isochron: Calculate and plot isochrons

In pvermees/IsoplotR: Statistical Toolbox for Radiometric Geochronology

Calculate and plot isochrons

Description

Plots cogenetic U-Pb, Ar-Ar, K-Ca, Pb-Pb, Th-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, titterington or ludwig function, and computes the corresponding isochron age, including decay constant uncertainties.

Usage

isochron(x, ...)

## Default S3 method:
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  xlab = "x",
  ylab = "y",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  title = TRUE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  ...
)

## S3 method for class 'other'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  xlab = "x",
  ylab = "y",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  title = TRUE,
  model = 1,
  wtype = 1,
  anchor = 0,
  flippable = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  ...
)

## S3 method for class 'UPb'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  joint = TRUE,
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  type = 1,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  show.ellipses = 1 * (model != 2),
  anchor = 0,
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  y0option = 1,
  taxis = FALSE,
  ...
)

## S3 method for class 'PbPb'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = TRUE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  growth = FALSE,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  ...
)

## S3 method for class 'ArAr'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = TRUE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'ThPb'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = FALSE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'KCa'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  inverse = FALSE,
  ci.col = "gray80",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  bratio = 0.895,
  ...
)

## S3 method for class 'RbSr'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = FALSE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'ReOs'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = FALSE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'SmNd'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = FALSE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'LuHf'
isochron(
  x,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  inverse = FALSE,
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  taxis = FALSE,
  ...
)

## S3 method for class 'UThHe'
isochron(
  x,
  sigdig = 2,
  oerr = 3,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  model = 1,
  wtype = 1,
  anchor = 0,
  show.ellipses = 2 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  ...
)

## S3 method for class 'ThU'
isochron(
  x,
  type = 2,
  oerr = 3,
  sigdig = 2,
  show.numbers = FALSE,
  levels = NA,
  clabel = "",
  ellipse.fill = c("#00FF0080", "#FF000080"),
  ellipse.stroke = "black",
  ci.col = "gray80",
  line.col = "black",
  lwd = 1,
  plot = TRUE,
  exterr = FALSE,
  model = 1,
  wtype = "a",
  show.ellipses = 1 * (model != 2),
  hide = NULL,
  omit = NULL,
  omit.fill = NA,
  omit.stroke = "grey",
  y0option = 4,
  ...
)

Arguments

x	EITHER a matrix with the following five columns: X: the x-variable sX: the standard error of X Y: the y-variable sY: the standard error of Y rXY: the correlation coefficient of X and Y OR an object of class ArAr, KCa, PbPb, UPb, ThPb, ReOs, RbSr, SmNd, LuHf, UThHe or ThU.
...	optional arguments to be passed on to scatterplot
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	the number of significant digits of the numerical values reported in the title of the graphical output
show.numbers	logical flag (TRUE to show grain numbers)
levels	a vector with additional values to be displayed as different background colours within the error ellipses.
clabel	label for the colour scale
xlab	text label for the horizontal plot axis
ylab	text label for the vertical plot axis
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.col=NA
ellipse.stroke	the stroke colour for the error ellipses. Follows the same formatting guidelines as ellipse.fill
ci.col	the fill colour for the confidence interval of the intercept and slope.
line.col	colour of the isochron line
lwd	line width
plot	if FALSE, suppresses the graphical output
title	add a title to the plot?
model	construct the isochron using either: 1: Error-weighted least squares regression 2: Total least squares regression 3: Error-weighted least squares with overdispersion term
wtype	controls the parameter responsible for the overdispersion in model-3 regression. If x has class PbPb, ArAr or PD, wtype can have one of two values: 1: attribute the overdispersion to variability in the non-radiogenic component, as controlled by settings('iratio',...) 2: attribute the overdispersion to variability in the age, i.e. to diachronous closure of the isotope system. otherwise, wtype can have one of four values: 1: attributes the overdispersion to the y-intercept of the equivalent conventional isochron. 2: attributes the overdispersion to the slope of the equivalent conventional isochron. 'A': only available if x has class ThU and x$format is 1 or 2. Attributes the overdispersion to the authigenic ^{230}Th/^{238}U-intercept of the isochron. 'B': only available if x has class ThU and x$format is 1 or 2. Attributes the overdispersion to the ^{230}Th/^{232}Th-slope of the isochron.
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].
show.ellipses	show the data as: 0: points 1: error ellipses 2: error crosses
hide	vector with indices of aliquots that should be removed from the plot.
omit	vector with indices of aliquots that should be plotted but omitted from the isochron 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.
flippable	controls if generic data (where x has class other and x$format is either 4 or 5) should be treated as inverse isochrons (flippable=1) or as conventional isochrons (flippable=2). If flippable=0 (which is the default value), then the data are passed on to isochron.default.
joint	logical. Only applies to U-Pb data formats 4 and above. If TRUE, carries out three dimensional regression. If FALSE, uses two dimensional isochron regression. The latter can be used to compute {}^{207}Pb/{}^{235}U isochrons, which are immune to the complexities of initial {}^{234}U/{}^{238}U disequilibrium.
type	if x has class UPb and x$format=4, 5 or 6: 1: ^{204}Pb/^{206}Pb vs. ^{238}U/^{206}Pb 2: ^{204}Pb/^{207}Pb vs. ^{235}U/^{207}Pb if x has class UPb and x$format=7 or 8: 1: ^{208}Pb{}_\circ/^{206}Pb vs. ^{238}U/^{206}Pb 2: ^{208}Pb{}_\circ/^{207}Pb vs. ^{235}U/^{207}Pb 3: ^{206}Pb{}_\circ/^{208}Pb vs. ^{232}Th/^{208}Pb 4: ^{207}Pb{}_\circ/^{208}Pb vs. ^{232}Th/^{208}Pb if x has class ThU, and following the classification of Ludwig and Titterington (1994), one of either: 1: ‘Rosholt type-II’ isochron, setting out ^{230}Th/^{232}Th vs. ^{238}U/^{232}Th 2: ‘Osmond type-II’ isochron, setting out ^{230}Th/^{238}U vs. ^{232}Th/^{238}U 3: ‘Rosholt type-II’ isochron, setting out ^{234}U/^{232}Th vs. ^{238}U/^{232}Th 4: ‘Osmond type-II’ isochron, setting out ^{234}U/^{238}U vs. ^{232}Th/^{238}U
exterr	propagate external sources of uncertainty (J, decay constant)?
y0option	controls the type of y-intercept or activity ratio that is reported along with the isochron age. Only relevant to U-Pb data and Th-U data formats 1 and 2. For U-Pb data: y0option=1 reports the common Pb composition, y0option=2 reports the initial ^{234}U/^{238}U activity ratio. y0option=3 reports the initial ^{230}Th/^{238}U activity ratio, For Th-U data: y0option=1 reports the authigenic ^{234}U/^{238}U activity ratio, y0option=2 reports the detrital ^{230}Th/^{232}Th activity ratio, y0option=3 reports the authigenic ^{230}Th/^{238}U activity ratio, y0option=4 reports the initial ^{234}U/^{238}U activity ratio.
taxis	logical. If TRUE, replaces the x-axis of the inverse isochron with a time scale. Only applies if inverse is TRUE or, when x has class U-Pb, if x$format is 4 or higher.
inverse	toggles between normal and inverse isochrons. If the isochron plots Y against X, and If inverse=TRUE, then X = {}^{204}Pb/{}^{206}Pb and Y = {}^{207}Pb/{}^{206}Pb (if x has class PbPb), or X = {}^{232}Th/{}^{208}Pb and Y = {}^{204}Pb/{}^{208}Pb (if x has class ThPb), or X = {}^{39}Ar/{}^{40}Ar and Y = {}^{36}Ar/{}^{40}Ar (if x has class ArAr), or X = {}^{40}K/{}^{40}Ca and Y = {}^{44}Ca/{}^{40}Ca (if x has class KCa), or X = {}^{87}Rb/{}^{87}Sr and Y = {}^{86}Sr/{}^{87}Sr (if x has class RbSr), or X = {}^{147}Sm/{}^{143}Nd and Y = {}^{144}Nd/{}^{143}Nd (if x has class SmNd), or X = {}^{187}Re/{}^{187}Os and Y = {}^{188}Os/{}^{187}Os (if x has class ReOs), or X = {}^{176}Lu/{}^{176}Hf and Y = {}^{177}Hf/{}^{176}Hf (if x has class LuHf). If inverse=FALSE, then X = {}^{206}Pb/{}^{204}Pb and Y = {}^{207}Pb/{}^{204}Pb (if x has class PbPb), or X = {}^{232}Th/{}^{204}Pb and Y = {}^{208}Pb/{}^{204}Pb (if x has class ThPb), or X = {}^{39}Ar/{}^{36}Ar and Y = {}^{40}Ar/{}^{36}Ar (if x has class ArAr), or X = {}^{40}K/{}^{44}Ca and Y = {}^{40}Ca/{}^{44}Ca (if x has class KCa), or X = {}^{87}Rb/{}^{86}Sr and Y = {}^{87}Sr/{}^{86}Sr (if x has class RbSr), or X = {}^{147}Sm/{}^{144}Nd and Y = {}^{143}Nd/{}^{144}Nd (if x has class SmNd), or X = {}^{187}Re/{}^{188}Os and Y = {}^{187}Os/{}^{188}Os (if x has class ReOs), or X = {}^{176}Lu/{}^{177}Hf and Y = {}^{176}Hf/{}^{177}Hf (if x has class LuHf).
growth	add Stacey-Kramers Pb-evolution curve to the plot?
bratio	the ^{40}K branching ratio.

Details

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 total 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 Ar-Ar, K-Ca, Pb-Pb, Th-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. Finally, the constrained maximum likelihood algorithms of Ludwig (1998) and Vermeesch (2020) are used for isochron regression of U-Pb data. 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:

1. 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}.

2. Ignore the analytical uncertainties and perform a total least squares regression.

3. 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.

Value

If x has class PbPb, ThPb, ArAr, KCa, RbSr, SmNd, ReOs or LuHf, or UThHe, returns a list with the following items:

a

the intercept of the straight line fit and its standard error.

b

the slope of the fit and its standard error.

cov.ab

the covariance of the slope and intercept

df

the degrees of freedom of the linear fit (df=n-2 for non-anchored fits)

y0

a two- or three-element list containing:

y: the atmospheric ^{40}Ar/^{36}Ar or initial ^{40}Ca/^{44}Ca, ^{187}Os/^{188}Os, ^{87}Sr/^{87}Rb, ^{143}Nd/^{144}Nd, ^{176}Hf/^{177}Hf or ^{208}Pb/^{204}Pb ratio.

s[y]: the standard error of y

disp[y]: the standard error of y enhanced by \sqrt{mswd} (only applicable if model=1).

age

a three-element list containing:

t: the ^{207}Pb/^{206}Pb, ^{208}Pb/^{232}Th, ^{40}Ar/^{39}Ar, ^{40}K/^{40}Ca, ^{187}Os/^{187}Re, ^{87}Sr/^{87}Rb, ^{143}Nd/^{144}Nd or ^{176}Hf/^{177}Hf age.

s[t]: the standard error of t

disp[t]: the standard error of t enhanced by \sqrt{mswd} (only applicable if model=1).

mswd

the mean square of the residuals (a.k.a 'reduced Chi-square') statistic (omitted if model=2).

p.value

the p-value of a Chi-square test for linearity (omitted if model=2)

w

the overdispersion term, i.e. a two-element vector with the standard deviation of the (assumed) normally distributed geological scatter that underlies the measurements, and its standard error (only returned if model=3).

ski

(only reported if x has class PbPb and growth is TRUE) the intercept(s) of the isochron with the Stacey-Kramers mantle evolution curve.

OR, if x has class ThU:

par

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.

cov

the covariance matrix of par.

df

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

a

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.

b

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.

cov.ab

the covariance between a and b.

mswd

the mean square of the residuals (a.k.a 'reduced Chi-square') statistic.

p.value

the p-value of a Chi-square test for linearity.

y0

a three-element vector containing:

y: the initial ^{234}U/^{238}U-ratio

s[y]: the standard error of y

disp[y]: the standard error of y enhanced by \sqrt{mswd}.

age

a two (or three) element vector containing:

t: the initial ^{234}U/^{238}U-ratio

s[t]: the standard error of t

disp[t]: the standard error of t enhanced by \sqrt{mswd} (only reported if model=1).

w

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 its standard error.

d

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.

xlab

the x-label of the isochron plot

ylab

the y-label of the isochron plot

OR if x has class UPb:

par

if model=1 or 2, a three element vector containing the isochron age and the common Pb isotope ratios. If model=3, adds a fourth element with the overdispersion parameter w.

cov

the covariance matrix of par

logpar

the logarithm of par

logcov

the logarithm of cov

n

the number of analyses in the dataset

df

the degrees of freedom for the linear fit, i.e. 2n-3

a

the y-intercept and its standard error

b

the isochron slope and its standard error

cov.ab

the covariance between a and b.

mswd

the mean square of the residuals (a.k.a 'reduced Chi-square') statistic.

p.value

the p-value of a Chi-square test for linearity.

y0

a two or three-element vector containing:

y: the initial ^{206}Pb/^{204}Pb-ratio (if type=1 and x$format=4,5 or 6); ^{207}Pb/^{204}Pb-ratio (if type=2 and x$format=4,5 or 6); ^{208}Pb/^{206}Pb-ratio (if type=1 and x$format=7 or 8); ^{208}Pb/^{207}Pb-ratio (if type=2 and x$format=7 or 8); ^{206}Pb/^{208}Pb-ratio (if type=3 and x$format=7 or 8); or ^{207}Pb/^{208}Pb-ratio (if type=4 and x$format=7 or 8).

s[y]: the standard error of y

disp[y]: the standard error of y enhanced by \sqrt{mswd} (only returned if model=1)

y0label

the y-axis label of the isochron plot

age

a two (or three) element vector containing:

t: the isochron age

s[t]: the standard error of t

disp[t]: the standard error of t enhanced by \sqrt{mswd} (only reported if model=1).

xlab

the x-label of the isochron plot

ylab

the y-label of the isochron plot

OR, if x has class other and x$format is either 4 or 5 and flippable is not 0, returns

Dd: the ratio of the inherited radiogenic daughter to its nonradiogenic sister isotope

DP: the ratio fo the radiogic daughter to its radioactive parent

cov.DdDP: the covariance between Dd and DP.

In the remaining types of other data, the intercept a and b are returned along with their covariance.

References

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.

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

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.

Vermeesch, P., 2020. Unifying the U-Pb and Th-Pb methods: joint isochron regression and common Pb correction, Geochronology, 2, 119-131.

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.

See Also

york, titterington, ludwig

Examples

attach(examples)
isochron(RbSr)

fit <- isochron(ArAr,inverse=FALSE,plot=FALSE)

dev.new()
isochron(ThU,type=4)

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