Ratio estimation in SIMS analysis

The determination of an isotope ratio by secondary ion mass spectrometry (SIMS) traditionally involves averaging a number of ratios collected over the course of a measurement. We show that this method leads to an additive positive bias in the expectation value of the estimated ratio that is approximately equal to the true ratio divided by the counts of the denominator isotope of an individual ratio. This bias does not decrease as the number of ratios used in the average increases. By summing all counts in the numerator isotope, then dividing by the sum of counts in the denominator isotope, the estimated ratio is less biased: the bias is approximately equal to the ratio divided by the summed counts of the denominator isotope over the entire measurement. We propose a third ratio estimator (Beale's estimator) that can be used when the bias from the summed counts is unacceptably large for the hypothesis being tested. We derive expressions for the variance of these ratio estimators as well as the conditions under which they are normally distributed. Finally, we investigate a SIMS dataset showing the effects of ratio bias, and discuss proper ratio estimation for SIMS analysis.