Radiocarbon calibration

Willard Libby, the inventor of radiocarbon dating, pointed out as early as 1955 the possibility that the atmospheric ratio might have varied over time. Discrepancies began to be noted between measured ages and known historical dates for artefacts, and it became clear that corrections would need to be applied to radiocarbon ages to obtain calendar dates. ==Construction of a curve==

To produce a curve that can be used to relate calendar years to radiocarbon years, a sequence of securely-dated samples is needed, which can be tested to determine their radiocarbon age. Dendrochronology, or the study of tree rings, led to the first such sequence: tree rings from individual pieces of wood show characteristic sequences of rings that vary in thickness due to environmental factors such as the amount of rainfall in a given year. Those factors affect all trees in an area and so examining tree-ring sequences from old wood allows the identification of overlapping sequences. In that way, an uninterrupted sequence of tree rings can be extended far into the past. The first such published sequence, based on bristlecone pine tree rings, was created in the 1960s by Wesley Ferguson. Hans Suess made radiocarbon measurements on the bristlecone pine tree rings to publish the first calibration curve for radiocarbon dating in 1967. The curve showed two types of variation from the straight line: a long-term fluctuation with a period of about 9,000 years, and a shorter-term variation, often referred to as "wiggles", with a period of decades. Suess said that he drew the line showing the wiggles by "cosmic schwung", or freehand. It was unclear for some time whether the wiggles were real or not, but they are now well-established. They were superseded by the INTCAL series of curves, beginning with INTCAL98, published in 1998, and updated in 2004, 2009, 2013 and 2020. The improvements to these curves are based on new data gathered from tree rings, varves, coral, and other studies. Significant additions to the datasets used for INTCAL13 include non-varved marine foraminifera data, and U-Th dated speleothems. The INTCAL13 data includes separate curves for the Northern and Southern Hemispheres, as they differ systematically because of the hemisphere effect. There is also a separate marine calibration curve, as radiocarbon concentrations differ between the ocean and atmosphere. The calibration curve for the southern hemisphere is known as the SHCal as opposed to the IntCal for the northern hemisphere; the most recent version was published in 2020. There is also a curve for the period after 1955, where radiocarbon levels were artificially inflated due to atomic bomb testing, varying with latitude, known as Bomb Cal. == Methods ==

Probabilistic Modern methods of calibration take the original normal distribution of radiocarbon age ranges and use it to generate a histogram showing the relative probabilities for calendar ages. This has to be done by numerical methods rather than by a formula because the calibration curve is not describable as a formula. In the example CALIB output shown at left, the input data is 1270 BP, with a standard deviation of 10 radiocarbon years. The curve selected is the northern hemisphere INTCAL13 curve, part of which is shown in the output; the vertical width of the curve corresponds to the width of the standard error in the calibration curve at that point. A normal distribution is shown at left; this is the input data, in radiocarbon years. The central darker part of the normal curve is the range within one standard deviation of the mean; the lighter grey area shows the range within two standard deviations of the mean. The output is along the bottom axis; it is a trimodal graph, with peaks at around 710 AD, 740 AD, and 760 AD. Again, the 1σ confidence ranges are in dark grey, and the 2σ confidence ranges are in light grey. :\sigma_\text{total} = \sqrt{\sigma_\text{sample}^2 + \sigma_\text{calib}^2 } Example t1, in green on the graph, shows this procedure—the resulting error term, σtotal, is used for the range, and this range is used to read the result directly from the graph itself without reference to the lines showing the calibration error. The technique is not restricted to tree rings; for example, a stratified tephra sequence in New Zealand, known to predate human colonization of the islands, has been dated to 1314 AD ± 12 years by wiggle-matching. == Combination of calibrated dates ==

When several radiocarbon dates are obtained for samples which are known or suspected to be from the same object, it may be possible to combine the measurements to get a more accurate date. Unless the samples are definitely of the same age (for example, if they were both physically taken from a single item) a statistical test must be applied to determine if the dates do derive from the same object. This is done by calculating a combined error term for the radiocarbon dates for the samples in question, and then calculating a pooled mean age. It is then possible to apply a T test to determine if the samples have the same true mean. Once this is done the error for the pooled mean age can be calculated, giving a final answer of a single date and range, with a narrower probability distribution (i.e., greater accuracy) as a result of the combined measurements. Bayesian statistical techniques can be applied when there are several radiocarbon dates to be calibrated. For example, if a series of radiocarbon dates is taken from different levels in a given stratigraphic sequence, Bayesian analysis can help determine if some of the dates should be discarded as anomalies, and can use the information to improve the output probability distributions. ==References==