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.     

Seismic Attenuation Computed from GLIMPCE Reflection Data and Comparison with Refraction Results

—Instantaneous frequency matching has been used to compute differential t* values for seismic reflection data from the Great Lakes International Multidisciplinary Program on Crustal Evolution (GLIMPCE) experiment. The differential attenuation values were converted to apparent Q ^?1 models by a fitting procedure that simultaneously solves for the interval Q ^?1 values using non-negative least squares. The bootstrap method was then used to estimate the variance in the interval Q ^?1 models. The shallow Q ^?1 structure obtained from the seismic reflection data corresponds closely with an attenuation model derived using instantaneous frequency matching on seismic refraction data along the same transect. This suggests that the effects of wave propagation and scattering on the apparent attenuation are similar for the two data sets. The Q ^?1 model from the reflection data was then compared with the structural interpretation of the reflectivity data. The highest interval Q ^?1 values (>0.01) were found near the surface, corresponding to the sedimentary rock sequence of the upper Keweenawan. Low Q ^?1 values (<0.0006) are found beneath the Midcontinent rift’s central basin. In addition to structural interpretation, seismic attenuation models derived in this way can be used to correct reflection data for dispersion, frequency and amplitude effects, and allow for improved imaging of the subsurface.     

A Dislocation Model of Seismic Wave Attenuation and Micro-creep in the Earth: Harold Jeffreys and the Rheology of the Solid Earth

—A microphysical model of seismic wave attenuation is developed to provide a physical basis to interpret temperature and frequency dependence of seismic wave attenuation. The model is based on the dynamics of dislocation motion in minerals with a high Peierls stress. It is proposed that most of seismic wave attenuation occurs through the migration of geometrical kinks (micro-glide) and/or nucleation/migration of an isolated pair of kinks (Bordoni peak), whereas the long-term plastic deformation involves the continuing nucleation and migration of kinks (macro-glide). Kink migration is much easier than kink nucleation, and this provides a natural explanation for the vast difference in dislocation mobility between seismic and geological time scales. The frequency and temperature dependences of attenuation depend on the geometry and dynamics of dislocation motion both of which affect the distribution of relaxation times. The distribution of relaxation times is largely controlled by the distribution in distance between pinning points of dislocations, L, and the observed frequency dependence of Q, Q, Q∝ω^α is shown to require a distribution function of P(L)∝L ^-m with m=4-2α The activation energy of Q ^?1 in minerals with a high Peierls stress corresponds to that for kink nucleation and is similar to that of long-term creep. The observed large lateral variation in Q ^?1 strongly suggests that the Q ^?1 in the mantle is frequency dependent. Micro-deformation with high dislocation mobility will (temporarily) cease when all the geometrical kinks are exhausted. For a typical dislocation density of ～ 10⁸ m^?2, transient creep with small viscosity related to seismic wave attenuation will persist up to the strain of ～ 10^?6, thus even a small strain (～ 10^?6?10^?4) process such as post-glacial rebound is only marginally affected by this type of anelastic relaxation. At longer time scales continuing nucleation of kinks becomes important and enables indefinitely large strain, steady-state creep, causing viscous behavior.     

Elastic modulus and internal friction in enstatite,forsterite and peridotite at seismic frequencies and high temperatures

Spectra of internal friction between 2 and 8 Hz were studied in a single crystal of enstatite, in a polycrystal of synthetic forsterite and in several samples of natural peridotite. Measurements of Q^?1 and μ were performed in vacuum (10^?6 torr), from room temperature up to 1100°C. For these experimental conditions no peak was observed in the polycrystalline undeformed forsterite, but the background attenuation irregularly increased from 5 · 10^?3 to 10^?2.A peak Q^?1 = 7 · 10^?2 appears in a deformed peridotite at 930°C. It is reduced of 60% after 5 h of annealing at 1100°C. But the background attenuation persists. In the single crystal of enstatite, a peak is observed at 760°C (Q^?1 = 6 · 10^?2). A mechanism involving dislocations is suggested as a possible explanation for the peak obtained with the peridotite samples. If this hypothesis is right, the observed effect would be diffusion controlled so that one can expect pressure to translate it towards higher temperature. This mechanism could therefore appear in the upper mantle. Background attenuation could be the result of intergranular thermal losses.     

Great-circle Rayleigh wave attenuation and group velocity,Part IV: Regionalization and pure-path models for shear velocity and attenuation

Pure-path averages for group velocities and specific attenuation have been calculated from individual observations and from path averages for two regionalizations; one original to this study and the other previously devised by Wu. Both are based on four upper-mantle provinces: ocean basin, continent, island arc and mid-ocean ridge. Pure-path group velocities and specific attenuation have also been calculated for combinations of regions and provide well separated regional measurements for such composite regions.Shear-velocity models for pure and combined regions have been derived by a controlled Monte Carlo inversion procedure and indicates that a low-velocity zone is required beneath the oceans, but is not required beneath continents. Models have been produced for pure and combined ocean, ocean-ridge, continent and continent-arc provinces.Q^?1_R determined from pure-path average group velocities and attenuation coefficients has been regionalized successfully for 2- and 3-region combinations. The resulting pure-path Q^?1_R for continents is much lower than that for ocean basins and ocean-ridge provinces. Inversion of Q^?1_R for ocean-ridge provinces shows that the average Q_β for the upper 200 km of these regions is between 85 and 100.     

SEISMIC ATTENUATION IN THE VICINITY OF THE GEOTHERMAL ANOMALY AT URACH OBTAINED FROM NEAR-VERTICAL REFLECTION PROFILES1

The attenuation in the vicinity of the geothermal anomaly at Urach was determined by means of two near-vertical reflection profiles. The attenuation in the sediments and in the upper crust (3-4 km depth) was estimated by interpretation of the first (refracted) arrivals. For calculating the attenuation, the amplitude decay with respect to distance was used. Corrections for the spread factor, i.e. the geometric amplitude divergence was deduced from the traveltime curves. Below the anomaly, higher attenuation values (Q^?1～ 0.008) were observed compared with those in the undisturbed crust (Q^?1～ 0.002). This effect is probably due to the cracks and fissures in the upper part of the crystalline basement. The attenuation in the middle and lower crust was determined using near-vertical reflections from this depth interval. The use of the spectral ratio method leads to higher values of the effective attenuation Q^?1_eff below the heat flow anomaly compared to those of the‘ normal’crust. This zone of high Q^?1_eff coincides with the low velocity body below the heat flow anomaly. Both effects, the higher attenuation and the lower velocities, could be caused by high temperatures, cracks and fissures in the crust.     

Frequency-dependent Attenuation of High-frequency P and S Waves in the Upper Crust in Western Nagano, Japan

—Borehole seismograms from local earthquakes in the aftershock region of the 1984 western Nagano Prefecture, Japan earthquake were analyzed to measure the frequency-dependent characteristics of P- and S-wave attenuation in the upper crust. The records from a three-component velocity seismometer at the depth of 145m exhibit high S/N-ratio in a wide frequency range up to 100 Hz. Extended coda normalization methods were applied to bandpass-filtered seismograms of frequencies from 25 to 102 Hz. For the attenuation of high-frequency P and S waves, our measurements show Q _P ^-1? 0.052?^-0.66 and Q _S ^-1? 0.0034?^-0.12 respectively. The frequency dependence of the quality factor of S waves is very weak as compared with that of P waves. The ratio of Q _P ^-1/Q _S ^-1 is larger than unity in the entire analyzed frequency range.     

Ultrasonic measurements of Vp and Qp: relaxation spectrum of complex modulus on basalt melts

Ultrasonic compressional wave velocity V_p and quality factor Q_p have been measured in alkali basalt, olivine basalt and basic andesite melts in the frequency range of 3.4–22 MHz and in the temperature range of 1100–1400°C. Velocity and attenuation of the melts depend on frequency and temperature, showing that there are relaxation mechanisms in the melts. Complex moduli are calculated from the ultrasonic data. The results fit well a complex modulus of Arrhenius temperature dependence with log-normal Gaussian distribution in relaxation times of attenuation. The analysis yields average relaxation time, its activation energy, relaxed modulus, unrelaxed modulus and width of Gaussian distribution in relaxation times. Relaxed modulus is smaller (17.5 GPa) for basic andesite melt of high silica and high alumina contents than for the other two basalt melts (18.1–18.4 GPa). The most probable relaxation times decrease from ～ 3 × 10^?10 s for basic andesite to ～ 10^?11 s for alkali basalt at 1400°C. Activation energies of attenuation, ranging from 270 to 340 kJ mol^?1 in the three melts, are highest in basic andesite. Longitudinal viscosity values and their temperature dependences are also calculated from V_p and Q_p data. The volume viscosity values are estimated from the data using the shear viscosity values. Longitudinal, volume and shear viscosities and their activation energies are highest in the basic andesite melt of the most polymerized structure.     

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.     

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.     

Q-structure beneath the Tibetan Plateau from the inversion of Love- and Rayleigh-wave attenuation data

The fundamental mode Love and Rayleigh waves generated by ten earthquakes and recorded across the Tibet Plateau, at QUE, LAH, NDI, NIL, KBL, SHL, CHG, SNG and HKG are analysed. Love- and Rayleigh-wave attenuation coefficients are obtained at time periods of 5–120 s using the spectral amplitudes of these waves for 23 different paths. Love wave attenuation coefficient varies from 0.0021 km^?1, at a period of 10 s, to 0.0002 km^?1 at a period of 90 s, attaining two maxima at time periods of 10 and 115 s, and two minima at time periods of 25 and 90 s. The Rayleigh-wave attenuation coefficient also shows a similar trend. The very low value for the dissipation factor, Q_β, obtained in this study suggests high dissipation across the Tibetan paths. Backus-Gilbert inversion theory is applied to these surface wave attenuation data to obtain average Q_β^?1 models for the crust and uppermost mantle beneath the Tibetan Plateau. Independent inversion of Love- and Rayleigh-wave attenuation data shows very high attenuation at a depth of ～50–120 km ($\text{Q}{}_{\mathrm{\beta }}\text{? 10}$). The simultaneous inversion of the Love and Rayleigh wave data yields a model which includes alternating regions of high and low Q_β^?1 values. This model also shows a zone of high attenuating material at a depth of ～40–120 km. The very high inferred attenuation at a depth of ～40–120 km supports the hypothesis that the Tibetan Plateau was formed by horizontal compression, and that thickening occurred after the collision of the Indian and Eurasian plates.     

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.     

Near-source attenuation of high-frequency body waves beneath the New Madrid Seismic Zone

Attenuation characteristics in the New Madrid Seismic Zone (NMSZ) are estimated from 157 local seismograph recordings out of 46 earthquakes of 2.6?≤?M?≤?4.1 with hypocentral distances up to 60 km and focal depths down to 25 km. Digital waveform seismograms were obtained from local earthquakes in the NMSZ recorded by the Center for Earthquake Research and Information (CERI) at the University of Memphis. Using the coda normalization method, we tried to determine Q values and geometrical spreading exponents at 13 center frequencies. The scatter of the data and trade-off between the geometrical spreading and the quality factor did not allow us to simultaneously derive both these parameters from inversion. Assuming 1/R ^1.0 as the geometrical spreading function in the NMSZ, the Q _P and Q _S estimates increase with increasing frequency from 354 and 426 at 4 Hz to 729 and 1091 at 24 Hz, respectively. Fitting a power law equation to the Q estimates, we found the attenuation models for the P waves and S waves in the frequency range of 4 to 24 Hz as Q _P?=?(115.80?±?1.36) f ^{(0.495?±?0.129)} and Q _S?=?(161.34?±?1.73) f ^{(0.613?±?0.067)}, respectively. We did not consider Q estimates from the coda normalization method for frequencies less than 4 Hz in the regression analysis since the decay of coda amplitude was not observed at most bandpass filtered seismograms for these frequencies. Q _S/Q _P?>?1, for 4?≤?f?≤?24 Hz as well as strong intrinsic attenuation, suggest that the crust beneath the NMSZ is partially fluid-saturated. Further, high scattering attenuation indicates the presence of a high level of small-scale heterogeneities inside the crust in this region.     

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.     

Estimation of Q p and Q s of Kinnaur Himalaya

The attenuation characteristics of the Kinnaur area of the North West Himalayas were studied using local earthquakes that occurred during 2008–2009. Most of the analyzed events are from the vicinity of the Panjal Thrust (PT) and South Tibetan Detachment Thrust, which are well-defined tectonic discontinuities in the Himalayas. The frequency-dependent attenuation of P and S waves was estimated using the extended coda normalization method. Data from 64 local earthquakes recorded at 10 broadband stations were used. The coda normalization of the spectral amplitudes of P and S waves was done at central frequencies of 1.5, 3, 6, 9, and 12 Hz. Q _p increases from about 58 at 1.5 Hz to 706 at 12 Hz, and Q _s increases from 105 at 1.5 Hz to 1,207 at 12 Hz. The results show that the quality factors for both P and S waves (Q _p and Q _s) increase as a function of frequency according to the relation Q?=?Q _o f ⁿ , where Q _o is the corresponding Q value at 1 Hz frequency and “n” is the frequency relation parameter. We obtained Q _p?=?(47?±?2)f ^(1.04±0.04) and Q _s?=?(86?±?4)f ^(0.96±0.03) by fitting power law dependency model for the estimated values of the entire study region. The Q ₀ and n values show that the region is seismically very active and the crust is highly heterogeneous. There was no systematic variation of values of Q _p and Q _s at different frequencies from one tectonic unit to another. As a consequence, average values of these parameters were obtained for each frequency for the entire region, and these were used for interpretation and for comparison with worldwide data. Q _p values lie within the range of values observed for some tectonically active regions of the world, whereas Q _s values were the lowest among the values compared for different parts of the world. Q _s/Q _p values were >1 for the entire range of frequencies studied. All these factors indicate that the crust is highly heterogeneous in the study region. The high Q _s/Q _p values also indicate that the region is partially saturated with fluids.     

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).     

Comparison of Sesimic Body Wave and Coda Wave Measures of Q

—Measurements of seismic attenuation (Q ^?1) can vary considerably when made from different parts of seismograms or using different techniques, particularly at high frequencies. These discrepancies may be methodological, or may reflect earth processes. To investigate this problem, we compare body wave with coda Q ^?1 results utilizing three common techniques i) parametric fit to spectral decay, ii) coda normalization of S waves, and iii) coda amplitude decay with lapse time. Q ^?1 is measured from both body and coda waves beneath two mountain ranges and one platform, from recordings made at seismic arrays in the Caucasus and Kopet Dagh over paths ≤ 4° long. If Q is assumed frequency independent, spectral decay fits show Q _s and Q _coda near 700–800 for both mountain paths and near 2100–2200 for platform paths. Similar values are determined with the coda normalization technique. However, frequency-dependent parameterizations fit the data significantly better, with Q _s ?(1 Hz) and Q _coda?(1 Hz) near 200–300 for mountain paths and near 500–600 for platform paths. Lapse decay measurements are close to the frequency-dependent values, showing that both spectral and lapse decay methods can give similar results when Q has comparable parameterizations. Above 6 Hz, coda measurements suggest some enrichment relative to body waves, perhaps due to scattering, but intrinsic absorption appears to dominate at lower frequencies. All approaches show sharp path differences between the Eurasian platform and adjacent mountains, and all are capable of resolving spatial variations in Q.     

Scattering of Seismic Waves by Cracks with the Boundary Integral Method

— We develop a new scheme to compute 2-D SH seismograms for media with many flat cracks, based on the boundary integral method. A dry or traction-free boundary condition is applied to crack surfaces although other kinds of cracks such as wet or fluid-saturated cracks can be treated simply by assigning different boundary conditions. While body forces are distributed for cavities or inclusions to express scattered wave, dislocations (or displacement discontinuities between the top and the bottom surfaces of each crack) are used as fictitious sources along crack surfaces. With these dislocations as unknown coefficients, the scattered wave is expressed by the normal derivative of Green's function along the crack surface, which is called “double-layer potentials” in the boundary integral method, while we used “single-layer potentials” for cavities or inclusions. These unknowns are determined so that boundary conditions or crack surfaces are satisfied in the least-squared sense, for example, traction-free for dry cracks. Seismograms with plane-wave incidence are synthesized for homogeneous media with many cracks. First, we check the accuracy of our scheme for a medium with one long crack. All the predicted phases such as reflected wave, diffraction from a crack tip and shadow behind the crack are simulated quite accurately, under the same criterion as in the case for cavities or inclusions. Next, we compute seismograms for 50 randomly distributed cracks and compare them with those for circular cavities. When cracks are randomly oriented, waveforms and the strength of scattering attenuation are similar to the cavity case in a frequency range higher than k d $\simeq$ 2 where the size of scatterers d (i.e., crack length or cavity diameter) is comparable with the wavelength considered (k is the wavenumber). On the other hand, the scattering attenuation for cracks becomes much smaller in a lower frequency range (k d<2) because only the volume but not detail geometry of scatterers becomes important with wavelength much longer than each scatterer. When all the cracks are oriented in a fixed direction, the scattering attenuation depends strongly on the incident angle to the crack surface as frequency increases (k d>2): scattering becomes weak for cracks oriented parallel to the direction of the incident wave, while it gets close to the cavity case for cracks aligned perpendicular to the incident wave.     

含混合裂隙、孔隙介质的纵波衰减规律研究

地下多孔介质中的孔隙类型复杂多样,既有硬孔又有扁平的软孔.针对复杂孔隙介质,假设多孔介质中同时含有球型硬孔和两种不同产状的裂隙(硬币型、尖灭型裂隙),当孔隙介质承载载荷时,考虑两种不同类型的裂隙对于孔隙流体压力的影响,建立起Biot理论框架下饱和流体情况含混合裂隙、孔隙介质的弹性波动方程,并进一步求取了饱和流体情况下仅由裂隙引起流体流动时的含混合裂隙、孔隙介质的体积模量和剪切模量,随后,在此基础上讨论了含混合裂隙、孔隙介质在封闭条件下地震波衰减和频散的高低频极限表达式.最后计算了给定模型的地震波衰减和频散,发现地震波衰减曲线呈现"多峰"现象,速度曲线为"多频段"频散.针对该模型分析讨论了渗透率参数、裂隙纵横比参数以及流体黏滞性参数对于地震波衰减和频散的影响,表明三个参数均为频率控制参数.     

Q P andQ S in Beijing and Jianchuan,Yunnan areas

Events from the December 1982 Huairou County, Beijing, and the July 1982 Jianchuan, Yunnan earthquake series were recorded at one station in Beijing City and at four stations in Eryuan area, Yunnan, respectively. Dividing the spectra (for P and S waves) from the smaller events in the series by spectra (for corresponding P and S waves) from the larger events in cach region, we have determined the high frequency source spectral decay rate to be ω^?1in both cases. Through trial and error method, we are able to determine the appropriate constantQ _P andQ _S that corrects the individual spectra to the proper high frequency decay rate. It is found that aQ _P of 800 and aQ _S of 550 can adequately compensate for the attenuation of the waves in Beijing area. For Jianchuan, Yunnan area the corresponding values are 900 and 400.