One-dimensional wave propagation in a highly discontinuous medium

Burridge, Robert; Papanicolaou, George; White, Bebo

doi:10.1016/0165-2125(88)90004-2

Cited by 100 publications

(58 citation statements)

References 11 publications

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“…The effective equation for the wave front has in this case been obtained by several authors [5,6,10,15,23,27]. The pulse propagation is characterized by a random time shift and a deterministic spreading, that are of the same order.…”

Section: Randommentioning

confidence: 99%

Pulse Propagation in Random Media with Long-Range Correlation

Garnier¹,

Sølna²

2009

Multiscale Model. Simul.

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Abstract. This paper analyses wave propagation in a one-dimensional random medium with long-range correlations. The asymptotic regime where the fluctuations of the medium parameters are small and the propagation distance is large is studied. In this regime pulse propagation is characterized by a random time shift described in terms of a fractional Brownian motion and a deterministic spreading described by a pseudo-differential operator. This operator is characterized by a frequency-dependent attenuation that obeys a power law with an exponent ranging from 1 to 2 that is related to the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. This frequency-dependent attenuation is associated with a frequency-dependent phase, which ensures causality of the filter that realizes the approximation. A discussion is provided showing that the mean-field theory cannot capture the correct attenuation rate, this is because it also averages the random time delay. Numerical results are given to illustrate the accuracy of the asymptotic theory.Key words. wave propagation, random media, long-range processes.1. Introduction. Wave propagation in multiscale and rough media, with longrange fluctuations, has recently attracted a lot of attention, as more and more data collected in real environments confirm that this situation can be encountered in many different contexts, such as in geophysics [11] or in laser beam propagation through the atmosphere [13,16,25]. Recently it has been shown that the main effect of such fluctuations of the medium parameters is a random time shift for the wave front, that obeys a Gaussian statistics described in terms of a fractional Brownian motion [22]. Here we observe the wave front along its random characteristics and we show that the wave front also experiences a deterministic shape modification, that can be described in terms of a pseudo-differential operator that depends on the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. These results extend to general long-range media the ones derived in the context of a discrete Goupillaud medium in [26].The effective pseudo-differential operator obtained in this paper gives rise to a frequency-dependent attenuation that obeys a power law with an exponent ranging from 1 to 2. This exponent will be shown to be related to the exponent of the power decay rate of the autocorrelation function of the fluctuations of the medium parameters. Frequency-dependent attenuation has been observed in a wide range of applications in acoustics [4,29], and also in other domains, such as seismic wave propagation [7,8]. Experimental observations show that the attenuation of plane acoustic waves has a frequency dependence of the form E = E 0 exp(−γ(ω)z), where E denote the amplitude of an acoustic variable such as velocity or pressure and ω is the frequency. The damping coefficient has been seen to obey the empirical power law γ(ω) = γ 0 |ω| γ1 where γ 0 ∈ (0, ∞) and γ 1 ∈ (0, 2) are parameters ...

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Section: Randommentioning

confidence: 99%

Pulse Propagation in Random Media with Long-Range Correlation

Garnier¹,

Sølna²

2009

Multiscale Model. Simul.

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“…Resnick et al [34] present an interesting alternative derivation of the formula, and were the first to approach the problem from an invariant imbedding point of view. However, the first rigorous account for the stabilization phenomenon was given by Burridge et al in [11]. Here they derive the version of the formula which applies to an equal travel time discretized medium using an averaging technique.…”

Section: C(s) Dsmentioning

confidence: 99%

Ray theory for a locally layered random medium

Sølna

Papanicolaou

2000

Waves in Random Media

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We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modeled by a smooth three dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered.We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convoluted with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness, it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.

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“…These apparent phase changes with depth appear to be frequency dependent; i.e., we observe dispersion. Burridge et al (1988) show that thinbed scattering can lead to apparent dispersion with an effective wavelet that broadened about the pulse centroid, according to a diffusion equation, while propagating through many thin layers. Based on Figure 12a, their apparent phases appear to change little below approximately 1300 m depth.…”

Section: Phase Estimates Of Synthetic First Arrivals: Preparationsmentioning

confidence: 99%

“…However, across many thin layers, one can observe only the product of all the (frequency-independent) transmission losses, which has the cumulative effect of removing the high frequencies from the onset of the forward-scattered waveform and moving them into the coda. Theoretical approaches were developed later to describe this phenomenon in relation to the statistics of a discontinuous medium (e.g., Banik et al, 1985;Resnick et al, 1986;Burridge et al, 1988;Shapiro and Treitel, 1997). Burridge (1990) and de Hoop et al (1991) study the phenomenon of signal broadening with a pulse that traveled through a highly discontinuous medium comprising many homogeneous, horizontal layers.…”

Section: Introductionmentioning

confidence: 99%

Effect of flood basalt stratigraphy on the phase of seismic waveforms recorded offshore Faroe Islands

2015

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The generation of short-period multiples between highly heterogeneous layers of basalt flows can strongly alter transmitted seismic wavefields. These layers filter and modify penetrating waves, producing apparent attenuation and phase changes in the observed waveforms. We investigated the waveform and apparent phase changes of the primary seismic signal using mainly the maximum kurtosis approach. We compared the seismic recordings from two short-offset vertical seismic profiles (VSPs) with synthetic seismograms, generated from sonic logs in the same wells, and we found that short-period multiples cause a rapid broadening of the primary arrivals and strong apparent phase changes within a short depth interval below the top of the basalt flows. Relatively large uncertainties were associated with estimating constant phase shifts of the seismic arrivals within the topmost 250 m of the basalt sequences, where complex scattering occurred. Within this interval of the Brugdan I well, a phase-only compensation of the first arrivals with a frequency-independent, combined scattering, and intrinsic attenuation operator was unfeasible. At a greater depth, we found that the phase shifts, predicted by a VSP-derived effective Q value, were similar to those estimated from the VSP signals using the kurtosis method. Thus, phase-only compensation with a combined scattering and intrinsic attenuation operator could work well depending on the seismic signal bandwidth and the distribution, depth, and magnitude of the impedance contrasts in the basalt sequence.

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One-dimensional wave propagation in a highly discontinuous medium

Cited by 100 publications

References 11 publications

Pulse Propagation in Random Media with Long-Range Correlation

Pulse Propagation in Random Media with Long-Range Correlation

Ray theory for a locally layered random medium

Effect of flood basalt stratigraphy on the phase of seismic waveforms recorded offshore Faroe Islands

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