- Author: Sean R. Mulcahy
- License: MIT
- Source code on Github
nlsdif.r is an R function that performs non-linear least squares fitting of measured diffusion couple profiles using the equation of Crank, 1975. The function takes as input measured compositions along a traverse, starting values for the upper and lower concentrations and returns best fit values and uncertainties for each parameter and solves for the composite parameter of diffusivity and time (Dt).
We used a similar approach in the paper by Renne et al., 2012 to fit Ba diffusion profiles in feldspar in order to model temperature-time histories and argon retention in magmatic phenocrysts.
Download or fork the nlsdif.r code and dataset.csv file. Set the working directory and within R:
df <- read.table("dataset.csv", header = T, sep = ",")
nlsdif(df$x, df$c, df$sd, 0.02, 0.07)
The function will return the best fit non-linear least squares results:
Formula: c ~ 0.5 * (c2 + c1) + 0.5 * (c2 - c1) * erf((x + Dx)/(2 * sqrt(Dt)))
Parameters:
Estimate Std. Error t value Pr(>|t|)
c1 0.0222907 0.0003162 70.501 < 2e-16 ***
c2 0.0674114 0.0006946 97.049 < 2e-16 ***
Dt 2.4561664 0.3861678 6.360 7.93e-09 ***
Dx 0.0669055 0.1333017 0.502 0.617
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.7069 on 91 degrees of freedom
Number of iterations to convergence: 7
Achieved convergence tolerance: 8.107e-06
And plot the fitted diffusion profile:
- Crank, J., 1975. The Mathematics of Diffusion, Oxford, 421 p.
- Renne, P.R., Mulcahy, S.R., Morgan, L.E., Cassata, W.S., Kelley, S.P., Hlusko, L., and Njau, J., 2012 Retention of inherited Ar by alkali feldspar xenocrysts in a magma: Kinetic constraints from Ba zoning profiles: Geochimica et Cosmochimica Acta, v. 93, p. 129-142. (doi)