Fast migration/inversion with multivalued ray fields: Part 2—Applications to the 3D SEG/EAGE salt model

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1320-1328 ◽  
Author(s):  
Sheng Xu ◽  
Gilles Lambaré ◽  
Henri Calandra
Keyword(s):  
Three Dimensional ◽  
Velocity Model ◽  
Three Dimensions ◽  
Complex Media ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Convenient Approach ◽  
First Arrival ◽  
Subsalt Imaging ◽  
Lateral Variations

Three‐dimensional prestack depth migration is the convenient approach for seismic imaging in the case of strong lateral variations of the velocity. Because of computing limitations, it has been limited to single‐arrival kinematic Kirchhoff migration until recently. This approach fails in the case of complex media characterized by multiarrival traveltimes. We present numerical strategies for extending in three dimensions first‐arrival kinematic Kirchhoff migration to multiarrival quantitative ray‐based migration (preserved amplitude migration). We rely on wavefront construction in a smooth velocity model to compute the multivalued traveltime and amplitude maps, and the CPU efficiency of migration itself is ensured by efficient and robust interpolation or extrapolation strategies. We present an application to the synthetic 3D SEG/EAGE salt model. Taking into account multiarrivals clearly improves subsalt imaging at the price of quite limited computing costs (a 20% increase in our case, with respect to a preserved‐amplitude single‐arrival migration).

Multiarrival Kirchhoff beam migration

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB109-WB118 ◽  
Author(s):  
Jonathan Liu ◽  
Gopal Palacharla
Keyword(s):  
Model Updating ◽  
Plane Waves ◽  
Beam Width ◽  
Velocity Model ◽  
Model Complexity ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Optimal Beam ◽  
Local Plane ◽  
Gaussian Beam Migration

Kirchhoff-type prestack depth migration is the method most popular for outputting offset gathers for velocity-model updating because of its flexibility and efficiency. However, conventional implementations of Kirchhoff migration use only single arrivals. This limits its ability to image complex structures such as subsalt areas. We use the beam methodology to develop a multiarrival Kirchhoff beam migration. The theory and algorithm of our beam migration are analogs to Gaussian beam migration, but we focus on attaining kinematic accuracy and implementation efficiency. The input wavefield of every common offset panel is decomposed into local plane waves at beam centers on the acquisition surface by local slant stacking. Each plane wave contributes a potential single-arrival in Kirchhoff migration. In this way, our method is able to handle multiarrivals caused by model complexity and, therefore, to overcome the limitation of conventional single-arrival Kirchhoff migration. The choice of the width of the beam is critical to the implementation of beam migration. We provide a formula for optimal beam width that achieves both accuracy and efficiency when the velocity model is reasonably smooth. The resulting structural imaging in subsalt and other structurally complex areas is of better quality than that from single-arrival Kirchhoff migration.

Adaptive focusing window for seismic angle migration

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. S1-S10 ◽  
Author(s):  
Mathias Alerini ◽  
Bjørn Ursin
Keyword(s):  
Synthetic Data ◽  
Main Idea ◽  
Three Dimensions ◽  
User Friendliness ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Migration Operator ◽  
Local Geology ◽  
Original Approach

Kirchhoff migration is based on a continuous integral ranging from minus infinity to plus infinity. The necessary discretization and truncation of this integral introduces noise in the migrated image. The attenuation of this noise has been studied by many authors who propose different strategies. The main idea is to limit the migration operator around the specular point. This means that the specular point must be known before migration and that a criterion exists to determine the size of the migration operator. We propose an original approach to estimate the size of the focusing window, knowing the geologic dip. The approach benefits from the use of prestack depth migration in angle domain, which is recognized as the most artifact-free Kirchhoff-type migration. The main advantages of the method are ease of implementation in an existing angle-migration code (two or three dimensions), user friendliness, ability to take into account multiorientation of the local geology as in faulted regions, and flexibility with respect to the quality of the estimated geologic dip field. Common-image gathers resulting from the method are free from migration noise and can be postprocessed in an easier way. We validate the approach and its possibilities on synthetic data examples with different levels of complexity.

Can we image complex structures with first‐arrival traveltime?

Geophysics ◽  
1993 ◽  
Vol 58 (4) ◽  
pp. 564-575 ◽  
Author(s):  
Sébastien Geoltrain ◽  
Jean Brac
Keyword(s):  
Eikonal Equation ◽  
Geological Structure ◽  
Complex Structures ◽  
Complex Velocity ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Finite Differencing ◽  
Explicit Reference ◽  
First Arrival ◽  
Kirchhoff Depth Migration

We experienced difficulties when attempting to perform seismic imaging in complex velocity fields using prestack Kirchhoff depth migration in conjunction with traveltimes computed by finite‐differencing the eikonal equation. The problem arose not because of intrinsic limitations of Kirchhoff migration, but rather from the failure of finite‐differencing to compute traveltimes representative of the energetic part of the wavefield. Further analysis showed that the first arrival is most often associated with a marginally energetic event wherever subsequent arrivals exist. The consequence is that energetic seismic events are imaged with a kinematically incorrect operator and turn out mispositioned at depth. We therefore recommend that first‐arrival traveltime fields, such as those computed by finite‐differencing the eikonal equation, be used in Kirchhoff migration only with great care when the velocity field hosts multiple transmitted arrivals; such a situation is typically met where geological structure creates strong and localized velocity heterogeneities, which partition the incident and reflected wavefields into multiple arrivals; in such an instance, imaging cannot be strictly considered a kinematic process, as it must be performed with explicit reference to the relative amplitudes of multiple arrivals.

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.

Kirchhoff migration using eikonal equation traveltimes

Geophysics ◽  
1994 ◽  
Vol 59 (5) ◽  
pp. 810-817 ◽  
Author(s):  
Samuel H. Gray ◽  
William P. May
Keyword(s):  
Ray Tracing ◽  
Data Storage ◽  
Eikonal Equation ◽  
Synthetic Data ◽  
Velocity Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Regular Grid ◽  
Data Set ◽  
Kirchhoff Migration

The use of ray shooting followed by interpolation of traveltimes onto a regular grid is a popular and robust method for computing diffraction curves for Kirchhoff migration. An alternative to this method is to compute the traveltimes by directly solving the eikonal equation on a regular grid, without computing raypaths. Solving the eikonal equation on such a grid simplifies the problem of interpolating times onto the migration grid, but this method is not well defined at points where two different branches of the traveltime field meet. Also, computational and data storage issues that are relatively unimportant for performance in two dimensions limit the applicability of both schemes in three dimensions. A new implementation of a gridded eikonal equation solver has been designed to address these problems. A 2-D version of this algorithm is tested by using it to generate traveltimes to migrate the Marmousi synthetic data set using the exact velocity model. The results are compared with three other images: an F-X migration (a standard for comparison), a Kirchhoff migration using ray tracing, and a Kirchhoff migration using traveltimes generated by a commonly used eikonal equation solver. The F-X‐migrated image shows the imaging objective more clearly than any of the Kirchhoff migrations, and we advance a heuristic reason to explain this fact. Of the Kirchhoff migrations, the one using ray tracing produces the best image, and the other two are of comparable quality.

Ultrashallow depth imaging of a channel stratigraphy with first-arrival traveltime inversion and prestack depth migration: A case history from Bull Creek, Oklahoma

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. B87-B96 ◽  
Author(s):  
Ammanuel Fesseha Woldearegay ◽  
Priyank Jaiswal ◽  
Alexander R. Simms ◽  
Hanna Alexander ◽  
Leland C. Bement ◽  
...  
Keyword(s):  
Velocity Model ◽  
Depth Image ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Imaging ◽  
Traveltime Inversion ◽  
Time Processing ◽  
Depth Migration ◽  
First Arrival ◽  
Bull Creek

Depth imaging in ultrashallow ([Formula: see text]) environments presents twofold challenge: (1) coda available for depth migration is very limited; and (2) conventional time processing with limited coda generally fails to estimate reliable velocity models for depth migration. We studied the combining of first-arrival traveltime inversion and prestack depth migration (PSDM) for depth imaging of ultrashallow paleochannel stratigraphy associated with the Bull Creek drainage system, Oklahoma. Restricted by a limited number of geophones (24) we acquired data for inversion and migration through two coincident profiles. The first profile for inversion has a wider survey-aperture (115-m maximum shot-receiver spacing) and consequently sparse CMP spacing (2.5 m), whereas the second profile for PSDM has denser CMP spacing (1 m) and consequently a narrower survey aperture (46-m maximum shot-receiver spacing). We also found that the velocity model from traveltime inversion of the wider-aperture data set is more preferable for depth-migration than the velocity model from time processing of the denser data set. The preferred depth image showed three episodes of incision whose chronological order is resolved through radio-carbon dating of terrace sediments. Results suggested that even with limited geophones, depth imaging of ultrashallow targets can be achieved by combining first-arrival traveltime inversion and PSDM through coincident wide- and narrow-aperture acquisitions.

Prestack Gaussian‐beam depth migration

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1240-1250 ◽  
Author(s):  
N. Ross Hill
Keyword(s):  
Gaussian Beam ◽  
Velocity Structure ◽  
Saddle Points ◽  
Three Dimensional ◽  
Seismic Energy ◽  
Data Set ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Beam Depth ◽  
Gaussian Beam Migration

Kirchhoff migration is the most popular method of three‐dimensional prestack depth migration because of its flexibility and efficiency. Its effectiveness can become limited, however, when complex velocity structure causes multipathing of seismic energy. An alternative is Gaussian beam migration, which is an extension of Kirchhoff migration that overcomes many of the problems caused by multipathing. Unlike first‐arrival and most‐energetic‐arrival methods, which retain only one traveltime, this alternative method retains most arrivals by the superposition of Gaussian beams. This paper presents a prestack Gaussian beam migration method that operates on common‐offset gathers. The method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points. Gaussian beam migration of the two‐dimensional Marmousi test data set demonstrates the method’s effectiveness for structural imaging in a case where there is multipathing of seismic energy.

Geometrical spreading corrections of offset reflections in a laterally inhomogeneous earth

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 1054-1063 ◽  
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Three Dimensional ◽  
Velocity Model ◽  
Earth Model ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Depth Migration ◽  
Avo Analysis ◽  
Amplitude Versus Offset ◽  
One Step

Compressional primary seismic nonzero offset reflections are the most essential wavefield attributes used in seismic parameter estimation and imaging. We show how the determination of angle‐dependent reflection coefficients can be addressed from identifying such events for arbitrarily curved three‐dimensional (3-D) subsurface reflectors below a laterally inhomogeneous layered overburden. More explicitly, we show how the geometrical‐spreading factor along a reflected primary ray with offset can be calculated from the identified (i.e., picked) traveltimes of offset primary reflections. Seismic traces in which all primary reflections are corrected with the geometrical‐spreading factor are, as is well‐known, referred to as true‐amplitude traces. They can be constructed without any knowledge of the velocity distribution in the earth model. Apart from possibly finding a direct application in an amplitude‐versus‐offset (AVO) analysis, the theory developed here can be of use to derive true‐amplitude time‐ and depth‐migration methods for various seismic data acquisition configurations, which pursue the aim of performing the wavefield migration (based upon the use of a macro‐velocity model) and the AVO analysis in one step.

Efficient 2.5-D true‐amplitude migration

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 943-950 ◽  
Author(s):  
Joe A. Dellinger ◽  
Samuel H. Gray ◽  
Gary E. Murphy ◽  
John T. Etgen
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Three Dimensions ◽  
Additional Source ◽  
Standard Methods ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Out Of Plane ◽  
Kirchhoff Method ◽  
Kirchhoff Depth Migration

Kirchhoff depth migration is a widely used algorithm for imaging seismic data in both two and three dimensions. To perform the summation at the heart of the algorithm, standard Kirchhoff migration requires a traveltime map for each source and receiver. True‐amplitude Kirchhoff migration in 2.5-D υ(x, z) media additionally requires maps of amplitudes, out‐of‐plane spreading factors, and takeoff angles; these quantities are necessary for calculating the true‐amplitude weight term in the summation. The increased input/output (I/O) and computational expense of including the true‐amplitude weight term is often not justified by significant improvement in the final muted and stacked image. For this reason, some authors advocate neglecting the weight term in the Kirchhoff summation entirely for most everyday imaging purposes. We demonstrate that for nearly the same expense as ignoring the weight term, a much better solution is possible. We first approximate the true‐amplitude weight term by the weight term for constant‐velocity media; this eliminates the need for additional source and receiver maps. With one small additional approximation, the weight term can then be moved entirely outside the innermost loop of the summation. The resulting Kirchhoff method produces images that are almost as good as for exact true‐amplitude Kirchhoff migration and at almost the same cost as standard methods that do not attempt to preserve amplitudes.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

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