R: Concordia diagram

concordia {IsoplotR}

R Documentation

Concordia diagram

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

concordia(
  x = NULL,
  tlim = NULL,
  type = 1,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  concordia.col = "darksalmon",
  exterr = FALSE,
  show.age = 0,
  oerr = 3,
  sigdig = 2,
  common.Pb = 0,
  ticks = 5,
  anchor = 0,
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  ...
)

Arguments

`x`	an object of class `UPb`
`tlim`	age limits of the concordia line
`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.fill`	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.fill=NA`
`ellipse.stroke`	the stroke colour for the error ellipses. Follows the same formatting guidelines as `ellipse.fill`
`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 `\sqrt{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.
`oerr`	indicates whether the analytical uncertainties of the output are reported in the plot title as: `1`: 1`\sigma` absolute uncertainties. `2`: 2`\sigma` absolute uncertainties. `3`: absolute (1-`\alpha`)% confidence intervals, where `\alpha` equales the value that is stored in `settings('alpha')`. `4`: 1`\sigma` relative uncertainties (`\%`). `5`: 2`\sigma` relative uncertainties (`\%`). `6`: relative (1-`\alpha`)% confidence intervals, where `\alpha` equales the value that is stored in `settings('alpha')`.
`sigdig`	number of significant digits for the concordia/discordia age
`common.Pb`	common lead projection: `0`:none `1`: use the Pb-composition stored in `settings('iratio','Pb207Pb206')` (if `x$format<4`); `settings('iratio','Pb206Pb204')` and `settings('iratio','Pb207Pb204')` (if `3<x$format<7`); or `settings('iratio','Pb208Pb206')` and `settings('iratio','Pb208Pb207')` (if `x$format>6`). `2`: use the isochron intercept as the initial Pb-composition. If `show.age>1`, the data are projected along the isochron line, but the isochron itself is based on the uncorrected data. `3`: use the Stacey-Kramers two-stage model to infer the initial Pb-composition.
`ticks`	either a scalar indicating the desired number of age ticks to be placed along the concordia line, OR a vector of tick ages.
`anchor`	control parameters to fix the intercept age or common Pb composition of the isochron fit. This can be a scalar or a vector. If `anchor[1]=0`: do not anchor the isochron. If `anchor[1]=1`: fix the common Pb composition at the values stored in `settings('iratio',...)`. If `anchor[1]=2`: force the isochron line to intersect the concordia line at an age equal to `anchor[2]`. If `anchor[1]=3`: anchor the non-radiogenic component to the Stacey-Kramers mantle evolution model.
`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.fill`	fill colour that should be used for the omitted aliquots.
`omit.stroke`	stroke colour that should be used for the omitted aliquots.
`...`	optional arguments passed on to `scatterplot`

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 format 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 two-or three-element vector with:
t: the concordia_age (in Ma)
s[t]: the standard error of t
disp[t]: the standard error of t augmented by \sqrt{mswd} to account for any overdispersion.

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 on or two rows:

s: the standard errors of the parameter estimates

disp: the standard errors of the parameter estimates augmented by \sqrt{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).

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

attach(examples)
concordia(UPb,show.age=2)

dev.new()
concordia(UPb,type=2,xlim=c(24.9,25.4),
          ylim=c(0.0508,0.0518),ticks=249:254,exterr=TRUE)

dev.new()
concordia(UPb,show.age=2,anchor=c(2,260))

[Package IsoplotR version 6.2 Index]