Exponential asymptotically bounded velocity model: Part II — Ray tracing

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. T67-T85 ◽  
Author(s):  
Igor Ravve ◽  
Zvi Koren
Keyword(s):  
Ray Tracing ◽  
Constant Velocity ◽  
Critical Angle ◽  
Velocity Model ◽  
Velocity Variation ◽  
Infinite Depth ◽  
Source Point ◽  
Shallow Zone ◽  
Takeoff Angle ◽  
2D And 3D

A ray-tracing procedure is derived for the new exponential asymptotically bounded (EAB) velocity model introduced in Part I of this paper. The model inherits the properties of a medium with linear-velocity variation in depth in the shallow zone and of a medium with constant velocity in the deep zone. Two types of rays departing from the source point on the earth’s surface exist in this model, depending on the takeoff angles. The rays of the first kind are symmetric arcs that return up to the earth’s surface and have a limited maximum depth of propagation. The rays of the second kind propagate down to infinite depth. In the shallow region, they are curved lines, but at large depth they become asymptotically straight. The form of the ray is governed by the takeoff angle at the source point, where a critical angle splits the two kinds of rays. This critical angle depends only on the ratio between the velocity at the source point and the asymptotic velocity of the EAB model. We derive the formulae required to calculate the two kinds of rays and solve the inverse problem of two-point ray tracing. Finally, we construct the 2D- and 3D-isochron surfaces for a finite offset.

scholarly journals A Fast Ray-tracing Method for Locating Mining-Induced Seismicity by Considering Underground Voids

2020 ◽  
Vol 10 (19) ◽  
pp. 6763
Author(s):  
Pingan Peng ◽  
Yuanjian Jiang ◽  
Liguan Wang ◽  
Zhengxiang He ◽  
Siyu Tu
Keyword(s):  
Travel Time ◽  
Ray Tracing ◽  
Induced Seismicity ◽  
Constant Velocity ◽  
Velocity Model ◽  
Surface Model ◽  
Location Accuracy ◽  
Ray Tracing Method ◽  
Mining Induced Seismicity ◽  
Tracing Method

The accurate localization of mining-induced seismicity is crucial to underground mines. However, the constant velocity model is used by traditional location methods without considering the great difference in wave velocity between rock mass and underground voids. In this paper, to improve the microseismicity location accuracy in mines, we present a fast ray-tracing method to calculate the ray path and travel time from source to receiver considering underground voids. First, we divide the microseismic monitoring area into two categories of mediums—voids and non-voids—using a flexible triangular patch to model the surface model of voids, which can accurately describe any complicated three-dimensional (3D) shape. Second, the nodes are divided into two categories. The first category of the nodes is the vertex of the model, and the second category of the nodes is arranged at a certain step length on each edge of the 3D surface model to improve the accuracy of ray tracing. Finally, the set of adjacent nodes of each node is calculated, and then we obtain the shortest travel time from the source to the receiver based on the Dijkstra algorithm. The performance of the proposed method is tested by numerical simulation. Results show that the proposed method is faster and more accurate than the traditional ray-tracing methods. Besides, the proposed ray-tracing method is applied to the microseismic source localization in the Huangtupo Copper and Zinc Mine. The location accuracy is significantly improved compared with the traditional method using the constant velocity model and the FMM-based location method.

Postmap migration of crosswell reflection seismic data

Geophysics ◽  
2002 ◽  
Vol 67 (1) ◽  
pp. 135-146 ◽  
Author(s):  
Joongmoo Byun ◽  
James W. Rector III ◽  
Tamas Nemeth
Keyword(s):  
Constant Velocity ◽  
Guided Waves ◽  
Synthetic Data ◽  
Velocity Model ◽  
Lateral Resolution ◽  
Velocity Variation ◽  
Coherent Noise ◽  
Image Domain ◽  
Seismic Reflection Data ◽  
Specific Implementation

Vertical seismic profiling/common depth point (VSP‐CDP) mapping is often preferred to crosswell migration when imaging crosswell seismic reflection data. The principal advantage of VSP‐CDP mapping is that it can be configured as a one‐to‐one operation between data in the acquisition domain and data in the image domain and therefore does not smear coherent noise such as tube waves, guided waves, and converted waves as crosswell migration could. However, unlike crosswell migration, VSP‐CDP mapping cannot collapse diffractions; therefore, the lateral resolution of reflection events suffers. We present a migration algorithm that is applied to the crosswell data after they have been mapped. By performing crosswell migration in two distinct steps—mapping followed by diffraction stacking—noise events can be identified and filtered in the mapped domain without smearing effects commonly associated with conventional crosswell migration operators. Tests on noise‐free synthetic crosswell data indicate that the two‐step migration yields results nearly identical with conventional crosswell migration. Our specific implementation of the two‐step migration algorithm maps the data using an estimate of the interwell velocity field and then performs diffraction stacking using a constant‐velocity assumption. The migrated results are confined to the mapped region to reduce edge effects commonly associated with conventional crosswell migration. Results from synthetic data indicate that the constant‐velocity assumption used for diffraction stacking is remarkably robust, even for models with large vertical velocity variation. It is, however, important that the data are mapped with the correct interwell velocity model. After applying postmap migration to two field data sets mapped by VSP‐CDP mapping, better fault resolution was achieved and the lateral resolution was improved significantly.

Finite‐difference calculation of direct‐arrival traveltimes using the eikonal equation

Geophysics ◽  
2002 ◽  
Vol 67 (4) ◽  
pp. 1270-1274 ◽  
Author(s):  
Le‐Wei Mo ◽  
Jerry M. Harris
Keyword(s):  
Finite Difference ◽  
Ray Tracing ◽  
Finite Differences ◽  
Eikonal Equation ◽  
Velocity Model ◽  
Uniform Grid ◽  
Square Grid ◽  
Kirchhoff Migration ◽  
Difference Calculation

Traveltimes of direct arrivals are obtained by solving the eikonal equation using finite differences. A uniform square grid represents both the velocity model and the traveltime table. Wavefront discontinuities across a velocity interface at postcritical incidence and some insights in direct‐arrival ray tracing are incorporated into the traveltime computation so that the procedure is stable at precritical, critical, and postcritical incidence angles. The traveltimes can be used in Kirchhoff migration, tomography, and NMO corrections that require traveltimes of direct arrivals on a uniform grid.

Marchenko multiple elimination and full wavefield migration in a resonant pinch-out model

Geophysics ◽  
2021 ◽  
pp. 1-59
Author(s):  
Evert Slob ◽  
Lele Zhang ◽  
Eric Verschuur
Keyword(s):  
Constant Velocity ◽  
Local Minimum ◽  
Velocity Model ◽  
Dominant Frequency ◽  
Data Sampling ◽  
Multiple Reflections ◽  
Linear Inversion ◽  
Acoustic Reflection ◽  
Reflection Data ◽  
Multiple Elimination

Marchenko multiple elimination schemes are able to attenuate all internal multiple reflections in acoustic reflection data. These can be implemented with and without compensation for two-way transmission effects in the resulting primary reflection dataset. The methods are fully automated and run without human intervention, but require the data to be properly sampled and pre-processed. Even when several primary reflections are invisible in the data because they are masked by overlapping primaries, such as in the resonant wedge model, all missing primary reflections are restored and recovered with the proper amplitudes. Investigating the amplitudes in the primary reflections after multiple elimination with and without compensation for transmission effects shows that transmission effects are properly accounted for in a constant velocity model. When the layer thickness is one quarter of the wavelength at the dominant frequency of the source wavelet, the methods cease to work properly. Full wavefield migration relies on a velocity model and runs a non-linear inversion to obtain a reflectivity model which results in the migration image. The primary reflections that are masked by interference with multiples in the resonant wedge model, are not recovered. In this case, minimizing the data misfit function leads to the incorrect reflector model even though the data fit is optimal. This method has much lower demands on data sampling than the multiple elimination schemes, but is prone to get stuck in a local minimum even when the correct velocity model is available. A hybrid method that exploits the strengths of each of these methods could be worth investigating.

Fast salt boundary interpretation with optimal path picking

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. O45-O53 ◽  
Author(s):  
Xinming Wu ◽  
Sergey Fomel ◽  
Michael Hudec
Keyword(s):  
Model Building ◽  
Optimal Path ◽  
Velocity Model ◽  
Maximization Problem ◽  
Boundary Extraction ◽  
Initial Curve ◽  
Seismic Attribute ◽  
Human Interactions ◽  
Band Area ◽  
2D And 3D

Salt boundary interpretation is a crucial step for velocity-model building in seismic migration, but it remains a highly labor-intensive task for manual interpretation and a big challenge for automatic methods. We have developed a semiautomatic method to efficiently and accurately extract 2D and 3D complicated salt boundaries from a seismic attribute image that highlights salt boundaries. In 2D salt boundary extraction, we first pick a few points to interpolate an initial curve that is close to the true salt boundary. These points are picked near the salt boundary but are not required to be exactly on the boundary, which makes human interactions convenient and efficient. We then resample the salt boundary attribute image in a band area centered at the initial curve to obtain a new image in which the true salt boundary is an open curve extending laterally. We then extract the salt boundary in the new image using an optimal-path picking algorithm, which is robust enough to track a stable salt boundary from highly discontinuous attribute values by solving a global maximization problem. We finally map the picked path back to the original image to obtain a final salt boundary. In 3D salt boundary extraction, we apply the 2D method to recursively pick 2D salt boundaries in a sequence of inline or crossline slices and then use these 2D boundaries to fit an implicit (level-set) surface of the 3D salt boundary. In our recursive picking, human interactions are greatly reduced by using a salt boundary picked in the previous slice as an initial curve for picking in a followed slice. The effectiveness of our method is confirmed with 2D and 3D real seismic images.

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>

Fast acoustic velocity tomography of focusing operators

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. R121-R131 ◽  
Author(s):  
Hu Jin ◽  
George A. McMechan
Keyword(s):  
Constant Velocity ◽  
Null Space ◽  
Focal Point ◽  
Grid Point ◽  
Synthetic Data ◽  
Velocity Model ◽  
The Stability ◽  
Starting Model ◽  
Velocity Tomography ◽  
Focus Point

A 2D velocity model was estimated by tomographic imaging of overlapping focusing operators that contain one-way traveltimes, from common-focus points to receivers in an aperture along the earth’s surface. The stability and efficiency of convergence and the quality of the resulting models were improved by a sequence of ideas. We used a hybrid parameterization that has an underlying grid, upon which is superimposed a flexible, pseudolayer model. We first solved for the low-wavenumber parts of the model (approximating it as constant-velocity pseudo layers), then we allowed intermediate wavenumbers (allowing the layers to have linear velocity gradients), and finally did unconstrained iterations to add the highest wavenumber details. Layer boundaries were implicitly defined by focus points that align along virtual marker (reflector) horizons. Each focus point sampled an area bounded by the first and last rays in the data aperture at the surface; this reduced the amount of computation and the size of the effective null space of the solution. Model updates were performed simultaneously for the velocities and the local focus point positions in two steps; local estimates were performed independently by amplitude semblance for each focusing operator within its area of dependence, followed by a tomographic weighting of the local estimates into a global solution for each grid point, subject to the constraints of the parameterization used at that iteration. The system of tomographic equations was solved by simultaneous iterative reconstruction, which is equivalent to a least-squares solution, but it does not involve a matrix inversion. The algorithm was successfully applied to synthetic data for a salt dome model using a constant-velocity starting model; after a total of 25 iterations, the velocity error was [Formula: see text] and the final mean focal point position error was [Formula: see text] wavelength.

Improving input/output performance in 2D and 3D angle-domain common-image gathers from reverse time migration

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. S65-S77 ◽  
Author(s):  
Hu Jin ◽  
George A. McMechan ◽  
Bao Nguyen
Keyword(s):  
Velocity Model ◽  
Normal Direction ◽  
Propagation Direction ◽  
Input Output ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Step ◽  
Phase Gradient ◽  
Time Migration ◽  
2D And 3D

We have developed a new method of extracting angle-domain common-image gathers (ADCIGs) from prestack reverse time migration (RTM) that has minimal intermediate storage requirements. To include multipathing, we applied an imaging condition for prestack RTM that uses multiple excitation image times. Instead of saving the full-source snapshots at all time steps, we picked and saved only a few of the highest amplitude arrivals, and their corresponding excitation times, of the source wavefield at each grid point, and we crosscorrelated with the receiver wavefield. When extracting the ADCIGs from RTM, we calculated the source propagation direction from the gradient of the excitation times. The result was that the source time snapshots do not have to be saved or reconstructed during RTM or while extracting ADCIGs. We calculated the receiver propagation direction from Poynting vectors during the receiver extrapolation at each time step and the reflector normal direction by the phase-gradient method. With a new strategy that uses three direction vectors (the source and receiver propagation directions as well as the reflector normal direction), we provided more reliable ADCIGs that are free of low-wavenumber artifacts than any two of them do separately, when the migration velocity model was near to the correct velocity model. The 2D and 3D synthetic tests demonstrated the successful application of the new algorithms with acceptable accuracy, improved storage efficiency, and without an input/output bottleneck.

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