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calculate_model_ages.R
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calculate_model_ages.R
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#' Calculate lead isotope age models
#'
#' @author Thomas Rose, \email{[email protected]}
#'
#' This function provides a convenient way to calculate the different age
#' models implemented with \code{\link{Stacey_Kramers_1975}},
#' \code{\link{Cumming_Richards_1975}}, and \code{\link{Albarede_et_al_2012}}
#' (see \emph{References} for the respective publications). The updated
#' 238U/235U ratio of 137.79 (Goldmann et al. 2015) is used in all functions
#' instead of the original value 138.88 because this is the now commonly
#' accepted value.
#'
#' The used model is indicated in the column names of the output by the
#' abbreviations given above.
#'
#' Although not necessary for the model of Cumming and Richards 1975, the
#' 208Pb/204Pb ratio must also be provided for consistency of the input with
#' the other age model functions.
#'
#' @param ratio_206_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#' @param ratio_207_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#' @param ratio_208_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#' @param model Character string with the abbreviation of the model to
#' calculate: \itemize{ \item \code{"SK75"} for Stacey and Kramers 1975 \item
#' \code{"CR75"} for Cumming and Richards 1975 \item \code{"AJ84"} for
#' Albarède and Juteau 1984 \item \code{"ADB12"} for Albarède et al. 2012
#' \item \code{"all"} for all models at once.}
#'
#' @return A data frame.
#' @export
#'
#'
#' @references Albarède, F., Desaulty, A.-M. and Blichert-Toft, J. (2012) A
#' geological perspective on the use of Pb isotopes in Archaeometry,
#' Archaeometry, 54(5), pp. 853‒867.
#' \url{https://dx.doi.org/10.1111/j.1475-4754.2011.00653.x}.
#'
#' Albarède, F. and Juteau, M. (1984) Unscrambling the lead model ages,
#' Geochimica et Cosmochimica Acta, 48(1), pp. 207‒212.
#' \url{https://dx.doi.org/10.1016/0016-7037(84)90364-8}.
#'
#' Cumming, G.L. and Richards, J.R. (1975) Ore lead isotope ratios in a
#' continuously changing earth, Earth and Planetary Science Letters, 28(2),
#' pp. 155‒171. \url{https://dx.doi.org/10.1016/0012-821X(75)90223-X}.
#'
#' Goldmann, A., Brennecka, G., Noordmann, J., Weyer, S. and Wadhwa, M. (2015)
#' The uranium isotopic composition of the Earth and the Solar System,
#' Geochimica et Cosmochimica Acta, 148,
#' pp. 145–158.\url{https://dx.doi.org/10.1016/j.gca.2014.09.008}
#'
#' Stacey, J.S. and Kramers, J.D. (1975) Approximation of terrestrial lead
#' isotope evolution by a two-stage model, Earth and Planetary Science
#' Letters, 26(2), pp. 207‒221.
#' \url{https://dx.doi.org/10.1016/0012-821X(75)90088-6}.
#'
LI_model_age <- function(ratio_206_204, ratio_207_204, ratio_208_204, model = c("SK75", "CR75", "AJ84", "ADB12", "all"))
{
if (!is.numeric(ratio_206_204)) {stop("Wrong input: `", substitute(ratio_206_204), "` must be numeric.")}
if (!is.numeric(ratio_207_204)) {stop("Wrong input: `", substitute(ratio_207_204), "` must be numeric.")}
if (!is.numeric(ratio_208_204) && model != "CR75") {stop("Wrong input: `", substitute(ratio_208_204), "` must be numeric.")}
switch (model,
SK75 = Stacey_Kramers_1975(ratio_206_204, ratio_207_204, ratio_208_204),
CR75 = Cumming_Richards_1975(ratio_206_204, ratio_207_204),
AJ84 = Albarede_Juteau_1984(ratio_206_204, ratio_207_204, ratio_208_204),
ADB12 = Albarede_et_al_2012(ratio_206_204, ratio_207_204, ratio_208_204),
all = cbind(Stacey_Kramers_1975(ratio_206_204, ratio_207_204, ratio_208_204),
Cumming_Richards_1975(ratio_206_204, ratio_207_204),
Albarede_Juteau_1984(ratio_206_204, ratio_207_204, ratio_208_204),
Albarede_et_al_2012(ratio_206_204, ratio_207_204, ratio_208_204)),
stop("This model is not supported.")
)
}
# Age models --------------------------------------------------------------
#' Calculate different lead isotope age model parameter
#'
#' @author Thomas Rose, \email{[email protected]}
#'
#' These functions calculate the age model and their respective mu and kappa
#' values according to the publications they are named after (see
#' \emph{References}. The updated 238U/235U ratio of 137.79 (Goldmann et al.
#' 2015) is used in all functions instead of the original value 138.88 because
#' this is the now commonly accepted value.
#'
#' The used model is indicated in the column names of the output by the
#' initials of the author's last names and the publication year (e. g. SK75
#' for Stacey & Kramers, 1975).
#'
#' The ratio of 208Pb/204Pb is not necessary for \link{Cumming_Richards_1975}.
#' The function takes it as argument only to be consistent with the input of
#' the other age model functions. If provided, it will be ignored.
#'
#' The function for the age model of Albarède & Juteau 1984 is based on the
#' MATLAB-script of F. Albarède (version 2020-11-06).
#'
#' @param ratio_206_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#' @param ratio_207_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#' @param ratio_208_204 A character vector or data frame column with the
#' 206Pb/204Pb ratio.
#'
#' @return A data frame.
#'
#' @export
#'
#' @references Albarède, F., Desaulty, A.-M. and Blichert-Toft, J. (2012) A
#' geological perspective on the use of Pb isotopes in Archaeometry,
#' Archaeometry, 54(5), pp. 853‒867.
#' \url{https://dx.doi.org/10.1111/j.1475-4754.2011.00653.x}.
#'
#' Albarède, F. and Juteau, M. (1984) Unscrambling the lead model ages,
#' Geochimica et Cosmochimica Acta, 48(1), pp. 207‒212.
#' \url{https://dx.doi.org/10.1016/0016-7037(84)90364-8}.
#'
#' Cumming, G.L. and Richards, J.R. (1975) Ore lead isotope ratios in a
#' continuously changing earth, Earth and Planetary Science Letters, 28(2),
#' pp. 155‒171. \url{https://dx.doi.org/10.1016/0012-821X(75)90223-X}.
#'
#' Goldmann, A., Brennecka, G., Noordmann, J., Weyer, S. and Wadhwa, M. (2015)
#' The uranium isotopic composition of the Earth and the Solar System,
#' Geochimica et Cosmochimica Acta, 148,
#' pp. 145–158.\url{https://dx.doi.org/10.1016/j.gca.2014.09.008}
#'
#' Stacey, J.S. and Kramers, J.D. (1975) Approximation of terrestrial lead
#' isotope evolution by a two-stage model, Earth and Planetary Science
#' Letters, 26(2), pp. 207‒221.
#' \url{https://dx.doi.org/10.1016/0012-821X(75)90088-6}.
#'
#' @name age_models
#' @aliases Stacey_Kramers_1975
#' @aliases Cumming_Richards_1975
#' @aliases Albarede_et_al_2012
#' @aliases Albarede_Juteau_1984
Stacey_Kramers_1975 <- function(ratio_206_204, ratio_207_204, ratio_208_204)
{
# Defining constants
t0 <- 3.7 * 10^9 # start of second stage in years
a0 <- 11.152 # 206Pb/204Pb at start of second stage
b0 <- 12.998 # 207Pb/204Pb at start of second stage
c0 <- 31.23 # 208Pb/204Pb at start of second stage
l238 <- 0.155125 * 10^-9 # decay constant 238U
l235 <- 0.98485 * 10^-9 # decay constant 235U
l232 <- 0.049475 * 10^-9 # decay constant 232Th
U238235 <- 137.79
# Defining model age function
Model_age_func <- function(x,y) {
stats::uniroot(function(X, a, b) {
((exp(l235*t0)-exp(l235*X))/(U238235*(exp(l238*t0)-exp(l238*X))))-((b0-b)/(a0-a))
},
a = x, b = y,
interval = c(-10000*10^6, t0),
extendInt ="yes", f.lower=-1*10^-2, f.upper = 1*10^-2, tol=10^-12)$root
}
# Calculation and clean-up
Model_Age <- mapply(Model_age_func, ratio_206_204, ratio_207_204)
Model_Age <- replace(Model_Age, Model_Age <= -10000*10^6 + 1*10^6 | Model_Age >= t0- 1*10^6, NA)
mu <- (ratio_206_204 - a0)/(exp(l238*t0)-exp(l238*Model_Age))
kappa <- (ratio_208_204 - c0)/(mu*(exp(l232*t0)-exp(l232*Model_Age)))
result <- data.frame("Model_Age_SK75" = Model_Age * 10^-6, "mu_SK75" = mu, "kappa_SK75" = kappa)
result <- round(result, 3)
result
}
#' @rdname age_models
#' @export
#'
Cumming_Richards_1975 <- function(ratio_206_204, ratio_207_204, ratio_208_204) # ratio 208Pb/204Pb not needed, just for consistency
{
# Defining constants
a0 = 9.307 # 206Pb/204Pb of the CDT
b0 = 10.294 # 207Pb/204Pb of the CDT
c0 = 29.476 # 208Pb/204Pb of the CDT
t0 = 4509 * 10^6 # age of the CDT
l238 <- 0.155125 * 10^-9 # decay constant 238U
l235 <- 0.98485 * 10^-9 # decay constant 235U
l232 <- 0.049475 * 10^-9 # decay constant 232Th
Vp <- 0.07797 # present day 235U/204Pb
Wp <- 41.25 # present day 232Th/204Pb
e1 <- 0.05 * 10^-9 # epsilon parameter (growth rate mu)
e2 <- 0.037 * 10^-9 # epsilon apostrophe parameter (growth rate kappa)
U238235 <- 137.79
# Defining model age function
Model_age_func <- function(x,y) {
stats::optimize(function(X, a, b) {
(a0 - a + U238235*Vp*((exp(l238*t0)*(1 - e1*(t0 - 1/l238)))-(exp(l238*X)*(1 - e1*(X - 1/l238)))))^2 + (b0 - b + Vp*((exp(l235*t0)*(1 - e1*(t0 - 1/l235)))-(exp(l235*X)*(1 - e1*(X - 1/l235)))))^2
},
a = x, b = y,
interval = c(-10000*10^6,t0),
tol=10^-12)$minimum
}
# Calculation and clean-up
Model_Age <- mapply(Model_age_func, ratio_206_204, ratio_207_204)
Model_Age <- replace(Model_Age, Model_Age <= -10000*10^6 + 1*10^6 | Model_Age >= t0- 1*10^6, NA)
mu <- U238235*Vp*(1-e1*Model_Age)
kappa <- Wp*(1-e2*Model_Age)/mu
result <- data.frame("Model_Age_CR75" = Model_Age * 10^-6, "mu_CR75" = mu, "kappa_CR75" = kappa)
result <- round(result, 3)
result
}
#' @rdname age_models
#' @export
#'
Albarede_Juteau_1984 <- function(ratio_206_204, ratio_207_204, ratio_208_204)
{
# Defining constants
xstar <- 18.750 # 206Pb/204Pb of modern common Pb
ystar <- 15.63 # 207Pb/204Pb of modern common Pb
zstar <- 38.86 # 208Pb/204Pb of modern common Pb
T0 <- 3.8 * 10^9 # age of the Earth
mustar <- 9.66 # mu of modern common Pb
kappastar <- 3.90 # kappa of modern common Pb
l238 <- 0.155125 * 10^-9 # decay constant 238U
l235 <- 0.98485 * 10^-9 # decay constant 235U
l232 <- 0.049475 * 10^-9 # decay constant 232Th
U238235 <- 137.79
xstar0 <- xstar - mustar * (exp(l238 * T0) - 1)
ystar0 <- ystar - mustar / U238235 * (exp(l235 * T0) - 1)
zstar0 <- zstar - mustar * kappastar * (exp(l232 * T0) - 1)
# Defining model age function
Model_age_func <- function(x,y) {
if (!is.na(x) && !is.na(y)) {
rootSolve::multiroot(
function(x, parms) {
c(
F1 = xstar0+x[2]*(exp(l238*T0)-exp(l238*x[1]))-parms[1],
F2 = ystar0+x[2]*(exp(l235*T0)-exp(l235*x[1]))/U238235-parms[2]
)
},
start = c("Tmod" = 10*10^6, "mu" = 8),
parms = c(x, y), ctol = 10^-12)$root
} else {
c(NA, NA)
}
}
# Calculation and clean-up
roots <- mapply(Model_age_func, ratio_206_204, ratio_207_204)
kappa <- (ratio_208_204 - zstar0)/(exp(l232*T0)-exp(l232*roots[1,]))/roots[2,]
result <- data.frame("Model_Age_AJ84" = roots[1,] * 10^-6, "mu_AJ84" = roots[2,], "kappa_AJ84" = kappa)
result <- round(result, 3)
result
}
#' @rdname age_models
#' @export
#'
Albarede_et_al_2012 <- function(ratio_206_204, ratio_207_204, ratio_208_204)
{
# Defining constants
ap <- 18.750 # 206Pb/204Pb of modern common Pb
bp <- 15.63 # 207Pb/204Pb of modern common Pb
cp <- 38.83 # 208Pb/204Pb of modern common Pb
t0 <- 4.43 * 10^9 # age of the Earth
mup <- 9.66 # mu of modern common Pb
kappap <- 3.90 # kappa of modern common Pb
l238 <- 0.155125 * 10^-9 # decay constant 238U
l235 <- 0.98485 * 10^-9 # decay constant 235U
l232 <- 0.049475 * 10^-9 # decay constant 232Th
U238235 <- 137.79
# Defining model age function
Model_age_func <- function(x,y) {
stats::uniroot(function(X, a, b) {
((b-bp)/(a-ap)) - ((1/U238235)*((exp(l235*t0)-exp(l235*X))/((exp(l238*t0)-exp(l238*X))))) -
(((mup*(exp(l238*X)-1))/(U238235*(a-ap)))*(((exp(l235*t0)-exp(l235*X))/(exp(l238*t0)-exp(l238*X))) -
((exp(l235*X)- 1)/(exp(l238*X)- 1))))
},
a = x, b = y,
interval = c(-10*10^9, t0),
extendInt ="yes", f.lower=-1*10^-2, f.upper = 1*10^-2, tol=10^-12)$root
}
# Calculation and clean-up
Model_Age <- mapply(Model_age_func, ratio_206_204, ratio_207_204)
Model_Age <- replace(Model_Age, Model_Age <= -10000*10^6 + 1*10^6 | Model_Age >= t0- 1*10^6, NA)
mu <- (ratio_206_204 - ap + (mup*(exp(l238*Model_Age)-1)))/(exp(l238*t0)-exp(l238*Model_Age)) + mup
kappa <- (ratio_208_204 - cp + (mup*kappap*(exp(l232*t0)-1)))/(mu*(exp(l232*t0)-exp(l232*Model_Age)))
result <- data.frame("Model_Age_ADB12" = Model_Age * 10^-6, "mu_ADB12" = mu, "kappa_ADB12" = kappa)
result <- round(result, 3)
result
}