# GSA Bulletin

## Abstract

The advent of new procedures for the rapid collection of large amounts of detrital zircon age data has created significant challenges to the representation, quantitative interpretation, and analysis of these results. Principal among these is the development of the most efficient and informative methods to accurately present and quantitatively compare numerous age data from large numbers of samples in a statistically valid, geologically meaningful, and easily assessable manner. Many detrital zircon studies have adopted a subjective-graphic approach to these tasks, primarily through the examination of vertically stacked probability density plots (PDPs). Although this methodology allows for quick visual assessment of changes in zircon age frequencies in space and/or time, it is fundamentally qualitative, and becomes rather impractical as numbers of analyses become large. More quantitative approaches to the comparison of PDPs (e.g., Kolmogorov-Smirnov test and multi-dimensional scaling) have also become more popular, but are rather computationally intense.

In order to examine degrees of sameness among detrital zircon age populations in geographic and/or stratigraphic contexts, we develop a “likeness” metric that quantifies the degree of overlap between pairs of PDPs. We compare this metric to several others that have been proposed previously, and then evaluate its usefulness in describing source-to-sink changes in sample age populations by analyzing data from four published studies of detrital zircon age frequencies. Likeness among 26 Paleozoic samples from the Grand Canyon (southwestern United States) decreases with increasing stratigraphic separation, but stratigraphically contiguous sample likeness exhibits statistically insignificant upsection change. The degree to which these patterns are representative of other stratigraphic successions awaits further evaluation. Likenesses among 15 samples of modern fluvial sand from the Amazon River (South America), 11 stream samples from the French Broad River in the southern Appalachians (eastern United States), and 10 Quaternary coastal sand samples from eastern Australia decrease with increasing distance between samples; as might be expected, greater geographic disjunction results in greater dissimilarity. In contrast, likenesses among spatially contiguous fluvial and littoral sample ages all exhibit trends reflecting a greater down-system increase in likeness. These results suggest that sediment grain ages may exhibit increasing homogenization with transport in both fluvial and littoral systems. The extent to which these source-to-sink changes may characterize lateral variations in other sedimentary systems also awaits additional assessment. Nonetheless, each of these four examples demonstrates that the determination of sample-pair likenesses, and evaluation of their geographic and/or stratigraphic variation, serve to effectively highlight those aspects of differences in detrital zircon age frequencies that ultimately record the histories of geologic and geomorphic processes in both space and time.

## INTRODUCTION

Since the earliest pioneering efforts to study detrital zircon ages (e.g., Gaudette et al., 1981; Dodson et al., 1988), procedures for zircon analysis have become ever more sophisticated, with single-grain analytical techniques becoming the foremost methodology to collect detrital zircon age data. The most widespread procedures rely on measurements made using secondary ion mass spectrometry, laser ablation inductively coupled mass spectrometry, or thermal ionization mass spectrometry. In spite of significantly higher precision afforded by thermal ionization mass spectrometry, in situ microbeam approaches have become the preeminent tools for detrital zircon age analysis because of the inherent advantage of high spatial resolution and the ability to rapidly collect large amounts of data. Recent studies have typically incorporated data from many sediment samples, each producing many tens of zircon analyses or even individual samples that produce thousands of individual U-Pb dates (e.g., Gehrels et al., 2012). This considerable amount of data has led to several interpretational challenges which include determining: (1) the best ways to effectively present large amounts of age data from multiple samples, each of which contains multiple age populations of zircons, and (2) the best ways to visualize and quantitatively compare data from different samples within any individual study or with those previously published. Typically, sample ages are summarized utilizing a combination of age frequency distributions (AFDs), probability density plots (PDPs; e.g., Ludwig, 2003), and/or kernel density estimates (KDEs; e.g., Sircombe and Hazelton, 2004; Vermeesch, 2012). Comparison of samples is normally accomplished by visual examination of vertically stacked and unitized PDPs with a common age axis (e.g., DeCelles et al., 2007; Martin et al., 2008; Amato and Pavlis, 2010; among many others). The obvious problem with such a methodology is that it relies largely on visual inspection and interpretation, an approach that is at least in part dependent on biases and preconceptions of the analyst. In order to fully exploit the wealth of information contained in such data, it is critical to meaningfully visualize and easily quantify differences among detrital zircon age populations.

Determining the nature of spatial and temporal variations in detrital zircon age frequency distributions from within a single time horizon or across a single stratigraphic succession potentially allows for evaluation of lateral provenance heterogeneity and its evolution through time. In this regard, several modern systems have been investigated to assess such variation, including littoral (Sircombe, 1999), eolian (Pell et al., 1999; Pullen et al., 2011), and fluvial (Amidon et al., 2005a, 2005b; Hietpas et al., 2011; Niemi, 2012; Saylor et al., 2013) environments. Moreover, the nature of sediment transport in these systems is sensitive to a combination of climatic, geographic, and geologic factors. Zircon ages within sediment samples may therefore reflect variable influences of: (1) differing source rock compositions, (2) the extents of areal exposure of different source lithologies, (3) the lateral and temporal variation in rates of source rock erosion, and (4) changes in zircon age distributions imposed by hydrologic processes during source rock weathering and subsequent sediment transport. While all of these factors might serve to either fractionate or homogenize sediment zircon ages, the nature of changes in differences among PDP pairs across any particular sedimentary system is poorly understood.

It might be presumed, for example, that zircon ages would be more heterogeneous among samples collected close to sediment donor regions owing to the fact that within-system transport and sediment mixing would serve to homogenize differences with increasing transport distance. However, it could also be maintained that grain ages might be more homogeneous among samples collected in close proximity to source regions, and that the down-system confluence of tributaries and addition of age populations from other more distal regions would serve to impart greater differences between samples with increasing distance from sediment sources. In spite of a growing literature that assesses spatial and temporal variation in detrital mineral age spectra, there are several questions that remain largely unresolved: (1) which (if either) of the two described scenarios is the more dominant, (2) is a similar pattern observable among different systems of sediment transport, and (3) how do these factors vary in time as well as space during long-term sediment accumulation? The reasons for this lack of understanding are certainly multifaceted, and undoubtedly in part reflect the fact that rapid microbeam techniques, which now allow for such questions to be addressed, are relatively new. However, it also seems apparent that evaluations of spatial and temporal differences among zircon ages also necessitate a more widely accepted metric for measuring such differences. In order to demonstrate the utility of comparing zircon age frequencies through stratigraphic and geographic space, we examine four previously published sets of detrital zircon data; each represents sampling that spans either a significant range of geologic time at a single locality or a significant extent of geographic space along one time horizon.

## NUMERICAL APPROACHES

With the expansion of detrital zircon data sets, both with respect to the number of ages measured per sample, as well as the number of sediment samples analyzed per study, the utility of stacking and visually interpreting PDPs becomes increasingly limited. In an effort to more objectively quantify and evaluate detrital zircon data, several approaches have been proposed. These include methodologies such as mixture modeling (Sambridge and Compston, 1994), hierarchical cluster analysis (Sircombe and Hazelton, 2004), principal component analysis (Sircombe, 2000; Fedo et al., 2003), and multi-dimensional scaling (Vermeesch, 2013); such methods are primarily directed toward better interpretation of age frequency distributions and derivative (PDP, KDE) metrics. Although these methods are certainly of merit, here we primarily focus on approaches that are intended to highlight and compare the differences between two sample age distributions. A variety of techniques have been used to compare the “sameness” between two detrital zircon U-Pb age populations. These include the Kolmogorov-Smirnov test, measures of PDP overlap and similarity, correlation of cumulative age frequencies (CAFs) and PDPs, and degrees of common coincidence among PDP pairs. The following sections briefly summarize these metrics.

### Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov (K-S) test is a method to determine if a statistically significant difference exists between two populations (Press et al., 1986). In the context of detrital zircon ages, the probability calculated by the K-S test represents the probability that two populations could have been selected at random from the same parent population. This approach has been widely applied to the interpretation of detrital zircon data (e.g., Berry et al., 2001; DeGraaff-Surpless et al., 2003; Dickinson et al., 2010; Dickinson and Gehrels, 2009; Miller et al., 2013; Schoenborn et al., 2012; and many others). Its extensive use is due in part to the ease with which the test can be performed using the free software provided by the University of Arizona’s LaserChron Center (Guynn and Gehrels, 2010). The K-S test is based on the maximum difference between two empirical cumulative distribution functions (ECDFs). As part of K-S analysis, a *p*-value is generated that relates to the probability that the observed maximum difference is due to random sampling error versus a true difference between the populations. If *p* is high (typically *p* > 0.05, at 95% level of confidence), it is unlikely that the two samples are from different populations; it cannot determine if two distributions are the same (Guynn and Gehrels, 2010). The reason that the K-S statistic is not widely used to describe degrees of difference between two populations arises from the fact that the test was designed to evaluate the probability that two samples are drawn from the same population; it does not readily translate into a measure of degrees of dissimilarity. Vermeesch (2013) provided a detailed discussion as to why *p*-values may be unsuitable measures of sameness; this is primarily because they combine the effects of differences in ECDFs as well as in sample size. Vermeesch (2013), however, did suggest that “sample effect size” of the K-S test, the maximum difference between two ECDFs, is a useful measure of sameness among sample zircon ages, and employed this metric and age data from Chinese fluvial and loess samples to produce a multi-dimensional scaling “map” on which “similar” samples cluster. As discussed below, the K-S statistic increases nonlinearly with other measures of PDP similarity. As such, the utilization of the K-S *p*-value as a measure of PDP dissimilarity is awkward at best.

### Overlap and Similarity

The concept of PDP overlap was introduced by Gehrels (2000) as a way to quantify resemblance among detrital zircon age spectra of the western Laurentian miogeocline. Overlap represents the proportion of age intervals (at some temporal scale of resolution) that contain some number of zircon ages from either population. It is insensitive to numbers of grain ages that might occur within any age bin; a value of 100% indicates that all age intervals are represented by each population, while a value of 0% indicates that no occupied age intervals are common to the populations.

Similarity is a measure of resemblance between portions of two populations with overlapping ages (Gehrels, 2000). Calculation begins as with overlap, but instead of counting common age intervals, it provides a weight based on the number of grain ages common to the two samples. It is calculated by summing, over the time interval of interest and at some temporal scale of resolution, the square root of the product of each pair of common probabilities. It yields a value of 100% for identical curves, and lower values for populations with very low proportions of overlap. The approach is similar to that introduced by Sircombe and Hazelton (2004) which assess the “distance” between age distributions based on their KDEs.

The use of the overlap and similarity metrics to assess relative differences between two age distributions is more straightforward than that of the K-S statistic, and change in one metric correlates with an approximate linear change in the other. However, the methodology has not been widely embraced. This perhaps is because they entail somewhat different attributes, overlap being a measure of PDP age resemblance (abscissa variation), and similarity being a measure of degree of overlapping of PDP density (ordinate variation). Moreover, unlike the K-S statistic which has been widely employed in a large number of scientific fields, the geological implications of differing overlap and similarity values are less familiar.

### Cumulative Distributions and Probability Densities

Several other approaches to measuring sample sameness employ the determination of degrees of correlation (often Pearson product-moment correlation) between sample ages, either through cross plots of sample cumulative distribution functions (CDFs; Wilk and Gnanadesikan, 1968; Saylor et al., 2013), or cross plots of sample probability densities (Saylor et al., 2012). Both approaches ascertain degrees of correlation (r^{2} values) among quantiles of sample pair CDFs or PDFs (cumulative or probability density functions), thus providing a quantitative measure of similarity between samples. For identical samples, the coefficient will be 1, while for those that share no common ages, the coefficient will be 0. Because the cross plotting of cumulative distributions CDFs constrain densities to range from 0 to 100%, r^{2} values are significantly higher than for PDP cross plots, regardless of sample age sameness.

### Percent Area Mismatches

In examining detrital mineral populations from the mixing of U-Pb zircon ages in Himalayan rivers, Amidon et al. (2005a, 2005b) developed an “areal percent mismatch” metric determined by summing, at some temporal scale of resolution, the absolute value of differences between all “coeval” PDP values. Because the absolute values of all differences between two completely dissimilar curves (overlap and similarity = 0) with subjacent areas of unity will sum to 2, this summation is then divided by 2 resulting in a value of 100% for completely non-overlapping PDPs. This approach is also similar to that proposed by Sircombe and Hazelton (2004) which measures the “distance” between age frequency distributions based on their KDEs.

### Likeness

While all of these approaches are of merit, none have been widely accepted for comparing detrital zircon age data. Here we propose the use of a measure of PDP similarity we call “likeness”. It is somewhat simpler and more easily calculated than many of the measures noted above, and is not unlike the similarity metric of Gehrels (2000), and the areal percent mismatch of Amidon et al. (2005a, 2005b). However, our metric utilizes all information in any pair of PDPs, and combines both common age and associated probability density values. In addition, we desire a metric in which higher values denote greater similarity. Likeness, *L*, represents the percent of “sameness” between two unitized PDPs, and is calculated as one minus the summation of absolute differences between all PDP values divided by two:where N is the number of ages in both PDPs, and a and b are the y-axis values for any given age i. Values approaching 100% correspond to a high degree of likeness between two distributions (see the GSA Data Repository^{1}). It is essentially the inverse of the areal percent mismatch metric of Amidon et al. (2005a, 2005b).

## EVALUATION OF THE LIKENESS METRIC—CAVEATS AND VARIATION IN STRATIGRAPHIC SPACE

Detrital zircon studies have been effectively applied to a wide number of geologic problems, including placing constraints on ages of siliciclastic sequences, identifying disconformities (particularly in Precambrian successions), and provenance analysis and associated paleogeographic reconstructions. Although all of these investigations entail some sort of comparison between different samples, most rely on qualitative inspection of PDPs, and the quantitative nature of patterns of temporal and/or lateral difference are therefore not readily determined or exploited. Here, we propose that determination of likeness between sample pairs provides an effective tool for gauging degrees of change among such data sets that span variable scales of geographic space and geologic time. Given the variety of measures that have been employed to gauge differences among sample zircon ages, there is a notable and somewhat surprising lack of studies in which this has been attempted; most of these are noted in the foregoing discussion. Moreover, our primary interest here is in determining the utility of PDP pair likenesses, particularly with respect to differences in sample ages along transects in both time and space. To this end, we first examine temporal differences in measures of sample age sameness at one locality, the Paleozoic Grand Canyon succession (southwestern United States) (Gehrels et al., 2011), and then examine spatial differences in samples from fluvial sands along the Amazon River (South America) (Mapes, 2009), alluvial samples collected along the French Broad River as it transverses the southern Appalachian orogen (eastern United States) (Hietpas et al., 2011), and Quaternary coastal sands along eastern Australia (Sircombe, 1999).

### Example 1: Paleozoic Grand Canyon Succession

A number of studies have emphasized the importance of assessing temporal variation in detrital zircon age distributions for samples from a specific locality in order to determine the nature of evolving source contributions (e.g., Cawood et al., 2000; Hallsworth et al., 2000; DeGraaff-Surpless et al., 2002). Here, we focus on data from the study by Gehrels et al. (2011), which consists of 2529 zircon ages from 26 samples (average n = 97 per sample) of Cambrian through Permian sandstones exposed in the Grand Canyon of the southwestern United States. Collectively, these span ∼1086 m of Paleozoic strata, and represent accumulation over 260 m.y. of geologic time (Fig. 1A).

Before discussing likenesses among Grand Canyon detrital zircon samples, it is important to first generally comment on the nature of these data. The aggregate frequency distribution of all 2529 ages (Fig. 1B) embodies several clusters centered at ca. 465, 1085, 1445, 1710, and 2705 Ma, groupings readily ascribed to derivation from various age-specific source provenances across North America (e.g., Dickinson and Gehrels, 2009, and references therein). However, at least some Grand Canyon zircon ages occur in most of the successive age bins that collectively span the last 4 b.y. When reported mean analytical ages are binned over 5 m.y. intervals, these ages represent >61% of the span of geologic time represented by this range of ages. This continuum is perhaps unexpected as detrital zircon data are commonly characterized as comprising a quasi-magmatic “barcode” (e.g., Link et al., 2005) in which grain ages compose discreet “populations” or “tectonic event bins”, such that any single zircon grain age should belong to a distinct group, a categorical view generally accepted in attempts to determine how many grains need to be dated in order that all fractions of some parent population be represented (e.g., Vermeesch, 2004; Andersen, 2005). If detrital zircon age frequencies indeed were such that most ages could be categorized as belonging to several well-defined age brackets representing detritus derived from only a finite number of discrete age provinces of igneous rocks, then similarity measures based on degrees of such categorical membership might be more appropriate. However, available data suggest that the distribution of Grand Canyon ages is perhaps better perceived as comprising a more-or-less continuous series of ages, albeit of varying frequency. It is unclear if this continuum reflects a real temporal distribution of zircon-generating events in geologic time, the mixing of discontinuous ages in zoned zircon crystals, or other analytical artifacts. However, differences between these two points of view about the temporal distributions of detrital grain ages are critically important in attempting to measure degrees of difference between PDP pairs. Numbers of grain ages within different 5 m.y. time intervals are not only a continuum, they also define a power law relation in which age frequency (*F*) smoothly decreases with deceasing number of ages per bin (*A*) as: *F* = 229*A*^{–1.46}, with r^{2} = 0.87 (Fig. 1C). Alone, such a pattern largely refutes “barcode” views in which grain ages comprise distinct tectonic populations; these zircon ages comprise a more-or-less continuous succession, albeit of varying frequency, over much of the past 4 b.y.

Visual inspection of each of the 26 Grand Canyon PDPs (Fig. 2A) indicates an overall smoothing with increasing age and, as pointed out by Gehrels et al. (2011), a rather dramatic loss of density associated with the Proterozoic peaks (at ca. 1445 and 1710 Ma) between Mississippian samples 9 and 10. Moreover, what appears to be an approximately continuous group in the Phanerozoic aggregate (Fig. 1B) may in fact comprise several smaller groups in the upper part of the section (samples 10–26, Figs. 2A and 2B). Given the ∼97 zircon ages per sample, it is also possible that sample-to-sample variation apparent within this particular time span may largely represent sampling biases.

#### Measures of Sameness among Grand Canyon Samples

The availability of zircon ages from 26 Grand Canyon samples permits the consideration of three interrelated questions relative to comparisons among the 325 sample pairs present in these data: (1) what are the relative merits of determining pair likenesses utilizing sample age frequency distributions versus sample probability density plots, (2) what is the influence of varying numbers of sample ages on values of pair likeness, and (3) how do measures of likeness compare to measures of sample sameness employing each of the alternative metrics noted above?

#### Age Frequencies versus PDPs

In order to determine which representation of age data is more appropriate for comparing sample populations, likeness values were determined for each of the 325 Grand Canyon sample pairs utilizing AFDs or PDPs normalized to unity, and determined at temporal resolutions of 1, 5, 10, 50, and 100 m.y. Comparisons of AFD-derived likenesses over this range of durations demonstrate that lower values result from smaller time intervals (Fig. 3A). This is perhaps not surprising because, if taken to an extreme time span of a single ∼4 b.y. bin, all normalized age frequencies would compose one identical AFD class. Clearly, likenesses between binned sample ages, even when normalized to a common sample size, are strongly dependent on the temporal resolution employed in their calculation. In contrast, comparisons of PDP-derived likenesses over this same range of durations demonstrate that similar values characterize each sample pair regardless of the time resolution employed in construction of the PDPs (Fig. 3B). Owing to the robustness of this result, and the fact that the average analytical error of the Grand Canyon zircon ages is 32 ± 19 m.y., we determine all subsequent likeness values at a temporal resolution of 5 m.y.

#### Impact of the Number of Zircon Analyses per Sample

As is the case with any measurement of sameness, the numbers of dated zircon grains in each sample will influence the determined values of PDP pair likenesses. Population similarities are dependent both on numbers of dated grains and on the complexity of the populations of sample ages being compared. As one end member example, we might posit some parent zircon grain population, all yielding only one narrowly defined age. In this case, an age determination from only one grain from each of two subsamples would result in a likeness value of 1.0; by definition, all subsample PDPs would be identical. On the other hand we might imagine some very complex and heterogeneous parent population with ages, for example, spanning, say, 4000 m.y., and being equally represented by ∼400 narrowly defined grain age groups separated by ∼10 m.y. intervals. In this case, the dating of many thousands of grains from each subsample would be necessary to even approach a likeness value of 1.0.

In order to gain better insight into the effect of numbers of dated grains on real-world PDP pair likenesses, here we consider the five samples (1 through 5, Fig. 1A) spanning the terrigenous portion of the basal Sauk sequence (Tapeats Sandstone, Bright Angel Shale) in the Grand Canyon. Collectively, these comprise 486 dated grains (Fig. 4A). If we theorize that these five samples indeed comprise a homogeneous parent population of grain ages, the influence of numbers of dated grains on calculated likeness values can be evaluated by bootstrapping. Specifically, we can specify some number of dated grains to be sampled, and then determine likenesses for some large number of PDP pairs (here we use 1000 pairs), each containing that specified number of dated grains, by repeatedly resampling the measured Sauk zircon age population. Likenesses among PDP pairs, determined for some specified number of ages drawn at random from the Sauk population of 486 grains, define Gaussian distributions (Fig. 4B). Moreover, likenesses among PDP pairs determined over a range of numbers of subsample ages demonstrates that mean likeness exhibits an approximately logarithmic relation to numbers of grain ages under consideration, increasing from an average of 61% ± 9% for subpopulations of 50 grain ages to an average of 79% ± 4% for subpopulations of 150 grain ages. Importantly, for samples of ∼100 ages drawn from the *same* parent population (average Grand Canyon sample size = 97 ages), average likeness should be 72% ± 6% (Fig. 4C). On average, this is the maximum likeness we might expect between two PDPs if sampled from the same identical parent population. Age differences imposed by variable source rock compositions, extents of source rock exposure, lateral and temporal variation in rates of source rock erosion, and changes in grain age populations imposed by hydrologic processes during weathering and sediment transport will serve to lower likeness values among PDP pairs.

#### Other Measures of Sameness

The availability of zircon age data from the 26 Grand Canyon samples also allows for a comparison of values of likeness to other measures of sample sameness for each of the 325 sample pairs in this data set. All comparisons yield similar measures of pair resemblance. Overlap and similarity values (Gehrels, 2000) exhibit a logarithmic relation to likeness (Figs. 5A, 5B). Closest agreement to likeness is exhibited by similarity (r^{2} = 0.96), an agreement undoubtedly reflecting the fact that the two metrics are alike in their derivation. Resemblances based on correlation of sample pair CDFs (Wilk and Gnanadesikan, 1968; Saylor et al., 2013) or PDFs (Saylor et al., 2012) are also closely related to those measured as likeness, with r^{2} values of 0.79 and 0.77, respectively (Figs. 5C, 5D). Likenesses among the 325 Grand Canyon pairs are also related to K-S parameters. The maximum difference between two ECDFs (e.g., Vermeesch, 2013) exhibits an inverse correlation with pair likeness (r^{2} = 0.85; Fig. 5E). While K-S *p*-values themselves are at best unwieldy measures of sameness, they do exhibit an approximately power-law relation to likeness values (r^{2} = 0.82; Fig. 5F), probably reflecting a similarity in sizes of the Grand Canyon samples (97 ± 10).

#### Likenesses among Grand Canyon Clastic Samples

In order to examine changes in PDP character across stratigraphic space, we determined degrees of likeness between all 325 sample pairs relative to their net spatial stratigraphic separation. Degrees of likeness decrease with increasing stratigraphic distance (*S*) as: *L* = 70.8% –0.00035*S* (Table 1). For each meter of stratigraphic separation between samples, PDP likeness decreases by 0.035% from an average of ∼71%; sample pairs spanning greater stratigraphic ranges are increasingly dissimilar (Fig. 6A). As noted above, a likeness intercept at a value of ∼71% is about what would be expected from the somewhat imperfect duplication of an essentially unknown parent population of ages that make up the terrigenous sedimentary unit under consideration, given that the Grand Canyon data consist of 2529 grain ages divided among 26 samples with an average of 97 ± 10 ages each. On average, samples drawn from exactly the same population of ages will be no more alike than ∼71%.

PDP pair likeness values among the contiguous samples collected vertically through the Grand Canyon exhibit little overall change upsection, but exhibit a sharp minimum in the Mississippian Surprise Canyon Formation (sample pairs 9–10 and 10–11; Fig. 6B). Reasons for this reduction are apparent when values of the 26 PDPs are interpolated between the samples relative to their stratigraphic positions (Fig. 2B); the abrupt decrease in likeness primarily reflects a rather sudden Mississippian change in PDPs within the thin Surprise Canyon Formation, with ca. 1445 and 1710 Ma ages dominating below, and Grenville (ca. 1085 Ma) and younger ages dominating above. These findings are consistent with those of Gehrels et al. (2011) who concluded that these Mississippian strata record a major change in sand provenance. Below this level, zircon ages indicate derivation from local basement rocks of the Mazatzal and Yavapai provinnces and the Amarillo-Wichita uplift. Zircon grains in younger units record ages that are interpreted to have been derived predominantly from the Ancestral Rocky Mountains and the central Appalachian orogen. High values of pair likenesses below and above this interval serve as a quantitative measure of this dramatic change.

## EVALUATION OF THE LIKENESS METRIC—VARIATION IN GEOGRAPHIC SPACE

In order to achieve a better understanding of spatial differences among the zircon age populations in modern systems, we now examine data from three modern siliciclastic environments. These data are from samples collected along the Amazon River from the Andes to the Atlantic Ocean, along the French Broad River as it transverses the southern Appalachian orogen, and from Quaternary sand along the eastern Australian coast.

### Example 2: Modern Amazon River Transect

Detrital zircon ages from the modern Amazon River are described in Mapes (2009). This set of data consists of 1387 U-Pb detrital zircon ages for 15 samples collected along the Amazon/Solimões River system. These samples, each with an average of ∼92 ages, span a distance of ∼2742 km, extending from the eastern Andes in Ecuador to the Amazon River estuary near Santarém, Brazil (Fig. 7A). When grain frequency is plotted as a function of age, it is clear that there are apparent broad age clusters centered at ca. 260 and 1080 Ma (Fig. 7B), groups undoubtedly related to derivation from various age-specific source provenances within the Amazon River drainage. As noted above, detrital zircon ages are often seen as composing largely discrete groups reflecting derivation from distinct geographic and chronologic source provenances. However, at a temporal resolution of 5 m.y., ∼56% of the past 4 b.y. of Earth’s history is evidently recorded by least some portion of the Amazon River zircon AFD (Fig. 7B). In this context, presence predominates over absence. Moreover, like in the Grand Canyon data, the number of grain ages that occupy different 5 m.y. time intervals define a power law relation in which frequency (*F*) decreases with numbers of grain age bins (*A*) as: *F* = 259*A*^{–1.66}, with r^{2} = 0.93 (Fig. 7C). Although several broad clusters of ages are indeed apparent among these data, grain ages exhibit an approximately power-law abundance decrease.

Visual inspection of the 15 Amazon River PDPs (Fig. 8A) suggests a general decrease and smoothing of peak densities with increasing age, a variation in part related to larger analytical error associated with older ages. In addition, there is a less prevalent ca. 1080 Ma peak in the extreme upstream and downstream samples, and persistence of several Phanerozoic groups throughout all 15 samples (Fig. 8). Close inspection of the Phanerozoic interval also demonstrates that it may consist of a series of smaller subgroups. Given that each sample contains ∼92 grain ages, it is possible that this apparent Phanerozoic variation may represent a sampling bias of what indeed is an approximately continuous group of Phanerozoic ages.

In order to examine changes in PDP character across this geographic space, we determined degrees of likeness between all possible pairs among the 15 samples (n = 105 pairs). When plotted relative to geographic separation (in kilometers), it is apparent that percent likeness (*L*) decreases with increasing geographic separation (*S*) as: *L* = 69.8% –0.00015*S* (Table 1); for each kilometer of separation between sample sites, PDP likeness decreases by 0.015% from an average of ∼70% (Fig. 9A). Calculation of changes in PDP pair likenesses of contiguous samples collected along the Amazon River indicate that likenesses for more-upstream pairs are at a minimum, and become higher (0.03% per kilometer) in a downstream direction (Fig. 9B). When the 15 PDPs are interpolated relative to their geographic positions, it becomes clear that this pattern of change in likeness largely arises from the fact that the sample closest to the Andean range (sample 1) is dominated by a mode centered at ca. 180 Ma, that this population is largely restricted to this sample, and that ages in sample 2 are widely dispersed over the past ∼3 b.y. (Fig. 8B). Because the 13 more-downstream samples possess similar age groups to that in sample 2, likenesses are higher and geographic variation in pair likeness values decreases. This observation is consistent with that of Mapes (2009) who noted that upstream detrital zircon ages predominantly reflect Andean sources, that the predominance of Proterozoic grains increases downstream, and that, even in the lowermost river reaches, a significant portion of zircon grains are of Andean origin.

### Example 3: Modern French Broad River Transect

Hietpas et al. (2011) studied alluvial samples collected along the French Broad River as it transverses the southern Appalachian orogen (Fig. 10A). Their data consist of 11 samples comprising 836 zircon ages (mean n = 70 per sample) that collectively span ∼84 km. All 11 PDPs are dominated either by a ca. 450 Ma Taconian peak or a broad ca. 1550–1175 Grenvillian peak (Fig. 10B). Grenvillian ages dominate the most upstream sample and the 5 most downstream samples (Fig. 10B). Comparison of all 55 possible PDP pair likenesses versus their geographic separation show that likeness (*L*) decreases with increasing spatial separation (*S*) as: *L* = 71% – 0.0028*S* (Table 1); for each kilometer of separation between sample sites, PDP likeness decreases by 0.28% (Fig. 10C). Moreover, changes in contiguous pair likenesses are readily appreciated in a context of mapped downstream change in age density (Fig. 10B). Greatest dissimilarity occurs between samples 1 and 2 as Taconic ages come to predominate over Grenville ages and between samples 6 and 7 where this dominance is reversed. Between these transitions, however, downstream likenesses (Fig. 10D) generally increase at rates between 0.83% per kilometer (samples 1–6) and 0.91% per kilometer (samples 7–11). Although in the same direction, these downstream increases (which span a few tens of kilometers) are much greater than that observed among contiguous Amazon River samples (0.029% per kilometer) which span many hundreds of kilometers (Table 1). Moreover, while these changes perhaps reflect the progressive downstream mixture of grain age populations, each run of increasing likeness only spans six samples; it is therefore entirely possible that apparent downstream likeness increases are largely spurious.

### Example 4: Quaternary Australian Coastal Transect

Sircombe (1999) determined 716 zircon ages among 10 Quaternary coastal sand samples (average n = ∼72 grains per sample) collected along ∼1601 km of the eastern Australia shore. Here, strong southeasterly waves and a narrow continental shelf result in littoral drift from south to north (Fig. 11A). Because of the high-energy character of this coast, one might presuppose that zircon ages would be homogenized. However, as noted by Sircombe (1999), sample PDPs exhibit strong regional differences, with zircon from the Lachlan orogen (ca. 405 Ma) predominating in the south, those from an eastern Antarctic orogen (ca. 550 Ma) prevailing across the central area, and those from the New England orogen (ca. 230 Ma) dominating in the north (Fig. 11B).

As is the case with the Amazon River, Grand Canyon, and French Broad River samples, assessment of all 45 possible PDP pair likenesses versus their geographic separation shows that littoral sand likeness (*L*) decreases with increasing spatial separation (*S*) as: *L* = 60.5% – 0.00015*S* (Table 1); for each kilometer of separation between sample sites, PDP likeness decreases by 0.015% (Fig. 11C). Representation of changes in contiguous pair likenesses as a function of lateral position along the eastern Australian coast is obfuscated because two locations were sampled more than once. Samples 6 and 7 were collected near Newcastle, New South Wales (Fig. 10A), from each of adjacent twin barriers that exist along parts of this coast, and samples 2, 3, and 4 are all from bayhead beaches along a large embayment (Twofold Bay) on the southern coast of New South Wales; samples 3 and 4 were collected 50 months apart from the same locality. As a result, downdrift changes in PDP pair likenesses comprise 8 localities, 6 representing possible average likenesses among 5 samples noted above, and 2 between localities with one sample each. PDP pair likenesses among contiguous eastern Australian samples generally increase in a downdrift direction, but only a fraction of the variance in likenesses correlates with geographic position. While samples 3 and 4, which were collected from the same location, also exhibit the highest (∼77%) likeness value (Fig. 11D), likenesses between these and sample 2, which was also collected along Twofold Bay, are among the lowest (∼51% and 54%, respectively). This suggests that localized processes such as the influx of river sediment and effects of mass wasting are sufficiently important so as to largely overwhelm any homogenizing influence that might result from processes associated with regional-scale littoral transport.

## DISCUSSION

The primary goal of this study has been to develop and apply a methodology for assessing temporal and spatial change in resemblances among detrital zircon age distributions. The quantification of likeness between pairs of detrital zircon age populations serves to highlight their similarities and differences. When interpreted in the context of spatial and/or temporal variation, this metric additionally serves to focus attention on various aspects of age frequencies that ultimately shed light on questions concerning siliciclastic provenance, sedimentary basin evolution, and complex tectonic processes.

Here we assert that, owing to several practical and esthetic qualities, the likeness metric is perhaps the better of the several that have been proposed to quantify differences and samenesses among detrital zircon age populations. However, save *p*-values of K-S, which exhibits an approximately power-law relation to other measures (Fig. 5F), most of the other measures of sameness (e.g., Fig. 5) could be applied in a comparable manner. Perhaps the most important point to be made here is that some attempt should be made to apply some measure of similarity among population PDPs. Above and beyond specific choice of likeness metric, it is significantly more important that an effort be made to quantify spatial and temporal changes in differences among sample grain ages; this will become increasingly true with the continued development of new procedures for the rapid collection of large amounts of detrital age data.

The results from our study demonstrate that stratigraphic and geographic separation among sample sites serves to impart a greater amount of dissimilarity among sample PDPs than can be explained by the random sampling of a master population. This separation is readily quantified using the likeness metric. However, we only considered three modern sets of data here; spatial decrease in likenesses among fluvial (Amazon and French Broad Rivers) and littoral (eastern Australia) PDPs occur at rates of 0.015% per kilometer, 0.275% per kilometer, and 0.015% per kilometer, respectively (Table 1). As might be expected, greater geographic separation among samples results in greater dissimilarity. While this range exceeds an order of magnitude, the lateral distances among samples represented by these systems (2742, 84, and 1601 km, respectively) exhibit a similar range such that the net decrease in pair likenesses over each of the geographic spans in question are similar (42.0%, 23.2%, and 24.2%, respectively; Table 1). The degree to which these values are representative of other sedimentary systems awaits further evaluation.

Among the three Quaternary sedimentary systems examined in this study, likenesses among contiguous pairs of PDPs all exhibit trends suggesting down-system increases in sameness. Source-toward-sink increases in likenesses among fluvial (Amazon and French Broad Rivers) and littoral (eastern Australia) system PDPs occur at rates of 0.029% per kilometer, 0.871% per kilometer, and 0.005% per kilometer, respectively (Table 1). Given the lateral distances represented by each set of samples, total increases in pair likenesses are comparable for the two fluvial systems (36.5% and 65.7%, respectively), and are somewhat lower for the single littoral system (5.8%). Although it seems obvious that such changes should also be critically dependent on degrees of provenance heterogeneity and zircon abundances that characterize any sediment source region, these results at least hint that sediment ages may become more homogenized in fluvial than in littoral settings. The degree to which these values are representative of other sedimentary systems also awaits further evaluation.

Finally, the succession of samples from the Grand Canyon exhibits a trend of decreasing pair likenesses with increasing stratigraphic separation that is similar to that seen with increasing geographic separation for sediment samples from the Amazon and French Broad Rivers and the eastern Australian coast. Not surprisingly, greater sample proximity in space and in time is manifest as greater likenesses between pair age frequencies. Conversely, contiguous sample pair likenesses exhibit a statistically insignificant change upsection in the Grand Canyon. However, given that the Grand Canyon succession comprises no less than three continent-wide sedimentary sequences (Meyers and Peters, 2011; Sloss, 1964) separated by significant unconformable surfaces (Fig. 1A), it may be that the density of samples is simply insufficient to capture any systematic up-sequence changes in PDP likenesses that may exist.

In addition to their value in the assessment of sediment provenances, detrital zircon ages offer great potential in understanding the dynamics of sediment transport in a variety of depositional systems. Studies such as those by Amidon et al. (2005a, 2005b) on fluvial sediment mixing in modern Himalayan rivers, by Niemi (2012) on Miocene rivers of Death Valley (western United States), and by Saylor et al. (2013) on modern rivers in the Colombian foreland are excellent examples of this approach. We conclude by suggesting that both in the cases of provenance assessment as well as those directed toward understanding processes of sediment transport, there is a need for quantitatively measuring relative similarities and differences among zircon age populations both in space and in time. The development of new procedures for the generation of very large amounts of detrital zircon age data is probably just now in its infancy. As of June 2013, the American Geosciences Institute’s GeoRef database lists some 890 peer-reviewed papers with the term “detrital zircon” appearing in their titles. Virtually all deal with grain U-Pb ages; virtually all have appeared over the past several decades; and their numbers comprise a nearly perfect logarithmic increase, doubling in number every ∼3.4 years. In the not-far-distant future, it should be possible to truly quantify variations in such data sets in a real three-dimensional framework of temporal and geographic coordinates. In so doing, it will be increasingly necessary to routinely quantify resemblances among zircon age populations. Differences in unitized PDPs represented as sample pair likeness afford a straightforward approach to this need.

## Acknowledgments

Many individuals have aided in many ways during the formulation and completion of this study. We sincerely thank George Gehrels for clarification in the computation of his overlap and similarity metrics. We thank Marion (Pat) Bickford, David Gombosi, Greg Hoke, and Linda Ivany for discussions, comments, and/or critical evaluations of early drafts of the manuscript. This paper has benefited greatly from substantial reviews by Bill Dickinson, Luke Beranek, Quentin Crowley, and Nancy Riggs. This research was in part supported by the National Science Foundation (EAR #0635643 and EAR #0635688).

## Footnotes

↵* Current address: Department of Geoscience, University of Wisconsin–Madison, Madison, Wisconsin

↵

^{1}GSA Data Repository item 2013331, Excel file exemplifying calculation of the likeness metric from sample data, is available at http://www.geosociety.org/pubs/ft2013.htm or by request to editing{at}geosociety.org.Science Editor: Nancy Riggs

Associate Editor: Quentin G. Crowley

- Received 17 March 2013.
- Revision received 4 June 2013.
- Accepted 7 August 2013.

- © 2013 Geological Society of America