the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
An algorithm for U–Pb geochronology by secondary ion mass spectrometry
Secondary ion mass spectrometry (SIMS) is a widely used technique for in situ U–Pb geochronology of accessory minerals. Existing algorithms for SIMS data reduction and error propagation make a number of simplifying assumptions that degrade the precision and accuracy of the resulting U–Pb dates. This paper uses an entirely new approach to SIMS data processing that introduces the following improvements over previous algorithms. First, it treats SIMS measurements as compositional data using logratio statistics. This means that, unlike existing algorithms, (a) its isotopic ratio estimates are guaranteed to be strictly positive numbers, (b) identical results are obtained regardless of whether data are processed as normal ratios (e.g. ^{206}Pb $/$ ^{238}U) or reciprocal
ratios (e.g. ^{238}U $/$ ^{206}Pb), and
(c) its uncertainty estimates account for the positive skewness of
measured isotopic ratio distributions. Second, the new algorithm
accounts for the Poisson noise that characterises secondary electron
multiplier (SEM) detectors. By fitting the SEM signals using the
method of maximum likelihood, it naturally handles lowintensity ion
beams, in which zerocount signals are common. Third, the new
algorithm casts the data reduction process in a matrix format and
thereby captures all sources of systematic uncertainty. These
include significant interspot error correlations that arise from
the Pb $/$ U–UO_{(2)} $/$ U calibration curve. The new
algorithm has been implemented in a new software package called
simplex
. The simplex
package was written in R
and can
be used either online, offline, or from the command line. The programme
can handle SIMS data from both Cameca and SHRIMP instruments.
Secondary ion mass spectrometry (SIMS) combines high sensitivity with
high mass resolution (Williams, 1998). This allows the technique
to obtain precise U–Pb dates on nanogramsized samples whilst resolving
isobaric interferences on ^{204}Pb to a degree that is
currently unachievable by other techniques such as laser ablation
inductively coupled plasma mass spectrometry (LAICPMS). There are
some other differences between LAICPMS and SIMS as well. LAICPMS
instrumentation is built by numerous manufacturers, whose data files
are compatible with a rich ecosystem of data reduction codes such as
Iolite
, Glitter
, and LADR
. This facilitates
the intercomparison of different laboratories, different instrument
designs, and so forth. In contrast, the SIMS U–Pb world is dominated
by just two manufacturers, whose SHRIMP (Sensitive High Resolution Ion
MicroProbe) and Cameca instruments use completely separate data
reduction protocols. Most SHRIMP laboratories use Squid
(Ludwig, 2000; Bodorkos et al., 2020), which is incompatible with Cameca
data. In contrast, Cameca data tend to be processed by inhouse
software such as Martin John Whitehouse's NordSIM spreadsheet, which are
incompatible with SHRIMP data.
This paper introduces a unified algorithm for SIMS U–Pb data reduction that aims to address five problems with existing data reduction methods. The first three of these problems are the following.
 1.
Accuracy: existing algorithms give (slightly) different results depending on whether the raw data are processed as ^{206}Pb $/$ ^{238}U ratios or as ^{238}U $/$ ^{206}Pb ratios.
 2.
Precision: current data reduction routines produce symmetric confidence intervals, which are unrealistic for lowintensity ion beams.
 3.
Systematic uncertainties: current data reduction protocols use a hierarchical error propagation approach, in which random uncertainties and systematic uncertainties are propagated separately. However, such a clean separation is not always possible, and this can complicate higherorder data processing steps such as isochron regression and averaging.
Sections 2–4 will provide further details about these problems using synthetic examples. Section 5 will show that problems 1, 2, and 3 can be solved by treating the U–Pb system as a compositional data space using logratio statistics. However, logratio statistics do not solve the remaining two problems.
 4.
Backgrounds: background correction of lowintensity signals such as ^{204}Pb sometimes exceeds 100 %, producing physically impossible negative isotope ratios.
 5.
Zeros: Pb isotopes are usually measured by secondary electron multipliers (SEMs), which record ions as counts. For ^{204}Pb and other lowintensity ion species, it is not uncommon to register zero counts during any given analytical cycle. This causes problems if the zero count appears in the denominator of an isotopic ratio.
These problems are discussed in more detail in
Sect. 6. Section 7 addresses them by
incorporating multinomial counting statistics into the compositional
data framework. Sections 8–11 recast the U–Pb processing chain in this mixed multinomial–compositional form. Section 12 applies the new data
reduction paradigm to two datasets produced by Cameca and SHRIMP
instruments, and Sect. 13 introduces a computer code
called simplex
that generated these results. Finally, further
details about the implementation of the new algorithm are reported in
the Appendix to this paper.
The common Pbcorrected ^{206}Pb $/$ ^{238}U age (t) depends on the relative abundances of the three nuclides ^{204}Pb, ^{206}Pb, and ^{238}U:
where the subscript c marks the common lead composition, and λ_{238} is the decay constant of ^{238}U. Equation (1) does not depend on the absolute amounts of ^{204}Pb, ^{206}Pb, and ^{238}U, but only on their ratios. Unfortunately, the statistical analysis of the ratios of strictly positive numbers is full of potential pitfalls, as will be illustrated with an example that was inspired by McLean et al. (2016). Consider a simple dataset of 10 synthetic U–Pb measurements (Table 1).
Let us calculate the ^{206}Pb $/$ ^{238}U ratios for these data,
which are needed to solve Eq. (1). Comparing these
ratios with their reciprocals yields two new sets of 10 numbers.
The elementary rules of mathematics dictate that $\mathrm{1}/(y/x)=x/y$ for any two real numbers x and y. In other words, the reciprocal of the reciprocal ratio ought to equal that ratio. Indeed, for our example it is easy to see that $\mathrm{1}/\mathrm{0.085}=\mathrm{11.70}$ and so forth. However, when we take the arithmetic means of the (reciprocal) ratios
and
then we find that
So the reciprocal of the mean reciprocal ratio does not equal the mean of that ratio! This is a counterintuitive and clearly wrong result. Unfortunately, current algorithms for SIMS data reduction average ratios using the arithmetic mean or perform (linear) regression through ratio data, which causes similar problems (Ogliore et al., 2011). Inaccurate ^{206}Pb $/$ ^{238}U ratios inevitably result in inaccurate U–Pb dates, with the degree of inaccuracy scaling with the relative standard deviation of the measured data. Therefore, the numerical example shown in this section is deeply troubling for isotope geochemistry in general and SIMS U–Pb geochronology in particular.
Traditionally, the precision of isotopic data used in U–Pb
geochronology has been calculated as symmetric confidence
intervals. Unfortunately, this is fraught with similar problems as
those discussed in Sect. 2. For example, take the
arithmetic mean ($\stackrel{\mathrm{\u203e}}{x}$) and standard deviation (s_{x}) of the
^{206}Pb $/$ ^{204}Pb ratios in Table 1 and construct a Studentised 95 % confidence interval
for $\stackrel{\mathrm{\u203e}}{x}$.
Here, ${t}_{\mathrm{df}}^{\mathit{\alpha}}$ is the α percentile of a t distribution with df degrees of freedom (e.g.
${t}_{\mathrm{9}}^{\mathrm{2.5}}=\mathrm{2.262}$). Then the lower limit of the confidence interval
is negative. This nonsensical result is yet another indication that
there are some fundamental problems with the application of
“conventional” statistical operations to isotopic data. Negative
isotope ratios can also arise from background corrections, but this
will be further discussed in Section 6.
The statistical uncertainty of analytical data can be classified into two components (Renne et al., 1998).

Random (or internal) errors are caused by electronic noise in the ion detectors, counting statistics, and/or temporal variability of the background as a result of changes in the lab environment, for example. They are independent for different aliquots of the same sample and can be quantified by taking replicate measurements. The standard error of these measurements ($\mathit{\sigma}/\sqrt{N}$, where σ is the standard deviation of N replicate measurements) is a measure of their precision. The standard error can be reduced to arbitrarily low levels by simply averaging more measurements (i.e. by increasing N). For example, the precision of SIMS ^{206}Pb $/$ ^{204}Pb ratio measurements can be increased by simply extending the duration of the primary ion bombardment.

Systematic (or external) errors include the effects of decay constant uncertainty and the ^{206}Pb $/$ ^{238}U ratio of reference materials. Getting these constants wrong causes bias in some or all of the measurements and thus affects the accuracy of the age determinations. As their name suggests, the systematic uncertainties are not independent but correlated between different aliquots and samples. They cannot be reduced by simple averaging.
Great care must be taken in choosing which sources of uncertainty should or should not be included in the error propagation. In some cases, intersample comparisons of SIMS U–Pb data may legitimately ignore systematic uncertainties. However, when comparing data acquired by different methods, both random and some systematic uncertainties must be accounted for. For example, when comparing U–Pb and ^{40}Ar $/$ ^{39}Ar data, the decay constant uncertainty must be propagated, and when comparing SIMS and TIMS U–Pb data, SIMS primary standard age uncertainty and the TIMS tracer uncertainty must be propagated. The conventional way to tackle intersample comparisons is called “hierarchical” error propagation (Renne et al., 1998; Min et al., 2000; Horstwood et al., 2016). Under this paradigm, the random uncertainties are processed first and the systematic uncertainties afterwards.
Hierarchical error propagation is straightforward in principle but not always in practice. Some processing steps are of a hybrid nature, including both systematic and random uncertainties. ^{206}Pb $/$ ^{238}U calibration for SIMS is a good example of this. ^{206}Pb $/$ ^{238}U ratios are sensitive to elemental fractionation in SIMS analysis (see Sect. 10 for further details). These fractionation effects are captured by the following power law (Williams, 1998; Jeon and Whitehouse, 2015):
where the subscript m stands for the measured signal ratio, which is generally different from the atomic ratio, and the subscript (2) refers to the fact that the calibration normally uses uranium oxide for SHRIMP and uranium dioxide for Cameca instruments. The atomic ^{206}Pb $/$ ^{238}U log ratio of a sample is determined by (1) determining the intercept (A) and slope (B) of a reference material (r) of known Pb $/$ U ratio^{1} and (2) using this calibration curve to estimate the equivalent Pb $/$ U log ratio of the reference material corresponding to the UO_{(2)} $/$ U log ratio of the sample (s). Then,
where the subscript “a” stands for the estimated (for the sample) or known (for the reference material) atomic log ratios. The analytical uncertainty of $\mathrm{ln}\phantom{\rule{0.125em}{0ex}}{\left[{}^{\mathrm{206}}\mathrm{Pb}/{}^{\mathrm{238}}\mathrm{U}\right]}_{\mathrm{a}}^{\mathrm{s}}$ depends on the analytical uncertainties of both the intercept (A) and slope (B) of the reference material. But it does not necessarily do so to the same degree for all aliquots. Analytical spots that have similar UO_{(2)} $/$ U ratios as the reference material will be less affected by the uncertainty of the reference material fit than spots that have very different UO_{(2)} $/$ U ratios (Fig. 1). Hence, it is not possible to make a clean separation between random and systematic uncertainties.
Section 2 pointed out that the U–Pb age equation (Eq. 1) does not depend on the absolute amounts of ^{204}Pb, ^{206}Pb, and ^{238}U, but only on their relative abundances. Thus, we could normalise the ^{204}Pb, ^{206}Pb, and ^{238}U measurements of Table 1 to unity and plot them on a ternary diagram. The same is true for other geochronometers such as U–Th–He and ^{40}Ar $/$ ^{39}Ar (Vermeesch, 2010, 2015). In mathematics, the ternary sample space is known as a twodimensional simplex. Data that live within this type of space are called compositional data.
Ternary systems are common in igneous petrology (e.g. the A–F–M diagram) and sedimentary petrography (e.g. the Q–F–L diagram). Geologists have long been aware of the problems associated with averages, confidence regions, and linear regression in these closed data spaces (Chayes, 1949, 1960). But a general solution to this conundrum was not found until the 1980s, when the Scottish statistician John Aitchison published a landmark paper and book on the subject (Aitchison, 1982, 1986).
In this work, Aitchison proved that all the problems associated with the statistical analysis of compositional data can be solved by mapping those data from the simplex to a Euclidean space by means of an additive logratio (ALR) transformation. For example, given the ternary system $\mathit{\{}x,y,z\mathit{\}}$, we can define two new variables {u,v} so that
and
In this space, Aitchison showed, one can safely calculate averages and confidence limits. Once the statistical analysis of the transformed data has been completed, the results can then be mapped back to the simplex by means of an inverse logratio transformation:
and
For example, the ^{204,6}Pb–^{238}U
system of Table 1 can be mapped from the ternary diagram
to a bivariate
$\mathrm{ln}({}^{\mathrm{204}}\mathrm{Pb}/{}^{\mathrm{238}}\mathrm{U})$–$\mathrm{ln}({}^{\mathrm{206}}\mathrm{U}/{}^{\mathrm{238}}\mathrm{U})$ space.
Alternatively, we could also use ^{206}Pb as the
denominator isotope.
Calculating the average of the transformed data and mapping the
results back to the simplex using the inverse logratio transformation
yields the geometric mean of the ratios:
which is an altogether more satisfying result than in
Sect. 2. Moving on to the 95 % confidence intervals
of the ^{206}Pb $/$ ^{204}Pb ratios, we
first determine the conventional confidence limits for the log ratios.
After the inverse logratio transformation, these values produce an asymmetric 95 % confidence interval for the geometric mean ^{206}Pb $/$ ^{204}Pb ratio of ${\mathrm{305}}_{\mathrm{212}}^{+\mathrm{695}}$. This interval contains only strictly positive values, solving the problem of Sect. 3. The ALR transformation of Eq. (4) can easily be generalised to more than three components. For example, if ^{207}Pb is added to the mix, then the fourcomponent ${}^{\mathrm{204}\left\mathrm{6}\right\mathrm{7}}$Pb–^{238}U system can be mapped to the threecomponent $\mathrm{ln}({}^{\mathrm{204}\left\mathrm{6}\right\mathrm{7}}\mathrm{Pb}/{}^{\mathrm{238}}\mathrm{U})$ space (Fig. 2).
The compositional nature of isotopic data embeds a covariant structure into the very DNA of geochronology: in a Kcomponent system, increasing the absolute amount of one of the components automatically lowers the relative amount of the remaining (K−1) components (Chayes, 1960). To deal with this phenomenon, it is customary in compositional data analysis to process data in matrix form using the full covariance matrix. For example, four components yield three log ratios, which require a 3×3 covariance matrix to describe the uncertainties and uncertainty correlations.
This approach is now widely used in sedimentary geology, geochemistry, and ecology (e.g. Weltje, 2002; Vermeesch, 2006; PawlowskyGlahn and Buccianti, 2011), and it has recently been adopted for geochronological applications as well (Vermeesch, 2010, 2015; McLean et al., 2016). The logratio covariance matrix approach is also uniquely suited to capture the systematic uncertainties (i.e. the interspot error correlations) that are produced by the SIMS U–Pb calibration procedure (Sect. 4). In that case, the covariance matrix must be expanded to accommodate multiple spots (e.g. Vermeesch, 2015). For example, the covariance structure of three spots in a fourcomponent system can be captured in a 9×9 matrix.
In conclusion, the logratio transformation solves the statistical woes described in Sects. 2, 3, and 4 of this paper. However, there are two additional problems that require further remediation.
Many mathematical operations are easier in logarithmic space than in linear space: multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. These mathematical operations are very common in mass spectrometer data processing chains (Vermeesch, 2015). However, there are exceptions. For example, background correction does not involve the multiplication but subtraction of two signals. For lowintensity ion beams such as ^{204}Pb, it is possible, by chance, that the background exceeds the signal. This results in negative values of which one cannot take the logarithm.
The subtraction problem can be solved by using a different logarithmic change of variables:
where x and y are the signals and b is the background. The infinite space of ${\mathit{\beta}}_{x}^{y}$ covers all possible values of x, y, and b for which x>b and y>b. Thus, background correction should be done in β space, given an appropriate error model as described in Sect. 7.
One caveat to this background correction method is that it does not account for isobaric interferences, which may result in “overcounted” signals. The high mass resolution of SIMS instruments removes most but not all isobaric interferences. For example, spurious HfSi, REE dioxide, or longchain hydrocarbon ions can interfere with ^{204}Pb, which is generally the rarest isotopic species detected. If unaccounted for, these interferences lead to the proportion of nonradiogenic Pb being overestimated (and the proportion of radiogenic Pb underestimated), resulting in excessive common Pb corrections and underestimated dates. The accuracy of the background measurements can be monitored via the use of isotopically homogeneous reference materials (Black, 2005). A correction can then be applied by choosing the “session blank” that brings the common Pbcorrected ^{207}Pb $/$ ^{206}Pb ratios in alignment with the reference values.
Another issue arises when SIMS signals are registered by secondary electron multiplier (SEM) detectors. These record ion beams as discrete counts, i.e. as integers, which are incompatible with the logratio transformation. For example, it is not uncommon for SEM detectors to register zero counts for lowintensity ion beams such as ^{204}Pb. These zero counts blow up the logratio transformation because $\mathrm{ln}\left(\mathrm{0}\right)=\mathrm{\infty}$.
This and other issues are diagnostic of a fundamental difference between compositional data and counting data that has been previously recognised and solved in fissiontrack dating (Galbraith, 2005) and in sedimentary petrography (Vermeesch, 2018b). The same solutions can be applied to mass spectrometric count data in general and to U–Pb geochronology in particular (Sect. 7).
Standard data reduction procedures for geochronology assume normally distributed residuals. In compositional data analysis, these are replaced by logistic normal distributions. However, neither the normal nor the logistic normal distribution is perfectly suited to dealing with discrete count data. The multinomial distribution is a simple alternative that seems more appropriate for the task at hand. Before we proceed, let us define the following variables.

ϕ_{x} and ϕ_{b} are the normalised true ion beam intensities (in counts per second) of mass x (from a set of monitored masses 𝕩) and the normalised background signal, respectively, so that
$$\begin{array}{}\text{(7)}& {\mathit{\varphi}}_{b}+\sum _{x\in \mathbb{x}}{\mathit{\varphi}}_{x}=\mathrm{1}\end{array}$$ 
d_{x} and d_{b} are the dwell times of mass x and the background b

θ_{x} and θ_{b} are the normalised expected beam counts of the ions and the background so that
$$\begin{array}{}\text{(8)}& {\mathit{\theta}}_{x}={\displaystyle \frac{{\mathit{\varphi}}_{x}{d}_{x}}{{\mathit{\varphi}}_{b}{d}_{b}+{\sum}_{y\in \mathbb{x}}{\mathit{\varphi}}_{y}{d}_{y}}}\end{array}$$and
$$\begin{array}{}\text{(9)}& {\mathit{\theta}}_{b}+\sum _{x\in \mathbb{x}}{\mathit{\theta}}_{x}=\mathrm{1}\end{array}$$
Using the ^{204}Pb $/$ ^{206}Pb ratio as an example, the probability of observing n_{4} counts at mass 204, n_{6} counts at mass 206, and n_{b} counts of background is given by
Whereas the observations n_{4}, n_{6}, and n_{b} are integers, the parameters θ_{4}, θ_{6}, and θ_{b} are decimal numbers that are constrained to a constant sum. In other words, they belong to the simplex. Thus, we can map the three multinomial parameters to two logratio parameters, thereby establishing a natural link between counting data and compositional data. For example,
and
${\mathit{\beta}}_{\mathrm{6}}^{\mathrm{4}}$ and ${\mathit{\beta}}_{\mathrm{6}}^{b}$ can be estimated from n_{4}, n_{6}, and n_{b} by maximising the likelihood defined by Eq. (10).
The normal and logistic normal distributions are controlled by two sets of parameters: location parameters and shape parameters. In the case of the normal distribution, the location parameter is the mean and the shape parameter is the standard deviation (or covariance matrix). In contrast, the multinomial distribution has only one set of (θ) parameters. The precision of multinomial counts is governed by the number of observed counts ($\mathit{\sigma}\left[n\right]=\sqrt{n}$). More sophisticated models are possible when the observed dispersion of the data exceeds that which is expected from the multinomial counting statistics (e.g. Galbraith and Laslett, 1993; Vermeesch, 2018b). However, in this paper we will assume that such overdispersion is absent from the reference materials and from the singlespot analyses.
It takes a few tens of nanoseconds for a secondary electron multiplier to record the arrival of an ion. During this “dead time”, the detector is unable to register the arrival of additional ions. This phenomenon can significantly bias isotope ratio estimates that include ion beams of contrasting intensity. Fortunately, the deadtime effect can be easily corrected. It suffices that the dwell times are adjusted by the cumulative amount of time that the detectors were incapacitated. Let ${d}_{x}^{\prime}$ be the “effective dwell time” of ion beam x:
where δ_{x} is the (nonextended) dead time of the detector that measures x. Then the expected normalised beam counts can be redefined as
and the normalised beam intensities as
The difference between this approach and existing SIMS data reduction approaches is that conventional data reduction algorithms apply the deadtime correction to the raw data before calculating isotopic data, whereas the new method applies the deadtime correction to ${\mathit{\theta}}_{x}^{\prime}$ and ${\mathit{\varphi}}_{x}^{\prime}$, which are unknown parameters that must be estimated from the data.
Thus far we have assumed that all ions are measured synchronously, which is the case in multicollector configurations. However, in singlecollector experiments, the measurements are made asynchronously. This can cause biased results if the signals drift over time. In SHRIMP data processing, it is customary to correct this drift by normalising to a secondary beam monitor (SBM) signal^{2}, followed by double interpolation of numerator and denominator isotopes (Bodorkos et al., 2020; Dodson, 1978). A unified data reduction algorithm for SHRIMP and Cameca instruments requires a different approach, in which the time dependency of the signals is parameterised using a loglinear model. For Faraday detectors,
where ${n}_{x}^{i}$ is the ion beam intensity of the ith integration for mass x of element X evaluated at time ${\mathit{\tau}}_{x}^{i}$, σ is the standard deviation of the normally distributed data scatter around the best fit line, and “bkg” is the background signal. This is usually a nominal value for Cameca instruments and an actual set of measurements (${n}_{b}^{i}$) for SHRIMP data. Note that the intercept parameters (α_{x}) are ionicmassspecific, whereas the slopes (γ_{X}) are elementspecific. See the Appendix for an example. For SEM detectors, the scatter of the data around the loglinear fit is controlled by Poisson shot noise:
The backgroundcorrected signal ratio (evaluated at ${\mathit{\tau}}_{x}^{i}$) for two ion beams x and y of different elements X and Y can then be driftcorrected as follows:
where ${\mathit{\varphi}}_{x}^{i}$ and ${\mathit{\varphi}}_{y}^{i}$ are the deadtimecorrected normalised beam intensities for the ith integration of masses x and y, respectively, and ${\mathit{\varphi}}_{b}^{i}$ is the corresponding background value. See the Appendix for further details.
Mass spectrometer signals are recorded in volts (for Faraday detectors) or hertz (or secondary electron multipliers). The age equation, however, requires atomic ratios. In general, signal ratios do not equal atomic ratios because they are affected by two types of fractionation.

Massdependent fractionation. The Pb isotopes span a range of four mass units, with ^{208}Pb being 2 % heavier than ^{204}Pb. Both the production and detection efficiency of secondary ions vary with atomic mass, and significant errors can potentially occur if the resulting mass fractionation is uncorrected for. Mass fractionation can be quantified by comparing the measured signal ratios of a reference material with its known isotopic ratio. This is easy to do in a logratio context (Vermeesch, 2015).

Elemental fractionation. The fractionation between the Pb isotopes is caused by (slight) differences in their physical properties, i.e. their mass. As briefly mentioned in Sect. 4, much stronger fractionation effects tend to occur between the isotopes and Pb and U because they are not only physically but also chemically different. These chemical differences affect the complex processes that occur when the primary ion beam interacts with the target material (Williams, 1998).
In the context of SIMS U–Pb geochronology, massdependent fractionation is commonly ignored (but not always, e.g. Stern et al., 2009) because the most important isochemical ratio is that between ^{206}Pb and ^{207}Pb, which lie within 0.5 % mass units of each other. This is unresolvable given typical analytical uncertainties. The mass fractionation is greater for ^{204}Pb, but so is its analytical uncertainty. Therefore, the atomic ^{204}Pb $/$ ^{206}Pb, ^{207}Pb $/$ ^{206}Pb, and ^{208}Pb $/$ ^{206}Pb ratios can be directly estimated from the (driftcorrected) ^{204}Pb $/$ ^{206}Pb, ^{207}Pb $/$ ^{206}Pb, and ^{208}Pb $/$ ^{206}Pb signal ratios. This is not the case for the ^{206}Pb $/$ ^{238}U and ^{208}Pb $/$ ^{232}Th ratios, which are affected by strong elemental fractionation effects. This fractionation expresses itself in two ways.

Withinspot fractionation
Over the course of a SIMS spot analysis, the Pb $/$ U and Pb $/$ Th ratio changes as a function of time. This elemental fractionation can be modelled using a loglinear model that is similar to that used for the withinspot drift correction:
$$\begin{array}{}\text{(19)}& {}^{i}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}{=}^{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}+{\mathit{\gamma}}_{X}^{Y}{\mathit{\tau}}_{x}^{i},\end{array}$$where ${}^{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}$ is the inferred log ratio of the backgroundcorrected signals at “time zero”, which can be found using the method of maximum likelihood (see the Appendix). Note that ${\mathit{\gamma}}_{X}^{Y}=\mathrm{0}$ if X=Y, and, if the sensitivity drift of the data is smooth and closely follows the trends defined by Eqs. (16) and (17), then ${}^{\mathrm{0}}{\mathit{\beta}}_{x}^{y}\approx {\mathit{\alpha}}_{y}{\mathit{\alpha}}_{x}$ and ${\mathit{\gamma}}_{X}^{Y}\approx {\mathit{\gamma}}_{Y}{\mathit{\gamma}}_{X}$.
Using Eq. (19), the isotopic log ratios can be interpolated (or extrapolated) to any point in time (τ):
$$\begin{array}{}\text{(20)}& {}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}{=}^{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}+{\mathit{\gamma}}_{X}^{Y}\mathit{\tau}.\end{array}$$The most precise values of ${}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}$ are obtained when τ is chosen in the middle of the analytical sequence. These values can be used for subsequent calculations. Alternatively, we can also use the timezero intercepts ${}^{\mathrm{0}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{x}^{y}$.

Betweenspot fractionation
The Pb $/$ U and Pb $/$ Th signal ratios may vary between adjacent spots on the same isotopically homogenous reference material. This fractionation obeys the powerlaw relationship given by Eq. (2). Expressing this formula in terms of corrected signal ratios gives
$$\begin{array}{}\text{(21)}& \begin{array}{rl}& \mathrm{ln}\phantom{\rule{0.125em}{0ex}}\left[{\displaystyle \frac{({\mathit{\varphi}}_{\mathrm{6}}^{\prime}{\mathit{\varphi}}_{b}^{\prime})({\mathit{\varphi}}_{\mathrm{4}}^{\prime}{\mathit{\varphi}}_{b}^{\prime})(\mathrm{6}/\mathrm{4}{)}_{\mathrm{c}}}{{\mathit{\varphi}}_{u}^{\prime}{\mathit{\varphi}}_{b}^{\prime}}}\right]=\\ & \phantom{\rule{1em}{0ex}}A+B\mathrm{ln}\phantom{\rule{0.125em}{0ex}}\left[{\displaystyle \frac{{\mathit{\varphi}}_{o}^{\prime}{\mathit{\varphi}}_{b}^{\prime}}{{\mathit{\varphi}}_{u}^{\prime}{\mathit{\varphi}}_{b}^{\prime}}}\right],\end{array}\end{array}$$where $(\mathrm{6}/\mathrm{4}{)}_{\mathrm{c}}$ stands for the ^{206}Pb $/$ ^{204}Pb ratio of the common Pb (see the footnote of Sect. 4). Recasting Eq. (21) in terms of the interpolated logratio estimates gives
$$\begin{array}{}\text{(22)}& \mathrm{ln}\phantom{\rule{0.125em}{0ex}}\left[\mathrm{exp}{(}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{\mathrm{u}}^{\mathrm{6}})\mathrm{exp}{(}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{\mathrm{u}}^{\mathrm{4}})(\mathrm{6}/\mathrm{4}{)}_{\mathrm{c}}\right]=A+B{}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{\mathrm{u}}^{\mathrm{o}},\end{array}$$where “o” stands for the uranium oxide (^{238}U^{16}O${}_{\mathrm{2}}^{+}$ or ^{238}U^{16}O^{+}) and “u” stands for ^{238}U^{+}.
Having applied Eq. (22) to a reference material with a known atomic ^{206}Pb $/$ ^{238}U ratio ${\left[{}^{\mathrm{206}}\mathrm{Pb}/{}^{\mathrm{238}}\mathrm{U}\right]}_{\mathrm{a}}^{\mathrm{r}}$, the atomic ^{206}Pb $/$ ^{238}U ratio of the sample is given by
where ${}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{\mathrm{u}}^{\mathrm{6}}\left(s\right)$ and ${}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{\mathrm{u}}^{\mathrm{o}}\left(s\right)$ are the interpolated logratio estimates of the sample. The ^{206}Pb $/$ ^{238}U age is then obtained by plugging ${\left[{}^{\mathrm{206}}\mathrm{Pb}/{}^{\mathrm{238}}\mathrm{U}\right]}_{\mathrm{a}}^{\mathrm{s}}$ into the age equation. Uncertainties are obtained by standard error propagation (see the Appendix).
The following paragraphs will illustrate the SIMS U–Pb data reduction process using two datasets.

Dataset 1 was acquired by Yang Li at IGGCAS Beijing using a Cameca 1280HR instrument with five SEMs and two Faraday detectors run in singlecollector mode using an SEM for all mass stations. The dataset uses Temora2 zircon (416.8 ± 1.1 Ma, Black et al., 2004) as a reference material and 91500 zircon (1062.4 ± 0.2 Ma, Wiedenbeck et al., 1995) as a sample. Measurements consist of seven cycles through a set of 11 mass stations per singlespot measurement for ^{90}Zr_{2}O (0.48 s dwell time per cycle), ^{92}Zr_{2}O (0.08 s), mass 200.5 (background, 4.00 s), ^{94}Zr_{2}O (0.32 s), ^{204}Pb (4.96 s), ^{206}Pb (2.96 s), ^{207}Pb (6.00 s), ^{208}Pb (2.00 s), ^{238}U (2.96 s), ThO_{2} (2.96 s), and UO_{2} (2.96 s).

Dataset 2 was acquired by Simon Bodorkos at Geoscience Australia using a singlecollector SHRIMPII instrument, employing an SEM for all mass stations. This dataset also uses Temora2 as a reference material, as well as 91500 zircon, OG.1 (3440.7 ± 3.2 Ma, Stern et al., 2009), and M127 (524.36 ± 0.16 Ma, Nasdala et al., 2016) as samples. Measurements consist of six cycles through a set of 10 mass stations per singlespot measurement for ^{90}Zr_{2}O (2.0 s dwell time per cycle), ^{204}Pb (20 s), mass 204.04091 (background, 20 s), ^{206}Pb (15 s), ^{207}Pb (40 s), ^{208}Pb (5 s), ^{238}U (5 s), ThO (2 s), UO (2 s), and UO_{2} (2 s).
Figure 3 shows the timeresolved SEM counts of one representative spot measurement for each dataset. Sidebyside comparison of these two datasets reveals some interesting similarities and differences. All of the highintensity signals exhibit clear transient behaviour, which is caused by changes in oxygen availability that occur during primary ion bombardment (Magee et al., 2017). The transience of the individual SEM signals biases the isotopic ratios. For example, there are 109 s between the ^{204}Pb and ^{208}Pb measurements in each SHRIMP cycle. During these 109 s, the ^{208}Pb signal drops on average by 2 %, resulting in an equivalent bias of the ^{204}Pb $/$ ^{208}Pb ratio (Sect. 9). A similar drop per cycle is observed for the Cameca data.
However, there is a key difference between the Cameca and SHRIMP datasets. For the Cameca data, the U and Pb signals both drift in the same direction, resulting in positive slopes for the example of Fig. 3. But for the SHRIMP data, the U and Pb drift in opposite directions: the U signal exhibits an increase in sensitivity with time, whereas the Pb signals decrease in intensity with time. This marked difference in behaviour between the two instruments reflects a difference in their design, causing a difference in the energy window of the secondary ions analysed (Ireland and Williams, 2003).
For Faraday detectors, the withinspot drift correction uses a generalised linear model with a lognormal link function (Eq. 16). For SEM data, it uses a Poisson link function (Eq. 17). In either case, the model enforces strictly positive isotopic abundances. Isotopes of the same element (such as ${}^{\mathrm{204}\left\mathrm{6}\right\mathrm{7}\mathrm{8}}$Pb) have the same slope parameter but different intercepts. Isotopes of different elements are free to have different slopes (Fig. 4).
Figure 5 applies another loglinear function (Eq. 19) to model the withinspot fractionation of SHRIMP Temora2 spot 11. This function models the driftcorrected log ratios as a linear function of analysis time. The slopes of the loglinear functions are a function of the withinspot fractionation between the numerator and denominator elements. Because there is no fractionation between two isotopes of the same element, the slope of the Pb $/$ Pb ratios is zero.
The ability of logratio statistics to avoid negative ratios is apparent from the first panel of Fig. 5. Even though some of the the backgroundcorrected ratio signals are zero or negative (because the background exceeded the signal), the generalised linear fit is strictly positive. The natural ability of compositional data analysis to rule out negative ratios avoids many problems further down the data processing chain.
The righthand side of Fig. 5 maps the four (log) ratios back to five equivalent raw signals (one for each isotope). The last two panels of the figure show how the logratio approach manages to effectively capture subtle fluctuations of the U and UO signal intensities.
The logratio intercepts obtained by Eq. (19) form a linear array of calibration data (Eq. 22). Figure 6a fits a straight line through these points using the linear regression algorithm of York et al. (2004). Alternatively, instead of fitting a calibration line through the logratio intercepts (τ=0, Fig. 6a), it is also possible to interpolate or extrapolate the logratio composition to any other point in time. For example, the green ellipses in Fig. 6b show the inferred logratio compositions at 544 s (i.e. τ=544), which represents the midpoint of the analyses. The slope of this calibration line is notably different than that obtained by fitting a line through the compositions at 0 s. This change in slope reflects the different mechanisms that are responsible for elemental fractionation within and between SIMS spots.
Figure 7 applies the Pb $/$ U calibration (Eq. 23) to 91500 zircon using the Temora2 data for calibration. It shows only the purely random errors, i.e. ignoring the uncertainty of the calibration. Including the calibration errors not only inflates the uncertainties, but also causes interspot error correlations (Fig. 1). To demonstrate this phenomenon, let us revisit the Cameca data and swap the sample and reference materials around. Thus, we use 91500 for the calibration curve and Temora2 as a sample (Fig. 8). Table 2 shows the uncertainty budget of four selected aliquots from this sample.
Propagating the calibration curve uncertainty increases the error estimates (Table 2) by different amounts for different spots. For aliquot c, which is located immediately below the mean of the 91500 data, the calibration uncertainty only mildly increases the standard error from 0.30 % to 0.36 %. However, for aliquot a, which is horizontally offset from the mean of the calibration data, the systematic calibration uncertainty more than doubles the standard error from 0.32 % to 0.70 %. The calibration error also causes the standard errors of the various aliquots to be correlated with each other. For example, the total uncertainties of aliquots a and b are positively correlated ($r[a,b]=\mathrm{0.62}$) because their UO_{2} $/$ U ratios are both offset from the mean of the calibration in the same direction. In contrast, the uncertainties of aliquots a and d are negatively correlated ($r[a,d]=\mathrm{0.34}$) because they are offset in opposite directions from the mean of the calibration data.
The interspot error correlations are important when calculating
weighted means (Vermeesch, 2015) and isochrons. Taking into
account the full covariance structure of the data affects both the
accuracy and the precision of any derived age information. For
example, the conventional weighted mean can be replaced with a matrix
expression that accounts for correlated uncertainties
(Eq. 92 of Vermeesch, 2015). Applying this modified
algorithm to the positively correlated aliquots a and b of
Table 2 changes the weighted mean from −2.6922
to −2.6854 and increases the standard error of that mean from 0.37 %
to 0.44 %. Applying the same calculation to the negatively correlated
aliquots a and d changes their weighted mean from −2.7083 to
−2.7076 and reduces uncertainty from 0.43 % to 0.35 %. To take
full advantage of the covariance matrix will require the development
of a new generation of highlevel data reduction software. For
example, future versions of IsoplotR
(Vermeesch, 2018a)
will be modified to handle these rich data structures. A
comprehensive discussion of this topic falls outside the scope of this
paper.
R
package simplex
The SIMS data processing algorithm presented in this paper is implemented in an R
package called simplex
. The programme
can be run in three modes: online, offline, and from the command
line. The online version can be accessed at
https://isoplotr.es.ucl.ac.uk/simplex/ (last access: 16 August 2022). It contains three example U–Pb datasets, including the two datasets used in this paper, plus a Cameca monazite U–Th–Pb dataset.
The simplex
package currently accepts raw data as Cameca .asc
and SHRIMP .op
and .pd
files. Support for SHRIMP
.xml
files will be added later. The online version is a good
place to try the look and feel of the software. However, it is
probably not the most practical way to process lots of large data
files. For a more responsive user experience, simplex
can
also be run natively on any operating system (Windows, Mac OS, or
Linux). To this end, the user needs to install R
on their
system (see https://rproject.org/ for details, last access: 16 August 2022). Within
R
, the simplex
package can be installed from the
GitHub
codesharing platform using the remotes
package by entering the following commands at the console.
install.packages("remotes") remotes::install_github("pvermees/simplex")
Once installed, the simplex
graphical user interface (GUI) can
be started by entering the following command at the console.
simplex::simplex()
A third and final way to use simplex
is from the command
line. This allows advanced users to create automation scripts and
extend the functionality of the package. The simplex
package comes with
an extensive API (Application Programming Interface) of fully
documented user functions. An overview of all these functions can be
obtained by typing the following command at the console.
help(package="simplex")
This paper introduced a new algorithm for SIMS U–Pb geochronology, in which raw mass spectrometer signals are processed using a combination of logratio analysis and Poisson counting statistics. In contrast with existing data reduction protocols, which handle each aliquot of an analytical sequence separately, the new algorithm simultaneously processes all of them in parallel. It thereby produces an internally consistent set of isotopic ratios and their associated covariance matrix. This covariance matrix is a rich source of information that captures both random and systematic uncertainties, including interspot error correlations that have hitherto been ignored in geochronology.
The example data of Sect. 12 showed that these
interspot error correlations can be either positive or negative
(see also Vermeesch, 2015; McLean et al., 2016). Ignoring them affects
both the accuracy and precision of highend data processing steps such
as isochron regression and concordia age calculation. Unfortunately,
existing postprocessing software such as Isoplot
(Ludwig, 2003) and IsoplotR
(Vermeesch, 2018a) does
not yet handle interspot error correlations. Further work is needed
to extend these codes and take full advantage of the new
algorithm. IsoplotR
was designed with such future upgrades in
mind: its input window contains a large number of spare columns that
will accommodate covariance matrices in a future update. Once the
aforementioned highend data reduction calculations have been updated,
it will be possible to fully quantify the gain in precision and
accuracy of the new algorithm compared to the previous generation of
SIMS data reduction software.
The data reduction principles laid out in this paper are applicable
not only to U–Pb geochronology, but also to other SIMS applications
such as stable isotope analysis. In fact, simplex
already
handles such data for multicollector Cameca instruments. It is worth
mentioning that the stable isotope functionality can also be used to
correct ^{207}Pb $/$ ^{206}Pb ratio
measurements for massdependent isotope fractionation, as was briefly
discussed in Sect. 10. Future updates of the
massdependent fractionation correction will also address the
overcounted background problem that was mentioned in
Sect. 6.
Besides U–Pb geochronology and stable isotopes, the new data
reduction paradigm can also be adapted to other chronometers and other
mass spectrometer designs, such as thermal ionisation mass
spectrometry (TIMS, Connelly et al., 2021), noble gas mass
spectrometry (Vermeesch, 2015), and LAICPMS (McLean et al., 2016).
The simplex
package already includes a function to export data to
IsoplotR
. Adding similar functionality to other data
processing software will improve geochronologists' ability to
integrate multiple datasets whilst keeping track of systematic
uncertainties, including those associated with reference materials and
decay constants.
This section provides further algorithmic details for the new U–Pb data processing workflow. It assumes that ions are recorded on SEM detectors, which is by far the most common configuration. The case of Faraday collectors is similar and, in fact, simpler.
Withinspot drift is modelled using a loglinear function (Eq. 17) with a distinct intercept (α_{x}) for each ion channel (x) and shared slopes (γ_{X}) between isotopes of the same element (X). To illustrate this concept, consider the case of ${}^{\mathrm{204}\left\mathrm{6}\right\mathrm{7}}$Pb (inclusion of ^{208}Pb is a trivial extension). Let ${\widehat{n}}_{x}^{i}$ be the timedependent parameter of the shot noise for ^{20x}Pb, where $x\in \mathit{\{}\mathrm{4},\mathrm{6},\mathrm{7}\mathit{\}}$:
Then the loglikelihood function of the parameters given the data is defined as
where N represents the number of cycles. The parameters ${\mathit{\alpha}}_{\mathit{\left\{}x\mathit{\right\}}}=\mathit{\{}{\mathit{\alpha}}_{\mathrm{4}},{\mathit{\alpha}}_{\mathrm{6}},{\mathit{\alpha}}_{\mathrm{7}}\mathit{\}}$ and γ_{Pb} are estimated by maximising ℒℒ_{d} with respect to them. Only γ_{Pb} is used in subsequent calculations. The intercepts α_{{x}} are discarded.
The next step of the data reduction extracts log ratios from the raw data using a loglinear model that is similar to the withinspot drift correction (Eq. 19). Here, in contrast with the drift correction, the intercepts are just as important as the slopes. For isochemical ratios such as ^{207}Pb $/$ ^{206}Pb, the slope of the driftcorrected log ratios is zero and we only need to estimate the intercept. For multichemical ratios such as ^{238}U $/$ ^{206}Pb, both the slope and the intercept are nonzero. In order to keep track of covariances, it is useful to process all the isotopes together using a common denominator. For example, using ^{206}Pb (6) as a common denominator and ^{204}Pb (4), ^{207}Pb (7), ^{238}U (u), and UO (o) as numerators,
where X stands for Pb if $x\in \mathit{\{}\mathrm{4},\mathrm{6},\mathrm{7}\mathit{\}}$, for U if x=u, and for UO if x=o. Then the normalised ion counts are given by
and
for $y\in \mathit{\{}\mathrm{4},\mathrm{7},u,o\mathit{\}}$, with
in which $z\in \mathit{\{}\mathrm{4},\mathrm{6},\mathrm{7},u,o,b\mathit{\}}$. Then the loglikelihood is calculated as
where ${\mathit{\gamma}}_{\mathrm{6}}^{\mathrm{4}}={\mathit{\gamma}}_{\mathrm{6}}^{\mathrm{7}}=\mathrm{0}$ because there is no elemental fractionation between the Pb isotopes. From the log ratios with a common denominator, it is easy to derive any other log ratio:
One of the main advantages of the new data reduction method is its ability to keep track of the full covariance structure of the data, including interspot error correlations. This ability is derived from the fact that all parameters are derived by the method of maximum likelihood, which stipulates that the approximate covariance matrix of any set of estimated parameters can be obtained by inverting the negative matrix of second derivatives (i.e. the Hessian matrix) of the loglikelihood function with respect to said parameters.
For example, to estimate the covariance matrix of the logratio slopes and intercepts for a singlespot analysis, the Hessian is a 6×6 matrix that includes the second derivatives of ℒℒ_{l} with respect to ${\mathit{\beta}}_{\mathrm{6}}^{\mathrm{4}}$, ${\mathit{\beta}}_{\mathrm{6}}^{\mathrm{7}}$, ${\mathit{\beta}}_{\mathrm{6}}^{\mathrm{u}}$, ${\mathit{\beta}}_{\mathrm{6}}^{\mathrm{o}}$, ${\mathit{\gamma}}_{\mathrm{6}}^{\mathrm{u}}$, and ${\mathit{\gamma}}_{\mathrm{6}}^{\mathrm{o}}$. Computing this matrix is tedious to do by hand but straightforward to do numerically.
Given the covariance matrix of the log ratios, subsequent data reduction steps propagate the analytical uncertainties by a conventional firstorder Taylor approximation. Thus, if y=f(x), then
where J_{f} is the Jacobian matrix (and ${\mathbf{J}}_{f}^{\mathrm{T}}$ its transpose) of partial derivatives of f with respect to x. For example, to estimate the m×m covariance matrix of m fractionationcorrected ^{206}Pb $/$ ^{238}U ratios, error propagation of Eq. (23) would involve an $m\times (\mathrm{2}m+\mathrm{3})$ Jacobian matrix and a $(\mathrm{2}m+\mathrm{3})\times (\mathrm{2}m+\mathrm{3})$ covariance matrix containing the uncertainties of A, B, and $\mathrm{ln}\phantom{\rule{0.125em}{0ex}}{\left[\frac{{}^{\mathrm{206}}\mathrm{Pb}}{{}^{\mathrm{238}}\mathrm{U}}\right]}_{\mathrm{a}}^{\mathrm{r}}$, as well as ${}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{u}^{\mathrm{6}}\left(j\right)$ and ${}^{\mathit{\tau}}\phantom{\rule{0.125em}{0ex}}{\mathit{\beta}}_{u}^{o}\left(j\right)$ (for j from 1 to m).
The source code, installation instructions, and example datasets for simplex
can be accessed at https://github.com/pvermees/simplex/ (last access: 16 August 2022; https://doi.org/10.5281/zenodo.6954203, Vermeesch, 2022).
The author is a member of the editorial board of Geochronology.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The idea for this work was born during an academic visit of the author
to the Institute of Geology and Geophysics (IGG) at the Chinese
Academy of Sciences (CAS) in Beijing. With no prior experience in
handling SIMS data, the author relied on the expertise of Yang Li
and LiGuang Wu (IGGCAS) to introduce him to the world of Cameca data processing, as well as of Simon Bodorkos, Charles Magee, and Andrew Cross (Geoscience Australia) for SHRIMP data processing. The test data were kindly shared by Yang Li and Simon Bodorkos, who also provided detailed feedback on the paper prior to submission. Software development for the simplex
data reduction programme was supported by NERC standard grant
NE/T001518/1 (“Beyond Isoplot
”), which aims to develop an
“ecosystem” of interconnected geochronological data processing
software. The graphical user interface for simplex
makes
extensive use of the shinylight
and dataentrygrid
packages developed by software engineer Tim Band, who is employed on
this NERC grant.
This research has been supported by the Natural Environment Research Council (grant no. NE/T001518/1).
This paper was edited by Noah M. McLean and reviewed by Morgan Williams and Nicole Rayner.
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 Abstract
 Introduction
 Accuracy
 Precision
 Systematic errors
 U–Pb geochronology as a compositional data problem
 Background correction
 Dealing with count data
 Deadtime correction
 Withinspot drift correction
 Fractionation
 U–Pb age calculation
 Examples

The
R
packagesimplex
 Discussion
 Appendix A
 Code and data availability
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
compositional data, which means that the relative abundances of ^{204}Pb, ^{206}Pb, ^{207}Pb, and ^{238}Pb are processed within a tetrahedral data space or
simplex. The new method is implemented in an eponymous computer programme that is compatible with the two dominant types of SIMS instruments.
 Abstract
 Introduction
 Accuracy
 Precision
 Systematic errors
 U–Pb geochronology as a compositional data problem
 Background correction
 Dealing with count data
 Deadtime correction
 Withinspot drift correction
 Fractionation
 U–Pb age calculation
 Examples

The
R
packagesimplex
 Discussion
 Appendix A
 Code and data availability
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References