The Distribution of Amplitude with Time in Fluctuation Noise

Landon, V.D.

doi:10.1109/jrproc.1941.233980

Cited by 49 publications

(15 citation statements)

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“…2 r For these distributions, the most likely value of envelope amplitude is r, the mean is 1.252r, and the standard deviation of envelope amplitude is .655r (Landon, 1942). This means that the standard deviation of envelope amplitude is .655r ‫ס‬ .655M /1.242 ‫ס‬ .527M where M is the amplitude of the mean envelope.…”

Section: Ensemble Average Envelopesmentioning

confidence: 99%

“…The distribution of trace and envelope amplitudes for waveforms composed of a superposition of oscillations having the same frequency but random phase was studied by Rayleigh (1880) and later by Landon (1941Landon ( , 1942. Landon (1942) shows that the distribution of instantaneous amplitudes of a waveform composed of a sum of waves of the same frequency but random phase is given by 1 2 2 ‫מ‬g /2r…”

Section: Ensemble Average Envelopesmentioning

confidence: 99%

“…Landon (1942) shows that the distribution of instantaneous amplitudes of a waveform composed of a sum of waves of the same frequency but random phase is given by 1 2 2 ‫מ‬g /2r…”

Section: Ensemble Average Envelopesmentioning

confidence: 99%

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Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Fehler¹

2000

Bulletin of the Seismological Society of America

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Observations of seismic waves from earthquakes at depths between 100 and 200 km beneath Japan show that the initial portion of the S-wave arrival has greater duration than can be accounted for by earthquake source-time duration. The observed long duration of S waves has been explained as being caused by multiple forward scattering around the ray path between source and receiver. Array observations of Lg waveforms have also shown that multiple forward scattering along the path between the source and receiver are important influences on the character of Lg waveforms and that the scattering cannot be explained only by vertical variations in velocity. Multiple forward scattering in 3D has been modeled using the Markov approximation for the parabolic-wave equation, which allows the calculation of seismogram envelopes in statistically characterized random media. To test the range of validity of the Markov approximation-derived solutions, we made 2D numerical calculations of wavefields in random media using approximations to the parabolic wave equation, which only models forward scattering, and by finite-difference solution of the scalar-wave equation, which gives complete wavefields. Media of background velocity 4 km/sec are characterized using a Gaussian autocorrelation function with a 5 km correlation distance and 5% rms fractional fluctuation. To compare with envelopes obtained from the Markov approximation, we calculated wavefields for source-receiver distances ranging from 50 to 200 km for several statistically identical realizations of random media. We obtain ensemble average envelopes by averaging envelopes from the realizations. We find a good agreement between ensembleaverage envelopes obtained from the numerical solution of the parabolic-wave equation and envelopes obtained from the Markov approximation. The later portion of the ensemble-average envelopes calculated using finite difference have larger amplitudes than those from the Markov approximation, which is probably due to the late-arriving energy that has been scattered at wide angles from the global propagation direction between the source and the receiver. We observe that the variations among the numerically calculated envelopes of individual realizations of random media are well fit by a Rayleigh distribution, which describes the distribution of envelope amplitude when signals of one frequency but random phase are summed. Our results show that the Markov approximation provides reliable information about envelope shapes for forward-scattered wavefields but that the influence of wide-angle scattering and backscattering have some influences on envelope shapes and should be considered when analyzing data using random media models.

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Section: Ensemble Average Envelopesmentioning

confidence: 99%

Section: Ensemble Average Envelopesmentioning

confidence: 99%

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Envelope Broadening of Outgoing Waves in 2D Random Media: A Comparison between the Markov Approximation and Numerical Simulations

Fehler¹

2000

Bulletin of the Seismological Society of America

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“…As with Landon in [36], the form of his proof has a Bayesian interpretation allowing a simple conversion from Likelihoodist to Bayesian thinking.…”

Section: White's Proofmentioning

confidence: 98%

“…What is important is the distinction between methods available when there is one error in the decision process versus two errors as in normal equity trading. Additionally, the proof by Landon in [36] allows a direct linkage between the subjective decisions of individual actors and the calculation of the distribution that exists in equilibrium Because of the relationship between subjective personal realities and objective market relationships, there will be a slight notation change. For the personal calculated rewards of a subjective actor, the notation will be i t r at time t while the equilibrium return will be t r .…”

Section: The Distribution Of Mergers For Cashmentioning

confidence: 99%