INTRODUCTION
Conventions for reporting of radiocarbon data have been established in the seminal paper by Stuiver and Polach (Reference Stuiver and Polach1977) and later slightly revised and clarified by Mook and van der Plicht (Reference Mook and van der Plicht1999) and Reimer et al. (Reference Reimer, Brown and Reimer2004). However, heretofore no conventions have been established for reporting 14C disequilibria or age offsets between contemporaneous carbon reservoirs despite their necessity for calendar age determinations and broad use in reconstructing past carbon cycle dynamics. This lack of conventions may explain miscalculations that can be found in the scientific literature. This note aims to formalize the conventions for reporting of reservoir 14C disequilibria and age offsets. We advocate the use of new metrics (F14R and δ14R) as conservative isotopic tracers to characterize the 14C disequilibrium between contemporaneous reservoirs. From these metrics, we derive the corresponding reservoir age offset: R.
REPORTING OF RESERVOIR 14C DISEQUILIBRIA
General Framework
The measured δ13C-normalized fraction modern (Fm
x
) of an environmental sample (Stuiver and Polach Reference Stuiver and Polach1977; Mook and van der Plicht Reference Mook and van der Plicht1999; Reimer et al. Reference Reimer, Brown and Reimer2004) may be used to reconstruct that of its carbon source (e.g. reservoir x) at the time of its formation (T, yr BP) via the Cambridge half-life (5730 yr) and the law of radioactive decay, i.e.
$$Fm_{x}^{T} {\rm {\equals}}Fm_{x} .{\rm exp}(T/8267)$$
. Therefore, the ratio of Fm values from two contemporaneous carbon reservoirs (x and y) at time T [i.e.
$Fm_{x}^{T} /Fm_{y}^{T} $
] is equal to that of two corresponding samples measured today and is defined here as the reservoir’s “relative enrichment” (F14R
x–y
):
$$F^{{14}} R_{{x{\minus}y}} {\rm {\equals}}{{Fm_{x} } \over {Fm_{y} }}$$
The reservoir’s relative enrichment (Equation 1) is conserved with the passage of time and therefore a fundamental measure of the relative disequilibrium between the 14C inventories of two contemporaneous reservoirs. By convention, F14R is dimensionless and ranges from 0 to 1 by placing the more commonly enriched reservoir (y) in the denominator. For instance, under natural circumstances (pre-bomb epoch) the atmosphere is always enriched compared to all other carbon reservoirs and therefore would typically serve as reservoir y. Likewise, the surface ocean could serve as reservoir y when evaluating disequilibria with the deep ocean reservoir. Alternatively, the reservoir’s relative enrichment can be expressed as the relative difference between the 14C contents of reservoirs x and y, defined here as the reservoir’s “relative deviation” (δ14R x–y ):
$${\rdelta} ^{{14}} R_{{x{\minus}y}} ={\rm (}F^{{14}} R_{{x{\minus}y}} {\minus}1{\rm )}{\times} 1000‰$$
Finally, the reservoir age offset (R x–y ) between two contemporaneous carbon reservoirs x and y can be easily calculated from F14R and the Libby half-life (5568 yr), and expressed in 14C yr:
$$R_{{x{\minus}y}} {\rm {\equals}}{\minus}\!8033{\rm {\times}ln(}F^{{14}} R_{{x{\minus}y}} {\rm )}$$
The Atmospheric Reference
Comparing relative disequilibria through time and space (e.g. in paleoceanography) requires a common reference. The atmosphere is the most logical reference because it is the most uniform and 14C-enriched global carbon reservoir, with a 14C concentration that is quite precisely known for the past 14,000 calendar years, and reasonably well known until 50,000 calendar years ago (Reimer et al. Reference Reimer, Bard, Bayliss, Beck, Blackwell, Bronk Ramsey, Buck, Cheng, Edwards, Friedrich, Grootes, Guilderson, Haflidason, Hajdas, Hatté, Heaton, Hoffmann, Hogg, Hughen, Kaiser, Kromer, Manning, Niu, Reimer, Richards, Scott, Southon, Staff, Turney and van der Plicht2013a). Hence, in most cases, a reservoir’s relative enrichment should be calculated relative to the atmosphere, thereby permitting unambiguous comparisons of reservoir 14C disequilibria and age offsets through time and space:
$$F^{{14}} R_{{x{\minus}atm}} {\rm {\equals}}{{Fm_{x} } \over {Fm_{{atm}} }}$$
The Case of Speleothems
Speleothem (S) 14C contents are usually lower than that of the contemporaneous atmosphere, mainly due to the incorporation of bedrock-derived 14C-free (“dead”) carbon during formation. The speleothem 14C contents must be corrected for this dead carbon contribution in order to reflect the actual atmospheric 14C content. A common correction, the dead carbon proportion (dcp) (Genty and Massault Reference Genty and Massault1997) or the equivalent dead carbon fraction (dcf) (Fohlmeister et al. Reference Fohlmeister, Kromer and Mangini2011), can be defined using the F14R:
$$dcp={\rm (}1{\minus}F^{{14}} R_{{S{\minus}atm}} {\rm )}{\times} 100\,\%\,$$
DISCUSSION
The reservoir age offset metric (R) is almost always used to characterize reservoir 14C disequilibria. This is historically linked to the fact that reservoir age offsets are extensively used to adjust 14C dates to the atmospheric reservoir for various calibration purposes, e.g. construction of the IntCal calibration curves (corals and speleothem data, see Reimer et al. Reference Reimer, Bard, Bayliss, Beck, Blackwell, Bronk Ramsey, Brown, Buck, Edwards, Friedrich, Grootes, Guilderson, Haflidason, Hajdas, Hatté, Heaton, Hogg, Hughen, Kaiser, Kromer, Manning, Reimer, Richards, Scott, Southon, Turney and van der Plicht2013b) or obtaining calendar chronologies from lacustrine/marine 14C-dated archives (e.g. Toucanne et al. Reference Toucanne, Soulet, Freslon, Jacinto, Dennielou, Zaragosi, Eynaud, Bourillet and Bayon2015). However, the metrics proposed here (F14R, δ14R, R) are also well suited for studying carbon dynamics and chemical processes in soils (Trumbore Reference Trumbore2000), inland waters (Soulet et al. Reference Soulet, Ménot, Garreta, Rostek, Zaragosi, Lericolais and Bard2011; Keaveney and Reimer Reference Keaveney and Reimer2012), the ocean (Broecker et al. Reference Broecker, Mix, Andree and Oeschger1984; DeVries and Primeau Reference DeVries and Primeau2010), groundwater (Boaretto et al. Reference Boaretto, Thorling, Sveinbjörnsdottir, Yechieli and Heinemeier1998), and caves (Genty and Massault Reference Genty and Massault1997; Fohlmeister et al. Reference Fohlmeister, Kromer and Mangini2011).
The F14R, δ14R, and R metrics are easy to calculate, conserved with time, and thus clearer measures of both past and present reservoir 14C disequilibria. For example, reservoir age offsets traditionally calculated as 14C age differences are unsuitable for post-bomb samples (Fm >1) because the corresponding ages are reported qualitatively as “>modern” by convention (Stuiver and Polach Reference Stuiver and Polach1977). Thus, post-bomb reservoir age offsets must be calculated directly from the fraction modern values using the reservoir’s relative enrichment (F14R) and Equation 3 [see also Burr et al. (Reference Burr, Beck, Corrège, Cabioch, Taylor and Donahue2009) and Keaveney and Reimer (Reference Keaveney and Reimer2012)]. Likewise, ∆14C nomenclature permits quantitative reporting of post-bomb 14C measurements, but they, too, should be normalized to the contemporaneous atmosphere in order to unambiguously quantify temporal changes in disequilibria. As an example, the Δ14C values of dissolved inorganic carbon (DIC) in surface waters of the Black Sea were similar in 1988 (57.3‰; Jones and Gagnon Reference Jones and Gagnon1994) and 2004 (62.5‰; Fontugne et al. Reference Fontugne, Guichard, Bentaleb, Strechie and Lericolais2009), whereas the contemporaneous atmospheric Δ14C values were very different (175.0‰ and 70.4‰, respectively; Levin and Kromer Reference Levin and Kromer2004). Thus, despite similar DIC Δ14C values, the surface Black Sea was depleted by 100‰ relatively to the atmosphere in 1988 (δ14R BS-atm =–100‰; R BS-atm =850 14C yr) but nearly equilibrated with the atmosphere in 2004 (δ14R BS-atm =–7‰; R BS-atm =60 14C yr), suggesting two very different geochemical states.
Other measures of reservoir 14C disequilibria have been proposed, such as the ΔΔ notation that reports differences between the ∆ values of a reservoir and the atmosphere (Thornalley et al. Reference Thornalley, Barker, Broecker, Elderfield and McCave2011; Burke and Robinson Reference Burke and Robinson2012). However, unlike F14R, δ14R, or R, the ΔΔ metric will take different values for a given level of isotopic disequilibrium (
$\delta ^{{14}} R_{{x{\minus}atm}} $
), depending on the initial atmospheric 14C concentration (
$$Fm_{{atm}}^{T} $$
) since actually
$$\rDelta \rDelta _{{x{\minus}atm}} =Fm_{{atm}}^{T} {\times}\delta ^{{14}} R_{{x{\minus}atm}} $$
. It is for this reason that recent papers advocated the use of the “atmosphere normalized Δ14C” (Δ14Catm normalized; Burke et al. Reference Burke, Stewart, Adkins, Ferrari, Jansen and Thompson2015) or the “initial Δ14C corrected to a world with atmospheric Δ14Catm=0” (Δ14C0,adj; Cook and Keigwin Reference Cook and Keigwin2015), both of which correspond to the reservoir’s relative deviation (δ14R). Thus, F14R and its derived metrics (Equations 1–4) would provide a clear and unified framework for expressing a host of marine 14C “ventilation metrics” that are found in the paleoceanographic literature, including e.g. B-P (benthic-planktonic) offsets, B-Atm (benthic-atmosphere) offsets, Δ
x
, and ΔΔ
x-atm
. Similarly, the dead carbon proportion dcp (Equation 5), which is currently exclusively applied to speleothems, would be equally useful as a measure of the hardwater effect, which is actually a dilution of the inorganic 14C pool by bedrock-derived dead carbon in lakes and rivers (Deevey et al. Reference Deevey, Gross, Hutchinson and Kraybill1954; Keaveney and Reimer Reference Keaveney and Reimer2012) rather than the result of limited exchange with the atmospheric carbon pool.
Finally, we have been careful not to overlap our metrics with the marine ΔR metric (Stuiver et al. Reference Stuiver, Pearson and Braziunas1986) that expresses the difference between the reservoir age offset of a regional part of the ocean and the expected value derived from the oceanic box model used to build the marine calibration curve (Stuiver and Braziunas Reference Stuiver and Braziunas1993; and e.g. Reimer et al. Reference Reimer, Bard, Bayliss, Beck, Blackwell, Bronk Ramsey, Buck, Cheng, Edwards, Friedrich, Grootes, Guilderson, Haflidason, Hajdas, Hatté, Heaton, Hoffmann, Hogg, Hughen, Kaiser, Kromer, Manning, Niu, Reimer, Richards, Scott, Southon, Staff, Turney and van der Plicht2013a):
$$\rDelta R{\rm {\equals}}R_{{x{\minus}atm}} {\minus}R_{{MarineXX{\minus}IntcalXX}} $$
. By definition, ΔR is useful to calibrate marine 14C ages using the marine calibration curve. However, unlike R, the definition of marine ΔR depends on the ocean box model used and its parameterization, including in particular the assumption of constant ocean circulation and carbon cycling (Stuiver et al. Reference Stuiver, Pearson and Braziunas1986). Hence, akin to Jull et al. (Reference Jull, Burr and Hodgins2013), reporting the actual measured values of R (i.e. R
x-atm
, or the related metrics F14R
x-atm
and δ14R
x-atm
defined above) would help to avoid any ambiguity.
CONCLUDING REMARKS AND RECOMMENDATIONS
This note presents a common framework for reporting 14C disequilibria that is based upon the fundamental “relative enrichment” (F14R) between two contemporaneous reservoirs. As the use of these metrics is appropriate to a broad range of environmental sciences, we advocate quantifying 14C disequilibria as a reservoir’s relative enrichment (F14R), relative deviation (δ14R), or reservoir age offset (R), with a clearly reported reference (e.g. “ocean-atmosphere relative enrichment,” etc.) and a cautiously discussed causality [for reviews about various causes, see Jull et al. (Reference Jull, Burr and Hodgins2013) and Philippsen (Reference Philippsen2013)]. The equations used to calculate these metrics are summarized in Table 1, and their uncertainties are detailed in the Appendix.
Table 1 Summary of metrics used to report 14C disequilibria between contemporaneous reservoirs x and y.
ACKNOWLEDGMENTS
GS acknowledges the Postdoctoral Scholar Program at the Woods Hole Oceanographic Institution with funding provided by the National Ocean Sciences Accelerator Mass Spectrometry Facility (OCE-1239667), and warmly thanks Ann P McNichol and Bill Jenkins for their support during his 2013–2015 stay at NOSAMS. SRB acknowledges Dean Minghua Zhang and Provost Dennis Assanis of Stony Brook University for financial support.
SUPPLEMENTARY MATERIAL
To view supplementary material for this article, please visit http://dx.doi.org/10.1017/RDC.2015.22
APPENDIX
This short appendix provides the equations to be used to calculate the uncertainties of F14R, δ14R, R, and dcp. These metrics implicitly assume strict synchrony between reservoirs. Two cases have to be considered.
Case 1: Pair of contemporaneous 14 C ages. For example, benthic and planktonic foraminifera picked from the same sediment layer may, in some cases, be assumed to be contemporaneous 14C records of two distinct reservoirs. A similar example would be the 14C dating of a shell and a piece of wood embedded in the same sediment layer. In such cases, the uncertainties (σ) on the reservoir’s relative enrichment (F14R), relative deviation (δ14R), dead carbon proportion (dcp), and reservoir age offset (R), are simple functions of the measured Fm values and their associated uncertainties (σ x and σ y ):
$$\sigma _{{F^{{14}} R}} =F^{{14}} R {\times}\sqrt {\left( {{{\sigma _{x} } \over {Fm_{x} }}} \right)^{2} {\plus}\left( {{{\sigma _{y} } \over {Fm_{y} }}} \right)^{2} } $$
$$\sigma _{{\delta ^{{14}} R}} =\sigma _{{F^{{14}} R}} {\times}1000‰$$
$$\sigma _{{dcp}} =\sigma _{{F^{{14}} R}} {\times}100\,\%\,$$
$$\sigma _{R} =8033 {\times} \sqrt {\left( {{{\sigma _{x} } \over {Fm_{x} }}} \right)^{2} {\plus}\left( {{{\sigma _{y} } \over {Fm_{y} }}} \right)^{2} } $$
Case 2: Paired 14 C age and calendar age. This case is encountered when the 14C age of the reservoir is associated with a calendar age that has significant measurement uncertainty. This is generally the case for speleothem dcp calculations (Southon et al. Reference Southon, Noronha, Cheng, Edwards and Wang2012), when dealing with 14C and U/Th-dated corals (Druffel et al. Reference Druffel, Robinson, Griffin, Halley, Southon and Adkins2008), or 14C and chronostratigraphically dated foraminifera (Skinner et al. Reference Skinner, Fallon, Waelbroeck, Michel and Barker2010). To calculate the F14R, δ14R, and R, and their associated uncertainties, the use of the atmospheric calibration curve is required along with a methodology that propagates the uncertainties of the (i) 14C measurements, (ii) calendar ages, (iii) and atmospheric calibration curve, as well as the calibration curve structures. As such, the resulting F14R, δ14R, dcp, and R probability density functions are not necessarily Gaussian. Instead, they may be asymmetric and multimodal. This uncalibration-convolution process has been recently developed for reservoir age offset calculations, and has been coded as the freely available ResAge program (Soulet Reference Soulet2015) for the R statistical platform. The ResAge package has been updated and now includes functions for F14R, δ14R and dcp calculations (available in the online Supplemental Material).
