Statistics for Fission-Track Thermochronology 6 Pieter Vermeesch Abstract 6.1 Introduction This chapter introduces statistical tools to extract geolog- fi ically meaningful information from ssion-track 238U is the heaviest naturally occurring nuclide in the solar (FT) data using both the external detector and system. Like all nuclides heavier than 208Pb, it is physically fi 238 LA-ICP-MS methods. The spontaneous ssion of U unstable and undergoes radioactive decay to smaller, more is a Poisson process resulting in large single-grain age stable nuclides. 99.9998% of the 238U nuclei shed weight by uncertainties. To overcome this imprecision, it is nearly disintegrating into eight He-nuclei (a-particles) and a 206Pb always necessary to analyse multiple grains per sample. atom, forming the basis of the U–Pb and (U–Th)/He clocks. The degree to which the analytical uncertainties can The remaining 0.0002% of the 238U undergoes spontaneous explain the observed scatter of the single-grain data can fission, forming the basis of FT geochronology (Price and be visually assessed on a radial plot and objectively Walker 1963; Fleischer et al. 1965). Because spontaneous fi fi quanti ed by a chi-square test. For suf ciently low values fission of 238U is such a rare event, the surface density of fi of the chi-square statistic (or suf ciently highp values), fission tracks (in counts per unit area) is 10–11 orders of the pooled age of all the grains gives a suitable magnitude lower than the atomic abundances of 238U and description of the underlying‘true’ age population. 4He, respectively. So whereas the U–Pb and (U–Th)/He Samples may fail the chi-square test for several reasons. methods are based on mass spectrometric analyses of bil- fi A rst possibility is that the true age population does not lions of Pb and He atoms, FT ages are commonly based on consist of a single discrete age component, but is manual counts of at most a few dozen features. Due to these characterised by a continuous range of ages. In this case, low numbers, the FT method is a low precision technique. a‘random effects’ model can constrain the true age Whereas the analytical uncertainty of U–Pb and (U–Th)/He distribution using two parameters: the‘central age’ and ages is expressed in % or‰-units, it is not uncommon for the‘(over)dispersion’. A second reason why FT data sets single-grain FT age uncertainties to exceed 10% or even might fail the chi-square test is if they are underlain by 100% (Sect. 6.2). Early attempts to quantify these uncer- multimodal age distributions. Such distributions may tainties (McGee and Johnson 1979; Johnson et al. 1979) consist of discrete age components, continuous age were criticised by Green (1981a, b), who subsequently distributions, or a combination of the two. Formalised engaged in a fruitful collaboration with two statisticians— statistical tests such as chi-square can be useful in Geoff Laslett and Rex Galbraith—to eventually solve the fi preventing over tting of relatively small data sets. problem. Thanks to the combined efforts of the latter two However, they should be used with caution when applied people, it is fair to say that the statistics of the FT method are to large data sets (including length measurements) which better developed than those of any other geochronological fi generate suf cient statistical‘power’ to reject any simple technique. Several statistical tools that were originally yet geologically plausible hypothesis. developed for the FT method have subsequently found applications in other dating methods. Examples of this are the radial plot (Sect. 6.3), which is routinely used in lumi- nescence dating (Galbraith 2010b), random effects models P. Vermeesch (&) London Geochronology Centre, University College London, London, UK e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2019 109 M. G. Malusà and P. G. Fitzgerald (eds.), Fission-Track Thermochronology and its Application to Geology, Springer Textbooks in Earth Sciences, Geography and Environment, https://doi.org/10.1007/978-3-319-89421-8_6 110 P. Vermeesch 238 (Sect. 6.4.2), which have been generalised to (U–Th)/He wherek D is the total decay constant of U −10 −1 (Vermeesch 2010) and U–Pb dating (Rioux et al. 2012), and (1.55125 10 year ; Jaffey et al. 1971),k f is thefission − − finite mixture models (Sect. 6.5), which were adapted for decay constant (7.9–8.7 10 17 year 1; Holden and Hoff- – 1 detrital U Pb geochronology (Sambridge and Compston 1994). man 2000) ,q s is the density (tracks per unit area) of the The statistical analysis offission tracks is a rich and spontaneousfission tracks on an internal crystal surface, fi diverse eld of research, and this short chapter cannot pos- 238U is the current number of 238U atoms per unit volume, ½ sibly cover all its intricacies. Numerate readers are referred andR is etchable range of thefission tracks, which is half the to the book by Galbraith (2005), which provides a com- 238 equivalent isotropic FT length. U can be determined by prehensive, detailed and self-contained review of the subject, ½ irradiating the (etched) sample with thermal neutrons in a from which the present chapter heavily borrows. The chapter reactor. This irradiation induces syntheticfission of 235U in comprisesfive sections, which address statistical issues of the mineral, producing tracks that can be monitored by progressively higher order. Section 6.2 introduces the FT attaching a mica detector to the polished mineral surface and age equation using the external detector method (EDM), etching this monitor subsequent to irradiation. Using which offers the most straightforward and elegant way to this external detector method (EDM), Eq. 6.1 can be estimate single-grain age uncertainties, even in grains fi rewritten as: without spontaneous ssion tracks. Section 6.3 compares and contrasts different ways to visually represent multi-grain 1 1 qs assemblages of FT data, including kernel density estimates, t ln1 kDfqd 6:2 ¼ kD þ 2 qi ð Þ cumulative age distributions and radial plots. Section 6.4 reviews the various ways to estimate the‘average’ age of wheref is a calibration factor (Hurford and Green 1983),q i such multi-grain assemblages, including the arithmetic mean is the surface density of the inducedfission tracks in the age, the pooled age and the central age. This section will also mica detector andq d is the surface density of the induced introduce a chi-square test for age homogeneity, which is fission tracks in a dosimeter glass of known (and constant) used to assess the extent to which the scatter of the U-concentration. The latter value is needed to‘recycle’ the single-grain ages exceeds the formal analytical uncertainties calibration constant from one irradiation batch to the next, as obtained from Sect. 6.2. This leads to the concept of neutronfluences might vary through time, or within a sample ‘overdispersion’ (Sect. 6.4.2) and more complex distribu- stack.q s,q i andq d are unknown but can be estimated by tions consisting of one or several continuous and/or discrete counting the number of tracksN * over a given areaA * age components. Section 6.5 discusses three classes of (where‘*’ is either‘s’ for‘spontaneous’,‘i’ for‘induced’ or mixed effects models to resolve discrete mixtures, continu- ‘d’ for‘dosimeter’): ous mixtures and minimum ages, respectively. It will show Ns Ni Nd that these models obey the classic bias–variance trade-off, q^ ; q^ and q^ 6:3 s A i A d A which will lead to a cautionary note regarding the use of ¼ s ¼ i ¼ d ð Þ formalised statistical hypothesis tests for FT interpretation. It is customary for the spontaneous and inducedfission Finally, Sect. 6.6 will give the briefest of introductions to tracks to be counted over the same area (i.e.A A ), either s ¼ i some statistical aspects of thermal history modelling, a more using an automated microscope stage (Smith and comprehensive discussion of which is provided in Chap. 3, Leigh-Jones 1985; Dumitru 1993) or by simply reposition- (Ketcham 2018). In recent years, severalfission-track labo- ing the mica detector on the grain mount after etching ratories around the world have abandoned the elegance and (Jonckheere et al. 2003). Using these measurements, the robustness of the EDM for the convenience of estimated FT age (^t) is given by ICP-MS-based measurements. Unfortunately, the statistics of the latter is less straightforward and less well developed 1 1 ^ Ns ^t ln1 kDfq^d 6:4 than that of the EDM. Section 6.7 presents an attempt to ¼ kD þ 2 Ni ð Þ address this problem. where ^f is obtained by applying Eq. 6.4 to an age standard and rearranging. Equations 6.2 and 6.4 assume that the ratio 6.2 The Age Equation of the etchable range (R) between the grain and the mica detector is the same for the sample and the standard. The fundamental FT age equation is given by: ! 1The uncertainty associated with thefission decay constant vanishes 1 kD qs t ln1 238 6:1 when kf is folded into thef-calibration constant. This is one of the ¼ kD þ kf U R ð Þ main reasons why thef-method was developed (see Chap. 1, Hurford ½ 2018). 6 Statistics for Fission-Track Thermochronology 111 Violation of this assumption leads to apparent FT ages of Note that this equation breaks down ifN 0. There are s ¼ unclear geological significance. This is an important caveat two solutions to this problem. The easiest of these is to as samples with shortened tracks are very common.