concordia: Concordia diagram

In pvermees/IsoplotR: Statistical Toolbox for Radiometric Geochronology

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))

pvermees/IsoplotR documentation built on Nov. 18, 2024, 1:02 a.m.