Do traveltimes in pulse‐transmission experiments yield anisotropic group or phase velocities?

Geophysics ◽  
1994 ◽  
Vol 59 (11) ◽  
pp. 1774-1779 ◽  
Author(s):  
Joe Dellinger ◽  
Lev Vernik
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Elastic Constants ◽  
Near Field ◽  
Frequency Dispersion ◽  
Transversely Isotropic ◽  
Sh Waves ◽  
Phase Velocities ◽  
Pulse Transmission ◽  
Transmission Technique

The elastic properties of layered rocks are often measured using the pulse through‐transmission technique on sets of cylindrical cores cut at angles of 0, 90, and 45 degrees to the layering normal (e.g., Vernik and Nur, 1992; Lo et al., 1986; Jones and Wang, 1981). In this method transducers are attached to the flat ends of the three cores (see Figure 1), the first‐break traveltimes of P, SV, and SH‐waves down the axes are measured, and a set of transversely isotropic elastic constants are fit to the results. The usual assumption is that frequency dispersion, boundary reflections, and near‐field effects can all be safely ignored, and that the traveltimes measure either vertical anisotropic group velocity (if the transducers are very small compared to their separation) or phase velocity (if the transducers are relatively wide compared to their separation) (Auld, 1973).

Wall Pressure Phase Velocity Measurements in a Turbulent Boundary Layer

2012 ◽  
Author(s):  
Teresa S. Miller ◽  
Mark J. Moeller
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Phase Velocity ◽  
Group Velocity ◽  
Noise Source ◽  
Wall Pressure ◽  
Interior Noise ◽  
Convection Velocity ◽  
Phase Velocities ◽  
Wall Pressure Fluctuation

The turbulent boundary layer that forms on the outer surfaces of vehicles can be a significant source of interior noise. In automobiles this is known as wind noise, and at high speeds it dominates the interior noise. For airplanes the turbulent boundary is also a dominant noise source. Because of its importance as a noise source, it is desirable to have a model of the turbulent wall pressure fluctuations for interior noise prediction. One important parameter in building the wall pressure fluctuation model is the convection velocity. In this paper, the phase velocity was determined from the streamwise pressure measurements. The phase velocity was calculated for three separation distances ranging from 0.25 to 1.30 boundary layer thicknesses. These measurements were made for a Mach number range of 0.1 < M < 0.6. The phase velocity was shown to vary with sensor spacing and frequency. The data collapsed well on outer variable normalization. The phase velocities were fit and the group velocity was calculated from the curve fit. The group velocity was consistent with the array measured convection velocities. The group velocity was also estimated by a band limited cross correlation technique that used the Hilbert transform to find the energy delay. This result was consistent with the group velocity inferred from the phase velocities and the array measured convection velocity. From this research, it is suggested that the group velocity found in this study should be used to estimate the convection velocity in wall pressure fluctuation models.

scholarly journals Combined spatiotemporal and frequency-dependent shear wave elastography enables detection of vulnerable carotid plaques as validated by MRI

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
David Marlevi ◽  
Sharon L. Mulvagh ◽  
Runqing Huang ◽  
J. Kevin DeMarco ◽  
Hideki Ota ◽  
...  
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Shear Wave ◽  
Shear Wave Elastography ◽  
Intraplaque Hemorrhage ◽  
Frequency Dependent ◽  
Invasive Approach ◽  
Carotid Plaques ◽  
Phase Velocities ◽  
Luminal Narrowing

AbstractFatal cerebrovascular events are often caused by rupture of atherosclerotic plaques. However, rupture-prone plaques are often distinguished by their internal composition rather than degree of luminal narrowing, and conventional imaging techniques might thus fail to detect such culprit lesions. In this feasibility study, we investigate the potential of ultrasound shear wave elastography (SWE) to detect vulnerable carotid plaques, evaluating group velocity and frequency-dependent phase velocities as novel biomarkers for plaque vulnerability. In total, 27 carotid plaques from 20 patients were scanned by ultrasound SWE and magnetic resonance imaging (MRI). SWE output was quantified as group velocity and frequency-dependent phase velocities, respectively, with results correlated to intraplaque constituents identified by MRI. Overall, vulnerable lesions graded as American Heart Association (AHA) type VI showed significantly higher group and phase velocity compared to any other AHA type. A selection of correlations with intraplaque components could also be identified with group and phase velocity (lipid-rich necrotic core content, fibrous cap structure, intraplaque hemorrhage), complementing the clinical lesion classification. In conclusion, we demonstrate the ability to detect vulnerable carotid plaques using combined SWE, with group velocity and frequency-dependent phase velocity providing potentially complementary information on plaque characteristics. With such, the method represents a promising non-invasive approach for refined atherosclerotic risk prediction.

On: “Weak elastic anisotropy,” by L. Thomsen (GEOPHYSICS, 52, 1954–1966, October, 1986).

Geophysics ◽  
1988 ◽  
Vol 53 (4) ◽  
pp. 558-559 ◽  
Author(s):  
Franklyn K. Levin
Keyword(s):  
Elastic Constants ◽  
Vertical Velocity ◽  
Elastic Waves ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Sh Waves ◽  
The Individual ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

In a paper whose importance seems to have escaped notice, Thomsen (1) derived equations that give the moveout velocities of P, SV, and SH-waves when solids are weakly transversely isotropic and (2) tabulated experimentally determined elastic constants for a large number of rocks, crystals, and a few other solids. For rocks, one of the constants, delta, differed from zero by as much as 0.73 and −0.27. Delta is the fraction by which P-wave moveout velocity deviates from the vertical velocity [Thomsen’s equation (27a)]. Although some deltas indicated deviations from the vertical velocity smaller than 1 or 2 percent, most were larger and positive. Until the publication of Thomsen’s data, most of us concerned with elastic waves traveling in earth sections that act as transversely isotropic solids because the sections consist of thin beds had assumed the individual beds were isotropic solids, all with the same Poisson’s ratios. That assumption results in a zero value for delta and a moveout velocity equal to the vertical velocity. The validity of the assumption is now doubtful.

Ray-direction velocities in VTI media

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. F1-F5 ◽  
Author(s):  
Vladimir Grechka
Keyword(s):  
Ray Tracing ◽  
Group Velocity ◽  
Transversely Isotropic ◽  
Polynomial Equation ◽  
Wave Velocities ◽  
Phase Velocities ◽  
Computer Codes ◽  
Group And Phase Velocities ◽  
Vti Media ◽  
Ray Bending

Two-point ray tracing in anisotropic media requires the group and phase velocities to be calculated along ray directions available at each step of a ray bending algorithm. This computation, usually done iteratively or through velocity tables, becomes exceedingly involved for shear-waves that have multivalued group-velocity surfaces, such as in the presence of triplications on the SV wavefronts in vertically transversely isotropic (VTI) media. The difficulties encountered in computing the SV-wave velocities for a given ray direction can be circumvented by solving a polynomial equation whose real-valued roots provide the phase directions of the P- and either one or three SV-waves propagating along a selected ray; those phase directions then allow the group and phase velocities to be computed in a standard fashion. I construct the polynomial and supply computer codes implementing its solution, the codes that can be used in two-point ray-tracing software to improve its performance.

Elastic constants of transversely isotropic media from constrained aperture traveltimes

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 658-667 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Elastic Constants ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
System Of Equations ◽  
Sh Waves ◽  
Velocity Models ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
P And Sv Waves ◽  
Homogeneous Media

The elastic constants that control P‐ and SV‐wave propagation in a transversely isotropic media can be estimated by using P‐ and SV‐wave traveltimes from either crosswell or VSP geometries. The procedure consists of two steps. First, elliptical velocity models are used to fit the traveltimes near one axis. The result is four elliptical parameters that represent direct and normal moveout velocities near the chosen axis for P‐ and SV‐waves. Second, the elliptical parameters are used to solve a system of four equations and four unknown elastic constants. The system of equations is solved analytically, yielding simple expressions for the elastic constants as a function of direct‐ and normal‐moveout velocities. For SH‐waves, the estimation of the corresponding elastic constants is easier because the phase velocity is already elliptical. The procedure, introduced for homogeneous media, is generalized to heterogeneous media by using tomographic techniques.

scholarly journals Rational approximation of P-wave kinematics — Part 2: Orhorhombic media

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C175-C186 ◽  
Author(s):  
Mohammad Mahdi Abedi
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Group Velocity ◽  
Transversely Isotropic ◽  
P Wave ◽  
Propagation Modeling ◽  
Wave Kinematics ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Wave Propagation Modeling

Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.

scholarly journals Rational approximation of P-wave kinematics — Part 1: Transversely isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C163-C173 ◽  
Author(s):  
Mohammad Mahdi Abedi
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Continued Fraction ◽  
Transversely Isotropic ◽  
P Wave ◽  
Algebraic Complexity ◽  
Wave Kinematics ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Phase Direction

In seismic data processing and several wave propagation modeling algorithms, the phase velocity, group velocity, and traveltime equations are essential. To have these equations in explicit form, or to reduce algebraic complexity, approximation methods are used. For the approximation of P-wave kinematics in acoustic transversely isotropic media, we have developed a new flexible 2D functional equation in a continued fraction form. Using different orders of the continued fraction, we obtain different approximations for (1) phase velocity as a function of phase direction, (2) group velocity as a function of group direction, and (3) traveltime as a function of offset. Then, we use them in the approximation of the group direction as a function of phase direction, and phase direction as a function of group direction. The proposed approximations have a rational form, which is considered algebraically simple and computationally efficient. The used continued fraction form rapidly converges to exact kinematics. By introducing the optimal ray into our approximations and using it for parameter definition, the convergence becomes faster, so the accuracy of the existing most accurate approximations is available by the third order, and new most accurate approximations are obtained by the fourth order of the proposed general form. The error of the most accurate version of the proposed approximations is below 0.001% for moderate anisotropic models with an anellipticity parameter up to 0.3. This high accuracy is considered to be attractive in practical implementations that use the kinematic equations and their derivatives.

Compressional‐wave velocities in attenuating media: A laboratory physical model study

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1162-1167 ◽  
Author(s):  
Joseph B. Molyneux ◽  
Douglas R. Schmitt
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Wave Velocity ◽  
Propagation Distance ◽  
Compressional Wave ◽  
Arrival Times ◽  
Wave Velocities ◽  
Amplitude Maximum ◽  
Phase And Group Velocities ◽  
Group Velocities

Elastic‐wave velocities are often determined by picking the time of a certain feature of a propagating pulse, such as the first amplitude maximum. However, attenuation and dispersion conspire to change the shape of a propagating wave, making determination of a physically meaningful velocity problematic. As a consequence, the velocities so determined are not necessarily representative of the material’s intrinsic wave phase and group velocities. These phase and group velocities are found experimentally in a highly attenuating medium consisting of glycerol‐saturated, unconsolidated, random packs of glass beads and quartz sand. Our results show that the quality factor Q varies between 2 and 6 over the useful frequency band in these experiments from ∼200 to 600 kHz. The fundamental velocities are compared to more common and simple velocity estimates. In general, the simpler methods estimate the group velocity at the predominant frequency with a 3% discrepancy but are in poor agreement with the corresponding phase velocity. Wave velocities determined from the time at which the pulse is first detected (signal velocity) differ from the predominant group velocity by up to 12%. At best, the onset wave velocity arguably provides a lower bound for the high‐frequency limit of the phase velocity in a material where wave velocity increases with frequency. Each method of time picking, however, is self‐consistent, as indicated by the high quality of linear regressions of observed arrival times versus propagation distance.

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