Plot U-Th(-Sm)-He data on a (log[He/Th] vs. log[U/He]) logratio plot or U-Th-He ternary diagram
helioplot(x, logratio = TRUE, model = 1, show.central.comp = TRUE,
show.numbers = FALSE, alpha = 0.05, contour.col = c("white", "red"),
levels = NA, clabel = "", ellipse.col = c("#00FF0080", "#0000FF80"),
sigdig = 2, xlim = NA, ylim = NA, fact = NA, ...)an object of class UThHe
Boolean flag indicating whether the data should be shown on bivariate log[He/Th] vs. log[U/He] diagram, or a U-Th-He ternary diagram.
choose one of the following statistical models:
1: weighted mean. This model assumes that the scatter between
the data points is solely caused by the analytical uncertainty. If
the assumption is correct, then the MSWD value should be
approximately equal to one. There are three strategies to deal with
the case where MSWD>1. The first of these is to assume that the
analytical uncertainties have been underestimated by a factor
\(\sqrt{MSWD}\).
2: unweighted mean. A second way to deal with over- or
underdispersed datasets is to simply ignore the analytical
uncertainties.
3: weighted mean with overdispersion: instead of attributing
any overdispersion (MSWD > 1) to underestimated analytical
uncertainties (model 1), it can also be attributed to the presence
of geological uncertainty, which manifests itself as an added
(co)variance term.
show the geometric mean composition as a white ellipse?
show the grain numbers inside the error ellipses?
probability cutoff for the error ellipses and confidence intervals
two-element vector with the fill colours to be assigned to the minimum and maximum age contour
a vector with additional values to be displayed as different background colours within the error ellipses.
label of 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.
number of significant digits for the central age
optional limits of the x-axis (log[U/He]) of the
logratio plot. If xlim=NA, the axis limits are
determined automatically.
optional limits of the y-axis (log[Th/He]) of the
logratio plot. If ylim=NA, the axis limits are
determined automatically.
three-element vector with scaling factors of the
ternary diagram if fact=NA, these will be determined
automatically
optional arguments to the generic plot function
U, Th, Sm and He are compositional data. This means that it is not so much the absolute concentrations of these elements that bear the chronological information, but rather their relative proportions. The space of all possible U-Th-He compositions fits within the constraints of a ternary diagram or `helioplot' (Vermeesch, 2008, 2010). If Sm is included as well, then this expands to a three-dimensional tetrahaedral space (Vermeesch, 2008). Data that fit within these constrained spaces must be subjected to a logratio transformation prior to statistical analysis (Aitchison, 1986). In the case of the U-Th-He-(Sm)-He system, this is achieved by first defining two (or three) new variables:
\(u \equiv \ln[U/He]\) \(v \equiv \ln[Th/He]\) \((, w \equiv \ln[Sm/He] )\)
and then performing the desired statistical analysis (averaging, uncertainty propagation, ...) on the transformed data. Upon completion of the mathematical operations, the results can then be mapped back to U-Th-(Sm)-He space using an inverse logratio transformation:
\([He] = 1/[e^{u}+e^{v}+(e^{w})+1]\), \([U] = e^{u}/[e^{u}+e^{v}+(e^{w})+1]\) \([Th] = e^{v}/[e^{u}+e^{v}+(e^{w})+1]\), \(([Sm] = e^{w}/[e^{u}+e^{v}+(e^{w})+1])\).
where \([He] + [U] + [Th] (+ [Sm]) = 1\). In the context of
U-Th-(Sm)-He dating, the central age is defined as the age
that corresponds to the arithmetic mean composition in logratio
space, which is equivalent to the geometric mean in compositional
dataspace (Vermeesch, 2008). IsoplotR's helioplot
function performs this calculation using the same algorithm that is
used to obtain the weighted mean U-Pb composition for the
concordia age calculation. Overdispersion is treated
similarly as in a regression context (see isochron).
Thus, there are options to augment the uncertainties with a factor
\(\sqrt{MSWD}\) (model 1); to ignore the analytical uncertainties
altogether (model 2); or to add a constant overdispersion term to
the analytical uncertainties (model 3). The helioplot
function visualises U-Th-(Sm)-He data on either a ternary diagram
or a bivariate \(\ln[Th/U]\) vs. \(\ln[U/He]\) contour
plot. These diagrams provide a convenient way to simultaneously
display the isotopic composition of samples as well as their
chronological meaning. In this respect, they fulfil the same
purpose as the U-Pb concordia diagram and the
U-series evolution plot.
Aitchison, J., 1986, The statistical analysis of compositional data: London, Chapman and Hall, 416 p.
Vermeesch, P., 2008. Three new ways to calculate average (U-Th)/He ages. Chemical Geology, 249(3), pp.339-347.
Vermeesch, P., 2010. HelioPlot, and the treatment of overdispersed (U-Th-Sm)/He data. Chemical Geology, 271(3), pp.108-111.
# NOT RUN {
data(examples)
helioplot(examples$UThHe)
dev.new()
helioplot(examples$UThHe,logratio=FALSE)
# }
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