U-Th-He as a ternary system

The age-equations 2, 3 and 5 do not specify the measurement units of [U], [Th] and [He]. These can be expressed in moles or moles/g, but they can also be non-dimensionalized by normalization to a constant sum: [U'] $\equiv$ [U]/([U]+[Th]+[He]), [Th'] $\equiv$ [Th]/([U]+[Th]+[He]) and [He'] $\equiv$ [He]/([U]+[Th]+[He]) so that [U'] + [Th'] + [He'] = 1. Therefore, U, Th and He form a ternary system, can be plotted on a ternary diagram, and are subject to the peculiar mathematics of the ternary dataspace. In a three-component system (A+B+C=1), increasing one component (e.g., A) causes a decrease in the two other components (B and C). Another consequence of so-called data closure is that the arithmetic mean of compositional data has no physical meaning (Weltje, 2002).

Plotting the (U-Th)/He age equation on ternary diagrams

Following the nomenclature of Aitchison (1986), the ternary diagram is a 2-simplex ($\Delta_2$) (also see Weltje, 2002). The very fact that it is possible to plot ternary data on a two-dimensional sheet of paper tells us that the sample space really has only two, and not three dimensions. As a solution to the compositional data problem, Aitchison (1986) suggested to transform the data from $\Delta_2$ to $\mathbb{R}^2$ using the logratio transformation. After performing the desired (``traditional'') statistical analysis on the transformed data in $\mathbb{R}^2$, the results can be transformed back to $\Delta_2$ using the inverse logratio transformation (Figure 3). Implementation details about the logratio transformation will be given in Section 4.

Ternary diagrams and logratio plots are useful tools for visualizing U-Th-He data and the (U-Th)/He age equation. Thus, it can be shown that the linear age-equation is accurate to better than 1% for ages up to 100 Ma (Figure 4.a) whereas the equation of Meesters and Dunai (2005) reaches the same accuracy at 1 Ga (Figure 4.b). Figure 4.c represents a warning against applying the $\alpha$-ejection correction after, rather than before the age-calculation. This causes a partial ``linearization'' of the age-equation and results in a loss of accuracy. For example, dividing an uncorrected (U-Th)/He age by an $\alpha$-retention factor F$_t$ of 0.7 results in a misfit that is 30% of the linear age equation misfit. To take full advantage of the accuracy of the exact age equation, one must divide [He] by F$_t$ before calculating the (U-Th)/He age.

Figure 3: Ternary U-Th-He data can be mapped to two-dimensional logratio-space without loss of information.

Figure 4: (a) Relative misfit of the linear approximation (Equation 3) to the exact age equation (Equation 2); (b) misfit of the direct solution of Meesters and Dunai (2005) (Equation 5); (c) misfit caused by making the $\alpha$-ejection correction after the age calculation (with F$_t$=0.7).

(a) (b) (c)

The central age

The logratio transformation is useful for more than just the purpose of visualization. It provides a fourth and arguably best way to calculate the average age of a population of single-grain (U-Th)/He measurements. The central age is calculated from the ``average'' U-Th-He composition of the dataset, where ``average'' is defined as the geometric mean of the single grain U, Th and He measurements. The geometric mean of compositional data equals the arithmetic mean of logratio transformed data.

How important is the difference between the arithmetic mean age and the central age? To simplify this question, consider the special case of a sample with only one radioactive parent, say Th. Assume that W = ln([He]/[Th]) is normally distributed with mean $\mu$ and standard deviation $\sigma$. Using the linearized age equation for clarity, the central age t$_c$ is given by:

$$t_c = C ~ e^{\mu}$$ (7)

with C = 1/(6 $\lambda_{232}$) for Th. Using the first raw moment of the lognormal distribution (Aitchison and Brown, 1957), the arithmetic mean age t$_m$ is:

$$t_m = C ~ e^{\mu + \sigma^2/2}$$ (8)

so that the relative difference between t$_m$ and t$_c$ is:

$$\frac{t_m - t_c}{t_c} = e^{\sigma^2/2} - 1$$ (9)

Using the second central moment of the lognormal distribution (Aitchison and Brown, 1957), the variance of the single-grain ages is given by:

$$\sigma_t^2 = C^2 ~ \left(e^{\sigma^2}-1\right)e^{2\mu + \sigma^2}$$ (10)

Plotting $(t_m - t_c)/t_c$ versus $\sigma_t$/t$_c$ reveals that the central age is systematically younger than the mean age. Fortunately, the difference is small. For example, for a typical external reproducibility of $\sim$ 25% (e.g. $\sigma_t/t$ = 11% for Stock et al., 2006), the expected difference is $<$ 1% (Figure 5). Finally, it is interesting to note that the geometric mean of the log-normal distribution equals its median. Therefore, the central age asymptotically converges to the median age. However, typical numbers of replicate analyses are not sufficient for this approach to be truely beneficial.

Figure 5: Expected difference between the mean age t$_c$ and the central age t$_m$ plotted against the relative spread of the single-grain ages.

Application to HF-treated Naxos apatites

We now return to the sample of HF-treated inclusion-bearing apatites from Naxos that was previously used to illustrate the pooled and isochron age (Figures 1 and 2). The raw data and the different steps of the central age calculation are given in Table 1. We will now walk through the different parts (labeled a, b and c) of this table.

(a) The upper left part of Table 1 lists the U, Th and He abundances of 11 single-grain analyses. Their respective single-grain ages (t) were calculated using the exact age equation, although the linear age approximation (Equation 3) is accurate to better than 0.1% for such young ages (Figure 4.a). The pooled U, Th and He abundances are obtained by simple summation of the constituent grains. Note that the helium abundances are corrected for $\alpha$-ejection prior to being pooled. A nominal $\sigma$=15% statistical uncertainty is associated with F$_t$, assuming randomly distributed mineral inclusions (Vermeesch et al., 2007). The pooled abundances were normalized to unity to facilitate comparison with the geometric mean composition (see below).

(b) To calculate the isochron age, the abundances are first rescaled to units of concentration (e.g. in nmol/g). This removes the bias towards large grains, which can dominate the pooled age calculation. The $\alpha$-production rate P is given by Equation 4. The linear regression (Figure 2) was done using Isochron 3.0 (Ludwig, 2003), yielding a slope of 12.0 $\pm$ 4.2 Ma with an intercept of -0.05 $\pm$ 0.45 nmol/g He.

(c) Central ages are somewhat more complicated to calculate than arithmetic mean ages, pooled ages or isochron ages. Therefore, these calculations will be discussed in more detail. First, transform each of the n single grain analyses to logratio space:

$$V_i = ln\left(\frac{[U_i]}{[He_i]}\right),~ W_i = ln\left(\frac{[Th_i]}{[He_i]}\right)$$ (11)

For i = 1,...,n. Note that this transformation can be done irrespective of whether the U, Th and He measurements are expressed in abundance units or in units of concentration. Following standard error propagation, the (co)variances of these quantities are estimated by:

$$\hat{\sigma}_{V_i}^2 = \left(\frac{\sigma_{U_i}}{U_i}\right)^2 + ... ...i}}{He_i}\right)^2,~ cov_{V_i,W_i} = \left(\frac{\sigma_{He_i}}{He_i}\right)^2$$ (12)

Next, calculate the arithmetic mean of the logratio transformed data:

$$\overline{V} = \frac{1}{n} \sum_{i=1}^n V_i,~ \overline{W} = \frac{1}{n} \sum_{i=1}^n W_i,~$$ (13)

With the following (co)variances:

$$\hat{\sigma}_{\overline{V}}^2 = \frac{1}{n^2} \sum_{i=1}^n \hat{\... ...2,~ cov_{\overline{V},\overline{W}} = \frac{1}{n^2} \sum_{i=1}^n cov_{V_i,W_i}$$ (14)

Note that equation 14 only propagates the internal (i.e. analytical) uncertainty, and not the external error. Single grain (U-Th)/He ages tend to suffer from overdispersion with respect to the formal analytical precision for a number of reasons (Fitzgerald et al., 2006; Vermeesch et al., 2007). Therefore, it may be better to use an alternative equation propagating the external error:

$$\hat{\sigma}_{\overline{V}}^2 = \frac{\sum_{i=1}^n (V_i - \overli... ...line{W}} = \frac{\sum_{i=1}^n (V_i - \overline{V})(W_i - \overline{W})}{n(n-1)}$$ (15)

Error-weighting can be done by trivial generalizations of equations 13, 14 and 15, which are implemented in the web-calculator. The geometric mean composition is given by the inverse logratio transformation (Aitchison, 1986; Weltje, 2002):

$$\overline{[U]} = \frac{e^{\overline{V}}}{e^{\overline{V}}+e^{\ove... ...rline{W}}+1},~ \overline{[He]} = \frac{1}{e^{\overline{V}}+e^{\overline{W}}+1}$$ (16)

With variances:

$$\left( \begin{array}{c} \hat{\sigma}_{\overline{U}}^2 \\ \hat... ...{\overline{W}}^2 \\ cov_{\overline{V},\overline{W}} \\ \end{array} \right)$$ (17)

where

$$a = \frac{e^{\overline{V}}(e^{\overline{W}}+1)} {(e^{\overline{V}}+e^{\overline{W}}+1)^2} b = \frac{-e^{\overline{V}+\overline{W}}} {(e^{\overline{V}}+e^{\overline{W}}+1)^2}$$

$$c = \frac{e^{\overline{W}}(e^{\overline{V}}+1)} {(e^{\overline{V}}+e^{\overline{W}}+1)^2} d = \frac{-e^{\overline{V}}} {(e^{\overline{V}}+e^{\overline{W}}+1)^2}$$

$$e = \frac{-e^{\overline{W}}} {(e^{\overline{V}}+e^{\overline{W}}+1)^2} ~$$

The central age is then simply calculated by plugging $\overline{[U]}$, $\overline{[Th]}$ and $\overline{[He]}$ and their uncertainties into equation 2, 3 or 5.

As predicted (Figure 5), the arithmetic mean age is older than the central age. There is less than 2% disagreement between the arithmetic mean age ($\sim$ 11.58 Ma) and the central age ($\sim$ 11.38 Ma), and 7% difference between the pooled age ($\sim$ 11.28 Ma) and the isochron age ($\sim$ 12.0 Ma).

Figure 6: (a) Ternary diagram of the Naxos data (Table 1). (b) The same data plotted in logratio-space. Error ellipes are 2$\sigma$.

(a) (b)