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

## Description

Plots U-Pb data on Wetherill, Tera-Wasserburg or U-Th-Pb concordia diagrams, calculates concordia ages and compositions, evaluates the equivalence of multiple (^{206}Pb/^{238}U-^{207}Pb/^{235}U, ^{207}Pb/^{206}Pb-^{206}Pb/^{238}U, or ^{208}Th/^{232}Th-^{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 = NULL, tlim = NULL, alpha = 0.05, type = 1, 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 = 5, 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 `type` one of `1`: Wetherill – {}^{206}Pb/{}^{238}U vs. {}^{207}Pb/{}^{235}U `2`: Tera-Wasserburg – {}^{207}Pb/{}^{206}Pb vs. {}^{238}U/{}^{206}Pb `3`: U-Th-Pb concordia – {}^{208}Pb/{}^{232}Th vs. {}^{206}Pb/{}^{238}U (only available if `x\$format=7` or `8`) `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` Fill colour for the error ellipses. This can either be a single colour or multiple colours to form a colour ramp. Examples: a single colour: `rgb(0,1,0,0.5)`, `'#FF000080'`, `'white'`, etc.; multiple colours: `c(rbg(1,0,0,0.5)`, `rgb(0,1,0,0.5))`, `c('#FF000080','#00FF0080')`, `c('blue','red')`, `c('blue','yellow','red')`, etc.; a colour palette: `rainbow(n=100)`, `topo.colors(n=100,alpha=0.5)`, etc.; or a reversed palette: `rev(topo.colors(n=100,alpha=0.5))`, etc. For empty ellipses, set `ellipse.col=NA` `concordia.col` colour of the concordia line `exterr` show decay constant uncertainties? `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` common lead correction: `0`:none `1`: use the Pb-composition stored in `settings('iratio','Pb206Pb204')` (if `x\$format<4`); `settings('iratio','Pb206Pb204')` and `settings('iratio','Pb207Pb204')` (if `3

## 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). Alternatively, for data formats 7 and 8, it is also possible to plot ^{208}Pb/^{232}Th against the ^{206}Pb/^{238}U. 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.

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`).

par

a vector with the upper and lower intercept ages (if `type=1`) or the lower intercept age and common Pb intercept(s) (if `type=2`). If `show.age=4`, includes an overdispersion term as well.

cov

the covariance matrix of the elements in `par`.

logpar

the logarithm of `par`

logcov

the logarithm of `cov`

err

a 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`).

fact

the (1-α/2)-percentile of a t-distribution with `df` degrees of freedom.

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``` ```data(examples) concordia(examples\$UPb,show.age=2) dev.new() concordia(examples\$UPb,type=1,xlim=c(24.9,25.4), ylim=c(0.0508,0.0518),ticks=249:254,exterr=TRUE) dev.new() concordia(examples\$UPb,type=2,show.age=2,anchor=list(TRUE,0)) ```

pvermees/IsoplotR documentation built on Jan. 8, 2020, 7:03 p.m.