results from frozen and moving surface ray models allows effects of a moving surface to be investigated. The surface movement effects can be seen with the difference between channel impulse responses from the frozen and moving surface ray models. For an investigation of ray path dependence of the surface movement effects, delay times of surface reflective paths from the two ray models are observed according to transmitted ping time. As the ray path from the source to surface gets longer, difference of travel time results from the two ray models increase. This fact indicates that surface movement effects depend on ray path, in particular travel time until a ray meets a surface.

Examination of the optimum XY value by ray tracing

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1151-1156 ◽  
Author(s):  
Tak Ming Leung
Keyword(s):  
Ray Tracing ◽  
Lateral Variation ◽  
Velocity Models ◽  
Different Types ◽  
First Arrival ◽  
Reciprocal Method ◽  
Lateral Variations

The ability to recognize undetected layers in the generalized reciprocal method (GRM) depends on the optimum XY value. The determination of the optimum XY value, in turn, requires that some lateral variations exist in the refractor. Since lateral subsurface variations may not be just on the refractor, different locations and aspects of lateral variations may invalidate the usefulness of the optimum XY value. The optimum XY value is examined through velocity models having different types of lateral variations. Accurate traveltime data are generated by a first‐arrival ray‐tracing routine. The results of the study indicate that the integrity of the optimum XY value may be violated because of the various types of lateral variation.

Tsunami Ray Tracing Method for Shortest Travel-Time Path

2021 ◽  
Author(s):  
Tung-Cheng Ho ◽  
Shingo Watada ◽  
Kenji Satake
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Tsunami Wave ◽  
Wave Theory ◽  
Ray Tracing Method ◽  
Tsunami Early Warning ◽  
Time Path ◽  
Long Wave ◽  
Ray Path ◽  
Tracing Method

<p>We propose a ray-tracing method to solve the two-point boundary value problem for tsunamis based on the long-wave theory. In the long-wave theory, the tsunami wave velocity is proportional to the square root of water depth, which is available from global bathymetric atlases. Our method computes the shortest travel times starting from each of the two given points and calculates the local ray direction to trace the ray path. We utilize an explicit, non-iterative tracing scheme that exhibits robust results and applies to any tsunami-accessible locations, and the global-shortest travel-time path is derived. In simple and real bathymetry cases, our method demonstrates stable results with neglectable low uncertainties. The ray-tracing method is then applied to analyze the path of tsunamis from different directions to four important bays in Japan. The result shows that tsunami ray paths are significantly influenced by local bathymetry, and some crucial structures, such as trench and trough, behave as the primary routes of this region. Deploying stations near these routes will be most beneficial for tsunami early warning. The existing tsunami-observing system off the Honshu area works well for tsunamis from the east side but slightly deficient for tsunamis from the west side. The far-field ray tracing shows that tsunamis traveling from Chile to Japan through two main routes—one via north Hawaii and the other via the south— depending on the location of the source.</p>

scholarly journals Imaging the Mediterranean upper mantle by p- wave travel time tomography

1997 ◽  
Vol 40 (4) ◽  
Author(s):  
C. Piromallo ◽  
A. Morelli
Keyword(s):  
Travel Time ◽  
Upper Mantle ◽  
Three Dimensional ◽  
Velocity Model ◽  
Structural Heterogeneity ◽  
Linear Interpolation ◽  
P Wave ◽  
Travel Times ◽  
P Waves ◽  
Tectonic Lineament

Travel times of P-waves in the Euro-Mediterranean region show strong and consistent lateral variations, which can be associated to structural heterogeneity in the underlying crust and mantle. We analyze regional and tele- seismic data from the International Seismological Centre data base to construct a three-dimensional velocity model of the upper mantle. We parameterize the model by a 3D grid of nodes -with approximately 50 km spacing -with a linear interpolation law, which constitutes a three-dimensional continuous representation of P-wave velocity. We construct summary travel time residuals between pairs of cells of the Earth's surface, both inside our study area and -with a broader spacing -on the whole globe. We account for lower mantle heterogeneity outside the modeled region by using empirical corrections to teleseismic travel times. The tomo- graphic images show generai agreement with other seismological studies of this area, with apparently higher detail attained in some locations. The signature of past and present lithospheric subduction, connected to Euro- African convergence, is a prominent feature. Active subduction under the Tyrrhenian and Hellenic arcs is clearly imaged as high-velocity bodies spanning the whole upper mantle. A clear variation of the lithospheric structure beneath the Northem and Southern Apennines is observed, with the boundary running in correspon- dence of the Ortona-Roccamonfina tectonic lineament. The western section of the Alps appears to have better developed roots than the eastern, possibly reflecting à difference in past subduction of the Tethyan lithosphere and subsequent continental collision.

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