The U–Pb method consists of two paired-decay systems, in which two isotopes of the same radioactive parent (238U and 235U) decay to two isotopes of the same radiogenic daughter (206Pb and 207Pb, respectively). This paired-decay system provides a powerful internal consistency check for the method, which is absent from other chronometers. By “double dating” samples with the 206Pb/238U and 207Pb/235U methods (or, equivalently, the 206Pb/238U and 207Pb/206Pb methods) it is possible to verify whether the isotopic system is free of primary or secondary disturbances. The most reliable age constraints are obtained from samples whose 206Pb/238U, 207Pb/235U, and 207Pb/206Pb ages are statistically indistinguishable from each other. U–Pb compositions that fulfil this requirement are “concordant”. Those that fail to meet it are “discordant”.

Discordance can be caused by a number of mechanisms, including (a) the presence of non-radiogenic (“common”) lead; (b) initial disequilibrium between the short-lived nuclides of the 238U-206Pb and 235U-207Pb decay chains; (c) partial loss of radiogenic lead during high grade metamorphism; and (d) mixing of different age domains during micro-analysis . These complicating effects can often be diagnosed and remediated when multiple cogenetic crystals are available from the same sample. If the aliquots form an isochron (or “discordia”) line in U–Pb isotope space, then this line can be used to recover robust chronologies from discordant data .

Unfortunately, this procedure is rarely or never possible for detrital samples, in which crystals of datable minerals are not guaranteed to be cogenetic. Without a universal mechanism to identify the cause of U–Pb discordance and remove its effects, detrital geochronologists have no choice but to accept some discordant analyses and somehow incorporate them into their age spectra. There exists a lack of consensus among the detrital zircon geochronology community on how to do this. Two outstanding questions are as follows:

Which age estimate should be used? It is widely recognized that 206Pb/238U age estimates offer the optimal accuracy and precision at the young end of the age spectrum, whereas the 207Pb/206Pb method is better suited for older samples. However the cutoff between the two clocks varies between studies, with values ranging from 800 Ma to 1.5 Ga .

This paper addresses both of these issues. Section  advocates the use of single-grain concordia ages as a way to avoid the arbitrary cutoff between the 206Pb/238U and 207Pb/206Pb methods. Although previous workers have argued for the use of single-grain concordia ages before seefor a recent example, this study uses a semi-analytical model, rather than purely empirical arguments, to demonstrate the superior precision of this hybrid chronometer.

Section  compares and contrasts existing discordance filters based on age disparity and p values. It shows that the relative age definition strongly favours older samples over young ones and that the p value definition, which has gained popularity in recent years, hurts both the accuracy and precision of detrital geochronology. The age disparity and p value definitions are heuristic by nature and are not based on firm statistical or geological arguments. Although they are the two most popular definitions of discordance in use today, they are by no means the only two possible options.

Section  addresses the inherent biases of the existing discordance definitions by proposing three new definitions, which are based directly on U–Pb compositions rather than on the ages calculated therefrom. The first new definition assumes that the discordance is caused by the presence of common lead. The other two new definitions treat U–Pb discordance as a compositional data problem sensu. Isotope ratios are strictly positive quantities and log contrasts are the “natural” way to quantify “distances” between them. Section  introduces two log ratio definitions of discordance, ignoring and accounting for analytical uncertainty, respectively.

Although the new definitions are arguably more attractive than the old ones from a theoretical point of view, this does not guarantee that they produce more sensible results. To test their performance on real data, Sect.  applies the discordance filters to a compilation of zircon U–Pb data. Although the true age distribution of this dataset is unknowable, the results suggest that the log-ratio-based discordance filters produce the most accurate and most easily interpretable results. The relative age definition fares the worst.

The U–Pb method is based on three separate chronometers: 206Pb/238U, 207Pb/235U, and 207Pb/206Pb. The half-life of 235U is more than 6 times shorter than that of 238U, and 235U is more than 100 times less abundant than 238U. For these two reasons, little 207Pb has been produced during the last billion years of Earth history compared to 206Pb. Consequently, the 207Pb/235U and 207Pb/206Pb methods are less precise than the 206Pb/238U method during the Phanerozoic and Neoproterozoic.

However, during earlier stages of Earth's history, 235U was significantly more abundant than it is today. The 238U/235U ratio was ∼60 at 1 Ga, ∼26 at 2 Ga, ∼11 at 3 Ga, and ∼5 at 4 Ga. Due to the greater abundance of 235U in this past and because it decays much faster than 238U, the precision of the 207Pb/235U and 207Pb/206Pb clocks exceeds that of the 206Pb/238U method during the Palaeoproterozoic and Archaean. The gradual shift in sensitivity between the two chronometers is visible in the slope of a Tera–Wasserburg concordia line, which is steep at old ages (high 207Pb/238U gradient with respect to time) and shallow at young ages (low 207Pb/238U gradient with respect to time).

Most published detrital zircon U–Pb studies switch from 206Pb/238U to 207Pb/206Pb at some point during the Proterozoic. Unfortunately there are two problems with such a switch. First, it requires the selection of a discrete discordance cutoff between the two methods. If this cutoff differs between two studies (which it often does), then this complicates the intercomparison of their respective age spectra. Second, the sudden switch between the 206Pb/238U and 207Pb/206Pb clocks is often marked by a discrete step in the age spectrum . This step is entirely artificial and obscures any geologically significant events that might occur around the same time.

Both of these problems can be solved by using “hybrid” concordia ages instead of “pure” 206Pb/238U and 207Pb/206Pb ages. Concordia ages are defined by as the “most likely” (in a statistical sense) U–Pb age given the isotopic ratio composition and its analytical uncertainty (Fig. ). Let r86 and r76 be the measured 238U/206Pb and 207Pb/206Pb ratios, respectively; let σ[r86]2, σ[r76]2, and σ[r86,r76] be their (co)variances; and let R68(t)=exp⁡(λ238t)-1, R75(t)=exp⁡(λ235t)-1, and R58=235U/238U. Then the concordia age tc is obtained by numerically minimizing the sum of squares S

The same calculation can also be performed in Wetherill space and is actually easier there.

The single-grain concordia age combines the chronometric power of the 206Pb/238U and 207Pb/206Pb systems. For young (<1 Ga) samples, the concordia age is nearly identical to the 206Pb/238U age. For old samples (>2 Ga) it approaches the 207Pb/206Pb age. Using concordia ages removes the need for an arbitrary cutoff between the two chronometers. An additional advantage is that the concordia age offers better precision than the 206Pb/238U and the 207Pb/206Pb chronometer (or the 207Pb/235U matter for that matter). Figure  quantifies this effect using a semi-analytical mass spectrometry simulation whose algorithm is provided in Appendix .

The most common definition of discordance uses the relative difference between the 206Pb/238U and 207Pb/206Pb age estimate : 2dr=1-t68/t76.

However other definitions are possible as well. For example, one could also define discordance in terms of absolute age differences : 3dt=t76-t68.

A third option is to define discordance in terms of U–Pb compositions rather than ages. advocate using p values to assess concordance. In the context of single-grain concordia ages, the p value is the probability that the sum of squares S (Eq. ) exceeds the observed value under a chi-square distribution with 2 degrees of freedom: 4dp=Probs>S|S∼χ22.

Zircon U–Pb data can be filtered by removing all measurements whose discordance values exceed a certain threshold value. Typical cutoff values for dr are 10 %–30 % , whereas dp is generally set to 5 % . Different discordance criteria produce different U–Pb age spectra. For example, a relative age cutoff will preferentially remove young grains, whereas an absolute age cutoff is comparatively more likely to remove old grains (Fig. ).

The p value definition affects grains differently depending on their analytical precision . For example, consider a 1.5 Ga zircon that is dr=1 % discordant. If this grain were analysed by laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) with an analytical precision of 2 %, say, then it would pass the chi-square test and be accepted as being concordant. However, if that same grain were analysed by thermal ionization mass spectrometry (TIMS) with a precision of 0.2 %, then the p value criterion would reject it as being discordant. It seems fundamentally wrong that an imprecise analytical method would be favoured over a precise one (Fig. ). This is a pertinent problem because technical innovations are increasing the precision of all analytical approaches to U–Pb geochronology. As precision improves, so does the ability to detect ever small degrees of discordance. Using the p value criterion, there may come a time when no zircon passes this filter.

A final argument against the p value discordance criterion is that it biases against old U–Pb ages. This is because old zircon contains more radiogenic Pb than young zircon does. Therefore the analytical precision of the isotopic ratio measurements tends to be better for old grains than it is for young ones. Consequently, the chi-square test has greater power sensu to reject them. In conclusion, p-value-based discordance filters are fundamentally flawed. Despite their appeal as “objective” tools for statistical decision making, formalized hypothesis tests such as chi-square are rarely useful in geology. For the same reason, the widely used MSWD mean square of the weighted deviates; statistic (which is just S/2 in this case) should be used with caution. This is because, like p values, MSWD cutoffs also punish precise datasets in favour of imprecise ones. Note that this caveat also goes against the recommendations of .

Section  reviewed three existing discordance definitions. This section will introduce three new ones. None of the definitions discussed thus far encode any information about the geological mechanisms behind the discordance. As explained in Sect. , common Pb is one of the most likely causes of discordance. Using a mantle evolution model e.g. to approximate the isotopic composition of this common Pb, discordance can be defined as 5dsk=1-r86/r86*, where r86* is the 238U/206Pb ratio of the intersection between concordia and a straight line connecting the 238U/206Pb-207Pb/206Pb measurement to the inferred mantle composition (Fig. ).

The common-Pb definition of discordance is more forgiving for young grains than it is for old ones. Importantly, if the discordance is caused by common Pb, then the 206Pb/238U, 207Pb/206Pb, and concordia age estimates are all positively biased with respect to the true age. However this bias can be removed by applying a common-Pb correction after the data have been filtered.

Even though Eq. () is mathematically able to produce negative discordance values, such values lack a geologically meaningful interpretation because it is impossible for minerals to inherit negative amounts of common Pb. Thus it is sensible to set a minimum cutoff of dsk>0 when using the Stacey–Kramers filter.

Each discordia definition that we have studied thus far is expressed in different units. For the absolute age definition, degrees of discordance are expressed in units of time (ranging from 0 to 4.5 Ga). The relative age definition uses fractions of time (ranging from -∞ to 1). The p value definition expresses discordance in terms of probability (ranging from 0 to 1). And the definition uses fractions of ratios (ranging from -∞ to 1). None of these scales is particularly intuitive or natural. They certainly do not match the usual definition of distance in the geographical sense of the word.

To address this issue, it is useful to subject the U–Pb isotopic ratio data to a logarithmic transformation. So instead of analysing the data on a conventional Tera–Wasserburg concordia diagram, all calculations can be done in ln⁡(207Pb/206Pb) vs. ln⁡(238U/206Pb) space. The advantage of this transformation is that it produces values that are free to range from -∞ to +∞. Within this infinite data space, the Euclidean distance metric can be safely applied.

There exists a vast body of statistical literature detailing the theoretical and practical advantages of log ratio analysis. A deeper discussion of this topic falls outside the scope of this paper, but the interested reader is referred to and for further information. The Euclidean distance between log ratios is also known as the “Aitchison distance”. Discordance can be redefined as the Aitchison distance from the measured log ratios to the concordia line. We introduce two ways to do so here. A first option is to simply measure the distance along a perpendicular line to the concordia curve (Fig. ): 6da=dx(t76)sin⁡arctan⁡dy(t68)dx(t76), where 7dx(t)=ln⁡r86+ln⁡R68(t)anddy(t)=ln⁡r76-ln⁡R58R75(t)R68(t).

This definition produces a parallel band around the concordia line in logarithmic Tera–Wasserburg space. In contrast with dr, dt, and dsk, the da criterion is less strict at both the young and old extremes of the geological timescale and more strict during the Proterozoic eon, when the U–Pb method is most reliable.

The perpendicular Aitchison distance criterion does not take into account the analytical precision of the isotopic measurements. To address this issue, we can also measure the Aitchison distance along a line connecting the measured log ratio and the maximum likelihood composition on the concordia line: 8dc=sgn[t76-t68]dx(tc)2+dy(tc)2, where sgn[*] stands for “the sign of *”, which produces positive values for measurements that plot above the concordia line and negative values for measurements that plot below it.

It is difficult to ascertain the mechanism causing discordance in any particular zircon grain. Therefore, it is unclear which of the definitions in Sects.  and  is “correct”. All we can do is apply the methods to real samples and investigate their outcomes. This section will apply the discordance filters to a dataset of 70 869 zircon U–Pb analyses that were acquired by Sensitive High Resolution Ion Micro-Probe (SHRIMP) mass spectrometry and compiled by Simon Bodorkos of Geoscience Australia.

The dataset includes 1665 sedimentary, igneous, and metamorphic samples, mostly from Australia but including some other locations as well. The data were acquired by a variety of instruments (including SHRIMP-1, SHRIMP-2, and SHRIMP-RG) using a range of different reference materials and processed on a range of different types of data reduction software (including Squid-1, Squid-2, and Prawn/Lead). The data were not subjected to any common-Pb correction or other filters and were saved in a Tera–Wasserburg format with zero error correlation between the 238U/206Pb and 207Pb/206Pb ratios.

Figure  shows the frequency distribution of the complete, unfiltered dataset as a kernel density estimate. The 207Pb/235U, 206Pb/238U, and concordia age spectra all look similar. However, the 207Pb/206Pb age distribution deviates from the other three chronometers. It reduces the prominence of the young age components and inflates the old end of the age spectrum.

Figure  applies five of the six discordance filters to this database (the p value filter was omitted for reasons given in Sect. ). In order to emphasize the difference between the discordance definitions whilst treating them on an equal footing, each of the filters was adjusted until half of the data were removed. This was achieved by discordance cutoffs of -18.6<dt<46.0 Myr, -1.4<dr<3.66 %, -0.11<dsk<0.27 %, -0.78<da<1.94 %, and -0.91<dc<2.20 %.

There are noticeable differences between the density estimates. As expected from the theoretical considerations laid out in Sects.  and , the relative age filter greatly suppresses the younger age components (<1.5 Ga) relative to the older parts of the age spectrum (>1.5 Ga). The filter has the opposite effect. It suppresses the Archaean age component by ∼50 % whilst further increasing the prominence of the Neoproterozoic and Phanerozoic modes.

The discordance definitions based on the absolute age difference and log ratio distances have a comparatively minor effect on the shape of the age spectrum. The change in shape between the age spectrum of the full (unfiltered) dataset and the age spectra of the filtered datasets can be visually assessed on quantile–quantile plots and quantified using the Kolmogorov–Smirnov (KS) statistic . If the KS misfit is taken as a measure of success, then the concordia distance filter (dc) is the most effective discordance criterion. It “sharpens” the spectrum without changing the relative prominence of the modes at 400, 1200, 1800, and 2500 Ma.

Figure  removed 50 % of the data, in order to emphasize the differences between the discordance filters. In real applications, less stringent discordance filters are usually applied. As mentioned in the introduction, most current detrital zircon studies apply a 10 %–30 % relative age cutoff. Using the test data, we can evaluate the equivalent values for the dt, dsk, da, and dc criteria (Table ). For example, a relative age filter of 10 % removes the same fraction of the test data as an absolute age filter with dt=97 Myr, a Stacey–Kramers filter with dsk=0.62 %, a perpendicular Aitchison filter with da=4.1 %, or a concordia distance filter with dc=4.6 %.

The p value discordance filter has been omitted from this comparison for two reasons. First, the use of this filter is discouraged for reasons given in Sect. . Second, the p value cutoffs that are equivalent to any given relative age difference are highly laboratory dependent, with precise equipment requiring different dp cutoffs than imprecise instruments. The other five discordance filters are more universally applicable. So using a different set of test data should only make a modest difference to the values in Table .

This paper compared four U–Pb clocks and six discordance filters.

The 206Pb/238U clock is most precise at the young end of the geologic timescale.

The six discordance filters include three existing ones and three new ones (Table ).

The relative age discordance dr is the most widely used criterion today. It is more likely to remove young grains than old ones and strongly skews the age distribution towards old age components as a result.

All the discordance filters presented in this paper (both old and new) have been implemented in IsoplotR , a geochronological toolbox written in the R language. Further details about this implementation are provided in Appendix .