e-journal

Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff

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Abstract

Earth and environmental variables are commonly taken to have multivariate Gaussian or heavy-tailed distributions in space and/or time. This is based on the observation that univariate frequency distributions of corresponding samples appear to be Gaussian or heavy-tailed. Of particular interest to us is the well-documented but heretofore little noticed and unexplained phenomenon that whereas the frequency distribution of log permeability data often seems to be Gaussian, that of orresponding increments tends to exhibit heavy tails. The tails decay as powers of -a where 1\a\2 is either constant or grows
monotonically toward an asymptote with increasing separation
distance or lag. We illustrate the latter phenomenon
on 1-m scale log air permeabilities from pneumatic tests in
6 vertical and inclined boreholes completed in unsaturated
fractured tuff near Superior, Arizona. We then show theoretically
and demonstrate numerically, on synthetically generated signals, that whereas the case of constant a is consistent with a collection of samples from truncated sub-Gaussian fractional Le´vy noise, a random field (or process) subordinated to truncated fractional Gaussian noise, the case of variable a is consistent with a collection of samples from truncated sub-Gaussian fractional Le´vy motion (tfLm), a random field subordinated to truncated fractional Brownian motion. Whereas the first type of signal is relatively regular and characterized by Le´vy index a, the second is highly irregular (punctuated by spurious spikes) and characterized by the asymptote of a values associated with its increments. We describe a procedure to estimate the parameters of univariate distributions characterizing such signals and apply it to our log air permeability data. The latter are found to be consistent with a collection of samples from tfLm with a slightly smaller than 2, which is easily confused with a Gaussian field (characterized by
constant a = 2). The irregular (spiky) nature of this signal
is typical of observed fractured rock properties. We propose
that distributions of earth and environmental variable
be inferred jointly from measured values and their increments
in a way that insures consistency between these two sets of data.

Keywords : Air permeabilities, Fractured tuff, Heavy-tailed distributions, Parameter estimation, Nonlinear scaling, Power law

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Informasi Detail

Judul Seri

Stoch Environ Res Risk Assess (2013) 27:195–207

No. Panggil

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Penerbit

: Springer-Verlag., 2013

Deskripsi Fisik

Stoch Environ Res Risk Assess (2013) 27:195–207

Bahasa

English

ISBN/ISSN

1436-3259

Klasifikasi

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Tipe Isi

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Tipe Media

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Tipe Pembawa

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Edisi

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Subjek

ILMU LINGKUNGAN

Info Detail Spesifik

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Pernyataan Tanggungjawab

Ety

Versi lain/terkait

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