Pulse propagation in media with frequency‐dependent Q

Brennan, B.J.

doi:10.1029/gl007i003p00211

Cited by 10 publications

(5 citation statements)

References 18 publications

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“…In developing the method it is assumed that Q is constant, or nearly so, with frequency. If Q exhibits a powei-law dependency on frequency as described by Brennan (1980) the results are still valid.…”

Section: Discussionmentioning

confidence: 85%

“…Kjartansson (1979) shows that his constant Q model satisfies equation (1) giving C = 0.485, for Q greater than 20. Brennan (1980) considers a frequency-dependent Q model in which the frequency dependence is described by a parameter a. For (Y = 1 (constant Q) he evaluates C to be 0.485.…”

Section: Rise Time $Nd Attenuationmentioning

confidence: 99%

“…5 Frequency-dependent Q The method developed above for estimating t* from the rise or fall time of a pulse has assumed that the specific quality factor Q is independent of frequency for teleseismic body waves. Brennan (1980) asserts that there is increasing evidence (e.g. Anderson 9r Minster 1979;Berckhemer, Auer & Drisler 1979) that Q in the Earth has a power-law dependency on frequency and he presents results for the propagation of a seismic pulse through such a Q model.…”

Section: Fall Time and Attenuationmentioning

confidence: 99%

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Q and the rise and fall of a seismic pulse

Stewart

1984

Geophys J Int

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Gladwin & Stacey and others have shown that if Q (the specific quality factor) is independent of frequency, or nearly so, then the rise time of a seismic pulse is proportional to the anelastic attenuation it has experienced during propagation. In this paper the use of rise times to estimate the attenuation of seismic body wavesis examined numerically and the method is extended to include the fall time of a pulse. Although the proportionality of both rise and fall time with attenuation is shown here to be valid only for an impulse source function it is possible to estimate the maximum attenuation experienced by a seismic pulse from the source to the receiver with no a prion assumptions about the shape of the radiated pulse.The method, developed for a constant Q model, can still be applied if Q has an acceptable power-law dependency on frequency. Application to teleseismic body waves from earthquakes and an underground nuclear explosion indicates that transmission paths exist over which the attenuation is much lower than is often assumed in seismogram modelling.

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Section: Discussionmentioning

confidence: 85%

Section: Rise Time $Nd Attenuationmentioning

confidence: 99%

Section: Fall Time and Attenuationmentioning

confidence: 99%

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Q and the rise and fall of a seismic pulse

Stewart

1984

Geophys J Int

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“…Brennan [1980] discussed pulse propagation in media for which Q had a simple power law dependence upon frequency given by Q = Qslfl • and suggested that 0.1 < 0• < 0.3 is consistent with experimental data. Brennan [1980] discussed pulse propagation in media for which Q had a simple power law dependence upon frequency given by Q = Qslfl • and suggested that 0.1 < 0• < 0.3 is consistent with experimental data.…”

Section: Non-constant-q Modelsmentioning

confidence: 60%

Seismic source influence in pulse attenuation studies

Blair¹,

Spathis²

1984

J. Geophys. Res.

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Analysis of pulse rise time data produced from repetitive piezoelectric and magnetostrictive seismic sources shows that the change in rise time as a function of distance is source dependent. Thus a linear rise time law of the form τ = τ0 + CT/Q, with C source independent, cannot be used to assess the attenuation of rock in situ. In fact, application of such a law to the data results in a significant overestimate in the value of Q. A numerical example is also presented to show that in terms of filter theory a linear rise time law is ill founded. An alternative, source independent method of assessing the attenuation is employed for both constant‐Q and non‐constant‐Q models, and it is found that either model may be made to give a reasonable fit to the data. However, it is shown that if a non‐constant‐Q mechanism is operating, then there is probably less than a 5% change in Q over the frequency range 0.80–70.OkHz.

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“…The difference between % and o is of the order of 0.1%, and a correction is required if the group velocity is the quantity that has been determined, which, for instance, is the case with body wave travel times. Brennan [1980] has demonstrated that the degradation of an initially sharp pulse as it propagates im a medium which satisfies (88) is described by a simple scaling process. The shape of a plane pulse which is initially a delta function at the origin (so that •(0, w) in (73) [1979] suggest that a value for a of 0.7 may be appropriate.…”

Section: Q(to) = O(tor)lto/tor[ '-" (90)mentioning

confidence: 99%

Linear viscoelasticity and dispersion in seismic wave propagation

Brennan¹,

Smylie²

1981

Reviews of Geophysics

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The theoretical background to the application of linear viscoelasticity in describing the anelastic behavior of rocks is developed in this review. The constraints imposed on the behavior of the impulse response and system function of a linear system by the requirements that the system be causal and passive are examined. These include the existence of dispersion relations for the system function for causal systems and the role of the concepts of positive reality and minimum phase in passive systems. The uses of these linear system concepts in linear viscoelasticity and in the propagation of plane waves in an isotropic viscoelastic medium are considered. Finally, applications to seismology are presented, in particular the significance of the dispersion associated with a quality factor Q that is constant or varies as a power of frequency in comparisons of seismic data for different frequency ranges and in the evolution of the shape of propagating pulses. INTRODUCTIONIn studies relating to the anelasticity of rocks one of the most important considerations has been whether the anelastic behavior is or is not linear, in the sense that the principle of superposition of signals is valid. In seismological applications it has been common practice, if only for the obvious convenience, to assume that the anelasticity is linear, and a large body of theoretical seismology is founded, apparently successfully, on this assumption. Until recently, the experimental evidence was not convincing, and the validity of this approach has been questioned in the past [e.g., Knopoff,, 1964;Stacey et al., 1975]. Laboratory experiments by McKavanagh and Stacey [1974], which directly observed the stress-strain hysteresis loops for rocks subjected to a sinusoidal axial load, yielded loops that were cusped at the ends and were thus indicative of nonlinear artelasticity. On the other hand, the creep experiments of Pandit and Savage [ 1973] were consistent with the linear assumption. Less direct methods such as the analysis by Wuenschel [1965] of pulse degradation in Pierre shale and the observations of ultrasonic pulse rise times by Gladwin and Stacey [1974] also yielded results that were consistent with that assumption. Recent results indicate that at the strain levels involved in seismic waves the anelasticity of rocks is likely to be linear. Brennan and Stacey [1977] conducted experiments, similar to those of McKavanagh and Stacey [1974], in which granite and basalt rocks at room temperature and atmospheric pressure were subjected to sinusoidal shear. At the strain amplitudes involved (less than 3 x 10 -6) it was found that the hysteresis loops were elliptical, as is expected for linear anelasticity, and in addition, the variation of the real part of the compliance was of the form predicted by linear theory. Experiments conducted since then have demonstrated that similar results are observed in a sandstone specimen at a strain amplitude of 10 -6 . The implication is that the previous observations of hysteresis in rocks by McKavanagh and Stacey ...

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Pulse propagation in media with frequency‐dependent Q

Cited by 10 publications

References 18 publications

Q and the rise and fall of a seismic pulse

Q and the rise and fall of a seismic pulse

Seismic source influence in pulse attenuation studies

Linear viscoelasticity and dispersion in seismic wave propagation

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