Although rigorous uncertainty reporting on (U–Th)

The (U–Th)

One current challenge to comprehensive uncertainty propagation is that,
although individual laboratories have derived the methods for propagating
uncertainty components into single-grain (U–Th)

Comprehensive uncertainty accounting in individual (U–Th)

Here we explain how analytical and

The currently quantifiable uncertainties in single-grain (U–Th)

In the (U–Th)

Because of the kinetic energy associated with alpha decay, individual alpha
particles (i.e.,

For this work, it is assumed that the amount and uncertainty of each nuclide
have been constrained. The natural U isotopic ratio (137.818

Uncertainty and systematic error in

Several additional sources of dispersion in (U–Th)

Here, (U–Th)

The initial value for iterative age calculation is obtained by calculating
an approximated noniterative solution of the (U–Th)

Using the resulting date approximation as an initial guess (

Here we provide a method of calculating date uncertainty using linear
propagation of uncertainty. We apply the general formula for uncertainty
propagation through a function

Applying the uncertainty propagation equation to the (U–Th)

While solving the (U–Th)

Monte Carlo uncertainty propagation is based on the approach of combining
the uncertainty in measured parameters with any given probability
distribution (including non-Gaussian distributions as may be caused by
compositional zoning; Hourigan et
al., 2005) by randomly sampling each distribution a large number of times
and propagating those randomly generated parameters through some function of
interest (Eq. 2; Fig. 1). This method yields a probability
density histogram that describes the true uncertainty to arbitrary precision
depending on the number of simulations run
(Anderson, 1976; Possolo and Iyer,
2017). As such, the application of Monte Carlo techniques is mathematically
straightforward, in this case requiring no knowledge beyond that required to
calculate a (U–Th)

Here, Monte Carlo uncertainty modeling of (U–Th)

A conceptual diagram of Monte Carlo uncertainty modeling for the
(U–Th)

The number of Monte Carlo simulations dictates the precision of the results
because Monte Carlo analysis is a numerical approximation of uncertainty
(e.g., the lower panels in Fig. 1 become progressively smoother with an
increasing number of simulations). Therefore, separate from the probability
distribution describing date uncertainty, there is a predictable level of
variation in uncertainty estimates and other parameters describing the
probability distribution (e.g., its mean) given a certain number of total
Monte Carlo simulations (Wübbeler et al., 2010).
Specifically, the standard error of the standard deviation of a Monte Carlo
model is dependent on the uncertainty in the value itself and the number of
simulations:

Example HeCalc inputs

In this section we describe the implementation of the above methods of date
and uncertainty calculation in the new HeCalc software
(Martin, 2022). For ease of access and to best provide this
software as a resource to the (U–Th)

The input for HeCalc is designed to be straightforward and flexible (Table 1). Input files may be in Excel (.xls/.xlsx), comma-separated value (.csv),
or tab-delimited text (.txt) format. In addition to data input through a
file, HeCalc users may manually input values to calculate a date and
uncertainty for a single set of data by clicking on the “Manual” tab. If
importing data through a file, the file must contain columns for sample
name, U, Th, Sm, He, and all

Example HeCalc outputs produced by Table 1 inputs.

The order in which these columns appear is unimportant as long as the uncertainty associated with each value follows that value. Extraneous columns with differing headers also will not interfere with the code's execution. Additionally, if an input Excel file has multiple sheets, the first sheet will be read in by default. If this sheet does not contain the required column headers, the program will ask for the name of the sheet to use instead. In this way, HeCalc ideally allows for input of any given lab's standard data reduction spreadsheet or other typical data product with no or minimal alteration, allowing it to be integrated seamlessly into a lab's existing workflow.

In addition to data input, several further options are provided. The number
of decimals included in the output is determined by the user (this option
affects only output and does not impact the statistical aspects of the
code). The user can also select whether to perform linear uncertainty
propagation, Monte Carlo uncertainty propagation, both, or neither. If Monte
Carlo uncertainty propagation is selected, the desired precision of the mean
is specified in percent as described above. In practice, the precision of
the mean date need be no better than the number of significant figures
present the in data; for common (U–Th)

There are two main outputs from HeCalc: the results of the date calculation
and uncertainty propagation and the histograms of the Monte Carlo results
for each sample (Table 2). At a minimum, the sample name, raw date, and
corrected date are saved to an Excel sheet titled “Uncertainty Output”
that includes a header with the input file's name and directory. The raw and
corrected dates in these columns are calculated using each exact input value
(e.g., mol

Histograms of percent relative uncertainty in

If the user chooses to include histograms in the output, an Excel sheet
titled “Histogram Output” is added to the workbook, with columns for the
center of each histogram bin (i.e., the individual intervals in the
histogram) and number of simulations in that bin as

Here we use the methods described above to calculate the dates and
uncertainties for a compilation of real apatite and zircon (U–Th)

To assess the uncertainty budget in real (U–Th)

Histograms of percent relative uncertainty for corrected (U–Th)

Percent uncertainties in absolute amounts of

We analyze the uncertainty in these data with and without propagating

Percent uncertainty in corrected (U–Th)

Histograms of skew for

We also analyze the compilation of real data in Fig. 2 for skew in Monte
Carlo-generated date probability distributions (i.e., asymmetric
uncertainty). Skew refers to the extent of asymmetry in the “tails” of a
distribution (Figs. 1, D1). “Skewness” is a statistical concept that
strictly refers to the third standardized moment of a population, which is a
unitless and generally nonintuitive metric. Here we additionally report a
percent skew to more intuitively convey the distribution asymmetry,
calculated by taking the percent difference between the positive and
negative 68 % confidence intervals with respect to the date. This
asymmetry would most accurately be reported as separate positive and
negative uncertainty values referring to the 68 % confidence interval
rather than the more typical 1

In our data compilation, positive skew is common and can be significant
(Table 3). This is consistent with analysis of theoretical data revealing that
skew increases with relative input uncertainty and varies with age (Appendix D; Figs. D1–D3). For the real dataset, with only analytical uncertainty
included, the median skew in apatites and zircons is 0.050 and 0.020 (or
4.4 % and 3.2 % skew), respectively. The inclusion of

General practice in (U–Th)

Histograms of the percent difference between averaged
Monte Carlo-derived confidence intervals and linear uncertainty propagation
for

Finally, we compare the uncertainties derived from linear uncertainty
propagation with the averaged 68 % confidence intervals from Monte Carlo
propagation for the compiled dataset. For nearly all analyses, the
uncertainties yielded by the two methods are within 1 % of each other,
regardless of the amount of

Here we publish fully traceable end-to-end calculations of uncertainty in
(U–Th)

For the compiled dataset, the asymmetry in the 68 % confidence interval
can be significant, especially for dates with less precise input
uncertainty. With 2 % uncertainty included in

The methods presented here allow for more rigorous inter-laboratory data
comparisons and retrospective data analyses by providing a consistent means
of quantifying the uncertainty budget of a given (U–Th)

Here we provide a set of equations that allows for propagation of directly quantified of

Negative dates are permitted in the probability distributions produced by
HeCalc; this is because the input distributions are presumed to be Gaussian,
meaning that if the input variables have high relative errors, negative
molar amounts of U, Th, Sm, and He are possible. This behavior is formally
correct for Gaussian uncertainties, albeit nonphysical. For low count rates
associated with high relative uncertainty, a Poisson distribution (rather
than Gaussian distribution) is appropriate and would prevent negative input
values. However, high relative input uncertainties are generally due to a
measurement being near or below background rather than low count rates for which
the underlying poisson distribution of the data is not well-approximated by
a Gaussian. As a result, there are potential instances of negative molar
amounts included in the Monte Carlo calculations. In some rare instances
when a negative amount of a given parent nuclide is produced in the
generation of random data, the (U–Th)

We examined the overall behavior of date uncertainty from 0 to 4.6 Ga as a
function of relative input uncertainties of 1 %, 5 %, and 20 % for

For individual input uncertainties, at young dates the input and output
relative uncertainties are similar. If all uncertainty is in the

Corrected (U–Th)

Corrected (U–Th)

The relative date uncertainty decreases with increasing absolute date for
constant relative input uncertainties. For example, while uncertainty in

Decreasing uncertainty with increasing date is also observed for multiple
input uncertainties. Figure C2 illustrates examples of combining
uncertainties with differing magnitudes in quadrature. At zero age, the
uncertainty in the date introduced by each input parameter combines roughly
in quadrature to provide the uncertainty in the date. For example, at zero
age, a 5 % uncertainty in all parameters (

The magnitude of skew correlates directly with the magnitude of input
uncertainty (Figs. D1–D2). For low percent input uncertainties in all
parameters, the magnitude of skew is low. For example, uncertainties of
1 % for all inputs yield 0.06 skewness for dates from 0 to 4.6 Ga (Fig. D2a–c, top panels). Only when the input uncertainties are larger does the
effect of skew on the dates become substantial (Fig. D2a–c, middle and
bottom panels). In the case of larger uncertainty in

When uncertainty is included in multiple input parameters, the overall skew is a combination of the skew resulting from individual input uncertainties (Fig. D3). Unlike date uncertainty, which combines individual inputs in quadrature, the combination of skew from individual inputs does not follow an easily predictable trend.

An illustration of how differing uncertainty affects the skew of
date probability distributions for inputs yielding a date of 15.1 Ma
(assuming a typical apatite Th

Skew for dates from 0 to 4.6 Ga with input uncertainty in only one parameter and all others held at zero.

To compare linear and Monte Carlo error propagation derived uncertainties,
we average the two 68 % confidence intervals to determine uncertainty from
both methods at the 1

As linear uncertainty propagation relies on an arithmetic calculation rather
than random sampling, this method provides predictable and repeatable
results for uncertainty calculations and is amenable to encoding in
spreadsheet programs, facilitating the inclusion of the equations provided
in Sect. 3.2 in existing spreadsheet-based workflows. However, the presence
of skew in (U–Th)

The percent error introduced by using linear uncertainty
propagation instead of Monte Carlo uncertainty propagation for dates from 0
to 4.6 Ga. Plots of percent difference between the Monte Carlo and linear uncertainty propagation results vs. corrected date for uncertainty only in

Version 1.0.1 of the HeCalc software is available at

The data used in this paper are from an anonymized compilation of samples run in the CU TRaIL at the University of Colorado Boulder. These data are dominated by independent contract samples and are published at the discretion of the individuals who paid for the analyses.

PEM, RMF, and JRM conceptualized the project; JRM curated the data; PEM performed the formal analyses; RMF and JRM acquired funding; PEM, RMF, and JRM performed the investigation; PEM developed the methodology and wrote the software; RMF provided supervision; PEM wrote the original draft, and RMF and JRM reviewed and edited the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Noah McLean for numerous and very helpful discussions while
developing HeCalc and writing this paper. HeCalc and this paper were
greatly improved following discussions at the 17th International Conference
on Thermochronology, in particular with Danny Stockli, Florian Hoffman,
Marissa Tremblay, and Kip Hodges. We appreciate helpful reviews by Ryan Ickert and an anonymous reviewer that helped to clarify and streamline this
paper. The (U–Th)

This research has been supported by the National Science Foundation (grant nos. EAR-1126991, EAR-1559306, and EAR-1920648).

This paper was edited by Brenhin Keller and reviewed by Ryan Ickert and one anonymous referee.