Rayleigh Wave Attenuation Research Articles

Propagation and attenuation of Rayleigh waves in a partially-saturated porous solid with impervious boundary

Abstract Propagation of Rayleigh waves is studied in a three-phase porous solid half-space, which is bounded above by an impervious plane surface. In this dissipative medium, Rayleigh wave propagates as an inhomogeneous wave, which decays with distance from the stress-free plane boundary. The impervious boundary restricts the flow of pore-fluids to the interior of porous solid only. This is ensured by fixing the fluid-pressure gradient in pores at boundary or with the sealing of surface pores. In either case, the existence and propagation of inhomogeneous wave are represented by a dispersion equation, which happens to be complex and irrational. This equation is rationalized into an algebraic equation of degree 24, which is solved for a numerical example. Solutions of the dispersion equation are checked to represent an inhomogeneous wave decaying with depth. Each qualified solution is resolved to define the phase velocity and attenuation coefficient of a Rayleigh wave in the medium. Numerical example compares the velocity and attenuation of Rayleigh wave in porous sandy loam for the two representations of the impervious boundary, one with sealed pores and other with no fluid-pressure gradient. Effects of saturation degree, porosity, capillary pressure, pore-fluids viscosity and frame anelasticity are observed on the propagation characteristics of Rayleigh waves. Existence of second Rayleigh wave is checked numerically. Such a wave is possible only when the porous frame is highly anelastic and saturated enough.

Propagation and attenuation of Rayleigh waves in generalized thermoelastic media

This study considers the propagation of Rayleigh waves in a generalized thermoelastic half-space with stress-free plane boundary. The boundary has the option of being either isothermal or thermally insulated. In either case, the dispersion equation is obtained in the form of a complex irrational expression due to the presence of radicals. This dispersion equation is rationalized into a polynomial equation, which is solvable, numerically, for exact complex roots. The roots of the dispersion equation are obtained after removing the extraneous zeros of this polynomial equation. Then, these roots are filtered out for the inhomogeneous propagation of waves decaying with depth. Numerical examples are solved to analyze the effects of thermal properties of elastic materials on the dispersion of existing surface waves. For these thermoelastic Rayleigh waves, the behavior of elliptical particle motion is studied inside and at the surface of the medium. Insulation of boundary does play a significant role in changing the speed, amplitude, and polarization of Rayleigh waves in thermoelastic media.

A 2.5D boundary element formulation for modeling damped waves in arbitrary cross-section waveguides and cavities

A regularized 2.5D boundary element method (BEM) is proposed to predict the dispersion properties of damped stress guided waves in waveguides and cavities of arbitrary cross-section. The wave attenuation, induced by material damping, is introduced using linear viscoelastic constitutive relations and described in a spatial manner by the imaginary component of the axial wavenumber. The discretized dispersive wave equation results in a nonlinear eigenvalue problem, which is solved obtaining complex axial wavenumbers for a fixed frequency using a contour integral algorithm. Due to the singular characteristics and the multivalued feature of the wave equation, the requirement of holomorphicity inside the contour region over the complex wavenumber plane is fulfilled by the introduction of the Sommerfeld branch cuts and by the choice of the permissible Riemann sheets. A post processing analysis is developed for the extraction of the energy velocity of propagative guided waves. The reliability of the method is demonstrated by comparing the results obtained for a rail and a bar with square cross-section with those obtained from a 2.5D Finite Element formulation also known in literature as Semi Analytical Finite Element (SAFE) method. Next, to show the potential of the proposed numerical framework, dispersion properties are predicted for surface waves propagating along cylindrical cavities of arbitrary cross-section. It is demonstrated that the attenuation of surface waves approaches asymptotically the attenuation of Rayleigh waves.

Estimation of near-surface quality factors by constrained inversion of Rayleigh-wave attenuation coefficients

Quality factors (Q) of near-surface materials are as important as velocities of the materials in many applications. Only phase information of surface-wave data is utilized when high-frequency (≥ 2 Hz) surface-wave data are routinely inverted to determine near-surface shear (S)-wave velocities. Amplitude information of high-frequency surface-wave data can be used to determine quality factors of near-surface materials. Given S-wave velocity, compressional (P)-wave velocity, and Rayleigh-wave phase velocities , it is feasible to solve for S-wave quality factor Q S and P-wave quality factor Q P (for some specific velocity models) down to 30 m below the ground surface in many settings by inverting high-frequency Rayleigh-wave attenuation coefficients in a layered earth model. Amplitude of seismic data is an exponential function of attenuation coefficients. When calculating attenuation coefficients from changes in amplitude, this nonlinear nature would result in that small variations in amplitude cause huge changes in attenuation coefficients. This result suggests data (attenuation coefficients) that normally possess large errors could eventually transfer to a model (quality factors); therefore, constraints (or a priori information) on models are necessary. Because an inversion system to solve this problem is unstable, a regularization parameter must be introduced into an inversion algorithm to stabilize the inversion. These characteristics of the inversion problem allow us to solve the problem as a constrained and regularized linear system . Usually, a set of models that meet the defined constraints can be obtained by solving the system. Based on the linear nature of the inversion system, a smooth model can be selected from the set of models as a solution of the inversion using the L-curve method. This approach is a trade-off solution between data misfit and model length. Several real-world examples demonstrate the importance of constraints in finding acceptable realistic quality factors from empirical data. ► Amplitude of seismic data is an exponential function of attenuation coefficients. ► Small variations in amplitude cause huge changes in attenuation coefficients. ► Large errors in attenuation coefficients could transfer to quality factors. ► Constraints (or a priori information) on models are necessary. ► A realistic model can be selected using the L-curve method.

Comments on paper “Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium” by W.-C. Lo [Adv. Water Resour. 31 (2008) 1399–1410

Comments on paper “Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium” by W.-C. Lo [Adv. Water Resour. 31 (2008) 1399–1410

On the reliability of attenuation measurements from ambient noise cross-correlations

[1] We compare spatially averaged Rayleigh wave attenuation between 10 and 18 sec period observed on the symmetric component of ambient noise cross-correlations with regional seismic event measurements observed by the USArray Transportable Array across the western US. The ambient noise attenuation measurements are shown to be consistent with attenuation observed following an earthquake in Nevada and a mining blast in Wyoming. We demonstrate that common ambient noise data processing procedures such as temporal normalization and spectral whitening can be retained as long as the amplitudes of the cross-correlations are corrected for (1) the duration of the ambient noise cross-correlation, (2) geometrical spreading, and (3) the azimuthal variation in the strength of ambient noise sources. Correction for time-series length can be achieved accurately by dividing the empirical Green's function by the squared root-mean-squared (rms) amplitude of the trailing noise. These results provide strong justification for the ability to constrain seismic attenuation using ambient noise. However, further study of the expected asymmetry in attenuation for waves approaching (incoming) or receding from (outgoing) a central station is needed to understand the effect of uneven noise source distribution prior to estimation of local variations in attenuation.

Application of geophysical measurements for assessing the role of fissures in water infiltration within a clay landslide (Trièves area, French Alps)

AbstractRainfall and its consequences are usually considered as the major factor triggering slope instabilities within clay deposits. The link between rainfall, water infiltration and landslide activity may be complex, and comprehensive sets of data are still necessary to understand how water infiltrates in cohesive material. On unstable slopes made of fine‐grained sediments, gravitational deformation generates superficial shear ruptures and tension fissures whose continuous opening could be amplified by shrinkage during dry periods. This fissure network at the surface can be very dense and strongly controls the water infiltration process. However, it remains difficult to assess the fissure evolution at depth and its eventual relationship with shallow slip surfaces. This work presents the results of a geophysical study which aimed to characterize the fissures that are observed along the unstable clay slopes of the Trièves area (French Alps). The site was first investigated combining geomorphological analysis, drilling, borehole logging, geotechnical tests and geophysical experiments. This investigation evidenced the presence of at least three rupture surfaces (at 5, 10–15 and 42 m) with a water flow at the second one. In a second step, permanent instruments were installed (piezometers and soil water content probes). Results indicate high water infiltration velocities, down to the water table located at 2·5 m depth, that are likely to be linked with sub‐vertical preferential paths, such as fissures. Geophysical parameters (electrical resistivity, S‐wave velocities and Rayleigh waves attenuation) were monitored. Geophysical results suggest that fissures are permanently open, that they serve as preferential infiltration paths, and that they can reach, at least, a depth of 2 m. The methods employed in this research put forward that the system of imbricated fissures drains water from the surface down to the shallow slip surfaces at 5 and 10–15 m. This work highlights the role of fissures network in water infiltration in the numerous clay landslides of the Trièves area. Copyright © 2011 John Wiley & Sons, Ltd.

Rayleigh Waves in Generalized Magneto‐Thermo‐Viscoelastic Granular Medium under the Influence of Rotation, Gravity Field, and Initial Stress

The surface waves propagation in generalized magneto‐thermo‐viscoelastic granular medium subjected to continuous boundary conditions has been investigated. In addition, it is also subjected to thermal boundary conditions. The solution of the more general equations are obtained for thermoelastic coupling. The frequency equation of Rayleigh waves is obtained in the form of a determinant containing a term involving the coefficient of friction of a granular media which determines Rayleigh waves velocity as a real part and the attenuation coefficient as an imaginary part, and the effects of rotation, magnetic field, initial stress, viscosity, and gravity field on Rayleigh waves velocity and attenuation coefficient of surface waves have been studied in detail. Dispersion curves are computed numerically for a specific model and presented graphically. Some special cases have also been deduced. The results indicate that the effect of rotation, magnetic field, initial stress, and gravity field is very pronounced.

Mechanical properties estimation of functionally graded materials using surface waves recorded with a laser interferometer

An approach is presented here to invert surface wave dispersion and attenuation relative to the depth-dependence of the visco-elastic parameters of functionally graded materials (FGMs). The particularity of this method lies in allowing visco-elastic parameters to vary continuously with depth and in properly incorporating the continuous nature of these variations into both the forward problem (the calculation of dispersion and attenuation) and the inverse problem (evaluation of visco-elastic parameters). The forward problem solves the equation of elastic motion using a Runge–Kutta integration scheme, while the inverse problem is solved with the nonlinear solution for continuous inverse problems developed by Tarantola and Valette (1982) [32]. Viscoelasticity is treated as a first-order perturbation to the elastic structure. Testing on a synthetic example shows that the procedure is able to closely reproduce the S-wave velocity and attenuation profiles. As expected, the variations in P-wave velocity are not resolved, yet they do not introduce any significant bias into the S-wave velocity profile. The Rayleigh wave phase velocity and attenuation, measured by laser ultrasonic experiments, are used to infer the depth-dependence of S-wave velocity and of attenuation on mortar samples. This depth-dependence compares well with the depth-dependence derived from the sample density inferred from gamma-densitometry.

On the Causes of Frequency-Dependent Apparent Seismological Q

Variability of the Earth’s structure makes a first-order impact on attenuation measurements which often does not receive adequate attention. Geometrical spreading (GS) can be used as a simple measure of the effects of such structure. The traditional simplified GS compensation is insufficiently accurate for attenuation measurements, and the residual GS appears as biases in both Q 0 and η parameters in the frequency-dependent attenuation law Q(f) = Q 0 f η . A new interpretation approach bypassing Q(f) and using the attenuation coefficient χ(f) = γ + πf/Q e(f) resolves this problem by directly measuring the residual GS, denoted γ, and effective attenuation, Q e. The approach is illustrated by re-interpreting several published datasets, including nuclear-explosion and local-earthquake codas, Pn, and synthetic 50–300-s surface waves. Some of these examples were key to establishing the Q(f) concept. In all examples considered, χ(f) shows a linear dependence on the frequency, γ ≠ 0, and Q e can be considered frequency-independent. Short-period crustal body waves are characterized by positive γ SP values of (0.6–2.0) × 10−2 s−1 interpreted as related to the downward upper-crustal reflectivity. Long-period surface waves show negative γ LP ≈ −1.9 × 10−5 s−1, which could be caused by insufficient modeling accuracy at long periods. The above γ values also provide a simple explanation for the absorption band observed within the Earth. The band is interpreted as apparent and formed by levels of Q e ≈ 1,100 within the crust decreasing to Q e ≈ 120 within the uppermost mantle, with frequencies of its flanks corresponding to γ LP and γ SP. Therefore, the observed absorption band could be purely geometrical in nature, and relaxation or scattering models may not be necessary for explaining the observed apparent Q(f). Linearity of the attenuation coefficient suggests that at all periods, the attenuation of both Rayleigh and Love waves should be principally accumulated at the sub-crustal depths (~38–100 km).

Propagation and attenuation of Rayleigh waves in a semi-infinite unsaturated poroelastic medium

An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [Lo WC, Sposito G, Majer E. Wave propagation through elastic porous media containing two immiscible fluids. Water Resour Res 2005;41:W02025]. The dispersion equation obtained is complex-valued due to viscous dissipation resulting from the relative motion of the solid to the pore fluids. As an excitation frequency is stipulated, the dispersion equation that is a cubic polynomial is numerically solved to determine the phase speed and attenuation coefficient of Rayleigh waves in Columbia fine sandy loam permeated by an air–water mixture. Our numerical results show that, corresponding to three dilatational waves, there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10 Hz–10 kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest.

Attenuation of a Stoneley wave and higher Lamb modes due to the scattering by two-dimensional irregularities of the walls of a fluid-filled borehole

Attenuation of Stoneley waves and higher Lamb modes propagating along an irregular surface of a fluid-filled borehole is investigated. This problem generalizes the problem on the attenuation of Rayleigh waves by an irregular surface of an empty borehole [10]. The technique used to evaluate the attenuation coefficient is based on the perturbation method (surface irregularity heights are considered to be small in comparison with the wavelength) and the mean field method. As a result, an expression is obtained for the partial coefficients of the eigenmode attenuation due to the scattering of eigenmodes by the irregularities of the borehole walls into the same or other eigenmodes, as well as into the bulk longitudinal and transverse waves. The frequency-dependent behavior of the partial attenuation coefficients of both Stoneley waves and higher modes is analyzed against the ratio between the irregularity correlation length and the borehole radius for different correlation functions of irregularities.

The Anelastic Attenuation of Surface Waves Beneath Southwestern Arabia and the Southern Red Sea

The seismic attenuation coefficients of surface waves in southwestern Arabian Peninsula and the southern Red Sea were inves- tigated using the station ratio method. Love and Rayleigh waves data recorded by two GSN stations RAYN and ATD are used to compute Love and Rayleigh waves attenuation coefficients (γ L and γ R ), and Love and Rayleigh waves internal friction (Quality factors, Q L and Q R ) in the region between the two GSN seismic stations RAYN and ATD. RAYN lies at the center of Arabia, while ATD is located on the western coast of the southern Red Sea on the Afar depression. γ R is found to vary from a low of 4.03 × 10 -4 km -1 at 27 s and a high of 22.9 × 10 -4 km -1 at 44 s. γ L , on the other hand, varies between a mini- mum of -2.28 × 10 -4 km -1 at 6 s and a maximum of 10.81 × 10 -4 km -1 at 23 s. The average quality factors Q R and Q L for the upper crust are estimated to be 100 and 160, respectively. Surface wave attenuation below the upper crust is extremely high. The high attenua- tion of the crust and upper mantle may be explained in the frame work of the ongoing tectonic process of the Red Sea and the Afar depres- sion. High temperature associated with partial melting beneath the crust can result in increasing the attenuation which is observed at long periods, while at short periods (less than 27 s), the level of attenuation is less. However, the attenuation is remarkably high. The results of this confirm previous indications that the Arabian plate, unlike other stable regions, has unusually higher attenuation characteristics for seismic waves in the crust and upper mantle comparable to many tectonically active regions of the world.

Rayleigh wave attenuation due to scattering by stationary defects

We obtain expressions for the energy spectrum widths of Rayleigh waves arising because of their scattering by point and distributed defects of the surface, as well as by the edge dislocations on the surface and by the grooves of a random lattice in the surface plane. The calculations are valid when the defect density is small. Under certain conditions, our results coincide with the results of other authors who studied the scattering of Rayleigh waves by point defects and by the grooves of a random lattice. The calculations are based on the Keldysh diagram technique modified for the case of semibounded media.

Investigation of Rayleigh wave attenuation via annular lenses in acoustic microscopy

AbstractUsing annular lenses in scanning acoustic microscopy, the attenuation coefficients for several materials were determined from the calculated V(z) interference‐free curves. The dependence of such coefficients on Rayleigh velocities as well as on operating frequencies was then investigated to establish empirical relations between these parameters. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Frequency-dependent Shear-wave Q Models for the Crust of China and Surrounding Regions

— Starting with fundamental-mode Rayleigh-wave attenuation coefficient values (γ R ) predicted by previously determined frequency-independent models of shear-wave Q (Q μ ), we have obtained frequency-dependent Q μ models that explain measured values of γ R as well as of Lg coda Q and its frequency dependence at 1 Hz (Q o and η, respectively) for China and some adjacent regions. The process combines trial-and-error selection of a model for the depth distribution of the frequency dependence parameter (ζ) for Q μ with a formal inversion for the depth distribution of Q μ at 1 Hz. Fifteen of the derived models have depth distributions of ζ that are constant, or nearly constant, between the surface and a depth of 30 km. ζ distributions that vary with depth in the upper 30 km are necessary to explain the remaining seven models. ζ values for the depth-independent models vary between 0.4 and 0.7 everywhere except in the western portion of the Tibetan Plateau where they range between about 0.1 and 0.3 for three paths. These low ζ values lie in a region where QLg and crustal Q μ are very low and suggest that they should also be low for high-frequency propagation. The models in which ζ varies with depth all show a decrease in that value ranging between 0.55 and 0.8 in the upper 15 km of the crust and (with two exceptions where ζ=0.0) between 0.3 and 0.55 in the depth range 15–30 km. The distribution of ζ values between 0.6 and 0.8 (the higher part of the range) in the upper crust indicates that high-frequency waves will propagate most efficiently, relative to low-frequency waves, in a band that includes, and strikes north-northeastward from the path between event 212/97 and KMI to the path between event 180/95 and station HIA in the north.

The 20‐s Rayleigh wave attenuation tomography for central and southeastern Asia

We conducted a tomographic inversion of 20‐s Rayleigh wave spectral amplitudes to obtain a two‐dimensional (2‐D) attenuation model for central and southeastern Asia. We designed an amplitude‐measuring procedure, making use of the phase match filtering technique and available source and path information, to minimize the measurement error. The resulting amplitude measurements showed much reduced scatter with consistent average attenuation estimates. The average quality factor Q from the measurements is about 220, lower than most global estimates. We took a two‐step approach in our tomographic inversion. We first inverted the two‐station amplitude ratios for a coarse‐grid attenuation model. We then used this model as the a priori model and inverted the single‐station spectral amplitudes to obtain the final attenuation model along with source and site response terms. The attenuation model from the inversion has a broad correlation with the geology and tectonics of the region. Low attenuation is seen in stable cratonic regions. High attenuation correlates with tectonically active regions. Compared with 1‐D distance corrections, the use of 2‐D attenuation model for path correction in Ms calculations reduced the station magnitude scatter by 16–18% on the average.

Rayleigh wave attenuation due to the scattering by two-dimensional irregularities of the walls of an empty borehole

Attenuation of the Rayleigh waves propagating along an irregular surface of an empty borehole is investigated. This problem generalizes the problem on the attenuation of Rayleigh waves by an irregular boundary of a half-space. The technique used to evaluate the attenuation coefficient is based on the perturbation method and the mean field method. As a result, an expression is obtained that relates the partial attenuation coefficients of the surface Rayleigh wave to the scattering by the irregular surface of an empty borehole into the bulk longitudinal and transverse waves (the R → P and R → S processes) and into the surface Rayleigh waves (the R → R processes). The frequency-dependent behavior of the partial attenuation coefficients is analyzed for different correlation functions of irregularities.

Vehicle-induced seismic effects at a gravitational wave observatory

The seismic signature of a vehicle traveling on the site of the Australian International Gravitational Observatory has been studied using a two-axis seismometer. The main laboratory strongly attenuated the vehicle-induced seismic noise across all frequencies. We find that the magnitude of the vehicle-induced seismic signal followed a d−γ power law, with d the source–detector distance and γ≈1.7±0.4. The results are indicative of strong attenuation of seismic disturbances and are consistent with other measurements of low-amplitude Rayleigh wave attenuation in similar sandy soil types, a highly desirable property for the site of a gravitational wave observatory. This analysis provides a tool for estimating the magnitude of vehicle-induced seismic noise at the laboratory buildings from more distant sources on and around the site.

Dispersion and attenuation of Rayleigh waves at a one-dimensional random roughness of the free surface of a hexagonal crystal

Analytical expressions are derived for dispersion and attenuation of Rayleigh waves propagating along the statistically rough free surface of a hexagonal crystal (Z cut). The roughness under consideration is one-dimensional (the profile function of the roughness depends on one coordinate) and has the form of hollows of a random lattice. The results obtained earlier in the solution of an analogous problem for a two-dimensional roughness are used in the one-dimensional case. The relationships derived for the dispersion and attenuation of Rayleigh waves are treated analytically and numerically over the entire range of frequencies acceptable in the framework of the perturbation theory. It is shown that the dispersion and attenuation of Rayleigh waves are qualitatively similar to those observed in an isotropic medium.