On the visualisation of detrital age distributions

Pieter Vermeesch
Birkbeck, University of London
Malet Street, London WC1E 7HX
+44 (0)20 7679 2418
p.vermeesch [at] ucl.ac.uk


Since the development of SIMS and LA-ICP-MS technology in the 1980s and 1990s, single grain U-Pb dating of detrital zircon has quickly become the most popular technique for sedimentary provenance studies. Currently by far the most widespread method for visualising detrital age distributions is the so-called Probability Density Plot (PDP), which is calculated by summing a number of Gaussian distributions whose means and standard deviations correspond to the individual ages and their respective analytical uncertainties. Unfortunately, the PDP lacks a firm theoretical basis and can produce counter-intuitive results when data quantity (number of analyses) and/or quality (precision) is high. As a more robust alternative to the PDP, this paper proposes a standard statistical technique called Kernel Density Estimation (KDE), which also involves summing a set of Gaussian distributions, but does not explicitly take into account the analytical uncertainties. The Java-based DensityPlotter program (http://densityplotter.london-geochron.com) was developed with the aim to facilitate the adoption of KDE plots in the context of detrital geochronology. ____________________________________________________________

1. Introduction

Zircon (ZrSiO4) is a common accessory mineral in acidic igneous rocks, which form the main proto-sources of the siliciclastic sediments. It contains up to a percent of radioactive U and Th, which decay to various isotopes of lead with known half lives, forming the basis of the U-Th-Pb geochronometer. Zircon is a very durable mineral that undergoes minimal chemical alteration or mechanical abrasion. Therefore, zircon crystals can be considered time capsules carrying the igneous and metamorphic history of their proto-sources. The probability distribution of a representative sample of zircon U-Pb ages from a detrital population can serve as a characteristic fingerprint that may be used to trace the flow of sand through sediment routing systems. As a provenance tracer, zircon U-Pb data are less susceptible to winnowing effects than conventional petrographic techniques (Garzanti et al.2009). Using modern microprobe technology, it is quite easy to date, say, a hundred grains of zircon in a matter of just a few hours. Due to the robustness of zircons as a tracer of sedimentary provenance, and the relative ease of dating them, the use of detrital zircon U-Pb geochronology has truly exploded in recent years. A literature survey using the keywords ‘detrital’, ‘zircon’, and ‘provenance’ indicates that the proliferation of detrital zircon studies has followed an exponential trend, with the number of publications doubling roughly every five years over the past two decades. At present, nearly a thousand detrital zircon publications appear each year.

To address the technical and methodological issues arising from the increased popularity of the technique, a detrital zircon geochronology workshop was held immediately prior to the 2011 Goldschmidt conference in Prague. During this workshop, a concensus was reached that the statistical tools used by the community to analyse the vast amounts of U-Pb data produced by today’s SIMS and LA-ICP-MS laboratories are inadequate. Particularly the use of so-called Probability Density Plots (PDPs), which are ubiquitous in the detrital zircon geochronology literature, was identified as a cause for concern. It was noted that, in spite of its name, the PDP lacks any theoretical basis as a probability density estimator, although some maintained that it may still serve a purpose as a data visualisation tool. Galbraith (19982010) eloquently pointed out some problems with the PDP in the context of fission track and luminescence dating, but unfortunately this work was published in specialised journals and has not been picked up by the detrital zircon U-Pb dating community. Furthermore, Galbraith (1990)’s alternative for the PDP, the so-called radial plot, is not ideally suited for displaying large datasets of relatively precise zircon U-Pb ages. This paper proposes the Kernel Density Estimator (KDE), which has a similar appearance as the PDP, but is built on solid theoretical foundations, as a statistically more robust alternative to the PDP (Section 2). In order to aid in the adoption of this method by the detrital zircon U-Pb dating community, a Java-based data visualisation tool called DensityPlotter was developed for plotting detrital age distributions (Section 3).

2. KDE vs. PDP


Figure 1: (a) The ‘Kernel Density Estimator’ (KDE) is a function that stacks a Gaussian ‘bell curve’ on top of each measurement, whose standard deviation is determined by the local probability density. (b) The ‘Probability Density Plot’ (PDP) is a function that stacks a Gaussian ‘bell curve’ on top of each measurement whose standard deviation is determined by the analytical precision.

In contrast with the geochronology of igneous and metamorphic rocks, in which one seeks to date an individual geological event, detrital studies generally aim to characterise an entire population of ages by means of a large number of single grain analyses. In statistical terms, the goal of such studies is to estimate the so-called Probability Density Function (PDF), which gives the relative likelihood of the different ages in the population. By definition, the probability for a randomly selected grain to fall within a given range is then given by the integral of the PDF over this range:

P(t1 ≤ t ≤ t2) =     PDF  (t)dt

2.1. Kernel Density Estimation

A large body of statistical literature has been written on the subject of probability density estimation (e.g., Silverman1986Devroye1987). The most widely used tools are the traditional histogram, which is discrete and discontinuous, and the Kernel Density Estimator (KDE), which is a smooth and continuous alternative (Silverman1986). A KDE is produced by arranging the measurements along a line and stacking a so-called ‘kernel’ (i.e., a box, triangle, Gaussian, etc.) of a certain width (the ‘bandwidth’) on top of each of them (Figure 1.a). Given a set of n measurements xi (i=1n), the kernel density estimator can be written as follows:

                      (       )
             1  ∑n      x - xi
KDE   (x) = ---    K    ------
            nh  i=1       h

where K() is the ‘kernel’ and h is the ‘bandwidth’. Some popular choices for the kernel are (Silverman1986):

Epanechnikov: K(t) = max[    (        )     ]
    3      1 2   √ --
 0, -- 1 - -t   ∕  5
    4      5 (3)
Triangular: K(t) = max[0, 1 -∥t] (4)
Gaussian: K(t) = --1--
√2-π-e-t22 (5)

It turns out that the choice of kernel is not important (other than in determining the smoothness characteristics of the density estimate), but the bandwidth is. Picking the wrong bandwidth may lead to density estimates that are either over- or undersmoothed (Figure 2). The bandwidth can either be constant (‘kernel density estimation’), or vary according to the local density (‘adaptive kernel density estimation’). If the distribution is unimodal and smooth, the optimal bandwidth may be calculated analytically (Silverman1986):

h = 1.06σn -1∕5

where σ is the standard deviation of the data set. Most detrital age distributions, however, are not smooth or unimodal, and Equation 6 is therefore rarely appropriate, yielding oversmoothed density estimates. Adaptive kernel density estimates vary the bandwidth according to the local density. This means that in areas where data density is sparse, a large bandwidth is used, and the density estimate is smooth. But where lots of data are available, a narrower bandwidth is used, allowing the KDE to provide a high resolution estimate in those parts of the distribution. A large number of methods are available for the selection of the adaptive bandwidth. These include ‘balloon estimators’ (Loftsgaarden and Quesenberry1965), in which the bandwidth is made proportional to the density at the evaluation point (‘x’ in Equation 2), and ‘sample-point estimators’ (Breiman et al.1977), which vary the bandwidth based on the data points (‘xi’ in Equation 2).


Figure 2: The Normal distribution is just one of many possible ‘kernels’. The three columns of this figure show that the choice of kernel has a relatively minor effect on the shape of the KDE of 145 concordant detrital zircon U-Pb ages from Hong Kong. The bandwidth h is far more important than the choice of kernel. The KDEs on the middle row use the optimal bandwidth calculated according to Botev et al. (2010). Choosing a different bandwidth can lead to estimates that are either oversmoothed (top row) or undersmoothed (bottom row).

2.2. Probability Density Plots

KDEs are commonly used in many branches of Science, but not in the context of detrital geochronology. Instead, a similar-looking, but fundamentally different tool named ‘Probability Density Plot’ (PDP, Ludwig, 2003, also called Probability Density Distribution by Sircombe, 2004), has gained significant popularity. The PDP was first proposed by Hurford et al. (1984), in an attempt to account for the variable analytical precision (a.k.a. ‘heteroscedasticity’) of (detrital fission track) data. Just like the KDE, the PDP is also produced by stacking a Normal distribution on top of each measurement whose bandwidth, however, is not determined by the density, but by the analytical precision (Figure 1.b), so as to reduce the importance of imprecise measurements and to emphasize the precise measurements. Although this may seem like a sensible strategy at first glance, it may actually degrade results in certain situations, as illustrated with the following two synthetic examples.

  1. Consider a bimodal distribution of ‘true ages’ (i.e., the probability distribution of an exhaustive sample of infinitely precise measurements) where the first peak (500±50 Ma) contains one third and the second peak (2000±50 Ma) two thirds of the detrital population (Figure 3.a). Suppose that our measurements are infinitely precise. In this case, the KDE of a representative sample of single grain measurements is an unbiased estimator of the true age distribution (Figure 3.b). The PDP, however, breaks down into a sequence of ‘spikes’ (Figure 3.c).
  2. In reality, geochronological measurements are always convoluted with analytical uncertainty, making it very hard to recover the true age distribution (Carroll and Hall1988). A more realistic aim of detrital studies is to estimate the ‘measurement distribution’, which is a smoothed version of the true age distribution. As a second example, suppose that the age measurements of the aforementioned synthetic population are heteroscedastic, with the two modes having analytical uncertainties of 50Ma and 200Ma, respectively. As a result of the heteroscedasticity, the second mode will be smoothed to a greater extent than the first, resulting in a reversal of the apparent peak heights (Figure 3.d). Once again, the KDE converges to the correct solution with increasing number of age measurements (Figure 3.e). The PDP, however, oversmooths the measurement distribution and gives the wrong answer, even after 10,000 measurements (Figure 3.f). This is because the PDP ‘punishes’ the imprecise ages twice (once by sampling/measuring, and once by smoothing). The key point here is that it is not necessary to explicitly account for the analytical precision in the density estimation because Nature, and the analytical process itself, have already taken care of this.

Although cumulative distributions are an easy and effective way to circumvent the double-smoothing problem (Vermeesch2007), probability density estimates are often easier to visually interpret and are therefore preferred by many workers. This paper proposes that the KDE be used for this purpose instead of the PDP (or variants thereof, such as those proposed by Brandon1996). Part of the reason why PDPs have become so popular is that they have been implemented in user-friendly software packages such as Isoplot (Ludwig2003) and AgeDisplay (Sircombe2004). To aid in the adoption of the KDE by the geochronological community, a Java-based data visualisation tool called DensityPlotter was developed.


Figure 3: Illustration of two problems with PDPs. The first row of this figure shows how the PDP (black curves) breaks down when data quality (i.e., analytical precision) is high. Left (a): a synthetic example of a bimodal probability density function (PDF, green) of ‘true ages’, i.e. the age distribution which one would observe by measuring an infinite number of randomly sampled zircon U-Pb ages with infinite precision. Middle (b): the KDE of a sample of 117 randomly selected measurements from this distribution is an unbiased estimator of the PDF. Right (c): the PDP of the same measurements is a nonsensical set of discrete spikes. The bottom row of the figure makes the point that the PDP is biased when data quantity (i.e., sample size) is high. Left (d): the probability distribution of the measurements with finite analytical precision (blue) is a smoothed version of the PDF of the ‘true ages’ shown in (a). In this example, the extent of the smoothing is greater for the second mode, due to the lower analytical precision of the latter. Middle (e): the KDE converges to the true measurement distribution with increasing sample size n. Right (f): The PDP is a biased estimator of the measurement distribution because it ‘double-smooths’ the imprecise measurements. Bias is therefore worst for the second mode.

3. DensityPlotter

DensityPlotter is a computer program developed with the aim to facilitate the adoption of the KDE by the geochronological community. It is based on RadialPlotter which, as the name suggests, implements Rex Galbraith’s radial plots for the fission track and luminescence communities (Vermeesch2009). Both RadialPlotter and DensityPlotter were programmed in Java and are therefore perfectly platform independent. DensityPlotter can be downloaded from http://densityplotter.london-geochron.com. To calculate the optimal KDE bandwidth, the program implements an algorithm based on the diffusion equation, which can be shown to be both fast and accurate (Botev et al.2010). In addition to KDEs (and radial plots), DensityPlotter also provides functionality to plot histograms and PDPs, as well as basic mixture models (Sambridge and Compston1994Galbraith and Green1990). Data can be copied and pasted to-and-from Microsoft Excel and saved in a simple comma-separated variable (.csv) format. An example of such an input file is provided on the website. The graphical output can be saved either as bitmap or vector images, in a .png or .pdf format, respectively. Although DensityPlotter was designed to produce publication-ready figures and includes many options to customise the look and feel of the density estimates, the pdf file can also be further edited in vector-graphics applications such as Illustrator, CorelDRAW, or Inkscape. Because the analytical precision of many geochronological datasets is proportional to the age, the relative errors generally show less grain-to-grain variability than the absolute errors. To take advantage of this, DensityPlotter includes an option to log-transform the data (Brandon1996). The algorithm used for the mixture modelling is a hybrid one using both deterministic parts (taken from Galbraith and Green1990) and Markov Chain Monte Carlo (MCMC) ingredients to maximise robustness and reproducibility. The reader is referred to the Help pages for further implementation details about the program. For access to the source code, see the website.

4. A concluding example

This paper showed that the PDP lacks the firm theoretical basis of the KDE. The former produces noisy results when applied to high precision data, and yields oversmoothed density estimates when analytical precision is low and/or sample size is large. The question then arises why the PDP has become so popular in geochronology in the first place. To answer this question, consider the dataset of zircon U-Pb ages from a Mesozoic sandstone from Hong Kong shown in Figure 4 and posted on the DensityPlotter website. Figure 4.a shows the density estimates of the full dataset of 145 concordant U-Pb ages, while Figure 4.b shows a subset of 35 randomly selected ages from the same sample. The KDE (blue, filled) of the full dataset looks similar to that of the subsample, featuring a first major peak at ~1000 Ma and a second mode at ~2500 Ma. The PDPs (black, empty) of the two datasets, however, have quite a different appearance. Although the PDP of the full dataset is undeniably undersmoothed in comparison with the KDE, it does a decent job at capturing both of the major peaks, as well as most of the smaller scale features. The PDP of the subsample, however, has broken down into an assemblage of spikes bearing little or no resemblance to the KDE. This PDP may still serve a purpose as a way to visualise the data. Unfortunately PDPs are seldom used in this way. Because of their very name, they are erroneously used and interpreted as estimators of probability density.


Figure 4: KDE (blue shaded area), PDP (black line), and histogram (light grey rectangles) of (a) the entire Hong Kong dataset of Figure 2 and (b) a random subset of 35 ages from the same sample. In both cases, the PDP suffers from undersmoothing. Both KDEs adequately capture the bimodality of the distribution, and so does the PDP of the full dataset. The PDP of the subset, however, breaks down into a number of spikes marking the individual age measurements, while obscuring the underlying probability distribution.

In order for a PDP to match the corresponding KDE and be useful as a probability density estimator, the analytical precision of the ages would have to equal the bandwidth of the KDE. For this to happen, the underlying age distribution would have to be made of a number of peaks which are slightly wider than the (2σ) measurement errors. In statistical terms, the variance of the different age components of the detrital population would have to be twice the variance of the age measurements, so that the observed scatter of the age measurements around the different age components would be overdispersed with respect to the formal analytical precision by a factor of √ --
  2. It appears that this condition is approximately and fortuitously fulfilled by typical U-Pb datasets featuring ~100 ages of 2-5% analytical precision. The ‘reasonable’ appearance of many published PDPs is therefore caused by chance and not by design, and this is the likely explanation for their enduring popularity in the geological literature. Despite the fact that PDPs may be considered ‘good enough’ as a visual aid in many situations, in other cases they may be noninformative at best, and misleading at worst. The KDE is a valid alternative with solid theoretical underpinnings which combines the best features of the PDP (namely a continuous curve that is easy to interpret) without its failings when data quality or quantity are high. It is hoped that DensityPlotter will aid in the adoption of the KDE by the Earth Sciences community, and the user is invited to contact the author with remarks or requests for additional features.


The author wishes to thank Andy Carter for providing the Hong Kong dataset and Rex Galbraith for proofreading the manuscript and expert advice.


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