# concordia: Concordia diagram In IsoplotR: Statistical Toolbox for Radiometric Geochronology

## Description

Plots U-Pb data on Wetherill and Tera-Wasserburg concordia diagrams, calculate concordia ages and compositions, evaluates the equivalence of multiple (^{206}Pb/^{238}U-^{207}Pb/^{235}U or ^{207}Pb/^{206}Pb-^{206}Pb/^{238}U) compositions, computes the weighted mean isotopic composition and the corresponding concordia age using the method of maximum likelihood, computes the MSWD of equivalence and concordance and their respective Chi-squared p-values. Performs linear regression and computes the upper and lower intercept ages (for Wetherill) or the lower intercept age and the ^{207}Pb/^{206}Pb intercept (for Tera-Wasserburg), taking into account error correlations and decay constant uncertainties.

## Usage

 ```1 2 3 4 5 6``` ```concordia(x, tlim = NULL, alpha = 0.05, wetherill = TRUE, show.numbers = FALSE, levels = NA, clabel = "", ellipse.col = c("#00FF0080", "#FF000080"), concordia.col = "darksalmon", exterr = FALSE, show.age = 0, sigdig = 2, common.Pb = 0, ticks = NULL, anchor = list(FALSE, NA), hide = NULL, omit = NULL, omit.col = NA, ...) ```

## Arguments

 `x` an object of class `UPb` `tlim` age limits of the concordia line `alpha` probability cutoff for the error ellipses and confidence intervals `wetherill` logical flag (`FALSE` for Tera-Wasserburg) `show.numbers` logical flag (`TRUE` to show grain numbers) `levels` a vector with `length(x)` values to be displayed as different background colours within the error ellipses. `clabel` label for the colour legend (only used if `levels` is not `NA`. `ellipse.col` a vector of two background colours for the error ellipses. If `levels=NA`, then only the first colour is used. If `levels` is a vector of numbers, then `ellipse.col` is used to construct a colour ramp. `concordia.col` colour of the concordia line `exterr` show decay constant uncertainty? `show.age` one of either: `0`: plot the data without calculating an age `1`: fit a concordia composition and age `2`: fit a discordia line through the data using the maximum likelihood algorithm of Ludwig (1998), which assumes that the scatter of the data is solely due to the analytical uncertainties. In this case, `IsoplotR` will either calculate an upper and lower intercept age (for Wetherill concordia), or a lower intercept age and common (^{207}Pb/^{206}Pb)-ratio intercept (for Tera-Wasserburg). If `mswd`>0, then the analytical uncertainties are augmented by a factor √{mswd}. `3`: fit a discordia line ignoring the analytical uncertainties `4`: fit a discordia line using a modified maximum likelihood algorithm that includes accounts for any overdispersion by adding a geological (co)variance term. `sigdig` number of significant digits for the concordia/discordia age `common.Pb` apply a common lead correction using one of three methods: `1`: use the Stacey-Kramer two-stage model to infer the initial Pb-composition `2`: use the isochron intercept as the initial Pb-composition `3`: use the Pb-composition stored in `settings('iratio','Pb206Pb204')` and `settings('iratio','Pb207Pb204')` `ticks` an optional vector of age ticks to be added to the concordia line to override `IsoplotR`'s default spacing, which is based on `R`'s `pretty` function. `anchor` control parameters to fix the intercept age or common Pb composition of the discordia fit. This is a two-element list. The first element is a boolean flag indicating whether the discordia line should be anchored. If this is `FALSE`, then the second item is ignored and both the common Pb composition and age are estimated. If the first element is `TRUE` and the second element is `NA`, then the common Pb composition is fixed at the values stored in `settings('iratio',...)`. item If the first element is `TRUE` and the second element is a number, then the discordia line is forced to intersect the concordia line at an age equal to that number. `hide` vector with indices of aliquots that should be removed from the concordia diagram `omit` vector with indices of aliquots that should be plotted but omitted from concordia or discordia age calculation `omit.col` colour that should be used for the omitted aliquots. `...` optional arguments to the generic `plot` function

## Details

The concordia diagram is a graphical means of assessing the internal consistency of U-Pb data. It sets out the measured ^{206}Pb/^{238}U- and ^{207}Pb/^{235}U-ratios against each other (‘Wetherill’ diagram) or, equivalently, the ^{207}Pb/^{206}Pb- and ^{206}Pb/^{238}U-ratios (‘Tera-Wasserburg’ diagram). The space of concordant isotopic compositions is marked by a curve, the ‘concordia line’. Isotopic ratio measurements are shown as 100(1-`alpha`)% confidence ellipses. Concordant samples plot near to, or overlap with, the concordia line. They represent the pinnacle of geochronological robustness. Samples that plot away from the concordia line but are aligned along a linear trend form an isochron (or ‘discordia’ line) that can be used to infer the composition of the non-radiogenic (‘common’) lead or to constrain the timing of prior lead loss.

## Value

if `show.age=1`, returns a list with the following items:

x

a named vector with the (weighted mean) U-Pb composition

cov

the covariance matrix of the (weighted mean) U-Pb composition

mswd

a vector with three items (`equivalence`, `concordance` and `combined`) containing the MSWD (Mean of the Squared Weighted Deviates, a.k.a the reduced Chi-squared statistic) of isotopic equivalence, age concordance and combined goodness of fit, respectively.

p.value

a vector with three items (`equivalence`, `concordance` and `combined`) containing the p-value of the Chi-square test for isotopic equivalence, age concordance and combined goodness of fit, respectively.

df

a three-element vector with the number of degrees of freedom used for the `mswd` calculation. These values are useful when expanding the analytical uncertainties if `mswd>1`.

age

a 4-element vector with:

`t`: the concordia age (in Ma)

`s[t]`: the estimated uncertainty of `t`

`ci[t]`: the studentised 100(1-α)\% confidence interval of `t` for the appropriate degrees of freedom

`disp[t]`: the studentised 100(1-α)\% confidence interval for `t` augmented by √{mswd} to account for overdispersed datasets.

if `show.age=2`, `3` or `4`, returns a list with the following items:

model

the fitting model (`=show.age-1`).

x

a two-element vector with the upper and lower intercept ages (if `wetherill=TRUE`) or the lower intercept age and ^{207}Pb/^{206}Pb intercept (if `wetherill=FALSE`).

cov

the covariance matrix of the elements in `x`.

err

a `[2 x 2]` or `[3 x 2]` matrix with the following rows:

`s`: the estimated standard deviation for `x`

`ci`: the studentised 100(1-α)\% confidence interval of `x` for the appropriate degrees of freedom

`disp[t]`: the studentised 100(1-α)\% confidence interval for `x` augmented by √{mswd} to account for overdispersed datasets (only reported if `show.age=2`).

df

the degrees of freedom of the concordia fit (concordance + equivalence)

p.value

p-value of a Chi-square test for age homogeneity (only reported if ` type=3`).

mswd

mean square of the weighted deviates – a goodness-of-fit measure. `mswd > 1` indicates overdispersion w.r.t the analytical uncertainties (not reported if `show.age=3`).

w

three-element vector with the standard deviation of the (assumedly) Normal overdispersion term and the lower and upper half-widths of its 100(1-α)\% confidence interval (only important if `show.age=4`).

n

the number of aliquots in the dataset

## References

Ludwig, K.R., 1998. On the treatment of concordant uranium-lead ages. Geochimica et Cosmochimica Acta, 62(4), pp.665-676.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```data(examples) concordia(examples\$UPb,show.age=2) dev.new() concordia(examples\$UPb,wetherill=FALSE, xlim=c(24.9,25.4),ylim=c(0.0508,0.0518), ticks=249:254,exterr=TRUE) dev.new() concordia(examples\$UPb,wetherill=FALSE,show.age=2,anchor=list(TRUE,0)) ```

IsoplotR documentation built on Dec. 9, 2018, 1:04 a.m.