note on the variance of a background-corrected OSL count

Abstract

_____________________________________________________________________________________________ It is common practice to calculate the relative standard error of a background-corrected optically stimulated luminescence (OSL) count by assuming Poisson errors. This note corrects a formula given by Banerjee et al. (2000) and suggests alternative formulae for use when the variation in background counts is larger than that implied by the Poisson distribution. For moderately bright samples, the contribution to the relative standard error from estimating the background rate is small, whichever formula is used. The usual scenario is as follows. Optical stimulation of an aliquot of quartz produces a series of counts-a number of recorded photons for each of N equal length consecutive time intervals (channels). For example, Banerjee et al. (2000) used a stimulation period of 60 s with counts in N = 250 channels each lasting 0.24 s. The OSL "signal'' is measured from the total count in the first n channels minus an estimate of the contribution to this count from background sources. Often n is taken to be quite small, for example n = 5, corresponding to the first 1.2 s of stimulation. The background emission rate is assumed to be constant over the whole 60 s, and is estimated from counts near the end of this period, where the contribution from the signal is assumed to be negligible. Mathematically, the above may be expressed as follows. Let y i denote the OSL count from channel i, for i = 1, 2, …, N, and let be the total count over the first n channels. Write ∑ = = n i i y Y 1 0 Y 0 = S 0 + B 0 where S 0 and B 0 are the contributions to Y 0 from the signal (or source of interest) and background respectively. Of course S 0 and B 0 are not observed directly. Assume that S 0 and B 0 are independent random quantities with expectations µ S and µ B , and variances σ 2 S and σ 2 B , respectively. Then the observed count Y 0 will have expectation µ S + µ B and variance σ 2 S + σ 2 B. An estimate of the signal µ S is thus obtained by subtracting an estimate of µ B from Y 0 , i.e., B S µ Y µ ˆ ˆ 0 − = We want to calculate the relative standard error of this estimate. An estimate of µ B is usually obtained from the average OSL count over the last m channels, for some suitable m chosen so that the contribution from the signal is negligible. It is useful to choose m be a multiple of n: let m = nk, say. For example, Banerjee et al. (2000) used the last m = 25 channels (6 s) of the series, corresponding to k=5 when n=5. Then let Y 1 , Y 2 , …, Y k denote the total counts in the last k sets of n channels, i.e., Y j = N−jn+n ∑ i=N−jn+1 y i for j = 1, 2, …, k. Thus Y 1 , Y 2 , …, Y k are all counts over n channels (the same as for Y 0) and we assume that they are independent random quantities from the same distribution as that of B 0 (i.e., the signal is negligible). In particular, each has expectation µ B and variance σ 2 B. The estimate of µ B may then be written as ∑ = = = k j j B Y k Y 1