Statistical age determination of tree rings

Dendrochronology, the study of annual rings formed by trees and woody plants, has important applications in research of climate and environmental phenomena of the past. Since its inception in the late 19th century, dendrochronology has not had a way to quantify uncertainty about the years assigned to each ring (dating). There are, however, many woody species and sites where it is difficult or impossible to delimit annual ring boundaries and verify them with crossdating, especially in the lowland tropics. Rather than ignoring dating uncertainty or discarding such samples as useless, we present for the first time a probabilistic approach to assign expected ages with a confidence interval. It is proven that the cumulative age in a tree-ring time series advances by an amount equal to the probability that a putative growth boundary is truly annual. Confidence curves for the tree stem radius as a function of uncertain ages are determined. A sensitivity analysis shows the effect of uncertainty of the probability that a recognizable boundary is annual, as well as of the number of expected missing boundaries. Furthermore, we derive a probabilistic version of the mean sensitivity of a dendrochronological time series, which quantifies a tree's sensitivity to environmental variation over time, as well as probabilistic versions of the autocorrelation and process standard deviation. A computer code in Mathematica is provided, with sample input files, as supporting information. Further research is necessary to analyze frequency patterns of false and missing boundaries for different species and sites.