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. See replaceN andN withN 1=2 andN 1=2, respectively s i s þ i þ Sect. 6.6 and Chap. 3 (Ketcham 2018) for further details on (Galbraith 2005, p. 80). A second (and preferred) approach how to deal with this situation. The standard errors ^t of the is to calculate exact (and asymmetric) confidence intervals. ½  single-grain age estimate is given by standardfirst-order See Galbraith (2005, p. 50) for further details about this Taylor expansion: procedure.         ^ 2 ^ 2 ^ 2 ^ 2 2 @t ^ 2 @t 2 @t 2 @t 2 s ^t s f s Ns s Ni s Nd ½   @^f ½  þ @Ns ½  þ @Ni ½  þ @Nd ½  6.3 Fission-Track Plots 6:5 ð Þ The single-grain uncertainties given by Eq. 6.8 tend to be where it is important to point out that all covariance terms very large. For example, a grain containing just four spon- are zero because ^f,N ,N andN are independent variables.2 fi s i d taneous ssion tracks (i.e.N ffiffiffi s = 4) is associated with an To simplify the calculation of the partial derivatives, we note analytical uncertainty of p4=4 50% even ignoring the that ln 1 x x ifx 1 so that, for reasonably lowN =N ¼ ð þ Þ  s i analytical uncertainty associated with thef-calibration con- values, Eq. 6.4 reduces to stant, the dosimeter glass, or the induced FT count. The single-grain age precision of the FT method, then, is orders 1 ^ Ns ^t fq^d 6:6 of magnitude lower than that of other established  2 Ni ð Þ geochronometers such as 40Ar/39Ar or 206Pb/238U, which Using this linear approximation, it is easy to show that achieve percent or permil level uncertainties. To overcome Eq. 6.5 becomes: this limitation and‘beat down the noise’, it is important that ! multiple grains are analysed from a sample and averaged  2 2       s ^t s ^f s N 2 s N 2 s N 2 using methods described in Sect. 6.4. Multi-grain assem- ½  ½  ½ s ½ i ½ d ^t  ^f þ Ns þ Ni þ Nd blages of FT data are also very useful for sedimentary provenance analysis and form the basis of a newfield of 6:7 ð Þ research called‘detrital thermochronology’ (Bernet 2018; The standard error of the calibration constant ^f is obtained Carter 2018). Irrespective of the application, it is useful for fi by repeated measurements of the age standard and will not be any multi-grain FT data set to rst be assessed visually. This section will introduce three graphical devices to do this: discussed further. The standard errors ofN s,N i andN d are governed by the Poisson distribution, whose mean equals its cumulative age distributions, (kernel) density estimates and variance. This crucial property can be illustrated with a radial plots. To illustrate these graphical devices as well as physical example in which a mica print attached to a dosimeter the different summary statistics of Sect. 6.4, consider the glass is subdivided into a number of equally sized squares four different geological scenarios shown in Fig. 6.2: (Fig. 6.1, left). Counting the number of inducedfission tracks I. A rapidly cooled volcanic rock extruded at 15 Ma. Nd in each square yields a skewed frequency distribution whose mean indeed equals its variance (Fig. 6.1, right). II. A slowly cooled intrusive rock exhibiting a range of 2 Cl/F ratios resulting in a 150 Ma± 20% range of Applying this fact to Eq. 6.7, we can replaces N withN , ½ s s apparent FT ages. s N 2 withN ands N 2 withN to obtain the following ½ i i ½ d d III. A detrital sample collected from a river draining two expression for the standard error of the estimated FT age: volcanic layers extruded at 15 and 75 Ma, respectively. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! IV. A detrital sample collected from a river draining u 2 t s ^f 1 1 1 lithologies I and II. s ^t ^t ½  6:8 ½  ^f þ Ns þ Ni þ Nd ð Þ

6.3.1 The Cumulative Age Distribution (CAD)

2 Ns,N i are independent within a single grain, but of course not The cumulative distribution function cdf(x) describes the between different grains of the same sample, as the spontaneous and fraction of the detrital age population whose age is less than induced track counts both depend on the U-concentration, which or equal tox: tends to vary significantly from grain to grain (McGee and Johnson 1979; Johnson et al. 1979; Green 1981b; Galbraith 1981; Carter cdf x P t x 6:9 1990). ð Þ¼ ð  Þ ð Þ 112 P. Vermeesch

Fig. 6.1 Left: inducedfission tracks recorded in a mica detector number of tracks is only due to Poisson statistics. Fission tracks were attached to a dosimeter glass. Numbers indicate the number of tracks counted with FastTracks image recognition software (see Chap. 4, counted in 48 150 150lm-sized areas. Colours indicate single Gleadow 2018). Right: the frequency distribution of the FT counts, (yellow), double (blue) and triple (red) etch pits. Dosimeter glasses which has a mean of 3.7 and a variance of 3.5 counts per graticule, exhibit a uniform U-concentration so that the observed variation in the consistent with a Poisson distribution

Under Scenario I, the cdf consists of a simple step age distribution (i.e. the cdfs) from the measurement distri- function, indicating that 0% of the grains are younger, and bution (CADs). 100% are older than the extrusive age (Fig. 6.2, I-a). Under Scenario II, the cdf is spread out over a wider range, so that 90% of the ages are between 90 and 210 Ma (Fig. 6.2, II-a). 6.3.2 (Kernel) Density Estimates (KDEs) Under Scenario III (Fig. 6.2, III-a), the cdf consists of two discrete steps at 15 and 75 Ma, the relative heights of which A probability density function (pdf) is defined as thefirst depend on the hypsometry of the river catchment and the derivative of the cdf: spatial distribution of erosion (Vermeesch 2007). Finally, under Scenario IV, the cdf consists of a discrete step from 0  Zx  d cdf y  to 50% at 15 Ma, followed by a sigmoidal rise to 100% at pdf x ½ ð Þ cdf x pdf y dy 6:11 75 Ma (Fig. 6.2, IV-a). ð Þ¼ dy x, ð Þ¼ ð Þ ð Þ In reality, the cdfs of Scenarios I–IV are, of course, 1 unknown and must be estimated from sample data, by means Under Scenario I, the pdf is a discrete peak of zero width of an empirical cumulative distribution function (ecdf), which and infinite height, marking the timing of the volcanic may be referred to as a cumulative age distribution (CAD) in a eruption (Fig. 6.2, I-b). In contrast, under Scenario II, the geochronological context (Vermeesch 2007). A CAD is pdf is a smooth (a)symmetric bell curve reflecting the spread simply a step function in which the single-grain age estimates in closing temperatures and, hence, ages, associated with the (^tj, forj=1 n) are plotted against their rank order: range of Cl/F-ratios present in apatites of this slowly cooled ! pluton (Fig. 6.2, II-b). Under Scenario III, the pdf consists of Xn two discrete spikes corresponding to the two volcanic events CAD x 1 ^tj x =n 6:10 ð Þ¼ j 1 ð  Þ ð Þ (Fig. 6.2, III-b). Finally, under Scenario IV, the pdf effec- ¼ tively combines those of Scenarios I and II (Fig. 6.2, IV-b). where 1(TRUE) = 1 and 1(FALSE) = 0. In contrast with the The pdfs, like the cdfs discussed before, are unknown but true cdfs, the measured CADs are invariably smoother, as the can be estimated from sample data. analytical uncertainties spread the dates out over a greater There are several ways to do this. Arguably the simplest of range. Because the uncertainties of FT ages are so large, the these is the histogram, in which the observations are grouped difference between the measured CADs and the true cdfs is into a number of discrete bins. Kernel density estimates very significant. Sections 6.4 and 6.5 of this chapter present (KDEs) are a continuous alternative to the histogram, which several algorithms to extract the key parameters of the true are constructed by arranging the measurements from young 6 Statistics for Fission-Track Thermochronology 113 ) 6.5 and 6.4 hoeia ds(e)and (red) pdfs b — theoretical locations; distribution sampling age cumulative four synthetic the and c — radial red) data; at (cdf, synthetic function samples same distribution the cumulative the for theoretical of blue) (KDEs, blue) (Sects. estimates fi ts (CAD, density model kernel with sample data synthetic synthetic the of plots rapidly I — a locations: sampling different four featuring (hill) landscape Hypothetical lwycoe ltncrc containing rock plutonic cooled slowly II — a Ma; 15 at extruded rock volcanic cooled i.6.2 Fig. oenrvrdann ihlg n nodr(5Ma) (75 older an these and of I each lithology For draining II. river and I modern lithologies III — a a — draining apatites; devices: river graphical 20% different second ± three Ma IV — a on 150 and plotted and deposit; generated volcanic were data synthetic scenarios, 114 P. Vermeesch sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to old along the time axis, adding a Gaussian‘bell curve’ (or Nsj 3=8 any other symmetric shape) on top of them and then summing zj arcsin þ 6:12 ¼ N j N j 3=4 ð Þ those to create one continuous curve (Silverman 1986; Ver- s þ i þ meesch 2012). The standard deviation of the Gaussian‘ker- and nel’ is called the‘bandwidth’ of the estimator and may be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 chosen through a host of different approaches, a proper dis- rj 6:13 ¼ 2 Nsj N ij 1=2 ð Þ cussion of which falls outside the scope of this review þ þ (Abramson 1982; Silverman 1986; Botev et al. 2010; Ver- Precise measurements plot towards the right-hand side of meesch 2012). An important feature of all these algorithms is the radial plot while imprecise measurements plot closer to the that the bandwidth monotonically decreases with increasing origin. A single-grain age may be read off by extrapolating a sample size. Please note that the so-called probability density line from the origin (0,0) of the radial plot through the sample plot (PDP, not to be confused with pdf!), in which the ana- point (xj,y j) to a radial scale plotted at some convenient dis- lytical uncertainty (or 0.6 times the analytical uncertainty, tance. Similarly, the analytical uncertainty can be obtained by Brandon 1996) is used as a‘bandwidth’ does not possess this extrapolating lines from the origin to the radial scale through feature. Therefore, PDPs are not proper density estimates, the top and the bottom of an imaginary 2r-error bar added to and consequently, their use is not recommended (Galbraith each sample point. Finally, drawing two parallel lines at 2r 1998; Vermeesch 2012). Like the CAD, which is a smooth distances from either side of the origin allow the analyst to version of the cdf, KDEs (and histograms) are smooth ver- visually assess whether all the single-grain ages within a sions of the pdf. But whereas the CAD has only been sample agree within the analytical uncertainties. smoothed once, histograms and KDEs are smoothed twice, Revisiting Scenario I of Fig. 6.2, the data points plot within once by the analytical uncertainties, and once by the width of a 2r band on the radial plot, consistent with a single discrete the bins or kernels. Because the analytical uncertainties of FT age component (Fig. 6.2, I-c). Under Scenario II, the data are data are so big, the components of FT age distributions are more dispersed and scatter beyond the 2r band, reflecting the often spread out very widely, resulting in poorly resolved dispersion of the underlying geological ages (Fig. 6.2, II-c). KDEs (blue curves in Figs. 6.2b). Under Scenario III, the data are randomly scattered along two linear trajectories which represent the two volcanic events (Fig. 6.2, III-c). Finally, Scenario IV combines the radial 6.3.3 Radial Plots patterns of Scenarios I and II, as expected (Fig. 6.2, IV-c). Of all the summary plots in Fig. 6.2, the radial plot contains the Single-grainfission-track age uncertainties are not only largest amount of quantitative information about the age large, but generally also variable (‘heteroscedastic’). Due to measurements and about the underlying geological ages. a combination of Poisson sampling statistics and variable Using the graphical design principles of Tufte (1983), the U-concentrations, the analytical uncertainties propagated radial plot exhibits a far higher‘ink-to-information ratio’ than using Eq. 6.8 may vary over an order of magnitude within the CAD, KDE or histogram. We will therefore use it as a basis the same sample. Neither CADs nor KDEs (let alone PDPs) from which to introduce the summary statistics discussed in are able to capture this uncertainty. The radial plot is a the next section of this chapter. graphical device that was specifically designed to address this issue (Galbraith 1988, 1990; Dunkl 2002; Vermeesch 6.4 Summary Statistics 2009). Givenj=1…n numerical values z j and their analytical uncertaintiesr j, the radial plot is a bivariate The previous sections have shown that the presence of large xj;y j scatterplot setting out a standardised estimate ð Þ and highly variable analytical uncertainties can easily yj zj z 0 =rj , wherez is some reference value) against ð ¼ ½   Þ 0 obscure the underlying age distribution and all the geologi- the single-grain precision (x 1= ). For FT data using the j r j cally meaningful information encoded by it. The next two 3 ¼ fi EDM, it is convenient to use the following de nitions for zj sections will introduce some useful summary statistics which andr j (Galbraith 1990): can be used to disentangle that geologically meaningful information from the random noise produced by the Poisson counting uncertainties.

6.4.1 The Pooled Age

3The remainder of this and the next three sections of this chapter will focus on the EDM. Alternative equations for ICP-MS-based Let us begin with the single discrete age component in fission-track data are provided in Sect. 6.7. Scenario I of the previous section. Several approaches can be 6 Statistics for Fission-Track Thermochronology 115 used to estimate this age from a set of noisy sample data. under this distribution is called thep value and can be used Panels I-a, I-b and I-c of Fig. 6.2 show that the single-grain to formally test the assumption of zero dispersion. age estimates follow an asymmetric probability distribution A 0.05 cut-off is often used as a criterion to abandon the (symmetric when plotted on a logarithmic scale) which is single-grain age model of Scenario I and, hence, the pooled skewed towards older ages. This is a consequence of the fact age. that, ifN sj andN ij are sampled from two independent Poisson distributions with expected valuesq s andq i, respectively, then the conditional probability ofN sj on 6.4.2 Central Ages and‘Overdispersion’ Nsj N ij follows a binomial distribution: þ   A more meaningful age estimate for Scenario II is obtained N N P N N N sj ij Nsj 1 Nij f using a two-parameter‘random effects’ model, in which the sj sj ij Nþ h h j h ð j þ Þ ¼ sj ð  Þ  ð Þ trueq s=qi-ratio is assumed to follow a log-normal distribu- 6:14 tion with location parameterµ and scale parameterr (Gal-   ð Þ braith and Laslett 1993): a fi whereh q s= qs q i and is the binomial coef - 2  ð þ Þ b ln qs=qi l;r 6:17 ð Þ Nð Þ ð Þ cient. Given a sample of n sets of FT counts, this leads to the This model gives rise to a two-parameter log-likelihood following (log-)likelihood function forh: function: Xn Xn h lnf j h 6:15 ; 2 lnf ; 2 6:18 Lð Þ ¼ j 1 ð Þ ð Þ l r j l r ¼ Lð Þ ¼ j 1 ð Þ ð Þ ¼ wheref j h is the probability mass function for thejth grain 2 ð Þ where the probability mass functionf j l;r is defined as: defined in Eq. 6.14. As afirst approach to obtaining an ð Þ     ‘average’ age, one might be tempted to simply take the Z N j N j 1 ebNsj 1 e b  s  i arithmetic mean of the single-grain age estimates. Unfortu- 2 Nsj N ij ffiffiffiffiffiffi fj l;r þ þ 2 db N j b l = 2r2 nately, the arithmetic mean does not cope well with outliers ð Þ ¼ s rp2peð  Þ ð Þ and asymmetric distributions and therefore yields poor 1 6:19 estimates of the geological age. The geometric mean fares ð Þ much better. It is closely related to the‘central age’, which is in which the FT ratios are subject to two sources of varia- discussed in Sect. 6.4.2. The‘pooled age’ is obtained by tion: the Poisson uncertainty described by Eq. 6.15 and an maximising Eq. 6.15 to obtain a‘maximum likelihood’ ‘(over)dispersion’ factorr. Maximising Eq. 6.18 results in ^ h^ estimateP (h), and substitutingPe forN s=Ni in Eq. 6.2, where two estimates l^ and r^ and their respective standard errors. n n ^l Ns j 1 Nsj andN i j 1 Nij. This is equivalent to Substitutinge forN s=Ni in Eq. 6.2 produces the so-called ¼ ¼ ¼ ¼ taking the sum of all the spontaneous and induced tracks, central age. r^ quantifies the excess scatter of the single-grain respectively, and treating these as if they belonged to a ages that cannot be explained by the Poisson counting single crystal. This procedure yields the correct age if the statistics alone. This dispersion can be just as informative as true ages are indeed derived from a single discrete age the central age itself, as it encodes geologically meaningful component (i.e. Scenario I). However, if there is any dis- information about the compositional heterogeneity and persion of the true FT ages, as is the case under Scenario II, cooling history of the sample. In the absence of excess dispersion (i.e. if r^ 0) the central age equals the pooled then the pooled age will be biased towards values that are far ¼ too old. Whether this is the case or not can be verified using age. a formalised statistical hypothesis test. Galbraith (2005, p. 46) shows that in the absence of excess dispersion, the following statistic: 6.5 Mixture Models

1 Xn N N N N 2 A FT data set may fail the chi-square test introduced in the c2 sj i ij s 6:16 ð  Þ previous section for different reasons. The true ages may ¼ NsNi j 1 Nsj N ij ð Þ ¼ þ exhibit excess scatter according to Eq. 6.18. Or it may be so follows a chi-square distribution withn− 1 degrees of that there are more than one age component (Galbraith and freedom. The probability of observing a value greater thanc 2 Green 1990; Galbraith and Laslett 1993). These components 116 P. Vermeesch could either be discrete age peaks (Scenario III) or they log-likelihood ratio test is provided in Sect. 6.5.2. An could be any combination of discrete and continuous age alternative approach is to maximise the so-called Bayes components (Scenario IV). Information Criterion (BIC), which is defined as

BIC 2 p ln n 6:21 ¼ L max þ ð Þ ð Þ 6.5.1 Finite Mixtures where is the maximum log-likelihood of a model L max comprisingp parameters andn grains. A worked example of Finite mixture models are a generalisation of the discrete age this method is omitted for brevity, and the reader is referred model of Scenario I in which the true ages are not derived to Galbraith (2005, p. 91) for further details. from a single, but from multiple age populations (Galbraith and Green 1990). Scenario III is an example of this with two such components. In contrast with the common age model of 6.5.2 Continuous Mixtures Scenario I, which is completely described by a single parameter (h, or the pooled age), and the random effects So far we have considered pdfs consisting of a single dis- model, which comprises two parameters (µ andr, or the crete age peak (Scenario I), a single continuous age distri- central age and overdispersion), thefinite mixture of Sce- bution (Scenario II) and multiple discrete age peaks nario III requires three parameters. These are the age of the (Scenario III). The logical next step is to consider multiple first component, the age of the second component and the continuous age distributions (Jasra et al. 2006). In principle proportion of the grains belonging to thefirst component. such models can be obtained by maximising the following The proportion belonging to the second component is simply likelihood function the complement of the latter value. Generalising toN com- "# ponents, the log-likelihood function becomes: Xn XN "# ; ; 2;k 1...N ln f ; 2 Xn XN pk l k r k pk j lk r k Lð ¼ Þ¼ j 1 k 1 ð Þ pk;h k;k 1...N ln pkfj hk ¼ ¼ Lð ¼ Þ¼ ð Þ XN 1 j 1 k 1  ¼ ¼ 6:20 withp N 1 pk XN 1 ð Þ  ¼  k 1 withp N 1 pk ¼ ¼  k 1 6:22 ¼ ð Þ 2 wheref j hk is given by Eq. 6.14. Equation 6.20 can be wheref j l ;r is given by Eq. 6.19. However, in reality ð Þ ð k kÞ solved numerically. Applying it to the single component data this is often impractical due to the high number of param- set of Scenario I again yields the pooled age as a special eters involved, which require exceedingly large data sets. In case. The detrital FT ages in Scenario III clearly fall into two detrital geochronology, the analyst rarely knows that the data groups so it is quite evident that there are two age compo- are underlain by a continuous mixture and so it is tempting nents. Unfortunately, the situation is not always this clear. to reduce the number of unknown parameters by simply Due to the large single-grain age uncertainties discussed in assuming a discrete mixture. Unfortunately, this is fraught Sect. 6.2, the boundaries between adjacent age components with problems as well since there is no upper bound on the are often blurred, making it difficult to decide how many number of discrete age components tofit to a continuous ‘peaks’ tofit. Several statistical approaches may be used to data set. To illustrate this point, let us reconsider the data set answer this question. One possibility is to use a of Scenario II, this time applying afinite mixture model log-likelihood ratio test. Suppose that we have solved rather than the random effects model of Sect. 6.4.2. Eq. 6.20 for the case ofN = 2 age components and denote For a small sample ofn = 10 grains, the chi-square test for the corresponding maximum log-likelihood value as 2. We age homogeneity yields ap value of 0.47, which is above the L then consider an alternative model withN = 3 components. 0.05 cut-off and thus provides insufficient evidence to reject This results in two additional parameters (p2 andh 3) and a the common age model (Table 6.1; Fig. 6.3a). Increasing the new maximum log-likelihood value, 3. We can assess sample size ton = 25 results in ap value of 0.03, justifying L whether the three-component model is a significant the addition of extra model parameters (Fig. 6.2, I-c). Further improvement over the two-componentfit by comparing increasing the sample size ton = 100 reduces the likelihood twice the difference between 3 and 2 to a chi-square of the common age model (Eq. 6.15) and results in ap value L L distribution with two degrees of freedom (because we have of 0.0027, well below the 0.05 cut-off. Let us now replace the added two additional parameters) and calculating the corre- common age model with a two-componentfinite mixture spondingp value like before. An illustration of the model. For the same 100-grain sample, this increases the 6 Statistics for Fission-Track Thermochronology 117

Table 6.1 Application of the log-likelihood ratio test to afinite mixturefitting experiment shown in Fig. 6.3. Rows mark different sample sizes (withn marking the number of grains) drawn from Scenario II. Columns labelled as N show the log-likelihood of different modelfits, where 2 L N marks the number of components. Columns labelled asp v 2 list thep values of a chi-square test with two degrees of freedom, which can be used to assess whether it is statistically justified to incrementð theÞ number offitting parameters (N) by one

2 2 2 1 p v 2 p v 3 p v 4 L ð 2Þ L ð 2Þ L ð 2Þ L n = 10−422.4 0.67−422.0 1.00−422.0 1.00−422.0 n = 100−4598.3 0.001−4591.4 0.45−4590.6 1.00−4590.6 n = 1000−45,030.7 0.00−44,966.5 0.00003−44,956.1 0.67−44,955.7

Fig. 6.3 Application offinite mixture modelling to the continuous sample size, from 150 Ma for sample (a) to 94 Ma for sample (c), and mixture of Scenario II (Fig. 6.2). Increasing sample size from left (a) to is therefore not a reliable estimator of the minimum age.p v 2 marks right (c) provides statistical justification tofit more components using thep value of the chi-square test for age homogeneity andð notÞ the the log-likelihood ratio approach of Table 6.1. Note that the age of the log-likelihood ratio tests of Table 6.1 youngest age component gets progressively younger with increasing log-likelihood from−4598.3 to−4591.4 (Table 6.1). Using distributions (pdfs) may be approximately (log)normal as in the log-likelihood ratio test introduced in Sect. 6.5.1, that Scenario I, but they are never exactly so. Given a sufficiently corresponds to a chi-square value of large sample, formalised statistical hypothesis tests such as 2 (4598.3− 4591.4) = 13.8 and ap value of 0.001, chi-square are always able to detect even the most minute  lending support to the abandonment of the single age model deviation from any hypothetical age model and thereby in favour of the two-parameter model (Fig. 6.3b). However, provide statistical justification to add further parameters. This doing the same calculation for a three-component model is important in the common situation where one is interested yields a log-likelihood of−4590.6 and a chi-square value of in the youngest age component of afission-track age distri- 2 (4591.4− 4590.6) = 1.6, resulting in an insignificant bution, for example when one aims to calculate‘lag times’  p value of 0.45 (Table 6.1). Thus, the 100-grain sample does and estimate exhumation rates (Garver et al. 1999; Bernet not support the three-component model. It is only when the 2018). It would be imprudent to estimate the lag-time by sample size is increased from 100 to 1000 grains that the applying a general purpose multi-component mixture model chi-square test gains enough‘power’ to justify the and simply picking the youngest age component. This would three-componentfinite mixture model (Fig. 6.3c). It is easy provide a biased estimate of the minimum age, which would to see that this trend continues ad infinitum: with increasing steadily drift towards younger values with increasing sample sample size, it is possible to add ever larger numbers of size (Fig. 6.3). Instead, it is better to use a simpler but more components (Fig. 6.3). stable and robust model employing three4 or four parameters One might object to this hypothetical example by noting to explicitly determine the minimum age component: that thefinite mixture model is clearly inappropriate for a data set that is derived from a continuous mixture. But the key point is that all statistical models are inappropriate to some 4Eq. 6.23 may be simplified by imposing the requirement that l e h, degree. Even the random effects model is a mathematical which significantly benefits numerical stability, while having only¼ a abstraction that does not exist in the real world. True age minor effect on the accuracy ofh. 118 P. Vermeesch

Xn h i distribution (Corrigan 1991; Lutz and Omar 1991; Gallagher 2 2 p;h;l;r lnpf j h 1 p f j0 l;r 6:23 1995; Willett 1997; Ketcham et al. 2000; Ketcham 2005; Lð Þ ¼ j 1 ð Þþð  Þ ð Þ ð Þ ¼ Gallagher 2012). Then, these‘forward model’ predictions 2 are compared with the measured values and the‘best’ mat- wheref j h is given by Eq. 6.14 andf 0 l;r is a truncated ð Þ j ð Þ ches are retained for geological interpretation. All the‘in- version of Eq. 6.19 (Galbraith and Laslett 1993). Applying verse modelling’ software that has been developed over the this model to the synthetic example of Scenario IV correctly years for the purpose of thermal history reconstructions yields the age of the youngest volcanic unit regardless of essentially follows this same recipe. The most important sample size (Fig. 6.2, IV-c). In conclusion, statistical difference between these algorithms is how they assess the hypothesis tests such as chi-square can be used to prevent goodness offit and decide which candidatet–T paths to overinterpreting perceived‘clusters’ of data that may arise retain and which to reject. from random statistical samplingfluctuations. But they must One class of software, including the popular HeFTy be used with caution, bearing in mind the simplifying program and its predecessor AFTSolve (Ketcham et al. assumptions which all mathematical models inevitably make 2000; Ketcham 2005), uses thep value of formalised and the dependence of test statistics andp values on sample hypothesis tests like the chi-square test described in the size. Ignoring this dependence may lead to statistical models previous sections, to decide whether the measured FT length that might make sense in a mathematical sense, but have distribution is a‘good’(p > 0.5),‘acceptable’ little or no geological relevance. This note of caution applies (0.5 >p > 0.05) or‘poor’(p < 0.05)fit. The problem with not only to mixture modelling but even more so to thermal this approach is that, due to the dependence ofp values on history modelling, as will be discussed next. sample size, it inevitably breaks down for large data sets. This is because the statistical‘power’ of statistical hypoth- 6.6 Thermal History Modelling esis test to resolve even the tiniest disagreement between the measured and the predicted length distribution, monotoni- cally increases with increasing sample size (Vermeesch and So far in this chapter, we have made the implicit assumption Tian 2014). (in Eq. 6.2) that allfission tracks have the same length. In A second class of inverse modelling algorithms (includ- reality, however, this is not the case and the length of (ap- ing QTQt, Gallagher 2012) does not employ formalised atite)fission tracks varies anywhere between 0 and 16lm as hypothesis tests orp values, but aims to extract the‘most a function of the thermal history and chemical composition likely’ thermal history models among all possiblet–T paths of a sample (Gleadow et al. 1986). If the compositional (Gallagher 1995; Willett 1997). These methods do not effects (notably the Cl/F ratio, Green et al. 1986) are well ‘break down’ when they are applied to large data sets. On characterised, then the measured length distribution of hor- the contrary, large data sets are‘rewarded’ in the form of izontally confinedfission tracks can be used to reconstruct tighter‘credibility intervals’ and higher resolutiont–T paths. the thermal history of a sample. Laboratory experiments Furthermore, they are easily extended to multi-sample and show that the thermal annealing offission tracks in apatite multi-method data sets. However, with great power also obeys a so-called fanning Arrhenius relationship, in which comes great responsibility. Vermeesch and Tian (2014) the degree of shortening logarithmically depends on both the show that QTQt always produces a‘bestfitting’ thermal amount and duration of heating (Green et al. 1985; Laslett history even for physically impossible data sets. To avoid et al. 1987; Laslett and Galbraith 1996; Ketcham et al. 1999, this potential problem, it is of paramount importance that the 2007): model predictions are shown alongside the FT data (Gal- pffiffiffiffiffiffiffiffiffiffi lagher 2012, 2016, Vermeesch and Tian 2014). K ln t ln t c ln1 L=L0 c 0 c 1 ð Þ ð Þ 6:24  ¼  1=T 1=T c ð Þ  whereL 0 andL are the initial and measured track length, 6.7 LA-ICP-MS-Based FT Dating t andT are time and absolute temperature, respectively, and c0,c 1,K,t c andT c arefitting parameters (Laslett and Gal- The EDM outlined in Sect. 6.2 continues to be the most braith 1996; Ketcham et al. 1999). Using these laboratory widely used analytical protocol in FT dating. However, over results, thermal history reconstructions are a two-step pro- the past decade, an increasing number of laboratories have cess. First, a large number of random thermal histories are abandoned it and switched to LA-ICP-MS as a means of generated, and for each of these, the fanning Arrhenius determining the uranium concentration of datable minerals, relationship is used to predict the corresponding FT length thus reducing sample turnover time and removing the need 6 Statistics for Fission-Track Thermochronology 119 to handle radioactive materials (Hasebe et al. 2004, 2009; the U/Ca-ratio measurement, say), which can be estimated Chew and Donelick 2012; Soares et al. 2014; Abdullin et al. using two alternative approaches as discussed in Sect. 6.7.2. 2016; Vermeesch 2017). The statistical analysis of Equation 6.27 can also be applied to the‘absolute’ dating ICP-MS-based FT data is less straightforward and less well method (Eq. 6.25) by simply settings ^f =^f 0. developed than that of the EDM. As described in Sect. 6.2, ½  ¼ the latter is based on simple ratios of Poisson variables, and forms the basis of a large edifice of statistical methods which 6.7.2 Error Propagation of LA-ICP-MS-Based cannot be directly applied to ICP-MS-based data. This sec- Uranium Concentrations tion provides an attempt to address this issue. Uranium-bearing minerals such as apatite and zircon often exhibit compositional zoning, which must either be removed 6.7.1 Age Equation or quantified in order to ensure unbiased ages.

The FT age equation for ICP-MS-based data is based on 1. The effect of compositional zoning can be removed by Eq. 6.1: covering the entire counting area with one large laser   spot (Soares et al. 2014) or a raster (Hasebe et al. 2004). 1 k N ^t ln1 D s 6:25 s U^ is then simply given by the analytical uncertainty of ¼ kD þ kf U^ A R q ð Þ ½  ½  s the LA-ICP-MS instrument, which typically is an order of magnitude lower than the standard errors of induced whereN is the number of spontaneous tracks counted over s track counts in the EDM. an areaA s,q is an‘efficiency factor’(*0.93 for apatite and 2. Alternatively, the uranium heterogeneity can be quanti- *1 for zircon, Iwano and Danhara 1998; Enkelmann and fied by analysing multiple spots per analysed grain Jonckheere 2003; Jonckheere 2003; Soares et al. 2013) and (Hasebe et al. 2009). In this case, it is commonly found U^ is the 238U-concentration (in atoms per unit volume) ½  that the variance of the different uranium measurements measured by LA-ICP-MS. Equation 6.25 requires an explicit within each grain far exceeds the formal analytical value fork f and assumes that the etchable range (R) is uncertainty of each spot measurement. The following accurately known (Soares et al. 2014). Alternatively, these paragraphs will outline a method to measure that dis- factors may be folded into a calibration factor akin to the persion, even if some of the grains in a sample were only EDM (Eq. 6.4): visited by the laser once.   1 1 ^ Ns ^t ln1 kDf 6:26 The true statistical distribution of the U-concentrations ¼ kD þ 2 A U^ ð Þ s½  within each grain is unknown but is likely to be log-normal: in which ^f is determined by analysing a standard of known 2 ln U^jk l ;r 6:28 FT age (Hasebe et al. 2004). Note that, in contrast with the ½  Nð j j Þ ð Þ ‘absolute’ dating method of Eq. 6.25, thef-calibration where U^jk is thekth out ofn j uranium concentration mea- method of Eq. 6.26 allows U^ to be expressed in any con- 2 ½  surements, andl j andr j are the (unknown) mean and centration units (e.g. ppm or wt% of total U) or could even variance of a normal distribution. Unfortunately, it is diffi- be replaced with the measured U/Ca-, U/Si- or U/Zr-ratios cult to accurately estimate these two parameters from just a produced by the ICP-MS instrument. The standard error of handful of spot measurements and it is downright impossible the estimated age is given by ifn j 1. This problem requires a simplifying assumption vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ u ! j u 2  2 such asr j r . In that case, we can estimate the param- t s ^f s U^ 1 ¼ 8 s ^t ^t ½  ½  6:27 eters of Eq. 6.28 as follows: ½  ^f þ U^ þ Ns ð Þ Xnj l^ ln U^jk =nj 6:29 for thef-calibration approach (Eq. 6.26), wheres U^ is the j ¼ ½  ð Þ ½  k 1 standard error of the uranium concentration measurement (or ¼

6 Statistics for Fission-Track Thermochronology 121

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