Determination of lead isotope endmembers

Introduction

Lead isotope analysis is a widely employed technique for determining the provenance of ancient artefacts by comparing the lead (Pb) isotope ratios of an artefact with those of known ore deposits. However, a significant limitation of this method arises from the complexity of ancient metallurgy. It is improbable that all metal at a given archaeological site originated from a single ore source, and the historical practice of recycling and mixing metals further complicates analysis. Consequently, when metals from two distinct ore sources are combined, the resulting Pb isotope ratios deviate from those of the original sources. This phenomenon is graphically represented as a “mixing line,” which plots the isotopic composition of the mixed metal between the two or more parent ore sources (Stos-Gale 2001).

To mitigate the complexities introduced by metal mixing, (Eshel et al. 2019) proposed a methodology utilizing “endmembers” from a sample group. Endmembers are defined as samples that lie at the extremes of a mixing line and are presumed to represent unmixed metals with the most direct relationship to their parent ore sources. In cases where two primary ore sources are involved in the metal mixture, this approach allows for the clear distinction of three groups: the two end-member groups, which correspond to the unmixed parent materials, and a third group representing the samples that are a result of the mixing process between the two.

While defining endmembers as distinct groups is not always precise, it is preferable to analysing individual data points separately because it provides a more structured and meaningful interpretation of isotopic variability (Eshel et al. 2019). Then, assuming that each endmember reflect a single ore deposit or mining district, provenance is suggested based on the cluster with the highest probability, considering all results (Shnyr et al. 2026).

Use of pb_iso_endmembers()

A major challenge for reproducibility has been the visual identification of endmember clusters. To address this, (Albarede et al. 2024) proposed a method using principal component analysis (PCA) on ²⁰⁴Pb normalized data to determine the number of endmembers in a sample set. This approach maps the maximum and minimum values of the principal component to the isotope vectors.

Trends in the three dimensional space of ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb are determined by the principal component analysis scores of each vector $\vec{p}_{i}$ as $[c_1, c_2, c_3]$ using the stat::prcomp() function. The collection of vectors is considered to represent mixing between two endmembers, $c_a$ and $c_b$, when $c_1$ explains 95% of the variance of the group, where:

$$(c_{a} = \min c_{1}) \mapsto \vec{p_{a}} \;and\; (c_{b} = \max c_{1}) \mapsto \vec{p_{b}}$$

This is achieved using the stats::prcomp() function, which is incorporated into the pb_iso_endmembers() function. In addition, the PCA values are subjected to the Shapiro and Wilk normality using the stats::shapiro.test() function. If PC2 and PC3 are not normally distributed, the function generates a message informing the user that PC2 and PC3 may not be attributed to random noise.

The ASTR object containing the Pb isotope ratios ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb is appended with an end_membr column identifying each sample as either belonging to an endmember group or the mixing group.

pb_iso_endmembers(tel_dor)
#> Warning in .calc_pb_iso_endmembers(x, iso_cols = .pb_iso_cols(), tolerance =
#> tolerance, : PC2 or PC3 are not normally distributed. This may indicate that
#> the variation is not random noise.
#> ASTR table
#> Analytical columns: 206Pb/204Pb, 207Pb/204Pb, 208Pb/204Pb
#> Contextual columns: end_membr 
#> # A data frame: 32 × 5
#>       ID `206Pb/204Pb` `207Pb/204Pb` `208Pb/204Pb` end_membr
#>    <int>         <dbl>         <dbl>         <dbl> <chr>    
#>  1     1          19.0          15.7          39.0 group2   
#>  2     2          19.0          15.7          39.0 group2   
#>  3     3          19.0          15.7          39.1 group2   
#>  4     4          19.0          15.7          39.0 group2   
#>  5     5          19.0          15.7          39.0 group2   
#>  6     6          17.9          15.6          38.0 group1   
#>  7     7          17.9          15.6          38.0 groupmix 
#>  8     8          17.9          15.6          38.0 group1   
#>  9     9          17.9          15.6          38.0 group1   
#> 10    10          19.0          15.7          39.4 group2   
#> # ℹ 22 more rows

However, this method is limited as it only identifies two points, which are unlikely to fully represent all endmembers. While $\vec{p_{a}}$ and $\vec{p_{b}}$ represent the most distinct ends of the set, additional endmembers can be identified using $\vec{p_i}$.

Samples from the same ore body usually share the same geological age and consequently align along a single isochron. While the two-stage model (Stacey and Kramers 1975) provides an imperfect representation of isotopic age, the true distribution of samples from an endmember set follows the slope of the geochron. Therefore, a more accurate definition of endmembers involves identifying them as a set of vectors that lie near a line parallel to the geochron, passing through the principal vector mapped to either the minimum or maximum values of the first principal component of the entire data set. This refined method enables the identification of a comprehensive set of samples that constitute an endmember group.

A vector $\vec{e_i}$, as defined above, is considered an endmember if it lies within a tolerance of $\varepsilon$ from the line $\overline{p_i}$ which passes through $\vec{p_i}$ parallel to the geochron in the Cartesian plane of $\left(\frac{^{206}\text{Pb}}{^{204}\text{Pb}},\frac{^{207}\text{Pb}}{^{204}\text{Pb}}\right)$

$$f(\vec{e}, \vec{p}) = |e_2 - (p_1 - (S\times p_2))| < \varepsilon$$

where $S$ is the slope of the geochron. The values for $\varepsilon$ was chosen as 0.01 by default. This value was manually set for optimal results based on trial and error, and can be further refined based on the data.

In addition to $\varepsilon$, endmember points may be further constrained by clamping their distance from the principal point using the Euclidean distance. This may be required if two distant ores sources share the same geochron.

$$f(\vec{e}, \vec{p}) = ||\vec{e} - \vec{p}|| < d$$

where $d$ is the Euclidean distance. The default parameter is chosen to be Inf, as in most cases this clamping may not be required.

# Example with no clamping
no_clamp <- pb_iso_endmembers(tel_dor)
#> Warning in .calc_pb_iso_endmembers(x, iso_cols = .pb_iso_cols(), tolerance =
#> tolerance, : PC2 or PC3 are not normally distributed. This may indicate that
#> the variation is not random noise.
no_clamp[no_clamp$end_membr == "group2", ]
#> ASTR table
#> Analytical columns: 206Pb/204Pb, 207Pb/204Pb, 208Pb/204Pb
#> Contextual columns: end_membr 
#> # A data frame: 12 × 5
#>       ID `206Pb/204Pb` `207Pb/204Pb` `208Pb/204Pb` end_membr
#>    <int>         <dbl>         <dbl>         <dbl> <chr>    
#>  1     1          19.0          15.7          39.0 group2   
#>  2     2          19.0          15.7          39.0 group2   
#>  3     3          19.0          15.7          39.1 group2   
#>  4     4          19.0          15.7          39.0 group2   
#>  5     5          19.0          15.7          39.0 group2   
#>  6    10          19.0          15.7          39.4 group2   
#>  7    24          18.9          15.7          38.9 group2   
#>  8    26          19.0          15.7          39.1 group2   
#>  9    27          19.0          15.7          39.1 group2   
#> 10    29          19.0          15.7          39.0 group2   
#> 11    30          19.0          15.7          39.0 group2   
#> 12    31          19.0          15.7          39.0 group2

# Example with clamping
clamp <- pb_iso_endmembers(tel_dor, clamp = c(Inf, 0.1))
#> Warning in .calc_pb_iso_endmembers(x, iso_cols = .pb_iso_cols(), tolerance =
#> tolerance, : PC2 or PC3 are not normally distributed. This may indicate that
#> the variation is not random noise.
#> Warning in .calc_pb_iso_endmembers(x, iso_cols = .pb_iso_cols(), tolerance =
#> tolerance, : Endmember group has less than two points. The likelihood of a
#> point being an endmember is low.
clamp[clamp$end_membr == "group2", ]
#> ASTR table
#> Analytical columns: 206Pb/204Pb, 207Pb/204Pb, 208Pb/204Pb
#> Contextual columns: end_membr 
#> # A data frame: 1 × 5
#>      ID `206Pb/204Pb` `207Pb/204Pb` `208Pb/204Pb` end_membr
#>   <int>         <dbl>         <dbl>         <dbl> <chr>    
#> 1    10          19.0          15.7          39.4 group2

This results in the creation of two sets of endmembers, $E_a$ and $E_b$, where:

$$E_i = \left\{\vec{p_i}, \vec{e_i^1},\ldots, \vec{e_i^n}\right\}$$

In the function output $E$ is represented as group1 and group2.

However, if $|E_i| < 2$, then it is difficult to consider only $\vec{p_i} \in E_i$ as an endmember, and so $p_i$ is considered only a possible endmember. When this happens, the package informs the user with a message.

Reference

Albarede, Francis, Gillan Davis, Liesel Gentelli, et al. 2024. “Bullion Mixtures in Silver Coinage from Ancient Greece and Egypt.” Journal of Archaeological Science 162 (February): 105918. https://doi.org/10.1016/j.jas.2023.105918.

Eshel, T, Y. Erel, N. Yahalom-Mack, O. Tirosh, and A. Gilboa. 2019. “Lead Isotopes in Silver Reveal Earliest Phoenician Quest for Metals in the West Mediterranean.” Proceedings of the National Academy of Sciences 116 (13): 6007–12. https://doi.org/10.1073/pnas.1817951116.

Shnyr, Evgeny, Tsvi Kuflik, Karan Desai, and Tzilla Eshel. 2026. “Determining the Origins of Phoenician Silver: Exploring the Potential of Machine Learning for Lead Isotope Analysis.” Journal of Archaeological Science 188 (April): 106–499. https://doi.org/10.1016/j.jas.2026.106499.

Stacey, J. S., and J. D. Kramers. 1975. “Approximation of Terrestrial Lead Isotope Evolution by a Two-Stage Model.” Earth and Planetary Science Letters 26 (2): 207–21. https://doi.org/10.1016/0012-821X(75)90088-6.

Stos-Gale, Zofia Anna. 2001. “The Impact of the Natural Sciences on Studies of Hacksilber and Early Silver Coinage.” Hacksilber to Coinage: New Insights into the Monetary History of the Near East and Greece, 53–76.