Relationship between phase velocities and polarization in transversely isotropic media

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1578-1583 ◽  
Author(s):  
J. de Parscau
Keyword(s):  
Incidence Angle ◽  
Transversely Isotropic ◽  
Multiple Source ◽  
P Wave ◽  
Accurate Estimation ◽  
Polarization Angle ◽  
Exact Model ◽  
Phase Velocities ◽  
Wave Phase ◽  
Anisotropy Parameters

Most techniques used to estimate anisotropy from multiple‐source offset VSP data assume angles measured from particle motion as an incidence angle. However, the difference between P‐wave polarization and the propagation direction for an anisotropic medium can be higher than 8 degrees. This difference provides a nonnegligible error in the estimation of anisotropy parameters from phase velocities. An exact model, proposed to describe P‐ and SV‐phase velocity variations for a transversely isotropic medium (TIM), takes into account the polarization angles. This model is a function of two anisotropy parameters (η and τ), of the vertical P‐ and SV‐wave phase velocities and of the polarization angle γ. However, η and τ can be used to express the polarization angle equation in a much simpler way. To quantify the error in estimated anisotropy parameters due to the assumption that the polarization angle is equal to the incidence angle, I study five TIMs. Each medium has an anisotropy that is representative of those observed in seismic surveying. The anisotropy parameters are recovered by inverting the P‐ and SV‐wave phase velocities for different incidence angles, and these incidence angles are assumed to be equal to the corresponding polarization angles. The mean error in estimated parameters is about 10 percent. This error is about the same as the one that would be obtained for velocities with uncertainties in their measurements. Unfortunately, the inversion of phase velocities measured from a real multiple‐source offset VSP to estimate anisotropy parameters needs, for calculating the misfit function, to add both errors in velocities due to hypothesis for angles and errors in velocity measurements due to uncertainties in data. In this case an exact model eliminates errors due to the assumption for the model and provides a more accurate estimation of anisotropy parameters.

Measurements of Rayleigh-wave phase velocities in Nevada: Implications for explosion sources and the Massachusetts Mountain earthquake

1982 ◽  
Vol 72 (4) ◽  
pp. 1329-1349
Author(s):  
H. J. Patton
Keyword(s):  
Rayleigh Wave ◽  
Rayleigh Waves ◽  
Moment Tensor ◽  
Test Site ◽  
P Wave ◽  
Stress Axis ◽  
Phase Velocities ◽  
Wave Phase ◽  
Single Station ◽  
Source Phase

abstract Single-station measurements of Rayleigh-wave phase velocity are obtained for paths between the Nevada Test Site and the Livermore broadband regional stations. Nuclear underground explosions detonated in Yucca Valley were the sources of the Rayleigh waves. The source phase φs required by the single-station method is calculated for an explosion source by assuming a spherically symmetric point source with step-function time dependence. The phase velocities are used to analyze the Rayleigh waves of the Massachusetts Mountain earthquake of 5 August 1971. Measured values of source phase for this earthquake are consistent with the focal mechanism determined from P-wave first-motion data (Fischer et al., 1972). A moment-tensor inversion of the Rayleigh-wave spectra for a 3-km-deep source gives a horizontal, least-compressive stress axis oriented N63°W and a seismic moment of 5.5 × 1022 dyne-cm. The general agreement between the results of the P-wave study of Fischer et al. (1972) and this study supports the measurements of phase velocities and, in turn, the explosion source model used to calculate φs.

Effective elastic properties of rocks with transversely isotropic background permeated by a set of vertical fracture cluster

Geophysics ◽  
2021 ◽  
pp. 1-78
Author(s):  
Da Shuai ◽  
Alexey Stovas ◽  
Jianxin Wei ◽  
Bangrang Di ◽  
Yang Zhao
Keyword(s):  
Standard Deviation ◽  
Elastic Wave ◽  
Significant Influence ◽  
Transversely Isotropic ◽  
Azimuth Angle ◽  
Effective Medium ◽  
P Wave ◽  
Wave Velocities ◽  
Anisotropy Parameters ◽  
Vertical Fractures

The linear slip theory is gradually being used to characterize seismic anisotropy. If the transversely isotropic medium embeds vertical fractures (VFTI medium), the effective medium becomes orthorhombic. The vertical fractures, in reality, may exist in any azimuth angle which leads the effective medium to be monoclinic. We apply the linear slip theory to create a monoclinic medium by only introducing three more physical meaning parameters: the fracture preferred azimuth angle, the fracture azimuth angle, and the angular standard deviation. First, we summarize the effective compliance of a rock as the sum of the background matrix compliance and the fracture excess compliance. Then, we apply the Bond transformation to rotate the fractures to be azimuth dependent, introduce a Gaussian function to describe the fractures' azimuth distribution assuming that the fractures are statistically distributed around the preferred azimuth angle, and average each fracture excess compliance over azimuth. The numerical examples investigate the influence of the fracture azimuth distribution domain and angular standard deviation on the effective stiffness coefficients, elastic wave velocities, and anisotropy parameters. Our results show that the fracture cluster parameters have a significant influence on the elastic wave velocities. The fracture azimuth distribution domain and angular standard deviation have a bigger influence on the orthorhombic anisotropy parameters in the ( x2, x3) plane than that in the ( x1, x3) plane. The fracture azimuth distribution domain and angular standard deviation have little influence on the monoclinic anisotropy parameters responsible for the P-wave NMO ellipse and have a significant influence on the monoclinic anisotropy parameters responsible for the S1- and S2-wave NMO ellipse. The effective monoclinic can be degenerated into the VFTI medium.

Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1855-1866 ◽  
Author(s):  
Jack K. Cohen
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Function ◽  
Transversely Isotropic Media ◽  
Wave Phase Velocity ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Vti Media ◽  
Ray Parameter

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P‐wave observations. They substantiated this result by using the weak‐anisotropy approximation and exploited it to develop a time‐domain processing sequence that takes into account vertical transverse isotropy. In this study, the two‐parameter Alkhalifah‐Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time‐domain P‐wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero‐dip NMO velocity—one of the Alkhalifah‐ Tsvankin parameters. The two‐parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear‐wave vertical velociy. The analytic results derived here are based on new representations for both the P‐wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second‐order expansions in the anisotropy parameters for both the P‐wave phase‐velocity function and NMO‐velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.

Decoupling approximation of P- and S-wave phase velocities in orthorhombic media

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Bowen Li ◽  
Alexey Stovas
Keyword(s):  
Phase Velocity ◽  
P Wave ◽  
Wave System ◽  
S Wave ◽  
Phase Velocities ◽  
Wave Phase ◽  
Algebraic Expressions ◽  
Wave Phase Velocity ◽  
Independent Parameters ◽  
Velocity Approximation

Characterizing the kinematics of seismic waves in elastic orthorhombic media involves nine independent parameters. All wave modes, P-, S1-, and S2-waves, are intrinsically coupled. Since the P-wave propagation in orthorhombic media is weakly dependent on the three S-wave velocity parameters, they are set to zero under the acoustic assumption. The number of parameters required for the corresponding acoustic wave equation is thus reduced from nine to six, which is very practical for the inversion algorithm. However, the acoustic wavefields generated by the finite-difference scheme suffer from two types of S-wave artifacts, which may result in noticeable numerical dispersion and even instability issues. Avoiding such artifacts requires a class of spectral methods based on the low-rank decomposition. To implement a six-parameter pure P-wave approximation in orthorhombic media, we develop a novel phase velocity approximation approach from the perspective of decoupling P- and S-waves. In the exact P-wave phase velocity expression, we find that the two algebraic expressions related to the S1- and S2-wave phase velocities play a negligible role. After replacing these two algebraic expressions with the designed constant and variable respectively, the exact P-wave phase velocity expression is greatly simplified and naturally decoupled from the characteristic equation. Similarly, the number of required parameters is reduced from nine to six. We also derive an approximate S-wave phase velocity equation, which supports the coupled S1- and S2-waves and involves nine independent parameters. Error analyses based on several orthorhombic models confirm the reasonable and stable accuracy performance of the proposed phase velocity approximation. We further derive the approximate dispersion relations for the P-wave and the S-wave system in orthorhombic media. Numerical experiments demonstrate that the corresponding P- and S-wavefields are free of artifacts and exhibit good accuracy and stability.

Multiparameter TTI tomography of P-wave reflection and VSP data

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC51-WC63 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Model Updating ◽  
Transversely Isotropic ◽  
P Wave ◽  
Ocean Bottom ◽  
Symmetry Direction ◽  
Direction Parallel ◽  
The North ◽  
Vsp Data ◽  
Anisotropy Parameters

Transversely isotropic models with a tilted symmetry axis (TTI media) are widely used in depth imaging of complex geologic structures. Here, we present a modification of a previously developed 2D P-wave tomographic algorithm for estimating heterogeneous TTI velocity fields and apply it to synthetic and field data. The symmetry-direction velocity [Formula: see text], anisotropy parameters [Formula: see text] and [Formula: see text], and symmetry-axis tilt [Formula: see text] are defined on a rectangular grid. To ensure stable reconstruction of the TTI parameters, reflection data are combined with walkaway vertical seismic profiling (VSP) traveltimes in joint tomographic inversion. To improve the convergence of the algorithm, we develop a three-stage model-updating procedure that gradually relaxes the constraints on the spatial variations of the anisotropy parameters, while the symmetry axis is kept orthogonal to the reflectors. Only at the final stage of the inversion are the parameters [Formula: see text], [Formula: see text], and [Formula: see text] updated on the same grid. We also incorporate geologic constraints into tomography by designing regularization terms that penalize parameter variations in the direction parallel to the interfaces. First, we examine the performance of the regularized joint tomography of reflection and VSP data for two sections of the BP TTI model that contain an anticline and a salt dome. All three TTI parameters in the shallow part of both sections (down to 5 km) are well resolved by the proposed model-updating process. Then, the algorithm is applied to a 2D section from 3D ocean-bottom seismic data acquired at Volve field in the North Sea. The inverted TTI model produces well-focused reflectors throughout the section and accurately positions the key horizons, which is confirmed by the available well markers.

Exact elastic impedance matrices for transversely isotropic medium

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. C1-C15 ◽  
Author(s):  
Feng Zhang ◽  
Xiang-Yang Li
Keyword(s):  
Shale Gas ◽  
Isotropic Medium ◽  
Seismic Reflection ◽  
Transversely Isotropic ◽  
Seismic Inversion ◽  
P Wave ◽  
Reflection Coefficients ◽  
Angle Of Incidence ◽  
Transversely Isotropic Medium ◽  
Anisotropy Parameters

Conventional elastic impedances are derived as scalars by means of the integration of reflectivity. In this sense, they are attributes of the seismic reflection but not the intrinsic physical property of the subsurface media. The derivation of these expressions shares the same assumptions as the reflectivity approximations, such as weak impedance contrast, small angle of incidence, or weak anisotropic media, and thus it limits the accuracy and interpretation capability. The exact P/SV impedance matrices relating the stress and strain represent the mechanical property of the subsurface media and yield reflection coefficients at an arbitrary angle of incidence. We have extended the impedance matrices to a transversely isotropic medium. The resulting elastic impedances (stress/velocity ratios) can be used to characterize those unconventional reservoir formations with strong seismic anisotropy, such as shale-gas and coal-bed methane. Our numerical analyses determined their variations with the phase angle and anisotropy parameters. The exact expressions of the P- and S-wave elastic impedances are used to model the seismic reflection coefficients, and thus they can be inverted simultaneously if all of the types of reflection waves are available. We then derive approximations of quasi-P-wave elastic impedances for seismic inversion of anisotropy parameters and further interpretation. Our applications on real logs and seismic data for a turbidite fan reservoir and a shale-gas reservoir determined the robust interpretation capability of derived elastic impedances in lithology characterizations.

Q values and wave inhomogeneity parameters of reflected inhomogeneous P and S waves at the free surface of an effective Biot solid

2020 ◽  
Vol 222 (2) ◽  
pp. 919-939 ◽  
Author(s):  
Xu Liu ◽  
Stewart Greenhalgh ◽  
Bing Zhou ◽  
Zhengyong Ren ◽  
Huijian Li
Keyword(s):  
Free Surface ◽  
Incidence Angle ◽  
Complex Form ◽  
Dissipation Factor ◽  
Seismic Wave Propagation ◽  
Energy Balance Equation ◽  
P Wave ◽  
Biot Theory ◽  
Reflected Waves ◽  
Phase Velocities

SUMMARY In this study, new methods are developed to estimate the dissipation factors, inhomogeneity parameters and phase velocities of the reflected waves at the free surface of a poro-viscoelastic solid in which the seismic wave propagation is described by effective Biot theory. The Christoffel equations of an effective Biot medium are solved for a general harmonic plane wave and the three complex velocities obtained corresponding to the shear wave (SV), fast-P wave and slow-P wave, together with their polarizations. Based on the complex form of the energy balance equation in an effective Biot material, expressions are derived for the energy ratios at the free surface. Moreover, the equations for the inhomogeneity parameters are derived as functions of the complex slowness or the unit polarization vectors. Based on the implicit and the explicit dissipation factor expressions, two methods are developed to obtain the dissipation factors, the inhomogeneity parameters and the phase velocities of mode-converted waves. These methods are illustrated by numerical examples which show that the dissipation factors, inhomogeneity parameters and phase velocities of reflected waves can strongly depend on the incidence angle (also reflected angle), the incident wave inhomogeneity parameter and the wave frequency. Ignoring these dependencies and using dissipation factors only valid for homogeneous waves can cause discrepancies in computed phase velocities and dissipation factors for interface generated (reflected/transmitted) inhomogeneous waves.

scholarly journals Helmholtz tomography of ambient noise surface wave data to estimate Scholte wave phase velocity at Valhall Life of the Field

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. WA99-WA109 ◽  
Author(s):  
Aurélien Mordret ◽  
Nikolaï M. Shapiro ◽  
Satish S. Singh ◽  
Philippe Roux ◽  
Olav I. Barkved
Keyword(s):  
Phase Velocity ◽  
Ambient Noise ◽  
Waveform Inversion ◽  
Wave Model ◽  
P Wave ◽  
Data Set ◽  
Phase Velocities ◽  
Wave Phase ◽  
Wave Phase Velocity ◽  
Scholte Wave

We applied the Helmholtz tomography technique to 6.5 hours of continuous seismic noise record data set of the Valhall Life of Field network. This network, that has 2320 receivers, allows us to perform a multifrequency, high-resolution, ambient-noise Scholte wave phase velocity tomography at Valhall. First, we computed crosscorrelations between all possible pairs of receivers to convert every station into a virtual source recorded by all other receivers. Our next step was to measure phase traveltimes and spectral amplitudes at different periods from crosscorrelations between stations separated by distances between two and six wavelengths. This is done in a straightforward fashion in the Fourier domain. Then, we interpolated these measurements onto a regular grid and computed local gradients of traveltimes and local Laplacians of the amplitude to infer local phase velocities using a frequency dependent Eikonal equation. This procedure was repeated for all 2320 virtual sources and final phase velocities were estimated as statistical average from all these measurements at each grid points. The resulting phase velocities for periods between 0.65 and 1.6 s demonstrate a significant dispersion with an increase of the phase velocities at longer periods. Their lateral distribution is found in very good agreement with previous ambient noise tomography done at Valhall as well as with a full waveform inversion P-wave model computed from an active seismic data set. We put effort into assessing the spatial resolution of our tomography with checkerboard tests, and we discuss the influence of the interpolation methods on the quality of our final models.

scholarly journals Estimation of in situ anisotropy parameters from two perpendicular walkaway VSP lines in South Pars field

2019 ◽  
Vol 16 (5) ◽  
pp. 926-938 ◽  
Author(s):  
Mohammad Mahdi Abedi ◽  
Mahtab Rashidi Fard ◽  
Mohammad Ali Riahi
Keyword(s):  
Wave Propagation ◽  
P Wave ◽  
Reasonable Assumption ◽  
Polarization Angle ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Estimated Parameters ◽  
Walkaway Vsp ◽  
Anisotropy Parameters

Abstract Vertical phase slowness and polarization angle are the in situ parameters of P-wave propagation that can be derived from walkaway vertical seismic profiling (VSP) data. To use these data for estimating anisotropy parameters, we obtain an explicit equation of vertical slowness as a function of polarization angle for P-wave propagation in transversely isotropic with vertical symmetry axis (VTI) media. We use this equation to estimate anisotropy parameters of a target layer in the South Pars field, Iran. This field is one of the world's largest gas fields. We show that the orthorhombic symmetry is a reasonable assumption for this layer, providing some geological and petrophysical information. Two walkaway VSP lines along the symmetry axes of the presumed orthorhombic layer are used to estimate its parameters. Seven is the maximum number of parameters that can be estimated using P-wave data in this acquisition pattern. Of those estimated parameters are six of the Tsvankin style parameters for orthorhombic media, plus an approximate combination of two others that define vertical S-wave splitting. We show that a previous method, which is based on weak anisotropy approximation, leads to considerable errors, even in models where the magnitude of anisotropy parameters do not exceed 0.1. We design a numerical experiment to study the reliability of the estimated parameters by the exact approach and show the importance of acquisition pattern in this regard. To show applicability, these parameters are used to estimate the in situ fracture properties of the studied layer.

Decoupled approximation and separate extrapolation of P- and SV-waves in transversely isotropic media

Geophysics ◽  
2021 ◽  
pp. 1-57
Author(s):  
Bowen Li ◽  
Alexey Stovas
Keyword(s):  
Phase Velocity ◽  
Wave Velocity ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Wave Phase ◽  
Wave Phase Velocity ◽  
Tti Media ◽  
Vti Media ◽  
Acoustic Approximation

Characterizing the kinematics of seismic waves in elastic vertical transversely isotropic (VTI) media involves four independent parameters. To reduce the complexity, the acoustic approximation for P-waves reduces the number of required parameters to three by setting the vertical S-wave velocity to zero. However, since only the SV-wave phase velocities parallel or perpendicular to the symmetry axis are indirectly set to zero, the acoustic approximation leads to coupled P-wave components and SV-wave artifacts. The new acoustic approximation suggests setting the vertical S-wave velocity as a phase angle-dependent variable so that the SV-wave phase velocity is zero at all phase angles. We find that manipulating this parameter is a valid way for P-wave approximation, but doing so inevitably leads to zero- or non-zero-valued spurious SV-wave components. Thus, we have developed a novel approach to efficiently approximate and thoroughly separate the two wave modes in VTI media. First, the exact P- and SV-wave phase velocity expressions are rewritten by introducing an auxiliary function. After confirming the insensitivity of this function, we construct a new expression for it and obtain simplified P- and SV-wave phase velocity expressions, which are three- and four-parameter, respectively. This approximation process leads to the same reasonable error for both wave modes. Accuracy analysis indicates that for the P-wave, the overall accuracy performance of our approach is comparable to that of some existing three-parameter approximations. We then derive the corresponding P- and SV-wave equations in tilted transversely isotropic (TTI) media and provide two available solutions, the hybrid finite-difference/pseudo-spectral scheme and the low-rank approach. Numerical examples illustrate the separability and high accuracy of the proposed P- and SV-wave simulation methods in TTI media.

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