For over half a century, radiocarbon (14C) dating has been applied to deep-sea sediment archives. The material that is typically analysed from these archives consists of the calcareous tests of foraminifera. The minimum amount of material required for viable 14C analysis has meant that researchers have had to pick tens to hundreds of individual foraminifera specimens (depending on specimen size) from a single discrete-depth core interval (typically 1 cm of core depth) and combine these into a single sample for analysis. Such multi-specimen samples are likely to be heterogeneous in 14C activity (i.e. combine individual specimens of varying true age). The 14C laboratory measurement (and reported machine error) applied to such an amalgamated multi-specimen sample will simply represent the mean 14C activity of the total carbon of all individual specimens. Consequently, the true intra-sample 14C age heterogeneity of a sample is concealed from the researcher. Failure to consider the actual 14C age heterogeneity of multi-specimen samples can lead to downcore 14C age artefacts when post-depositional processes mix foraminifera with differing 14C activities, which is especially pronounced during periods of dynamic Δ14C. Furthermore, one must also take into consideration that younger specimens within a sample contribute exponentially more to the sample's mean 14C activity than older specimens do, a process referred to as the isotope mass balance effect (Erlenkeuser, 1980; Keigwin and Guilderson, 2009), due to 14C being a radioactive isotope (specimen 14C activity decreases exponentially with the passing of time).

Systematic bioturbation has long been recognised as an inherent feature of deep-sea sediment archives (Bramlette and Bradley, 1942; Arrhenius, 1961; Olausson, 1961). Long-established mathematical models of bioturbation in deep-sea sediment archives consider the uppermost ∼10 cm of a sediment archive to be uniformly mixed due to active bioturbation – the bioturbation depth (BD) (Berger and Heath, 1968; Berger and Johnson, 1978; Berger and Killingley, 1982). The presence of such a BD has been supported by the detection of a uniform mean age in the uppermost intervals of sediment archives (Peng et al., 1979; Trauth et al., 1997; Boudreau, 1998; Teal et al., 2008) and suggested by the 14C analysis of single foraminifera (Lougheed et al., 2018). The total range of single-specimen ages mixed within the BD is dependent upon two main factors: the depth of the BD itself and the sediment accumulation rate (SAR), both of which can exhibit spatio-temporal variation due to environmental and biological factors (Müller and Suess, 1979; Trauth et al., 1997). The presence of uniform mixing within the BD throughout the sedimentation history of a deep-sea sediment archive ultimately results, in the case of temporally constant SAR and BD, in the single-specimen population of discrete sediment intervals being characterised by an exponential probability density function (PDF) for true age, with a maximum probability for younger ages and a long tail towards older ages. The existence of such a distribution has been supported by the post-depositional mixing of tephra layers (Bramlette and Bradley, 1942; Nayudu, 1964; Ruddiman and Glover, 1972; Abbott et al., 2018) and the smoothing out of the downcore mean signal (Guinasso and Schink, 1975; Pisias, 1983; Schiffelbein, 1984; Bard et al., 1987; Löwemark et al., 2008; Trauth, 2013), the smoothing of which can change downcore in tandem with foraminiferal abundance changes (Ruddiman et al., 1980; Peng and Broecker, 1984; Paull et al., 1991; Löwemark et al., 2008). If SAR, BD and the Δ14C history of the planet were all to be temporally constant, then the idealised 14C activity PDF of each discrete depth (expressed as, for example, the 14C/12C ratio or normalised as fraction modern [F14C]) would, therefore, exhibit the combination of two exponential functions (the exponential PDF of true age plus the exponential PDF of 14C activity vs. time predicted by the half-life of 14C). However, the distribution of the 14C activity PDF is further complicated by the fact that 14C activity vs. time is not always the exact exponential function that would be predicted by the radioactive half-life of 14C, seeing as the Earth's carbon reservoirs exhibit a dynamic Δ14C history, as demonstrated by temporal changes in atmospheric 14C activity (Suess, 1955, 1965; de Vries, 1958; Reimer et al., 2013). These changes are brought about by changes in 14C production in the atmosphere in combination with climatic and oceanic influence upon the carbon cycle (Craig, 1957; Damon et al., 1978; Siegenthaler et al., 1980). Furthermore, non-uniform mixing of the oceans can contribute to temporal changes in local water 14C activity at a given coring site, further affecting the idealised PDF shape.

When applying the 14C method to sediment core material, researchers represent the 14C activity of a discrete-depth interval using a normal (Gaussian) distribution, based on the conventional mean 14C age (a reporting convention for 14C activity) and measurement error reported by the 14C laboratory (Stuiver and Polach, 1977). In some cases, this 14C age normal distribution is widened by researchers to also incorporate a reservoir age uncertainty, but it remains a normal distribution. This normal distribution of 14C age is subsequently calibrated using a suitable reference record of past Δ14C (e.g. those produced by the IntCal group), allowing researchers to arrive at an estimation of the discrete-depth interval's true (i.e. calendar) age. Such an approach inherently excludes the effects of bioturbation, because one would not expect a normal 14C age distribution to be representative of a discrete-depth interval for the reasons described in the previous paragraph. Currently, systematic investigation is lacking into whether neglecting to include the effects of bioturbation has significant impact upon the interpretative accuracy of 14C dating as it is currently applied in palaeoceanography, i.e. if it may ultimately lead to spurious geochronological interpretations.

Here, we take advantage of computer modelling to construct an ideal experimental design whereby we can evaluate how the current 14C state of the art within palaeoceanography would work in the case of best-case sediment conditions. Such best-case conditions do not exist in the field, meaning that a computer modelling environment can uniquely be used to create such a best-case scenario, which is ideal for testing the current state of the art. We use the Δ14C-enabled, single-specimen SEdiment AccuMUlation Simulator (SEAMUS) (Lougheed, 2020). This model uses the long-established understanding of bioturbation as included in existing bioturbation models (Trauth, 2013; Dolman and Laepple, 2018), but it differs in that it explicitly simulates the accumulation and bioturbation of single foraminifera, each with individually assigned 14C activities, to create a synthetic sediment archive history. Subsequently, current palaeoceanographic subsampling and 14C dating practices are virtually applied to the 1 cm discrete-depth sediment intervals of the model's outputted synthetic archive, resulting in discrete-depth 14C ages and calibrated ages that are representative of the existing palaeoceanographic state of the art. These results are subsequently compared to the actual discrete-depth true age distributions within the model, allowing us to quantitatively evaluate contemporary palaeoceanographic 14C measurement and calibration techniques. By keeping multiple model input parameters constant, we can construct an experimental environment whereby we have full control over the degrees of freedom. This modelling approach allows us to test, at a most fundamental level, the accuracy of the current 14C dating state of the art as applied to deep-sea sediments.

The SEAMUS model (Lougheed, 2020) synthesises n single foraminifera raining down from the water column per simulation time step, whereby n is the capacity of the synthetic sediment archive being simulated (analogous to sediment core radius) scaled to the SAR of the time step as predicted by an inputted age–depth relationship (Lougheed, 2020). To provide good statistics, all simulations use a time step of 5 years and 104 synthetic foraminifera per centimetre of core depth. An abundance of 104 specimens per centimetre is also similar to a best-case scenario value for a particular sample in the field (Broecker et al., 1992).

In each time step, all newly created single foraminifera are assigned an age (corresponding to the time step), a sediment depth (according to the age–depth input), and a 14C age (in 14C BP) and normalised 14C activity (in F14C) based on Marine13 (Reimer et al., 2013) after the application of a prescribed reservoir age for the time step. For older sections of the Marine13 calibration curve, where only 10-year time steps are available, linear interpolation is used to provide a 5-year 14C activity time step resolution. Within SEAMUS, all single foraminifera older than the oldest available age within the chosen calibration curve (in this case Marine13) are assigned the same 14C activity: that of the analytical blank, which must be set in the simulation. In this way, the model incorporates the principles of 14C dating, whereby individual very old foraminifera contained within a sample will contribute a 14C signal equivalent to the analytical blank. Here, we choose to set the simulation's analytical blank value to 46 806 14C BP (more precisely the F14C equivalent thereof), which corresponds to the lowest activity level in the Marine13 calibration curve. The analytical blank activity in most laboratories is somewhat lower (e.g. >50 000 14C BP), but we have no way of accurately applying an activity to single foraminifera older than the oldest value contained within Marine13. Rather than infer a Δ14C history beyond the limit of Marine13, we simply set the analytical blank in our simulation to 46 806 14C BP. In some scenarios we wish to investigate parameters within an experimental construct with temporally constant Δ14C, and in such scenarios we assign 14C activity (as F14C) as follows: F14C(t)=e([t+R(t)]/-8267), where t is the single foraminifera age in years before 1950 CE, and R(t) is the reservoir age for age t.

After the creation of all new single foraminifera within the synthetic core for a specific time step, bioturbation is simulated. Specifically, for each time step the depth values corresponding to all simulated foraminifera within the contemporaneous BD are each assigned a new depth by way of uniform random sampling of the BD interval. In this way, uniform mixing of foraminifera within the BD is simulated by following the established understanding of bioturbation (Berger and Heath, 1968; Trauth, 2013). All of the aforementioned processes are repeated for every simulation time step until such point that the end of the age–depth input (i.e. the final core top) is reached. All simulations are initiated at 70 ka (in true age) in order to confidently exclude the influence of model spin-up effects upon our period of interest (0–45 ka), given the possibility of a given centimetre of sediment to have a long tail of older foraminifera specimens. While SEAMUS can in principle be run on a local machine, to save time multiple simulations were run in parallel on a computing cluster provided by the Swedish National Infrastructure for Computing (SNIC) at the Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX).

After the completion of the synthetic core simulation, synthetic foraminifera (and corresponding values for true age, F14C and 14C age) are picked from each discrete 1 cm interval of the sediment core. In this study, we assume best-case scenarios where it is possible to pick all whole foraminifera contained within the sediment intervals. Subsequently, each of these picked 1 cm samples also undergoes a synthetic 14C determination analogous to a perfect accelerator mass spectrometry (AMS) measurement, whereby it is assumed that the AMS determination perfectly reproduces the mean 14C activity (in F14C) of the sample. Within the discrete-depth subsampling simulation, this mean 14C activity is calculated by taking the mean of all F14C values of all the single foraminifera contained within the picked sample. As mentioned in Sect. 2.1, the analytical blank is already included when assigning 14C to single foraminifera, meaning that the influence of the analytical blank upon sample AMS measurements is incorporated.

Using the Libby half-life, a sample's mean F14C value is also reported as a conventional 14C age determination (in 14C yr). All such synthetic determinations are assigned a synthetic 1σ measurement error analogous to a best-case scenario laboratory counting error for a large sample. The prescribed synthetic measurement error ranges from 30 14C yr in the case of near-modern samples to 500 14C yr in the case of samples nearing the blank value. Specifically, when assigning measurement errors to synthetic AMS determinations, a 14C determination of 1.0 F14C is assumed to have a measurement error of 30 14C yr, and a determination with the F14C value e(blank value-1)/-8033 (i.e. one 14C yr younger than the blank value) is assumed to have a measurement error of 500 14C yr. Errors (in 14C yr) for intermediate dates are linearly interpolated to F14C.

The synthetic laboratory 14C determinations and associated measurement uncertainties for each 1 cm discrete-depth sample are subsequently converted to calibrated years within SEAMUS using the embedded MatCal (v 2.6) 14C calibration software (Lougheed and Obrochta, 2016), the Marine13 calibration curve (Reimer et al., 2013) and a prescribed reservoir age (according to the scenario – see following sections) to produce a calibrated age probability density function (PDF) and 95.4 % highest posterior density (HPD) credible interval(s) for every centimetre core depth, i.e. analogous to what would be typically produced using contemporary palaeoceanography methods in the case of every discrete centimetre of core depth being exhaustively 14C dated. The MatCal software calibrates ages in F14C space, resulting in an accurate calibration, especially in the case of older samples or samples with large uncertainty.

Radiocarbon analysis focuses on determining the mean 14C activity of a particular sample, which is reported together with an associated analytical error. This mean activity of samples is often considered in the literature as conventional 14C age in 14C BP. Conventional 14C age, a unit of convenience, is linear vs. time, whereas 14C activity is actually exponential vs. time, due to 14C being a radioactive isotope. Therefore, with increasing age heterogeneity of a sample, we can expect an increased offset between the AMS conventional 14C age of a sample (the mean measured activity of the homogenised sample reported as conventional age) and the idealised mean of the conventional 14C ages of all single foraminifera within the sample. In Fig. 1, we compare the simulated AMS mean conventional 14C age calculated for each discrete depth to the idealised mean 14C age (based on the mean value of all single foraminifera conventional 14C ages contained within a sample). The resulting offset can help shed light upon how the measurement of age-heterogenous material is inherently biased towards younger (higher 14C activity) specimens contained within the sample. We find that the AMS mean 14C age is generally younger than the idealised mean 14C age in all cases. This effect can be attributed to the fact that younger foraminifera within a heterogeneous sample contribute exponentially more to a sample's mean 14C activity (what the measurement process is actually analysing) than older foraminifera do. This bias towards younger foraminifera is most apparent in cases with large intra-sample heterogeneity, such as in scenarios with lower SAR (Fig. 1a), and it is also reduced somewhat in the case of more broken foraminifera (Fig. 3a), due to lesser older foraminifera being picked, thus reducing the age heterogeneity. In the case of the highest SAR scenarios (>40 cm kyr-1) the aforementioned bias is insignificant in a practical sense in that it falls within the typical 14C measurement error. For all scenarios, superimposed upon the general bias are artefacts of the Earth's dynamic Δ14C history, caused by foraminifera from times of markedly differing Δ14C to be mixed together into a single sample, thus altering a sample's 14C activity distribution and causing downcore dynamic offsets between AMS mean 14C age and idealised mean 14C age. The most pronounced example of these artefacts can be seen during known periods of dynamic Δ14C, such as during the Laschamp geomagnetic event (ca. 40–41 ka) (Guillou et al., 2004; Laj et al., 2014), when a large spike in atmospheric 14C production occurred (Muscheler et al., 2014). We note that our simulations assign single foraminifera 14C activity using the Marine13 calibration curve, while newer records of Δ14C (Cheng et al., 2018) suggest that the Laschamp Δ14C excursion may have been of greater magnitude than was previously thought. A larger excursion would generate even more pronounced 14C artefacts in the downcore, multi-specimen, discrete-depth record. Furthermore, there may exist as yet undiscovered short-lived excursions in Δ14C (Miyake et al., 2012, 2017; Mekhaldi et al., 2015).

We can also visualise how well a sample's 14C activity probability distribution function (PDF) is represented by a distribution based on its mean AMS-measured 14C activity and 1σ measurement error. This visualisation is shown on the vertical axes of Figs. 1d–i and 2d–i for a number of simulated discrete depths for the different SAR scenarios with a BD of 10 cm. It can be clearly seen that the normal distribution derived from a sample's AMS mean measurement and associated uncertainty is a poor representation of a sample's actual 14C activity distribution.

When estimating a true age distribution for a particular sample, researchers calibrate a normal distribution of 14C age using suitable calibration curve (in this case Marine13). As discussed in the previous section, the aforementioned normal distribution of 14C activity derived from the measurement mean and machine error is not a faithful representation of the actual 14C activity distribution for a particular discrete depth. Such a misrepresentation has the potential to be further amplified during the calibration process itself, potentially resulting in a poor estimation of a discrete depth's 95.4 % age range and/or median age, the latter of which is often used to calculate, for example, sedimentation rates or represents the region of highest probability which will steer age–depth modelling routines. In Fig. 1b (0 % broken foraminifera) and Fig. 3b (10 % broken foraminifera), we show the offset between each discrete depth's true median age, and the corresponding median age derived from the 14C calibration process. We find large offsets for all constant SAR scenarios, ranging from ∼200 years in the case of the 60 cm kyr-1 scenario up to ∼700 years in the case of the 5 cm kyr-1 scenario. In certain low-SAR scenarios that coincide with intervals of the calibration curve that are highly resolved (e.g. the late Holocene), the discrete-depth true median age can consistently fall outside the 68.2 % age range predicted by the 14C measurement and calibration processes. A 68.2 % certainty suggests that, statistically, the true median will fall outside of the 68.2 % calibrated age range in only 31.8 % of cases, but, in the case of the 5 cm kyr-1 scenario (Fig. S1), the true median falls outside of the 68.2 % calibrated age range for 84 % of the discrete depths spanning the 5 to 0 ka period. In the case of 10 % broken foraminifera, this effect is reduced.

All offsets for all scenarios vary dynamically downcore, meaning that they can potentially cause spurious interpretations of changes in SAR. Furthermore, as these offsets occur during periods of dynamic Δ14C, which can be caused by large-scale changes in the carbon cycle caused by climate shifts (such as during the last deglaciation), it is possible that some apparent changes in SAR in the palaeoceanographic literature may have been erroneously attributed to climate processes, when they may be (partially) an artefact of the current application of 14C measurement and calibration within palaeoceanography.

Using the simulation output, it is also possible to quantitatively estimate how well the current 14C measurement and calibration state of the art applied within palaeoceanography estimates the true age range contained within discrete-depth sediment intervals. The offset between the calibrated 95.4 % age range and the true 95.4 % age range for each discrete depth for all SAR scenarios is shown in Fig. 1c (0 % broken foraminifera) and Fig. 3c (10 % broken foraminifera), and it is further visualised for all scenarios in Figs. S1–S10. For the lower SAR scenarios, the current application of 14C dating within palaeoceanography significantly underestimates the total age range contained within each discrete depth by many thousands of years. The underestimation is less in the case of the scenario with 10 % broken foraminifera. In the case of higher-SAR scenarios, the discrete-depth 95.4 % age range predicted by the 14C calibration process is similar to that of the discrete-depth 95.4 % age range of the sediment itself. In some cases with very high SAR, the 14C calibration process actually overestimates the 95.4 % age range (e.g. Figs. 1e, 3e, S5 and S10).

A general consequence of bioturbation and the subsequent mixing of single foraminifera specimens is that older foraminifera become systematically mixed upwards throughout the sedimentation history of a sediment archive. This general mixing can have a particular consequence near the analytical limit of the 14C method in that foraminifera with a 14C activity that is lower than a laboratory-based analytical sensitivity can become mixed into samples. Samples with a 14C age that is equal to or older than the established 14C blank value (i.e. the samples 14C activity falls below the detection limit of the analytical process) are commonly referred to as “14C-dead”. Within older intervals of heterogeneous deep-sea sediment archives, it is possible that a sample with an apparent measured 14C age younger than the 14C blank value can already contain a significant proportion of 14C-dead foraminifera. The presence of these 14C-dead specimens within a sample will bias the sample's apparent measured 14C age towards a value that is too young, because they will contribute a 14C activity to the sample that is equivalent to the laboratory's analytical blank. Such artefactually young 14C ages could ultimately erroneously be interpreted as age–depth features. In Table 1, the very first downcore occurrence of at least one simulated 14C-dead foraminifer is detailed for each of the aforementioned constant SAR scenarios introduced in Sect. 3. In the case of low-SAR scenarios with 0 % broken foraminifera, 14C-dead foraminifera are already present in discrete-depth samples with apparent AMS ages that would normally be considered well above the 14C blank value, e.g. an apparent AMS age of 22 647 14C BP in the case of 5 cm kyr-1, and 33 747 14C BP in the case of 10 cm kyr-1. However, the contribution of 14C-dead foraminifera at these levels may still be insignificant. The exact percentage contribution of 14C-dead foraminifera to discrete-depth AMS determinations is, therefore, detailed in Fig. 4a, c, e, g and i. From this analysis, it transpires that the first occurrence of at least 1 % contribution of 14C-dead foraminifera to discrete-depth AMS determinations occurs in the case of AMS ages of 39 158 and 43 601 14C BP, respectively, for the 5 and 10 cm kyr-1 scenarios. The percentage increases quickly further downcore. In the case of scenarios involving 10 % broken foraminifera, older foraminifera within discrete-depth sediment intervals are no longer whole, and therefore they are not picked for samples by a palaeoceanographer preferring whole specimens. The consequence of this effect is that the first occurrence of picked 14C-dead whole foraminifera occurs much further downcore (Table 1, Fig. 4b, d, f, h and j). This finding further underlines the importance of understanding foraminifera preservation conditions for particular species and/or water chemistry, and the associated consequences for 14C dating.

As motivated in the methods section, for practical reasons we have set the 14C analytical blank value to 46 806 14C BP within our model simulations. The laboratory blank value in most laboratories is around ∼50 000 14C BP, or even greater, depending on sample size, preparation conditions and measurement capability. For such greater blank values, essentially the same curves as shown in Fig. 4 would apply (i.e. assuming there are no, as of yet undiscovered, large Δ14C excursions around the period of the blank age) but shifted further to the right on the x axis. In other words, researchers interested in interpreting Fig. 4 in the case of an analytical blank of 50 000 14C BP should simply shift the curves to the right such that the 100 % 14C-dead contribution exactly coincides with 50 000 14C BP on the x axis.