Doug Keenan’s long essay about the IPCC treatment of surface temperature trends that I discussed earlier had a digression into his work on the calibration of radiocarbon dates that he published in 2012 in the open access EGU journal Nonlinear Processes in Geophysics. This caught my attention as I have a research interest in radiocarbon dating and constructing age-depth models. Radiocarbon dates need calibrating because the relationship between calendar age and radiocarbon age is non-linear. This is due to fluctuations in the concentration of radiocarbon in the atmosphere caused, for example, by changes in solar activity and hence variability in the flux of cosmic radiations that produces radiocarbon. The Holocene part of the calibration curve is derived from radiocarbon analyses of tree-rings of known age. Corals and other materials are used for the late-Pleistocene section. There have been several versions of the calibration curve, beginning, I think, in 1986 with Stuiver and Kra (1986). The current version of the calibration curve is IntCal13.
Many excellent statisticians have worked both on constructing the calibration curve and on developing methods to use it for over two decades. To pick some names: Caitlin Buck, Christopher Bronk Ramsey, Andrew Millard and Marian Scott. These people are no fools, so Keenan’s brief abstract is a surprise.
The calibration of a radiocarbon age to a calendar date is reviewed. It is shown that the commonly-used programs for calibration sometimes give results that are signiﬁcantly in error
Unfortunately, Nonlinear Processes in Geophysics is one of only two EGU journals that does not have an open peer-review process, so we cannot read what the reviewers made of this paper. In his essay, Keenan writes that 25 potential reviewers had to be approached before enough could be found who were prepared to review the paper. No reviewer recommended accepting the paper, until an “eminent” statistician was asked to review the paper and declared the paper to be “obviously correct”. So let’s have a look at the paper. Without explaining the traditional method of calibrating dates, Keenan proposes an alternative procedure that gives very different results, as shown by calibrating 4530 ± 50 14C BP using his method.
The shape of the probability distribution function is very different from that calculated by OxCal above. What’s happening? Which procedure is correct? The traditional procedure is explained in the calib manual
The probability distribution P(R) of the radiocarbon ages R around the radiocarbon age U is assumed normal with a standard deviation equal to the square root of the total sigma [combined error of radiocarbon date and calibration curve]. Replacing R with the calibration curve g(T), P(R) is defined as
To obtain P(T), the probability distribution along the calendar year axis, the P(R) function is transformed to calendar year dependency by determining g(T) for each calendar year and transferring the corresponding probability portion of the distribution to the T axis.
Yes, I know that’s difficult to grasp, but it is actually very easy to implement, even in Excel (but don’t bother). Consider figure 1 above. Starting with 5600 BP, the calibration curve at this calendar date is ~4800 14C BP, far away from the radiocarbon date at 4530 ± 50 14C BP, so the probability density of the radiocarbon date at 4800 14C BP is low. This probability density is assigned to 5600 BP. The procedure is then repeated every year (or five years) along the time axis to get the probability density for each year. The probability densities are then rescaled to sum to one. If there is a plateau in the calibration curve, many calendar dates are given approximately the same probability. With the date 4530 ± 50 14C BP, the peak of the radiocarbon date’s Gaussian distribution is close to the calibration curve between ~5300 and 5100 BP, so all these dates are assigned a high probability. So although the radiocarbon date has a Gaussian distribution, the effect of this procedure is to weight parts of that distribution that are close to a plateau more heavily than parts of the distribution where the calibration curve is steep. This is what Keenan appears to object to. Keenan wants the Gaussian distribution to be preserved. He does this by downweighting the probability of dates that lie on a plateau by the length of the plateau. Σ(probability sample has age a) = (probability year y has age a)/(# of years that have age a) This seems superficially reasonable: the Gaussian distribution is known from the laboratory. But it makes the implicit assumption that the every radiocarbon date is a priori equally likely. It is easy to demonstrate that this is not true by taking the calibration curve and for each calendar date, finding the probability of each radiocarbon age. I assume a measurement error is 50 years. The probabilities assigned for each radiocarbon age are then summed. This analysis gives the distribution of radiocarbon ages expected when dating objects with calendar ages 0BP, 5BP, 10BP, …, 50000BP. Here I focus on the Holocene.
intcal09<-read.table("o:/curves/intcal09.14c", skip=11, sep=",") names(intcal09)<-c("CAL.BP","C14.age","Error","Delta.14C","Sigma") decal<-sapply(intcal09$C14.age, function(x) dnorm(seq(0,11500,5),x,50)# ignore calibration-curve error ) plot(seq(0,11500,5), rowSums(decal), type="l", xlim="Radiocarbon age", ylim="Summed probability")
This curve is obviously not flat. By inspecting the calibration curve, it is obvious that peaks in summed probability coincide with plateaux in the radiocarbon calibration curve. For example, the peak at 4480 14C BP is the plateau in figure 1. Conversely, minima in the curve correspond with especially steep parts of the calibration curve. This shows that radiocarbon dates corresponding to plateaux in the calibration curve are inherently more likely than those that correspond to steep parts of the calibration curve. Therefore, it is entirely correct that dates from the plateaux are not downweighted by the length of the plateaux as suggested by Keenan. Keenan’s method is invalid. This conclusion is perhaps not a surprise given the number and calibre of scientists who have worked with radiocarbon calibration.