ROUTINE TO MODEL K-FELDSPAR AGE SPECTRA

by Oscar M. Lovera

The autoage-free.f routine was develop to automatically estimate a family of thermal histories that satisfactorily fit a given k-feldspar age spectrum (40-Ar/39-Ar data), once the diffusion and distribution parameters of thesample had been estimated by modelling its 39-Ar data. Solutions obtained by this program are not constrained to monotonic cooling as it does in our monotonic version, autoage-mon.f . It implies the lack of a unique solution (see NG17ksp thermal histories ) and thus a family of them is more representative of the sample previous thermal history. The program starts an initial guessed thermal history (TH). This initial thermal history is approached using an expansion in Chebyshev polynomials. Subsequent better estimates of the TH 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 program then make another ramdon estimated of the TH based on the parameters of our initial TH and the iterative process begin again until reached another solutions. At the begining of the program the user will be asked for the desired number of solutions that will comprise the family. Sometimes the program will fail to find a satisfactory solution for a given initial TH, therefore the number of final solutions given by the program could be less than the number asked.

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 Cumultive % 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.
• fchist.in Definition of the initial guessed thermal history. Two column list of the intersection points of the many segment that define your thermal history. There is not limit to the number of segments but because it is only a rough estimate of the TH, usually a few segments (~ 5) are enough to define it. The guessed TH must start older than the age spectrum and at sufficiently high temperatures (~650 C). In this way, the program will have enough elasticity to adjust the TH to fit the age spectrum.

The following output files are created by the program.

• chist-inp.dat: Initial guessed CH for each run.
• chist-out.dat: Best fit CH obtained at each run.
• ages-inp.dat: Age spectrum obtained from each initial CH.
• ages-out.dat: Age spectrum obtained from each best fit CH.
• param-free.out: Parameters, (chi-squares, # of iteration, etc.)
• chisq.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.

RUNNING THE PROGRAM

To run the program follow these steps:

• Create the files age.in, sig.in and fchist.in
• Check that files temstep.in and arr-me.in exist from the previos 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-free) The program will prompt you to enter the number of runs. It means that you are allowed to test different initial guessed CH. The first run will use your proposed initial TH, after that, the program will create aleatories initial TH based on your initial guess to find the others solutions. Note, that because the there is not an unique minimum on the phase space of the solutions, then each different initial TH could lead the program to a different solution.
• Put the program to run in the background if you want to continue with other work.

HINTS and COMMENTS

• The program autoage-free.f is still in its experimental stage and had not been fully neither tested nor optimized. The running time of the program could be very large, from hours to days depending on the number of runnings you select (~ 2 hs per run). It is suggest that a few number of runnings (~5) be selected for modelling new samples.
• Steps with anomalous age (i.e. excess of argon, etc.), produce larges value of chi-square sometimes preventing the program of better fitting other more important portions of the age spectrum. When this occurs, a way to circumvet the problem is to increase the value of the uncertainty in that steps modifying the file sig.in, or by changing the file age.in so that these anomalous steps resembles the ages of the neigborgh steps. In general, the quoted 1-sigma age uncertainty does not represent very well the total uncertainty of the age step so large values of chi-square are obtained. We are currently working in a new method of calculating the error on the apparent age so it will fairly represent the total uncertainty of the age.