Modeling incomplete and heterogeneous bleaching of mobile grains partially exposed to the light: Towards a new tool for single grain OSL dating of poorly bleached mortars

Accepted Manuscript Modeling incomplete and heterogeneous bleaching of mobile grains partially exposed to the light: Towards a new tool for single grain OSL dating of poorly bleached mortars Pierre Guibert, Claire Christophe, Petra Urbanová, Guillaume Guérin, Sophie Blain PII:

S1350-4487(16)30314-6

DOI:

10.1016/j.radmeas.2017.10.003

Reference:

RM 5845

To appear in:

Radiation Measurements

Received Date: 19 October 2016 Revised Date:

18 September 2017

Accepted Date: 7 October 2017

Please cite this article as: Guibert, P., Christophe, C., Urbanová, P., Guérin, G., Blain, S., Modeling incomplete and heterogeneous bleaching of mobile grains partially exposed to the light: Towards a new tool for single grain OSL dating of poorly bleached mortars, Radiation Measurements (2017), doi: 10.1016/j.radmeas.2017.10.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Modeling incomplete and heterogeneous bleaching of mobile grains partially exposed to the light: towards a new tool for single grain OSL dating of poorly bleached mortars. Pierre Guiberta , Claire Christophea, Petra Urbanováa, Guillaume Guérina, Sophie Blainb a IRAMAT-CRP2A, « Institut de Recherche sur les ArchéoMATériaux – Centre de Recherche en

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Physique Appliquée à l’Archéologie », UMR5060 CNRS-Université Bordeaux Montaigne, Maison de l’Archéologie, Esplanade des Antilles, 33607 Pessac, France

b FRS-FNRS, Centre Européen d’Archéométrie- Bât B5A, Université de Liège, 17 Allée du 6 Août, 4000 Liège, Belgium

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Key-words: partial bleaching, exponential distribution, residual dose, single grain OSL dating, mortar

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Abstract

Dating archaeological lime mortars by single grain OSL is founded on the bleaching of grains during the mixing of lime and sand. However incomplete and heterogeneous optical zeroing of grains is often observed. In the view of evaluating reliable archaeological doses, we have developed nonstandard statistical tools that describe residual dose distributions. For these purposes, a model of the bleaching process, based on simple physical assumptions leads to an exponential decreasing

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distribution of light exposure among sand grains. Coupling this exposure distribution with both decay of OSL grains with light exposition, which is assumed to be exponential, and dose response growth curve of OSL, chosen in our case as a single saturating exponential, distributions of residual doses are obtained. Such distributions are then added to burial dose distributions (defined as the doses

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absorbed by the grains since the mortar making) to simulate equivalent dose distributions of archaeological materials. A very satisfactory agreement is obtained with known age reference

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samples; in particular, our model leads to more accurate ages than the commonly employed Minimum Age Model. Finally, we estimate that this model, to some extent, could be generalized to other bleaching processes involving a mobility of grains as a major source of random exposure of grains to light.

Corresponding author: [email protected] (tel +33557124549, cell +33688615853)

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1. Introduction Dating archaeological mortar by OSL is one of the challenging steps in obtaining the chronology of ancient constructions. Amongst building materials, mortar appears to be more convenient than

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bricks to date masonries, since the latter can be reused from older buildings or structures (Goedicke, 2002, 2011; Zacharias et al., 2002; Gueli et al., 2010; Guibert et al., 2012). The target event to be dated is the preparation of mortar when grains of sand (including quartz) were extracted, possibly sieved, and then mixed with lime. In many cases, we observe that grains were heterogeneously and insufficiently bleached, leading to overestimated ages when classical multigrain OSL is implemented.

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To overcome this problem, it is necessary to use a single grain technique for OSL measurements to solve partial bleaching problems (e.g., Botter-Jensen et al., 2000; Jain et al., 2004; Urbanová et al.,

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2015).

The aim of this work is to propose a novel approach to the single grain OSL dating procedure and to apply it to archaeological mortars, but to some extent it can be applied to other situations including a natural movement of sand grains that results in heterogeneous and insufficient bleaching. For these purposes, modeling the bleaching process of grains during the manufacturing of mortar is a first necessary step that we will present in details. This model is based on simple physical considerations

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at the level of individual grains, and this mechanism leads to defining a general distribution function of light exposure among grains. To demonstrate the suitability of this theoretical approach, experimental distributions of individual doses measured from mortar samples of known age are

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compared with simulated ones according to our model. The purpose of this comparative approach is to determine archaeological doses corrected for residuals resulting from incomplete bleaching.

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The examples (Urbanová, 2015; Urbanová et al., 2015, 2016; Urbanová and Guibert, 2017; Blain et al., in prep.) that illustrate our research are taken from a variety of sites of various known ages: Gallo-Roman mortars from the Palais-Gallien (Bordeaux), the foundations of the Picasso Museum at Antibes (Alpes Maritimes, France), medieval mortars from the Oudenaarde abbey (Belgium). 2. Modeling the residual dose distribution 2.1. A demonstartion example A typical equivalent dose (ED) distribution of one of the poorly bleached mortar samples is given in Fig. 1. We distinguish a peak with a maximum around 3-4 Gy at low doses (close to the expected dose, near 3Gy), and a tail in the high dose region. The highest dose measured for this sample, extracted from the Gallo-roman amphitheater in Bordeaux (Palais-Gallien), is around 300 Gy (this

ACCEPTED MANUSCRIPT dose value was obtained by extrapolation of the growth curve, the maximum regeneration dose given for this sample was 40Gy) with a minimum around 1.8 Gy giving a dynamic scale of more than 2 orders of magnitude in the distribution of individual doses. Urbanová et al. (2015) already discussed the origin of such variability in equivalent doses of individual quartz grains. To summarize this published study, the distribution of individual ED values results from a combination of (i)

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variations of dose rates at the grain size scale, mainly due to the heterogeneous beta radiation field inside mortar, (ii) intrinsic dispersion of dose measurements by OSL on single grains of quartz, and (iii) individual variations in the bleaching of the grains. As shown by these authors, for this known age monument, the dose calculated using the Minimum Age Model (MAM: Galbraith et al., 1999)

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systematically overestimated the expected age of these poorly bleached materials. Fig. 1

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2.2. Principle of the modeling

The idea is to reconstruct, by a numerical approach, the observed distribution of ED values on the basis of (supposedly) realistic assumptions on the OSL behavior of grains. The very starting point is the physical meaning of an individual dose Di measured with the ith grain of a series extracted from a mortar sample. We can write:

where

i

i

A + Ri , (eq.1)

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Di =

is the annual dose to which the ith grain has been exposed in the mortar, A is the time

elapsed since the making of mortar (since the end of the last exposure of the collection of grains, i.e.

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the age of the mortar), and Ri the residual dose of the grain. The distribution of Di is then a linear combination of two distributions that will be considered

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independently: on the one side, the distribution of dose-rates (or more precisely the distribution of ‘burial’ doses acquired by grains since the target event, arising from the variability in single grain dose rates) and on the other side, the distribution of residual doses. Although the distribution of dose rates at the millimeter scale can be difficult to model precisely, the extent of its variations can be estimated by e.g., beta imaging systems (Rufer and Preusser, 2009; Guérin et al., 2012a), Monte Carlo (Nathan et al., 2003; Guérin et al., 2012b, 2015a; Martin et al., 2015) in combination with numerical simulations (Mayya et al., 2006) based on radioelement mapping and petrographic images. In this study, we focused on the characterization of the initial resetting state of quartz grains of ancient mortars, and thus that of residual doses, in order to determine the age A.

ACCEPTED MANUSCRIPT Bailey and Arnold (2006) performed numerical simulations to evaluate and understand the variability of OSL properties observed on individual grains of fluvial sediments. Our approach is to some extent similar, however, instead of assuming that the time of exposure of grains is distributed according a uniform rectangular distribution, our work consists of searching the form of the exposure distribution based on elementary assumptions. After recalling simple physical considerations about the behavior

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of grains, we propose a model for the exposure time that will be exploited to estimate, by Monte Carlo calculation, the distribution of residual doses. 2.3. Some basic physical considerations

First of all, we consider that the quartz grains studied and selected for OSL measurements for a given

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sample are independent, meaning that the bleaching state of one of these grains is not correlated to that of another grain of the studied series. Furthermore, we consider that the observed distribution

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results from a random selection of grains amongst an ‘infinite’ (= very large number) collection of grains used by the ancient builders for the construction.

In addition, during the manufacture of mortar by mixing lime, sand and water, individual quartz grains are moving and at any moment some of them are exposed to the ambient light for a short while until returning to the inside of the mixture, shielded from light. The possibility of bleaching

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stops when the fresh mortar is set in masonry. 2.4. Modeling the time of exposure

We assume that the probability for a grain to be exposed to light does not depend on its previous

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exposure. In other words, for each grain, there is no memory effect that could affect its probability to be exposed again. It is well-known that any memoryless continuous random variable follows an exponential distribution (see for instance, NIST, 2016; Pishro-Nik, 2016). We deduce that the time of

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exposure of a grain follows an exponential law, ε(λ), where λ is the inverse of the mean duration of exposure. The duration of exposure, denoted ϕ, leads to the expectation value E(ϕ) that is equal to 1/λ, according to the following density of probability f, with t being the time relative to the mortar mixing process:

for t > 0,

( ) =

, (eq. 1)

This is at the origin of the name of this model: “exponential exposure distribution”, denoted EED model herafter. 2.5. Relation between exposure and OSL signal attenuation

ACCEPTED MANUSCRIPT We assume that the OSL signal of interest (or its fast component, in case the studied samples displays other OSL components) decays exponentially with light exposure. The following equation gives the relationship between the OSL signal of a grain and its exposure time ϕ:

L(ϕ) = L0 exp(- a ϕ),

(eq.2)

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where L(ϕ) is the OSL intensity of a grain after light exposure, L0 its initial OSL intensity before light exposure, and a is an efficiency parameter (analogous to the inverse of a time) that depends on common factors shared by all grains of the sample studied: The light intensity and spectrum during the mortar making,

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The interaction cross section between photons and the optical center of interest

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(corresponding to the fast component of quartz OSL).

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2.6. Relationship between exposure time and residual dose

To determine an expression of residual doses as a function of exposure time, we may first define the relationship between OSL and dose. Among the most used dose response functions, we have chosen in this work a single saturating exponential:

(eq.3)

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L(D) = Lm[1 – exp(-D/D0)],

where L(D) is the OSL intensity after irradiation by a dose D, D0 the curvature parameter (or the dose saturation parameter), and Lm the maximum value of the OSL signal. This general expression of the growth function is supposed to be commonly shared by all grains within the same mortar sample. In

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practice, as we are interested in dating ‘young’ samples with low doses, the age of which is less than or around 2 ky, this function describes correctly the experimental growth of OSL. We note that the

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choice of another growth function does not change the principle of the residual dose determination, but the growth function type both used for dose measurements and for residual dose calculation must be the same, for self-consistency. We need an expression of the residual dose, R, as a function of the exposure term (a ϕ) to build the distribution of residual doses from the exposure one. R is thus deduced from the growth function by equating D as R in eq.3, and L(R) as L0 exp(- a ϕ). It comes: L(R)/Lm = L0 exp(- a ϕ)/Lm That can be written also as L(R)/Lm = I0 exp(- a ϕ), with I0 = L0/Lm

ACCEPTED MANUSCRIPT I0 represents the ratio between the initial level of OSL signal of a grain before bleaching (L0) and its saturation value (Lm). Thus, the initial relative value of OSL, I0, belongs to the interval [0, 1] and its maximum value corresponds to the saturation state. The combination of Eqs (2) and (3) leads to the following result, allowing to get rid of the individual variability of OSL sensitivity of grains:

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R(ϕ) = - D0 ln[1 – I0 exp(- a ϕ)] (eq. 4) This is the relationship that we used in our simulations to generate individual residual doses from a distribution of exposure times. 2.6. Simulation of residual dose distributions

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The calculation process used is of the Monte-Carlo type, since it permits to simulate easily the production of experimental data, and in particular to estimate the uncertainty arising due to the

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limited number of quartz grains analyzed in single grain OSL.

Production of an exponential distribution and distribution of residual doses We used a random function which is available for many computing languages and softwares, in particular with the EXCEL© package, the one we used in this work. This function, Rand() (runif() in R package, as random uniform), generates a pseudo random number u following a uniform distribution

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in the interval from 0 to 1. The random time variable ϕ, that represents the time of exposure, given by

ϕ = - ln[1 –u] /

(eq. 5)

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is distributed according to an exponential distribution with rate λ (ln: natural logarithm).

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For the sake of simplicity, in view of Eq. (2), the exposure term (a ϕ ) can be calculated as (a ϕ) = - κ ln(1 – u)

(eq. 6)

where κ = a / λ.

κ (kappa) is an adjustable positive dimensionless parameter associated with all grains coming from the same sample of mortar. κ is what we can call the “exposure parameter” and can be considered as a characteristic of the bleaching state of a given sample. The greater this parameter κ, the greater is the global exposure of sand grains. According to the previous equations, the residual dose distribution is obtained as follows:

ACCEPTED MANUSCRIPT R = D0 ln( 1 – I0 [1-u]κ) (eq. 7) As an illustration, Fig. 2 shows examples of residual dose distributions obtained with different κ parameters (with I0 = 1, D0 = 120Gy). A series of 4000 values of exposure was calculated for each example. Withκ = 10, more than 77% of the residual doses are below 10Gy, and more than 46% below 0.1Gy. With a very poor bleaching state (κ =2), these proportions are about 30% and 3%,

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respectively. For roman and medieval monuments, the residuals are a crucial factor and it is important to realize that even a residual dose as low as 0.1Gy corresponds to an overestimation of 50 years in age (for a dose-rate to quartz grains 2Gy/ka).

To summarize, the residual distributions are obtained by assuming that all grains have the same type

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of OSL dose response curve and sensitivity to light exposure but were exposed to light for variable durations during the mortar making and wall erection processes (extraction of sand, storage and

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transport, mixing with lime, wall making). Under these – in our view – reasonable assumptions, the residual dose distribution resulting from a movement of grains cannot be approximated by a Gaussian/log-normal or a truncated Gaussian/log-normal function, contrary to what is assumed by the MAM (Galbraith et al., 1999). It should also be noted here that, mathematically speaking, there is

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no part of the ED distribution that is strictly unaffected by residuals, even for the lowest ED values.

Conversely to natural sediment deposition mechanisms, we do not expect any ‘post-depositional’ mixing of grains coming from other parts of the building under study because of the mechanical hardness of the mortar samples that makes mortar comparable to a solidified sediment. If

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restorations or modifications of architectural structures are possible in ancient monuments, they are generally visible and are detected during archaeological investigations or sampling campaigns. These

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changes in the environment of the structure under study can affect the gamma dose-rate, but not the integrity of the subsequent mortar sample. Fig. 2

3. From residual dose to the archaeological dose distribution 3.1. Modelling individual dose rate distributions In most sediment samples, at least part of the radioactivity is heterogeneously distributed, at the sub-mm scale, in certain minerals such as K-feldspar or heavy minerals (e.g., zircon, monazite end other rare earth phosphates). As a result, distributions of beta dose rates to single grains of quartz

ACCEPTED MANUSCRIPT are positively skewed and can be described by log-normal distributions (e.g., Mayya et al., 2006; Cunningham et al., 2012; Guérin et al., 2015a).

To check the suitability of this kind of distribution, we have examined individual EDs of well bleached

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samples from the Antibes gallo-roman wall. The SG-OSL ages obtained using the CAM (Central Age Model) procedure are in agreement with both standard multi-grain OSL ages and with the archaeological approach of its chronology (Urbanová et al., 2016). Thus, we assumed that the relative distributions of ED matched those of dose-rates, except for the fact that the experimental distributions implicitly comprise measurement errors, both statistical errors (including those arising

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from counting statistics, instrument reproducibility and curve fitting uncertainties) and intrinsic overdispersion of dose measurements (this latter source of dispersion of individual doses was evaluated

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by single grain dose recovery experiments and give a minor contribution of around 10-15% for those samples). We note that with those mortar samples, the distribution of EDs is then well represented by a log-normal distribution (Fig. 3). We have to point out that in our approach we have considered only individual EDs the relative error of which is below 30% in order to constrain the impact of highly imprecise and insignificant EDs on histograms and subsequent calculations of archaeological doses. In the future, we intend to enter as input parameters more realistic dose distributions than log-

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normal ones, which we can obtain for instance from Geant 4 simulations (Martin et al. , 2015) or from beta auto-radiography imaging (Rufer and Preusser, 2009). At present, we assume that the lognormal approximation for the dose rate distribution is a convenient approach that permits to

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describe a large number of situations. We must keep in mind that definite functions as the one we use in this work have the status of numerical approximations only and are not the exact

Fig 3:

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representations of real configurations of beta dose rate heterogeneities.

Production of a dose-rate distribution In our model, the production of a reference list of thousands of dose rates is obtained by random sampling from a log-normal distribution. To do so, a series of random variables, zj, is generated following a centered normalized Gaussian distribution. Then, by using Eq. (8), the random variable dj, representing a dose rate, follows a log-normal distribution: di = exp( m + zj s )

(eq. 8)

ACCEPTED MANUSCRIPT where m and s represent respectively the mean and the standard deviation of the logarithm of individual dose rates. 3.2. Generation of a list of simulated archaeological doses The final step is the calculation of a series of archaeological doses. For a given age A, all simulated

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individual doses are calculated according to Eq. (1) that refers to the ith element of both reconstructed data sets. This series of doses is called the simulated list and it will be compared to the experimental one.

4. Reconstructing distributions of ED values from the experimental data sets and determination

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of archaeological dose and age

In the following section, we first compare reconstructed dose distributions with experimental ED

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distributions for different archaeological known age mortar samples. Then, we compare the ages obtained with our model with the independently known ages. 4.1. Selection of experimental data

The experimental data consist of lists of measured ED values with their analytical uncertainties (including counting statistics, curve fitting errors and instrumental reproducibility) using the SAR

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protocol (Murray and Wintle, 2000) performed by Urbanová et al. (2015). Only ED values with an uncertainty of less than 30 % were selected for analysis; the risk of applying such a selection criterion is to favor grains with higher OSL signals, which may lead to an over-representation of poorly bleached grains (the higher the ED, the higher the signal), in particular for very young and low

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sensitivity samples. A possible way to check the influence of the selection criteria is to change the threshold of the selection criteria and compare results.

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The dose-rate received by the 200-250 µm quartz grains from the mortar samples was determined by classical means, in Bordeaux, by high resolution gamma spectrometry for the beta dose rate(Guibert et al, 2009), evaluation of the archaeological moisture, environmental Al2O3 OSL dosimetry for the gamma dose rates (Richter et al., 2010; Blain et al, 2011), and ICP-MS for the internal radioactivity of grains (Urbanová et al., 2015). Analytical data were then converted into dose-rates with the updated dose rate conversion factors from Guérin et al. (2011) and beta attenuation factors (Guérin et al., 2012b). Examinations under a microscope of thin sections of mortar, beta auto-radiography, as well as multiple potassium concentration measurements of areas of polished sections with a SEM-EDS device centered on 200-250 µm quartz grains (as in Urbanová et al., 2015), can give some clues about

ACCEPTED MANUSCRIPT the potential variation of the beta dose rates to single grains, but until now the results of this characterization step are mainly qualitative. As a result, the experimentally derived dose rate data are considered as mean values of individual dose-rates to single grains. 4.2. Sketch of the simulation process and list of parameters to be determined

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In addition to the experimental data, including the annual dose-rate, the following parameters must be integrated to the calculation in order to reconstruct the ED distribution: -

The expected relative standard deviation of dose rates to single grains; assuming that this is the major source of intrinsic variability, denoted σ. σ is related to the standard deviation s in

-

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(Eq.8) by ln σ = s.

D0 , the curvature parameter (in Gy) of the normalized dose response curve applied to all grains,

I0, the OSL signal intensity normalized level characteristic of the sample before manufacture

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-

of the mortar (0≤ I0 ≤1), we recall that I0 = 1 corresponds to the saturation state of OSL. -

A, the age of the mortar (in yr),

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κ (kappa) the light exposure parameter (dimensionless parameter).

Starting with some input initial values for these parameters denoted

,

0,

0,

,

̃ , we then

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iteratively update the parameters to fit the simulated distribution to the experimental one (more practical details are presented in the following sections, and particularly in section 4.4). 4.3. Modes of comparison of experimental and simulated data

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A prototype of data treatment was achieved with the Excel© package and allows adjusting these parameters by a step by step process. A series of 10000 equivalent doses is generated during any

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iteration according to Eq. (1) and forms the simulated data (using the current values of the parameters that are updated before every interation cycle). Then, a first mode of comparison that can be used at any step of the simulations is the comparison of histograms of both experimental and calculated ED values (Fig. 4). But more precise comparisons based on cumulative means are preferably used in order to determine the above-mentioned parameters (section 4.2). For this purpose, we describe hereafter the two-step comparative process we used. Figure 4 Step 1: cumulative mean

ACCEPTED MANUSCRIPT A comparison between experimental and calculated data is performed by examining the variation of the cumulative mean of ED values previously sorted in ascending order. The jth cumulative arithmetic mean of the simulated ED values is obtained using the following formula

j

(eq. 9)

gives the simulated, uncorrected (i.e. including residual doses) mean dose. In

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We note that

= ∑

parallel, the corresponding mean value of the simulated residual for the j first simulated equivalent doses is also calculated and is denoted

j.

The experimental ED values are also sorted in ascending order. In the same way as the simulated

calculated as follows:

(eq. 10)

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= ∑! " !

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data were computed, the cumulative mean values of experimental ED (arithmetic mean) are

!

where xi is the ith value of the sorted list of individual measured ED (1≤k≤n); From a practical perspective, the comparison of experimental and simulated data is facilitated by plotting both

j

as a function of Dj and Xk as a function of xk.

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Step 2: computation of cumulative means corrected for residual doses A calculation of the residual dose to be subtracted from the mean value of experimental ED is implemented. The practical problem to solve is that the number of experimental data and that of the simulated ones are very different, so we may not consider the absolute number of data (k) taken into

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account but the proportion they represent in the whole series of the n experimental data: this proportion is equal to (k/n). To calculate the residual dose to be subtracted to the kth mean value Xk,

-

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the following correction procedure is performed: we consider the same proportion (k/n) of the sorted list of the N simulated ED values and we

calculate

j(k)

the mean value of the corresponding residual doses, with j(k) the number of

simulated data to be taken into account (j(k) = Integer[(k/n)N]).

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The net mean value of the dose Yk corresponding to the kth experimental element of the cumulative mean is then calculated as: Yk = # –

j(k)

(Eq. 11)

Yk represents the mean equivalent dose corrected from residuals. Yk and Xk are plotted as a function of xk and compared to both calculated net and uncorrected mean value of ED respectively. Figure 5

ACCEPTED MANUSCRIPT As seen in Fig. 5, the mean equivalent doses corrected for residuals of the first k data of the experimental series, both quantities Yk and (

− j), reach a plateau when the sorted individual ED

values correspond to the tail of the dose rate distribution and thus the residual component of the OSL signal becomes preponderant compared to the burial part. In other words, adding the subsequent data to the cumulative corrected mean does not change its value. We note that, at the (!)

−

j(k)

) with j(k) = Integer[(k/n)N] should be

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end of the iterative process, the ratio Yk / (

constant and as close to 1 as possible within the variations due to statistical uncertainties.

4.4. Some comments on practical aspects

To introduce this new tool, we will give hereafter some comments about our present

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experience on mortar dating (Urbanová and Guibert, 2017). We have to note that all parameters of the simulation are interlinked more or less closely, and we will discuss their general effect on the

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calculated distributions. In practice σ, I0, κ, and the age A are the parameters that have the strongest effects on the simulated ED distributions. These parameters are unknown a priori and must be estimated, in particular the age A which is our final aim.

Determination of I0, the initial level of OSL

In general, the order of magnitude of the age of a structure being dated is known, so it can be set

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initially at a working value and re-evaluated more precisely once the other parameters have been determined. In practice, we fix D0 at a standard value, 120 Gy, constant for any grain. Indeed, the first parameter to be adjusted is I0, initial level of OSL, because it gives the upper limit of the dose distribution in case of poorly bleached samples. An initial value,

0,

can be set based on Eq. (3),

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replacing D by Dmax, the maximal individual dose of the experimental distribution: 0

= [1 – exp(-Dmax/D0)]

(eq. 12)

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If, like at the Palais-Gallien, the maximum individual measured ED is around 300 Gy, this parameter is chosen close to 1 (with D0=120Gy). For other samples, like that from Oudenaarde in Belgium (Table 3), the natural distribution of ED is limited to 30 Gy at high values, so that, with D0=120Gy, the corresponding I0 level is ~0.21. The precise value of this parameter is progressively defined by a series of iterative tests, until the histogram of simulated ED values matches the natural one for high dose values. We must keep in mind that for totally or well bleached samples, there is no need for a precise determination of I0 since this parameter has no importance in the reconstruction of dose distributions that can be approached by classical means like the CAM calculation (in practice, for well-bleached samples, a high value for the light exposure parameter κ will mask the value of the I0 parameter).

ACCEPTED MANUSCRIPT Determination of kappa: the use of a plateau test. In the case of insufficiently bleached materials, our principal concern, a consistent evaluation of the κ parameter is crucial. When a suitable value of κ is found, a plateau of Yk (cumulative mean of corrected values of the k first sorted archaeological doses) as a function of xk (kth value of the sorted individual archaeological doses) is obtained. When κ is taken too low in comparison with the real exposure of the grains, the plateau drops quickly towards low and insignificant values. Inversely,

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when κ is taken too high, no plateau is observed and the curve of mean net ED values monotonically increases with measured ED and tends to the uncorrected one. As an illustration, Fig. 6 shows a series of plateaus obtained using different values of κ. We noted also that in the high dose region, the random calculation can give highly uncertain values because the high residual doses are generally

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not numerous and scatter on a large scale. So we must consider that the plateau technique is mainly significant in the low dose region when poorly bleached samples are under study.

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Figure 6 Determination of sigma

In our model, the σ parameter represents the variability of dose rates to which single grains were exposed. As it is generally unknown, it can be adjusted from an initial value around 0.2 (20%) that is the minimal overdispersion (OD) value observed for well bleached mortars (Urbanová, 2015). Although the OD refers to a quadratic combination of intrinsic variability of OSL measurements of

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doses (intrinsic overdispersion, Thomsen et al., 2005) and burial dose variability, it affects the dose rates in our actual model, since in general, the dose rate dispersion is the dominant term of OD in well bleached samples (Guérin et al., 2015b). Sigma values must then be tried to best fit the experimental data set and it is not necessary to fix a specific and arbitrary value of σ as it is necessary

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with the IEU model (Thomsen et al., 2007). In the low dose region of the cumulative mean plots, the growth of the experimental mean with the dose should be the same as that of the simulated one

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when a suitable value of σ is assigned (see Fig. 5b).

Final values of sigma and kappa are determined according to a least square calculation. To do so, we examine the relative variances of the ratio Yk / (

(!)

−

j(k)

) (see § 4.3) for diverse values of kappa

and sigma and we look for their respective minima for a given number of data taken into account (k).

About the number of experimental data to take into account, and the calculation of statistical uncertainties The question that arises at that point is the definition of the dose interval to be taken into account for an accurate evaluation of the archaeological dose, or in other words, how many sorted ED

ACCEPTED MANUSCRIPT experimental data are necessary to assess the mean dose. The answer depends on the sample characteristics and it is difficult to specify a minimum number of data or a minimum proportion of the data series, because it depends on the bleaching state of the sample, except that the number of grains must be as high as possible. In fact what seems to us a good indicator of the number of data to be taken into account is deduced from the variation of the statistical uncertainty with the number of

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sorted EDs integrated into the calculation. Let km be that number (km ≤ n, n being the total number of data available of the experimental series). Figure 7 shows an example of such a variation with the Palais-Gallien sample BDX 15544. The statistical error of the mean corrected dose comes from 2 different sources the values of which are quadratically added:

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i) The errors on individual dose measurements (as given by the software Analyst), the effect of which decreases approximatively in proportion of the inverse of the square root of the number of data

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taken into account (km),

ii) The limited number of grains of the experimental series generates a statistical uncertainty which is the second component. To evaluate its contribution to the overall statistical error, successive drawings of simulated sets of data are performed, (about 50 independent drawings), the number of data in each set being equal to the number of grains of the experimental series (n): for a given number of data taken into account, km, this component is taken equal to the standard deviation of

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the values of the mean corrected ED obtained by the successive drawings already mentioned.

In fig. 7 with the sample cited we note that the statistical uncertainty attains a minimum level around

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ca 8 Gy. At low doses, integrating an increasing number of data, as commonly expected, enhances the precision of the mean value, but at higher doses, the residuals are preponderant in the OSL signal, and the uncertainty tends to increase, due to the fact that residual doses are both highly

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variable (exponential distribution of the exposition to the light of grains) and scarce. All samples investigated exhibit such a variation in case of partial bleaching. Beyond the region of doses corresponding to the minimum uncertainty the increasing number of data does not compensate for the important increase of the variability of residual doses. Finally, we limit the number of data (km) to the number of individual doses that corresponds to the minimal uncertainty. The number of data to be taken into account is then defined and we are able to determine the best fit values of sigma and kappa. At the end we get the mean archaeological dose value that we use to determine the archaeological age in a very classical way. Figure 7

ACCEPTED MANUSCRIPT About the role of the saturation parameter D0 The interplay between the parameters D0, I0 and κ for a given experimental distribution is such that different sets of these parameters may lead to similarly good fits of simulated to experimental data. If D0 is taken low, the range of variation of the residual doses is then reduced to a

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smaller interval of dose than with higher D0, and to reproduce the experimental scatter of data, κ will be found lower and accordingly I0 higher (below the upper limit of 1 that corresponds to the saturation level) to fit the experimental ED distribution. As an example, Table 1 reports different sets of parameters that satisfactorily fit the experimental distribution of the poorly-bleached PalaisGallien mortar samples, for two values of D0: 60 and 120 Gy.

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Table 1

These data show that different sets of triplets (D0, I0, κ) can satisfactorily describe the same residual

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dose distributions, but that eventually, the corresponding calculated mean corrected dose remains constant. Reciprocally, taking a fixed value for D0 instead of a variable one has no meaningful consequence on the evaluation of the archaeological dose, with our mortar samples, probably because most of the OSL signals of grains of interest were far from the saturation level.

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5. Case studies

We distinguish two types of samples, those which were totally or well bleached in the mortar making process, and the others.

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5.1. Well bleached samples

For the gallo-roman foundations of the Grimaldi Castle at Antibes (Urbanová et al., 2016), we found that the κ factor ranges, from sample to sample, from 45 to 500 and we can consider that the

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bleaching state is rather satisfactory in a first approach. For the most exposed samples, there is no need for a correction of a residual dose. High residuals are very unlikely with such a state of bleaching, but the calculation predicts the existence of some rare high values (less than 0.5% of doses greater than 30Gy for κ =300) and this is what is observed with all samples (fig. 8). In general, for all Grimaldi Castle samples, the CAM doses agree with the EED doses (Table 2) except for BDX 16047, where a relative deviation of 10% is observed, the EED model giving, as we could expect, a lower dose (and age) than the CAM which integrates without correction the highest ED values. The bayesian approach of mean dose calculation from all OSL data leads to the same trend as observed with the CAM, except that a systematic deviation of ca. 100yr is observed, in the sense that bayesian ages are younger than CAM ones with that series of samples. For all samples except BDX 16047, the

ACCEPTED MANUSCRIPT less bleached one, we note that the values for σ (deviation in dose rates) determined by best fit in our simulation are close to the over-dispersion (OD) determined by the CAM. Indeed higher residuals are integrated as dispersive elements in the calculation of the central dose using the CAM. Finally, the average age of these five mortar samples obtained by the EED model is 1910 ± 120yr (AD 110 ± 120). It is close to that obtained with the CAM: 1950 ± 120yr (AD 65 ± 120), or with the

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bayesian calculation of mean dose: 1850 ± 120yr (AD 165 ± 120) due to a high bleaching state of the quartz grains that makes the level of residual OSL low, except for sample BDX 16047 as mentioned before. The archaeological date of the wall, based on the chrono-typology of ceramics associated with the foundations of the wall, is expected to be 60-110 AD. Our data treatment (EED) agrees with

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this archaeological attribution (more details in Urbanová et al., 2016). Figure 8, table 2

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5.2. Partially bleached samples

Partially bleached samples are the ones that initially led to this research on exposure distributions. We have different case studies: a sample from the medieval abbey of Oudenaarde in Belgium, and Gallo-roman samples of the Palais-Gallien in Bordeaux.

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Oudenaarde abbey sample

The abbey church of Maagdendale in Oudenaarde, Belgium, is well dated of the 14th century by historical sources (Debonne, 2011) and dendrochronology of the wood beams of the roof of the chancel and the transept: 1290-1310 (Haneca, 2010). A sample of mortar was taken with a chisel and

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a hammer from within the eastern internal wall of the northern transept arm, in direct archaeological link with the dated wood beams. This piece of mortar was selected as one of our

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reference samples to develop single grain OSL dating of mortar. The distribution of ED values for the Oudenaarde mortar is typical of very poorly bleached sand (Fig. 9). We note that even with this low bleaching state, the highest measured ED is ~30Gy. With a D0 value of 120Gy, as mentioned above, I0 is around 21%. We observe that experimental and reconstructed distributions match well. Due to the low exposure experienced by the sand, there is no remarkable accumulation of low ED values doses as it could be seen, for instance, on the ED histograms of Palais-Galien samples (Fig. 4). Nevertheless, we were able to determine residual doses and a mean corrected archaeological dose of this mortar sample. The EED SG OSL age of this sample is 730 ± 70yr (1 sigma uncertainty), which is in good agreement with the dendrochronological age of the masonry (710 ± 10yr). The large uncertainty of the OSL age

ACCEPTED MANUSCRIPT results both from high residuals to be subtracted from the mean ED value (the κ value is particularly low, κ =1.23), from the low number of points taken into consideration for the dose calculation (integration of ED values of less than 6.0 Gy, corresponding to the onset of the mean dose plateau) and from the fact that individual ED values have rather high uncertainties (from 10 to 20%). We also note that the MAM significantly overestimates the age of that sample; the EED approximation

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appears to be more appropriate for such poorly bleached samples (Table 3). Figure 9 Palais-Gallien samples

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All Palais-Gallien samples exhibit the same type of ED distribution as the one chosen as example in Fig. 1. Reconstructed dose distributions and experimental ones are very similar. The initial OSL of the sediment was close to the saturation with I0 varying from 0.70 to 0.95 according to samples (with

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D0=120Gy), perhaps indicating that a source of sand deposited very anciently was used by roman builders. We noticed that the log-normal distribution of dose rates deviates from the experimental distribution that appears flatter than modelled and might be bimodal. This should be taken into account in the future, when consistent distributions of dose rates derived from experimental data and local measurements of dose or radioelement concentrations are available. Finally, all EED SG-OSL

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ages are in better agreement with the known age (Urbanová, 2015; Urbanová and Guibert, 2017) than those provided by the standard MAM (Minimum Age Model) model that tends to give overestimates (Table 3). In addition, it is important to notice that the MAM ages were obtained by taking an arbitrary dispersion equal to 30%, value that is slightly higher than that determined by our EED

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approach. As an exercise, we also undertook a calculation of the archaeological dose by using the IEU approach. The input values of the expected dispersions that are supposed to describe correctly the scatter of equivalent doses in the well-bleached part of the population, were taken equal to those

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found by our EED approach (20 to 25% according to the sample). As seen in table 3, IEU ages obtained are also over-estimated, indicating that residual doses are not to be neglected even in the lower dose region of the individual ED distribution. Finally, after averaging the four EED SG OSL dates, an age of 2010 ± 110 yr (uncertainty given as 1 s.d.) is obtained and is in good agreement with the archaeological age of the monument (end of 1st- mid 2nd century, Urbanová et al., 2015). Table 3 6. Conclusion Figure 10

ACCEPTED MANUSCRIPT In this study, we have demonstrated that a simple assumption about the mobility of grains during the mixing process of lime and sand leads to and explains heterogeneous bleaching. As a result, the distribution of exposure times among the grains of interest is expected to follow a decreasing exponential distribution. The examination of the distribution of equivalent doses obtained for the known age mortars presented in this paper confirms that an exponential distribution of light

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exposure is a well-suited approach to describe the bleaching state of the quartz grains investigated. Concerning the burial dose acquired during the archaeological times, we noted that the log-normal distribution law for individual dose rates is appropriate, even if a detailed analysis of deviations between experimental data and calculated ones in the net dose plateau construction indicates that other distribution functions could be proposed, for example, on the basis of experimental data about

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spatial repartition of radioelement concentrations or beta imaging. In comparison with the MAM or IEU approach applied to poorly bleached materials, our calculation appears to be more robust and

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the calculated ages are in better agreement with known archaeological ages (fig. 10). In addition, in contrast to the cited approaches, there is no need to input arbitrary parameters about the expected dispersion of the ED distribution.

The type of calculation presented here can be applied as a technique aiming at correcting ED distributions from residuals. Further developments can be imagined in a near future, like a user friendly calculation routine for analyzing SG ED distributions from mortars, or from other kinds of

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sedimentary material, the main bleaching process of which is based on the mobility of grains in a partly exposed environment (fluvial sediments for instance).

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Acknowledgements

We are grateful to the following institutions which contributed to support this work: the CNRS, The University of Bordeaux Montaigne, the University of Liège and the labex LaScArBx (label of excellence

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Archaeological Sciences Bordeaux) according to the general program supported by the ANR (Agence Nationale de la Recherche) - n°ANR-10-LABX-5. Data were collected from mortars of different origins and we thank there the local authorities involved that allowed us to sample those materials (Mairie de Bordeaux, Mairie d’Antibes, Direction Régionale des affaires culturelles d’Aquitaine, Stad Oudenaarde). References Bailey, R. M., Arnold, L. J., 2006. Statistical modelling of single grain quartz De distributions and an assessment of procedures for estimating burial dose. Quaternary Science Reviews, 25, 24752502.

ACCEPTED MANUSCRIPT Blain, S., Guibert, P., Prigent, D., Lanos, P., Oberlin, C., Sapin, C., Bouvier, A., Dufresne, P., 2011. Dating methods combined to building archaeology: the contribution of thermoluminescence to the case of the bell tower of St Martin’s church, Angers (France). Geochronometria, 38-1, 5563. Bøtter-Jensen, L., Solongo, S., Murray, A.S., Banerjee, D., Jungner, H., 2000. Using OSL single-aliquot

dosimetry. Radiation Measurements, 32(5-6), 841-845.

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regenerative-dose protocol with quartz extracted from building materials in retrospective

Combès, B., Philippe, A., Lanos, Ph., Mercier, N., Tribolo, C., Guérin, G., Guibert, P., Lahaye, C., 2015.

Quaternary Geochronology, 28, 62-70.

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Cunningham, A. C., DeVries, D. J., Schaart, D. R., 2012. Experimental and computational simulation of beta-dose heterogeneity in sediment. Radiation Measurements, 47, 1060-1067. Debonne V., 2011, “Oudenaarde (prov. Oost-Vlaanderen), vml. abdijkerk van Maagdendale. Bouwhistorisch onderzoek van de dakkap”, Onroerend Erfgoed-Rapporten Bouwhistorisch Onderzoek 1, Brussels.

Düller, G.A.T., 2008. Single-grain optical dating of Quaternary sediments: why aliquot size matters in

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luminescence dating, Boreas, 37, 589–612. (DOI 10.1111/j.1502-3885.2008.00051.x). Galbraith, R.F., Roberts, R.G., Laslett, G.M., Yoshida, H., Olley, J.M., 1999. Optical dating of single and multiple grains of quartz from Jinmium Rock Shelter, Northern Australia: part I, experimental design and statistical models.

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Goedicke, G., 2002. Dating historical calcite mortar by blue OSL: results from known age samples. Radiation Measurements, 37, 409–415. Goedicke, G., 2011. Dating mortar by optically stimulated luminescence: a feasibility study. Geochronometria, 38 (1), 42-49. Gueli, A. M., Stella, G., Troja, S. O., Burrafato, G., Fontana, D., Ristuccia, G. M., Zuccarello, A. R., 2010. Historical buildings: Luminescence dating of fine grains from bricks and mortar. Il Nuovo cimento, 125 B. Guérin, G., Mercier, N. and Adamiec, G, 2011. Dose-rate conversion factors: update. Ancient TL, 29 (1), 5-8.

ACCEPTED MANUSCRIPT Guérin, G., Discamps, E., Lahaye, C., Mercier, N., Guibert, P., Turq, A., Dibble, H., McPherron, S., Sandgathe, D., Goldberg, P., Jain, M., Thomsen, K., Patou-Mathis, M., Castel, J.-C., Soulier, M.C., 2012a. Multi-method (TL and OSL), multi-material (quartz and flint) dating of the Mousterian site of the Roc de Marsal (Dordogne, France): correlating Neanderthals occupations with the climatic variability of MIS 5-3. Journal of Archaeological Science, 39,

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3071-3084. Guérin, G., Mercier, N., Nathan, R., Adamiec, A., Lefrais, Y, 2012b. On the use of the infinite matrix assumption and associated concepts: A critical review. Radiation Measurements, 47-9, 778785.

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Guérin, G, Mayank, J, Thomsen, K.J., Murray, A.S., Mercier, N., 2015a. Modelling dose rate to single grains of quartz in well-sorted sand samples: The dispersion arising from the presence of

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potassium feldspars and implications for single grain OSL. Quaternary Geochronology, 27, 5265.

Guérin, G., Combès, B., Lahaye, C., Thomsen, K.J., Tribolo, C., Urbanová, P., Guibert, P., Mercier, N., 2015b. Testing the accuracy of a single grain OSL Bayesian central dose model with known-age samples, Radiation Measurements, 81, 62-70.

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Guibert, P., Lahaye, C., Bechtel, F., 2009. The importance of U-series disequilibrium of sediments in luminescence dating: a case study at the Roc de Marsal cave (Dordogne, France). Radiation Measurements, 44, 223-231.

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Guibert, P., Bailiff, I.K., Baylé, M., Blain, S., Bouvier, A., Büttner, S., Chauvin, A., Dufresne, Ph., Gueli, A., Lanos, Ph., Martini, M., Prigent, D., Sapin, C., Sibilia, E., Stella, G., Troja, O., 2012. The use of dating methods for the study of building materials and constructions: state of the art and

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current challenges, Proceedings of the 4th International Congress on Construction History, Paris 3-7 July 2012, 469-480. Haneca K., 2010, “Verslag dendrochronologisch onderzoek. Dakkap van de voormalige kerk van de Abdij

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Natuurwetenschappelijk Onderzoek VIOE 10, Brussels. Jain, M., Thomsen, K.J., Bøtter-Jensen, L., Murray, A.S., 2004. Thermal transfer and apparent-dose distributions in poorly bleached mortar samples: results from single grains and small aliquots of quartz, Radiation Measurements, 38, 101-109, DOI 10.1016/j.radmeas.2003.07.002.

ACCEPTED MANUSCRIPT Martin, L., Mercier, N., Incerti, S., Lefrais, Y., Pecheyran, C., Guérin, G., Jarry, M., Bruxelles, L., Bon, F., Pallier, C., 2015. Dosimetric study of sediments at the beta dose rate scale: Characterization and modelization with the DosiVox software. Radiation Measurements, 81, 134-141. Mayya, Y.S., Morthekai, P., Murari, M.K. Singhvi, A.K., 2006. Towards quantifying beta microdosimetric effects in single-grain quartz dose distribution. Radiation Measurements 41,

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Nathan, R.P., Thomas, P.J., Jain, M., Murray, A.S., Rhodes, E.J., 2003. Environmental dose rate heterogeneity of beta radiation and its implications for luminescence dating: Monte Carlo modelling and experimental validation. Radiation Measurements 37, 305-313.

http://www.itl.nist.gov/div898/handbook/, october 2016.

Pishro-Nik, H., 2016, consultation of Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik, https://www.probabilitycourse.com/, special case about memoryless random processes at: https://www.probabilitycourse.com/chapter4/4_2_2_exponential.php,

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august 2016.

Richter, D., Dombrowski, H., .Neumaier, S., Guibert, P., Zink, A.C., 2010. Environmental γ-dosimetry with OSL of α-Al2O3:C for in-situ sediment measurements. Radiation Protection Dosimetry ; 141, 1, 27-35 (doi:10.1093/rpd/ncq146).

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Rufer, D., Preusser, F., 2009. Potential of autoradiography to detect spatially resolved radiation

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patterns in the context of trapped charge dating. Geochronometria 34, 1-13. Thomsen K. J., Murray A. S., Bøtter-Jensen L., 2005. Sources of variability in OSL dose measurements using single grains of quartz. Radiation Measurements, 39, 47-61. Thomsen, K. J., Murray, A. S., Bøtter-Jensen, L., Kinahan, J., 2007. Determination of burial dose in incompletely bleached fluvial samples using single grains of quartz. Radiation Measurements, 42/3, 370-379. Urbanová, P., 2015. Recherches sur la datation directe de la construction des édifices : Exploration des potentialités de la datation des mortiers archéologiques par luminescence optiquement stimulée (OSL). Supervision by Pierre Guibert, Defence on 27th november 2015, PhD in Physics of Archaeomaterials, University Bordeaux Montaigne, 284p.

ACCEPTED MANUSCRIPT Urbanová, P., Hourcade, D., Ney, C., Guibert, P., 2015. Sources of uncertainties in OSL dating of archaeological mortars: the case study of the Roman amphitheatre “Palais-Gallien” in Bordeaux, Radiation Measurements, 72, 100-110. Urbanová, P., Delaval, E., Lanos, P., Guibert, P., Dufresne, P., Ney, C., Thernot, R., Mellinand, Ph., 2016, submitted. Multi-method dating comparison of Grimaldi castle foundations in Antibes,

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France. Archéosciences, 40, 17-33.

Urbanová, P., Guibert, P., 2017, Methodological study on single grain OSL dating of mortars: comparison of five reference archaeological sites. Geochronometria, 44, 77-9.

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Zacharias N., Mauz B., Michael C.T., 2002. Luminescence quartz dating of lime mortars. A first

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research approach. Radiation Protection Dosimetry, 101, 379-382.

ACCEPTED MANUSCRIPT Figure captions Fig. 1: Histogram of individual equivalent doses of quartz grains (200-250µm) from a mortar sample of the foundations of the Palais Gallien, a Gallo-Roman amphitheater in Bordeaux. Only the 0.0-32.8 Gy region of the ED distributions is displayed. Actually, a mean dose around 3 Gy is expected with

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this 1900 year old monument. Fig. 2: Percentage of grains having residuals lower than 100, 10, 1 and 0.1 Gy as a function of the exposure factor κ. The actual range of κ with the mortar samples studied in Bordeaux is 2-500. With poorly bleached samples (κ <10) we cannot expect a majority of grains with low residual doses (below 0.1 Gy) and our challenging issue is thus to determine a precise distribution of residual doses,

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whatever the κ factor.

Fig 3: Distribution of EDs of a well bleached mortar sample taken from the foundations of the

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Chateau Grimaldi in Antibes (BDX 16048). A log-normal distribution of dose rates (and final burial doses) fits satisfactorily the experimental one.

Fig. 4: Sample BDX 15544, Palais-Gallien, Gallo-roman amphitheater in Bordeaux, comparison of histograms of experimental (in red) and simulated data (in blue). The simulated histogram (from 4000 different values of ED) was scaled to fit the number of experimental data (127 grains of

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interest) for an easy comparison, this is the reason why the histogram of simulated data refers to decimal values and not to integer ones.

Fig. 5: Sample BDX 15544, Palais-Gallien, Gallo-roman amphitheater in Bordeaux. Comparison of

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experimental and simulated cumulative means of uncorrected EDs (fig. 5a). Same comparison with the data corrected for residuals (fig. 5b): a plateau is obtained for doses beyond ca. 6Gy indicating that the integration of the whole distribution of burial doses was completed and that the κ

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parameter and the EED approximation describe correctly the residual dose distribution. Error bars in experimental data are 1 sigma uncertainty and are calculated from the statistical error of individual dose measurements only.

Fig. 6: Sample BDX 15544, Palais-Gallien, Gallo-roman amphitheater in Bordeaux. Plateau plot of cumulative mean doses as a function of individual doses. Different values of the k parameter were tested and the best plateau (the most horizontal one) is obtained with κ=2.75. The difference between the uncorrected mean (bold line) and that of the best plateau demonstrates the important amount of residual signals in the ED distribution, even at the lowest doses.

ACCEPTED MANUSCRIPT Fig. 7: Sample BDX 15544, Palais-Gallien, Gallo-roman amphitheater in Bordeaux. Statistical uncertainty of the corrected mean ED as a fonction of the maximal ED taken into account. The uncertainty attains a minimum around 7-8 Gy. Fig. 8: BDX 16047, Gallo-roman foundations of the Grimaldi Castle in Antibes (France). Compared histograms of experimental and simulated data. The distributions show high dose values (beyond ca

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15Gy) attributed to scarce high residuals as predicted by the EED model.

Fig. 9: Mortar from Oudenaarde abbey in Belgium (early 14th century). Compared histograms of experimental data and simulated ones (fig. 8a). Comparative study of cumulative means (fig. 8b). A

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plateau is observed for individual doses beyond ca 4Gy.

Fig. 10. Deviation of SG-OSL ages of mortars from known ages (ratio [OSL-age – known age/known age]) according to the data treatment, as a function of the exposure parameter kappa (mean

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exposure to the light). The EED approach gives relevant dates within the usual uncertainties of luminescence dating even with very poorly bleached mortars such as those from Oudenaarde abbey

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and the Gallo-roman amphitheater Palais-Gallien.

ACCEPTED MANUSCRIPT Table captions Table 1: Palais Gallien mortar samples, best fit parameters with two different D0. As expected I0 and D0 are linked, but we note that κ is not obviously linked with D0. The final mean corrected doses are not significantly affected by the change of D0.

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Table 2: Mortar samples from the foundations of the Chateau Grimaldi, in Antibes (France), comparative results of the EED, CAM and BaSAR procedures. The EED model gives the dose and then the age is classically calculated by dividing the dose by the dose-rate.

Table 3: EED results of poorly bleached samples, compared to previous results obtained with the

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MAM-3 model. For the Palais Gallien samples, the internal dispersion, σb, was taken equal to 25%, the closest value deduced from our own simulations. The MAM data presented here are deduced from Petra Urbanová’s calculation using different σb in the range 5-50% (in P. Urbanová, 2015, table

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V-19, p. 130). The EED approach for all samples is in agreement with the archaeological age. *About the number of grains taken into consideration: at the denominator, all grains having a relative standard deviation below 30% permit to determine the parameters of the ED distribution, and more specifically the number at the numerator is the number of individual dose values, sorted by ascending order, from which the mean archaeological dose is calculated after correction for

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residuals.

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BDX 15541 BDX 15542 BDX 15543 BDX 15544

D0

I0

κ

120 60 120 60 120 60 120 60

0.7 0.9 0.75 0.94 0.90 1.0 0.9 1.0

4.8 4.25 3.5 3.1 4.05 3.5 2.75 2.4

Mean corrected dose (Gy) 3.34±0.10 3.32±0.10 2.98±0.14 2.98±0.13 2.91±0.10 2.89±0.10 2.58±0.09 2.56±0.09

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Table 1

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σ (%)

I0

D0 (Gy)

κ

BDX 16045 BDX 16046 BDX 16047 BDX 16048 BDX 16049

33 33 24 27 32

0.15 0.45 0.39 0.22 0.15

120 120 120 120 120

100 120 45 500 120

Nb of grains taken into account 167 159 163 149 95

Archaeological dose EED (Gy) 3.79±0.08 3.55±0.10 4.17±0.06 3.68±0.08 3.82±0.10

OSL age EED (y) 2074±96 2060±106 1825±120 1776±103 1989±132

Archaeological dose CAM (Gy) 3.85±0.10 3.56±0.10 4.60±0.16 3.66±0.08 3.89±0.13

OSL age CAM (y) 2107±103 2066±106 2013±149 1776±103 2025±141

31 31 43 27 36

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Table 2

OD (%)

Archaeological dose – baSAR (Gy) 3.80±0.10 3.49±0.10 4.31±0.15 3.79±0.10 3.70±0.13

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Sample

OSL age baSAR (y) 2012±100 1894±115 1773±145 1680±125 1889±140

κ

Nb of grains taken into account*

Archaeological dose - EED model (Gy)

Oudenaarde medieval abbey (Belgium) Oude 30 0.21 120 1.23 19/71 Palais-Gallien, Gallo-roman amphitheater, Bordeaux (France) BDX 15541 20 0.70 120 4.80 42/75 BDX 15542 25 0.75 120 3.50 19/48 BDX 15543 23 0.90 120 4.05 37/84 BDX 15544 25 0.90 120 2.75 42/127

OSL age EED (y)

Archaeological dose - MAM-3 (Gy)

OD (%)

1.31±0.12

730±70

2.5±0.2

137

3.34±0.10 2.98±0.14 2.91±0.10 2.58±0.09

2200±120 1720±120 2040±100 1975±110

3.76±0.08 3.24±0.07 3.01±0.07 2.99±0.07

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Table 3

OSL age MAM-3 (y)

Archaeological dose – IEU (Gy)

σ (%)

OSL-age IEU (y)

1400±130

-

-

-

3.43±0.13 3.39±0.31 3.40±0.15 3.12±0.14

20 25 23 25

2260±130 1960±150 2390±140 2380±150

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I0

D0 (Gy)

121 129 148 144

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σ (%)

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2480±120 1870±100 2120±90 2280±120

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ACCEPTED MANUSCRIPT Guibert et al, Modeling incomplete and heterogeneous bleaching of mobile grains partially exposed to the light: towards a new tool for single grain OSL dating of poorly bleached mortars. HIGHLIGHTS

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A new model describes the bleaching process of sand grains during the making of lime mortar. The exposition of a grain to the light is assumed to be a memoryless process during the dynamic mixing of sand and lime.

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The statistical distribution of grains according to their exposure follows a decreasing exponential function. Known age archaeological structures are studied and experimental EDs and simulated ones are compared.

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A plateau procedure is implemented to determine the parameters of ED distribution and the final archaeological dose.

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EP

TE D

The EED model (Exposure Exponential Distribution) gives reliable results and opens new perspectives in SG OSL dating.