where \({u}\) denotes uncertainty (at given \(\sigma\) level), and \(\rho\) denotes correlation coefficient.
Caution
If the data point error correlation coefficients column is not provided, these correlations will be assumed equal to 0. This is not a good general assumption for young samples!
Disequilibrium concordia-intercept ages are calculated by solving Eq. (15) in the manuscript using Newton’s method. The initial guess for the numerical age solution can either be entered directly (under the ‘Disequilibrium’ tab of the main window) or set to the lower intercept equilibrium age by selecting the Useequilibriumage option.
In cases where present-day [\(^{234}\mathrm{U}\)/\(^{238}\mathrm{U}\)] or [\(^{230}\mathrm{Th}\)/\(^{238}\mathrm{U}\)] values are entered, it is possible to have two intercept age solutions in close proximity. In such cases, the algorithm uses a brute force method to search for all age solutions between user defined upper and lower age and activity ratio limits (set in set in the ‘Numerical’ tab of the Preferences window). Typically, the upper intercept will have a physically implausible initial activity ratio solution, and so the lower intercept is always selected by default, however, the brute force method is implemented to guard against the numerical solution converging to the wrong intercept age.
Fig. 5 Disequilibrium tab of the main window on Mac
If the assumeinitialeq. option is checked, a lower intercept initial equilibrium age will be computed from the standard U-Pb equations, using the algorithm in [POWELL2020]. Note, this algorithm converges rapidly for intercept ages less than ~1 Ga but may not converge for older ages and does not propagate decay constant errors.
For equilibrium ages, age uncertainties may optionally be computed using Monte Carlo methods. To do this, ensure the EquilibriumageMonteCarloerrors option is checked on the main window (note, for disequilibrium ages, Monte Carlo errors are always implemented whether this box is checked or not). Decay constant errors may be included if age uncertainties are computed by Monte Carlo simulation (see the ‘Monte Carlo’ tab in main window).
For concordia-intercept ages, two separate plots are output by default. The first is an “isochron” style plot showing the data points as 95% confidence ellipses along with the linear regression fit. The second plot is a “concordia-intercept plot” and shows an enlarged view of the intersection between the regression line and the (dis)equilibrium concordia curve.
where \(^{23y}\mathrm{U}\) is the primordial uranium isotope (either \(^{238}\mathrm{U}\) or \(^{235}\mathrm{U}\)), and \(^{20x}\mathrm{Pb}\) is the normalising isotope (either \(^{204}\mathrm{Pb}\) or \(^{208}\mathrm{Pb}\), with \(^{208}\mathrm{Pb}\) assumed to be a stable isotope and therefore, applicable to young ages only), \({u}\) denotes uncertainty (at given \(\sigma\) level), and \(\rho\) denotes correlation coefficient. Disequilibrium U-Pb isochron ages are calculated by solving the equations given in Sect. 3.2 of the manuscript using Newton’s method. As for concordia-intercept ages, the initial age guess can either be entered directly (under the Disequilibrium tab of the main window) or set to the equilibrium age. Again, age uncertainties may optionally be computed using Monte Carlo methods for equilibrium ages but are always implemented for disequilibrium ages.
Single aliquot Pb/U and \(^{207}\mathrm{Pb}\)-corrected ages
Typically, multiple co-genetic single aliquot Pb/U ages will be computed at once. For Pb/U ages, data points should be arranged with columns ordered as:
where \({u}\) denotes uncertainty (at given \(\sigma\) level). Each row relates to a separate aliquot (i.e., a separate mineral grain or spot analysis). For \(^{207}\mathrm{Pb}\)-corrected-corrected ages, data points should be arranged as Tera-Wasserburg diagram variables (i.e., columns ordered as above).
\(^{206}\mathrm{Pb}\)/\(^{238}\mathrm{U}\) and \(^{207}\mathrm{Pb}\)-corrected ages are either computed by assuming that (i) the partition coefficient ratio (i.e. \(\mathrm{Th}/\mathrm{U}\)) is constant for all mineral grains, but Th/U of the melt may be heterogenous, or (ii) Th/\(\mathrm{U}_{\mathrm{melt}}\) value is constant for all aliquots, but the partition coefficients may vary. For single aliquot \(^{207}\mathrm{Pb}\)/\(^{235}\mathrm{U}\) ages, only approach (i) is available. To implement approach i, ensure that Singlealiquotages is selected in the first tab of the main window along with the correct data type. Then under the ‘Disequilibrium’ tab, select `DThUconst.` in the active combo box. The value and its uncertainty may then be set in the fields to the left.
To implement approach ii, select Th/Umeltconst from the combo box. The Th/U value of the melt and its uncertainty may then be entered in the fields to the left. After clicking OK, a separate dialog pops up allowing either measured \({\mathrm{^{232}Th/^{238}U}}\) or \({\mathrm{^{208}Pb^*/^{206}Pb^*}}\) values and uncertainties to be selected from the spreadsheet. With this approach, Th/U of the mineral is inferred from these measured values and aliquot age, using an iterative procedure.
As for other age types, a single age guess can either be entered directly (under the ‘Disequilibrium’ tab of the main window) or set to the lower intercept equilibrium age solution(s).
To output a plot of data points on a Tera-Wasserburg diagram, select the Outputdatapointplotfor207Pb-correctedages option in ‘Plotting’ tab of the Preferences window. A disequilibrium concordia may also be plotted if the Th/U disequilibrium state is input as a constant \(\mathrm{Th}/\mathrm{U}\) value for all data points, depending on the settings in the Type-specific plot settings window.
To compute a weighted average age from multiple single-analysis ages, select either spine or classical from the Fit type combo box on the main window. To output a plot of the weighted average, ensure Outputweightedaverageplot is checked in the Plotting tab of the Preferences window.
Note
The Assumeinitialequilibrium option is not yet implemented for single aliquot Pb/U ages.
The concordant initial [\(^{234}\mathrm{U}\)/\(^{238}\mathrm{U}\)] routine computes an initial [\(^{234}\mathrm{U}\)/\(^{238}\mathrm{U}\)] value that results in agreement (i.e., “concordance”) between the \(^{238}\mathrm{U}\)-\(^{206}\mathrm{Pb}\) and \(^{235}\mathrm{U}\)-\(^{207}\mathrm{Pb}\) isochron ages following [ENGEL2019]. This routine requires two data selections, one to compute a \(^{235}\mathrm{U}\)-\(^{207}\mathrm{Pb}\) isochron ages, and another to compute a \(^{238}\mathrm{U}\)-\(^{206}\mathrm{Pb}\) isochron age. The columns for these data selections should be arranged as outlined above for U-Pb isochron ages above. After clicking Ok on the main window, a data point selection dialog will appear. The initial activity ratio state of isotopes other than \(^{234}\mathrm{U}\) may be specified in the ‘Disequilibrium’ tab of the main window. Typically, for carbonates [\(^{230}\mathrm{Th}\)/\(^{238}\mathrm{U}\)] and [\(^{231}\mathrm{Pa}\)/\(^{235}\mathrm{U}\)] will be set to zero. Uncertainties in the initial [\(^{234}\mathrm{U}\)/\(^{238}\mathrm{U}\)] value are computed using Monte Carlo methods.
