scholarly journals Analytic solutions for seismic travel time and ray path geometry through simple velocity models.

2007 ◽  
Author(s):  
Sanford Ballard
Keyword(s):  
Travel Time ◽  
Analytic Solutions ◽  
Velocity Models ◽  
Path Geometry ◽  
Ray Path

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>

Application of ray-path geometry modification to the design of a collimating structure for LEDs

2008 ◽  
Vol 281 (23) ◽  
pp. 5674-5682 ◽  
Author(s):  
A. Mandatori ◽  
A. Benedetti ◽  
C. Sibilia ◽  
M. Bertolotti
Keyword(s):  
Path Geometry ◽  
Ray Path

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.

Power-gradient velocity model

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. U13-U33 ◽  
Author(s):  
Alexey Stovas
Keyword(s):  
Velocity Distribution ◽  
Travel Time ◽  
Nonlinear Model ◽  
Velocity Model ◽  
Velocity Models ◽  
Kinematic Characteristics ◽  
Special Cases ◽  
Wide Range ◽  
Geometric Spreading ◽  
Time Parameters

A proposed power-gradient velocity model incorporates several well-known velocity models as special cases. The model covers a wide range of possible velocity distributions and has four parameters, giving more flexibility in velocity-model manipulation. For this nonlinear model, the kinematic characteristics — offset-traveltime parameteric equations, traveltime parameters, relative geometric spreading, and propagator phase — are computed. The characteristics are investigated with respect to a parameter responsible for nonlinearity of velocity distribution. The inversion of travel-time parameters was studied in three- and four-parameter frameworks.

scholarly journals Parking Pricing and Model Split under Uncertainty

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Chengjuan Zhu ◽  
Bin Jia ◽  
Linghui Han ◽  
Ziyou Gao
Keyword(s):  
Travel Time ◽  
Confidence Level ◽  
Time Budget ◽  
Route Choice ◽  
Analytic Solutions ◽  
User Equilibrium ◽  
Travel Times ◽  
Modal Split ◽  
Parking Pricing ◽  
The Impact

In order to investigate different route choice criteria in a competitive highway/park-and-ride (P&R) network with uncertain travel times on the road, a bilevel programming model for solving the problem of determining parking fees and modal split is presented. In the face of travel time uncertainty, travelers plan their trips with a prespecified on-time arrival probability. The impact of three route choice criteria: the mean travel time, the travel time budget, and mean-excess travel time, is compared for parking pricing and modal split. The model at user equilibrium is described as a minimization model. And the analytic solutions are given. Analytic solutions show that both flow and travel time at equilibrium are independent of the price difference of travel expense on money. The main findings from the numerical results are elaborated. While given a confidence level, the flow on the highway changed significantly with the criteria, although the differences of the travel times are small. Travelers can be guided to choose their modes coordinately by improving the quality of the transit service. The optimal parking fees can be affected markedly by the confidence level. Finally, the influence of the log-normal distribution parameters is tested and analyzed.

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