### 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 corresponding increments tends to exhibit heavy tails. The tails decay as powers of -α where 1 lt; α lt; 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 α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy noise, a random field (or process) subordinated to truncated fractional Gaussian noise, the case of variable α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy motion (tfLm), a random field subordinated to truncated fractional Brownian motion. Whereas the first type of signal is relatively regular and characterized by Lévy index α, the second is highly irregular (punctuated by spurious spikes) and characterized by the asymptote of α 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 α slightly smaller than 2, which is easily confused with a Gaussian field (characterized by constant α = 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.

Original language | English (US) |
---|---|

Pages (from-to) | 195-207 |

Number of pages | 13 |

Journal | Stochastic Environmental Research and Risk Assessment |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

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### Keywords

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

### ASJC Scopus subject areas

- Environmental Engineering
- Environmental Science(all)
- Environmental Chemistry
- Water Science and Technology
- Safety, Risk, Reliability and Quality

### Cite this

*Stochastic Environmental Research and Risk Assessment*,

*27*(1), 195-207. https://doi.org/10.1007/s00477-012-0576-y

**Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff.** / Riva, Monica; Neuman, Shlomo P; Guadagnini, Alberto.

Research output: Contribution to journal › Article

*Stochastic Environmental Research and Risk Assessment*, vol. 27, no. 1, pp. 195-207. https://doi.org/10.1007/s00477-012-0576-y

}

TY - JOUR

T1 - Sub-Gaussian model of processes with heavy-tailed distributions applied to air permeabilities of fractured tuff

AU - Riva, Monica

AU - Neuman, Shlomo P

AU - Guadagnini, Alberto

PY - 2013

Y1 - 2013

N2 - 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 corresponding increments tends to exhibit heavy tails. The tails decay as powers of -α where 1 lt; α lt; 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 α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy noise, a random field (or process) subordinated to truncated fractional Gaussian noise, the case of variable α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy motion (tfLm), a random field subordinated to truncated fractional Brownian motion. Whereas the first type of signal is relatively regular and characterized by Lévy index α, the second is highly irregular (punctuated by spurious spikes) and characterized by the asymptote of α 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 α slightly smaller than 2, which is easily confused with a Gaussian field (characterized by constant α = 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.

AB - 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 corresponding increments tends to exhibit heavy tails. The tails decay as powers of -α where 1 lt; α lt; 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 α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy noise, a random field (or process) subordinated to truncated fractional Gaussian noise, the case of variable α is consistent with a collection of samples from truncated sub-Gaussian fractional Lévy motion (tfLm), a random field subordinated to truncated fractional Brownian motion. Whereas the first type of signal is relatively regular and characterized by Lévy index α, the second is highly irregular (punctuated by spurious spikes) and characterized by the asymptote of α 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 α slightly smaller than 2, which is easily confused with a Gaussian field (characterized by constant α = 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.

KW - Air permeabilities

KW - Fractured tuff

KW - Heavy-tailed distributions

KW - Nonlinear scaling

KW - Parameter estimation

KW - Power law

UR - http://www.scopus.com/inward/record.url?scp=84871417952&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871417952&partnerID=8YFLogxK

U2 - 10.1007/s00477-012-0576-y

DO - 10.1007/s00477-012-0576-y

M3 - Article

AN - SCOPUS:84871417952

VL - 27

SP - 195

EP - 207

JO - Stochastic Environmental Research and Risk Assessment

JF - Stochastic Environmental Research and Risk Assessment

SN - 1436-3240

IS - 1

ER -