A periodical devoted to Luminescence and ESR dating Department of Geography and Earth Sciences, Aberystwyth University, Ceredigion SY23 3DB, United Kingdom and Department of Physics, East Carolina University 1000 East 5th Street, Greenville, NC 27858, USA Volume 32 No.2 December 2014 A simple Bayesian method for assessing the standard error of equivalent dose estimates J. Peng, Z. B. Dong ___________________________________ 17 What to do when carbonate replaced water: Carb, the model for estimating the dose rate of carbonate-rich samples B. Mauz, D. Hoffmann ________________________________ 24 Thesis abstracts ______________________________________ 33 Bibliography _________________________________________42 Announcements XIX INQUA, Nagoya _______________ __________________ 53 ISSN 0735-1348 Ancient TL Started by the late David Zimmerman in 1977 EDITOR G.A.T. Duller, Department of Geography and Earth Sciences, Aberystwyth University, Ceredigion SY23 3DB, United Kingdom ([email protected]) and R.DeWitt, Department of Physics, East Carolina University, Howell Science Complex, 1000 E. 5th Street Greenville, NC 27858, USA ([email protected]) EDITORIAL BOARD I.K. Bailiff, Luminescence Dosimetry Laboratory, Dawson Building, University of Durham, South Road, Durham DH1 3LE, United Kingdom ([email protected]) S.H. Li, Department of Earth Sciences, The University of Hong Kong, Hong Kong, China ([email protected]) R.G. Roberts, School of Geosciences, University of Wollongong, Wollongong, NSW 2522, Australia ([email protected]) REVIEWERS PANEL R.M. Bailey, Oxford University Centre for the Environment, Dyson Perrins Building, South Parks Road, Oxford OX1 3QY, United Kingdom ([email protected]) J. Faïn, Laboratoire de Physique Corpusculaire, 63177 Aubière Cedex, France ([email protected]) R. Grün, Research School of Earth Sciences, Australian National University, Canberra ACT 0200, Australia ([email protected]) T. Hashimoto, Department of Chemistry, Faculty of Sciences, Niigata University, Niigata 950-21, Japan ([email protected]) D.J. Huntley, Department of Physics, Simon Fraser University, Burnaby B.C. V5A1S6, Canada ([email protected]) M. Lamothe, Dépt. Sci. de la Terre, Université du Québec à Montréal, CP 8888, H3C 3P8, Montréal, Québec, Canada ([email protected]) N. Mercier, Lab. Sci. du Climat et de l’Environ, CNRS-CEA, Av. de la Terrasse, 91198, Gif sur Yvette Cedex, France ([email protected]) D. Miallier, Laboratoire de Physique Corpusculaire, 63177 Aubière Cedex, France ([email protected]) S.W.S. McKeever, Department of Physics, Oklahoma State University, Stillwater Oklahoma 74078, U.S.A. ([email protected]) A.S. Murray, Nordic Laboratory for Luminescence Dating, Risø National Laboratory, Roskilde, DK-4000, Denmark ([email protected]) N. Porat, Geological Survey of Israel, 30 Malkhe Israel St., Jerusalem 95501, Israel ([email protected]) D. Richter, Lehrstuhl Geomorphologie, University of Bayreuth, 95440 Bayreuth, Germany ([email protected])` D.C.W. Sanderson, Scottish Universities Environmental Research Centre, Scottish Enterprise Technology Park, East Kilbride G75 OQF, UK ([email protected]) A.K. Singhvi, Rm 203, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India ([email protected]) K.J. Thomsen, Radiation Research Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000, Roskilde, Denmark ([email protected]) Ancient TL A periodical devoted to Luminescence and ESR dating Web site: http://www.aber.ac.uk/ancient-tl will move to: http://www.ecu.edu/cs-cas/physics/Ancient-TL.cfm Department of Geography and Earth Sciences Aberystwyth University SY23 3DB United Kingdom Tel: (44) 1970 622606 Fax: (44) 1970 622659 E-mail: [email protected] future address and contact: Department of Physics, East Carolina University Greenville, NC 27858, USA Tel: +252-328-4980 Fax: +252-328-0753 E-mail: [email protected] Ancient TL Vol. 32 No.2 2014 17 A simple Bayesian method for assessing the standard error of equivalent dose estimates Jun Peng, ZhiBao Dong Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, 730000 Lanzhou, China *Corresponding author: [email protected] (Received 8 August 2014; in final form 17 November 2014) _____________________________________________________________________________________________ Abstract value and outlined two protocols (i.e. simple Estimating the equivalent dose (ED) value is transformation and Monte Carlo simulation) to assess critical to obtaining the burial dose for Optically its standard error. Berger (2010) outlined methods Stimulated Luminescence (OSL) dating. In this that incorporate the contribution of errors from the study, a simple Bayesian method is used to assess the characteristic parameters of a growth curve and their standard error of an ED value in a linear or an covariances to estimate standard errors of ED values. exponential model. An ED value is treated as a Classic numeric techniques such as non-linear stochastic node that depends on a random variable parameter optimization and interpolation are whose posterior distribution can be constructed and routinely employed in these procedures (Peng et al., sampled. The results show that the Bayesian 2013). In this study, a simple Bayesian method was approach may improve the precision of an ED constructed to estimate ED values and their standard estimate by avoiding the repeated curve-fitting errors using measured datasets. The Gibbs sampler procedure employed in the routine “parametric WinBUGS (Lunn et al., 2013) was employed to bootstrap” Monte Carlo method. perform the Bayesian Markov chain Monte Carlo Key words: Bayesian method; Markov chain sampling, and the estimates were compared to those Monte Carlo; equivalent dose; standard error assessed using the Analyst software (Duller, 2007b). Introduction Methods In the commonly adopted single aliquot We use xi and yi to denote the i-th regenerative regenerative-dose (SAR) protocol (Murray and dose and measured standardized (i.e. sensitivity Wintle, 2000; Murray and Wintle, 2003), the corrected) OSL, respectively, for each of the i=1 to n standardized natural OSL signal is projected onto the observations. A saturating exponential growth curve growth curve that is constructed using a series of may be described as: sensitivity-corrected regenerative OSL signals to calculate the corresponding ED value. A maximum yˆ a(1 ebxi ) c (Eqn. 1) likelihood method was used by Yoshida et al. (2000) i to fit the growth curve with an exponential-plus- linear model in which the true ED value was treated where yˆ i denotes the i-th fitted standardized OSL as an unknown parameter, and the standard error of and a,b,c are parameters to be optimized. Here a is the ED value was estimated through the profile the saturation level (maximum value) of the likelihood function. However, this method may result standardized OSL, b is the reciprocal of the saturation in unreliable estimates (Yoshida et al., 2003) as the dose, and c is an offset accounting for potential number of points is not many more than the “recuperation” effects. Let y0 and x0 denote the dimension of the problem under consideration natural standardized OSL and the corresponding ED (Galbraith and Roberts, 2012). This problem led value respectively. An estimate of the ED value can Yoshida et al. (2003) to use a “parametric bootstrap” be obtained by inversing Eqn. 1 analytically: method to simulate and fit a number of growth curves repeatedly to obtain a more reliable estimate of the ln[(a y c) / a] standard error of an ED value. Duller (2007a) gave a 0 x0 (Eqn. 2) detailed introduction about how to estimate an ED b 18 Ancient TL Vol. 32 No.2 2014 n n bxi Eqn. 2 is justified only if a + c > y0, that is, the wi yi a wi (1 e ) natural standardized OSL y0 must not exceed the c i1 i1 (Eqn. 4) saturation level of the growth curve. It should be n w noted that even if the natural standardized OSL y0 is i close to (but does not exceed) the saturation level, the i1 In a weighted nonlinear least-squares estimation randomly selected quantity Y0 ~N(y0,0) (see we wish to minimize 2 w (y yˆ )2 , where the below) may exceed the maximum due to random i i i 2 errors controlled by its standard error σ0. For this weight wi are equal to 1 / σi . The estimates obtained reason, it is best to ensure that the dose to be with a least-squares estimation will be identical to estimated does not exceed the double value of the those given by a maximum likelihood estimation if saturation dose (Wintle and Murray, 2006). we suppose that each of the i-th regenerative Software Analyst (Duller, 2007b) employs the standardized OSL is independent of the others and Levenberg-Marquardt algorithm to estimate the follows a normal distribution with mean and parameters of a saturating exponential growth curve. Once the parameters are determined, an ED value can standard error σi, i.e., yi ~ N(yˆi , i ) . Note that yˆ i be calculated with Eqn. 2 or an interpolation is the model based (fitted) standardized OSL and σi procedure using the natural standardized OSL. The is the standard error (based on photon counting “parametric bootstrap” Monte Carlo method that statistics and measurement error) for the measured yi. assesses the standard error of the ED value involves Combining Eqn. 1, 3 and 4, we can treat as a fitting the growth curve and calculating the ED value function of the observations