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.

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).

scholarly journals Incoherent scatter radar observations of AGW/TID events generated by the moving solar terminator

1998 ◽  
Vol 16 (7) ◽  
pp. 821-827 ◽  
Author(s):  
V. G. Galushko ◽  
V. V. Paznukhov ◽  
Y. M. Yampolski ◽  
J. C. Foster
Keyword(s):  
Group Velocity ◽  
Incoherent Scatter Radar ◽  
Incoherent Scatter ◽  
Phase Velocities ◽  
Scatter Radar ◽  
Cross Correlation Analysis ◽  
Waves And Tides ◽  
Solar Terminator ◽  
Group And Phase Velocities ◽  
Atmospheric Gravity

Abstract. Observations of traveling ionospheric disturbances (TIDs) associated with atmospheric gravity waves (AGWs) generated by the moving solar terminator have been made with the Millstone Hill incoherent scatter radar. Three experiments near 1995 fall equinox measured the AGW/TID velocity and direction of motion. Spectral and cross-correlation analysis of the ionospheric density observations indicates that ST-generated AGWs/TIDs were observed during each experiment, with the more-pronounced effect occurring at sunrise. The strongest oscillations in the ionospheric parameters have periods of 1.5 to 2 hours. The group and phase velocities have been determined and show that the disturbances propagate in the horizontal plane perpendicular to the terminator with the group velocity of 300-400 m s-1 that corresponds to the ST speed at ionospheric heights. The high horizontal group velocity seems to contradict the accepted theory of AGW/TID propagation and indicates a need for additional investigation.Key words. Ionosphere (wave propagation) · Meteorology and atmospheric dynamics (waves and tides)

Do dipole sonic logs measure group or phase velocity (revisited)?

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. C311-C322
Author(s):  
Stephen Horne ◽  
Richard T. Coates ◽  
Alexei Bolshakov
Keyword(s):  
Symmetry Axis ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Flexural Waves ◽  
S Wave ◽  
Slowness Surface ◽  
Phase Velocities ◽  
Group And Phase Velocities ◽  
Sonic Logs

We have revisited the debate about whether flexural waves from dipole sonic tools and standard processing algorithms measure group or phase velocities in anisotropic formations. We observe that much of the confusion arises from a failure to understand the different meanings of group and phase velocities. Using a transversely isotropic medium with a vertical axis of symmetry that exhibits a triplication in its S-wave group slowness surface, we generate synthetic flexural sonic waveforms corresponding to boreholes at angles of 0°–90° with respect to the anisotropy symmetry axis in 1° increments. We processed these synthetic data using standard time- and frequency-domain semblance methods. The results conclusively demonstrate that dipole sonic logs measure the group slowness for the group angle corresponding to the angle between the borehole and the anisotropic symmetry axis. In addition, data that we have evaluated suggest that current tool geometries and semblance processing may not always be sensitive enough to resolve all branches of the group slowness triplication surface.

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.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2082-2091 ◽  
Author(s):  
Bjørn Ursin ◽  
Ketil Hokstad
Keyword(s):  
Ray Tracing ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Analytical Approximation ◽  
Geometrical Spreading ◽  
S Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Amplitude Versus

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

Sign in / Sign up

Export Citation Format

Share Document