Nucl. Tracka Radiat. M¢0~., Vol. 13, No. 4, pp. 17"7-184, 1987 Int. J. :t,~_~. Appl. Inatrum., Part D Printed in Great Britain 0191-278X,88 $3.00 + 0.00 Perpmou Pre~ ptc REGRESSION AND ERROR ANALYSIS APPLIED TO THE DOSE-RESPONSE CURVES IN THERMOLUMINESCENCE DATING G. W. BEaOE2Z Geology Department, Western Washington University. Bcllingham, WA 98225, U.S.A. R. A. LOCKHART Statistics Department, University of Toronto, Toronto, Ontario. Canada M5S IAI and J. Kuo Mathematics Department, Simon Fraser University, Burnaby, B.C., Canada V5A IS6 (Received 20 August 1987) Abstract--In the dating of Quaternary geological materials by thermolumlinescence (TL). one frequently encounters sublinear TL vs applied dose curves for which accurate extrapolation is required. For dating the last exposure to sunlight of unheated Quaternary sediments, the intersection of two sublinear dose-- response curves is usually sought. Adequate methods for estimating the uncertainty in this intersection have not been previously described. Here we outline an iteratively reweighted least-~uares procedure for calculating the coefficients of second-order polynomials as well as saturating exponentials, and a variance-covariance method of error analysis appropriate for the intersection of two dose-response curves. Applications to several data sets are presented. The procedures described here are equally applicable, with trivial changes, to extrapolations of single sublinear dose-response curves (the additive dose technique). I. INTRODUCTION WITHIN the past several years thermoluminescence (TL) techniques (Wintle and Huntley, 1982; Aitken, 1985; Berger, 1986) have been applied to the dating of unheated Quaternary sediments with variable suc- cess. The event being dated is the last exposure to sunlight. Three of the principal causes of erroneous results have been: (I) uncertainty in the degree of resetting of the different TL mineral clocks at depo- sition; (2) failure to correct for a type of TL in- stability called anomalous fading; and (3) inap- propriate TL data processing. Means for dealing with the first two factors have been demonstrated (see lkrger, 1986, 1987; Berger et ai., 1987). In particular, it has been shown by these studies that the preferred TL method for measuring the equivalent dose (TL apparent age ~- equivalent dose/effective dose-rate) is the partial bleach or R-gamma procedure introduced by Wintle and Huntley (1980). With this procedure, accurate extrapolation of two dose-response curves to their intersection is required (Fig. 1). This is straightforward for young samples (<i0-20× 10~yr (ka)), which often exhibit linear response curves, and routine weighted least-squares fitting of a straight line (e.g. York, 1966) can be used. For young marine sediments, Berger et al. (1984) used such straight-line least-squares fits and an ad hoc procedure for estimating the variance in the inter- section point. However, depending upon several fac- tors, many sediments greater than 10-20 ka in age will exhibit sublinear dose-response curves. For these, a rigourous and accurate, yet simple, technique for calculating the variance in the intersection is required. Here we describe such a technique. For very old samples (e.g. greater than 100 ka) or at high applied doses (greater than a few kGy), the electron traps contributing to the TL signal generally become filled, and the simplest physically realistic model (first-order kinetics) describing the dose- response leads to a saturating exponential form (Huntley et al., 1987). Indeed, Berger 0987) has demonstrated that this model satisfies the data for a young loess given large artificial doses. Thus, for many samples, saturating exponential functions are generally appropriate for the required extrapolations and error analysis. Nevertheless, there are also many samples for which the dose--response curves are only somewhat sublinear over the range of useful doses. That is, in the expansion e-" = I - x + x2/2! - x3/3!+ .... x is small and therefore sufficiently accurate extrapo- lation and error analysis can be obtained by using a 177