Plots cogenetic Ar-Ar, Pb-Pb, Rb-Sr, Sm-Nd, Re-Os, Lu-Hf, U-Th-He
or Th-U data as X-Y scatterplots, fits an isochron curve through
them using the york function, and computes the corresponding
isochron age, including decay constant uncertainties.
isochron(x, ...)# S3 method for default
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
title = TRUE, model = 1, ...)
# S3 method for ArAr
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), inverse = TRUE,
line.col = "red", lwd = 2, plot = TRUE, exterr = TRUE, model = 1,
...)
# S3 method for PbPb
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), inverse = TRUE,
line.col = "red", lwd = 2, plot = TRUE, exterr = TRUE, model = 1,
...)
# S3 method for RbSr
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
plot = TRUE, exterr = TRUE, model = 1, ...)
# S3 method for ReOs
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
plot = TRUE, exterr = TRUE, model = 1, ...)
# S3 method for SmNd
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
plot = TRUE, exterr = TRUE, model = 1, ...)
# S3 method for LuHf
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
plot = TRUE, exterr = TRUE, model = 1, ...)
# S3 method for ThU
isochron(x, type = 2, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, levels = NA, clabel = "",
ellipse.col = c("#00FF0080", "#FF000080"), line.col = "red", lwd = 2,
plot = TRUE, exterr = TRUE, model = 1, ...)
# S3 method for UThHe
isochron(x, xlim = NA, ylim = NA, alpha = 0.05,
sigdig = 2, show.numbers = FALSE, line.col = "red", lwd = 2,
plot = TRUE, model = 1, ...)
EITHER a matrix with the following five columns:
the x-variable
the standard error of X
the y-variable
the standard error of Y
the correlation coefficient of X and Y
OR
an object of class ArAr, PbPb, ReOs,
RbSr, SmNd, LuHf, UThHe or ThU.
optional arguments to be passed on to the generic plot
function if model=2
2-element vector with the x-axis limits
2-element vector with the y-axis limits
confidence cutoff for the error ellipses and confidence intervals
the number of significant digits of the numerical values reported in the title of the graphical output
logical flag (TRUE to show grain numbers)
a vector with additional values to be displayed as different background colours within the error ellipses.
label for the colour scale
a vector of two background colours for the error
ellipses. If levels=NA, then only the first colour will
be used. If levels is a vector of numbers, then
ellipse.col is used to construct a colour ramp.
colour of the isochron line
line width
add a title to the plot?
construct the isochron using either:
Error-weighted least squares regression
Ordinary least squares regression
Error-weighted least squares with overdispersion term
if FALSE and x has class ArAr, plots
\(^{40}\)Ar/\(^{36}\)Ar vs. \(^{39}\)Ar/\(^{36}\)Ar.
if FALSE and x has class PbPb, plots
\(^{207}\)Pb/\(^{204}\)Pb vs. \(^{206}\)Pb/\(^{204}\)Pb.
if TRUE and x has class ArAr, plots
\(^{36}\)Ar/\(^{40}\)Ar vs. \(^{39}\)Ar/\(^{40}\)Ar.
if TRUE and x has class PbPb, plots
\(^{207}\)Pb/\(^{206}\)Pb vs. \(^{204}\)Pb/\(^{206}\)Pb.
if FALSE, suppresses the graphical output
propagate external sources of uncertainty (J, decay constant)?
following the classification of Ludwig and Titterington (1994), one of either:
`Rosholt type-II' isochron, setting out \(^{230}\)Th/\(^{232}\)Th vs. \(^{238}\)U/\(^{232}\)Th
`Osmond type-II' isochron, setting out \(^{230}\)Th/\(^{238}\)U vs. \(^{232}\)Th/\(^{238}\)U
`Rosholt type-II' isochron, setting out \(^{234}\)U/\(^{232}\)Th vs. \(^{238}\)U/\(^{232}\)Th
`Osmond type-II' isochron, setting out \(^{234}\)U/\(^{238}\)U vs. \(^{232}\)Th/\(^{238}\)U
If x has class PbPb, ArAr, RbSr,
SmNd, ReOs or LuHf, or UThHe, returns a
list with the following items:
the intercept of the straight line fit and its standard error.
the slope of the fit and its standard error.
the covariance of the slope and intercept
the degrees of freedom of the linear fit (\(df=n-2\))
a four-element list containing:
y: the atmospheric \(^{40}\)Ar/\(^{36}\)Ar or initial
\(^{207}\)Pb/\(^{204}\)Pb, \(^{187}\)Os/\(^{188}\)Os,
\(^{87}\)Sr/\(^{87}\)Rb, \(^{143}\)Nd/\(^{144}\)Nd or
\(^{176}\)Hf/\(^{177}\)Hf ratio.
s[y]: the propagated uncertainty of y
ci[y]: the studentised \(100(1-\alpha)\%\) confidence
interval for y.
disp[y]: the studentised \(100(1-\alpha)\%\) confidence
interval for y enhanced by \(\sqrt{mswd}\) (only
applicable if model=1).
a four-element list containing:
t: the \(^{207}\)Pb/\(^{206}\)Pb,
\(^{40}\)Ar/\(^{39}\)Ar, \(^{187}\)Os/\(^{187}\)Re,
\(^{87}\)Sr/\(^{87}\)Rb, \(^{143}\)Nd/\(^{144}\)Nd or
\(^{176}\)Hf/\(^{177}\)Hf age.
s[t]: the propagated uncertainty of t
ci[t]: the studentised \(100(1-\alpha)\%\) confidence
interval for t.
disp[t]: the studentised \(100(1-\alpha)\%\) confidence
interval for t enhanced by \(\sqrt{mswd}\) (only
applicable if model=1).
the \(100(1-\alpha/2)\%\) percentile of a
t-distribution with df degrees of freedom.
the mean square of the residuals (a.k.a `reduced
Chi-square') statistic (omitted if model=2).
the p-value of a Chi-square test for linearity
(omitted if model=2)
the overdispersion term, i.e. a two-element vector with
the standard deviation of the (assumedly) Normally distributed
geological scatter that underlies the measurements, and the
corresponding studentised \(100(1-\alpha)\%\) confidence interval
(only returned if model=3).
OR, if x has class ThU:
if x$type=1 or x$type=3: the best fitting
\(^{230}\)Th/\(^{232}\)Th intercept,
\(^{230}\)Th/\(^{238}\)U slope, \(^{234}\)U/\(^{232}\)Th
intercept and \(^{234}\)U/\(^{238}\)U slope, OR, if
x$type=2 or x$type=4: the best fitting
\(^{234}\)U/\(^{238}\)U intercept,
\(^{230}\)Th/\(^{232}\)Th slope, \(^{234}\)U/\(^{238}\)U
intercept and \(^{234}\)U/\(^{232}\)Th slope.
the covariance matrix of par.
the degrees of freedom for the linear fit, i.e. \((3n-3)\) if
x$format=1 or x$format=2, and \((2n-2)\) if
x$format=3 or x$format=4
if type=1: the \(^{230}\)Th/\(^{232}\)Th
intercept; if type=2: the \(^{230}\)Th/\(^{238}\)U
intercept; if type=3: the \(^{234}\)Th/\(^{232}\)Th
intercept; if type=4: the \(^{234}\)Th/\(^{238}\)U
intercept and its propagated uncertainty.
if type=1: the \(^{230}\)Th/\(^{238}\)U slope;
if type=2: the \(^{230}\)Th/\(^{232}\)Th slope; if
type=3: the \(^{234}\)U/\(^{238}\)U slope; if
type=4: the \(^{234}\)U/\(^{232}\)Th slope and its
propagated uncertainty.
the covariance between a and b.
the mean square of the residuals (a.k.a `reduced Chi-square') statistic.
the p-value of a Chi-square test for linearity.
the \(100(1-\alpha/2)\%\) percentile of a
t-distribution with df degrees of freedom.
a four-element vector containing:
y: the initial \(^{234}\)U/\(^{238}\)U-ratio
s[y]: the propagated uncertainty of y
ci[y]: the studentised \(100(1-\alpha)\%\) confidence
interval for y.
disp[y]: the studentised \(100(1-\alpha)\%\) confidence
interval for y enhanced by \(\sqrt{mswd}\).
a three (or four) element vector containing:
t: the initial \(^{234}\)U/\(^{238}\)U-ratio
s[t]: the propagated uncertainty of t
ci[t]: the studentised \(100(1-\alpha)\%\) confidence
interval for t
disp[t]: the studentised \(100(1-\alpha)\%\) confidence
interval for t enhanced by \(\sqrt{mswd}\) (only reported
if model=1).
the overdispersion term, i.e. a two-element vector with
the standard deviation of the (assumedly) Normally distributed
geological scatter that underlies the measurements, and the
corresponding \(100(1-\alpha)\%\) confidence interval (only
returned if model=3).
a matrix with the following columns: the X-variable for the isochron plot, the analytical uncertainty of X, the Y-variable for the isochron plot, the analytical uncertainty of Y, and the correlation coefficient between X and Y.
the x-label of the isochron plot
the y-label of the isochron plot
Given several aliquots from a single sample, isochrons allow the
non-radiogenic component of the daughter nuclide to be quantified
and separated from the radiogenic component. In its simplest form,
an isochron is obtained by setting out the amount of radiogenic
daughter against the amount of radioactive parent, both normalised
to a non-radiogenic isotope of the daughter element, and fitting a
straight line through these points by least squares regression
(Nicolaysen, 1961). The slope and intercept then yield the
radiogenic daughter-parent ratio and the non-radiogenic daughter
composition, respectively. There are several ways to fit an
isochron. The easiest of these is ordinary least squares
regression, which weighs all data points equally. In the presence
of quantifiable analytical uncertainty, it is equally
straightforward to use the inverse of the y-errors as weights. It
is significantly more difficult to take into account uncertainties
in both the x- and the y-variable (York, 1966). IsoplotR
does so for its U-Th-He isochron calculations. The York (1966)
method assumes that the analytical uncertainties of the x- and
y-variables are independent from each other. This assumption is
rarely met in geochronology. York (1968) addresses this issue with
a bivariate error weighted linear least squares algorithm that
accounts for covariant errors in both variables. This algorithm was
further improved by York et al. (2004) to ensure consistency with
the maximum likelihood approach of Titterington and Halliday
(1979).
IsoplotR uses the York et al. (2004) algorithm for its
\(^{40}\)Ar/\(^{39}\)Ar, Pb-Pb, Rb-Sr, Sm-Nd, Re-Os and Lu-Hf
isochrons. The maximum likelihood algorithm of Titterington and
Halliday (1979) was generalised from two to three dimensions by
Ludwig and Titterington (1994) for U-series disequilibrium dating.
Also this algorithm is implemented in IsoplotR. The extent
to which the observed scatter in the data can be explained by the
analytical uncertainties can be assessed using the Mean Square of
the Weighted Deviates (MSWD, McIntyre et al., 1966), which is
defined as:
\(MSWD = ([X - \hat{X}] \Sigma_{X}^{-1} [X - \hat{X}]^T)/df\)
where \(X\) are the data, \(\hat{X}\) are the fitted values,
and \(\Sigma_X\) is the covariance matrix of \(X\), and \(df
= k(n-1)\) are the degrees of freedom, where \(k\) is the
dimensionality of the linear fit. MSWD values that are far smaller
or greater than 1 indicate under- or overdispersed measurements,
respectively. Underdispersion can be attributed to overestimated
analytical uncertainties. IsoplotR provides three
alternative strategies to deal with overdispersed data:
Attribute the overdispersion to an underestimation of the analytical uncertainties. In this case, the excess scatter can be accounted for by inflating those uncertainties by a factor \(\sqrt{MSWD}\).
Ignore the analytical uncertainties and perform an ordinary least squares regression.
Attribute the overdispersion to the presence of `geological scatter'. In this case, the excess scatter can be accounted for by adding an overdispersion term that lowers the MSWD to unity.
Ludwig, K.R. and Titterington, D.M., 1994. Calculation of \(^{230}\)Th/U isochrons, ages, and errors. Geochimica et Cosmochimica Acta, 58(22), pp.5031-5042.
Nicolaysen, L.O., 1961. Graphic interpretation of discordant age measurements on metamorphic rocks. Annals of the New York Academy of Sciences, 91(1), pp.198-206.
Titterington, D.M. and Halliday, A.N., 1979. On the fitting of parallel isochrons and the method of maximum likelihood. Chemical Geology, 26(3), pp.183-195.
York, D., 1966. Least-squares fitting of a straight line. Canadian Journal of Physics, 44(5), pp.1079-1086.
York, D., 1968. Least squares fitting of a straight line with correlated errors. Earth and Planetary Science Letters, 5, pp.320-324.
York, D., Evensen, N.M., Martinez, M.L. and De Basebe Delgado, J., 2004. Unified equations for the slope, intercept, and standard errors of the best straight line. American Journal of Physics, 72(3), pp.367-375.
# NOT RUN {
data(examples)
isochron(examples$ArAr)
fit <- isochron(examples$PbPb,inverse=FALSE,plot=FALSE)
dev.new()
isochron(examples$ThU,type=4)
# }
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