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Noah McLean, 13 March 2019
UThPb is a graphical user interface to visualize and interpret U-Pb carbonate data with disequilibrium constraints. There are three main inputs:
- A set of U-Pb isotope ratios and uncertainties (entered as the ratios for a Tera-Wasserburg plot), a subset of which are assumed to be isochronous, and
- A measured (present-day) or assumed initial value for the 234U/238U activity ratio and uncertainty, and
- A measured (present-day) or assumed initial value for the 230Th/238U activity ratio and uncertainty.
UThPb assumes a sample has zero initial 226Ra and 231Pa, and will also let you change the sample 238U/235U ratio.
There are five main outputs provided by UThPb.
- A linear regression through the U-Pb data, with statistics including a MSWD and an upper intercept value and uncertainty (the initial 207Pb/206Pb ratio if the U-Pb data meet isochron assumptions)
- The disequilibrium-corrected age of the sample, taking into account the measured or assumed initial activities of 234U, 230Th, 226Ra, and 231Pa, and their impact on radiogenic ingrowth of 206Pb and 207Pb.
- If the user inputs a measured 234U/238U and/or 230Th/238U, the software will output the initial values for these activity ratios. Or, if the user inputs an assumed initial value for either of these, the software will output the calculated present-day values.
- Uncertainties for the age and calculated initial/measured 234U/238U and calculated initial/measured 230Th/238U, as 95% confidence intervals, and histograms and correlation plots for Monte Carlo trials.
- A "disequilibrium concordia" plot visualizing the U-Pb and U-Th data. The concordia curve is the locus of points that correspond to an U-Pb age (with no initial Pb) and the given disequilibrium constraints and 238U/235U. The intersection between the regression line (though a set of U-Pb data assumed to share the same initial 207Pb/206Pb ratio and the same age) and the concordia curve gives the age of the sample. The uncertainty in the age is a product of both the uncertainty in the regression line and in the 234U/238U and 230Th/238U constraints.
For detailed installation instructions as a standalone application on a Mac or a PC, see this repository's readme file.
If you have a MATLAB license and MATLAB installed on your computer then you can also download the src folder (click the green "Clone or Download" button at the top right of the UThPb repository page and choose Download ZIP). Run the UThPbII_GUI_v3.m script.
To run UThPb, navigate to the path where you installed it (the default is the Applications folder on a Mac or Program Files on a PC), then click the UThPb folder. Inside the application folder, double-click the UThPb app/exe to run the program. After the application loads (it may take a few seconds), you should see a window that looks like this:
The UThPb window contains several components: a U-Pb data table at the left, a U-Th data and results column in the middle, and a concordia plot on the right. If the margins of one or more of these components is not showing correctly (e.g., a missing U-Pb label at the top left, a missing y-axis label, etc.), you can generally make these re-appear with a small adjustment to the UThPb window size.
You can find some example data that demonstrate a range of concordia topologies on this Google Sheet.
The easiest way to input your U-Pb data is to copy and paste it from a spreadsheet. Your spreadsheet needs four adjacent columns: 238U/206Pb, ±1σ, 207Pb/206Pb, ±1σ. The uncertainties should be "one sigma" and can be either relative (%) uncertainties or absolute. There should be one row for each U-Pb analysis. Optionally, you can include a fifth column containing the correlation coefficients ("rho_xy") of the two isotope ratio uncertainties.
Copy all four or five columns (without any header rows) from your spreadsheet, then click on the top left cell in the U-Pb data table. Press Control + v to paste your data into the U-Pb data table.
If you do not have a column of correlation coefficients, a column of zeros will be created for you. This is usually not a horrible simplification for LA-ICPMS data. A column of 'checked' check-boxes will also appear on the right side of the table. This controls whether data points are included/excluded from the regression and age calculation -- uncheck a box to exclude that row's analysis. Use the radio button at the top of the table to specify whether your entered uncertainties are relative (%) or absolute.
UThPb will make plots and perform calculations consistent with either measured or assumed initial 234U/238U and 230Th/238U activity ratios. These activity ratios and their uncertainties are typed into the boxes in the top of the center column. If you don't have measured value, you must estimate an initial value and assign it an uncertainty. You can also edit the default 238U/235U value and uncertainty of 137.82 ± 0.02 (1σ abs).
Note that switching the 234U/238U activity ratio to 'assumed initial' inserts a default value of 1.00005497 instead of a more conventional value of 1 for a system at secular equilibrium. Use the default value, which is just higher than 1, if you assume that the 238U - 234U system was initially at equilibrium, and see the footnote on transient equilibrium for more details.
After entering the U-Pb and U-Th data, press the Plot button to perform a linear regression through the U-Pb data and plot the disequilibrium concordia curve that matches the input U-Th constraints. Next, press the Intercept button to calculate the intercept between the U-Pb regression line and the disequilibrium concordia curve. Finally, press the Unct button to begin a Monte Carlo calculation of the uncertainties in the intercept date and the calculated initial/modern 234U/238U and 230Th/238U activity ratios.
The regression and intercept calculations should display results in seconds or less. The Monte Carlo uncertainty propagation takes a minute or two on my 2014 MacBook Pro and may take longer on your computer.
The Results area at the bottom center of the UThPb window shows information about the calculations it performs.
The top four fields display the results of the linear regression, including MSWD -- the mean of the squared weighted deviations (Wendt and Carl, 1991) or reduced chi-squared to the non-geochronological sciences -- and the y-intercept and uncertainty. If the U-Pb analyses reflect repeat measurements of a system that formed at the same time, shared an initial common Pb IC, and remained a closed system since formation, then the y-intercept reflects the best estimate of the initial common Pb isotopic composition.
If the U-Th disequilibrium measurements or assumptions apply to the same system as the isochronous U-Pb measurements, then the x-y coordinates of the intersection between the U-Pb regression line and the disequilibrium concordia curve represents the isotopic composition of an initial Pbc-free system, and the corresponding age is the age of the system. This disequilibrium-corrected age is consistent with both the U-Pb data and the U-Th data, and if the 234U/238U or 230Th/238U are measured, it is independent of assumptions about their initial values. The intercept age in ka, if it exists, is given on the third line of the Results table when all data has been entered and the Intercept button is pressed.
Additionally, UThPb calculates an initial 234U/238U activity ratio, denoted [234U/238U]i when a measured 234U/238U is input, and present-day 234U/238U activity ratio, denoted [234U/238U]t when an assumed initial value is provided. Likewise, an initial 230Th/238U activity ratio is output when a measured 230Th/238U is input, and a present-day 230Th/238U activity ratio is output when an assumed initial value is provided. These values are output in the fields below the intercept age, and their labels (subscript i or t for initial or present day) change dynamically when the measured vs. assumed initial radio buttons are flipped for the 234U/238U and 230Th/238U ratios.
The output concordia plot uses the conventional Tera-Wasserburg plot axes (207Pb/206Pb on the y-axis, 238U/206Pb on the x-axis). The U-Pb data is plotted as blue 2σ uncertainty ellipses (~86% confidence intervals) with black dots at the measured value. The best fit regression line is plotted in black, and its 2σ uncertainty envelope in green. The disequilibrium concordia curve is plotted as a blue line, with open circles at numbered ages, expressed in ka, or thousand years ago. The disequilibrium curve terminates at a purple dot when the initial conditions required to plot the curve are impossible or unrealistic (for more information, see the FAQ).
Pressing the Unct button starts a suite of 104 Monte Carlo simulations, which randomly sample the probability distributions for the U-Pb regression parameters and the input 234U/238U, 230Th/238U, and 238U/235U. For each trial, the intercept age and the present or initial 234U/238U and 230Th/238U are recorded. Once the Monte Carlo trials complete, 95% confidence intervals are output to the Results portion of the UThPb window in square brackets. These intervals represent the shortest interval that contains 95% (9500) of the Monte Carlo trials, and may not be symmetric around the calculated "best" value.
The Monte Carlo trials are visualized on a three-parameter cross plot that pops up as a new window.
The first row and column correspond to the intercept age, the second to the 234U/238U (initial if the measured value is input and vice versa), and the third to the 230Th/238U (likewise initial if you input measured and vice versa). Along the diagonal of the 3 x 3 plot are histograms illustrating the distribution of Monte Carlo trials for each variable. These are estimates of the probability densities for each output variable. In the off-diagonal positions are scatter plots for Monte Carlo realizations of each pair of variables. The scatter plots can illustrate the degree of correlation between the outputs -- for instance, the example above shows a positive correlation between the uncertainty in the intercept age and in the estimated initial 234U/238U and a negative correlation between the intercept age and the initial 230Th/238U.
For systems with large uncertainties in any of the three output variables, the resulting histograms may deviate significantly from the shape of a normal distribution, The scatter plots showing the relationships between two variables also may look significantly non-linear, instead of the elliptical point-clouds usually associated with bivariate normal uncertainty ellipses. This is because the equations for calculating the intercept age and U-Th ratios from the input data are significantly non-linear.
Questions I hope will be frequently asked.
1. Why does the concordia curve end at a purple dot?
The concordia curve plotted by UThPb is truncated when the initial 234U/238U and/or 230Th/238U ratios required to produce the input measured/assumed values become impossible or unrealistic.
For instance, for Sample 1 provided on the test data Google Sheet, both 234U/238U and 230Th/238U are provided as measured, present-day values. Plotting the point on the concordia curve for a given date (e.g., 200 ka) requires back-calculating what the initial 234U/238U and 230Th/238U ratios must have been to produce the measured values after 200 kyr elapsed, then calculating forward from those conditions (and assumed zero initial 226Ra and 231Pa) the radiogenic 206Pb and 207Pb produced. For Sample 1, at ~240 ka, the back-calculated initial 230Th/238U is zero -- this is the conventional U-Th date of the sample. Older dates on the concordia curve would require negative initial 230Th/238U, and therefore are not plotted.
UThPb plots a purple dot and ends the concordia curve when either of these conditions occur:
- The initial 234U/238U or 230Th/238U required to plot a date is zero, or
- The initial 234U/238U activity ratio required to plot a date is greater than a 'reasonable' threshold, set by default to 200. The maximum initial 234U/238U activity ratio is set on line 26 of the function plotConc_v2.m
2. How do I delete rows or enter a new U-Pb dataset?
There is currently no way to delete a single row or to clear a U-Pb data table in the UThPb graphical user interface. You can un-check the checkbox at the right hand side of the data table to exclude an analysis from the linear regression but still plot it as a gray ellipse. To cut/paste in data from a new sample, the easiest thing to do is to open a new UThPb window (i.e., double-click on the app).
3. What do the slider bars do under the 234U/238U, 230Th/238U, and 238U/235U fields? What about the menu items for 'Calculation' and 'Plotting'?
These sliders currently do not do anything. The menu items have not yet been populated as of v0.1.1.
4. How do I change the ages at which the disequilbrium concordia curve is labelled, and how do I change the x-axis limits?
Neither of these interactions are possible presently.
n. Where do I submit suggestions and bug reports or get support for UThPb?
If you don't mind asking publicly: At the top of this page, click the Issues link, and in the page that pops up, click the green 'New Issue' button at the top right.
Otherwise, email me at noahmc<at>ku<dot>edu.
Note that switching the 234U/238U activity ratio to 'assumed initial' inserts a default value of 1.00005497 instead of a more conventional value of 1 for a system at secular equilibrium. This is because the definition of secular equilibrium makes the first-order approximation that the parent isotope's half life is very long compared to that of the daughter isotope, so the parent isotope does not decay appreciably over the timescale of interest. This approximation does not hold for the timescales considered by UThPb, where 238U decays over timescales of hundreds of thousands to millions of years.
Instead, over long timescales in a closed system, the 238U decay chain reaches transient equilibrium. Transient equilibrium is a concept more frequently used when considering decay chains of short-lived isotopes, as in a nuclear reactor.
In secular equilibrium, the 234U/238U atom ratio is
but at transient equilibrium, this atom ratio becomes
The λ238 in the denominator of the transient equilibrium expression accounts for the long-term radioactive decay of 238U. To transform this atom ratio into a more conventional activity ratio, multiply the atom ratio as usual by λ234/λ238
For the 238U and 234U decay constants of Jaffey et al. (1971) and Cheng et al. (2013), this expression is approximately 1.00005497.
Why does this matter? Using a value of 1 vs. 1.00005497 for the 234U/238U activity ratio makes only a small ~55 ppm difference in the calculated age if used as an initial ratio, but it can make a large impact if used as a 'measured' value. Using a value of 1 as a present-day 234U/238U means that the system is currently depleted in 234U relative to transient equilibrium, and therefore must have been even more depleted in the past. Calculated points on concordia will reflect this depletion in 234U with less ingrown radiogenic 206Pb for a given age, moving the concordia curve to the left on a T-W plot and making intercept ages systematically too young.
Also, because a present-day 234U/238U activity ratio of 1 is depleted in 234U relative to transient equilibrium, the 234U/238U must have been even more depleted farther back in time. Calculating the 234U/238U at previous times yields lower 234U/238U ratios, eventually reaching 0 at ~3.48 Ma. UThPb will not calculate and plot concordia curve ages older than 3.48 Ma for a present-day 234U/238U activity ratio of 1, since older ages would require negative initial 234U/238U ratios. See the FAQ for details.
The transient equilibrium 230Th/238U atom ratio looks a bit more complicated,
Expressed as an activity ratio using the Jaffey et al. (1971) and Cheng et al. (2013) decay constants, the transient equilibrium 230Th/238U is about 1.00007189.
How the disequilibrium concordia curve is calculated depends on whether the input U-Th data are measured or assumed initial activity ratios.
The goal is to calculate the present-day 207Pb/206Pb and 238U/206Pb for a set of times, t, and then connect those points to make the disequilibrum concordia curve. This begins by using any specified present-day activity ratios (always the 238U/235U, and sometimes one or both of the 234U/238U and 230Th/238U) to back-calculate initial abundances at the time t. These back-calculated abundances are then combined with assumptions about the initial state of the system (sometimes one or both of the 234U/238U and 230Th/238U, always an assumed initial 226Ra/238U and 231Pa/235U, and always zero initial 206Pb/238U and 207Pb/235U). This initial condition is then forward-calculated using the matrix exponential formulation and the resulting 207Pb/206Pb and 238U/206Pb ratios are plotted as the concordia point for the time t.
For further details on the matrix exponential, see the 2016 AGU poster, posted here. The matrix exponential functions in the code used for back-calculating initial isotopic abundances are called mxp.U3 (238U-234U-235U) and mxp.UTh4 (238U-234U-230Th-235U). The function used for forward-calculating present-day abundances, which is used to generate the 207Pb/206Pb and 238U/206Pb points plotted as the concordia curve, is mxp.UTh8. These functions are defined in the code makeMxps_v2.m in the src folder of the UThPb repository. All calculations are scaled so that the amount of 238U at the time of measurement is 1.
1. Input initial 234U/238U and initial 230Th/238U activity ratios.
This is the simplest scenario. For each time t on the concordia curve, the amount of 238U is back-calculated from a value of 1 at present-day, then the value of 235U is back-calculated using the input present-day 238U/235U ratio. This information, along with the input initial 234U/238U and 230Th/238U and the assumed initial 226Ra/238U and 231Pa/235U (equal to zero by default), and assumed initial 206Pb/238U = 0 and 207Pb/235U = 0 defines the initial state of the system. The initial state is then used in the matrix exponential (see the AGU poster) with the time elapsed t to calculate the present-day abundances of the isotopes.
In the source code, the calculation of initial abundances happens in line 43 of calcNi_UTh8_v2.m and the calculation of present abundances from initial abundances happens at line 100 of plotConc_v2.m.
2. Input measured 234U/238U and measured 230Th/238U activity ratios.
This scenario requires several calculation steps. First, the input data is enough to form a vector of present-day [238U 234U 230Th 235U] abundances. These are back-calculated to initial abundances for the time t at which the concordia point is being plotted. The initial abundances for these isotopes are then assembled with initial assumptions for 226Ra/238U and 231Pa/235U (equal to zero by default) and zero initial 206Pb/238U and 207Pb/235U. The initial state is then used in the matrix exponential (see the AGU poster) with the time elapsed t to calculate the present-day abundances of the isotopes.
In the source code, the back-calculation of initial abundances happens in line 25 of calcNi_UTh8_v2.m and the calculation of present abundances from initial abundances happens at line 100 of plotConc_v2.m.
3. Input measured 234U/238U activity ratio and assumed initial 230Th/238U activity ratio.
This scenario occurs when the 230Th/238U is no longer distinguishable from secular equilibrium but there is still measurable 234U/238U disequilibrium. The first step is to use the present-day 238U/235U and measured 234U/238U to back-calculate the initial abundances for these isotopes at time t. These abundances are then combined with the input assumed initial 230Th/238U, as assumed value for the 226Ra/238U and 231Pa/235U (equal to zero by default) and an initial 206Pb/238U = 0 and 207Pb/235U = 0. he initial state is then used in the matrix exponential (see the AGU poster) with the time elapsed t to calculate the present-day abundances of the isotopes.
In the source code, the back-calculation of initial abundances happens in line 36 of calcNi_UTh8_v2.m and the calculation of present abundances from initial abundances happens at line 100 of plotConc_v2.m.