Shear Attenuation Research Articles

A guided wave technique for the characterization of highly attenuative viscoelastic materials

The measurement of the acoustic properties of highly attenuative materials such as bitumen is very difficult. One possibility is to use measurements of the extent to which filling a cylindrical waveguide with the material affects the dispersion relationship of the cylinder. Torsional modes have been excited using piezoelectric transducers placed at one end of the cylinder, while the phase velocity and attenuation spectra have been measured by means of laser scanning. At each frequency, under the hypothesis of linear viscoelasticity, the phase velocity and attenuation of the fundamental torsional mode have been calculated as a function of the bulk shear velocity and the bulk shear attenuation of the inner core at that frequency. The resulting phase velocity and guided wave attenuation contour plots have been employed for deriving the unknown shear properties from the measured velocity and attenuation of the guided wave. The monochromaticity of the approach has not required a particular frequency dependence of the material properties to be assumed. Results for bitumen are given.

Scattering of elastic waves in damaged media.

The scattering of elastic waves in a medium with damage from microcracking is discussed. The influence of damage from penny-shaped microcracks within a homogeneous medium is considered. The microcracks are assumed to be randomly oriented and uniformly distributed. Explicit expressions are derived for the attenuation of longitudinal and shear elastic waves in terms of the damage parameter and the effective elastic moduli of the medium. A generalized tensor-based approach is used such that the results are coordinate free. The derivation is based upon diagrammatic methods. The problem is formulated in terms of the Dyson equation, which is solved for the mean field response within the limits of the first-order smoothing approximation. The longitudinal and shear attenuations are discussed in terms of their frequency dependence and damage dependence. In particular, the attenuations are shown to scale linearly with the damage parameter.

Mapping lateral variations in upper mantle attenuation by stacking P and PP spectra

We study the lateral variations in P wave attenuation in the upper mantle at frequencies between 0.16 and 0.86 Hz by analyzing the spectra from >18,000 P and >14,000 PP arrivals. We select seismograms from shallow earthquakes at epicentral distances of 40°–80° for P waves and 80°–160° for PP waves. Each spectrum is the product of source, receiver, and propagation response functions as well as local source‐ and receiver‐side effects. We correct each spectrum for average source and attenuation models. Since there are multiple receivers for each source and multiple sources for each receiver, we can approximate the source‐ and receiver‐side terms by stacking the appropriate P log spectra. The resulting source‐specific response functions include any remaining source spectrum and near‐source Q structure; the receiver stacks include the site response and near‐receiver Q structure. We correct the PP log spectra for the appropriate source‐ and receiver‐side stacks. Since attenuation in the lower mantle is small, the residual log spectrum approximates attenuation in the upper mantle near the PP bounce point and is used to estimate δ. We constrain the anomalies to the top 220 km of the mantle, as suggested by previous Q studies, and translate the δ measurements to variations in 1000/Qα. The patterns of more and less attenuating regions generally correlate with previously published shear attenuation models and surface tectonics. Continents are usually less attenuating than the global average, whereas oceanic regions tend to be more attenuating. There are interesting exceptions to this tectonic pattern, such as an attenuating region beneath southern Africa.

The Q structure of the upper mantle: Constraints from Rayleigh wave amplitudes

The three‐dimensional shear attenuation, or Qμ, structure of the upper mantle is still poorly understood despite the increasing accuracy and sophistication of velocity models of the same part of the Earth. Here we present a set of three‐dimensional Qμ models of the upper mantle based on amplitude measurement of minor and major arc Rayleigh waves in the period range 70–170 s. The models show that areas of low attenuation underlie continents to a depth of around 300 km and areas of high attenuation are associated with oceans and plate boundaries, particularly in the top 200 km. Below ∼350 km, the sensitivity of the data to attenuation structure decreases rapidly. It is well known that the amplitudes of surface waves can be strongly influenced by factors other than attenuation, in particular, focusing due to elastic structure and errors in source mechanisms. We consider the effects of focusing by discussing the implications of the results of Selby and Woodhouse [2000]. Comparisons with focusing predictions using phase velocity maps allow us to identify features due to focusing in maps of attenuation distribution. The influence of the source on the observed attenuation distribution is investigated by including a source term in the model inversion. We find that successfully combining the data at each frequency into one three‐dimensional model requires that this source term must vary with frequency, suggesting that a simple correction to the scalar moment of each event cannot explain the observations. Although the effects of focusing and the source can be significant, we find that the observed attenuation pattern is robust up to degree 8; however, shorter‐wavelength structure can be strongly influenced by other controls. Finally, we use one of the models to predict the degree 2 attenuation pattern for 250‐s Rayleigh waves. The observed pattern suggests that existing normal mode observations may be explainable in terms of structure above 350 km depth.

Sonic waveform attenuation in gas hydrate‐bearing sediments from the Mallik 2L‐38 research well, Mackenzie Delta, Canada

The Mallik 2L‐38 research well was drilled to 1150 m under the Mackenzie Delta, Canada, and penetrated a subpermafrost interval where methane hydrate occupies up to 80% of the pore space. A suite of high‐quality downhole logs was acquired to measure in situ the physical properties of these hydrate‐bearing sediments. Similar to other hydrate deposits, resistivity and compressional and shear sonic velocity data increase with higher hydrate saturation owing to electrical insulation of the pore space and stiffening of the sediment framework. In addition, sonic waveforms show strong amplitude losses of both compressional and shear waves in intervals where methane hydrate is observed. We use monopole and dipole waveforms to estimate compressional and shear attenuation. Comparing with hydrate saturation values derived from the resistivity log, we observe a linear increase in both attenuation measurements with increasing hydrate saturation, which is not intuitive for stiffening sediments. Numerical modeling of the waveforms allows us to reproduce the recorded waveforms and illustrate these results. We also use a model for wave propagation in frozen porous media to explain qualitatively the loss of sonic waveform amplitude in hydrate‐bearing sediments. We suggest that this model can be improved and extended, allowing hydrate saturation to be quantified from attenuation measurements in similar environments and providing new insight into how hydrate and its sediment host interact.

Effects of oxygen removal on ultrasonic attenuation and elastic behaviour of PrBa2Cu3Ox

This paper reports ultrasonic longitudinal and shear attenuation measurements performed on non-superconducting PrBa2Cu3Ox (x = 6.5 and 6.9) between 80 and 300 K. Longitudinal attenuation profile for orthorhombic PrBa2Cu3O6.9 showed attenuation peaks around 185, 205 and 240 K. No attenuation peaks were observed for tetragonal PrBa2Cu3O6.5 in the longitudinal mode. Shear attenuation in PrBa2Cu3O6.5 showed a broad attenuation peak centred around 140 K. The longitudinal attenuation peaks in PrBa2Cu3O6.9 are oxygen-related and are explained in relation to a possible phase transition involving oxygen ordering. In addition, we also report the longitudinal and shear attenuation and velocity measurements on pseudo-tetragonal PrBa2Cu3O6.6. A strong stiffening tendency was observed upon cooling from 270 to 220 K for the longitudinal mode. A thermally anelastic relaxation process involving oxygen hopping was suggested to cause the large change in velocity.

OPTIMAL MODEL FOR THE DIFFRACTION EFFECT IN THE ULTRASONIC FIELD OF PISTON TRANSDUCERS

In this work, the analytical and experimental examination of the problem of diffraction effect is treated. In the laboratory, the diffraction phenomena have been mainly due to the beam spread of the ultrasonic plane wave propagating through a viscoelastic material. In fact, this effect has been found to be related, essentially to the attenuation and dispersion losses on a viscoelastic material. In this work, a frequency domain system identification approach is applied to determine an optimal function correcting the beam spread effect in both the normal and oblique incidences for a large frequency band (300 kHz–3 MHz). The Maximum Likelihood Estimator is applied to the magnitude and phase of the measured beam patterns of the used transducers in order to determine the model parameters. The calibration procedure is also discussed. Once the proposed model is established, the propagation through a viscoelastic plate is described and a comparison with measurements is done to validate the investigated model. The obtained longitudinal and shear attenuation and dispersion of the ultrasound in the viscoelastic plate are compared with those obtained by applying the complex harmonic plane waves combination model.

Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation.

Noninvasive quantitation of the mechanical properties of tissue could improve early detection of pathology. Previously a method for detecting displacement from propagating shear waves using a phase-contrast MRI technique was developed. In this work it is demonstrated how a collection of data representing the full vector displacement field could be used to potentially estimate the full complex stiffness tensor. An algebraic inversion approach useful for piece-wise homogeneous materials is described in detail for the general isotropic case, which is then specialized to incompressible materials as a model for tissue. Results of the inversion approach are presented for simulated and experimental phantom data that show the technique can be used to obtain shear wave-speed and attenuation in regions where there is sufficient signal-to-noise ratio in the displacement and its second spatial derivatives. The sensitivity to noise is higher in the attenuation estimates than the shear wave-speed estimates. Magn Reson Med 45:299-310, 2001.

High‐temperature viscoelastic relaxation in iron and its implications for the shear modulus and attenuation of the Earth's inner core

The viscoelasticity of mildly impure polycrystalline iron has been studied over the temperature range 20–1300°C through a combination of seismic‐frequency torsional forced oscillation and microcreep tests. For all temperatures above ∼400°C, linear absorption band behavior is observed, both strain energy dissipationQ−1and shear modulus dispersion providing evidence of substantial departures from ideal elastic behavior. For 3 s oscillation period the temperature sensitivity of the shear modulus |∂G/∂Taverages 0.04 GPa K−1. An even larger derivative applies to the highest temperatures within the bcc field (600–800°C) and at longer periods. The isothermal variation ofQ−1with periodTois generally adequately described byQ−1∼Toα. Within the bcc field the exponent α, and hence the distributionD(τ) ∼ τα−1of relaxation times, are temperature‐independent, allowing the parameterizationQ−1∼ [Toexp(−E/RT)]α, with α = 0.20±0.02 and activation energyE= 280±30 kJ mol−1. Within the fcc field, the exponent α increases systematically with increasing temperature from ∼0.1 to ∼0.3 across a wide temperature interval, indicating that the distribution of relaxation times within the absorption band is strongly temperature‐dependent. Steady‐state viscosities inferred mainly from microcreep records for temperatures between 1000 and 1300°C, lie within the range (0.2–2)×1013Pa s. The observed mix of recoverable anelastic and viscous behavior (the latter becoming progressively more important with increasing temperature and time/period) is tentatively attributed to diffusional processes operative at grain boundaries in the relatively fine‐grained bcc‐Fe and to processes involving dislocations in the coarser‐grained specimens of the fcc phase. Relaxation of internal stresses caused by pronounced single‐crystal elastic anisotropy probably dominates the anelastic response of both phases. The marked elastic anisotropy of fcc‐Fe makes it an attractive alternative to the hcp structure widely favored for the dominant crystalline phase of the Earth's inner core. At seismic frequencies and homologous temperatures approaching unity in the inner core, solid‐state viscoelastic relaxation probably accounts for much of the observed seismic wave attenuation and contributes through the associated dispersion to the unusually high value of Poisson's ratio.

The effect of an equilibrated melt phase on the shear creep and attenuation behavior of polycrystalline olivine

The impact of a chemically and texturally equilibrated melt phase on the shear creep and attenuation behaviors of polycrystalline olivine has been measured experimentally at seismic‐to‐subseismic frequencies. The experiments were performed on aggregates that had a particularly uniform and fine grain size (∼3 µm). The effect of the melt phase (∼5 vol%) is to decrease the (Newtonian) viscosity by a factor of ∼6; there is no dramatic disaggregation effect or melt‐induced plummeting of the shear modulus. Both the melt‐free and melt‐bearing aggregates display an attenuation “band” of the form QG−1 ∝ f−1 4. This response cannot be attributed to a variation in microstructure; it is intrinsic to the diffusional creep behavior. There is no unique effect of the melt phase on the attenuation response: the slight increase in absorption of partial melt specimens is explained fully by the effect of the texturally equilibrated melt on aggregate viscosity.

Mixed‐mode crack surface interference under cyclic shear loads

A previous modelling analysis predicted that crack surface interference under cyclic shear loads is a combination of cyclic shear attenuation and cyclic wedge‐opening. In the present study, experimental evidence is provided on notched thin‐walled tubular specimens to evaluate the modelling predictions. Tests were carried out with varied static tensions superimposed on fully reversed (Rτ = −1) or pulsating (Rτ = 0) cyclic shear loads. The crack surface interference was measured by near‐tip strain gauge methods. Based on the single and dual strain gauge readings, the strains induced by the mode I and II interference are separately identified so that the cyclic wedge‐opening behaviour was noted as a companion of the cyclic shear attenuation. The crack surface interference under cyclic shear loads is compared with the influence from varied static tensions and shear stress ratios. A comparison of the mode I crack surface interference is also made between the conditions with cyclic shear loads and cyclic tensile loads. Finally, the characteristics of crack surface displacements are discussed, and the experimental results of effective mode I and II stress intensity ranges are compared with the modelling predictions.

Seismic characterization of the shore sand at Catania

In the framework of the Catania Project supported by GNDT (Gruppo Nazionale per la Difesa dai Terremoti) seismic measurements have been carried out in the shore sands of Catania to define their detailed shear wave velocity (V) profiles and attenuation properties (Q). The method used, FTAN, is based on the analysis of the dispersion curve extracted from signals generated by controlled sources. The computation of complete synthetic seismograms enables the estimation of the attenuation properties of the surface layers. Hereinafter we analyse the shear seismic characteristics of the shore sands as indicators of potential liquefaction phenomena.

Comparison between Time-domain and Frequency-domain Measurement Techniques for Mantle Shear-wave Attenuation

—We study the frequency- and time-domain techniques which have been used to measure shear attenuation in the mantle using long-period body waveforms. In the time-domain technique, waveform modeling is carried out and the attenuation model that best fits the data is chosen. In the frequency-domain technique, we solve for the attenuation model that best fits the spectra of the seismic waveforms. Though theoretically both these techniques are equivalent, modeling assumptions and measurement biases associated with each technique can give rise to different results. In this study, we compare these two techniques in terms of their accuracy in obtaining mantle shear attenuation. Specifically, we estimate the biases in constraining attenuation from differential SS−S and absolute S waveforms. We carry out these tests using realistic synthetic seismograms and we follow this with an analysis of recorded data to verify the results from the synthetic tests. For the SS−S waveforms, the primary biasing factors are interference with seismic phases due to mantle discontinuities and due to crustal reverberation under the SS bounce point. These factors can affect the t* measurements by up to 0.5 s in the frequency domain and more than 1.5 s in the time domain. For the S waveforms, the frequency-domain measurements are accurate to 0.3 s while the time-domain measurements can vary by more than 2.0 s from the predicted values. These errors are also manifested in the t* measurements made using teleseismically recorded waveforms and lead to comparatively larger noise levels in the time-domain measurements. Based on these results, we propose that in long-period body-wave attenuation studies, frequency-domain techniques should be the method of choice.

Low‐frequency shear attenuation in polycrystalline olivine: Grain boundary diffusion and the physical significance of the Andrade model for viscoelastic rheology

The high‐temperature (1200–1285°C) torsional dynamic attenuation (10−3–100 Hz) and unidirectional creep behavior of a fine, uniform grain sized (d ≈ 3 μm) olivine (∼Fo92) aggregate have been measured. In all cases, the material is found to be mechanically linear (i.e., γ(t), γ ∝ σxy1), indicating that diffusional processes dominate the deformation kinetics in these experiments. The creep response displays a large decelerating transient in the strain rate leading to a nominally constant “steady state.” The attenuation behavior displays a band in QG−1 that is moderately dependent on frequency (QG−1 ≈ f−0.35) and temperature with −1.5<log(QG−1)<0.5. The creep and attenuation behaviors are accurately represented by a compliance function based on the Andrade model of viscoelasticity, which incorporates a power law description of anelastic strain (i.e., γα ∝ tn with n ≈ ½), and its Laplace transform, respectively. The uniformity of the material and the nature of its dynamic response allow the argument that the power law transient has a physical interpretation: because the attenuation band is not associated with a range of grain sizes or a distribution of lattice dislocations, the transient term describes the intrinsic transient in diffusional creep, which arises due to the evolution of tractions on the grain boundaries and is effected by chemical diffusion within a diminishing potential. Employing the rheological model of Raj [1975] for the intrinsic transient, we demonstrate that the “high‐temperature background” absorption can be predicted from the creep response; a master curve description of the attenuation results. Comparison of these data to those of previous investigators, and contemplating their application to the upper mantle, raises the suggestion, explored in this paper, that the subgrain size may prove the critical microstructural variable effecting the broad attenuation band seen in all experiments as well as in the upper mantle.

Outstanding student awards

The AGU All‐Union Committee for Mineral and Rock Physics (M&RP) will present their 1998 Outstanding Student Awards to Tye Gribb and Abby Kavner during the M&RP and Nonlinear Geophysics Reception. The reception will be held during the AGU 1998 Fall Meeting in San Francisco, Calif., USA.Tye Gribb is being recognized for his innovative experiments to determine elastic attenuation in silicates and silicate partial melts. Tye received his B.S. in metallurgical engineering from the University of Wisconsin‐ Madison. He is presently completing his Ph.D. under the supervision of Reid Cooper at the University of Wisconsin‐Madison. Gribb is being honored for his experimental studies of the shear and Young's modulus attenuation responses of polycrystalline silicates and silicate partial melts. He is cited as an exceptionally talented experimentalist—one who can design an experiment to test a theory or observation, build the apparatus to carry out the experiment, fabricate appropriate materials for testing, and rigorously analyze the results.

Seismic attenuation of the inner core: Viscoelastic or stratigraphic?

Broadband velocity waveforms of PKIKP in the distance range 150° to 180° are inverted for inner core attenuation. A mean Qα of 244 is determined at 1 Hz from 8 polar and 9 equatorial paths. The scatter in measured Q−1 exceeds individual error estimates, suggesting significant variation in attenuation with path. These results are interpreted by (1) viscoelasticity, in which the relaxation spectrum has a low‐frequency corner near or slightly above the frequency band of short‐period body waves, and by (2) stratigraphic (scattering) attenuation, in which attenuation and pulse broadening are caused by the interference of scattered multiples in a velocity structure having rapid fluctuations along a PKIKP path. In the scattering interpretation, PKIKP attenuation is only weakly affected by the intrinsic shear attenuation measured in the free‐oscillation band. Instead, its frequency dependence, path variations, and fluctuations are all explained by scattering attenuation in a heterogeneous fabric resulting from solidification texturing of intrinsically anisotropic iron. The requisite fabric may consist of either single or ordered groups of crystals with P velocity differences of at least 5% and as much as 12% between two crystallographic axes at scale lengths of 0.5 to 2 km in the direction parallel to the axis of rotation and longer in the cylindrically radial direction, perpendicular to the axis of rotation.

Oxygen related ultrasonic attenuation and elastic anomalies in EuBa 2Cu 3O 7- δ ( δ∼0.02 and 0.45)

Ultrasonic shear velocity and attenuation measurements on superconducting ( δ∼0.02) and non-superconducting ( δ∼0.45) polycrystalline EuBa 2Cu 3O 7- δ have been performed at 7-8 MHz between 80 K and 220 K. Enhanced attenuation and softening tendency was observed near 170 K in the superconducting material. The attenuation peak is greatly suppressed and no softening tendency was observed in the non-superconducting sample. The reduction of oxygen content from O 6.98 to O 6.55 caused the absolute shear velocity to decrease. The O 6.98 sample showed a 2.25% change in shear velocity between 80 K and 220 K while the O 6.55 sample showed a 3.75% change within the same temperature range. These anomalies are discussed in terms of possible oxygen ordering. The acoustic Debye temperature of the superconducting sample in the void-free approximation is estimated to be Θ D∼457 K.

Geoacoustic Inversion for the Workshop '97 Benchmark Test Cases Using Simulated Annealing

This paper applies simulated annealing matched-field inversion to the Workshop '97 geoacoustic inversion benchmark test cases. The unknown parameters in these test cases include various combinations of sediment and basement geoacoustic properties (layer thickness, compressional and shear speeds, attenuations, and densities) and geometric parameters (water depth, and source range and depth). The sensitivity of the acoustic field data to the different parameters varies substantially. The simulated annealing inversion developed here minimizes a measure of mismatch based on the average Bartlett processor for acoustic data at a number of ranges. The replica pressure fields are computed by summing the modal components provided by an elastic normal mode model, which allows replica fields at multiple ranges to be calculated efficiently. A total of 15 different geoacoustic inversions are considered (three realizations each for five test cases). Excellent results are obtained in a reasonable amount of computation time for all cases. In particular, the final mismatch values are exceedingly small and relatively sensitive parameters (geometric parameters and sediment properties) are generally estimated to high precision. Relatively insensitive parameters (shear speeds and, in some cases, basement properties) are somewhat less well determined, but generally good estimates are obtained.

Shear attenuation and dispersion in MgO

An understanding of anelastic effects on seismic wave propagation is crucial for developing accurate seismic models of the mantle. The lower mantle is thought to contain a significant volume fraction, approximately 20%, of magnesiowüstite, (Mg,Fe)O, in addition to the dominant silicate perovskite. Weaker magnesiowüstite may dominate the rheological response of the lower mantle. It may also account for the shear attenuation observed in the lower mantle. We have measured the complex torsional modulus of a single crystal of the magnesiowüstite end member periclase, MgO, at small strains, seismic frequencies, and high temperature. The attenuation is approximately 0.030 ( Q approximately 33) at 0.003 Hz and 1500 K with a dislocation density of approximately 5 × 10 9 m −2. The corresponding dispersion reduces the shear velocity of MgO at 0.001 Hz and 1500 K by approximately 4% relative to ultrasonic values. To the extent that magnesiowüstite behaves similarly to MgO, it appears to be a good candidate to account for a major portion of lower-mantle attenuation.

Global lateral variations of shear wave attenuation in the upper mantle

We analyze several thousand high‐quality, globally recorded SS – S differential waveforms to constrain the lateral variation of shear wave attenuation (Qβ) in the upper mantle. We use a multitaper frequency domain technique to measure attenuation, parameterized by a t* operator, and implement a robust estimation technique to compute t* and its variance. The differential waveform technique minimizes the effect of factors such as finite source duration and structural complexity near the source and receiver so the differential SS – S waveforms are mainly sensitive to the shear attenuation in the upper mantle under the SS bounce point. We use seismograms recorded at ranges of 45° to 100° and compute the SS – S differential t* from the broadening of the SS waveform relative to the Hilbert transform of the S waveform. A careful choice of fitting windows allows us to reduce the biasing effects of interfering phases which can affect t* by up to 0.5 s. The t* residuals (with respect to preliminary reference Earth model (PREM)) vary by ±1.5 s with an average of ≈0.24 s. Our study suggests an average Qβ value of 112 (most of the lateral variations of Qβ are within 30% of this value) in the top 400 km of the mantle, slightly lower than the PREM value of 128. There is a qualitative correlation of t* residual with tectonic region with distinctly higher attenuation observed under young oceans compared to platforms and shields. Also, the lateral variations of the residuals are similar in trend to those observed in studies of the attenuation of SCS multiples. At long wavelengths, the Qβ map shows a modest correlation with shear wave attenuation maps computed from surface wave analyses and with the patterns of lateral variations of shear velocities at certain upper‐mantle depths predicted by the model S16B30. The correlation with the velocity model is highest at 300–500 km depth indicating that there may be a contribution to long‐wavelength attenuation from relatively deep regions. Formal inversion for an upper mantle Qβ model shows that while lateral resolution is quite good, depth resolution is poor as might be expected. Better depth resolution must await combined body wave and surface wave inversions.