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.

Effect of inertial coefficients in the propagation of plane waves in micropolar porous materials

2017 ◽  
Vol 06 (01) ◽  
pp. 1750007 ◽  
Author(s):  
R. Lianngenga
Keyword(s):  
Free Surface ◽  
Porous Materials ◽  
Plane Waves ◽  
Body Waves ◽  
Plane Body ◽  
Reflected Waves ◽  
Phase Velocities ◽  
Energy Ratios

The problem of phase velocities and attenuations of plane body waves and its reflection from a stress-free surface has been investigated in the micropolar porous materials. The amplitude and energy ratios of reflected waves are obtained analytically. The effect of inertial coefficients in the propagation of plane body waves is computed numerically for a particular model.

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.

An exact anelastic model for the free-surface relfection of P and S-I waves

1989 ◽  
Vol 79 (3) ◽  
pp. 842-859
Author(s):  
R. D. Borcherdt ◽  
G. Glassmoyer
Keyword(s):  
Free Surface ◽  
Volumetric Strain ◽  
Major Axis ◽  
P Wave ◽  
Reflection Coefficients ◽  
S Wave ◽  
Low Loss ◽  
Surface Reflection ◽  
Pierre Shale ◽  
Homogeneous Wave

Abstract Exact anelastic solutions incorporating inhomogeneous waves are used to model numerically S-I and P waves incident on the free surface of a low-loss anelastic half-space. Anelastic free-surface reflection coefficients are computed for the volumetric strain and displacement components of inhomogeneous wave fields. For the problem of an incident homogeneous S-I wave in Pierre shale, the largest strain and displacement amplitudes for the reflected P wave occur at angles of incidence for which the particle motion for the reflected inhomogeneous P wave is elliptical (minor/major axis = 0.6), the specific absorption (QP−1) is greater (300 per cent) and the velocity is less (25 per cent) than those for a corresponding homogeneous P wave, the direction of phase propagation is not parallel to the free surface, and the amplitude of the wave shows a significant increase with depth (6 per cent in one wavelength). Energy reflection coefficients computed for this low-loss anelastic model show that energy flow due to interaction of the incident and reflected waves reach maxima (30 per cent of the incident energy) near large but nongrazing angles of incidence. For the problem of an incident homogeneous P wave in Pierre shale, the inhomogeneity of the reflected S wave is shown not to contribute to significant variations in wave field characteristics over those that would be expected for a homogeneous wave.

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.

Three-component analysis of regional seismograms

1990 ◽  
Vol 80 (6B) ◽  
pp. 2032-2052 ◽  
Author(s):  
D. C. Jepsen ◽  
B. L. N. Kennett
Keyword(s):  
Free Surface ◽  
Time Window ◽  
Seismic Station ◽  
P Wave ◽  
Sh Waves ◽  
Near Surface ◽  
Source Discrimination ◽  
Decomposition Scheme ◽  
Phased Array Techniques ◽  
The Stability

Abstract Both phased array techniques for single-component sensors and vectorial analysis of three-component recordings can provide estimates of the azimuth and slowness of seismic phases. However, a combination of these approaches provides a more powerful tool to estimate the propagation characteristics of different seismic phases at regional distances. Conventional approaches to the analysis of three-component seismic records endeavor to exploit the apparent angles of propagation in horizontal and vertical planes as well as the polarization of the waves. The basic assumption is that for a given time window there is a dominant wavetype (e.g., a P wave) traveling in a particular direction arriving at the seismic station. By testing a range of characteristics of the three-component records, a set of rules can be established for classifying much of the seismogram in terms of wavetype and direction. It is, however, difficult to recognize SH waves in the presence of other wavetypes. Problems also arise when more than one signal (in either wavetype or direction) arrive in the same window. The stability and robustness of the classification scheme is much improved when records from an array of three-component sensors are combined. For a set of three-component instruments forming part of a larger array, it is possible to estimate the slowness and azimuth of arrivals from the main array and then extract the relative proportions of the current P-, SV-, and SH-wave contributions to the seismogram. This form of wavetype decomposition depends on a model of near-surface propagation. A convenient choice for hard-rock sites is to include just the effect of the free surface, which generates a frequency-independent operation on the three-component seismograms and which is not very sensitive to surface velocities. This approach generates good estimates of the character of the S wavefield, because the phase distortion of SV induced by the free surface can be removed. The method has been successfully applied to regional seismograms recorded at the medium aperture Warramunga array in northern Australia, and the two small arrays NORESS and ARCESS in Norway, which were designed for studies of regional phases. The new wavefield decomposition scheme provides results in which the relative proportions of P, SV, and SH waves as a function of time can be compared without the distortion imposed by free surface amplification. Such information can provide a useful adjunct to existing measures of signal character used in source discrimination.

scholarly journals A simple P-wave polarization analysis: its application to earthquake location

1994 ◽  
Vol 37 (5) ◽  
Author(s):  
B. Alessandrini ◽  
M. Cattaneo ◽  
M. Demartin ◽  
M. Gasperini ◽  
V. Lanza
Keyword(s):  
Ground Motion ◽  
Seismic Wave ◽  
Heterogeneous Media ◽  
Seismic Wave Propagation ◽  
Least Square ◽  
P Wave ◽  
Motion Direction ◽  
Arrival Times ◽  
Hypocentral Location ◽  
Ray Path

We present a method for hypocentral location which takes into account all three components of ground motion and not only the vertical one, as it is usually done by standard least-square techniques applied to arrival times. Assuming that P-wave particle motion direction corresponds to the propagation direction of the seismic wave, we carried out a simple statistical analysis of ground motion amplitudes, carefully using three-component records. We obtained the azimuth and the emersion angle of the seismic ray, which, added to Pg and Sg arrival times, allowed us to find reliable hypocentral coordinates of some local events by means of a ray-tracing technique. We compared our locations to those obtained using a least-square technique: our polarization method's dependence on the accuracy of the model used (on the contrary, the least-square technique proved to be quite stable with respect to changes in the model's velocity parameters) led us to conclude that polarization data provide coherent information on the true ray-path and can be successfully used for both location procedures and seismic wave propagation studizs in strongly heterogeneous media.

scholarly journals Vertical to Horizontal Spectral Ratio (VHSR) Response of Seismic Wave Propagation in a Homogeneous Elastic – Poroelastic Medium Using The Spectral Finite Element Method

2021 ◽  
Vol 11 (1) ◽  
pp. 95
Author(s):  
Sudarmaji Saroji ◽  
Budi Eka Nurcahya ◽  
Nivan Ramadhan Sugiantoro
Keyword(s):  
Finite Element Method ◽  
Elastic Medium ◽  
Seismic Waves ◽  
Low Frequency ◽  
Seismic Wave Propagation ◽  
P Wave ◽  
Poroelastic Medium ◽  
Compressional Waves ◽  
Spectral Finite Element Method ◽  
Spectral Finite Element

<p>Numerical modeling of 2D seismic wave propagation using spectral finite element method to estimate the response of seismic waves passing through the poroelastic medium from a hydrocarbon reservoir has been carried out. A hybrid simple model of the elastic - poroelastic - elastic with a mesoscopic scale element size of about 50cm was created. Seismic waves which was in the form of the ricker function are generated on the first elastic medium, propagated into the poroelastic medium and then transmitted to the second elastic medium. Pororoelastic medium is bearing hydrocarbon fluid in the form of gas, oil or water. Vertical and horizontal component of velocity seismograms are recorded on all mediums. Seismograms which are recorded in the poroelastic and second elastic medium show the existence of slow P compressional waves following fast P compressional waves that do not appear on the seismogram of the first elastic medium. The slow P wave is generated when the fast P wave enters the interface of the elastic - poroelastic boundary, propagated in the poroelastic medium and is transmited to the second elastic medium. The curves of Vertical to horizontal spectrum ratio (VHSR) which are observed from seismograms recorded in the poroelastic and the second elastic medium show that the peak of VHSR values at low frequency correlated with the fluid of poroelastic reservoir. The highest VHSR value at the low frequency which is recorded on the seismogram is above the 2.5 Hz frequency for reservoirs containing gas and oil in the second elastic medium, while for the medium containing water is the highest VHSR value is below the 2.5 Hz frequency.</p>

scholarly journals Explicit Q expressions for inhomogeneous P- and SV-waves in isotropic viscoelastic media

2019 ◽  
Vol 17 (2) ◽  
pp. 300-312 ◽  
Author(s):  
Xu Liu ◽  
Stewart Greenhalgh ◽  
Bing Zhou ◽  
Huijian Li
Keyword(s):  
Energy Density ◽  
Dissipation Factor ◽  
Energy Balance Equation ◽  
Dissipated Energy ◽  
P Waves ◽  
S Waves ◽  
Viscoelastic Media ◽  
Inhomogeneous Waves ◽  
Inhomogeneity Parameter ◽  
P And Sv Waves

Abstract We derive explicit expressions for the dissipation factors of inhomogeneous P and SV-waves in isotropic viscoelastic media. The Q−1 values are given as concise and simple functions of material parameters and the wave inhomogeneity parameter using two different definitions. Unlike homogenous waves, inhomogeneous waves may have significant differences in the values of dissipation factors because of different definitions. For example, under one of the three dissipation factor definitions that Q−1 is equal to the time-averaged dissipated-energy density divided by twice the time-averaged strain-energy density, it is found and proved that the dissipation factor of SV-waves is totally independent of the inhomogeneity parameter. For materials in which P-waves are normally more dissipative than S-waves (e.g. a porous reservoir), the dissipation factors of P-waves tend to decrease with increasing degree of inhomogeneity. Based on Buchan's classic real value energy balance equation, a parallel investigation is conducted for each step similar to that based on the Carcione equations, including derivation of explicit formulas (with inhomogeneity angle representing the degree of inhomogeneity of a plane wave), and dissipation curves calculations. We also obtain an inhomogeneity independent formula of $Q_{\, SV}^{ - 1}$, and exactly the same phase velocity and attenuation dispersion results for the example material.

The mechanism of vortex connection at a free surface

1999 ◽  
Vol 384 ◽  
pp. 207-241 ◽  
Author(s):  
CHIONG ZHANG ◽  
LIAN SHEN ◽  
DICK K. P. YUE
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Vortex Ring ◽  
Stokes Equations ◽  
Incidence Angle ◽  
Surface Boundary ◽  
Viscous Layer ◽  
Vertical Vorticity ◽  
Vorticity Component ◽  
Stress Boundary Conditions

Vortex connections at the surface are fundamental and prominent features in free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon that surface. We perform numerical simulations of the Navier–Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.

Scalar reverse‐time depth migration of prestack elastic seismic data

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1519-1527 ◽  
Author(s):  
Robert Sun ◽  
George A. McMechan
Keyword(s):  
Seismic Waves ◽  
Velocity Model ◽  
P Wave ◽  
Common Source ◽  
Reverse Time ◽  
S Wave ◽  
Wave Source ◽  
Reflected Waves ◽  
S Waves ◽  
Elastic Data

Reflected P‐to‐P and P‐to‐S converted seismic waves in a two‐component elastic common‐source gather generated with a P‐wave source in a two‐dimensional model can be imaged by two independent scalar reverse‐time depth migrations. The inputs to migration are pure P‐ and S‐waves that are extracted by divergence and curl calculations during (shallow) extrapolation of the elastic data recorded at the earth’s surface. For both P‐to‐P and P‐to‐S converted reflected waves, the imaging time at each point is the P‐wave traveltime from the source to that point. The extracted P‐wave is reverse‐time extrapolated and imaged with a P‐velocity model, using a finite difference solution of the scalar wave equation. The extracted S‐wave is reverse‐time extrapolated and imaged similarly, but with an S‐velocity model. Converted S‐wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources. Synthetic examples show that the algorithm gives satisfactory results for laterally inhomogeneous models.

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