Rethinking the AMS Counting Statistics Theory to Include Overdispersion: Evidence of Non-Poisson distribution in the Radiocarbon Data
Dr Gary Salazar1,2, Prof. Soenke Szidat1,2
1University Of Bern, Dept. of Chemistry, Biochemistry and Pharmaceutical Sciences, LARA Laboratory, Bern Bern, Switzerland , Bern, Switzerland, 2University of Bern, Oeschger Center for Climate Change Research, Bern, Switzerland
The Poisson distribution is the most fundamental principle for the counting statistics of Radiocarbon and other rare isotopes using AMS. It has been the golden standard for estimating the variance of the measurement for every single Radiocarbon publication. AMS Radiocarbon inherited this uncertainty estimation from radioactivity-decay counting methods. Nevertheless, as far as we know, the Poisson hypothesis has actually never been tested. If the Poisson model is true then the standard deviation of the whole population of AMS counts per cycle should be the root square of the mean counts per cycle (μ) as: σ pois = √μ.
In this work, we applied non-Poisson counting statistics models that account for overdispersion (σ counts > √μ) to our historical data from the analysis of samples and standards. These models were the quasipoisson, and the negative binomial I and II; and their standard deviation of the counts is defined as σ = D√μ where D is the dispersion parameter. The counts per cycle were queried from our database for each cycle of every pass. The query and analysis was done automatically using a script written in the R language. The statistics models were fitted with Generalized Linear Model packages of the R program. The counts were corrected for systematic drift or random current jumps from one pass to the other. The correction was done by multiplying the counts of each cycle by the global current mean divided by the mean current of the respective pass.
We discovered that, even though the correction for systematic effects, only a small fraction of our data (15%) presented Poisson-like distributions with D value of 1.0-1.1. The rest of the data presented D values of 1.1 to 3.0. Most of the standards that presented Poisson distribution were blanks. Monte Carlo simulations showed that the reason of the non-Poisson behavior was, in part, due to the scatter of the beam current within a pass. When the current oscillates or it is noisy, then the mean C14 counts is rather non-stationary. The Monte Carlo results were in agreement with the real data. The expanded counting statistics relative error was calculated as σ counts,rel=D/√Nt where Nt is the total counts. This allowed us to calculate an expanded quoted error. We also calculated the conventional quoted error using the Poisson relative counting error. The χ2 was calculated as the square of the ratio of the relative standard error of the passes ratios divided by the Poisson relative error. The average χ2 for all the standards and samples for a span of 1 year was 1.2±0.3.
Our average χ2 is quite acceptable; therefore, we think that it is not possible that all our data is somehow misleading. Rather, we conclude that the sputtering of the target causes the beam current to oscillate, to scatter and to become noisy. This makes the mean of the C14 counts to drift and to oscillate, thus many samples present non-Poisson distributions. Probably the same happens to other AMS instruments.
Gary Salazar studied his Ph.D. in Japan in the field of Mass Spectrometry. Later, he was a postdoctoral fellow at Purdue University with Prof. Graham Cooks. He entered the field of AMS under a postdoc position at Lawrence Livermore National Lab. Finally, he started a Sr. Scientist position at LARA laboratory at the University of Bern.