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El. knyga: Statistics for Fission Track Analysis

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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 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.

Recenzijos

... 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

List of tables ix
List of figures xi
Preface xiii
1 Introduction 1(12)
1.1 What are fission tracks?
1(1)
1.2 How are they observed?
1(2)
1.3 Why are they useful?
3(1)
1.4 Applications of fission track analysis
4(1)
1.5 Mathematical representation of fission tracks
5(1)
1.6 Fission track dating and provenance studies
5(1)
1.7 Thermal histories and track length distributions
5(2)
1.8 Sampling by plane section
7(1)
1.9 Initial formation of tracks
8(1)
1.10 Shortening of tracks by heat
8(3)
1.11 Properties of apatite
11(1)
1.12 Bibliographic notes
12(1)
2 The Poisson line segment model 13(14)
2.1 Joint distribution of length and orientation
14(3)
2.2 The number of tracks with a given attribute
17(1)
2.3 The expected number of tracks intersecting a plane
18(2)
2.4 Track density and equivalent isotropic length
20(1)
2.5 Tracks intersecting a prismatic face
21(1)
2.6 Effect of non-prismatic face on track density
22(1)
2.7 Track counts from a dosimeter glass
23(2)
2.8 Spatial and temporal variation
25(1)
2.9 Remarks
26(1)
2.10 Bibliographic notes
26(1)
3 Track counts and densities: fission track dating 27(30)
3.1 The mathematical basis of fission track dating
28(3)
3.2 The external detector method
31(3)
3.3 Observed and theoretical track densities
34(2)
3.4 A short digression
36(1)
3.5 Estimates of fission track age
37(3)
3.6 Inspection of single grain data
40(1)
3.7 Radial plot of single grain ages
41(5)
3.8 Chi-square age homogeneity test
46(1)
3.9 A measure of age dispersion
47(3)
3.10 A protocol for data analysis
50(1)
3.11 Dealing with small counts
50(2)
3.12 Practical considerations
52(2)
3.13 Remarks
54(1)
3.14 Historical note
55(2)
4 The population method 57(20)
4.1 Experimental method and data
57(2)
4.2 Theoretical and observed track densities
59(3)
4.3 An estimate of the uranium dispersion
62(1)
4.4 A uranium homogeneity test
62(1)
4.5 Estimates of fission track age
63(2)
4.6 Summarising and inspecting the data
65(4)
4.7 Age homogeneity test
69(1)
4.8 A measure of dispersion of true fission track ages
70(1)
4.9 Counts over unequal areas
71(1)
4.10 A protocol for data analysis
72(1)
4.11 Discussion
72(1)
4.12 The population-subtraction method
73(2)
4.13 Remarks
75(1)
4.14 Bibliographic notes
76(1)
5 Discrete mixtures of ages 77(18)
5.1 Maximum likelihood estimation of a common age
78(2)
5.2 Discrete mixture models
80(2)
5.3 Example: a synthetic mixture of two ages
82(2)
5.4 Example: apatite data from the Bengal Fan
84(2)
5.5 Maximum likelihood estimation formulae
86(4)
5.6 How many ages to fit?
90(1)
5.7 Example: zircon ages from Mount Tom
91(2)
5.8 Data from more than one irradiation
93(1)
5.9 Bibliographic notes
94(1)
6 Continuous mixtures of ages 95(20)
6.1 Example: Otway data from Australia
95(2)
6.2 General approach
97(1)
6.3 A random effects model with binomial errors
97(2)
6.4 Maximum likelihood estimation formulae
99(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


Galbraith, Rex F.
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