Recovering Acoustic and Elastic Parameters From Travel Times

2002 ◽  
Author(s):  
Gunther Uhlmann ◽  
So¨nke Hansen
Keyword(s):  
Phase Space ◽  
Anisotropic Media ◽  
Geometrical Optics ◽  
Index Of Refraction ◽  
Travel Times ◽  
Elastic Parameters ◽  
Space Setting ◽  
Additional Assumptions

The relation between travel times and waves in anisotropic media is explained using the geometrical optics method in a phase space setting. This approach also covers caustics and multiple arrivals. We than consider the question of whether one can determine an anisotropic index of refraction by measuring travel times. We show that this is indeed the case if the index of refraction satisfies some additional assumptions.

scholarly journals A new phase space method for recovering index of refraction from travel times

2007 ◽  
Vol 23 (1) ◽  
pp. 309-329 ◽  
Author(s):  
Eric Chung ◽  
Jianliang Qian ◽  
Gunther Uhlmann ◽  
Hongkai Zhao
Keyword(s):  
Phase Space ◽  
Index Of Refraction ◽  
Travel Times ◽  
New Phase ◽  
Space Method

Enabling isotropic reflectivity modeling and inversion in anisotropic media

2021 ◽  
Vol 40 (4) ◽  
pp. 267-276
Author(s):  
Peter Mesdag ◽  
Leonardo Quevedo ◽  
Cătălin Tănase
Keyword(s):  
Synthetic Data ◽  
Anisotropic Media ◽  
Forward Modeling ◽  
Seismic Inversion ◽  
Stratified Medium ◽  
Effective Parameters ◽  
Elastic Parameters ◽  
Pp Wave ◽  
Anisotropic Elastic ◽  
And Inversion

Exploration and development of unconventional reservoirs, where fractures and in-situ stresses play a key role, call for improved characterization workflows. Here, we expand on a previously proposed method that makes use of standard isotropic modeling and inversion techniques in anisotropic media. Based on approximations for PP-wave reflection coefficients in orthorhombic media, we build a set of transforms that map the isotropic elastic parameters used in prestack inversion into effective anisotropic elastic parameters. When used in isotropic forward modeling and inversion, these effective parameters accurately mimic the anisotropic reflectivity behavior of the seismic data, thus closing the loop between well-log data and seismic inversion results in the anisotropic case. We show that modeling and inversion of orthorhombic anisotropic media can be achieved by superimposing effective elastic parameters describing the behavior of a horizontally stratified medium and a set of parallel vertical fractures. The process of sequential forward modeling and postinversion analysis is exemplified using synthetic data.

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.

scholarly journals Brightness of synchrotron radiation from undulators and bending magnets

2015 ◽  
Vol 22 (2) ◽  
pp. 288-316 ◽  
Author(s):  
Gianluca Geloni ◽  
Vitali Kocharyan ◽  
Evgeni Saldin
Keyword(s):  
Electron Beam ◽  
Phase Space ◽  
Synchrotron Radiation ◽  
Wigner Distribution ◽  
Geometrical Optics ◽  
Single Electron ◽  
Beam Divergence ◽  
Photon Flux ◽  
Photon Flux Density ◽  
Beam Size

The maximum of the Wigner distribution (WD) of synchrotron radiation (SR) fields is considered as a possible definition of SR source brightness. Such a figure of merit was originally introduced in the SR community by Kim [(1986),Nucl. Instrum. Methods Phys. Res. A,246, 71–76]. The brightness defined in this way is always positive and, in the geometrical optics limit, can be interpreted as the maximum density of photon flux in phase space. For undulator and bending magnet radiation from a single electron, the WD function can be explicitly calculated. In the case of an electron beam with a finite emittance the brightness is given by the maximum of the convolution of a single electron WD function and the probability distribution of the electrons in phase space. In the particular case when both electron beam size and electron beam divergence dominate over the diffraction size and the diffraction angle, one can use a geometrical optics approach. However, there are intermediate regimes when only the electron beam size or the electron beam divergence dominate. In these asymptotic cases the geometrical optics approach is still applicable, and the brightness definition used here yields back once more to the maximum photon flux density in phase space. In these intermediate regimes a significant numerical disagreement is found between exact calculations and the approximation for undulator brightness currently used in the literature. The WD formalism is extended to a satisfactory theory for the brightness of a bending magnet. It is found that in the intermediate regimes the usually accepted approximation for bending magnet brightness turns out to be inconsistent even parametrically.

Optical Electromagnetics I

2006 ◽  
Author(s):  
Michael E. Thomas
Keyword(s):  
Electromagnetic Field ◽  
Optical Spectrum ◽  
Geometrical Optics ◽  
Theoretical Description ◽  
Index Of Refraction ◽  
Wave Number ◽  
Physical Optics ◽  
Electromagnetic Propagation ◽  
Optical Propagation ◽  
Precise Knowledge

In this chapter, the optical spectrum is defined and subdivided into many sub-bands, which are traditionally determined by transparency in various media. Propagation of the electromagnetic field in vacuum, as based on Maxwell’s equations, and basic notions of geometrical and physical optics, are covered. The theoretical and conceptual foundation of the remaining chapters is established in this chapter and the next. Optical electromagnetic propagation is generally and often accurately described by classical geometrical optics or ray optics. When diffraction or wave interference is of concern, then the more complete field of physical optics is used. Geometrical optics requires precise knowledge of the spatial and spectral dependence of the index of refraction. This requires electrodynamics, which is most appropriately described by quantum optics. These topics are covered in the first five chapters. The definitions of the optical spectrum and the various models for describing propagation are introduced in the following. The optical electromagnetic field covers the range of frequencies from microwaves to the ultraviolet (UV) or wavelengths from 10 cm to 100 nm. This is a very liberal definition covering six orders of magnitude, yet the description of propagation is very similar over this entire band, and distinct from radio-wave propagation and x-ray propagation. A listing of the nomenclature for the different spectral bands within the range of optical wavelengths is given in Table 1.1. Other commonly used units of spectral measure such as wave number, frequency, and energy are also listed in the table. These various quantities are related to wavelength by the following formulas: where c is the speed of light (c = 2.99792458 × 108 m/sec), λ is wavelength, f is frequency in hertz, E is energy, h is Planck’s constant (h = 6.6260755(40) × 10−34 J sec), and ν is frequency in wave numbers (the number of wavelengths per centimeter). Although wavelength is commonly used by applied scientists and engineers, frequency is the most appropriate unit for the theoretical description of light–matter interactions. Because of the importance of spectroscopy in the discussion of optical propagation, the spectroscopic unit of wave number will be consistently used.

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