3D mapping of kinematic attributes in anisotropic media

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C159-C170 ◽  
Author(s):  
Yuriy Ivanov ◽  
Alexey Stovas
Keyword(s):  
Anisotropic Medium ◽  
Heterogeneous Media ◽  
Homogeneous Medium ◽  
Anisotropic Media ◽  
Transformation Matrix ◽  
3D Mapping ◽  
Slowness Surface ◽  
Point Correspondence ◽  
Geometric Spreading ◽  
Point To Point

Based on the rotation of a slowness surface in anisotropic media, we have derived a set of mapping operators that establishes a point-to-point correspondence for the traveltime and relative-geometric-spreading surfaces between these calculated in nonrotated and rotated media. The mapping approach allows one to efficiently obtain the aforementioned surfaces in a rotated anisotropic medium from precomputed surfaces in the nonrotated medium. The process consists of two steps: calculation of a necessary kinematic attribute in a nonrotated, e.g., orthorhombic (ORT), medium, and subsequent mapping of the obtained values to a transformed, e.g., rotated ORT, medium. The operators we obtained are applicable to anisotropic media of any type; they are 3D and are expressed through a general form of the transformation matrix. The mapping equations can be used to develop moveout and relative-geometric-spreading approximations in rotated anisotropic media from existing approximations in nonrotated media. Although our operators are derived in case of a homogeneous medium and for a one-way propagation only, we discuss their extension to vertically heterogeneous media and to reflected (and converted) waves.

MCVM: MONTE CARLO MODELING OF PHOTON MIGRATION IN VOXELIZED MEDIA

2010 ◽  
Vol 03 (02) ◽  
pp. 91-102 ◽  
Author(s):  
TING LI ◽  
HUI GONG ◽  
QINGMING LUO
Keyword(s):  
Monte Carlo ◽  
Refractive Index ◽  
High Precision ◽  
Heterogeneous Media ◽  
Homogeneous Medium ◽  
Software Tool ◽  
Light Transport ◽  
Photon Migration ◽  
Monte Carlo Code ◽  
Monte Carlo Modeling

The Monte Carlo code MCML (Monte Carlo modeling of light transport in multi-layered tissue) has been the gold standard for simulations of light transport in multi-layer tissue, but it is ineffective in the presence of three-dimensional (3D) heterogeneity. New techniques have been attempted to resolve this problem, such as MCLS, which is derived from MCML, and tMCimg, which draws upon image datasets. Nevertheless, these approaches are insufficient because of their low precision or simplistic modeling. We report on the development of a novel model for photon migration in voxelized media (MCVM) with 3D heterogeneity. Voxel crossing detection and refractive-index-unmatched boundaries were considered to improve the precision and eliminate dependence on refractive-index-matched tissue. Using a semi-infinite homogeneous medium, steady-state and time-resolved simulations of MCVM agreed well with MCML, with high precision (~100%) for the total diffuse reflectance and total fractional absorption compared to those of tMCimg (< 70%). Based on a refractive-index-matched heterogeneous skin model, the results of MCVM were found to coincide with those of MCLS. Finally, MCVM was applied to a two-layered sphere with multi-inclusions, which is an example of a 3D heterogeneous media with refractive-index-unmatched boundaries. MCVM provided a reliable model for simulation of photon migration in voxelized 3D heterogeneous media, and it was developed to be a flexible and simple software tool that delivers high-precision results.

A ray expansion with matrix coefficients for sourcesin absorbing anisotropic media

1976 ◽  
Vol 15 (1) ◽  
pp. 151-163 ◽  
Author(s):  
J. A. Bennett
Keyword(s):  
Mode Coupling ◽  
Homogeneous Medium ◽  
Anisotropic Media ◽  
Far Field ◽  
Wave Polarization ◽  
Problem Method ◽  
Matrix Coefficients ◽  
Leading Term ◽  
Green’S Matrix ◽  
Complex Rays

A ray or quasi-optical approximation is developed, using complex rays. The ‘amplitude’ terms are matrices, rather than vectors that represent the wave polarization. Thus, the way the propagation resolves a source into various modes is described. The second term in the amplitude series is shown to include a type of inter-mode coupling. It is shown that initial values needed to integrate along the rays can be chosen so that the leading term of the approximation agrees with the far-field solution for localized sources in a homogeneous medium. By invoking the ‘canonical problem’ method, the result is extended to give an approximation for the Green's matrix in a slowly-varying medium.

Anisotropic Born scattering for the qP scalar wavefield using a low-rank symbol approximation

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Iury Araújo ◽  
Murillo Nascimento ◽  
Jesse Costa ◽  
Alan Souza ◽  
Jörg Schleicher
Keyword(s):  
Anisotropic Medium ◽  
Mathematical Formulation ◽  
Hamiltonian Formulation ◽  
Anisotropic Media ◽  
Low Rank ◽  
Direct Access ◽  
S Wave ◽  
Evolution Operators ◽  
Individual Parameter ◽  
Reflection Data

We present a procedure to derive low-rank evolution operators in the mixed space-wavenumber domain for modeling the qP Born-scattered wavefield at perturbations of an anisotropic medium under the pseudo-acoustic approximation. To approximate the full wavefield, this scattered field is then added to the reference wavefield obtained with the corresponding low-rank evolution operator in the background medium. Being built upon a Hamiltonian formulation using the dispersion relation for qP waves, this procedure avoids pseudo-S-wave artifacts and provides a unified approach for linearizing anisotropic pseudo-acoustic evolution operators. Therefore it is immediately applicable to any arbitrary class of anisotropy. As an additional asset, the scattering operators explicitly contain the sensitivity kernels of the Born-scattered wavefield with respect to the anisotropic medium parameters. This enables direct access to important information like its offset dependence or directional characteristics as a function of the individual parameter perturbations. For our numerical tests, we specify the operators for a mildly anisotropic tilted transversly isotropic (TTI) medium. We validate our implementation in a simple model with weak contrasts and simulate reflection data in the BP TTI model to show that the procedure works in a more realistic scenario. The Born-scattering results indicate that our procedure is applicable to strongly heterogeneous anisotropic media. Moreover, we use the analytical capabilities of the kernels by means of sensitivity tests to demonstrate that using two different medium parameterizations leads to different results. The mathematical formulation of the method is such that it allows for an immediate application to least-squares migration in pseudo-acoustic anisotropic media.

The wave front in a homogeneous anisotropic medium

1980 ◽  
Vol 70 (6) ◽  
pp. 2097-2101
Author(s):  
M. J. Yedlin
Keyword(s):  
Dispersion Relation ◽  
Wave Front ◽  
Group Velocity ◽  
Anisotropic Medium ◽  
Slowness Surface ◽  
Constant Frequency ◽  
Common Notion ◽  
The Common ◽  
Intuitive Method ◽  
Homogeneous Anisotropic Medium

abstract A simple geometric construction is derived for the shape of the wave front in a homogeneous anisotropic medium. It is shown to be equivalent to the intuitive method of constructing a wave front using Huygen's principle. Although this construction has been referred to and tersely described in the literature (Musgrave, 1970; Kraut, 1963; Duff, 1960), it is instructive to demonstrate its relationship to the common notion of the wave front obtained via consideration of the group velocity. The wave front is shown to be the polar reciprocal of the slowness surface (the dispersion relation at constant frequency). An appreciation of the pole-polar correspondence between the two surfaces allows quick inference of some of the important features of the wave front in a homogeneous anisotropic medium.

Disturbances in semi-infinite heterogeneous media generated by torsional sources. I

1972 ◽  
Vol 62 (2) ◽  
pp. 541-550
Author(s):  
R. S. Sidhu
Keyword(s):  
Heterogeneous Media ◽  
Formal Solution ◽  
Homogeneous Medium ◽  
Finite Depth ◽  
Free Boundaries ◽  
Axially Symmetric ◽  
Sh Waves ◽  
Reflected Waves ◽  
Buried Point ◽  
Heterogeneous Layer

abstract This paper studies the generation of axially symmetric transient SH waves in semi-infinite heterogeneous media in which μ and ρ vary with depth. The sources generating these waves are taken in the form of time-dependent torsional-body forces of finite dimensions. The solution is obtained using Hankel and Laplace transforms and Green's function. The disturbance from a buried point source of impulsive type is discussed in two cases, (a) μ = μo(1 + ɛz)2, ρ = ρo (1 + ɛz)2, (b) μ = μoe2az, ρ = ρoe2az. It is shown that, in contrast to the results for a homogeneous medium, in case (i), the wave reflected by the free surface generates secondary disturbances which trail behind the wave front and die out as t increases; the incident wave in this medium generates no such disturbance. In case (ii), however, both the incident as well as the reflected waves generate secondary disturbances. Formal solution for the disturbance in a heterogeneous layer of finite depth with stress-free boundaries is discussed in Appendix II.

scholarly journals Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

Seismic complex ray tracing in 2D/3D viscoelastic anisotropic media by a modified shortest-path method

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. T331-T342
Author(s):  
Xing-Wang Li ◽  
Bing Zhou ◽  
Chao-Ying Bai ◽  
Jian-Lu Wu
Keyword(s):  
Ray Tracing ◽  
Shortest Path ◽  
Anisotropic Medium ◽  
Wave Energy ◽  
Seismic Data ◽  
Seismic Waves ◽  
Effective Means ◽  
Anisotropic Media ◽  
The Real ◽  
Complex Ray

In a viscoelastic anisotropic medium, velocity anisotropy and wave energy attenuation occur and are often observed in seismic data applications. Numerical investigation of seismic wave propagation in complex viscoelastic anisotropic media is very helpful in understanding seismic data and reconstructing subsurface structures. Seismic ray tracing is an effective means to study the propagation characteristics of high-frequency seismic waves. Unfortunately, most seismic ray-tracing methods and traveltime tomographic inversion algorithms only deal with elastic media and ignore the effect of viscoelasticity on the seismic raypath. We have developed a method to find the complex ray velocity that gives the seismic ray speed and attenuation in an arbitrary viscoelastic anisotropic medium, and we incorporate them with the modified shortest-path method to determine the raypath and calculate the real and imaginary traveltime (wave energy attenuation) simultaneously. We determine that the complex ray-tracing method is applicable to arbitrary 2D/3D viscoelastic anisotropic media in a complex geologic model and the computational errors of the real and imaginary traveltime are less than 0.36% and 0.59%, respectively. The numerical examples verify that the new method is an effective and powerful tool for accomplishing seismic complex ray tracing in heterogeneous viscoelastic anisotropic media.

SOME EFFECTS OF FORMATION ANISOTROPY ON RESISTIVITY MEASUREMENTS IN BOREHOLES

Geophysics ◽  
1958 ◽  
Vol 23 (4) ◽  
pp. 770-794 ◽  
Author(s):  
K. S. Kunz ◽  
J. H. Moran
Keyword(s):  
Anisotropic Medium ◽  
Wide Class ◽  
Anisotropic Media ◽  
Resistivity Measurements ◽  
Potential Problems ◽  
Isotropic Media ◽  
Equivalent Problems ◽  
Resistivity Logging

It is shown that a wide class of potential problems involving anisotropic media can be transformed into equivalent problems involving only isotropic media. By means of such transformations it is possible, in a large number of cases, to determine the apparent resistivities which would be observed in anisotropic formations, using electrode‐type resistivity logging devices. Discussion is given of an infinite, anisotropic medium with and without borehole, of two semi‐infinite anisotropic beds (without borehole), and of a thin isotropic bed bounded by anisotropic adjacent formations (without borehole). An interpretation chart for the normal device is presented for thick, non‐invaded, anisotropic beds penetrated by a borehole.

scholarly journals Scanning anisotropy parameters in complex media

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. U13-U22 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropic Medium ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
Transient Behavior ◽  
Complex Media ◽  
Linear Partial Differential Equations ◽  
Anisotropy Parameters ◽  
Axis Of Symmetry ◽  
Ti Media ◽  
Taylor’S Series

Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.

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