First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. D65-D75 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Ray Tracing ◽  
Anisotropic Media ◽  
Order Accuracy ◽  
Weak Anisotropy ◽  
Higher Symmetry ◽  
First Order ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Order Quantities ◽  
Relative Errors

We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays. The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.

Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

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.

Calculation of slowness vectors from ray directions for qP-, qSV-, and qSH-waves in tilted transversely isotropic media

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. C153-C160 ◽  
Author(s):  
Lijiao Zhang ◽  
Bing Zhou
Keyword(s):  
Ray Tracing ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Seismic Wave Propagation ◽  
Previous Method ◽  
New Methods ◽  
Slowness Vector ◽  
Isotropic Media ◽  
Tti Media ◽  
Wave Modes

Kinematic ray tracing is an effective way to simulate the seismic wave propagation in isotropic and anisotropic media. It is essential to know the ray velocity when tracing seismic rays. But in anisotropic media, the ray velocity is a function of the direction of the slowness vector instead of the ray direction and it often deviates from the phase velocity. In this case, it causes a critical problem for ray tracing, which is how to calculate the ray velocity from a known ray direction. If we could calculate the phase slowness vector from ray directions, the ray velocity could be computed. We have evaluated a previous method in the first place. Then, we developed two new methods to solve two existing problems of the previous method: (1) It leads to complex and multiple solutions of the slowness vector and (2) it mixes up the qP- and qSV-wave modes. Our first method solves the two problems by applying eigenvalues to separate the wave modes and decrease the two unknowns ([Formula: see text] and [Formula: see text]) to only one unknown in two equations. Our second method is based on the general relationship between the slowness and ray-velocity vectors and shows that only one unknown is involved in one equation for tilted transversely isotropic (TTI) media. After obtaining the slowness vector, the ray velocity can be computed easily. A 2D model is designed to test the feasibility of the new methods. Using the results for the model, we found that the presented approaches were applicable for ray tracing in TTI media.

Fluid dependence of anisotropy parameters in weakly anisotropic porous media

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC137-WC145 ◽  
Author(s):  
Olivia Collet ◽  
Boris Gurevich
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Threshold Value ◽  
Seismic Velocities ◽  
Anisotropic Porous Media ◽  
Weak Anisotropy ◽  
Shear Moduli ◽  
Saturated Media ◽  
Anisotropy Parameters ◽  
Saturated Medium

Predicting seismic velocities in isotropic fluid-saturated rocks is commonly done using the isotropic Gassmann theory. For anisotropic media, the solution is expressed in terms of stiffness or compliance, which does not provide an intuitive understanding on how the fluid affects wave propagation in anisotropic media. Assuming weak anisotropy, we expressed the anisotropy parameters of transversely isotropic saturated media as a function of the anisotropy parameters in the dry medium, the bulk and shear moduli of the saturated and dry media, the grain and fluid bulk moduli, and the porosity. By deriving an approximation of the anellipticity parameter [Formula: see text], we discovered that if the dry medium was elliptical, the saturated medium was also elliptical but only if the porosity exceeded a certain threshold value. This result can provide a way of differentiating between stress- and fracture-induced anisotropy.

Approximate ray tracing for qP-waves in inhomogeneous layered media with weak structural anisotropy

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM47-SM60 ◽  
Author(s):  
Kaveh Dehghan ◽  
Véronique Farra ◽  
Laurence Nicolétis
Keyword(s):  
Ray Tracing ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Structural Anisotropy ◽  
First Order ◽  
Wave Ray ◽  
Order Quantities ◽  
Paraxial Ray ◽  
Approximate Equations

We present approximate equations for qP-wave ray tracing and paraxial ray tracing in inhomogeneous layered weakly transversely isotropic (TI) media. Inside layers, the symmetry axis direction of the TI medium is allowed to vary continuously. Approximate equations are based on perturbation theory in which deviations of anisotropy from isotropy are considered to be first-order quantities. For qP-waves propagating in a TI medium, the approximate ray-tracing and paraxial ray-tracing equations depend on three parameters and two angles defining the direction of the symmetry axis. We also present the boundary conditions at interfaces for first-order rays and paraxial rays and compute reflection/transmission coefficients to the first order. All the quantities required for evaluation of the Green’s function are calculated to the first order, except the traveltime that is calculated to the second order. The accuracy of the presented algorithm is verified on simple models with respect to exact ray-tracing results. For anisotropy of about [Formula: see text], considered in the examples presented, the relative errors in traveltime and amplitude are less than [Formula: see text] and [Formula: see text], respectively. We then consider media where the symmetry axis is conformable with the structure, named structural transverse isotropy (STI). The method is applied to simple STI models and to a realistic overthrust model, showing significant differences with vertical transverse isotropy (VTI) models not only for traveltimes, but also for amplitudes.

scholarly journals Application of Velocity Variation with Angle (VVA) Method on an Anisotropic Model with Thomsen Delta Anisotropy Parameters

2018 ◽  
Vol 16 (2) ◽  
pp. 6
Author(s):  
Waskito Pranowo ◽  
Sonny Winardhi
Keyword(s):  
Wave Velocity ◽  
Synthetic Data ◽  
Velocity Variation ◽  
Grid Search ◽  
Anisotropic Model ◽  
Analysis Method ◽  
Weak Anisotropy ◽  
Hockey Stick ◽  
Isotropic Media ◽  
Anisotropy Parameters

Anisotropic properties will influence seismic propagation, for example it will affect wave velocity. One of well-known anisotropi equation for Transversaly Isotropic media is weak anisotropy with Thomsen's notation. Supriyono [2011] tried to estimate all of these variables by using velocity variation with angle (VVA) attribute. This research uses synthetic data, which is CMP Gather to know limitations of VVA attribute, to identify the error values, and to determine the best indicator of anisotropic eect. This research also uses another analysis method, which is grid search inversion to estimate VP0. From this research, Both VVA and grid search invesion still produce signcant error. The effects which will appear because of anisotropic property's presence are hockey-stick and over NMO-stretching.

True-amplitude Kirchhoff depth migration in anisotropic media: The traveltime-based approach

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC33-WC39 ◽  
Author(s):  
Claudia Vanelle ◽  
Dirk Gajewski
Keyword(s):  
Ray Tracing ◽  
Reservoir Characterization ◽  
Seismic Anisotropy ◽  
Anisotropic Media ◽  
Weight Functions ◽  
Depth Migration ◽  
Dynamic Ray Tracing ◽  
Isotropic Media ◽  
Structural Image ◽  
Kirchhoff Depth Migration

True-amplitude Kirchhoff depth migration is a classic tool in seismic imaging. In addition to a focused structural image, it also provides information on the strength of the reflectors in the model, leading to estimates of the shear properties of the subsurface. This information is a key feature not only for reservoir characterization, but it is also important for detecting seismic anisotropy. If anisotropy is present, it needs to be accounted for during the migration. True-amplitude Kirchhoff depth migration is carried out in terms of a weighted diffraction stack. Expressions for suitable weight functions exist in anisotropic media. However, the conventional means of computing the weights is based on dynamic ray tracing, which has high requirements on the smoothness of the underlying model. We developed a method for the computation of the weight functions that does not require dynamic ray tracing because all necessary quantities are determined from traveltimes alone. In addition, the method led to considerable savings in computational costs. This so-called traveltime-based strategy was already introduced for isotropic media. We extended the strategy to incorporate anisotropy. For verification purposes and comparison to analytic references, we evaluated 2.5D migration examples for [Formula: see text] and [Formula: see text] reflections. Our results confirmed the high image quality and the accuracy of the reconstructed reflectivities.

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