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An Online Application for ΔR Calculation

Published online by Cambridge University Press:  29 December 2016

Ron W Reimer  and
Paula J Reimer
Ron W Reimer*
Affiliation:
14CHRONO Centre for Climate, the Environment and Chronology, Queen’s University Belfast, Belfast BT7 1NN, UK.
Paula J Reimer*
Affiliation:
14CHRONO Centre for Climate, the Environment and Chronology, Queen’s University Belfast, Belfast BT7 1NN, UK.
*
*Corresponding authors. Emails: r.reimer@qub.ac.uk; p.j.reimer@qub.ac.uk.
*Corresponding authors. Emails: r.reimer@qub.ac.uk; p.j.reimer@qub.ac.uk.
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Abstract

A regional offset (ΔR) from the marine radiocarbon calibration curve is widely used in calibration software (e.g. CALIB, OxCal) but often is not calculated correctly. While relatively straightforward for known-age samples, such as mollusks from museum collections or annually banded corals, it is more difficult to calculate ΔR and the uncertainty in ΔR for 14C dates on paired marine and terrestrial samples. Previous researchers have often utilized classical intercept methods that do not account for the full calibrated probability distribution function (pdf). Recently, Soulet (2015) provided R code for calculating reservoir ages using the pdfs, but did not address ΔR and the uncertainty in ΔR. We have developed an online application for performing these calculations for known-age, paired marine and terrestrial 14C dates and U-Th dated corals. This article briefly discusses methods that have been used for calculating ΔR and the uncertainty and describes the online program deltar, which is available free of charge.

Keywords

calibrationΔR calculationmarine reservoir corrections
Type
Method Development
Information
Radiocarbon , Volume 59 , Special Issue 5: 8th International Symposium, Edinburgh, 27 June – 1 July, 2016 Part 1 of 2 , October 2017 , pp. 1623 - 1627
Copyright
© 2016 by the Arizona Board of Regents on behalf of the University of Arizona 

INTRODUCTION

The marine reservoir age, R(t), is the difference between a marine radiocarbon age of a sample that derived its carbon from the marine reservoir in question and the atmospheric 14C age at the same time (t). A global marine surface mixed-layer calibration curve, Marine13 (Reimer et al. Reference Reimer, Bard, Bayliss, Beck, Blackwell, Bronk Ramsey, Buck, Cheng, Lawrence Edwards, Friedrich, Grootes, Guilderson, Haflidason, Hajdas, Hatté, Heaton, Hoffmann, Hogg, Hughen, Kaiser, Kromer, Manning, Niu, Reimer, Richards, Scott, Southon, Staff, Turney and van der Plicht2013), has been calculated for the Holocene using an ocean-atmospheric box model (Stuiver and Braziunas Reference Stuiver and Braziunas1993) and the Northern Hemisphere tree-ring based portion of the calibration curve (currently IntCal13). From 10.5 to 13.9 cal kBP, the curve is composed of foraminifera and corals data and from 13.9 to 50 cal kBP the IntCal13 curve offset by 405 yr was used.

Regional differences from the global curve are handled in calibration by including an offset ΔR(t), although in practice this value is often assumed to be constant. Although ΔR(t), the time-dependent regional offset from the global marine curve, was clearly defined in Stuiver et al. (Reference Stuiver, Pearson and Braziunas1986), there have been recent publications where calculations for samples with precisely known calendar age were made overly complicated and the results less precise (Alves et al. Reference Alves, Macario, Souza, Pimenta, Douka, Oliveira, Chanca and Angulo2015; Faivre et al. Reference Faivre, Bakran-Petricioli, Barešić and Horvatinčić2015) by inappropriately using phase models in OxCal (Bronk Ramsey Reference Bronk Ramsey2009). ΔR values calculated from independently dated samples, such as U-Th dated corals, have not always included the calendar age uncertainty (e.g. Toth et al. Reference Toth, Aronson, Cheng and Edwards2015). In addition, in more complex cases such as contemporaneous marine and terrestrial 14C samples, classical intercept methods have usually been used (cf. Southon et al. Reference Southon, Rodman and True1995; Reimer et al. Reference Reimer, McCormac, Moore, McCormick and Murray2002; Russell et al. Reference Russell, Cook, Ascough, Scott and Dugmore2011) which, because of “wiggles” in the calibration curve, provide poor estimates of the mean (e.g. Telford et al. Reference Telford, Heegaard and Birks2004) and can either overestimate or, more often, underestimate the uncertainty.

We have developed an online application for calculating ΔR for surface mixed-layer marine samples with (a) known calendar age, (b) independently derived (normally distributed) calendar ages such as U-Th dated corals, and (c) contemporaneous marine and terrestrial 14C ages. The method uses the full calibrated probability distributions to calculate the confidence ranges of the offset between the unknown sample and the marine calibration curve, currently Marine13 (Reimer et al. Reference Reimer, Bard, Bayliss, Beck, Blackwell, Bronk Ramsey, Buck, Cheng, Lawrence Edwards, Friedrich, Grootes, Guilderson, Haflidason, Hajdas, Hatté, Heaton, Hoffmann, Hogg, Hughen, Kaiser, Kromer, Manning, Niu, Reimer, Richards, Scott, Southon, Staff, Turney and van der Plicht2013). The mean and standard deviation of the 68% and 95% confidence ranges is given for practical purposes for use in calibration software.

METHODS

For calibration purposes, the uncertainty of ΔR does not include the marine calibration curve uncertainty since this is included in the calculation of the calibrated probability distribution (Stuiver and Reimer Reference Stuiver and Reimer1989). While this is not critical for recent ΔR values where the marine curve uncertainty is small, using the uncertainty twice for calibration of samples from further back in time where the curve uncertainty is larger would inflate the calibrated age ranges significantly.

Except for the simple case of known-age samples, the calculations in deltar make use of a convolution integral. This is an integral of the pointwise product of two probability density functions (pdfs), as a function of the amount of overlap between the two, as one is shifted relative to the other and is itself a probability density function. We calculate ranges of 68 and 95% probability from it in the same way that ranges are calculated from calibration probability density functions.

Known Calendar Age, Pre-Bomb Surface Mixed-Layer Samples

Known age, pre-bomb marine surface samples such as mollusk shells or coral can be used to calculate ΔR(t) relatively simply using Equation 1:

(1) $$\Delta {\rm R}\left( t \right){\equals}{}^{{14}}{\rm C}_{m} {\minus}{\rm Marine13C}\left( t \right)$$

where 14C m is the measured 14C age of the known-age sample and Marine13C(t) is the 14C age of Marine13 at time t.

The deltar application intersects the known calendar year of collection/growth with the marine calibration curve and determines the corresponding 14C age (reverse-calibrate). It then subtracts the reverse-calibrated age from the mean of the 14C age of the marine sample as illustrated in Figure 1. The uncertainty of ΔR is the uncertainty of the marine sample 14C measurement since the marine calibration curve uncertainty is included in the calibration process.

Figure 1 Illustration of ΔR and uncertainty calculation for samples with known age of collection or growth year (right) with resulting ΔR pdf and ranges on the left.

Independently Measured Calendar Ages

For marine samples such as corals that have a calendar age derived from radiometric measurements such as U-Th, the deltar application creates a normal distribution with the mean and standard deviation of the U-Th calendar age BP (Figure 2). It then reverse-calibrates discrete points on that distribution using the marine calibration curve. A convolution integral is used to determine a confidence interval for the offset between the 14C-dated marine sample and the uncalibrated probability density function of the U-Th age. Note that it is assumed that the U-Th calendar BP is corrected to 0 BP=AD 1950 rather than the year of measurement. Other type of measurements such as optically stimulated luminescence or varve counts could also be used as independent calendar ages if they can be approximated as normally distributed.

Figure 2 Illustration of ΔR and uncertainty calculation for samples with independently measured calendar age (e.g. U-Th) with resulting ΔR pdf and ranges on the left.

Contemporaneous Marine and Terrestrial Samples

Stuiver and Braziunas (Reference Stuiver and Braziunas1993) suggested calculating ΔR for contemporaneous (paired) marine and terrestrial material by intersecting with a combined marine and atmospheric calibration curve (i.e. marine vs. atmospheric 14C age). This method was further developed to include the uncertainty in ΔR (Reimer et al. Reference Reimer, McCormac, Moore, McCormick and Murray2002) and has been used in a number of studies (Russell et al. Reference Russell, Cook, Ascough, Scott and Dugmore2011; Dewar et al. Reference Dewar, Reimer, Sealy and Woodborne2012). An alternative method calibrated the terrestrial 14C age, then took the mean and standard deviation of the marine calibration curve 14C ages for the calibrated age ranges and subtracted this from the marine sample 14C age (Southon et al. Reference Southon, Rodman and True1995). Neither of these classical methods included the probability density function and therefore should be considered as approximations.

The deltar application does this as illustrated in Figure 3 by first calibrating the terrestrial 14C age with the appropriate Northern or Southern Hemisphere calibration curve, currently IntCal13 and SHCal13 (Hogg et al. Reference Hogg, Hua, Blackwell, Niu, Buck, Guilderson, Heaton, Palmer, Reimer, Reimer, Turney and Zimmerman2013), respectively. It then reverse-calibrates discrete points of the resulting pdf with the marine calibration curve. As for the case of U-Th ages, a convolution integral is used to determine a confidence interval for the offset between the 14C-dated marine sample and the reverse-calibrated pdf of the atmospheric sample.

Figure 3 Illustration of ΔR and uncertainty calculation for paired (contemporaneous) 14C-dated samples with (right) with resulting ΔR pdf and ranges on the left.

The resulting confidence interval will generally not be normally distributed. However, existing calibration programs are unable to handle non-normal distributions of ΔR, so the result will have to be approximated as a normal distribution. Note also that 14C ages that impinge on the end of the calibration curves will produce spurious ΔR results. A comparison of ΔR and uncertainties calculated using the classical intercept method and deltar for contemporaneous samples from South Africa (Dewar et al. Reference Dewar, Reimer, Sealy and Woodborne2012) is given in Table 1. While the differences in ΔR for these examples are not large (0–26 14C yr), the uncertainties are probably more realistic.

Table 1 Comparison of ΔR and uncertainties recalculated using the classical intercept method as described in Dewar et al. (Reference Dewar, Reimer, Sealy and Woodborne2012) using SHCal13 and Marine13 and calculated with deltar. The ΔR value calculated with deltar is taken as the midpoint of the 68% confidence interval.

DISCUSSION AND CONCLUSIONS

The deltar application calculates ΔR and the uncertainty for single samples. The uncertainty is more accurate than those provided by many other methods because it uses the full probability distribution functions rather than simple intercepts. The online program deltar is available free of charge at http://calib.org/deltar. For multiple contemporaneous samples, such as might occur in secure archaeological contexts, the standard error for predicted values has been proposed for determining the variability in ΔR (Russell et al. Reference Russell, Cook, Ascough, Scott and Dugmore2011) rather than using a simple standard deviation. For samples that are not strictly contemporaneous but come from within the same archaeological context, phase models in OxCal (Bronk Ramsey Reference Bronk Ramsey2009) have been effectively used to calculate ΔR for samples from shell middens (Macario et al. Reference Macario, Souza, Aguilera, Carvalho, Oliveira, Alves, Chanca, Silva, Douka, Decco, Trindade, Marques, Anjos and Pamplona2015). In sedimentary sequences, ΔR(t) can be calculated with depositional models in OxCal (Bronk Ramsey et al. Reference Bronk Ramsey, Staff, Bryant, Brock, Kitagawa, van der Plicht, Schlolaut, Marshall, Brauer, Lamb, Payne, Tarasov, Haraguchi, Gotanda, Yonenobu, Yokoyama, Tada and Nakagawa2012). For calculating the reservoir age, R, the Bayesian program ResAge (Soulet et al. Reference Soulet, Skinner, Beaupré and Galy2016) can be utilized.

Footnotes

Selected Papers from the 8th Radiocarbon & Archaeology Symposium, Edinburgh, UK, 27 June–1 July 2016

References

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Figure 0

Figure 1 Illustration of ΔR and uncertainty calculation for samples with known age of collection or growth year (right) with resulting ΔR pdf and ranges on the left.

Figure 1

Figure 2 Illustration of ΔR and uncertainty calculation for samples with independently measured calendar age (e.g. U-Th) with resulting ΔR pdf and ranges on the left.

Figure 2

Figure 3 Illustration of ΔR and uncertainty calculation for paired (contemporaneous) 14C-dated samples with (right) with resulting ΔR pdf and ranges on the left.

Figure 3

Table 1 Comparison of ΔR and uncertainties recalculated using the classical intercept method as described in Dewar et al. (2012) using SHCal13 and Marine13 and calculated with deltar. The ΔR value calculated with deltar is taken as the midpoint of the 68% confidence interval.