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.
isochron(x, ...)# S3 method for default
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 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = 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 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,
title = TRUE,
exterr = FALSE,
model = 1,
wtype = "a",
show.ellipses = 1 * (model != 2),
hide = NULL,
omit = NULL,
omit.fill = NA,
omit.stroke = "grey",
y0option = 4,
...
)
If x has class PbPb, ThPb,
ArAr, KCa, 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 (\(df=n-2\) for non-anchored fits)
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).
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).
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 (assumed) normally distributed
geological scatter that underlies the measurements, and its
standard error (only returned if model=3).
(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:
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.
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
\(\sqrt{mswd}\).
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).
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.
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
OR if x has class UPb:
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 \(w\).
the covariance matrix of par
the logarithm of par
the logarithm of cov
the number of analyses in the dataset
the degrees of freedom for the linear fit, i.e. \(2n-3\)
the y-intercept and its standard error
the isochron slope and its standard error
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.
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)
the y-axis label of the isochron plot
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).
the x-label of the isochron plot
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.
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
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').
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
text label for the horizontal plot axis
text label for the vertical plot axis
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
the stroke colour for the error
ellipses. Follows the same formatting guidelines as
ellipse.fill
the fill colour for the confidence interval of the intercept and slope.
colour of the isochron line
line width
if FALSE, suppresses the graphical output
add a title to the plot?
construct the isochron using either:
1: Error-weighted least squares regression
2: Total least squares regression
3: Error-weighted least squares with overdispersion term
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.
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 the data as:
0: points
1: error ellipses
2: error crosses
vector with indices of aliquots that should be removed from the plot.
vector with indices of aliquots that should be plotted but omitted from the isochron age calculation.
fill colour that should be used for the omitted aliquots.
stroke colour that should be used for the omitted aliquots.
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.
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.
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
propagate external sources of uncertainty (J, decay constant)?
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.
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.
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).
add Stacey-Kramers Pb-evolution curve to the plot?
the \(^{40}\)K branching ratio.
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:
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 a total 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.
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.
york,
titterington,
ludwig
attach(examples)
isochron(RbSr)
fit <- isochron(ArAr,inverse=FALSE,plot=FALSE)
dev.new()
isochron(ThU,type=4)
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