Time domain solution for multiple scattering and the coda envelopes

This article summarizes work on multiple scattering based on models of media with randomly distributed scatterers. The scatterers are isotropic and statistically uniform. Measuring distance in terms of mean-free pathL _s and time in terms of the mean-free timesL _s/V, whereV is the velocity of scattered waves, we have more convenient dimensionless distance and time. It can be shown that after the dimensionless time equals 0.65 energy contributed from multiple scattering becomes predominant. Thus the later coda reflects the effect of multiple scattering rather than single scattering. Treating the seismic record, including starting and tail parts, as a whole, the diffusion theory predicts that at a dense distribution of scatterers and a small distance between source and receiver, codas reflect mainly intrinsicQ _i. Of course, this conclusion is coincident with the presumption of the diffusion theory,Q _s>Q _i. However, from a new integral equation of multiple scattering, which deals with the scattered waves and primary waves separately, the conclusion is similar but clearer. This article quotes the new expression for coda energy in two-dimensional space. It shows that if the receiver is close to the source, the coda decay reflects only intrinsicQ _i, then as the distance increases, effects of scatteringQ _s, are involved in the decay feature. The theoretical plots of coda decay show that it seems in most cases in the earthQ _i should not be smaller than one tenth ofQ _s.Project Sponsored by the Joint Earthquake Science Foundation of China.     

The physical implication of coda amplitude ratio and its possible application

The physical implication of coda amplitude ratio is discussed in term of energy ratio. The digitized data recorded at the station of Beijing Telemetered Seismograph Network between 1989 and 1990 are used to calculate amplitude ratios of coda to direct S wave, and energy ratios. The spectral energy ratios are used to estimate the coda Q and mean free path l in the Beijing area, as well as the two quality factors Q _i and Q _S separately due to intrinsic absorption and scattering attenuation. The decay of seismic waves in their propagation seems mainly resulted from the intrinsic absorption in Beijing region. The temporal variations of amplitude ratio and energy ratio at Changli station during the above two years are inspected; some of them largely depart from their mean value. It may reflect the seismogenic process, but using the data lasting longer time with more case histories needs further study. This study is sponsored by the Key Project of State Science and Technology of China, No. 96-918.     

Unified approach to amplitude attenuation and coda excitation in the randomly inhomogeneous lithosphere

A unified model is proposed for explaining the frequency dependent amplitude attenuation and the coda wave excitation on the basis of the single scattering process in the randomly inhomogeneous lithosphere. Adopting Birch's law and a direct proportion between density and wave velocity, we statistically describe the inhomogeneous medium by one random function characterized by the von Karman autocorrelation function. We calculate the amplitude attenuation from the solid angle integral of scattered wave energy on the basis of the Born approxiimation after subtracting the travel-time fluctuation effect caused by slowly varying velocity inhomogeneities. This subtraction is equivalent to neglect energy loss by scattering within a cone around the forward direction. The random inhomogeneity of the von Karman autocorrelation function of order 0.35 with the mean square fractional fluctuation of 7.2×10^–3 1.3×10^–2 and the correlation distance of 2.15.1 km well explains observed backward scattering coefficientg and the ratioQ _P ^–1 /Q _S ^–1 , and observed and partially conjecturedQ _S ^–1 for frequencies between 0.5 Hz and 30 Hz.     

Regional characteristics of coda attenuation after the lancang-Gengma earthquake

In the light of the single scattering model of coda originating from local earthquakes, and based on the aftershock coda registered respectively at the 4 short period stations installed near the foci shortly after theM7.6 Lancang andM7.2 Gengma earthquakes, this paper has tentatively calculated the rate of amplitude attenuation and theQ _c-value of the coda in the Lancang and Gengma areas using a newly-founded synthetic determination method. Result of the study shows the rate of coda amplitude attenuation demonstrates remarkable regional differences respectively in the southern and northern areas. The southern area presents a faster attenuation (Q _c=114), whereas the northern area shows a slower attenuation (Q _c=231). The paper also discusses the reasons causing such differences. Result of the study also suggests a fairly good linear relation between the coda source factorA _o(f) and the seismic moment and the magnitude. Using the earthquake scaling law, the following formulas can be derived: lgM ₀=lgA ₀(f)+17.6,M _D=0.67lgA ₀(f)+1.21 and logM ₀=1.5M _D+15.79. In addition, the rates of amplitude attenuationβ _s andβ _m are respectively calculated using the single scattering and multiple scattering models, and the ratioβ _s/β_m=1.20−1.50 is found for the results respectively from the two models. Finally, the mean free pathL of the S-wave scattering in the southern and northern areas are determined to be 54 km and 122 km respectively by the relations which can distinguish between the inherentQ _i and scatteringQ _s, testify to this areas having lowQ-values correspond to stronger scatterings. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,14, 71–82, 1992. This study is partly supported by the Seismological Science Foundation of the State Seismological Bureau of China, and the present English version of the paper is translated from its Chinese original by Wenyi Xia, Seismological Bureau of Yunnan Province.     

Rock-mass characterization using intrinsic and scattering attenuation estimates at frequencies from 400 to 1600 Hz

Intrinsic and scatteringS-wave quality factors (Q ) were estimated using the Multiple Lapse Time Window Analysis (MLTWA) for microseismic events (M<–1) with source-sensor distances of 45 to 120 m, associated with an excavation at 630 m depth in Strathcona Mine, Sudbury, Canada. Additional information on the rock mass was provided by underground structural mapping data. IntrinsicQ values, at 800 Hz, were on the order of 140, similar to quality factor values obtained in previous studies using Spectral Decay and Coda-Q methods (120 to 170). The scattering quality factor at this frequency was about 520. An observed frequency dependence of the scattering attenuation suggested that a decrease in the density of scatterers, with scale lengths on the order of 2 m, exists at the site. Characteristic fracture scale lengths were considered to range from 4 to 6 m as identified in the mapping data. These observations were supported by the increase in scattering found for seismic waves with frequencies less than 1000 Hz. By assuming that the identified scatters are characteristic faults, these scatterers can then be considered to increase nonsimilar behavior in source scaling. Overall, our results suggest that MLTWA provides a practical method for remotely characterizing the quality of a rock mass when visual observations are not attainable.     

Frequency characteristics of coda and determination of medium parameters in Lingwu region, China

In this paper, considering the influences of source spectrum, the scattering property of medium and instrument response on the dominant frequency of coda, a method of using the coda of local earthquake to determine the correlation length of medium andQ-value is given. We find the following formula as: {fx719-1} wheret* =t/Q, f is the dominant frequency of coda,u ₁ andu ₂ are the parameters depend on the correlation length and the corner frequency of the source spectrum respectively,I(f) is a function of instrument response. If the source parameter is given, we can obtain the correlation length andQ-value by means of the inversion of observed curves off-t of coda. We processed the data of coda wave of more than 40 earthquakes from 1982 to 1989 in Lingwu region, China, determined the correlation length andQ-value, and preliminarily studied the temporal change of correlation length before and after moderately strong earthquakes. The result suggests that there are indications that the correlation length of medium decreases before the moderate earthquake. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,14, 62–70, 1992.     

CodaQ as a combination of scattering and intrinsic attenuation: Numerical simulations with the boundary integral method

Numerical modelling ofSH wave seismograms in media whose material properties are prescribed by a random distribution of many perfectly elastic cavities and by intrinsic absorption of seismic energy (anelasticity) demonstrates that the main characteristics of the coda waves, namely amplitude decay and duration, are well described by singly scattered waves in anelastic media rather than by multiply scattered waves in either elastic or anelastic media. We use the Boundary Integral scheme developed byBenites et al. (1992) to compute the complete wave field and measure the values of the direct waveQ and coda wavesQ in a wide range of frequencies, determining the spatial decay of the direct wave log-amplitude relation and the temporal decay of the coda envelope, respectively. The effects of both intrinsic absorption and pure scattering on the overall attenuation can be quantified separately by computing theQ values for corresponding models with (anelastic) and without (elastic) absorption. For the models considered in this study, the values of codaQ ^–1 in anelastic media are in good agreement with the sum of the corresponding scatteringQ ^–1 and intrinsicQ ^–1 values, as established by the single-scattering model ofAki andChouet (1975). Also, for the same random model with intrinsic absorption it appears that the singly scattered waves propagate without significant loss of energy as compared with the multiply scattered waves, which are strongly affected by absorption, suggesting its dominant role in the attenuation of coda waves.     

Scattering ofP andS waves by a spherically symmetric inclusion

Scattering of an arbitrary elastic wave incident upon a spherically symmetric inclusion is considered and solutions are developed in terms of the spherical vector system of Petrashen, which produces results in terms of displacements rather than displacement potentials and in a form suitable for accurate numerical computations. Analytical expressions for canonical scattering coefficients are obtained for both the cases of incidentP waves and incidentS waves. Calculations of energy flux in the scattered waves lead to elastic optical theorems for bothP andS waves, which relate the scattering cross sections to the amplitude of the scattered fields in the forward direction. The properties of the solutions for a homogeneous elastic sphere, a sphere filled by fluid, and a spherical cavity are illustrated with scattering cross sections that demonstrate important differences between these types of obstacles. A general result is that the frequency dependence of the scattering is defined by the wavelength of the scattered wave rather than the wavelength of the incident wave. This is consistent with the finding that the intensity of thePS scattering is generally much stronger than theSP scattering. When averaged over all scattering angles, the mean intensity of thePS converted waves is2V _p ² /V _s ⁴ times the mean intensity of theSP converted waves, and this ratio is independent of frequency. The exact solutions reduce to simple and easily used expressions in the case of the low frequency (Rayleigh) approximation and the low contrast (Rayleigh-Born) approximation. The case of energy absorbing inclusions can also be obtained by assigning complex values to the elastic parameters, which leads to the result that an increase in attenuation within the inclusion causes an increased scattering cross section with a marked preference for scatteredS waves. The complete generality of the results is demonstrated by showing waves scattered by the earth's core in the time domain, an example of high-frequency scattering that reveals a very complex relationship between geometrical arrivals and diffracted waves.     

Coda waves: A review

The analysis and interpretation of coda waves have received increasing attention since the early seventies. In the past few years interest in this subject has spread worldwide, and the study of high-frequency seismic coda waves has become a very important seismological topic. As a conclusion of the studies accomplished in this time, coda waves are considered the result of scattering processes caused by heterogeneities acting on seismic waves.P andS waves play a particularly important role in this interaction. The process introduces an attenuation which, added to the intrinsic absorption, gives the observed apparent attenuation. Therefore, coda waves constitute a thumbprint left by the heterogeneities on the seismograms. Coda waves offer decisive information about the mechanism of how scattering and attenuation take place. This review describes coda waves in detail, and summarizes the work done in this subject to 1986. The relation between coda waves and attenuation in the context of research on seismic scattering problems is stressed. Particular attention has been given to the application of coda waves to estimate source and medium parameters. The state-of-the-art of the temporal variations of coda wave shape, and the possible use of these variations as an earthquake precursor also are presented. Care has been taken to introduce the statistical models used to deal with the heterogeneities responsible for scattering.     

The Q value variations in the preparing process of rock rupture

In this paper, the "spectral amplitude ratio method" (SAR), "energy method" (EN) and "coda wave method" (CW) are used to calculate theQ value variations of gneiss in the preparing rupture process. The obtained results show that the variation state ofQ values by SAR features the shape of relative stability—gradual increment to the maximum—then decrement and final rupture. The variation state ofQ values by EN is just contrary to that by SAR, i. e. with the shape of stability—decrement—increment—and final rupture. The varation state ofQ values by CW is similar to that by EN, its main frequency features the shape of relatively high value—decrement to the minimum—increment—and final rupture. But to the high frequency (higher than the main frequency), the variation state ofQ values features the shape of the stable value-increment to the maximum-decrement-and final rupture. At the same time, the results by coda wave amplitude spectrum show that, when stress reaches 70% of rupture stress, the high frequency component of S wave rapidly reduces (Q _c increasing); at the time of impending the main rupture, the main frequency component reduces with a large scale (Q _c increasing again), this may be the reason which causes the different variation states of two codaQ values. The result of amplitude spectra of P, S (initial wave) waves also show that with the appearance of microcracks the frequency band of S wave turn to be narrow, the high frequency component is reduced quickly, i. e. the S wave spectra have different variation states with different frequency components. That is why theQ _s obtained by different methods have different variation characteristics.     

Coda wave analysis for distinguishing attenuation due to isotropic scattering from attenuation due to absorption

When the quality factorQ is taken into account in attenuation studies, it is necessary to know the relative losses of wave energy due to scattering and to anelastic absorption. The coda is the most important phenomenon now known which is related to elastic scattering of seismic waves. Utilizing coda, this study presents relationships which give theQ factors of the medium around the recording station and discriminate between attenuations arising from elastic scattering (under the assumption of isotropic scattering) and those arising from anelastic absorption. This work proposes a technique for separately determining the attenuation due to isotropic scattering and that due to absorption from the observed envelope of coda waves.     

Multiple scattering and energy transfer of seismic waves—Separation of scattering effect from intrinsic attenuation II. Application of the theory to Hindu Kush region

In order to separate the scattering effect from intrinsic attenuation, we need a multiple scattering model for seismic wave propagation in random heterogeneous media. In paper I (Wu, 1985), radiative transfer theory is applied to seismic wave propagation and the energy density distribution (or the average intensity) in space for a point source is formulated in the frequency domain. It is possible to separate the scattering effect and the absorption based on the measured energy density distribution curves. In this paper, the data from digital recordings in the Hindu Kush region are used as an example of application of the theory. We also discuss two approximate solutions of coda envelope in the time domain: the single scattering approximation and the diffusion approximation and discuss the relation with the frequency domain solution. We point out that in only two cases can the apparent attenuation be expressed as an exponential decay form. One is thedark medium case, i.e., whenB ₀0.5, whereB ₀ = _s /( _s + _a ) is the seismic albedo, _s is the scattering coefficient, _a is the absorption coefficient. In this case the absorption is dominant, the apparent attenuationb can be approximated by the coherent wave attenuationb = _s + _a . The other case is thediffuse scattering regime, i.e., whenB ₀0.5 (bright medium) andRL _s ,t _s , whereR andt are the propagation distance and lapse time,L _s and _s are the scattering lengths (mean free path) and scattering time (mean free time), respectively. However, in this case the envelope decays with a rate close to the intrinsic attenuation, while the intensity decreases with distance with a coefficientb d ₀( _s + _a ) d _{s s} , whered ₀ andd _s are the diffusion multipliers (0<d ₀,d _s <1).For the Hindu Kush region, by comparing the theory with data from two digital stations of 53 events distributed up to depths of 350 km, we find that the scattering is not the dominant factor for the measured apparent attenuation ofS waves in the frequency range 2–20 Hz. From the observation on high frequency (f>20 Hz) seismograms, we suggest the existence of a stron-scattering surface layer with fine scale heterogeneities in the crust, at least for this region.     

Propagation and attenuation characteristics of the crustal phaseLg

TheLg wave consists of the superposition ofS waves supercritically reflected, and thus trapped, in the crust. This mode of propagation explains the strong amplitude of this phase and the large distance range in which it is observed. The numerical simulation leads to successful comparison between observed seismograms in stable continental areas and synthetics computed for simple standard crustal models. In regions with strong lateral variations, the influence of large-scale heterogeneities on theLg amplitude is not yet clearly established in terms of the geometrical characteristics of the crustal structure.The analysis of the decay of amplitude ofLg with epicentral distance allows the evaluation of the quality factor ofS waves in the crust. The results obtained show the same trends as codaQ: a clear correlation with the tectonic activity of the region considered, both for the value ofQ at 1 Hz and for its frequency dependence, suggesting that scattering plays a prominent part among the processes that cause the attenuation.The coda ofLg is made up of scatteredS waves. The study of the spatial attenuation of the coda indicated that a large part of the arrivals that compose the coda propagate asLg. The relative amplitude of the coda is larger at sites located on sediments because, in these conditions, a part ofLg energy can be converted locally into lower order surface modes.     

Estimation of coda and shear wave attenuation in the volcanic area in SE Sabalan Mountain,NW Iran

The quality factors of coda and shear waves have been estimated for the SE Sabalan Mountain, geothermal region in northwestern Iran. We have analyzed 65 local earthquakes with magnitude of 2.8 to 6.1 and 2.8 to 5 for shear and coda wave quality factor estimation, respectively. These events were recorded on five stations installed by Building and Housing Research Center Network. Coda normalization and Spectral decay methods have been used to estimate the frequency dependence attenuation relation for shear wave, and single back-scattering method for coda waves. We have observed that the coda normalization method has supplied significantly higher Q _S values as compared to the spectral method. The results show that, in general, Q values are significantly smaller for the entire frequency range as compared to tectonically active areas and are close to the values for volcanic areas.     

Q c of three component seismograms of volcanic microearthquakes at Campi Flegrei volcanic area — Southern Italy

Digital recordings of three component microearthquake codas from shallow seismic events in the volcanic region of Campi Flegrei — Southern Italy — were used with an automatic technique to calculate the attenuation factorQ _c (codaQ) in the hypothesis of singleS toS backscattering.Results show the same value ofQ for each of the three components. This result is interpreted as due to isotropicS wave radiation pattern.A check of the coda method was performed using a single station method based on simple assumptions on the direct SH wave spectrum. Single stationQ was averaged over the stations and over the earthquakes. Results show that the two methods lead to comparable results.A frequency dependence quite different from that evaluated in active tectonic regions was found for coda attenuation, comparable to other volcanic areas throughout the world. This is interpreted as due to the presence of magma that affects anelasticity and scattering.     

Research on the regionalQ values of coda waves for the aftershocks in Luquan,Yunnan of 1985

Based on the scattering coda model by which local and regional earthquakes are interpreted (K. Aki, 1969), and using observational coda data of 68 aftershocks of the 1985 Luquan, Yunnan earthquake registered by the VGK seismographs installed at 12 stations in the Yunnan regional short-period network, theQ-values of coda waves are calculated respectively for 6 time intervals. It is observed that within the frequency range of 0.40–1.65 Hz of the observed data, theQ-values are closely related with the frequencies and the calculated codaQ ranges between 80–240 with the coefficient of frequency dependence η=0.45. The calculated source factorsB(f> _p) of the coda waves which indicate the scattering strength are mostly within the order 10^?23–10^?24. Areas with lowQ-values present high scattering. It should be noted that by comparing data obtained before and after the Luquan earthquake, clear changes can be detected in theQ-values measured at stations close to the epicentral region, and that theQ-values of the aftershock coda are less than about one half of the pre-shock values. It may be mentioned that the time-dependent regional variations of theQ-values might possibly bring about practical significance in earthquake prediction. Moreover, aftershock focal parameters are determined. Through discussions on the quantitative relations between the focal parameters, we get: 1gE=1.59M _L+ 11.335;E=(2.10 × 10^?5)M ₀; length of focal rupturea=0.40?0.80 km for 3.0≤M _L<5.0 events; stress drop Δσ=(6.0–130) ×10⁵ Pa. Through interpretation of the data, we have also learned the important characteristics that there is no linear relation between the stress drops and the earthquake magnitudes.     

基于CWI方法的门源MS6.9地震余震定位研究

门源发生M_S6.9地震后,余震目录中含有大量的单台事件,为了解决单台事件的定位不准确性,利用尾波干涉(CWI)的震源定位方法将门源台记录到震后8小时内的101次单台事件进行重定位,该方法主要通过计算事件对尾波部分的间距不确定性矩阵,从而估计聚类中具有相似震源机制的地震事件对之间的间距,通过优化这些事件对的间距来解决聚类中一组事件的相对位置。最后,对定位结果进行评估,得到相对可靠的定位结果。     

单次与多次散射对地方震尾波的作用

在Aki(1969)的单次反向散射理论和高龙生(1983)的多次散射理论基础上,提出了震源距不为零的单次与多次散射模型。在此模型下,研究了二维无限介质空间中,由统计上均匀分布的各向同性散射体引起的单次与多次散射对地方震尾波功率谱的作用。推导出了二维情况下单次散射和多次散射尾波功率谱的解析表达式,并在二维情况下对不同震源距的单次散射与多次散射对尾波功率谱的作用进行了比较。      

Scattering and anelastic attenuation of seismic energy in northern Greece

The relative contribution of scattering (Q _s ^–1 ) and intrinsic (Q _i ^–1 ) attenuation to the totalS-wave attenuation for the frequencies of 1.5, 3.0, 6.0 and 12.0 Hz has been studied by applying the radiative energy transfer theory, Data of local earthquakes which occurred in northern Greece and were recorded by the permanent telementered network of the Geophysical Laboratory of the University of Thessaloniki have been used. The results show that in this area the scattering attenuation is dominant over all frequencies while intrinsic attenuation is significantly lower. The estimatedQ _s ^–1 andQ _i ^–1 values have frequency dependences off ^–0.72 andf ^–0.45, respectively. The frequency dependence ofQ _s ^–1 is the same as that of the codaQ _c ^–1 , obtained by applying the single scattering model, which probably implies that the frequency dependence of the coda wave attenuation is attributed to the frequency dependence of the scattering attenuation.Q _c ^–1 values are very close to scattering attenuation for short lapse times, (10–20 sec), and intermediate between scattering and intrinsic attenuation for the longer lapse times, (50–100 sec). This difference is explained as the result of the depth-dependent attenuation properties and the multiple scattering effects.