Statistics For Fission Track Analysis

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ISBN: 9781584885337 | 1584885335
Cover: Hardcover
Copyright: 5/24/2005

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Statistical analyses of the numbers, lengths, and orientations of fission tracks etched in minerals yield dating and thermal history information valuable in geological applications, particularly in oil exploration. This book explores the statistical foundations and tools for the analysis and interpretation of fission track data--the first book to do so. The author sets out the theory of fission track dating and through numerical examples, presents methods for analyzing and interpreting fission track counts. Designed for broad accessibility, the book includes clear explanations of mathematical derivations and an appendix containing general notes on the relevant statistical concepts.

List of tables

List of figures

Preface

xiii

1 Introduction

(12)

1.1 What are fission tracks?

(1)

1.2 How are they observed?

(2)

1.3 Why are they useful?

(1)

1.4 Applications of fission track analysis

(1)

1.5 Mathematical representation of fission tracks

(1)

1.6 Fission track dating and provenance studies

(1)

1.7 Thermal histories and track length distributions

(2)

1.8 Sampling by plane section

(1)

1.9 Initial formation of tracks

(1)

1.10 Shortening of tracks by heat

(3)

1.11 Properties of apatite

(1)

1.12 Bibliographic notes

(1)

2 The Poisson line segment model

(14)

2.1 Joint distribution of length and orientation

(3)

2.2 The number of tracks with a given attribute

(1)

2.3 The expected number of tracks intersecting a plane

(2)

2.4 Track density and equivalent isotropic length

(1)

2.5 Tracks intersecting a prismatic face

(1)

2.6 Effect of non-prismatic face on track density

(1)

2.7 Track counts from a dosimeter glass

(2)

2.8 Spatial and temporal variation

(1)

2.9 Remarks

(1)

2.10 Bibliographic notes

(1)

3 Track counts and densities: fission track dating

(30)

3.1 The mathematical basis of fission track dating

(3)

3.2 The external detector method

(3)

3.3 Observed and theoretical track densities

(2)

3.4 A short digression

(1)

3.5 Estimates of fission track age

(3)

3.6 Inspection of single grain data

(1)

3.7 Radial plot of single grain ages

(5)

3.8 Chi-square age homogeneity test

(1)

3.9 A measure of age dispersion

(3)

3.10 A protocol for data analysis

(1)

3.11 Dealing with small counts

(2)

3.12 Practical considerations

(2)

3.13 Remarks

(1)

3.14 Historical note

(2)

4 The population method

(20)

4.1 Experimental method and data

(2)

4.2 Theoretical and observed track densities

(3)

4.3 An estimate of the uranium dispersion

(1)

4.4 A uranium homogeneity test

(1)

4.5 Estimates of fission track age

(2)

4.6 Summarising and inspecting the data

(4)

4.7 Age homogeneity test

(1)

4.8 A measure of dispersion of true fission track ages

(1)

4.9 Counts over unequal areas

(1)

4.10 A protocol for data analysis

(1)

4.11 Discussion

(1)

4.12 The population-subtraction method

(2)

4.13 Remarks

(1)

4.14 Bibliographic notes

(1)

5 Discrete mixtures of ages

(18)

5.1 Maximum likelihood estimation of a common age

(2)

5.2 Discrete mixture models

(2)

5.3 Example: a synthetic mixture of two ages

(2)

5.4 Example: apatite data from the Bengal Fan

(2)

5.5 Maximum likelihood estimation formulae

(4)

5.6 How many ages to fit?

(1)

5.7 Example: zircon ages from Mount Tom

(2)

5.8 Data from more than one irradiation

(1)

5.9 Bibliographic notes

(1)

6 Continuous mixtures of ages

(20)

6.1 Example: Otway data from Australia

(2)

6.2 General approach

(1)

6.3 A random effects model with binomial errors

(2)

6.4 Maximum likelihood estimation formulae

(1)

6.5 A random effects model with Normal errors

100

(1)

6.6 Examples

101

(2)

6.7 Finite mixtures of random effects models

103

(1)

6.8 A minimum age model

104

(1)

6.9 Data and statistical model with binomial errors

105

(1)

6.10 Maximum likelihood estimation formulae

106

(1)

6.11 A minimum age model with Normal errors

107

(1)

6.12 Example: apatite data from China

108

(4)

6.13 A synthetic mixture re-visited

112

(1)

6.14 Grain age distributions

113

(1)

6.15 Remarks

114

(1)

6.16 Bibliographic notes

114

(1)

7 Probability distributions of lengths and angles

115

(28)

7.1 All tracks having the same length

115

(4)

7.2 Each track having one of two lengths

119

(1)

7.3 Several different lengths

120

(1)

7.4 A general isotropic length distribution

121

(1)

7.5 A general anisotropic length distribution

122

(2)

7.6 Distributions on a prismatic face

124

(5)

7.7 Horizontal confined track lengths

129

(2)

7.8 Some explicit formulae

131

(3)

7.9 A two-component mixture of anisotropic lengths

134

(1)

7.10 Quantitative effects of anisotropy

135

(5)

7.11 Parametric models for length against angle

140

(2)

7.12 Bibliographic notes

142

(1)

8 Observational features of track measurements

143

(20)

8.1 Horizontal confined tracks

143

(5)

8.2 Length bias

148

(1)

8.3 The Loaded Dog experiments

149

(2)

8.4 Empirical verification of length bias

151

(2)

8.5 Fracture-thickness bias

153

(1)

8.6 Orientation bias

154

(3)

8.7 Surface proximity bias

157

(1)

8.8 Estimates of μ from horizontal confined tracks

158

(1)

8.9 Projected semi-track lengths and angles

158

(4)

8.10 Semi-track lengths and angles

162

(1)

8.11 Bibliographic notes

162

(1)

9 Further developments

163

(18)

9.1 Thermal history parameters

163

(2)

9.2 Combined likelihood for track measurements

165

(4)

9.3 Annealing experiments

169

(1)

9.4 Annealing data

170

(1)

9.5 Annealing models

171

(2)

9.6 Fitting annealing models

173

(3)

9.7 Calculating the length distribution from a thermal history

176

(2)

9.8 Inferring times and temperatures from lengths

178

(1)

9.9 Multi-compositional annealing models

179

(1)

9.10 Bibliographic notes

180

(1)

Appendix Notes on statistical methods

181

(20)

A.1 Poisson processes in one, two and three dimensions

181

(3)

A.2 Notes on the Poisson distribution

184

(1)

A.3 Relation between the binomial and Poisson distributions

185

(1)

A.4 Standard errors and confidence intervals

186

(2)

A.5 Components of error

188

(1)

A.6 Precision and accuracy

189

(1)

A.7 Statistical significance tests and p-values

189

(2)

A.8 Radial plots

191

(3)

A.9 Histograms and "probability density" plots

194

(1)

A.10 Parametric models and likelihood inference

195

(6)

References

201

(8)

Index

209

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Statistics For Fission Track Analysis

Summary

Table of Contents

Supplemental Materials