# helioplot: Visualise U-Th-He data on a logratio plot or ternary diagram In IsoplotR: Statistical Toolbox for Radiometric Geochronology

## Description

Plot U-Th(-Sm)-He data on a (log[He/Th] vs. log[U/He]) logratio plot or U-Th-He ternary diagram

## Usage

 1 2 3 4 5 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, hide = NULL, omit = NULL, omit.col = NA, ...) 

## Arguments

 x an object of class UThHe logratio 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. model 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 √{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.central.comp show the geometric mean composition as a white ellipse? show.numbers show the grain numbers inside the error ellipses? alpha probability cutoff for the error ellipses and confidence intervals contour.col two-element vector with the fill colours to be assigned to the minimum and maximum age contour levels a vector with additional values to be displayed as different background colours within the error ellipses. clabel label of the colour scale ellipse.col 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. sigdig number of significant digits for the central age xlim optional limits of the x-axis (log[U/He]) of the logratio plot. If xlim=NA, the axis limits are determined automatically. ylim optional limits of the y-axis (log[Th/He]) of the logratio plot. If ylim=NA, the axis limits are determined automatically. fact three-element vector with scaling factors of the ternary diagram if fact=NA, these will be determined automatically 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 central age calculation. omit.col colour that should be used for the omitted aliquots. ... optional arguments to the generic plot function

## Details

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 √{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.

## References

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.

radialplot
 1 2 3 4 data(examples) helioplot(examples$UThHe) dev.new() helioplot(examples$UThHe,logratio=FALSE)