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

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.

2015 ◽  
Vol 58 (5) ◽  
Author(s):  
Sankar N. Bhattacharya

<p>Sensitivity kernels or partial derivatives of phase velocity (<em>c</em>) and group velocity (<em>U</em>) with respect to medium parameters are useful to interpret a given set of observed surface wave velocity data. In addition to phase velocities, group velocities are also being observed to find the radial anisotropy of the crust and mantle. However, sensitivities of group velocity for a radially anisotropic Earth have rarely been studied. Here we show sensitivities of group velocity along with those of phase velocity to the medium parameters <em>V<sub>SV</sub>, V<sub>SH </sub>, V<sub>PV</sub>, V<sub>PH , </sub></em><em>h</em><em> </em>and density in a radially anisotropic spherical Earth. The peak sensitivities for <em>U</em> are generally twice of those for <em>c</em>; thus <em>U</em> is more efficient than <em>c</em> to explore anisotropic nature of the medium. Love waves mainly depends on <em>V<sub>SH</sub></em> while Rayleigh waves is nearly independent of <em>V<sub>SH</sub></em> . The sensitivities show that there are trade-offs among these parameters during inversion and there is a need to reduce the number of parameters to be evaluated independently. It is suggested to use a nonlinear inversion jointly for Rayleigh and Love waves; in such a nonlinear inversion best solutions are obtained among the model parameters within prescribed limits for each parameter. We first choose <em>V<sub>SH</sub></em>, <em>V<sub>SV </sub></em>and <em>V<sub>PH</sub></em> within their corresponding limits; <em>V<sub>PV</sub></em> and <em>h</em> can be evaluated from empirical relations among the parameters. The density has small effect on surface wave velocities and it can be considered from other studies or from empirical relation of density to average P-wave velocity.</p>


2019 ◽  
pp. 6-18
Author(s):  
V I Erofeev ◽  
D A Kolesov ◽  
V L Krupenin

We study the features of propagation of a longitudinal wave in an acoustic (mechanical) metamaterial, modeled as a one-dimensional chain, containing equal masses, connected by elastic elements (springs), and having the same rigidity. Each mass contains within itself a series connection of another mass and viscous element (damper). The mass-to-mass model is free from the drawbacks of a number of other mechanical models of metamaterials: i.e. it eliminates the need to have the property of a deformable body to possess a negative mass, density, and (or) a negative elastic modulus. It is shown that the model under consideration makes it possible to describe the dispersion and frequency-dependent attenuation of a longitudinal wave, the character of which essentially depends on the ratio of the external and internal mass of the metamaterial. The behavior of the phase and group velocities of the wave is studied, as well as the evolution of its profile, both in the low-frequency and high-frequency ranges. The mass ratios were found at which the phase velocity exceeds the group velocity (normal dispersion) in magnitude and those at which the group velocity exceeds the phase velocity (anomalous dispersion) in a wide frequency range. Having the same asymptotic values when the frequency tends to infinity, the phase and group velocities have significant differences in behavior, namely, that the phase velocity is a monotonic function of frequency, and the group velocity has a maximum. In addition, in the region of normal dispersion, the group velocity may be negative, i.e. the so-called “reverse wave” effect is true, when, despite the fact that the phase velocity is directed in the positive direction of the spatial axis, the energy in such a wave is transferred in the negative direction.


2020 ◽  
Vol 224 (1) ◽  
pp. 649-668
Author(s):  
A Wang ◽  
D Leparoux ◽  
O Abraham ◽  
M Le Feuvre

SUMMARY Monitoring the small variations of a medium is increasingly important in subsurface geophysics due to climate change. Classical seismic surface wave dispersion methods are limited to quantitative estimations of these small variations when the variation ratio is smaller than 10 per cent, especially in the case of variations in deep media. Based on these findings, we propose to study the contributions of the Rayleigh wave phase velocity derivative with respect to frequency. More precisely, in the first step of assessing its feasibility, we analyse the effects of the phase velocity derivative on the inversion of the fundamental mode in the simple case of a two-layer model. The behaviour of the phase velocity derivative is first analysed qualitatively: the dispersion curves of phase velocity, group velocity and the phase velocity derivative are calculated theoretically for several series of media with small variations. It is shown that the phase velocity derivatives are more sensitive to variations of a medium. The sensitivity curves are then calculated for the phase velocity, the group velocity and the phase velocity derivative to perform quantitative analyses. Compared to the phase and group velocities, the phase velocity derivative is sensitive to variations of the shallow layer and the deep layer shear wave velocity in the same wavelength (frequency) range. Numerical data are used and processed to obtain dispersion curves to test the feasibility of the phase velocity derivative in the inversion. The inversion results of the phase velocity derivative are compared with those of phase and group velocities and show improved estimations for small variations (variation ratio less than 5 per cent) of deep layer shear wave velocities. The study is focused on laboratory experiments using two reduced-scale resin-epoxy models. The differences of these two-layer models are in the deep layer in which the variation ratio is estimated as 16.4 ± 1.1 per cent for the phase velocity inversion and 17.1 ± 0.3 per cent for the phase velocity derivative. The latter is closer to the reference value 17 per cent, with a smaller error.


Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1627-1633 ◽  
Author(s):  
Bart W. Tichelaar ◽  
Klaas W. van Luik

Borehole sonic waveforms are commonly acquired to produce logs of subsurface compressional and shear wave velocities. To this purpose, modern borehole sonic tools are usually equipped with various types of acoustic sources, i.e., monopole and dipole sources. While the dipole source has been specifically developed for measuring shear wave velocities, we found that the dipole source has an advantage over the monopole source when determining compressional wave velocities in a very slow formation consisting of unconsolidated sands with a porosity of about 35% and a shear wave velocity of about 465 m/s. In this formation, the recorded compressional refracted waves suffer from interference with another wavefield component identified as a leaky P‐wave, which hampers the determination of compressional wave velocities in the sands. For the dipole source, separation of the compressional refracted wave from the recorded waveforms is accomplished through bandpass filtering since the wavefield components appear as two distinctly separate contributions to the frequency spectrum: a compressional refracted wave centered at a frequency of 6.5 kHz and a leaky P‐wave centered at 1.3 kHz. For the monopole source, the frequency spectra of the various waveform components have considerable overlap. It is therefore not obvious what passband to choose to separate the compressional refracted wave from the monopole waveforms. The compressional wave velocity obtained for the sands from the dipole compressional refracted wave is about 2150 m/s. Phase velocities obtained for the dispersive leaky P‐wave excited by the dipole source range from 1800 m/s at 1.0 kHz to 1630 m/s at 1.6 kHz. It appears that the dipole source has an advantage over the monopole source for the data recorded in this very slow formation when separating the compressional refracted wave from the recorded waveforms to determine formation compressional wave velocities.


Geophysics ◽  
1968 ◽  
Vol 33 (4) ◽  
pp. 584-595 ◽  
Author(s):  
A. Timur

Measurements of velocity of compressional waves in consolidated porous media, conducted within a temperature range of 26 °C to −36 °C, indicate that: (1) compressional wave velocity in water‐saturated rocks increases with decreasing temperature whereas it is nearly independent of temperature in dry rocks; (2) the shapes of the velocity versus temperature curves are functions of lithology, pore structure, and the nature of the interstitial fluids. As a saturated rock sample is cooled below 0 °C, the liquid in pore spaces with smaller surface‐to‐volume ratios (larger pores) begins to freeze and the liquid salinity controls the freezing process. As the temperature is decreased further, a point is reached where the surface‐to‐volume ratio in the remaining pore spaces is large enough to affect the freezing process, which is completed at the cryohydric temperature of the salts‐water system. In the ice‐liquid‐rock matrix system, present during freezing, a three‐phase, time‐average equation may be used to estimate the compressional wave velocities. Below the cryohydric temperature, elastic wave propagation takes place in a solid‐solid system consisting of ice and rock matrix. In this frozen state, the compressional wave velocity remains constant, has its maximum value, and may be estimated through use of the two‐phase time average equation. Limited field data for compressional wave velocities in permafrost indicate that pore spaces in permafrost contain not only liquid and ice, but also gas. Therefore, before attempting to make velocity estimates through the time‐average equations, the natures and percentages of pore saturants should be investigated.


2021 ◽  
Author(s):  
◽  
Holly Joanne Godfrey

<p>We use continuous seismic data from permanent and temporary, broadband and short-period stations that were operating during 2001 and 2008 to investigate the subsurface velocity structure of the Tongariro Volcanic Centre (TgVC) of New Zealand, particularly the highly active but poorly understood Ruapehu and Tongariro Volcanoes.  Stacks of cross-correlation of two concurrent ambient noise seismograms can be used to estimate the interstation Green's Function, i.e., the impulse response of the earth between the two receivers. The Green's Functions are used to retrieve the dispersion relation (frequency-dependent velocity) of surface waves at different periods, which reflects the shear-wave velocity structure in the Fresnel volume of the propagating surface waves. Several studies have used dispersion measurements from ambient noise cross-correlations to investigate the shallow subsurface shear-wave velocity structure of active volcanoes around the world. Most use vertical components to retrieve the Rayleigh waves, but it is becoming increasingly common to use the horizontal seismogram components in addition to the vertical, giving further constraints to Rayleigh-wave measurements and introducing data relating to Love waves.  We compute 1,048,968 daily cross-correlations for 955 viable station pairs across the two periods, including all nine-components of the cross-correlation tensor where possible. These daily functions are then stacked into 7458 full-stacks, of which we make group velocity dispersion measurements for 2641 RR-, RZ-, TT-, ZR- and ZZ-component stacks. Cross-correlation quality varies across the networks, with some station pairs possibly contaminated with timing errors.  We observe both the fundamental and rst higher-order modes within our database of dispersion measurements. However, correctly identifying the mode of some measurements is challenging as the range of group velocities measured reflects both presence of multiple modes and heterogeneity of the local velocity structure. We assign modes to over 1900 measurements, of which we consider 1373 to be high quality.  We invert fundamental mode Rayleigh- and Love-wave dispersion curves independently and jointly for one dimensional shear-wave velocity profiles at Ruapehu and Tongariro Volcanoes, using dispersion measurements from two individual station pairs and average dispersion curves from measurements within specifi c areas on/around the volcanoes. Our Ruapehu profiles show little velocity variation with depth, suggesting that volcanic edifice is made of material that is compacting and being hydrothermally altered with depth. At Tongariro, we observe larger increases in velocity with depth, which we interpret as different layers within Tongariro's volcanic system. Slow shear-wave velocities, on the order of 1-2 km/s, are consistent with both P-wave velocities from existing velocity pro files of areas within the TgVC, and the observations of worldwide studies of shallow volcanic systems that used ambient noise cross-correlation.  A persistent observation across the majority of our dispersion measurements is that group velocities of the fundamental mode Love-wave group velocity measurements are slower than those of fundamental mode Rayleigh-waves, particularly in the frequency range of 0.25-1 Hz. Similarly, first higher-order mode Love-wave group velocities are slower than first higher-mode Rayleigh-wave velocities. This is inconsistent with the differences between synthetic dispersion curves that were calculated using isotropic, layered velocity models appropriate for Ruapehu and Tongariro. We think the Love-Rayleigh discrepancy is due to structures such as dykes or cracks in the vertical plane having greater influence than horizontal layering on surface-wave propagation. However, several measurements where Love-wave group velocities are faster than Rayleigh-wave group velocities suggests that in some places horizontal layering is the stronger influence.  We also observe that the differences between the Love- and Rayleigh-wave dispersion curves vary with the azimuth of the interstation path across Ruapehu and Tongariro Volcanoes. Some significant differences between Rayleigh-wave velocities of measurements with different interstation orientations are also observed, as are differences between Love-wave velocities. This suggests a component of azimuthal anisotropy within the volcanic structures, which coupled with the radial anistropy makes the shear-wave velocity structures of Ruapehu and Tongariro Volcanoes anisotropic with orthorhombic symmetry. We suggest that further work to determine three-dimensional structure should include provisions for anisotropy with orthorhombic or lower symmetry.</p>


Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. R27-R39 ◽  
Author(s):  
Faranak Mahmoudian ◽  
G. F. Margrave ◽  
P. F. Daley ◽  
J. Wong ◽  
D. C. Henley

We acquired 3C ultrasonic transmission seismograms, measured group velocities associated with the three quasi-body wave types, and determined density-normalized stiffness coefficients ([Formula: see text]) over an orthorhombic physical (laboratory) model. The estimation of [Formula: see text] is based on an approximate relationship between the nine orthorhombic [Formula: see text] and group velocities. Estimation of the anisotropic [Formula: see text] is usually done using phase velocities with well-known formulas expressing their theoretical dependence on [Formula: see text]. However, on time-domain seismograms, arrivals are observed traveling with group velocities. Group velocity measurements are found to be straightforward, reasonably accurate, and independent of the size of the transducers used. In contrast, the accuracy of phase velocities derived from the [Formula: see text] transform analysis was found to be very sensitive to small differences in picked arrival times and to transducer size. Theoretical phase and group velocities, calculated in a forward manner from the [Formula: see text] estimates, agreed with the originally measured phase and group velocities, respectively. This agreement confirms that it is valid to use easily measured group velocities with their approximate theoretical relationship to the [Formula: see text] to determine the full stiffness matrix. Compared to the phase-velocity procedure, the technique involving group velocities is much less prone to error due to time-picking uncertainties, and therefore is more suitable for analyzing physical model seismic data.


2005 ◽  
Vol 297-300 ◽  
pp. 2213-2218 ◽  
Author(s):  
Jeong Ki Lee ◽  
Young H. Kim ◽  
Ho Chul Kim

The elastic waves in the isotropic plate are dispersive waves with the characteristics of Lamb wave, however, S0 symmetric mode is less dispersive in the frequency region less than the first cut-off frequency. In the anisotropic plates such as CFRP plates, the propagation velocities vary with the directions as well as the dispersion of the Lamb wave, and the phase velocity direction does not accord with the group velocity direction. The phase velocity direction is equivalent the wave vector direction, while the group velocity direction is equivalent the energy flow direction. In this work, the group velocity dispersion curves were obtained by the dispersion relation of the Lamb wave in unidirectional CFRP plate with an orthotropic structure. The group velocities of the S0 symmetric mode in the frequency region less than the first cut-off frequency were corrected by applying the slowness surface. The propagation velocities of Lamb wave were decided by measuring the arrival time of the Lamb wave signals received with the two pinducers varying the propagating direction in the laminated unidirectional CFRP plates of 8, 16 and 24 plies having a volume fraction of 67%. The measured velocities are better agreement with corrected group velocity curve, except near the fiber direction at the cusp region. When the propagating direction is not accorded with the principal axis, the direction of the group velocities inclines toward the fiber direction in the unidirectional CFRP plates, suggesting that the energy propagates preferentially toward fiber direction.


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