In practice, the PDF can never be precisely known, because that would require an exhaustive sampling of the detrital population. Therefore, we must work with estimates of the PDF based on a finite sample of detrital ages (typically tens to hundreds of ages). Besides limited data, measurement uncertainty is a second factor reducing the precision of density estimates. The most popular estimators of probability density are histograms and ``kernel density plots'' [e.g., Silverman, 1986]. Both of these methods apply some degree of ``smoothing'' to the data, either by binning them into a histogram, or by assigning a Gaussian uncertainty distribution to each measurement:
With N(t
) the normal distribution of t with mean
and standard deviation , and t and (t)
the measured ages and their respective 1 uncertainties. To
illustrate the different approaches to detrital thermochronological
density estimation, consider the degenerate case of a ``diving board''
hypsometry: all detrital grains are derived from a single elevation,
corresponding to a single ``true'' age t. The PDF of the true
ages is a deltafunction (spike at t, zero probability
elsewhere). For further simplification, all grains have identical,
Gaussian measurement uncertainties. In the following, t =
10Ma so all grains are 10Ma old and have Gaussian measurement
uncertainties of 1Ma.
Suppose we have access to an infinite number of measurement from this
detrital population. The PDF of these age measurements can then be
determined by a histogram with infinitessimal binwidth or a kernel
density estimate with infinitessimal . Note that the PDF of
the measurements is not the same as the PDF of the underlying
ages (Figure 3.a). Unless we deconvolve
the measurement uncertainties, the measurement distribution will
always be a ``smoothed'' version of the ``true'' age distribution. In
our toy example, the age distribution is a deltafunction at 10
Ma, whereas the measurement distribution is a Gaussian
distribution with mean 10 Ma and standard deviation 1 Ma (Figure
3.a).

Given a set of age data, the Gaussian kernel density estimator stacks
a bell curve on top of each measurement (Equation 2 and
Figure 3.b). Repeating this for a large number
of measurements drawn from our ``diving board'' hypsometry yields a
Gaussian distribution with mean 10 Ma and standard deviation
=
. Thus,
is ``doublesmoothed'': once by the measurement uncertainties, and a
second time by the construction of the kernel density estimator. The
amount of additional smoothing depends on the parameter .
Although it can be shown that = 0.6 is an optimal value [
Silverman, 1986; Brandon, 1996], previous studies by Brewer et al. [2003], Ruhl and Hodges [2005] and Stock
et al. [2006] have used = 1, and so does Figure
3.b. Ruhl and Hodges [2005] gave this
curve the name ``Synoptic Probability Density Function'' (SPDF).
These authors distinguish between three kinds of SPDF. SPDF is
the true underlying age distribution, in the hypothetical case of
errorless measurements (dashed lines in Figure
3). In our toy example, SPDF is a delta
function. SPDF is the kernel density estimator generated by
Equation 2 (gray lines in Figure
3). Finally, SPDF effectively is the
PDF of the measurements (black lines in Figure
3). Because SPDF is only smoothed
once, whereas SPDF is smoothed twice, SPDF is a biased estimator of SPDF (Figure
3.c).
One of the requirements for the application of Gaussian kernel density estimation is that the measurement uncertainties are normally distributed. This may be a reasonable assumption for Ar/Ar thermochronology, but not necessarily for fission tracks, which are governed by a Poisson process. However, by using the logistic transform, a set of fission track data can be recast in terms of a new parameter z [Brandon, 1996], which is estimated by
where is the decay constant of U (=1.5512510a), a (zeta) calibration factor measured on an AFT age standard, g a geometric factor (=0.5), N the number of spontaneous fission tracks, N the number of induced tracks in a mica detector and the induced track density of a glass standard that was irradiated along with the sample. N and N are Poisson variables, but is normally distributed with standard error