A function to calculate the expected sample specific fraction of saturation based on the model of Huntley (2006) using the approach as implemented in Kars et al. (2008) or Guralnik et al. (2015).

```
calc_Huntley2006(data, LnTn = NULL, rhop, ddot, readerDdot,
normalise = TRUE, fit.method = c("EXP", "GOK")[1],
lower.bounds = c(-Inf, -Inf, -Inf), summary = TRUE, plot = TRUE,
...)
```

data

data.frame (**required**):
A `data.frame`

with one of the following structures:

A

**three column**data frame with numeric values on a) dose (s), b) LxTx and and c) LxTx error.If a

**two column**data frame is provided it is automatically assumed that errors on LxTx are missing. A third column will be attached with an arbitrary 5 % error on the provided LxTx values.Can also be a

**wide table**, i.e. a data.frame with a number of colums divisible by 3 and where each triplet has the aforementioned column structure.

(optional) | dose (s)| LxTx | LxTx error | | [ ,1] | [ ,2]| [ ,3] | |---------|------|------------| [1, ]| 0 | LnTn | LnTn error | (optional, see arg 'LnTn') [2, ]| R1 | L1T1 | L1T1 error | ... | ... | ... | ... | [x, ]| Rx | LxTx | LxTx error |

**NOTE:** The function assumes the first row of the function to be the
`Ln/Tn`

-value. If you want to provide more than one `Ln/Tn`

-value consider
using the argument `LnTn`

.

LnTn

data.frame (**optional**):
This argument should **only** be used to provide more than one `Ln/Tn`

-value.
It assumes a two column data frame with the following structure:

| LnTn | LnTn error | | [ ,1] | [ ,2] | |--------|--------------| [1, ]| LnTn_1 | LnTn_1 error | [2, ]| LnTn_2 | LnTn_2 error | ... | ... | ... | [x, ]| LnTn_x | LnTn_x error |

The function will calculate a **mean** `Ln/Tn`

-value and uses either the
standard deviation or the highest individual error, whichever is larger. If
another mean value (e.g. a weighted mean or median) or error is preferred,
this value must be calculated beforehand and used in the first row in the
data frame for argument `data`

.

**NOTE:** If you provide `LnTn`

-values with this argument the data frame
for the `data`

-argument **must not** contain any `LnTn`

-values!

rhop

numeric (**required**):
The density of recombination centres (\(\rho\)') and its error (see Huntley 2006),
given as numeric vector of length two. Note that \(\rho\)' must **not** be
provided as the common logarithm. Example: `rhop = c(2.92e-06, 4.93e-07)`

.

ddot

numeric (**required**):
Environmental dose rate and its error, given as a numeric vector of length two.
Expected unit: Gy/ka. Example: `ddot = c(3.7, 0.4)`

.

readerDdot

numeric (**required**):
Dose rate of the irradiation source of the OSL reader and its error,
given as a numeric vector of length two.
Expected unit: Gy/s. Example: `readerDdot = c(0.08, 0.01)`

.

normalise

logical (*with default*):
If `TRUE`

(the default) all measured and computed LxTx values are
normalised by the pre-exponential factor A (see details).

fit.method

character (*with default*):
Fit function of the dose response curve. Can either be `EXP`

(the default)
or `GOK`

. Note that `EXP`

(single saturating exponential) is the original
function the model after Huntley (2006) and Kars et al. (2008) was
designed to use. The use of a general-order kinetics function (`GOK`

)
is an experimental adaption of the model and should be used
with great care.

lower.bounds

numeric (*with default*):
Only applicable for `fit.method = 'GOK'`

. A vector of length 3 that
contains the lower bound values for fitting the general-order kinetics
function using minpack.lm::nlsLM. In most cases, the default values
(c(`-Inf, -Inf, -Inf`

)) are appropriate for finding a best fit, but
sometimes it may be useful to restrict the lower bounds to e.g.
c(`0, 0, 0`

). The values of the vector are for parameters
`a`

, `D0`

and `c`

in that particular order (see details in
Luminescence::plot_GrowthCurve).

summary

logical (*with default*):
If `TRUE`

(the default) various parameters provided by the user
and calculated by the model are added as text on the right-hand side of the
plot.

plot

logical (*with default*):
enables/disables plot output.

...

Further parameters:

`verbose`

logical: Show or hide console output`n.MC`

numeric: Number of Monte Carlo iterations (default =`100000`

).**Note**that it is generally advised to have a large number of Monte Carlo iterations for the results to converge. Decreasing the number of iterations will often result in unstable estimates.

All other arguments are passed to plot and plot_GrowthCurve.

An '>RLum.Results object is returned:

Slot: **@data**

OBJECT |
TYPE |
COMMENT |

`results` |
data.frame | results of the of Kars et al. 2008 model |

`data` |
data.frame | original input data |

`Ln` |
numeric | Ln and its error |

`LxTx_tables` |
`list` |
A `list` of `data.frames` containing data on dose,
LxTx and LxTx error for each of the dose response curves.
Note that these do not contain the natural Ln signal, which is provided separately. |

`fits` |
`list` |
A `list` of `nls` objects produced by minpack.lm::nlsLM when fitting the dose response curves |

Slot: **@info**

OBJECT |
TYPE |
COMMENT |

`call` |
`call` |
the original function call |

`args` |
`list` |
arguments of the original function call |

0.4.0 (2018-03-09 13:48:48)

King, G.E., Burow, C. (2018). calc_Huntley2006(): Apply the Huntley (2006) model. Function version 0.4.0. In: Kreutzer, S., Burow, C., Dietze, M., Fuchs, M.C., Schmidt, C., Fischer, M., Friedrich, J. (2018). Luminescence: Comprehensive Luminescence Dating Data Analysis. R package version 0.8.6. https://CRAN.R-project.org/package=Luminescence

This function applies the approach described in Kars et al. (2008) or Guralnik et al. (2015),
which are both developed from the model of Huntley (2006) to calculate the expected sample
specific fraction of saturation of a feldspar and also to calculate fading
corrected age using this model. \(\rho\)' (`rhop`

), the density of recombination
centres, is a crucial parameter of this model and must be determined
separately from a fading measurement. The function analyse_FadingMeasurement
can be used to calculate the sample specific \(\rho\)' value.

**Kars et al. (2008) - Single saturating exponential**

To apply the approach after Kars et al. (2008) use `fit.method = "EXP"`

.

Firstly, the unfaded D0 value is determined through applying equation 5 of Kars et al. (2008) to the measured LxTx data as a function of irradiation time, and fitting the data with a single saturating exponential of the form:

$$LxTx(t*) = A x \phi(t*) x (1 - exp(-(t* / D0)))$$

where

$$\phi(t*) = exp(-\rho' x ln(1.8 x s_tilde x t*)^3)$$

after King et al. (2016) where `A`

is a pre-exponential factor,
`t*`

(s) is the irradiation time, starting at the mid-point of
irradiation (Auclair et al. 2003) and `s_tilde`

(3x10^15 s^-1) is the athermal
frequency factor after Huntley (2006).

Using fit parameters `A`

and `D0`

, the function then computes a natural dose
response curve using the environmental dose rate, `D_dot`

(Gy/s) and equations
`[1]`

and `[2]`

. Computed LxTx values are then fitted using the
plot_GrowthCurve function and the laboratory measured LnTn can then
be interpolated onto this curve to determine the fading corrected
De value, from which the fading corrected age is calculated.

**Guralnik et al. (2015) - General-order kinetics**

To apply the approach after Guralnik et al. (2015) use `fit.method = "GOK"`

.

The approach of Guralnik et al. (2015) is very similar to that of Kars et al. (2008), but instead of using a single saturating exponential the model fits a general-order kinetics function of the form:

$$LxTx(t*) = A x \phi(t*) x (1-(1+(1/D0) x t* x c)^(-1/c))$$

where `A`

, \(\phi\), `t*`

and `D0`

are the same as above and `c`

is a
dimensionless kinetic order modifier (cf. equation 10 in
Guralnik et al., 2015).

**Level of saturation**

The `calc_Huntley2006`

function also calculates the level of saturation (n/N)
and the field saturation (i.e. athermal steady state, (n/N)_SS) value for
the sample under investigation using the sample specific \(\rho\)',
unfaded `D0`

and `D_dot`

values, following the approach of Kars et al. (2008).

**Uncertainties**

Uncertainties are reported at 1 sigma and are assumed to be normally
distributed and are estimated using monte-carlo resamples (`n.MC = 1000`

)
of \(\rho\)' and LxTx during dose response curve fitting, and of \(\rho\)'
in the derivation of (n/N) and (n/N)_SS.

**Age calculated from 2*D0 of the simulated natural DRC**

In addition to the age calculated from the equivalent dose derived from
`Ln/Tn`

projected on the simulated natural dose response curve (DRC), this function
also calculates an age from twice the characteristic saturation dose (`D0`

)
of the simulated natural DRC. This can be a useful information for
(over)saturated samples (ie. no intersect of `Ln/Tn`

on the natural DRC)
to obtain at least a "minimum age" estimate of the sample. In the console
output this value is denoted by *"Age @2D0 (ka):"*.

Kars, R.H., Wallinga, J., Cohen, K.M., 2008. A new approach towards anomalous fading correction for feldspar IRSL dating-tests on samples in field saturation. Radiation Measurements 43, 786-790. doi:10.1016/j.radmeas.2008.01.021

Guralnik, B., Li, B., Jain, M., Chen, R., Paris, R.B., Murray, A.S., Li, S.-H., Pagonis, P., Herman, F., 2015. Radiation-induced growth and isothermal decay of infrared-stimulated luminescence from feldspar. Radiation Measurements 81, 224-231.

Huntley, D.J., 2006. An explanation of the power-law decay of luminescence. Journal of Physics: Condensed Matter 18, 1359-1365. doi:10.1088/0953-8984/18/4/020

King, G.E., Herman, F., Lambert, R., Valla, P.G., Guralnik, B., 2016. Multi-OSL-thermochronometry of feldspar. Quaternary Geochronology 33, 76-87. doi:10.1016/j.quageo.2016.01.004

**Further reading**

Morthekai, P., Jain, M., Cunha, P.P., Azevedo, J.M., Singhvi, A.K., 2011. An attempt to correct for the fading in million year old basaltic rocks. Geochronometria 38(3), 223-230.

# NOT RUN { ## Load example data (sample UNIL/NB123, see ?ExampleData.Fading) data("ExampleData.Fading", envir = environment()) ## (1) Set all relevant parameters # a. fading measurement data (IR50) fading_data <- ExampleData.Fading$fading.data$IR50 # b. Dose response curve data data <- ExampleData.Fading$equivalentDose.data$IR50 ## (2) Define required function parameters ddot <- c(7.00, 0.004) readerDdot <- c(0.134, 0.0067) # Analyse fading measurement and get an estimate of rho'. # Note that the RLum.Results object can be directly used for further processing. # The number of MC runs is reduced for this example rhop <- analyse_FadingMeasurement(fading_data, plot = TRUE, verbose = FALSE, n.MC = 10) ## (3) Apply the Kars et al. (2008) model to the data kars <- calc_Huntley2006(data = data, rhop = rhop, ddot = ddot, readerDdot = readerDdot, n.MC = 25) # } # NOT RUN { # You can also provide LnTn values separately via the 'LnTn' argument. # Note, however, that the data frame for 'data' must then NOT contain # a LnTn value. See argument descriptions! LnTn <- data.frame(LnTn = c(1.84833, 2.24833), LnTn.error = c(0.17, 0.22)) LxTx <- data[2:nrow(data), ] kars <- calc_Huntley2006(data = LxTx, LnTn = LnTn, rhop = rhop, ddot = ddot, readerDdot = readerDdot, n.MC = 25) # }