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Statistics for Fission Track Analysis

By Rex F. Galbraith

Series Editor: Byron J.T. Morgan, Niels Keiding, Peter Van der Heijden

Chapman and Hall/CRC – 2005 – 240 pages

Series: Chapman & Hall/CRC Interdisciplinary Statistics

Purchasing Options:

  • Add to CartHardback: $125.95
    978-1-58488-533-7
    May 23rd 2005

Description

Statistical analyses of the numbers, lengths, and orientations of fission tracks etched in minerals yield dating and thermal history information valuable in geological and geoscience applications, particularly in oil exploration. Fission tracks can be represented mathematically by a stochastic process of randomly oriented line segments in three dimensions, and this "line segment" model can describe and explain the essential statistical features of the data, providing a rigorous foundation for quantitative modelling and simulation studies.

Statistics for Fission Track Analysis explores the line segment model and its consequences for the analysis and interpretation of data. The author derives the equations for fission track data and the theoretical probability distributions for the number, orientation, and length measurements of the tracks. He sets out the theory of fission track dating and through numerical examples, presents methods for analyzing and interpreting fission track counts. Later chapters address statistical models for situations in which samples contain mixtures of fission track ages. These methods, along with observation features of the various measurements, are illustrated by real examples. Finally, the author brings together the theoretical and observation aspects to formulate a joint likelihood function of counts, lengths, and angles as a basis for parametric thermal history modelling. An appendix provides general notes on statistical concepts and methods.

Designed for broad accessibility, this is the first book to fully cover the statistical foundations of fission track analysis. Whether you work in a fission track lab, in archaeological, geological, or geochronological research, or in geological applications of statistics, you will find the background material and practical tools you need to optimize the use of fission track analysis in your work and to make further advances in the field.

Reviews

… The book collates the huge amount of high quality statistical work that the author has contributed to the analysis of fission track data, often in collaboration with G. M. Laslett, over a 20-year period. It seems to me to provide an object lesson in sound statistical practice, grounding the work in appropriate simple physical models, extending models empirically and fitting them by maximum likelihood, always being fully immersed in, and addressing, the context to hand, and providing a deep understanding of the limitations of the laboratory and statistical techniques used. … an essential handbook for all scientists who are involved with fission track analysis. … this is an excellent book and I congratulate the author on his contributions to the topic.

—M.C. Jones, Journal of the Royal Statistical Society, 2006, Volume 169, Issue 3

The book is a good survey for scientists who have worked in the field. …This would be an excellent introductory text for those wishing to enter this field, especially if they were also aware of the new statistical tools available.

—E. Enns, Short Book Reviews of the ISI

I strongly recommend this book to its intended audience and would use it in a course on statistical consulting.

—Thomas Burr, Los Alamos National Library, Technometrics, November 2008, Vol. 50, No. 4

Contents

INTRODUCTION

What are Fission Tracks?

How are they Observed?

Why are they Useful?

Applications of Fission Track analysis

Mathematical Representation of Fission Tracks

Fission Track Dating and Provenance Studies

Thermal Histories and Track Length Distributions

Sampling by Plane Section

Intitial Formation of Tracks

Shortening of Tracks by Heat

Properties of Apatite

Bibliographic Notes

THE POISSON LINE SEGMENT MODEL

Joint Distribution of Length and Orientation

The Number of Tracks with a Given Attribute

The Expected Number of Tracks Intersecting a Plane

Track Density and Equivalent Isotropic Length

Tracks Intersecting a Prismatic Face

Effect of Non-Prismatic Face on Track Density

Track counts from a Dosimeter Glass

Spatial and Temporal Variation

Remarks

Bibliographic Notes

TRACK COUNTS AND DENSITIES: FISSION TRACK DATING

The Mathematical Basis of Fission Track Dating

The External Detector Method

Observed and Theoretical Track Densities

A Short Digression

Estimates of Fission Track Age

Inspection of Single Grain Data

Radial Plot of Single Grain Ages

Chi-Square Age Homogeneity Test

A Measure of Age Dispersion

A Protocol for Data Analysis

Dealing with Small counts

Practical Considerations

Remarks

Historical Note

Bibliographic Notes

THE POPULATION METHOD

Experimental Method and Data

Theoretical and Observed Track Densities

An Estimate of the Uranium Dispersion

A Uranium Homogeneity Test

Estimates of Fission Track Age

Summarising and Inspecting the Data

Age Homogeneity Test

A Measure of Dispersion of True Fission Track Ages

Counts over Unequal Areas

A Protocol for data Analysis

Discussion

The Population-Subtraction Method

Remarks

Bibliographic Notes

DISCRETE MIXTURES OF AGES

Maximum Likelihood Estimation of a Common Age

Discrete Mixture Models

Example" A Synthetic Mixture of Two Ages

Example: Apatite Data from the Bengal Fan

Maximum Likelihood Estimation Formulae

How Many Ages to Fit?

Example: Zircon Ages from Mount Tom

Data from more than One Irradiation

Bibliographic Notes

CONTINUOUS MIXTURES OF AGES

Example: Otway Data from Victoria, Australia

General Approach

A Random Effects Model with Binomial Errors

Maximum Likelihood Estimation Formulae

A Random Effects Model with Normal Errors

Examples

Finite Mixtures of Random Effects Models

A Minimum Age Model

Data and Statistical Model with Binomial Errors

Maximum Likelihood Estimation Formulae

A Minimum Age Model with Normal Errors

Example: Apatite Data from China

A Synthetic Mixture Revisited

Grain Age Distributions

Remarks

Bibliographic Notes

PROBABILITY DISTRIBUTIONS OF LENGTHS AND ANGLES

All Tracks Having the Same Length

Each Track Having One of Two Lengths

Several Different Lengths

A General Isotropic Length Distribution

A General Anisotropic Length Distribution

Distributions on a Prismatic Face

Horizontal Confined Track Lengths

Some explicit Formulae

A Two-Component Mixture of Anisotropic Lengths

Quantitative Effects of Anisotropy

Parametric Models for Length Against Angle

Bibliographic Notes

OBSERVATIONAL FEATURES OF TRACK MEASUREMENTS

Horizontal Confined Tracks

Length Bias

The Loaded Dog Experiments

Empirical Verification of Length Bias

Fracture-Thickness Bias

Orientation Bias

Surface Proximity Bias

Estimate of m from Horizontal Confined Tracks

Projected Semi-Track Lengths and Angles

Semi-Track Lengths and Angles

Bibliographic Notes

FURTHER DEVELOPMENTS

Thermal History Parameters

Combined Likelihood for Track Measurements

Annealing Experiments

Annealing Data

Annealing Models

Fitting Annealing Models

Calculating the Length Distribution

Inferring Times and Temperatures from Lengths

Multi-Compositional Annealing Models

Bibliographic Notes

APPENDIX

Poisson Processes in One, Two, and Three Dimensions

Notes on the Poisson Distribution

Relation Between Binomial and Poisson Distribution

Standard Errors and Confidence Intervals

Components of Error

Statistical Significance Tests and p-Values

Radial Plots

Histograms and "Probability Density" Plots

Parametric Models and Likelihood Inference

Name: Statistics for Fission Track Analysis (Hardback)Chapman and Hall/CRC 
Description: By Rex F. GalbraithSeries Editor: Byron J.T. Morgan, Niels Keiding, Peter Van der Heijden. Statistical analyses of the numbers, lengths, and orientations of fission tracks etched in minerals yield dating and thermal history information valuable in geological and geoscience applications, particularly in oil exploration. Fission tracks can be...
Categories: Mathematics & Statistics for Engineers, Statistical Theory & Methods, Environmental Geology