Paraxial ray Kirchhoff migration

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1540-1546 ◽  
Author(s):  
T. H. Keho ◽  
W. B. Beydoun
Keyword(s):  
Ray Tracing ◽  
Green's Functions ◽  
Green’S Functions ◽  
Synthetic Data ◽  
Computation Time ◽  
Ray Method ◽  
Kirchhoff Migration ◽  
Order Of Magnitude ◽  
Vsp Data ◽  
Paraxial Ray

A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.

The paraxial ray method

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1639-1653 ◽  
Author(s):  
Wafik B. Beydoun ◽  
Timothy H. Keho
Keyword(s):  
Wave Field ◽  
Ray Tracing ◽  
Heterogeneous Media ◽  
Computation Time ◽  
Ray Method ◽  
Ray Tracing Method ◽  
Paraxial Ray ◽  
The Cost ◽  
Seismic Wave Field ◽  
Tracing Method

The paraxial ray method is an economical way of computing approximate Green’s functions in heterogeneous media. The method uses information from the standard dynamic ray‐tracing method to extrapolate the seismic wave field at receivers in the neighborhood of a ray so that two‐point ray tracing is not required. Applicability conditions are explicit: they define where asymptotic (high‐frequency) methods are valid, and how far away from the ray the extrapolation remains accurate. Increasing the density of the ray fan improves accuracy but increases computation time. However, since reasonable accuracy is obtained with relatively few rays, the method yields results similar to the two‐point ray‐tracing method, but at a fraction of the cost. Examples of wave‐field extrapolation from a ray to neighboring receivers show that traveltime extrapolation is more accurate than amplitude extrapolation. Accuracy, robustness, and efficiency tests, comparing paraxial ray synthetic seismograms with acoustic finite‐difference and elastic discrete‐wavenumber synthetics, are judged very satisfactory.

Prestack Gaussian-beam depth migration in anisotropic media

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S133-S138 ◽  
Author(s):  
Tianfei Zhu ◽  
Samuel H. Gray ◽  
Daoliu Wang
Keyword(s):  
Gaussian Beam ◽  
Ray Tracing ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Elastic Parameters ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Beam Depth ◽  
Dynamic Ray Tracing

Gaussian-beam depth migration is a useful alternative to Kirchhoff and wave-equation migrations. It overcomes the limitations of Kirchhoff migration in imaging multipathing arrivals, while retaining its efficiency and its capability of imaging steep dips with turning waves. Extension of this migration method to anisotropic media has, however, been hampered by the difficulties in traditional kinematic and dynamic ray-tracing systems in inhomogeneous, anisotropic media. Formulated in terms of elastic parameters, the traditional anisotropic ray-tracing systems aredifficult to implement and inefficient for computation, especially for the dynamic ray-tracing system. They may also result inambiguity in specifying elastic parameters for a given medium.To overcome these difficulties, we have reformulated the ray-tracing systems in terms of phase velocity.These reformulated systems are simple and especially useful for general transversely isotropic and weak orthorhombic media, because the phase velocities for these two types of media can be computed with simple analytic expressions. These two types of media also represent the majority of anisotropy observed in sedimentary rocks. Based on these newly developed ray-tracing systems, we have extended prestack Gaussian-beam depth migration to general transversely isotropic media. Test results with synthetic data show that our anisotropic, prestack Gaussian-beam migration is accurate and efficient. It produces images superior to those generated by anisotropic, prestack Kirchhoff migration.

Hybrid shortest path and ray bending method for traveltime and raypath calculations

Geophysics ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 648-653 ◽  
Author(s):  
Harm J. A. Van Avendonk ◽  
Alistair J. Harding ◽  
John A. Orcutt ◽  
W. Steven Holbrook
Keyword(s):  
Ray Tracing ◽  
Shortest Path ◽  
Velocity Structure ◽  
Seismic Velocity ◽  
Seismic Refraction ◽  
Computation Time ◽  
Hybrid Scheme ◽  
Seismic Velocity Structure ◽  
Order Of Magnitude ◽  
Ray Bending

The shortest path method (SPM) is a robust ray‐tracing technique that is particularly useful in 3-D tomographic studies because the method is well suited for a strongly heterogeneous seismic velocity structure. We test the accuracy of its traveltime calculations with a seismic velocity structure for which the nearly exact solution is easily found by conventional ray shooting. The errors in the 3-D SPM solution are strongly dependent on the choice of search directions in the “forward star,” and these errors appear to accumulate with traveled distance. We investigate whether these traveltime errors can be removed most efficiently by an SPM calculation on a finer grid or by additional ray bending. Testing the hybrid scheme on a realistic ray‐tracing example, we find that in an efficient mix ray bending and SPM account for roughly equal amounts of computation time. The hybrid method proves to be an order of magnitude more efficient than SPM without ray bending in our example. We advocate the hybrid ray‐tracing technique, which offers an efficient approach to find raypaths and traveltimes for large seismic refraction studies with high accuracy.

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.

Fresnel volume ray tracing

Geophysics ◽  
1992 ◽  
Vol 57 (7) ◽  
pp. 902-915 ◽  
Author(s):  
Vlastislav Červený ◽  
José Eduardo P. Soares
Keyword(s):  
Ray Tracing ◽  
High Frequency ◽  
Fresnel Zone ◽  
Body Wave ◽  
Low Frequency ◽  
Algebraic Manipulation ◽  
Ray Method ◽  
Propagator Matrix ◽  
Paraxial Ray ◽  
Fresnel Volumes

The concept of “Fresnel volume ray tracing” consists of standard ray tracing, supplemented by a computation of parameters defining the first Fresnel zones at each point of the ray. The Fresnel volume represents a 3-D spatial equivalent of the Fresnel zone that can also be called a physical ray. The shape of the Fresnel volume depends on the position of the source and the receiver, the structure between them, and the type of body wave under consideration. In addition, the shape also depends on frequency: it is narrow for a high frequency and thick for a low frequency. An efficient algorithm for Fresnel volume ray tracing, based on the paraxial ray method, is proposed. The evaluation of the parameters defining the first Fresnel zone merely consists of a simple algebraic manipulation of the elements of the ray propagator matrix. The proposed algorithm may be applied to any high‐frequency seismic body wave propagating in a laterally varying 2-D or 3-D layered structure (P, S, converted, multiply reflected, etc.). Numerical examples of Fresnel volume ray tracing in 2-D inhomogeneous layered structures are presented. Certain interesting properties of Fresnel volumes are discussed (e.g., the double caustic effect). Fresnel volume ray tracing offers numerous applications in seismology and seismic prospecting. Among others, it can be used to study the resolution of the seismic method and the validity conditions of the ray method.

Green’s function implementation of common‐offset, wave‐equation migration

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1813-1821 ◽  
Author(s):  
Andreas Ehinger ◽  
Patrick Lailly ◽  
Kurt J. Marfurt
Keyword(s):  
Wave Equation ◽  
Green's Functions ◽  
Wave Equations ◽  
Green’S Functions ◽  
Data Set ◽  
Kirchhoff Migration ◽  
Migration Velocity Analysis ◽  
Standard Wave ◽  
The Cost ◽  
Wavefield Extrapolation

Common‐offset migration is extremely important in the context of migration velocity analysis (MVA) since it generates geologically interpretable migrated images. However, only a wave‐equation‐based migration handles multipathing of energy in contrast to the popular Kirchhoff migration with first‐arrival traveltimes. We have combined the superior treatment of multipathing of energy by wave‐equation‐based migration with the advantages of the common‐offset domain for MVA by implementing wave‐equation migration algorithms via the use of finite‐difference Green’s functions. With this technique, we are able to apply wave‐equation migration in measurement configurations that are usually considered to be of the realm of Kirchhoff migration. In particular, wave‐equation migration of common offset sections becomes feasible. The application of our wave‐equation, common‐offset migration algorithm to the Marmousi data set confirms the large increase in interpretability of individual migrated sections, for about twice the cost of standard wave‐equation common‐shot migration. Our implementation of wave‐equation migration via the Green’s functions is based on wavefield extrapolation via paraxial one‐way wave equations. For these equations, theoretical results allow us to perform exact inverse extrapolation of wavefields.

Rapid VSP-CDP mapping of 3-D VSP data

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1631-1640 ◽  
Author(s):  
Genmeng Chen ◽  
Janusz Peron ◽  
Luis Canales
Keyword(s):  
Ray Tracing ◽  
Velocity Model ◽  
Mapping Method ◽  
Computation Method ◽  
Layered Model ◽  
Seismic Profiling ◽  
Vertical Seismic Profiling ◽  
Kirchhoff Migration ◽  
Vsp Data ◽  
Image Volume

Vertical seismic profiling‐common depth point (VSPCDP) mapping with rapid ray tracing in a horizontally layered velocity model is used to create 3-D image volumes using Blackfoot and Oseberg 3-D vertical seismic profiling (VSP) data. The ray‐tracing algorithm uses Fermat’s principle and is specially programmed for the layered model. The algorithm is about ten times faster than either a 3-D VSP-CDP mapping program with an eikonal traveltime computation method or a 3-D VSP Kirchhoff migration program. The mapping method automatically separates the image zone from the nonimage zone within the 3-D image volume. The Oseberg data example shows that the lateral extent of the image zone created by the 3D VSP-CDP mapping is larger than that created by 3-D VSP Kirchhoff migration. The same sample result also provides high‐frequency events at target zones. We include an analysis of the imaging error induced from using a horizontally layered model for the Oseberg data, indicating that the method is reliable in the presence of gently dipping structure.

scholarly journals Electromagnetic modeling of three‐dimensional bodies in layered earths using integral equations

Geophysics ◽  
1984 ◽  
Vol 49 (1) ◽  
pp. 60-74 ◽  
Author(s):  
Philip E. Wannamaker ◽  
Gerald W. Hohmann ◽  
William A. SanFilipo
Keyword(s):  
Integral Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Computation Time ◽  
Vector Current ◽  
Hankel Transform ◽  
Transverse Magnetic ◽  
Electromagnetic Modeling ◽  
Method Of Integral Equations

We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodies of identical cross‐section using plane wave incident fields is the only check available on our solution. Agreement is excellent; however, the length that a 3-D body must have before departures between 2-D transverse electric and corresponding 3-D signatures are insignificant depends strongly on the layering. The 2-D transverse magnetic and corresponding 3-D calculations agree closely regardless of the layered host.

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