Dispersion and attenuation of elastic waves due to multiple scattering from cracks

Multiple scattering from cracks is considered in the two-dimensional plane-strain condition. It is assumed that identical cracks are distributed uniformly in space and that the effective waves propagate normal to the crack surfaces. Then, the apparent dispersion and attenuation are calculated as functions of frequency for three independent modes of wave propagation: SV, P and SH.The calculated results show that, in each case, the attenuation coefficient Q^?1 takes a peak value when the wavelength is nearly twice the crack width, while phase velocity has a maximum deviation from the intrinsic value at a frequency lower than the peak frequency for Q^?1.     

Scattering of elastic waves by a fracture zone containing randomly distributed cracks

We theoretically study the scattering ofP, SV andSH waves by a zonal distribution of cracks, which simulates a fault fracture zone. An investigation is conducted how the geometrical properties of the crack distribution and the frictional characteristics of the crack surface are reflected in the attenuation and dispersion of incident waves, as well as in the amplitudes of the transmitted and reflected waves from the zone. If the crack distribution within the fault zone changes temporally during the preparation process of the expected earthquake, it will be important for earthquake prediction to monitor it, utilizing the scattering-induced wave phenomena.We consider the two-dimensional problem. Aligned cracks with the same length are assumed to be randomly distributed in a zone with a finite width, on which elastic waves are assumed to be incident. The distribution of cracks is assumed to be homogeneous and sparse. The crack surface is assumed to be stress-free, or to undergo viscous friction; the latter case simulates fluid-filled cracks. The opening displacement of the crack is assumed to be negligibly small. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.When the crack surface is stress-free, it is commonly observed for every wave mode (P, SV andSH) that the attenuation coefficientQ ^–1 peaks aroundka1, the phase velocity is almost independent ofk in the rangeka<1 and it increases monotonically withk in the rangeka>1, wherek is the intrinsicS wavenumber anda is the half length of the crack. The effect of the friction is to shift the peak ofQ ^–1 and the corner of the phase velocity curve to the low wavenumber range. The high wavenumber asymptote ofQ ^–1 is proportional tok ^–1 independently of model parameters and the wave modes. If the seismological observation thatQ ^–1 ofS waves has a peak at around 0.5 Hz in the earth's crust is combined with our results, the upper limit of crack size within the crust is estimated about 4 km. The information regarding the transmitted and reflected waves, such as the high wavenumber limit of the amplitude of the transmitted wave etc., allows estimation of the strength of the friction.     

Boundary element simulation of scattering of elastic waves by 3-D cracks

Numerical modelling techniques are now becoming common for understanding the complicated nature of seismic wave propagation in fractured rock. Here the Indirect Boundary Element Method (IBEM) is applied to study scattering of elastic waves by cracks. The problem addressed in this paper is the diffraction of P and S waves by open 3-D cracks of arbitrary shape embedded in a homogeneous isotropic medium. The IBEM yields the value of the jump of displacements between opposite surfaces of the crack, often called Crack Opening Displacement (COD). This is used to evaluate the solution away from the crack. We use a multi-regional approach which consists of splitting a surface S into two identical surfaces S⁺ and S⁻ chosen such that the crack lies at the interface. The resulting integral equations are not hyper-singular and wave propagation within media that contain open cracks can be rigorously solved. In order to validate the method, we compare results of displacements of a penny-shaped crack for a vertical incident P-wave with the classic results by Mal (1970) obtaining excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P or S waves upon cracks with various shapes are depicted and the salient aspects of the method are also discussed. Both frequency and time-domain results are included.     

Attenuation operators and complex wave velocities for scattering in random media

The concept of attenuation operators and complex velocities is applied to scattering attenuation in two and three dimensions, using the minimum-phase assumption for the attenuation operator. Acoustic 2D finite-difference computations of synthetic seismograms show, that the attenuation operator describes well the decay and lowpass filtering of the averaged wave form, which follows from averaging travel-time-corrected wave forms along the wave front. In the case of exponential random media, analytical forms of the attenuation operators and complex velocities are available. The complex velocities are incorporated into the reflectivity method. As an application, synthetic seismograms are presented for theS _n wave, attenuated by lithospheric velocity and density fluctuations. The limitations of attenuation operators and complex velocities for scattering are also discussed. With these quantities it is not possible to model phenomena related to the scattered waves themselves, such as amplitude and travel-time fluctuations along the wave front, codas and precursors.     

Multiple scattering ofSH waves in 2-D media with many cavities

The full waveform synthetic seismogram of multiple scatteredSH waves by many cylindrical cavities in two-dimensional homogeneous elastic media is computed. We used the so-called single-layer potential integral representation of the scattered field and a discretization scheme with line source distribution for each cavity. The total field is the sum of the incident wave plus the field radiated from all sources, each multiplied by an unknown complex constant representing its strength. These constants are determined by imposing the appropriate boundary conditions in the least-squares sense. Here we solve scattering problems involving one, two, four, twelve and fifty cavities regularly distributed in a half-space. The seismograms computed along the free-surface show regions where the incident wave is strongly attenuated, as well as the arrivals of all multiple scattered phases. The accuray of the method is estimated from the degree of agreement of our solution for one cavity with the corresponding analytical solution, and also from the magnitude of the residual tractions along the boundaries of two cavities separated at various distances. Finally we apply the method to compute the case of fifty cylindrical cavities, each of radiusa, randomly distributed in a region 80a wide by 30a deep in a half-space. The value of scattering loss is obtained from the amplitude decay of the primary wave with distance for wavelengths in the range from 1.7a to 13.3a, using the synthetic seismogram calculated for the same distribution of 50 cavities as above, but in full-space.     

Dispersion and attenuation of elastic waves due to multiple scattering from inclusions

The propagation of elastic waves in a medium containing many inclusions is considered. Under the assumption that the spatial distribution of inclusions is uniform, a general equation is derived for the determination of the velocity dispersion and attenuation coefficient of the effective waves. A simple example is presented where scatterers are infinitesimally thin cracks. The calculated results show that the attenuation coefficient Q^?1 takes a peak value for the wavelength nearly equal to twice the crack length.     

Estimation of frequency dependent coda wave attenuation structure at the vicinity of Cairo Metropolitan Area

Estimation of seismic wave attenuation in the shallow crust in terms of coda wave Q structure previously investigated in the vicinity of Cairo Metropolitan Area was improved using seismograms of local earthquakes recorded by the Egyptian National Seismic Network. The seismic wave attenuation was measured from the time decay of coda wave amplitudes on narrow bandpass filtered seismograms based on the single scattering theory. The frequency bands of interest are from 1.5 to 18 Hz. In general, the values obtained for various events recorded at El-Fayoum and Wadi Hagul stations are very similar for all frequency bands. A regional attenuation law Q _c = 85.66 f ^0.79 was obtained.     

Physical modeling observations of wave velocity and apparent attenuation in isotropic and anisotropic two-phase media

We study wave propagation through isotropic and anisotropic scatterer distributions in order to observe azimuthal variations in velocity and apparent attenuation. Using thin aluminum plates as physical models, we obtained seismograms for compressional and shear wave propagation through heterogeneous media. Three random distributions of scatterers are studied: circular scatterers in isotropic distributions (modeling circular scatterers), elongated scatterers in isotropic distributions (modeling randomly oriented elliptical scatterers), and elongated scatterers in anisotropic distributions (modeling aligned elliptical scatterers). All scatterers had approximately the same cross-sectional area and were filled with epoxy in order to reduce the impedance contrast. In addition to seismograms recorded for no scatterers, seismograms were recorded for several scatterer volume fractions. Azimuths were measured relative to the long axis of the aligned elongated scatterers. Velocities were calculated using travel times and phase shifts at low frequencies. The velocities measured from the data were compared to simple low-frequency average-velocity theories based on thin lamellae or on distributions of penny-shaped cracks. The apparent attenuation for different scatterer distributions was computed using spectral ratios.Comparisons of the results for circular and randomly oriented elongated scatterers were made to determine the effects of scatterer shape. As expected, the circular and randomly oriented elongated scatterers showed no systematic azimuthal variation in velocity. The velocity anomalies were systematically larger for the randomly oriented elongated scatterers than for the circular scatterers. Both methods of theoretical estimation for the isotropic velocities produced velocities significantly larger than those measured. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the circular scatterers.Comparisons of the results for the randomly oriented and aligned elongated scatterers were made to determine the effects of anisotropy in the scatterer distribution. Compressional waves for the aligned elongated scatterers with wave propagation parallel to the scatterers had larger velocities than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers for all velocity calculations. Shear wave velocities were complicated by an anomalous phase change in the shear wave seismograms for azimuths less than 40° and were not as conclusive. The general trend of the theoretical velocities is similar to the velocities calculated from the data. There are, however, what appear to be significant differences. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the aligned elongated scatterers with wave propagation parallel to the scatterers, and less attenuation than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers.     

P-wave dispersion and attenuation in fractured and porous reservoirs – poroelasticity approach

Natural fractures in hydrocarbon reservoirs can cause significant seismic attenuation and dispersion due to wave induced fluid flow between pores and fractures. We present two theoretical models explicitly based on the solution of Biot's equations of poroelasticity. The first model considers fractures as planes of weakness (or highly compliant and very thin layers) of infinite extent. In the second model fractures are modelled as thin penny-shaped voids of finite radius. In both models attenuation is a result of conversion of the incident compressional wave energy into the diffusive Biot slow wave at the fracture surface and exhibits a typical relaxation peak around a normalized frequency of about 1. This corresponds to a frequency where the fluid diffusion length is of the order of crack spacing for the first model and the crack diameter for the second. This is consistent with an intuitive understanding of the nature of attenuation: when fractures are closely and regularly spaced, the Biot's slow waves produced by cracks interfere with each other, with the interference pattern controlled by the fracture spacing. Conversely, if fractures are of finite length, which is smaller than spacing, then fractures act as independent scatterers and the attenuation resembles the pattern of scattering by isolated cracks. An approximate mathematical approach based on the use of a branching function gives a unified analytical framework for both models.     

An analysis method for elastic waves propagation in material involving cracks

In this paper, a new model to treat the problem of elastic wave travelling in rock mass that embeds cracks is proposed, in which, instead of modelling the crack directly the mechanical effects of discontinuities are translated into the equivalent nodal forces on a numerical procedure. Thus the response is physically obtained as the superposition of the incident wave field and the scattering wave field produced by the radiation of the reflected waves from the crack surfaces. Finally, a numerical example is carried out and it shows that the results by the proposed method have a very good agreement with those of the exact ones.     

Effective elastic properties of randomly fractured soils: 3D numerical experiments

This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.     

Attenuation and dispersion ofSH waves due to scattering by randomly distributed cracks

The effect of randomly distributed cracks on the attenuation and dispersion ofSH waves is theoretically studied. If earthquake ruptures are caused by sudden coalescence of preexisting cracks, it will be crucial for earthquake prediction to monitor the temporal variation of the crack distribution. Our aim is to investigate how the property of crack distribution is reflected in the attenuation and dispersion of elastic waves.We introduce the stochastic property, in the mathematical analysis, for the distributions of crack location, crack size and crack orientation. The crack size distribution is assumed to be described by a power law probability density (p(a) a ^– fora _minaa _max according to recent seismological and experimental knowledge, wherea is a half crack length and the range 13 is assumed. The distribution of crack location is assumed to be homogeneous for the sake of mathematical simplicity, and a low crack density is assumed. The stochastic property of each crack is assumed to be independent of that of the other cracks. We assume two models, that is, the aligned crack model and the randomly oriented crack model, for the distribution of crack orientation. All cracks are assumed to be aligned in the former model. The orientation of each crack is assumed to be random in the latter model, and the homogeneous distribution is assumed for the crack orientation. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.We observe the following features common to both the aligned crack model and the randomly oriented crack model. The attenuation coefficientQ ^–1 decays in proportion tok ^–1 in the high frequency range and its growth is proportional tok ² in the low frequency range, wherek is the intrinsic wave number. This asymptotic behavior is parameter-independent, too. The attenuation coefficientQ ^–1 has a broader peak as increases and/ora _min/a _max decreases. The nondimensional peak wave numberk _p a _max at whichQ ^–1 takes the peak value is almost independent ofa _min/a _max for =1 and 2 while it considerably depends ona _min/a _max for =3. The phase velocity is almost independent ofk in the rangeka _max<1 and increases monotonically ask increases in the rangeka _max>1. While the magnitude ofQ ^–1 and the phase velocity considerably depend on the orientation of the crack in the aligned crack model, the above feature does not depend on the crack orientation.The accumulation of seismological measurements suggests thatQ ^–1 ofS waves has a peak at around 0.5 Hz. If this observation is combined with our theoretical results onk _p a _max, the probable range ofa _max of the crack distribution in the earth can be estimated for =1 or 2. If we assume 4 km/sec as theS wave velocity of the matrix medium,a _max is estimated to range from 2 to 5 km. We cannot estimatea _max in a narrow range for =3.     

Seismic Wave Attenuation in the Greater Cairo Region, Egypt

In the present study, a digital waveform dataset of 216 local earthquakes recorded by the Egyptian National Seismic Network (ENSN) was used to estimate the attenuation of seismic wave energy in the greater Cairo region. The quality factor and the frequency dependence for Coda waves and S-waves were estimated and clarified. The Coda waves (Q _c) and S-waves (Q _d) quality factor were estimated by applying the single scattering model and Coda Normalization method, respectively, to bandpass-filtered seismograms of frequency bands centering at 1.5, 3, 6, 12, 18 and 24?Hz. Lapse time dependence was also studied for the area, with the Coda waves analyzed through four lapse time windows (10, 20, 30 and 40?s). The average quality factor as function of frequency is found to be Q _c?=?35?±?9f ^0.9±0.02 and Q _d?=?10?±?2f ^0.9±0.02 for Coda and S-waves, respectively. This behavior is usually correlated with the degree of tectonic complexity and the presence of heterogeneities at several scales. The variation of Q _c with frequency and lapse time shows that the lithosphere becomes more homogeneous with depth. In fact, by using the Coda Normalization method we obtained low Q _d values as expected for a heterogeneous and active zone. The intrinsic quality factor (Q _i ^?1 ) was separated from the scattering quality factor (Q _s ^?1 ) by applying the Multiple Lapse Time Domain Window Analysis (MLTWA) method under the assumption of multiple isotropic scattering with uniform distribution of scatters. The obtained results suggest that the contribution of the intrinsic attenuation (Q _i ^?1 ) prevails on the scattering attenuation (Q _s ^?1 ) at frequencies higher than 3?Hz.     

A new series solution method for two-dimensional elastic wave scattering along a canyon in half-space

This paper presents a semi-analytical method for studying the two-dimensional problem of elastic wave scattering by surface irregularities in a half-space. The new method makes use of the member of a c-completeness family of wave functions to construct the scattering fields, and then applies equal but opposite tractions to those of the foregoing constructed scattering fields on the horizontal surface of the half-space to produce additional scattering fields. These additional scattering fields are a series of Lamb's solutions. Thus the whole scattering field constructed in the series automatically satisfies the Navier equations, the condition of zero traction on the half-space surface, and the radiation boundary conditions at infinity. Using the traction-free conditions along the canyon surface, the coefficients of the series solutions are determined via a least-squares method. For incident P, SV, and Rayleigh waves, the numerical results are presented for the scattering displacements in the vicinity of a semi-circular canyon in the half-space.     

Estimation of the Intrinsic Absorption and Scattering Attenuation in Northeastern Venezuela (Southeastern Caribbean) Using Coda Waves

—Northeastern Venezuela has been studied in terms of coda wave attenuation using seismograms from local earthquakes recorded by a temporary short-period seismic network. The studied area has been separated into two subregions in order to investigate lateral variations in the attenuation parameters. Coda-Q ^?1 (Q _c ^?1) has been obtained using the single-scattering theory. The contribution of the intrinsic absorption (Q _i ^?1) and scattering (Q _s ^?1) to total attenuation (Q _t ^?1) has been estimated by means of a multiple lapse time window method, based on the hypothesis of multiple isotropic scattering with uniform distribution of scatterers. Results show significant spatial variations of attenuation the estimates for intermediate depth events and for shallow events present major differences. This fact may be related to different tectonic characteristics that may be due to the presence of the Lesser Antilles subduction zone, because the intermediate depth seismic zone may be coincident with the southern continuation of the subducting slab under the arc.     

The scattering of shear-waves in the crust

The two major sources of scattering for shear-waves in the crust, interactions with the topography at the surface and the effective anisotropy of aligned cracks throughout the rockmass, introduce first-order changes to the shear-wave particle-motion. At the surface, shear-waves are scattered by the topography within a wavelength or two of the recording site so that, unless the effective incidence angle is less than the critical angle sin^–1 V _S/V _P, the recorded waveforms may bear little relationship to the waveforms of the incident wave. Within the rockmass, shear-waves are scattered by extensive-dilatancy anisotropy (EDA), the distribution of stress-aligned fluid-filled cracks, microcracks, and preferentially oriented pore-space pervading most rocks in the crust. Analysis of this shear-wave splitting yields new information about the internal structure of thein situ rockmass which is not otherwise available.     

2D finite-difference viscoelastic wave modelling including surface topography

We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities.     

裂隙参数对衰减各向异性影响的数值模拟

Propagation through stress-aligned fluid-filled cracks and other inclusions have been claimed to be the cause of azimuthal anisotropy observed in the crust and upper mantle.This paper examines the behavior of seismic waves attenuation caused by the internal structure of rock mass,and in particular,the internal geometry of the distribution of fluid-filled openings Systematic research on the effect of crack parameters,such as crack density,crack aspect ratio(the ratio of crack thickness to crack diameter),pore fluid properties(particularly pore fluid velocity),VP/VS ratio of the matrix material and seismic wave frequency on attenuation anisotropy has been conducted based on Hudson’s crack theory.The result shows that the crack density,aspect ratio,material filler,seismic wave frequency,and P-wave and shear wave velocity in the background of rock mass,and especially frequency has great effect on attenuation curves.Numerical research can help us know the effect of crack parameters and is a good supplement for laboratory modeling.However,attenuation is less well understood because of the great sensitivity of attenuation to details of the internal geometry.Some small changes in the characteristics of pore fluid viscosity,pore fluids containing gas and liquid phases and pore fluids containing clay can each alter attenuation coefficients by orders of magnitude.Some parameters controlling attenuation are therefore necessary to make reasonable estimations,and anisotropic attenuation is worth studying further.     

Short-period PKP phases and the anelastic mechanism of the inner core

Short-period seismograms are synthesized for PKP phases in anelastic Earth models. The synthetics were constructed using a synthetic technique valid at grazing incidence, a source-time function appropriate for deep-focus earthquakes, and an instrument response for either a short-period WWSSN or SRO seismograph. The agreement between predicted and observed amplitudes and spectral ratios requires neither a low-Q_α zone at 0.2–2 Hz nor a low or negative P-velocity gradient at the bottom of the outer core. Thin low-Q_α zones beneath the inner core boundary fit spectral ratio data that sample the upper 200 km of the inner core but fail to fit data that sample the lower inner core. Only a model having Q_α^?1?[0.003, 0.004] at 0.2–2 Hz, nearly constant with depth in the inner core, satisfies all of the spectral ratio and amplitude data. The assumption of a bulk viscosity of 10-10³ Pa s for the liquid phase of a partially molten inner core combined with the observation of low shear attenuation in the inner core at frequencies less than 0.005 Hz limit the physical parameters associated with two possible attenuation mechanisms: (1) fluid flow and viscous relaxation due to ellipsoidally shaped inclusions of melt, and (2) the solid-liquid phase transformation induced by the stress change during the passage of a seismic wave. Both mechanisms require an order of 0.1% partial melt to reproduce the observed Q_α^?1. In the outer core, the time constant of the mechanism of phase transformation is predicted to be 10⁴–10⁶ s. Confirmation of small shear attenuation in the inner core in the frequency band of seismic body waves would favor the mechanism of phase transformation.     

Wave Generation from Explosions in Rock Cavities

—?We have developed a measurement method to monitor P- and S-waves generated from laboratory-scale explosions in meter-sized rock samples at a series of stations, as well as invented a device to drill spherical cavities in rock, with diameters up to 10?centimeters. We applied these to experiments in Bedford limestone in which spherical/cylindrical explosives (0.2 to 1.9?g) were centrally placed in 1.2- to 3-cm diameter cavities. Stress waves generated by the explosions were recorded within a radius of 25?cm. The radial stress wave records and post-explosion studies demonstrate that S-waves are generated from explosions in cavities as a result of both wave mode-conversion from the cavity wall and crack propagation in rocks. The experimental results of wave generation from the explosions in spherical and cylindrical cavities demonstrate the cavity geometrical effect on the resulting wave pattern. The P- and S-waves generated by explosions and crack propagation in rocks are analyzed. A simple analytic model for P-wave generation is proposed to explain the differences of P-wave-induced displacement histories between the observed waveforms and those predicted by a step-pressure source. Generally, the qualitative predictions of this model fit the observations. The present results demonstrate the importance of rock cracking and cavities in P- and S-wave generation.