Scaling and Extreme Value Statistics of Sub-Gaussian Fields with Application to Neutron Porosity Data

Persistent Link:
http://hdl.handle.net/10150/344220
Title:
Scaling and Extreme Value Statistics of Sub-Gaussian Fields with Application to Neutron Porosity Data
Author:
Nan, Tongchao
Issue Date:
2014
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Embargo:
Release after 3-Dec-2016
Abstract:
My dissertation is based on a unified self-consistent scaling framework which is consistent with key behavior exhibited by many spatially/temporally varying earth, environmental and other variables. This behavior includes tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that often decay with lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags, with breakdown in power-law scaling at small and/or large lags; linear relationships between log structure functions of successive orders at all lags, also known as extended self-similarity; and nonlinear scaling of structure function power-law exponents with function order. The major question we attempt to answer is: given data measured on a given support scale at various points throughout a 1D/2D/3D sampling domain, which appear to be statistically distributed and to scale in a manner consistent with that scaling framework, what can be said about the spatial statistics and scaling of its extreme values, on arbitrary separation or domain scales? To do so, we limit our investigation in 1D domain for simplicity and generate synthetic signals as samples from 1D sub-Gaussian random fields subordinated to truncated monofractal fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Such sub-Gaussian fields are scale mixtures of stationary Gaussian fields with random variances that we model as being log-normal or Lévy α/2-stable. This novel interpretation of the data allows us to obtain maximum likelihood estimates of all parameters characterizing the underlying truncated sub-Gaussian fields. Based on synthetic data, we find these samples conform to the aforementioned scaling framework and confirm the effectiveness of generation schemes. We numerically investigate the manner in which variables, which scale according to the above scaling framework, behave at the tails of their distributions. Ours is the first study to explore the statistical scaling of extreme values, specifically peaks over thresholds or POTs, associated with such families of sub-Gaussian fields. Before closing this work, we apply and verify our analysis by investigating the scaling of statistics characterizing vertical increments in neutron porosity data, and POTs in absolute increments, from six deep boreholes in three different depositional environments.
Type:
text; Electronic Dissertation
Keywords:
Heavy-tail distribution; Hydrogeology; Neutron porosity; Peaks over threshold; Scaling; Hydrology; Extreme Value Analysis
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Hydrology
Degree Grantor:
University of Arizona
Advisor:
Neuman, Shlomo P.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleScaling and Extreme Value Statistics of Sub-Gaussian Fields with Application to Neutron Porosity Dataen_US
dc.creatorNan, Tongchaoen_US
dc.contributor.authorNan, Tongchaoen_US
dc.date.issued2014-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.releaseRelease after 3-Dec-2016en_US
dc.description.abstractMy dissertation is based on a unified self-consistent scaling framework which is consistent with key behavior exhibited by many spatially/temporally varying earth, environmental and other variables. This behavior includes tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that often decay with lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags, with breakdown in power-law scaling at small and/or large lags; linear relationships between log structure functions of successive orders at all lags, also known as extended self-similarity; and nonlinear scaling of structure function power-law exponents with function order. The major question we attempt to answer is: given data measured on a given support scale at various points throughout a 1D/2D/3D sampling domain, which appear to be statistically distributed and to scale in a manner consistent with that scaling framework, what can be said about the spatial statistics and scaling of its extreme values, on arbitrary separation or domain scales? To do so, we limit our investigation in 1D domain for simplicity and generate synthetic signals as samples from 1D sub-Gaussian random fields subordinated to truncated monofractal fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Such sub-Gaussian fields are scale mixtures of stationary Gaussian fields with random variances that we model as being log-normal or Lévy α/2-stable. This novel interpretation of the data allows us to obtain maximum likelihood estimates of all parameters characterizing the underlying truncated sub-Gaussian fields. Based on synthetic data, we find these samples conform to the aforementioned scaling framework and confirm the effectiveness of generation schemes. We numerically investigate the manner in which variables, which scale according to the above scaling framework, behave at the tails of their distributions. Ours is the first study to explore the statistical scaling of extreme values, specifically peaks over thresholds or POTs, associated with such families of sub-Gaussian fields. Before closing this work, we apply and verify our analysis by investigating the scaling of statistics characterizing vertical increments in neutron porosity data, and POTs in absolute increments, from six deep boreholes in three different depositional environments.en_US
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectHeavy-tail distributionen_US
dc.subjectHydrogeologyen_US
dc.subjectNeutron porosityen_US
dc.subjectPeaks over thresholden_US
dc.subjectScalingen_US
dc.subjectHydrologyen_US
dc.subjectExtreme Value Analysisen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineHydrologyen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorNeuman, Shlomo P.en_US
dc.contributor.committeememberWinter, Larryen_US
dc.contributor.committeememberFerre, Paul A.en_US
dc.contributor.committeememberRiva, Monicaen_US
dc.contributor.committeememberNeuman, Shlomo P.en_US
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