Shear-wave group-velocity surfaces in low-symmetry anisotropic media

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Vladimir Grechka
Keyword(s):  
Group Velocity ◽  
Body Wave ◽  
Transverse Isotropy ◽  
Shear Waves ◽  
Anisotropic Media ◽  
The Body ◽  
Velocity Models ◽  
Phase And Group Velocities ◽  
Low Symmetry ◽  
Group Velocities

Shear waves excited by natural sources constitute a significant part of useful energy recorded in downhole microseismic surveys. In rocks, such as fractured shales, exhibiting symmetries lower than transverse isotropy (TI), the shear wavefronts are always multivalued in certain directions, potentially complicating the data processing and analysis. This paper discusses a basic tool — the computation of the phase and group velocities of all waves propagating along a given ray — that intends to facilitate the understanding of geometries of the shear wavefronts in homogeneous anisotropic media. With this tool, arbitrarily complex group-velocity surfaces can be conveniently analyzed, providing insights into possible challenges to be faced when processing shear waves in anisotropic velocity models that have symmetries lower than TI. Among those challenges are complicated multipathing and the presence of cones of directions, known as internal refraction cones, in which no fast shear waves propagate and the entire shear portion of the body-wave seismic data consists of several branches of the slow shear wavefronts.

Estimation of elastic stiffness coefficients of an orthorhombic physical model using group velocity analysis on transmission data

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. R27-R39 ◽  
Author(s):  
Faranak Mahmoudian ◽  
G. F. Margrave ◽  
P. F. Daley ◽  
J. Wong ◽  
D. C. Henley
Keyword(s):  
Physical Model ◽  
Group Velocity ◽  
Body Wave ◽  
Elastic Stiffness ◽  
Laboratory Model ◽  
Theoretical Dependence ◽  
Stiffness Coefficients ◽  
Phase Velocities ◽  
Phase And Group Velocities ◽  
Group Velocities

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.

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.

Narrow-angle representations of the phase and group velocities and their applications in anisotropic velocity-model building for microseismic monitoring

Geophysics ◽  
2011 ◽  
Vol 76 (6) ◽  
pp. WC127-WC142 ◽  
Author(s):  
Vladimir Grechka ◽  
Anton A. Duchkov
Keyword(s):  
Seismic Anisotropy ◽  
Model Building ◽  
Elastic Stiffness ◽  
Coordinate Frame ◽  
Gas Field ◽  
Angular Aperture ◽  
Data Set ◽  
Velocity Models ◽  
Phase And Group Velocities ◽  
Group Velocities

It is usually believed that angular aperture of seismic data should be at least 20° to allow estimation of the subsurface anisotropy. Although this is certainly true for reflection data, for which anisotropy parameters are inverted from the stacking velocities or the nonhyperbolic moveout, traveltimes of direct P- and S-waves recorded in typical downhole microseismic geometries make it possible to infer seismic anisotropy in angular apertures as narrow as about 10°. To ensure the uniqueness of such an inversion, it has to be performed in a local coordinate frame tailored to a particular data set. Because any narrow fan of vectors is naturally characterized by its average direction, we choose the axes of the local frame to coincide with the polarization vectors of three plane waves corresponding to such a direction. This choice results in a significant simplification of the conventional equations for the phase and group velocities in anisotropic media and makes it possible to predict which elements of the elastic stiffness tensor are constrained by the available data. We illustrate our approach on traveltime synthetics and then apply it to perforation-shot data recorded in a shale-gas field. Our case study indicates that isotropic velocity models are inadequate and accounting for seismic anisotropy is a prerequisite for building a physically sound model that explains the recorded traveltimes.

scholarly journals Dispersion and attenuation of a longitudinal wave propagating in a metamaterial defined as a mass-to-mass chain

2019 ◽  
pp. 6-18
Author(s):  
V I Erofeev ◽  
D A Kolesov ◽  
V L Krupenin
Keyword(s):  
Phase Velocity ◽  
Longitudinal Wave ◽  
Group Velocity ◽  
Mass Density ◽  
Deformable Body ◽  
Mass Model ◽  
Positive Direction ◽  
Normal Dispersion ◽  
Phase And Group Velocities ◽  
Group Velocities

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.

Weak elastic anisotropy by perturbation theory

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. D45-D58 ◽  
Author(s):  
Vigen Ohanian ◽  
Thomas M. Snyder ◽  
José M. Carcione
Keyword(s):  
Perturbation Theory ◽  
Group Velocity ◽  
Elastic Anisotropy ◽  
Anisotropic Media ◽  
Orthorhombic Symmetry ◽  
Wave Group ◽  
Wave Polarization ◽  
First Order ◽  
Christoffel Equation ◽  
Group Velocities

We demonstrate the advantages of adopting a wave-vector-based coordinate system (WCS) for the application of perturbation theory to derive and display approximate expressions for qP- and qS-wave polarization vectors, phase velocities, and group velocities in general weakly anisotropic media. The advantages stem from two important properties of the Christoffel equation when expressed in the WCS: (1) Each element of the Christoffel matrix is identical to a specific stiffness component in the WCS, and (2) the Christoffel matrix of an isotropic medium is diagonal in the WCS. Using these properties, one can easily identify the small components of the Christoffel matrix in the WCS for a weakly anisotropic medium. Approximate solutions to the Christoffel equation are then obtained by straightforward algebraic manipulations, which make our perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas. The form of the WCS phase-velocity expressions facilitates the derivation of closed-form, first-order expressions for qP- and qS-wave group-velocity vectors, providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media. Numerical evaluation of our group-velocity expressions demonstrates their accuracy. We discuss problems with the approximate qS-wave group velocities and polarizations in neighboring directions of singularities. Standard methods are used to transform our solutions from the WCS to the acquisition coordinates, as illustrated by application to orthorhombic symmetry.

The combined effects of transverse isotropy and inhomogeneity on Love waves

1973 ◽  
Vol 63 (1) ◽  
pp. 49-57
Author(s):  
V. Thapliyal
Keyword(s):  
Half Space ◽  
Transverse Isotropy ◽  
Frequency Equation ◽  
Love Waves ◽  
Anisotropy Factor ◽  
Wave Number ◽  
Elastic Parameters ◽  
Short Period ◽  
Phase And Group Velocities ◽  
Group Velocities

abstract The characteristic frequency equation for Love waves propagating in a finite layer overlying an anisotropic and inhomogeneous half-space is derived. This frequency equation takes into account the arbitrary variation of density, elastic parameters, and degree of anisotropy factor in the half-space. In fact, the problem of deriving the frequency equation has been reduced to finding the solution of the equation of motion subject to the appropriate boundary conditions. To illustrate the method, the author has derived the frequency equation for a generalized power law variation of density and elastic parameters with the depth, in the halfspace. As a step toward the systematic investigation of the effects of anisotropy and inhomogeneity, the relationship between the wave number and phase and group velocities has been worked out for increasing, uniform and decreasing anisotropy factor. The pronounced effects of anisotropy have been noticed in the long-period range compared to the short-period one. The numerical analysis shows that for a given phase velocity (or group velocity), the period of propagation depends on the sign and magnitude of power of variation of the density and anisotropy factor in the half-space. For the increased positive rate of variation of the anisotropy factor, the values of phase and group velocities have been found higher whereas the reverse is found true for an increasing negative rate of variation of the anisotropy factor.

3D-S wave velocity model of the Los Humeros geothermal field, Mexico, by ambient-noise tomography

2020 ◽  
Author(s):  
Joana Martins ◽  
Anne Obermann ◽  
Arie Verdel ◽  
Philippe Jousset
Keyword(s):  
Group Velocity ◽  
Ambient Noise ◽  
Cross Correlation ◽  
Geothermal Field ◽  
Seismic Network ◽  
Ambient Noise Tomography ◽  
Phase Velocities ◽  
Phase And Group Velocities ◽  
Velocity Anomalies ◽  
Group Velocities

<p>Since the successful retrieval of surface-wave responses from the ambient seismic field via cross-correlation, noise-based interferometry has been widely used for high-resolution imaging of the Earth’s lithosphere from all around the globe. Further applications on geothermal fields reveal the potential of ambient noise techniques to either characterize the subsurface velocity field or to understand the temporal evolution of the velocity models due to field operations.</p><p>Following the completion of the GeMEX<sup>*</sup> project, a European-Mexican collaboration to improve our understanding of two geothermal sites in Mexico, we present the results of ambient noise tomography (ANT) techniques over the Los Humeros geothermal field. We used the vertical component of the data recorded by the seismic network active from September 2017 to September 2018. The total network is composed of 45 seismometers from which 25 are Broadband (BB) and the remaining ones short-period stations. From the ambient noise recorded at the deployed seismic network, we extract surface-waves after the computation of the empirical Green’s functions (EGF) by cross-correlation and consecutive stacking. After the cross-correlations, we pick both phase and group velocity arrival times of the ballistic surface-waves for which we derive independent tomographic maps. Finally, using both the retrieved phase and group velocities, we jointly invert the tomographic results from frequency to depth.</p><p>We identify positive and negative velocity variations from an average velocity between -15% and 15% for group and between -10% and 10% for phase velocities in the frequency domain. While the velocity variations are consistent for both the phase and group velocities (with expected group velocities lower than the phase velocities), the group velocity anomalies are more pronounced than the phase velocity anomalies. Low-velocity anomalies fall mostly within the inner volcano caldera, the area of highest interest for geothermal energy. This is consistent with the surface temperatures measured at the Los Humeros caldera, indicating the presence of a heat source. Finally, we compare our results with other geophysical studies (e.g geodesy, gravity, earthquake tomography and magnetotelluric) performed during the GeMEX project within the same area.</p><p> </p><p> </p><p> </p><p>We thank the European and Mexican GEMex team for setting up the seismic network and station maintenance as well as data retrieval (amongst which Tania Toledo, Emmanuel Gaucher, Angel Figueroa and Marco Calo). We thank the Comisión Federal de Electricidad (CFE) who kindly provided us with access to their geothermal field and permission to install the seismic stations. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 727550 and the Mexican Energy Sustainability Fund CONACYT-SENER, project 2015-04-68074.</p><p> </p><p>* http://www.gemex-h2020.eu/index.php?option=com_content&view=featured&Itemid=101&lang=en</p>

scholarly journals Frequency derivative of Rayleigh wave phase velocity for fundamental mode dispersion inversion: parametric study and experimental application

2020 ◽  
Vol 224 (1) ◽  
pp. 649-668
Author(s):  
A Wang ◽  
D Leparoux ◽  
O Abraham ◽  
M Le Feuvre
Keyword(s):  
Phase Velocity ◽  
Rayleigh Wave ◽  
Group Velocity ◽  
Fundamental Mode ◽  
Deep Layer ◽  
Wave Phase Velocity ◽  
Phase And Group Velocities ◽  
Velocity Derivative ◽  
Rayleigh Wave Phase Velocity ◽  
Group Velocities

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.

Joint inversion for effective anisotropic velocity model and event locations using S-wave splitting measurements from downhole microseismic data

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. C133-C143 ◽  
Author(s):  
Duo Yuan ◽  
Aibing Li
Keyword(s):  
Joint Inversion ◽  
Transverse Isotropy ◽  
Velocity Model ◽  
Anisotropic Model ◽  
S Wave ◽  
Anisotropic Models ◽  
Velocity Models ◽  
Microseismic Data ◽  
Wave Splitting ◽  
Low Symmetry

An accurate velocity model is essential in microseismic imaging. Layered sediments and subvertical cracks from tectonic stress and hydraulic fracturing make velocity models more complicated than isotropy or simple anisotropy, such as vertical transverse isotropy. Downhole microseismic acquisition usually cannot achieve sufficient ray coverage that is required to develop a low-symmetry anisotropic model from traveltimes alone. To solve this problem, we have developed a new type of data, S-wave splitting parameters (the delay time of the slow S-wave and fast S-wave polarization direction), to determine anisotropic models. A genetic algorithm inversion is adopted to solve for the locations of events and the stiffness tensor of an anisotropic medium simultaneously. We applied this method to synthetic waveforms from numerical modeling and successfully recovered the input event locations and the velocity model. The effectiveness of this method is further demonstrated by using real microseismic data acquired in the Bakken shale reservoir. Compared with the inversion with an isotropic velocity model, event locations from an anisotropic model become well-aligned with natural fractures in the Bakken Formation. Our experiments have evidenced that adding full S-wave splitting parameters makes a significant improvement in constraining a low-symmetry anisotropic model from downhole microseismic data.

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