Extreme solar storms and the quest for exact dating with radiocarbon

Abstract

Radiocarbon (¹⁴C) is essential for creating chronologies to study the timings and drivers of pivotal events in human history and the Earth system over the past 55,000 years. It is also a fundamental proxy for investigating solar processes, including the potential of the Sun for extreme activity. Until now, fluctuations in past atmospheric ¹⁴C levels have limited the dating precision possible using radiocarbon. However, the discovery of solar super-storms known as extreme solar particle events (ESPEs) has driven a series of advances with the potential to transform the calendar-age precision of radiocarbon dating. Organic materials containing unique ¹⁴C ESPE signatures can now be dated to annual precision. In parallel, the search for further storms using high-precision annual ¹⁴C measurements has revealed fine-scaled variations that can be used to improve calendar-age precision, even in periods that lack ESPEs. Furthermore, the newly identified ¹⁴C fluctuations provide unprecedented insight into solar variability and the carbon cycle. Here, we review the current state of knowledge and share our insights into these rapidly developing, diverse research fields. We identify links between radiocarbon, archaeology, solar physics and Earth science to stimulate transdisciplinary collaboration, and we propose how researchers can take advantage of these recent developments.

Main

Since its existence was confirmed in 1940 (ref. ¹), radiocarbon has been both a cornerstone for dating the past 55,000 years and a fundamental tracer of Earth system processes. Here we draw together recent developments across multiple disciplines that offer new insights into the archaeological, Earth and space sciences. We discuss how the discovery of so-called Miyake events caused by ESPEs has created opportunities for more-precise radiocarbon calibration and stimulated research programmes to measure past ¹⁴C levels at annual resolution. We summarize how scientists can use ESPEs directly to obtain exact calendar dates using radiocarbon, and how they can leverage fine-scale variations in ¹⁴C identified by wider annual measurement programmes for calibration more broadly. We also survey the insights that ESPEs and consequent high-precision annual ¹⁴C measurements have provided regarding space weather and the global carbon cycle, and describe areas where further research is required to gain the greatest value from radiocarbon.

Introduction to radiocarbon

The approximately 5,700-year half-life of the radioactive ¹⁴C isotope of carbon (radiocarbon) and the ubiquitous nature of carbon in the biogeosphere make it extremely useful as a scientific tool to study the past 55,000 years. It is used both as the primary clock by which archaeologists and environmental scientists measure time, and as a probe to gain insight into key Earth system processes^2,3.

Radiocarbon dating^4,5 is based on the principle that while an organism is alive, it exchanges carbon with its surroundings, so it will have a ratio of ¹⁴C to stable carbon that is in equilibrium with that of its local environment; a similar situation also arises for some geological processes, such as the growth of speleothems. Once this exchange ceases, for example when an organism dies, the stable carbon incorporated into well-preserved chemical components will remain constant, but the amount of ¹⁴C will halve roughly every 5,700 years. Beyond ten half-lives (around 60,000 years), insufficient ¹⁴C remains to be measured reliably using current methods.

If the ¹⁴C concentration (the ratio of ¹⁴C to stable carbon) in the various reservoirs of the carbon cycle had been constant over time, obtaining a precise calendar age from radiocarbon dating would be straightforward. However, the ¹⁴C concentration has varied considerably over time and differs according to the carbon reservoir (for example, the atmosphere or ocean) being studied. These past fluctuations mean that all radiocarbon age measurements need to be adjusted to account for the ¹⁴C concentration when the sample stopped carbon exchange. This adjustment is known as radiocarbon calibration.

Calibrating a ¹⁴C measurement into an accurate calendar date requires comparison with reference material of known calendar age (for example, tree rings dated by dendrochronology) or with other independently dated ¹⁴C records (such as stalagmites, corals and lacustrine and marine sediments)³. These reference materials are combined into calibration curves that provide estimates of the ¹⁴C concentration of a sample that stopped exchanging in any individual calendar year^6,7,8. To calibrate a sample of unknown age, we calculate which calendar dates are consistent with the observed ¹⁴C concentration of the sample. This is typically achieved using a Bayesian approach (Fig. 1 and Supplementary Video 1). Calibration leads to calendar-age estimates that are uncertain and have potentially complex, asymmetric and multimodal probability distributions. The requirement for calibration is one of the main confounding factors in ¹⁴C dating, and many important archaeological debates, such as the date of the Minoan eruption of Thera in the second millennium bc, hinge on these calendar-age probability distributions^9,10,11.

Fig. 1: Radiocarbon calibration of the Minoan eruption of Thera.

The average of multiple key samples^9,11 indicates a ¹⁴C age of 3,350 ± 10 radiocarbon years before present (bp). To calibrate this ¹⁴C determination, shown in dark pink, we estimate which calendar dates are compatible with it. The IntCal20 calibration curve⁶ (with 1σ uncertainties) is shown in orange, and the compatible calendar dates are shown as a probability distribution in light pink. See Box 1 for definitions of notation. For an animation, see Supplementary Video 1.

Calibration and calendar-age precision

The need for calibration means that the calendar-age precision produced by radiocarbon dating arises from two distinct components: first, the measurement precision of the ¹⁴C content of the sample; and second, the calendar-age uncertainty introduced by the calibration process. Improvements in the former without consideration of how to reduce the latter offer only limited progress in increasing calendar-age precision. For some ¹⁴C users, typically those able to obtain just a single ¹⁴C determination, the imprecision introduced through calibration may be irreducible. However, for those with access to multiple potential ¹⁴C samples, careful consideration of how and what they choose to measure and calibrate offers substantial opportunities to reduce the final calendar-age uncertainties.

Archaeologists have been quick to embrace the potential for more-precise chronologies by incorporating independent information, either on relationships between the ¹⁴C samples or with external chronologies, during calibration^12,13. The routine application of Bayesian chronological modelling has led to the development of a step-by-step process to enable the selection of suites of samples for dating that can be interpreted together in formal statistical models¹⁴. Currently, relative dating known from stratigraphy, from either archaeology or environmental records, is the most widely used information incorporated into chronological models, although other kinds of sequence, such as artefact typology or seriation, are also used. This can be supplemented with tie-points provided, for example, by volcanic tephra¹⁵. Until recently, however, the decadal resolution of most of the data used to construct calibration curves has been a limitation on the types of independent archaeological and environmental information that can be used in chronological modelling¹⁶. The discovery of ESPEs has changed this.

Atmospheric ¹⁴C/¹²C variations

The variations in past atmospheric levels of ¹⁴C that necessitate calibration are a consequence of both changes in the rate of production over time and rearrangements in the global carbon cycle. Consequently, discoveries in the ¹⁴C record that lead to improvements in radiocarbon calibration also enable key insights in a broad range of other scientific fields.

Natural production of ¹⁴C occurs predominantly in the lower stratosphere and upper troposphere, and is driven by galactic cosmic rays (GCRs). These incoming GCRs initiate a cascade of nuclear collisions, eventually generating neutrons, which then react with atmospheric nitrogen to produce ¹⁴C. The number of GCRs that reach the atmosphere is modulated by the strength of the solar magnetic field and Earth’s geomagnetic field. Both partially shield Earth from cosmic radiation. When solar activity is high, and/or the geomagnetic field is strong, fewer GCRs reach Earth, leading to lower ¹⁴C production rates. Variations in the rate of ¹⁴C production resulting from solar activity tend to operate on short-to-moderate timescales (decadal to centennial), whereas those caused by changes in the geomagnetic field generally operate on much longer (centennial to millennial) timescales.

Until recently, no abrupt changes had been identified in the ¹⁴C record, and it was believed that the galactic cosmic-ray flux impinging on Earth varied only gradually in response to heliomagnetic and geomagnetic modulations^17,18,19. Consequently, measurement of reference ¹⁴C material at annual resolution was seen as a low priority. However, in 2012, analysis of a Japanese cedar identified an unexpected 12‰ increase in atmospheric Δ¹⁴C (a decrease of more than 100 ¹⁴C years) over the course of a single calendar year, ad 774–775²⁰. Such an abrupt increase in ¹⁴C levels required an entirely unprecedented spike in ¹⁴C production, with almost four times the yearly average amount of ¹⁴C being generated that year²¹.

Further analysis revealed that the cause of this ¹⁴C jump was an extreme solar storm that was an order of magnitude larger than any previously estimated²². Such a storm would have created a massive short-term burst of energetic solar particles (an ESPE) that led to a huge spike in ¹⁴C production.

Impact of the discovery of ESPEs

The occurrence of similar ESPEs today could have disastrous effects on our telecommunications, electricity grids and satellite systems^23,24. Understanding the frequency of such events and their possible size is therefore essential to prepare and build resilience into these systems. However, detailed instrumental measurements of the Sun’s behaviour are available only for the past few decades, during which time no ESPEs have occurred, so we are required to use the imprint they leave in cosmogenic isotope records. The discovery of the ad 774 ESPE initiated new and extensive programmes to measure ¹⁴C in tree rings at annual resolution and high precision throughout the Holocene epoch. These programmes have important implications for solar physics, enhancing our understanding of the Sun and space weather. They also provide opportunities for those seeking to calibrate radiocarbon determinations.

The ¹⁴C signatures left by ESPEs are unique. If one can obtain ¹⁴C samples from consecutive calendar years that bracket the ESPE (for example, by measuring individual growth rings in timbers), calibration to the exact year is possible. This step change in the precision obtainable through radiocarbon dating has the potential to extend the reach of the radiocarbon method into historical and other problems that require exact dating. Most immediately, such exact dating will be relevant to archaeologists. However, several fields of the Earth sciences also need high-precision dating when dealing with rare but dramatic events, such as earthquakes and tsunamis²⁵, extreme weather events such as hurricanes²⁶, or rainfall and flood events²⁷, all of which may have varied in strength and frequency in the past. Confirming the correct identification of at least a few singular events in distant archives would demonstrate the reliability of available records in this growing field of palaeo-extreme events.

Those who study periods lacking ESPEs (or without suitable items to bracket the ESPE) may not be able to obtain exact dating using ¹⁴C, but they can still take advantage of the increased resolution in radiocarbon calibration curves enabled by annual ¹⁴C measurement programmes. Such fine-scaled understanding of atmospheric ¹⁴C structure allows relative dating information from shorter ¹⁴C sequences and/or shorter-lived ¹⁴C samples to be used more effectively and reliably in statistical models. Harnessed optimally during calibration, this indirect consequence of the search for ESPEs will improve the dating accuracy and precision of ¹⁴C more broadly (Box 1).

Box 1 Notation

When discussing ¹⁴C, we denote conventional radiocarbon ages as bp (radiocarbon years before present)¹²⁹. Calendar years (such as those derived from dendrochronology or historical events) are expressed as ad or bc (elsewhere, sometimes as ce or bce), while calibrated radiocarbon measurements are expressed as cal bc and cal ad, or as cal bp (calibrated years before present). Here, ‘present’ is defined as the year ad 1950, such that, for example, 10 cal bp is the same as cal ad 1940. When discussing inference on the Earth system, we use Δ¹⁴C, which denotes depletion or enrichment of the (age-corrected and fractionation-corrected) ¹⁴C/¹²C ratio relative to an international standard expressed in per mil (‰)¹³⁰.

We define an ESPE^77,78 as a solar particle storm that can be statistically significantly resolved (at the 3σ level) in the annual cosmogenic-isotope data and where the flux of particles with kinetic energy greater than 200 MeV time-integrated over the entire event (the F₂₀₀ particle fluence) exceeds 10⁹ cm⁻².

Recent advances in ¹⁴C and calibration

Extreme solar particle (Miyake) events

Initially, the Miyake spike of ad 774 was proposed to be the consequence of a nearby supernova. It was soon shown, however, that the only plausible cause was the Sun, specifically an ESPE with an enormous flux of solar energetic particles (SEPs) producing a spike of cosmogenic isotopes in Earth’s atmosphere²². This solar event was later confirmed by similar peaks in Δ¹⁴C from other trees around the world, as well as in other cosmogenic isotopes, ¹⁰Be and ³⁶Cl, measured in ice cores^21,28. The use of different isotopes and sophisticated modelling made it possible to identify that the event had probably occurred during the Northern Hemisphere (NH) summer of ad 774. The reconstructed SEP energy spectrum^21,29 seemed to be similar to those of normal solar particle storms but several orders of magnitude greater in flux.

The identification of these huge spikes in ¹⁴C production has enabled ¹⁴C calibration at annual precision, but studying the spikes themselves is also important to better understand the behaviour of the Sun and the potential risks it poses to Earth and our technological society. The long-term behaviour of the Sun is poorly documented. Direct observational measurements of sunspots have been recorded for only a few centuries³⁰, and more-detailed instrumental measurements for only a few decades. The largest directly observed solar storm occurred on 1–2 September in ad 1859 and is known as the Carrington Event. It led to widespread disruption, including the destruction of telegraph machines and an aurora so bright that it seemed the Sun had begun to rise³¹. However, so far, no identifiable increase in either Δ¹⁴C or ¹⁰Be has been detected for this event (although a transient high-latitude offset in Δ¹⁴C a couple of years after the event has been tentatively proposed³²). The ad 774 Miyake Event was an entire order of magnitude greater in size than the Carrington Event, making understanding the risks of ESPEs even more important.

The discovery of the first Miyake Event stimulated research in solar physics and initiated a race for ¹⁴C measurement programmes at annual resolution to identify further ESPEs and investigate the potential for other fine-scaled solar behaviour. This work has been facilitated by advances in accelerator mass spectrometry (AMS) instrumentation^33,34,35, which means that high-precision ¹⁴C measurements (less than 2‰ in Δ¹⁴C, equivalently ± 16 bp) of samples containing just a few milligrams of carbon can now be achieved much more efficiently. So far, five ESPEs have been confirmed through comparison of multiple radionuclides (7176 bc, 5259 bc, 660 bc, ad 774 and ad 993) along with four further potential candidates (12350 bc, 5410 bc, ad 1052 and ad 1279) that are yet to be confirmed^{20,21,36,37,38,39,40,41,42}.

Annual-resolution ¹⁴C measurements

Although ¹⁴C measurements of single tree rings of known age have been available for use in ¹⁴C calibration for decades, they have focused only on the past few centuries, and their extension further back in time was not seen as urgent for calibration⁴³. Most of the reference ¹⁴C datasets used in calibration-curve construction were instead measurements of 5-, 10- or 20-year blocks of tree rings. The new measurement programmes have generated huge numbers of annual tree-ring ¹⁴C measurements, although primarily from the NH. Even in periods without ESPEs, they reveal short-term fluctuations in ¹⁴C concentrations that had been obscured in the legacy multi-year measurements. Such information can be used by anyone wishing to calibrate.

The IntCal20 NH curve⁶ is now based entirely on tree-ring measurements from 13,910 to 0 calendar years before present (cal bp). In this period, it draws on 10,713 ¹⁴C measurements from tree-ring samples. Of these, 5,866 now correspond to single-year measurements covering 2,734 individual calendar years (Fig. 2a). Currently, these new annual-resolution sections of calibration curve tend to concentrate on hot-topic periods, specifically the most recent millennium, the two original Miyake events, the date of the Minoan eruption of Thera (Figs. 1 and 2b) and the Younger Dryas climatic event. However, IntCal20 does not include the most recently issued datasets^37,38,40,41, and coverage of the Holocene with annual ¹⁴C measurements is expected to increase substantially.

Fig. 2: Construction of ¹⁴C calibration curves.

a, Distribution of block sizes in the tree-ring ¹⁴C measurements used for the IntCal20 curve⁶ back to 13,910 cal bp. Before the discovery of the ad 774 ESPE, most reference ¹⁴C samples were multi-annual tree rings (often ten-year blocks). Inset, calendar years with annual-resolution ¹⁴C data (orange vertical bars) and the confirmed (pink) and potential (green) ESPEs in the Holocene. b, The increased annual detail available with IntCal20 (ref. ⁶) compared with IntCal13 (ref. ¹²⁷) during the second millennium bc. c, The IntCal20 curve over the ad 774 ESPE plotted in Δ¹⁴C. This is shown alongside the preliminary MiyakeCal curve designed to better represent the abrupt increase in ¹⁴C production caused by the ESPE. d, IntCal20 and MiyakeCal plotted in radiocarbon age. For IntCal20 (b) and MiyakeCal (c,d), a sample of posterior curve realizations are plotted (lines of various colours) along with the summarized posterior mean (solid orange and blue lines) and 1σ intervals (shaded orange and blue). All observations (black dots) are shown with 1σ error bars on the ¹⁴C axis, and the horizontal bars represent the individual years represented in (multi-annual) measurements.

The current Southern Hemisphere SHCal20 curve⁸ has far fewer annual ¹⁴C measurements on which to draw. It is based on 2,091 direct ¹⁴C measurements of tree rings, covering just four distinct periods: 2,140–0, 3,520–3,453, 3,608–3,590 and 13,140–11,375 cal bp. Of these, only 262 correspond to single-year ¹⁴C measurements, covering just 150 individual calendar years. The remainder of the SHCal20 curve is based on modelling (applying a 36 ± 27 ¹⁴C year interhemispheric offset, to older ages, from IntCal20). Consequently, the SHCal20 curve must be considered with care if using short-term ¹⁴C fluctuations because they have mostly been inferred from the NH data.

Beyond 13,910 cal bp, the resolution of all calibration curves is much lower, owing to the limitations of our ¹⁴C archives. Here the calibration curves predominantly rely on measurements from the Hulu Cave stalagmites^44,45, which are smoothed as a result of the various natural processes involved. However, new tree-ring archives are becoming available^41,46,47, providing hope that annual-resolution calibration curves based on tree rings will soon extend further into the past.

Constructing calibration curves

The IntCal calibration curves^6,7,8 combine all the reference ¹⁴C measurements that pass the selection criteria⁴⁸. These includes both single-year and multi-year measurements, with the latter modelled as such. The current IntCal20 and SHCal20 curves are generated using Bayesian splines, generating a set of individual posterior realizations that are then summarized⁴⁹. The final curves are intended to take account of, and represent, the over-dispersion observed in ¹⁴C measurements within a hemisphere (observed ¹⁴C determinations in any calendar year are more widely spread than expected from their quoted laboratory uncertainties).

The current construction methods are designed to allow more detail to be retained in time periods with large volumes of single-year measurements (Fig. 2b). However, they still implicitly model the underlying processes as smooth over time. Consequently, over ESPEs, they tend to smooth out the ¹⁴C production spikes so that the increases in Δ¹⁴C (Fig. 2c,d) occur over more than a single year, although this is also partly a consequence of the seemingly different timing of some of the events in different set of trees (Box 2).

As a result of this smoothing, those seeking exact calibration over ESPEs have predominantly used the direct mean of annual ¹⁴C measurements from a reduced set of locations⁵⁰ instead of the IntCal curves. These direct-mean curves cover only short time periods and do not consider overdispersion. To demonstrate what could potentially be achieved if the international curves were modified to enable sharp ESPE discontinuities, we present a tentative MiyakeCal curve over the ad 774 event (Fig. 2c,d). MiyakeCal uses the same methodology as IntCal20 but allows an additional jump in Δ¹⁴C from ad 774–775 to be superimposed on the smooth spline processes.

Box 2 Case study I

Regional timing of the ad 993 signature and implications for exact dating

The ad 993 ESPE is recorded by a sharp increase in Δ¹⁴C in tree rings globally. However, the precise timing of this increase is less clear (Box 2 Fig. 1a). Current data from a high-latitude region in Sweden⁵⁰ (NH0, more than 60° N) indicate that the main increase occurred in ad 992–993. Alternatively, trees collected from Mongolia and Russia⁵⁰ (NH1, 40–60° N) indicate a two-year increase in ad 992–994. Finally, data from mid-latitude regions, specifically two locations in China and the United States⁵⁰ (NH2, 30–40° N) as well as locations in Japan^42,131, indicate that the Δ¹⁴C increase may have occurred in ad 993–994. The cause of these differences in phasing between tree-ring records is currently unknown. It has been suggested that the region (and specifically the latitude) in which the trees grew may be a key factor.

Box 2 Fig. 1 | Differences in the precise timing of the ad 993 ESPE in different NH tree rings and the impact on calibration over the event. a, Weighted-average of observed Δ¹⁴C (and 1σ error bars) for the tree rings in each NH zone⁵⁰ and the separate Japanese tree rings^42,131. Lines join the Δ¹⁴C averages to illustrate when the Δ¹⁴C increase occurs in each region. The shaded backgrounds correspond to ad 992–993 (blue) and ad 993–994 (yellow). Trees from different locations seem to record the ESPE spike in different years. b, Calibration of wooden item 4A 68 E2-2 from L’Anse aux Meadows⁵⁷ (located at 51º N) using the ¹⁴C ages of all 35 annually sampled growth rings (black dots with 1σ error bars, covering 51 calendar years) against a preliminary calibration curve (shown in blue with 1σ intervals, BüntgenNH1Cal) created using only the NH1 ¹⁴C datasets⁵⁰ in the region of the ESPE. c, Calibration of the same item against a Japanese-based calibration curve (pink, JapanCal) created using only Japanese ¹⁴C data^42,131. d, Calibration against the IntCal20 calibration curve⁶ (orange) created using all the ¹⁴C data in the Northern Hemisphere (from all NH zones⁵⁰ and the separate Japanese data^42,131).

The choice of which ¹⁴C reference data set to use to obtain exact ¹⁴C calibration is crucial. In the case of wooden item 4A 68 E2-2 (ref. ⁵⁷) from L’Anse aux Meadows (at 51° N), if we calibrate using the tree rings from either NH0 or NH1⁵⁰ as the reference, we obtain an approximately 97% posterior probability that the waney edge is from cal ad 1021. Box 2 Fig. 1b presents wiggle-matching of all 35 sampled annual growth rings of the item (covering 51 calendar years in total) against a calibration curve based on only NH1 ¹⁴C data⁵⁰. Conversely, if we calibrate against the reference Japanese tree-ring ¹⁴C data^42,131, we obtain an approximately 95% probability that the waney edge is from cal ad 1022 (Box 2 Fig. 1c). Similar differences in calibrated ages are found for the other items at L’Anse aux Meadows.

At present, the geographic coverage of reference ¹⁴C data over the ad 993 event is rather limited⁵⁰. More research, including sampling of ¹⁴C data with a greater geographical spread and improved modelling of ¹⁴C production and transport, is essential to understand the reasons behind the potential regional differences in the timing of the ad 993 ESPE signature. This will allow detailed guidance to be given to those seeking to use this event for exact calibration. When it is not known in which precise calendar year the ESPE will be recorded in sampled wood, to avoid potentially spurious precision it may be safer to calibrate against the IntCal20 curve, which combines all the NH zonal and Japanese reference data (Box 2 Fig. 1d). This provides an approximately 48% posterior probability that the waney edge is from cal ad 1021 and a 47% probability that it is from cal ad 1022.

Improving ¹⁴C calibration

Exact dating by ESPE wiggle-matching

Almost immediately after the discovery, it was realized that the substantial jumps in ¹⁴C production during ESPEs could provide exact calendar-age precision from ¹⁴C calibration if it was possible to find timber with growth rings that spanned an event. Measurement of ¹⁴C in consecutive annual samples to discover the corresponding jump in ¹⁴C concentration allows exact identification of the ring containing the ESPE and consequently its calendar year. Counting from this growth ring to the final waney edge in the timber then reveals the precise year in which the tree was felled. This process is known as ¹⁴C wiggle-matching and can be implemented in two different ways (Box 3).

The first application of these ideas, undertaken in 2014, provided an exact date for the felling of timber used in the Holy Cross chapel of the St John the Baptist convent, a UNESCO World Heritage site in Val Müstair, Switzerland⁵¹. This has been followed by further examples^{52,53,54,55,56,57,58}, including the exact anchoring of the stratigraphy of a Viking Age trade centre⁵⁶, identification of the precise year in which Vikings were present in North America⁵⁷ and evidence for a compound earthquake involving simultaneous ruptures across two fault zones near Seattle⁵⁸.

The principle has the potential to be extended to any ESPE of sufficient magnitude. So far, applications have focused on identification of the ad 774 or ad 993 events, although a 2023 study has used the 5259 bc ESPE to date a floating Neolithic tree-ring chronology in Dispilio, northern Greece⁵⁹. Users of the ad 993 event must proceed with care, however, because the exact timing its ¹⁴C signature in tree rings does not seem to be globally consistent (Box 2) and the choice of ¹⁴C reference data is crucial to obtaining reliable exact ¹⁴C calibration.

Box 3 Case study II

Bayesian wiggle-matching versus χ² testing: the timber lake fortress at Lake Āraiši

There are two approaches to calibrating ¹⁴C sequences and we illustrate them using the timber lake fortress at Lake Āraiši in Latvia⁵⁵. Single-year samples for 14 consecutive annual rings were obtained for a spruce timber recovered from the log-platform during excavation (Box 3 Fig. 1a). These annual measurements contain the ad 774–775 signature with a drop of more than 100 ¹⁴C years observed over the course of a single calendar year.

Box 3 Fig. 1 | Dating the Lake Āraiši timber fortress by ¹⁴C wiggle-matching. a, Remains of the Lake Āraiši timber fortress⁵⁵ (credit: J. Apals and Z. Apala). b, Bayesian wiggle-match of 14 consecutive annual ¹⁴C samples (black dots with 1σ error bars) against IntCal20 (orange with 1σ intervals). The posterior calendar-age estimate for the youngest ring in the sequence (n-53) is shown in red on the x axis. c, Series of test statistics for χ² wiggle-matching of nine Āraiši tree rings against the mean of a set of NH tree rings⁵⁰. The only calendar year fit that is not rejected at the 5% level is cal ad 782 (the dashed line shows the critical test value, χ²_0.95,9 = 15.5, above which a calendar year fit is rejected). d, Bayesian wiggle-match of 14 consecutive annual ¹⁴C samples (black dots) against MiyakeCal (blue). The insets in b and d show periods of increase in Δ¹⁴C (decline in radiocarbon age) over the ad 774 event. The light blue-shaded intervals show the year ad 774, in which the ESPE occurred. For an animation of Bayesian wiggle-matching, see Supplementary Video 2.

A frequently used way to obtain exact calibration over an ESPE is wiggle-matching using a series of χ² tests that assess the fit of the ¹⁴C sequence against the mean of known-age reference tree-ring datasets. Using the χ² approach, a series of independent hypothesis tests are performed, with each test shifting the ¹⁴C sequence by a single calendar year and assessing the fit to the reference. The resultant 95% confidence interval for the calibrated age of the sequence consists of all the calendar-age fits that are not rejected at the level of 5% significance.

With χ² tests, users are restricted to considering only those growth rings that overlap with the reference datasets for the chosen calendar fit. If the ¹⁴C sequence to be wiggle-matched is longer than the reference^55,57, a χ² approach will therefore result in the loss of valuable calendar-age information. For the Lake Āraiši fortress, to test each of six possible calendar fits (Box 3 Fig. 1c) against the standard reference⁵⁰, we are restricted to using only 9 of the 14 annual ¹⁴C determinations. This provides a 95% confidence interval for ring n-53 consisting of the single year cal ad 782. Users should also be aware that the current reference mean⁵⁰ does not incorporate the possibility of over-dispersion in the ¹⁴C measurements, and it is possible that no calendar-age fits will pass the test.

Outside of ESPE calibration, Bayesian wiggle-matching against the suitable IntCal curve⁶ is the standard method of calibrating ¹⁴C sequences. This allows wiggle-matching of ¹⁴C sequences of any length (although it does not provide a measure of the overall quality of fit). The IntCal20 curve has not been popular for ESPE calibration because, as a result of its construction, sharp increases in Δ¹⁴C are smoothed with the increase in Δ¹⁴C over the ad 774 event occurring over the course of four calendar years (Box 3 Fig. 1b, inset). It is, however, still possible to obtain exact calibration using IntCal20 if the ¹⁴C sequence to be calibrated is long enough (Box 3 Fig. 1b). Using all 14 annual ¹⁴C measurements from Lake Āraiši, we obtain a 99.3% probability that ring n-53 is from cal ad 782.

Finally, if calibration-curve construction is modified to allow an extra non-smooth jump in Δ¹⁴C in the year ad 774–775, we can obtain a calibration curve that better represents the sharp ESPE signature. A Bayesian wiggle-match of the 14 annual measurements in the Āraiši ¹⁴C sequence against the discontinuous MiyakeCal calibration curve provides a 99.9% probability that ring n-53 is from cal ad 782 (Box 3 Fig. 1d).

Our conclusions for Lake Āraiši are the same for all of these alternative methods. The waney edge is from 53 years after ring n-53, so the fortress platform can be identified as having been built from trees felled between the NH winter of cal ad 835 and the NH spring of cal ad 836.

Leveraging annual ¹⁴C calibration curves

For archaeologists and environmental scientists not studying periods with an ESPE, or who lack preserved wood on their sites, it is still possible to take advantage of the new annual-resolution detail in ¹⁴C calibration curves. Sampling strategies that allow stratigraphic and relative calendar information to be incorporated into calibration are key. Since the advent of routine chronological modelling, the identification and inclusion of sequences of samples that can be ordered independently has been essential for maximizing dating precision. Annual-resolution calibration datasets mean that shorter sequences, and sequences of short-lived samples, may now be included in models more effectively. This can substantially increase calendar-age precision, particularly on calibration plateaux.

Instead of submitting a single sample from a seed, one may aim to calibrate a sequence obtained from a charcoal fragment⁶⁰. Short wiggle-matches may also be obtained from timbers in historic or proto-historic buildings that are unsuitable for dendrochronology^61,62,63. Studies of human bone provide a further example, although modelling is complicated by the potential for dietary reservoir effects⁶⁴ and bone-collagen turnover⁶⁵, for which more research is sorely needed⁶⁶. Sampling single teeth from several individuals who died at the same time but were of different ages when they died enables a relative sequence of samples to be elicited from a group of burials⁶⁷. Differences in the formation times of different bones in a human skeleton, such as the petrous and rib, can also be used to produce a sequence of ¹⁴C measurements from a dated individual⁶⁸. Box 4 combines the sampling of teeth and bone from multiple individuals into a sequence of six ¹⁴C determinations with varying calendar-age gaps^69,70. Increasingly, studies of ancient DNA are not only providing phylogenetic trees⁷¹, but also kinship relationships between individuals⁷². These relationships can also provide short sequences for chronological models, based on either the degree of kinship⁷³ or reconstructed pedigrees^72,74.

Box 4 Case study III

Exploiting fine-scale structure in radiocarbon calibration: wiggle-matching people

Background: In 2013, the remains of 28 men were excavated from Palace Green in Durham, UK^69,70. Samples from four skeletons were dated, all of which were recovered from two mass graves thought to have been made at the same time. One grave was stratified below the Bishop’s stables, which are shown on a map of ad 1754, and, because the dentition of two individuals shows pipe facets (Box 4 Fig. 1b), the mass burial must have occurred after around ad 1620, when tobacco from the Virginia plantations made clay pipes widely available across Britain¹³².

Question: Could these graves contain prisoners from the Battle of Dunbar (3 September ad 1650)?

Relative dating information: For skeletons Sk12 and Sk21, it was possible to obtain ¹⁴C samples formed at different times of development. This allows relative dating to be incorporated into calibration. The upper half of the root of the first molar, which forms at 4–6 years of age, and the upper half of the root of the third molar, which forms at 14–16 years of age¹³³, were sampled from Sk12 and Sk21. Because these individuals died aged 19–23 years and 21–25 years, respectively, the calendar gap between the teeth can be incorporated in the model as 10 ± 1 years, followed by the gap to the individuals’ deaths as 6 ± 1 years and 8 ± 1 years, respectively. For Sk9, an individual who died aged 17, and Sk16A, who died aged 14, an allowance has been made for the turnover time of the sampled bones (a femur and a tibia, respectively)⁶⁵, equating to 5 ± 2 years and 4 ± 2 years, respectively. Combining the two molars for Sk12 and Sk21 with the two bones for Sk9 and Sk16A, and assuming a common date of death, provides a sequence of six ¹⁴C determinations to be wiggle-matched using variable gaps between each sample.

Dietary information: The proportion of marine foods in each individual’s diet was estimated from the ratio of stable carbon isotopes in the skeleton and a bespoke calibration curve for each individual constructed incorporating this¹³⁴. The marine component was modelled using the Marine20 calibration curve⁷ with a local ΔR₂₀ reservoir adjustment of −133 ± 33 bp, calculated from the 10 closest points to Durham in the Marine20 reservoir database (http://calib.org/marine/, accessed on 27 April 2023)¹³⁵. The remaining proportion of the diet was modelled using the terrestrial IntCal20 calibration curve⁶.

Box 4 Fig. 1 | Modelling and calibrating the Palace Green burials. a, Part of chronological model IID⁶⁹ for mass graves excavated on the Palace Green in Durham, UK (run in OxCal¹³). b, Skull of skeleton 21, showing pipe facets (highlighted) caused by the habitual use of a tobacco pipe (credit: J. Veitch/Department of Archaeology, Durham University).

Calibrating a single sample: A simple analysis of the graves might consider just calibration of Sk12’s third molar (SUERC-58784). Calibration of this single ¹⁴C measurement provides a date of death for Sk12 spanning cal ad 1480–1696, 1742–1761 and 1768–1806 (highest posterior density interval at 95% probability; see the light grey shaded distribution for R_Date SUERC-58784 in Box 4 Fig. 1a). This calibrated age range is potentially compatible with the Battle of Dunbar, but its wide breadth weakens any inference on the specific circumstances of death.

Chronological modelling incorporating archaeological information: Jointly calibrating all the skeletons in the grave and incorporating the available stratigraphic, historical and biological information enables a substantial increase in calendar-age precision. The resultant model (Box 4 Fig. 1a) estimates that the individuals buried in the mass graves died in cal ad 1630–1670 (highest posterior density interval at 95% probability; see red shaded distributions). This is a much narrower calendar range than is achievable by calibration of only a single sample and is clearly compatible with the Battle of Dunbar.

Future directions and opportunities

Understanding the Sun

For the period before telescopes were invented in the early seventeenth century, cosmogenic isotopes provide the only quantitative information on solar variability, particularly ¹⁴C owing to its high resolution. Models that link cosmogenic production and solar variability can be verified using direct instrumental data on the Sun obtained over the past four centuries^37,75,76.

All eight Holocene ESPEs identified so far were 20–100 times stronger than the strongest solar storm directly recorded over the past decades (on 23 February in ad 1956). The event of 12,350 bc still needs detailed analysis but has the largest Δ¹⁴C increase of all⁴¹. However, smaller ESPEs cannot be resolved in the existing ¹⁴C data series^77,78. Presently, only about one-sixth of the Holocene is covered by high-resolution ¹⁴C data^6,38. It is therefore likely that more ESPEs will be found as the coverage of precise annual ¹⁴C measurements increases.

Increasing annual coverage is crucial to fully understand ESPEs. It is still unclear whether they correspond to the extreme energetic tail of rare but normal solar storms, metaphorical Black Swan events⁷⁹, or whether they represent an entirely new solar eruptive process, metaphorical Dragon King events (Fig. 3). In 2012, super-flares on Sun-like stars that are 100–1,000 times stronger than all known solar flares were also discovered⁸⁰. It has been proposed that ESPEs and super-flares are related phenomena. However, a projection of super-flare results to the Sun does not match the ESPE statistics⁸¹. To resolve these questions, more high-precision ¹⁴C measurements, reproduced over multiple records at annual resolution, are needed.

Fig. 3: Identifying the solar processes underlying ESPEs.

Estimated probabilities that, in any given year, the annual F₂₀₀ fluence (defined in Box 1) will exceed specific values, as assessed from direct data for the space era (blue circles)¹²⁸ and cosmogenic isotope data for the Holocene (red diamonds). Data are shown with 90% confidence intervals. The vertical yellow bar indicates the current sensitivity/detection limit of the cosmogenic-isotope method for a single ¹⁴C record⁷⁷, with ESPEs lying to the right. The current lack of information near this limit prevents us from distinguishing between the Black Swan scenario, in which ESPEs are the tail of regular solar storms and all data can be fitted with a single distribution (Weibull, grey curve), and the Dragon King scenario, in which ESPEs and regular storms are distinct physical phenomena that follow separate distributions (dashed red and blue curves, respectively)⁷⁹. Further high-precision annual ¹⁴C measurements, reproduced over multiple records, will lower the detection limit and resolve this.

Solar activity is organized in approximately 11-year cycles of slightly variable length and highly variable amplitude⁸². These cycles are usually studied by sunspot number. Counting of sunspots began in ad 1610 and form the longest dataset of direct astronomical observations, covering 36 solar cycles including the Maunder minimum of activity (ad 1645–1715) and a Modern grand maximum (ad 1940–2008). This time series allows some empirical regularities of the solar cycle to be identified⁸² but is highly limited because the number of cycles is small.

Annual ¹⁴C data have been used to study solar cycles beyond the observational sunspot record for some time¹⁹, but were previously hampered by insufficient measurement resolution. In 2021, ¹⁴C measurements for the past millennium were obtained with sufficient precision to reconstruct individual solar cycles^37,75. This nearly tripled the number of known solar cycles to 96 and allowed some empirical rules to be confirmed. Further reconstruction of 11-year solar cycles on longer multi-millennial time scales, which requires high-precision annual ¹⁴C measurements, is critical to understanding the solar dynamo⁸³.

So far, excluding the 12,350 bc event⁴¹, ESPEs have been found only in Holocene subfossil trees. Solar variability beyond the Holocene remains terra incognita, although existing tree-ring ¹⁴C data and ice-core ¹⁰Be and ³⁶Cl are seen to exhibit high-frequency structures^46,84,85,86. Extending the solar record beyond the Holocene will require dedicated efforts to find relatively rare subfossil trees from Glacial and Late Glacial periods. Ancient New Zealand kauri trees (Agathis australis) show considerable promise beyond the Last Glacial Maximum^46,47. The AMS analyses of these trees will need to be optimized through careful controls on uncertainty⁸⁷ to detect small and rapid ¹⁴C changes. The Glacial and Late Glacial periods were characterized by rapid variations of the carbon cycle, as illustrated by changes in atmospheric ¹²CO₂ (ref. ⁸⁸) and ¹³CO₂ (ref. ⁸⁹). These findings emphasize the need to reconstruct abrupt ¹⁴C changes in trees from both hemispheres. Abrupt solar variations should be recorded in all trees, whereas rapid carbon-cycle changes may lead to changes in the interhemispheric ¹⁴C gradient^90,91.

Understanding environmental processes

Abrupt spikes in ¹⁴C production enable insight into Earth and environmental processes. Sub-annual ¹⁴C measurements across the ad 774 and ad 993 Miyake events from the mid to high latitudes of both hemispheres have helped to identify the seasonal timing and a latitudinal gradient of radiocarbon preserved in contemporary trees⁵⁰. Intriguingly, these early results indicate that there may be regional differences in the precise timings of some ESPE ¹⁴C signatures (Box 2) and in the interhemispheric ¹⁴C gradient between the events, possibly modulated by changes in the ocean–atmospheric circulation and the carbon cycle⁵⁰.

To improve our understanding of Earth system processes, high-precision annual (or even sub-annual) ¹⁴C measurements must be accompanied by the development of a carbon-cycle dynamic model with sufficient spatio-temporal accuracy to account for detailed processes, including latitudinal gradients, orography, proximity to the ocean and the growth period of trees. Although the standard carbon-cycle box model⁹² works well for slow decadal changes, large-scale ‘boxes’ that do not resolve uneven patterns of ¹⁴C production and transport may lead to inaccurate results for annual and sub-annual scales, especially during ESPE production spikes⁹³. One prospective direction is coupling a detailed climate-chemistry dynamic atmospheric model, such as SOCOL⁹⁴, ECHAM-HAM or GEOS-Chem⁹⁵, with the dynamic ocean⁹⁶ and a realistic biosphere, including growth and carbon uptake of specific tree species.

With the carbon cycle currently experiencing extreme disruption⁹⁷, inverse 3D-transport modelling is being used to identify CO₂ emission sources and sinks based on the time-varying spatial evolution of ¹²CO₂, ¹³CO₂ and ¹⁴CO₂ (refs. ^98,99,100). Application of this approach using ¹⁴C records of ESPE spikes and other rapid ¹⁴C changes at different locations may improve knowledge on natural ¹⁴C atmospheric dispersal and carbon sinks, extending the insights afforded by the 1960s nuclear ‘bomb spike’¹⁰¹. Complemented by an expanded global network of ¹⁴C measurements, such an approach would allow the use of observed latitudinal gradients in the amplitude of ESPEs⁵⁰. Similar chemical transport models could be used to simulate the dispersal and fallout of ¹⁰Be and ³⁶Cl on polar ice caps^95,102. This is crucial because aerosols from volcanic eruptions can increase the deposition of ¹⁰Be and ³⁶Cl, generating abrupt spikes that may be confused with ESPEs¹⁰³. Accurate geomagnetic records¹⁰⁴ are also necessary to model the amplitude of cosmogenic production associated with ESPEs.

Statistical considerations

Harnessing annual-resolution calibration datasets will require statistical hurdles to be overcome. Calibration-curve construction must be adapted to better represent abrupt ESPE ¹⁴C production changes. This is a challenge because regression methods typically assume that the underlying function is smooth¹⁰⁵ to prevent overfitting. Ensuring one does not overfit to spurious ¹⁴C artefacts is particularly important for calibration curves because the underlying reference data consist of overlapping trees from different locations, species and laboratories, all of which introduce considerable extra variability^106,107,108.

Such smoothness does not easily allow representation of ESPEs, however. One solution might incorporate extra ESPE-specific basis functions, placed at fixed times, into the regression to represent the temporal atmospheric Δ¹⁴C response of an ESPE. These functions could be informed by the atmospheric models discussed above^94,95. This will require agreement on where such additional basis functions should be located and, until coverage with annual ¹⁴C data becomes complete, leaves the possibility that some ESPE variations will be missed in the calibration curves³⁸.

With increasing calibration precision, it is essential to fully understand all sources of potential variations in the measurement of ¹⁴C samples, including shared offsets and biases¹⁰⁶. This is particularly relevant for annual measurements, owing to ¹⁴C differences during the growing season^50,108,109. Current tree-ring data exhibit overdispersion with ¹⁴C measurements in any calendar year more highly spread than their quoted laboratory-reported uncertainties should indicate⁴⁹. Furthermore, consistent ¹⁴C offsets have been observed between contemporaneous tree rings from different locations. In case study I (Box 2), for example, as well as a difference in ESPE timing, the ¹⁴C measurements from any region tend to stay above or below those of other regions. Understanding the causes of extra variations and shared biases in ¹⁴C is critical to accurate calibration, particularly when calibrating sequences of ¹⁴C samples (Fig. 4 and Box 2).

Fig. 4: Implications of a ¹⁴C offset when wiggle-matching over a plateau.

A sequence of ten ¹⁴C samples (black dots with 1σ error bars) beginning in 552 bc has been simulated, sharing an offset/bias of 16 ¹⁴C years from the IntCal20 curve (orange curve with 1σ intervals). The vertical dashed red line represents the true underlying calendar age. If we take the default approach to wiggle-matching, assuming measurement error is independent, the posterior calibrated age density (pink) is bimodal and inaccurate. However, if we perform a wiggle-match incorporating the possibility of an offset shared by all ten ¹⁴C measurements, the sequence calibrates accurately and precisely, providing the posterior calibrated age density shown in green.

The ¹⁴C research community has undertaken considerable work to assess and improve the accuracy and precision of their measurements through laboratory intercomparisons^87,110. This work has begun to apportion the sources of variation observed in ¹⁴C measurements, such as pre-treatment methods. Identification of offsets and their sources is challenging, however, because they are typically of a similar size to measurement precision. Furthermore, many offsets will not be constant over time¹⁰⁶. We require a dedicated programme of work, engaging with a broad set of laboratories, to improve our understanding of regional, altitudinal and laboratory sources of extra uncertainty, aided by high-precision AMS.

Calibration-curve usage could potentially be improved by the use of covariance information on the ¹⁴C ages in adjacent calendar years. This is currently lost when providing point-wise curve summaries^49,111. Furthermore, as calibration curves become wigglier, the possibility of highly multimodal fits when calibrating models increases substantially. The use of robust assessment metrics^{13,112,113,114,115} to ensure that calibration models have properly converged before inference is made will be even more important than it is now. More-advanced Markov chain Monte Carlo approaches, such as parallel tempering^49,116, slice sampling^117,118 or Hamiltonian Monte Carlo¹¹⁹, may be required.

To ensure uptake in the radiocarbon community, all these statistical considerations must be integrated into user-friendly software, accompanied by educational support.

Archaeological interpretations

Since ¹⁴C dating was first developed, the adoption of new developments by archaeologists has been rapid. The possibility of obtaining exact chronologies by fitting series of measurements on single-year tree rings to the abrupt changes in atmospheric ¹⁴C at ESPEs is the latest example^{51,52,53,54,55,56,57}. This methodology can provide precise calendar dating for particular sites that fits them exactly into historical narratives, thereby providing fixed points in wider archaeological discussion. It also has the potential to securely anchor floating historical sequences, such as those of the Near East and Central America, to the calendar scale¹²⁰. The search for key timbers, intact to waney edge and securely tied to these historical sequences, runs parallel with the hunt for new ESPEs.

For archaeologists studying sites or periods containing known ESPEs, dating a series of single tree rings containing the specific ESPE signature may provide an exact anchoring for that chronology, although the relationship between the dated sample and the archaeological question remains paramount^14,121. For long periods, however, there are no such events. As more single-year data become available, changes to the calibration curve in some periods will be substantial, particularly if there is a currently undetected ESPE³⁸. In most places, however, changes will be more modest, with small wiggles and steps becoming visible that cannot be resolved in the current coarser-resolution data⁸⁷. This extra structure, even on a small scale, can nevertheless still have a substantive impact on modelled chronologies.

Archaeologists have long focused on the in-built age and taphonomy of the material selected for dating¹²², and the advent of AMS brought a focus on dating short-life, single-entity samples¹²³. The current Bayesian process¹⁴ provides a framework to select suites of samples for dating that can be interpreted together in formal statistical models. This process will need to develop alongside a much more detailed annual calibration curve, in particular the possibility to more effectively leverage information from short ¹⁴C sequences and/or sequences of short-lived ¹⁴C samples (Box 4). In undertaking such studies, high-precision measurement of the unknown samples is vital¹²⁴. Increases in the modelling of suites of ¹⁴C samples may be supported by the development of model-selection approaches, building on existing individual and model agreement indices¹¹⁵. Such approaches could provide more-robust inference in the presence of model uncertainty^125,126 and help to select between multiple potential interpretations.

Finally, the increase in the detail and precision of calibration data mean it is essential to ensure the reproducibility of ¹⁴C modelling. Each update to the calibration datasets affects the calibrated dates in previous studies. There is a need to develop model repositories in which, as calibration curves are improved, previous studies can be recalibrated with the state-of-the-art knowledge. This will ensure that research remains comparable, and that archaeological and geoscientific knowledge is continuously updated.

Conclusions

The unexpected discovery of ESPEs in the ¹⁴C record has brought together archaeologists, environmental scientists and solar physicists in pursuit of common research objectives. ESPEs, and the extensive high-precision annual ¹⁴C measurement programmes their discovery initiated, are enabling substantial advances across numerous fields of endeavour. As well as improvements in the dating precision achievable using radiocarbon, including the potential calibration to an exact year, we are now gaining deep insights into the extreme behaviour of the Sun, societal evolution and the archaeological record, as well as the operation of the Earth system. Whether used for precise dating, solar physics, or as a tracer of carbon dynamics, radiocarbon has shown itself to remain fundamental to understanding the world in which we live.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.