ROUTINE TO MODEL K-FELDSPAR AGE SPECTRA

by Oscar M. Lovera

NEWS

Recent modifications:

1) We introduce the calculation of confidence intervals over the set of solutions. (See examples on the www page, RZT paper) The program now output a file name "confmed.dat" that contains the 90% confidence interval of the distribution and the 90% confidence interval of the mediam. Note: the program will not calculate the confidence intervals if the number of successful solutions is below 20.

2) The way the program guess each initial cooling history to start the iterative process that led to the final solution have changed. Now the temperature at the initial age is randomly pick between 350C to 600C. Accordingly, the requirement of starting the modelling at high temperature have been removed. However, it introduces the implicit assumption that the starting time of the model correspond to an age where the sample have been completely reset (zero initial argon concentration). Therefore, we suggest that the model be started at least 5Ma prior to the max. age shown in the sample age spectrum.

3) The program only model the heating steps that were degassing at temperatures below 1100C (below the onset of melting).

Program Description

The autoage-mon.f routine was develop to automatically estimate a cooling history from a k-feldspar age spectrum (40-Ar/39-Ar data), once the diffusion and distribution parameters of the sample had been estimated by modelling its 39-Ar data. The program starts using an initial guessed cooling history (CH). The first derivative of the cooling history is approached using an expansion in Chebyshev polynomials. Its allows to restrict the family of possible CH approximations to only monotonic cooling. Subsequent better estimates of the CH are obtained adjusting the coefficients of the Chebyshev polynomials by a iterative variational process (Levenberg-Marquardt method). The iteration stop when an acceptable minimum of chi-square (square difference between the model and the data) is obtained. The best fitting CH is recovered by integration of its first derivative.

The following input files are required by the program.

• temstep.in: Laboratory heat schedule. One column list of the temperatures (K) and time (minutes) for each step. Same file used as input in the autoarr program.
• age.in: Two columns list (X,Y) of the Cumulative % of 39-Ar released vs. the Apparent Age (Ma) for each step heating.
• sig.in One column list of 1-sigma uncertainty of the apparent age for each step heating.
• arr-me.in Diffusion and distribution parameters as estimated by the autoarr routine. Output file of the autoarr program.

The following output files are created by the program.

• mchist-inp.dat: Initial guessed CH for each run.
• mchist-out.dat: Best fit CH obtained at each run.
• confmed.dat: Distribution and median 90% confidence intervals of the set of best fit cooling histories.
• mages-inp.dat: Age spectrum obtained from each initial CH.
• mages-out.dat: Age spectrum obtained from each best fit CH.
• param-mon.dat: Parameters, (chi-squares, # of iteration, etc.)
• mchisq.dat: List of best chi-squares obtained for each run.

Cooling history and age data are given in a two column XY format. Each set of data (age or cooling history) is separated by '&' at the end of the set. Confidence interval data is given in a three column XY1Y2 format, X=age, Y1= 90% CI of the distribution, Y2=90% CI of the mediam.

RUNNING THE PROGRAM

To run the program follow these steps:

• Create the files age.in and sig.in
• Check that files temstep.in and arr-me.in exists from the previous running of the autoarr routine.
• Run the program from the directory where your input files are. NOTE: Sometimes is better to run the program in a less busy machine, To do that, first make a rlogin to that machine and change to the directory you are working, then run the program.
• Run the program: (i.e. enter autoage-mon) The program will prompt for two inputs:

  1) Enter the number of runs (solutions). Different kind of non-linear complications could led the program to stop a run before finding an acceptable solution. This run will count although the non-acceptable solution will be automatically discarded. Therefore, it is possible that you get an smaller number of successful solutions than the number of runs you selected. Remember that if you want to calculate confidence intervals a minimum of 20 successful solutions is required.

  2) Enter the "starting age" It is the age at which the model will be started. Remember that for consistency this age should be at least 5-10Ma prior to the max. age record in the sample age spectrum.

• Put the program to run in the background if you want to continue with other work.

HINTS and COMMENTS

• The program autoage-mon.f is still in its experimental stage and had not been fully neither tested nor optimized. The running could take a long time, from hours to days depending on the number of runnings you select. If you run the program for the first time or are not sure about a new sample, just start by checking the program with a few number of runnings (~5).
• Steps with anomalous age (i.e. excess of argon at initial steps), produce larges value of chi-square sometimes preventing the program of getting better fit over others more important portions of the age spectrum. A way to avoid this problem is to increase the value of the uncertainty in such anomalous steps modifying the file sig.in, or by changing the file age.in. In general, the absolute chi-square values are meaningless since not all source of errors are considered in the 1sd age uncertainty.