Sediment unmixing using detrital geochronology

Earth and Planetary Science Letters 477 (2017) 183–194

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Earth and Planetary Science Letters www.elsevier.com/locate/epsl

Sediment unmixing using detrital geochronology Glenn R. Sharman a,∗ , Samuel A. Johnstone b a b

Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin, Austin, TX, USA U.S. Geological Survey, Central Mineral and Environmental Resources Science Center, Denver, USA

a r t i c l e

i n f o

Article history: Received 19 April 2017 Received in revised form 18 July 2017 Accepted 25 July 2017 Available online xxxx Editor: A. Yin Keywords: detrital geochronology zircon sediment unmixing mixture modeling source to sink

a b s t r a c t Sediment mixing within sediment routing systems can exert a strong influence on the preservation of provenance signals that yield insight into the effect of environmental forcing (e.g., tectonism, climate) on the Earth’s surface. Here, we discuss two approaches to unmixing detrital geochronologic data in an effort to characterize complex changes in the sedimentary record. First, we summarize ‘topdown’ mixing, which has been successfully employed in the past to characterize the different fractions of prescribed source distributions (‘parents’) that characterize a derived sample or set of samples (‘daughters’). Second, we propose the use of ‘bottom-up’ methods, previously used primarily for grain size distributions, to model parent distributions and the abundances of these parents within a set of daughters. We demonstrate the utility of both top-down and bottom-up approaches to unmixing detrital geochronologic data within a well-constrained sediment routing system in central California. Use of a variety of goodness-of-fit metrics in top-down modeling reveals the importance of considering the range of allowable that is well mixed over any single best-fit mixture calculation. Bottom-up modeling of 12 daughter samples from beaches and submarine canyons yields modeled parent distributions that are remarkably similar to those expected from the geologic context of the sediment-routing system. In general, mixture modeling has the potential to supplement more widely applied approaches in comparing detrital geochronologic data by casting differences between samples as differing proportions of geologically meaningful end-member provenance categories. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The sedimentary record is an archive of global change over geologic time. It records the influence of tectonism, climate, sea level, and anthropogenic effects on the evolution of the Earth’s surface. Tracking variations in sediment provenance is an effective means of determining the influence of environmental change on sediment generation, transport, and deposition within sedimentary systems (Dickinson, 1974; Graham et al., 1986; Gehrels, 2014; Fildani et al., 2016; Mason et al., 2017). Temporal and spatial changes in sedimentary provenance may be signals of environmental change acting on the sediment routing system, including changing catchment drainage boundaries and/or differential erosion within existing drainage boundaries that reflect tectonic, climatic, and/or eustatic controls (Romans et al., 2016; and references within). In some cases, the presence of geologically distinctive materials in the sedimentary record makes interpreting changes in sedimentary provenance relatively straightforward. However, the

*

Corresponding author at: Department of Geosciences, University of Arkansas, Fayetteville, AR, USA. E-mail address: [email protected] (G.R. Sharman). http://dx.doi.org/10.1016/j.epsl.2017.07.044 0012-821X/© 2017 Elsevier B.V. All rights reserved.

complex process of sediment mixing that occurs during sediment transport can also obscure primary provenance signals, hampering efforts to decipher changes in sedimentary provenance (Fig. 1). Geochronology of detrital minerals (hereafter, detrital geochronology) has emerged as a leading method to investigate the provenance of the sand- and silt-sized fraction of sediment (e.g., Fedo et al., 2003; Gehrels, 2014). To the extent that sediment source areas produce unique detrital age distributions, downstream sediment that is well mixed will reflect the relative sediment contributions from each of these sediment source areas (Fig. 1). More specifically, a given daughter detrital age distribution (D o ) can be modeled as a linear mixture of the n parent detrital age distributions that contributed to that daughter:

Dm =

n 

φi P i

(1)

i =1

Where P i is the ith parent distribution and φi is the relative contribution, or ‘mixing coefficient’ of the ith parent, and D m is the modeled daughter distribution that best characterizes the actual, D o . Here, the mixing coefficients must sum to 1 to honor the assumption that the daughter was sourced entirely and only from

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Fig. 1. Hypothetical sediment routing system characterized by four source areas (blue, red, yellow, and green; modified from Romans et al., 2016). Although not depicted, windblown sediment can also constitute an important source of sediment in some systems. The relative frequencies of a generic categorical variable (nonnegative, sums to 1) are shown as probability density plots. Black arrows indicate longshore sediment transport. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

these parents. The distributions P and D describe the relative likelihood of a mineral grain of a particular age being found within a sample. Common practice is to estimate these distributions based on observed mineral ages in a geologic sample, often on the order of 60–300 individual observations. Detrital geochronologic distributions are commonly constructed by summing Gaussian distributions with a standard deviation that is defined by either the analytical uncertainty of the analysis (probability density plot, PDP) or a specified bandwidth (kernel density estimator, KDE) (Vermeesch, 2012). A number of approaches have been developed to unmix sedimentary data, with a particular focus on grain size distributions (Weltje, 1997; Orpin and Woolfe, 1999; Weltje and Prins, 2003, 2007; Dietze et al., 2012; Paterson and Heslop, 2015; Yu et al., 2016). However, approaches to sediment unmixing using detrital geochronologic and/or thermochronologic data have not been widely developed or applied, with some notable exceptions (e.g., Amidon et al., 2005a, 2005b; Enkelmann and Ehlers, 2015; Kimbrough et al., 2015; Mason et al., 2017), despite these data types becoming increasingly used in sedimentary provenance studies that aim to reconstruct ancient source regions and paleocatchment boundaries (Gehrels, 2014; Lawton, 2014). Compared to grain size data, detrital age distributions may be better suited to tracing sediment mixing due to the specificity with which geochronology can identify particular source terrains. For these reasons, detrital geochronology may provide a particularly useful, but underutilized, data type for sediment unmixing calculations. This study aims to review existing numerical approaches to sediment unmixing and provide recommendations for application of these techniques to detrital geochronologic datasets. In general, we find that unmixing calculations fall into two end-member categories: (1) ‘top-down,’ mixing models, and (2) ‘bottom-up,’ unmixing models (Fig. 2). To our knowledge, only top-down mixing models that forward model daughter populations by linear mixing of specified parent distributions have been applied to detrital geochronologic datasets (e.g., Amidon et al., 2005a, 2005b; Fletcher et al., 2007; Kimbrough et al., 2015; Mason et al., 2017). However, we contend that at least one bottom-up unmixing algorithm that was originally designed for grain size data (Paterson and Heslop, 2015) can also be effectively applied to detrital age distributions. This conclusion is supported by observations from Monte Carlo simulations of synthetic detrital age distributions. In the majority

Fig. 2. Schematic depiction of two general approaches to numerical unmixing of sedimentary data. Model inputs (typically measured distributions) are shown with solid coloration, and model outputs (modeled distributions) are shown as partially transparent with a dashed outline. Top-down approaches use defined age distributions of parents, or mixing end-members, to model the best-fit combination to daughter mixtures (e.g., Amidon et al., 2005a, 2005b). Bottom-up approaches attempt to use a number of daughter distributions to model end-member, or parent, distributions (e.g., Dietze et al., 2012). (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

of synthetic cases, comparisons between actual and modeled parent age distributions yield at least as good of agreement as was observed in 95% of cases where a distribution was compared to a derivative of itself constructed from random sampling. Finally, we provide a case study that demonstrates the utility of both top-down and bottom-up mixture modeling in a well-constrained, modern fluvial-to-marine sediment routing system along the margin of central California (Sickmann et al., 2016). 2. Review of approaches to sediment unmixing 2.1. Top-down sediment mixing models Top-down sediment mixing models use defined parent distributions to forward model daughter distributions. Mixing coefficients in this method are based on the best (or set of equally likely) values computed from an objective comparison between observed, D o , and modeled, D m , daughter distributions (e.g., a goodnessof-fit metric) (Equation (1), Fig. 2A). Many goodness-of-fit metrics have been defined for comparing detrital geochronologic data, including measures of the differences between cumulative distribution functions (e.g., D max and V max ) and the agreement between the shape of PDPs or KDEs (e.g., similarity, cross-correlation of PDPs) (see Saylor and Sundell, 2016, for a thorough review). In this approach, parent distributions must be known independently. In studies of modern sedimentary systems, parent distributions may be approximated by measurement of sediment within source area tributaries (e.g., Amidon et al., 2005a; Kimbrough et al., 2015; Mason et al., 2017) and/or measurement of bedrock distributions within the source area catchment (e.g., Amidon et al., 2005a; Saylor et al., 2013). In studies of ancient sedimentary systems, parent distributions cannot usually be measured directly and must

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instead be approximated from other sources of information (e.g., samples from the inferred bedrock source area and/or samples from upstream sedimentary units; Fletcher et al., 2007). In general, top-down sediment mixing models aim to answer the question: How much of each of these parents makes up this daughter? This approach is well suited to circumstances in which the character of the parent distributions is well known or can be independently estimated. A number of studies have used top-down sediment mixing models to identify the mixing coefficients that best produce a set of daughter distributions within modern and ancient fluvial systems (Amidon et al., 2005a, 2005b; Saylor et al., 2013; Kimbrough et al., 2015; Mason et al., 2017). For example, Amidon et al. (2005a) sampled basement and modern fluvial systems to characterize parent age distributions and zircon concentration, or fertility, within lithologic units in central Nepal. Differences between expected mixing coefficients (based on catchment area ( A) and a coefficient describing the concentration of the detrital mineral of interest in a unit volume (c), i.e., zircon fertility; Moecher and Samson, 2006) and observed mixing coefficients were accounted for by considering differences in erosion rate (d) among the tributary catchments (Amidon et al., 2005a, 2005b). For n tributary catchments, this relationship can be generalized as follows:

φi = A i d i c i

n 

A jd jc j

(2)

j =1

Kimbrough et al. (2015) build upon these efforts by developing a top-down mixing approach that evaluates a range of possible mixing coefficients in addition to the best-fit mixture to assess provenance changes between the modern and early Pliocene Colorado River in the southwestern United States. In this case, mixing calculations are based on a modification of the Kolgomorov– Smirnov (K–S) statistic that uses the maximum vertical separation between modeled and daughter cumulative age distributions (D max ) as a goodness-of-fit metric (Fletcher et al., 2007; Kimbrough et al., 2015). This approach yields a confidence interval based on a generalization of K–S statistic p-values that allows for an assessment of the range of possible parent mixtures that could produce the daughter age distribution. 2.2. Bottom-up sediment unmixing models Bottom-up sediment unmixing models are distinct from topdown approaches in that they aim to reproduce parent (endmember) distributions based solely on analysis of daughter distributions (Fig. 2). This approach has the advantage of not requiring independent knowledge of the nature of parent distributions. Thus, bottom-up unmixing models are best suited for circumstances when the nature of parent distributions are not well constrained and/or when a sufficient number of well-characterized daughter distributions are available. This approach seeks to answer the question: What parent distributions combined to form these daughters? However, while the parent distributions defined for the top-down approach contain an explicit geologic context, modeled parent distributions require interpretation to establish this geologic context. A large body of literature exists on numerical approaches to unmix sedimentary data using a bottom-up approach (e.g., see reviews in Heslop, 2015; Paterson and Heslop, 2015). Commonly applied to grain size distributions, these approaches have been termed “end-member modeling” (Weltje, 1997), “end-member modeling analysis” (Dietze et al., 2012, 2014; Yu et al., 2016) or “end-member analysis” (Paterson and Heslop, 2015). To our knowledge, this type of bottom-up unmixing analysis has not been applied to detrital geochronologic distributions despite the similarity

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with grain size distributions: both are compositional, non-negative data that sum to 1. Despite these similarities, there are also some important differences between these data types that may bear upon the applicability of existing bottom-up unmixing approaches to detrital geochronologic data. For example, detrital geochronologic datasets may be characterized by much lower sample to variable ratios than typical grain size datasets. Grain size distributions, typically expressed in phi (φ ) units, may typically contain 60–100 grain size bins, or variables (Dietze et al., 2012; Paterson and Heslop, 2015; Yu et al., 2016). Detrital geochronologic data are often summarized in distributions discretized in 1 Ma over the span of observed geochronologic ages (i.e., up to the age of the Earth). Thus depending on the span of detrital ages and the chosen discretization interval, detrital geochronologic datasets may typically have >500 variables. Low sample-to-variable ratios have been noted to lead to numerical instability in some bottom-up algorithms (e.g., Dietze et al., 2012). Grain size datasets are typically characterized by relatively continuous, smoothly varying distributions, whereas detrital geochronologic distributions commonly consist of complex, multimodal peaks that may be separated by regions with no grain ages (probability of zero). Furthermore, given the greater expense and effort of analyzing detrital mineral ages, the total number of daughter samples available may be relatively low when compared to typical grain size datasets that may be acquired efficiently through laser particle size analyzer technology. 3. Bottom-up unmixing of synthetic detrital age distributions 3.1. Overview of approach to Monte Carlo simulation of synthetic datasets To assess how well bottom-up unmixing approaches translate to the complicated, multi-modal distributions characteristic of detrital geochronologic data (Fig. 2), we performed Monte Carlo simulations of synthetic mixtures of detrital age distributions (Fig. 3). We used the nonparametric approach to end-member analysis outlined in Paterson and Heslop (2015) that casts the mixing of all daughters (e.g., Equation (1)) as the multiplication of matrices describing the mixing coefficients for each daughter and the parents. To solve this system of equations, Paterson and Heslop (2015) use an algorithm that employs hierarchical alternating least squares nonnegative matrix factorization (Lee and Seung, 1999; Chen and Guillaume, 2012) to estimate end-members from the data itself through an iterative process. The result is generation of a specified number of end-member distributions and estimation of the abundances of these end-members in each daughter (Fig. 3). The Paterson and Heslop (2015) algorithm was chosen based on preliminary tests that demonstrated this algorithm to be well suited for unmixing detrital age distributions. A variety of other approaches to end-member analysis also exist and may be useful for analyzing detrital geochronologic datasets. Although these models differ in detail, they are similar in their casting of daughter distributions as linear combinations of parents with mixing coefficients that are not dependent on any physical attributes of the sedimentary system (Weltje, 1997; Orpin and Woolfe, 1999; Weltje and Prins, 2003, 2007; Dietze et al., 2012; Paterson and Heslop, 2015; Yu et al., 2016). Synthetic parent age distributions were generated by randomly selecting a specified number of age peaks (2–15) from 0 to 4000 Ma (Fig. 3A). Age peaks are modeled as normal distributions with error that follows a power law relationship with age based on a modeled fit to a database of more than 100,000 detrital zircon U– Pb ages acquired through laser ablation inductively coupled plasma mass spectrometry spanning ages from ∼0 to 4000 Ma. Each parent is combined in a randomly defined mixture in 5% increments

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parametric end-member analysis approach of Paterson and Heslop (2015). We performed a sensitivity analysis in which we define a base case that has 4 parents (end-members), 5 to 7 age peaks per parent, 100 grains per daughter, and 20 daughters (Fig. 3). These parameters were chosen to model the ability of a typical geochronologic dataset to resolve a moderately complex mixture of parent age distributions. We then allow each of these parameters to vary independently from the base case over a range of reasonable scenarios, and the experiment is repeated 50 times (Fig. 4). We assess goodness-of-fit of the modeled daughters by comparing modeled φm and actual φe abundances (Fig. 3C). The r 2 PDP metric is used to assess the goodness-of-fit for the modeled parent distributions (Fig. 3A), following the recommendations of Saylor and Sundell (2016). We identified which modeled parent distribution corresponds to the actual parent by identifying the single best match and assigning that as a pair, removing that pair from the remaining options, identifying the next best match and assigning that as a pair, and repeating until all pairs have been assigned. How much variability in r 2 PDP can be attributed to random variation from a limited sample set? For each set of parameters (Fig. 4), we perform a second set of Monte Carlo experiments to assess the quality of r 2 PDP values. We (1) create a parent distribution following the procedure above; (2) draw the specified number of ages from it; (3) use the ages and errors of these grains to construct a new derived distribution; (4) compute the r 2 PDP between these synthetic and derived distributions; and (5) find the value of r 2 PDP that defines the lowest 5% of values from 105 iterations. In this way, we identify the r 2 PDP value that we expect to exceed in 95% of cases when comparing a PDP derived from a sampling of a distribution to the distribution itself, given the complexity of that 2 PDP. We refer to this significance criterion as rcrit . 3.2. Summary and interpretation of simulation results

Fig. 3. Example Monte Carlo simulation of bottom-up unmixing of synthetic detrital age distributions (Paterson and Heslop, 2015), using parameters from the base case. A) Four parent end-members are defined with 5–7 randomly defined age peaks (solid line). End-members (dashed line) are modeled using the nonparametric approach outlined in Paterson and Heslop (2015), and r 2 PDP values indicate the goodness-of-fit between the modeled and actual parent distributions. B) 20 daughter age distributions are generated by sampling 100 grains from random mixtures of the four parents. C) Comparison of true and modeled parent abundances, or mixing coefficients (φ ), in each of the 20 synthetic daughters (panel B). Black dashed line highlights a 1:1 relationship.

We present a summary of the Monte Carlo simulations in Fig. 4. Median modeled daughter abundances are within an absolute difference of 10% of the actual end-member abundance for all cases, except for scenarios with a low number of daughters (5 or 10). For the four base-case scenarios, median daughter abundance error is approximately 8% (Fig. 4). Parent age distributions are able to be reproduced with r 2 PDP values of ≥0.70 for all cases except the six-parent mixture scenario (median r 2 PDP of 0.63) and the five daughters scenario (median r 2 PDP of 0.52). The four basecase scenarios yielded median r 2 PDP between 0.82 and 0.84. For 2 the base case, we identified a value of rcrit of ∼0.8, with this value decreasing to ∼0.75 for distributions drawn with small numbers of grains and/or that are composed of large numbers of age peaks (Fig. 4). The median modeled end-member r 2 PDP value is approx2 imately equal to or better than rcrit in all cases but four (5 or 6 parents and 5 or 10 daughters) (Fig. 4). In other words, ≥50% of the runs for all but four scenarios have modeled end-members that are indistinguishable from the actual parent distribution. However, it is only in the case of low numbers of parents, or large numbers of daughters, that 5% or less of modeled distributions have r 2 PDP 2 above rcrit .

where the sum of the mixing coefficients equals 1. A number of ages are then drawn from the mixture according to its probability density to simulate the number of grains analyzed in a detrital geochronologic sample (Fig. 3B). This process is repeated for a number of daughters, each drawn from a different random mixture of the parent age distributions (Fig. 3B). Finally, end-member analysis of the synthetic daughters is performed using the non-

3.2.1. Number of parents and age peaks per parent Increasing the number of parents within the mixture results in a decrease in goodness-of-fit. The median absolute discrepancy between modeled and actual daughter end-member abundances ranges modestly from 6.6% (binary mixture) to 9.1% (six-parent mixture) (Fig. 4). The number of parent end-members has a relatively larger influence on parent r 2 PDP values, which decrease from 0.92 (binary mixture) to 0.63 (six-parent mixture). Median

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Fig. 4. Boxplots showing the results of Monte Carlo simulation of unmixing synthetic detrital age distributions using the nonparametric approach of Paterson and Heslop (2015). The top row displays the absolute difference between modeled φm and actual φe daughter end-member abundances. The bottom row displays the cross-correlation (r 2 ) of PDP between the modeled end-member and the actual parent age distribution. The base case scenarios are shaded in gray. Outliers are rejected and not plotted if 2 they are greater than 1.5 times the interquartile range (shown by whiskers) outside the 25th and 75 percentiles of the data. Dashed line in lower row shows rcrit . 2 parent r 2 PDP values are greater than rcrit for cases with four or fewer parents. Goodness-of-fit metrics for both daughters and parents show a moderate decline in accuracy with increasing number of age peaks, a proxy for the complexity of the parent detrital age populations. Absolute errors in daughter abundances range from 7.3% to 10%, and parent r 2 PDP values decrease moderately from 0.88 to 0.75 with increasing numbers of age peaks (Fig. 4). More than 50% of results from each scenario have parent r 2 PDP values that are 2 greater than rcrit .

3.2.2. Number of daughters and grains per daughter The number of daughters shows a relatively strong influence on the accuracy of the model, with cases with low numbers of daughters (5 or 10) yielding relatively poor goodness-of-fit for both parents and daughters (Fig. 4). The five-daughter case resulted in the worst performance of any modeled scenario with a median 2 daughter abundance error of 20% and r 2 PDP values < rcrit in over 75% of cases. Increasing the number of daughters to 20 (the base case) results in an improvement of the median abundance error to 2 7.6%, and median r 2 PDP values are greater than rcrit for cases with 20 or more daughters (Fig. 4). The number of grains per daughter shows only a moderate influence on modeled daughter and parent accuracy. Median daughter abundance error ranges from 6.6% to 10%, and median parent r 2 PDP values range from 0.75 to 0.86 with 50% or more cases hav2 for all tested scenarios. ing r 2 PDP values greater than rcrit

daughters and differences in the relative contributions of different parents that are necessary to extract the nature of parent distributions from a suite of daughters. Our synthetic modeling further suggests that the number of grains per daughter is of lesser importance in influencing accuracy (Fig. 4). A similar proportion of modeled r 2 PDP values are 2 less than rcrit whether 50 or 150 grains are drawn per daughter. While the number of samples required to characterize a distribution depends on the distribution itself, 50 ages is below typical recommendations for the number of grain analyses needed for a provenance study (Dodson et al., 1988; Vermeesch, 2004). The relatively minimal effect of the number of grains per daughter on our synthetic modeling results may be related to our base case still having 20 daughters. While 50 ages may be insufficient to accurately characterize typical detrital geochronologic distributions, when aggregated over 20 samples (effectively 1,000 total ages), the ‘bottom-up’ unmixing algorithm employed here is apparently able to resolve a relatively small number of parents (4) despite the uncertainty present in individual daughters. In light of these results, we contend that bottom-up unmixing approaches are likely to yield important insights into provenance mixing, particularly when a suitable number of daughter samples are available to characterize a sediment routing system that is expected to be relatively simple (e.g., only composed of a few parents) (Fig. 4). 4. Case study: modern sediment routing along the central California margin

3.3. Interpretation of results 4.1. Study area overview Our results demonstrate that bottom-up sediment unmixing approaches can be successfully applied to detrital geochronologic datasets under a range of reasonable parameters (Fig. 4). However, accuracy of modeled parent distributions and daughter abundances is strongly dependent on the ratio of the number of parents to daughters (e.g., Dietze et al., 2012). This conclusion is consistent with the recommendation of Paterson and Heslop (2015) that at least 10 daughters be used for end-member unmixing analysis. This is a natural conclusion as it is both the similarity between

To illustrate both top-down and bottom-up approaches to sediment unmixing using detrital geochronology, we consider a dataset from central California that provides an opportunity to examine sediment mixing in fluvial, littoral, and deep-water environments (Sickmann et al., 2016). The central California margin from Pigeon Point to Big Sur is characterized by steep river catchments that deliver sediment onto a narrow shelf that is locally incised by submarine canyons and gullies (Fig. 5). Sediment is

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Fig. 5. Overview of the Monterey and Sur sediment routing systems in central California (modified from Sickmann et al., 2016). The basemap depicts elevation and seafloor bathymetry overlain on a hillshade map (base from U.S. Geological Survey National Elevation Dataset 30-meter digital elevation model and from NOAA National Centers for Environmental Information, U.S. Coastal Relief Model). Drainage basin outlines of river samples are shown as black lines and rivers are shown as blue lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

delivered to the study area coastline from (1) longshore drift, particularly southward along the Pigeon Point coastline, and (2) river catchments of the San Lorenzo River (∼320 km2 ), Pajaro River (∼3,300 km2 ), Salinas River (∼10,800 km2 ), and Carmel River (∼640 km2 ) (Fig. 5). Due to the present high-stand in sea level, sediment is transported in the littoral zone in the direction of longshore drift until intercepted by the upper reaches of a submarine canyon (e.g., Paull et al., 2005; Covault et al., 2007; Sickmann et al., 2016) (Fig. 5). Within the study area, the Monterey Canyon, Carmel Canyon, and Sur-Partington canyon system have upper reaches in close proximity to the shoreline and are able to capture and route sediment into deep ocean environments (Fig. 5; Paull et al., 2005). Sickmann et al. (2016) describe two sediment routing systems present along this portion of the central California coastline (Fig. 5). (1) The Monterey sediment routing system receives sediment that is routed through the Monterey and Carmel submarine canyons, both of which have upper reaches that are in close proximity to the shoreline (Fig. 5). Although the Carmel Canyon is fed directly by the Carmel River, the Monterey Canyon lacks a large river directly upstream of its head and instead primarily receives

sediment intercepted from the littoral zone that is transported from both the north and south (Paull et al., 2005) (Fig. 5). (2) The Sur sediment routing system extends from the southern edge of the Monterey Peninsula southward along the Big Sur coastline. Sediment eroded from small, high-relief catchments is delivered to the coastline where south-directed longshore drift routes this sediment towards the Sur-Partington canyon system (Sickmann et al., 2016) (Fig. 5). 4.2. Daughter and parent samples Sickmann et al. (2016) used 12 sediment samples collected from beaches and submarine canyons to characterize the routing of sediment to the deep ocean (Figs. 5 and 6). Here, we model these samples as daughter mixtures of three general parent sources (P1–3) that are intended to characterize sediment input from different fluvial systems and longshore drift. Each parent was sourced from a distinct geologic province and contains a unique detrital zircon age distribution (Sickmann et al., 2016) (Fig. 6). Parent source P1 is characterized by a single sample (CAR) from the Carmel River that is sourced from the northwestern Santa Lu-

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cia Range (Fig. 5). P1 is distinct from P2 and P3 in that it contains only a small component of pre-105 Ma zircon (10%) and a much larger component of 90–75 Ma zircon (44%). The Carmel River empties in close proximity to the head of the Carmel Canyon, and thus represents a proxy for sediment entering the Carmel Canyon. Although Sickmann et al. (2016) did not collect samples to characterize sediment input into the Sur sediment routing system, the similarity of beach and submarine canyon samples from south of Carmel Canyon to the Carmel River sample (CAR) suggest that P1 is a good proxy for sediment delivered to the Big Sur coastline (Figs. 5 and 6). P2 is characterized by a sample from the Salinas River (SAL) that is sourced from both the western Diablo Range and the Santa Lucia Range (Fig. 5). P2 is distinct from P3 in that it has a smaller component of pre-105 Ma zircon (46%) and a small component of 90–75 Ma zircon (13%) that is nearly absent in P3. The Salinas River empties to the coastline south of the Monterey Canyon, and provides a proxy for sediment entering the upper reaches of the Monterey Canyon from the south (Fig. 5) (Sickmann et al., 2016). P3 comprises three samples that contain similar detrital zircon age distributions and represents a proxy for the character of sediment entering the Monterey Bay littoral cell north of Monterey Canyon (Figs. 5 and 6). This parent includes (1) beach sediments collected near Pigeon Point (sample ANB) as a proxy for longshore drift-transported sediment entering the system from the north, and (2) river sediments from the San Lorenzo (sample SNR) and Pajaro (sample PAR) Rivers that are sourced from the southern Santa Cruz Mountains and a portion of the central Diablo Range (Fig. 5). All three samples that constitute P3 share very similar detrital zircon age distributions, including a significant component of zircon ages older than 105 Ma (55–63%), that are recycled from non-basement units within the California Coast Ranges, and 105–90 Ma (33–42%) zircon that could reflect either recycling of sedimentary cover rock or first-cycle derivation from Cretaceous plutonic rocks of the Salinian block (Fig. 6) (Sharman et al., 2015; Sickmann et al., 2016). Because of the similarity of zircon ages in these samples, we ignore possible effects of unequal mixing of the sources represented by samples SNR, PAR, and ANB and aggregate the samples without weighting. Unlike P1 and P2, P3 lacks a significant component of 90–75 Ma zircon (1%) (Fig. 6). 4.3. Top-down sediment mixture modeling

Fig. 6. A) Probability density plots and histograms of detrital zircon U–Pb age distributions for samples from central California (modified from Sickmann et al., 2016). P1, P2, and P3 denote samples used as parents in top-down mixture modeling. The number (n) of grains displayed/total grains is indicated. B) Multi-dimensional scaling plot of the same samples shown in (A). Calculations are based on the approach outlined in Vermeesch (2013). Pie diagrams show the proportion of detrital analyses that fall within the specified detrital age populations. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

We perform top-down, forward sediment mixture modeling by conducting an exhaustive search of all mixing coefficients at a specified interval (1%) and identifying the set of mixing coefficients that yield the best match between the daughter and mixture age distributions (Equation (1), Fig. 2A). Rather than rely on a single goodness-of-fit metric, we consider a range of five metrics that are commonly used to assess the goodness-of-fit between detrital age distributions: likeness, similarity, r 2 PDP, D max , and V max (Saylor and Sundell, 2016; and references within). Fig. 7 displays a representative set of top-down mixture models for sample MAN (results for all samples are provided in Table DR1). Note that in some cases the best-fit mixture is defined by the maximum of the goodness-of-fit function (e.g., likeness); in other cases, it is the minimum (e.g., D max ). Although best-fit mixing coefficients range from 3–15% for P1, 0–31% for P2, and 86–66% for P3 (Fig. 7A), the resulting best-fit cumulative distribution functions are nearly identical (Fig. 7B). Fig. 8 presents a summary of mixture modeling results for each daughter sample. In addition to the best-fit mixtures from the five goodness-of-fit metrics described above, we also consider the 95% confidence mixing envelope from Kimbrough et al. (2015) (Fig. 8). In general, top-down mixing calculations tend to cluster within a single region (e.g., sample MLB) or along a linear trend (e.g., sam-

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Fig. 7. Examples of top-down, ternary forward mixture modeling results, using sample MAN as the daughter distribution. A) Ternary plots showing the distribution of five goodness-of-fit criteria for mixtures of P1, P2, and P3 (Fig. 6A). The white circle indicates the best-fit mixture. B) Cumulative distribution plots showing the parent and daughter inputs into the model (left), and the best-fit mixture for each of the five goodness-of-fit criteria in comparison with the daughter distribution (right). (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

ple DR584) (Fig. 8). Although seven of the daughter samples contain at least one best-fit mixture that lies outside of the Kimbrough et al. (2015) 95% confidence interval, the average of the best-fit mixtures all lie within or immediately adjacent to this confidence interval (Fig. 8). Sample GSB contains the widest range of best-fit mixtures, varying from 100% P1 to 100% P2, although it should be noted that GSB also displayed the worst modeled fit of any sample (e.g., this sample did not yield any mixtures within the 95% confidence interval of Kimbrough et al., 2015). This is likely a result of GSB containing a uniquely unimodal age population that is not well represented in any of the defined end-members (P1–3) (Fig. 6). 4.4. Bottom-up sediment unmixing We apply the bottom-up, nonparametric unmixing algorithm of Paterson and Heslop (2015) to the 12 daughter samples within the Monterey and Sur sediment routing systems (Fig. 6). We chose to model three end-members (EM1–3) to allow comparison with the three prescribed parents (P1–3) that represent major sediment inputs into the sediment routing system (Fig. 5) (Sickmann et al., 2016). End-member age distributions for a three-parent scenario are able to account for 97.9% of the total dataset variance. Because sample GSB is unique from all other samples, including samples that represent sediment inputs into the sediment routing system (Fig. 6), we consider an additional scenario that excludes this sample from the unmixing calculations to yield a slightly improved result of 98.3% of total dataset variance explained. To evaluate the results of the bottom-up unmixing calculations, we compare the bottom-up end-members (EM1–3) with the three independent sediment input proxies (P1–3) (Figs. 6 and 9). Bottom-up modeled end-member distributions share a number of similarities with the geologically chosen and empirically defined parents (Fig. 9). EM3 is nearly identical to the combination of samples ANB, SNR, and PAR (P3), which is a proxy for sediment routed into the upper reaches of the Monterey Canyon from the north (Figs. 5 and 8). EM2 has a very similar Cretaceous age population to the Salinas River sample (P2), including a peak age of 95 Ma,

particularly for the scenario in which sample GSB was excluded (Fig. 9). However, EM2 does have a lesser abundance of pre-105 Ma zircon (14%) as compared to the Salinas River sample (46%) (Fig. 9). Finally, EM1 shows a close match with the Carmel River sample (P1), with the exception of having a slightly greater proportion of pre-105 Ma zircon (23% versus 10%) (Fig. 9). To further assess the results of the bottom-up unmixing calculations, we plot the mixing coefficients of the bottom-up endmembers alongside the top-down mixing results (Fig. 8), using the axis of the geologically defined parent (P1–3) for which there is greatest correspondence, as shown in Fig. 9. Despite the bottomup end-members (EM1–3) and top-down parents (P1–3) not being exactly identical (Fig. 9), the bottom-up end-members generally plot within the same region as the top-down end-members (Fig. 8). 4.5. Interpretation of results Both top-down and bottom-up mixture modeling results are consistent with expected sediment routing patterns along the central California margin (Fig. 8). The spatial variation in modeled mixing coefficients suggests that the majority of sediment routed through the Monterey Canyon is derived from the Santa Cruz Mountains, Diablo Range, and Salinas River drainage basin (Fig. 8). Daughter samples from the proximal portions of the sediment routing system (beach or upper submarine canyon) have modeled parent (top-down) or end-member (bottom-up) abundances that largely match the expected sediment input proxy (P2-3) (Fig. 8). For example, beach samples from north of the Monterey Canyon (samples MAN and MLB) yield high abundances of P3, with the average top-down modeling result ranging from 77 to 83% (Fig. 8). The beach sample (MDP) south of the Monterey Canyon yields a high average top-down modeling result for P2 (48%), with D max and V max -based results of 65% for P2 (Fig. 8). As expected, samples from the Monterey Canyon show a predominance of P3 and P2 (Fig. 8). Modeled mixing coefficients in the Sur sediment routing system suggest sediment was supplied from locally adjacent ranges and,

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Fig. 8. Summary of both bottom-up and top-down sediment unmixing results from central California. Fluvial catchments and littoral cells are colored according to interpreted parent source areas (P1-yellow; P2-green; P3-blue). Pie diagrams show the average top-down mixing result for each daughter sample (Table DR1). See Fig. 5 and the manuscript text for additional explanation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to a lesser degree, from the Salinas River catchment (Fig. 8). For example, the upper Carmel Canyon sample (T1135) and a nearby beach sample (GSB) yield high values for P1 (71% and 66%, respectively) for the average top-down result. The two samples from the Sur-Partington canyon system (T958 and T978) both show high scores for P1 (62% and 77% average top-down results, respectively), which is consistent with the inference that P1 is a good proxy for sediment sources in the Big Sur region (Fig. 8). Interestingly, the sample from the lower Carmel Canyon (T693), collected approximately 100 m above its intersection with the thalweg of the Monterey Canyon, is more similar to the other samples from the Monterey sediment routing system than the proximal Carmel Canyon sample (T1135) (Fig. 6B). Our mixture modeling results thus support the interpretation of Sickmann et al. (2016) that sample T693 was predominantly derived from detritus routed through the upper Monterey Canyon and deposited within the lowermost reaches of the Carmel Canyon. Samples from the middle to lower reaches of the Monterey Canyon (samples DR585, DR591, DR584, and T755) show little variation in average end-member proportions, suggesting downsystem homogenization through mixing with P3 scores ranging from 83% to 54% (Sickmann et al., 2016) (Fig. 8). The predominance

of P3 over P2 suggests that a larger fraction of zircon-bearing sediment in the Monterey Canyon was derived from the Santa Cruz Mountains and Diablo Range than the Salinas Valley (Fig. 8). This could theoretically reflect either differences in the sediment load supplied by these regions or the fertility of zircon within the sediments supplied by each region (e.g., Equation (2), Amidon et al., 2005a, 2005b). Although the average mixing results are consistent with known sediment routing patterns, consideration of the range of best-fit mixtures for both top-down and bottom-up unmixing scenarios suggests that modeled mixing coefficients may have uncertainties typically in the range of 10–30% (Table DR1). This finding is consistent with Monte Carlo simulation of synthetic datasets that yield a similar range of daughter end-member abundance errors, and not surprising given the large degree of overlap in our defined parent distributions (Figs. 3 and 4). These uncertainties may help account for why some daughter samples have small components of parent end-members that are not expected to be present based on the geologic context of the sample. For example, sample MAN has an average top-down modeling result of 9% and 14% for P1 and P2, despite our prediction that this sample should be entirely composed of P3 (Fig. 8).

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5. Discussion 5.1. Top-down versus bottom-up approaches to mixture modeling Our analysis suggests that both top-down and bottom-up approaches can be well suited for analysis of detrital geochronologic data. In some cases, unmixing detrital age distributions into endmember components may allow a more refined understanding of changes in sedimentary provenance, spatially and/or temporally, than more traditionally used approaches. For example, proportions of user-defined age populations are commonly used to assess provenance changes (e.g., Sharman et al., 2015), but this approach may be insensitive to changes in source area distributions with distinct, yet overlapping, age populations and depends on the subjective selection of user-defined bin boundaries. This concept is illustrated by the central California margin where all three sediment input groups (P1–3) largely contain overlapping populations of Jurassic to Cretaceous zircon (Fig. 6). While other approaches have been developed to robustly assess the relative similarity or dissimilarity of samples, whether by a goodness-of-fit metric (e.g., r 2 PDP) or multi-dimensional scaling (Vermeesch, 2013) (Fig. 6B), these lack specificity regarding the underlying geologic reason that samples are similar or dissimilar. Mixture modeling has potential to supplement these approaches in casting differences between samples as differing proportions of geologically meaningful endmember provenance categories. However, the choice to apply topdown and/or bottom-up approaches to a given dataset requires consideration of the advantages and disadvantages of each approach. 5.1.1. Top-down mixture modeling: advantages and disadvantages Top-down mixture modeling is best suited for datasets where parent sediment mixtures are well constrained. Thus, this approach may be best suited for modern sedimentary systems where source area catchments can be directly sampled (e.g., Amidon et al., 2005a; Kimbrough et al., 2015). However, top-down mixture modeling may also be applied to ancient systems, provided independent constraints are available on the nature of parent age distributions that may be entering the sediment routing system (e.g., Fletcher et al., 2007). Regardless, our analysis is consistent with Kimbrough et al. (2015) in suggesting that any given best-fit mixture represents only one possible scenario, and there is likely a range of mixing coefficients that could produce any given daughter age distribution. Thus, we recommend avoiding reliance on any single best-fit mixture (Fig. 8). Future work to define a flexible approach for assessing the statistical significance of a particular mixture for different goodness-of-fit metrics would aid these efforts. Top-down mixture modeling also requires the assumption that parent distributions, however derived, adequately characterize the source area distribution. However, most detrital geochronologic datasets are constructed from a limited number of grains analyzed from a relatively small sample (typically a few kilograms) collected in the field. Significant work has gone into trying to understand how many grains need to be analyzed to appropriately characterize the sample (e.g., Vermeesch, 2004). However, it is less well understood, and perhaps harder to quantify, how well a sample from the bank of a river may characterize the entire source area catchment, or how well a lithified sample may represent the detrital age distribution of the formation from which it was derived. These uncertainties may help account for some of the scatter in mixture modeling results for the central California margin (Fig. 8). Additional work is needed to better understand these uncertainties, perhaps by studying well-constrained modern systems, and the influence that this uncertainty may have on top-down mixture modeling.

Fig. 9. Comparison of bottom-up end-members (EM1–3) with the parent source areas (P1–3) used in top-down mixture modeling (Fig. 6) that represent major points of sediment input into the sediment routing system (Figs. 5 and 8). The bottom-up unmixing scenario that includes sample GSB is shown as a dashed black line. Pie diagrams indicate the relative abundance of zircon age populations present in the sediment input proxies (P1–3) and bottom-up end-members (EM1–3) for the scenario that excludes sample GSB. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

5.1.2. Bottom-up mixture modeling: advantages and disadvantages Although bottom-up unmixing approaches have not been widely applied to detrital geochronologic data, we contend that this approach has significant potential for analysis of both modern and ancient sedimentary systems. In both synthetic experiments (Figs. 3 and 4) and a natural study (Fig. 9), bottom-up unmixing is generally able to reproduce expected parent distributions. Thus, bottom-up unmixing approaches have the significant benefit of not requiring independent knowledge of parent distributions, a particular advantage in analysis of ancient sedimentary systems where source areas are no longer preserved or known with confidence. In this regard it is also worth noting that the end-members, or parents, that are produced by bottom-up mixture models (e.g., Equation (1)), may differ slightly from the geologic concept of source areas. End-members in the context of a bottom-up mixture model are the resolvable contributors of unique detrital age distributions, rather than geologic terranes or geographic regions that normally define source areas. However, we anticipate that bottomup end-members may provide insight into the mixing proportions of different source areas, particularly when such source areas produce sediment with distinct detrital age distributions. However, bottom-up unmixing approaches will not likely be effective in circumstances where (1) daughters are well mixed, or homogenized, and little variation exists in parent mixing coefficients, (2) only a small number (e.g., <10) of daughter samples are available, and (3) outlier daughter samples are present in the dataset. Sample GSB provides a good example of the effect of an outlier on bottom-up modeling results (Figs. 6 and 9), as the uniquely narrow distribution of ages in this sample forced the creation of an end-member to describe this sample. In the case of this modern study, we had the context of its geologic setting and neighboring samples to recognize this sample as unique, but such context may not always be available in ancient settings. Furthermore, existing numerical approaches are largely deterministic and do not consider the range of allowable mixtures for a given set of end-members, or the range of allowable end-member distribu-

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tions for a given set of best-fit mixing coefficients. Currently, the decision of how many end-members to specify rests with the interpreter and their knowledge of the geologic system at hand. Alternatively, past efforts have attempted to address similar questions of how many components appropriately characterize geochronologic data and could provide guidance for defining objective, data-driven criteria (e.g., Sambridge and Compston, 1994). Thus, we recommend that bottom-up end-members be interpreted with caution by leveraging geologic insights into whether end-members could represent geologically plausible source area distributions. Ultimately, bottom-up mixture modeling provides a robust approach to analyzing detrital geochronologic datasets where daughter samples are cast in terms of a specified number of endmembers that are chosen to maximize the total amount of explained variance within the dataset (Paterson and Heslop, 2015). In a way, bottom-up unmixing can be viewed as a similar approach to multi-dimensional scaling (Vermeesch, 2013) but with the advantage of there being a logical linkage between end-member distributions and the geological processes of erosion, sediment transport, and mixing. While multi-dimensional scaling axes provide an excellent means of highlighting differences in populations, they lack an obvious geologic meaning (Fig. 6B) (Vermeesch, 2013). 6. Conclusions Sediment provenance may provide important insights into changing environmental conditions on the Earth’s surface over geologic time, but the complex process of sediment mixing during transport within a sediment routing system may hamper efforts to reconstruct such changes. We classify numerical approaches to sediment unmixing into two general categories: top-down and bottom-up mixture modeling. Although previous studies have used top-down approaches to analyze mixing of detrital geochronologic data, we contend that bottom-up approaches, borrowed from the grain size literature (e.g., Paterson and Heslop, 2015), can also be effectively applied to detrital age distributions. This conclusion is supported by Monte Carlo simulation of synthetic detrital geochronologic datasets where median modeled daughter endmember abundances are typically within an absolute difference of 10% of the actual abundance, and the majority of modeled endmember distributions yield r 2 PDP values in the expected range for indistinguishable populations. Application of both top-down and bottom-up sediment mixture modeling to a well-constrained sediment routing system in central California demonstrates the utility and agreement of both approaches. Using a variety of goodness-of-fit metrics in topdown modeling demonstrates a range of plausible best-fit mixtures, suggesting that it is best to avoid relying on any single best-fit mixture. Instead, we recommend considering the range of possible mixing coefficients that could produce a given daughter distribution. Bottom-up modeling of 12 daughter samples from beaches and submarine canyons yields results that are remarkably similar to expected sediment input proxies (parents) to the sediment routing system. Although not widely applied to detrital geochronologic datasets, we contend that bottom-up unmixing has significant potential for application to both modern and ancient sedimentary systems, particularly because this approach does not require independent knowledge of parent age distributions. In general, mixture modeling can provide a powerful tool for exploring detrital geochronologic and/or thermochronologic datasets by casting the differences between samples as differing abundances of geologically meaningful provenance endmembers.

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