SCALING THE ¹⁴C-EXCURSION SIGNAL IN MULTIPLE TREE-RING SERIES WITH DYNAMIC TIME WARPING

Abstract

A signal of rapid changes in ¹⁴C production is logged in annual series of ¹⁴C derived from tree rings, which can be associated with diverse effects of cosmic-ray fluxes, including solar burst and supernova events. These ¹⁴C signatures may vary in time and space. The intensity and structure of the ¹⁴C signal is multifaced, which complicates understanding of the forcing and attribution of the underlying astrophysical events. It was suggested that Δ¹⁴C in 1052/53 CE and 1054/55 CE signatures at a 4‰–6‰ range over two years could be caused by the Crab Nebula supernova (SN1054) or/and solar perturbation. The temporal incoherence of the signals in published ¹⁴C series is investigated with dynamic time warping (DTW), novel approach for matching time-behavioral patterns in multiple ¹⁴C datasets. DTW analysis of four ¹⁴C signatures from tree rings of California, Finland and England suggests that ¹⁴C spikes between 1052 CE and 1055 CE can be caused by a single event. The flickering fingerprint may result from cross-dating inconformity. Cross-checking of tree-ring records from distant locations is impossible sometimes due to large difference in environmental conditions limiting tree growth. The methodology helps to align the signals and can be applied to other ¹⁴C datasets.

INTRODUCTION

The annual rate of ¹⁴C production in the lower stratosphere-upper troposphere reflects the changes in cosmic-ray flux over time due to changes in the intensity of galactic cosmic rays and solar energetic particles, and solar-geomagnetic shielding (Burr 2007). Variations of the ¹⁴C production rate are present in time series that can be derived from various terrestrial archives. Yet, annual series of ¹⁴C from tree rings are considered superior tracers of the ¹⁴C signal in time and space, since trees continuously use carbon dioxide gas from the air and ground water during photosynthesis, and distribution of the carbon isotopes has been almost completely mixed and homogenized over the globe (Poluianov et al. 2016; Wu et al. 2018).

So far, three confirmed solar energetic particle events (SEP 774 CE, 993 CE, 660 BCE, and 7176 BCE), one supernova (SN1056) and proposed solar-proton event (5259 BCE) have been confirmed with annual ¹⁴C series from numerous geographical locations and different AMS radiocarbon laboratories (Jull et al. 2014; Fogtmann-Schulz et al. 2017; Büntgen et al. 2018; Terrasi et al. 2020; Sakurai et al. 2020; Brehm et al. 2021a; Paleari et al. 2022). Four other proposed ¹⁴C excursions (1279 CE, 813 BCE, 5480 BCE, and 5410 BCE) need to be corroborated with data from multiple locations or need to be confirmed (Miyake et al. 2017, 2021; Jull et al. 2018; Brehm et al. 2021b).

It is known that manifestations of the same astrophysical event in carbon-14 may occur with delays in timing and variation in intensity across different parts of the globe. Büntgen et al. (2018) clearly demonstrated the meridional decline of 11-year mean atmospheric ¹⁴C concentrations across the northern and southern hemispheres in a case study of SEP 774 CE and 993 CE. Studies of possible supernova events have been hampered by inconclusive patterns in the ¹⁴C signal response due to smaller effects than in the SEP cases and temporal discord in the variation of Δ¹⁴C (Dee et al. 2016; Eastoe et al. 2019; Brehm et al. 2021b). Yet, the increase of production rate in the ¹⁴C signatures around supernova appearances cannot be explained by the Schwabe cycle alone (Terrasi et al. 2020; Brehm et al. 2021b). It seems that the SN-signal in the rapid change in ¹⁴C production is less comprehensible and different from highly coherent SEP-signal in Δ¹⁴C series forced by solar proton radiation.

The γ-ray burst produced by a supernova explosion arrives to the Earth at the same time as the visible light, which is documented by historical archives describing the appearance of a guest or new star (Pavlov et al. 2013). The mean interval between supernova events in the Milky Way galaxy is not that long and is estimated at 40 years ± 10 yr (Tammann et al. 1994). Cosmic rays accelerated at supernova remnant shocks change the chemistry of the atmosphere through ionization and dissociation of O₃ and N₂ which lead to the formation of nitrogen oxide compounds (Thomas and Melott 2006). Damon et al. (1995) searched for the effects of γ-ray bursts produced by supernovae in ¹⁴C records and determined an upper limit of 6‰ in Δ ¹⁴C for a large SN event within our Galaxy (Damon et al. 1995). Back then the error of ¹⁴C measurements produced by beta-counting systems used to be high around 4‰ (Cook and van der Plicht 2013). The precision of modern AMS ¹⁴C measurements has significantly improved the errors and reduced the estimate for the background concentration to 1.3–2‰ (Wacker et al. 2010). The AMS records and the finer temporally resolved series of ¹⁴C from tree rings may assist in improving the attribution of SN signal in ¹⁴C series.

A recent investigation of the effect of the Crab Nebula supernova (SN1054) in the annual rate of ¹⁴C production found a Δ ¹⁴C excursion at 1054/55 CE of 4‰ at tree-ring archives from Sierra Nevada Mountains and Finnish Lapland (Terrasi et al. 2020). However, other studies of tree rings from Central England and another location in the Sierra Nevada Mountains (California) report much higher ¹⁴C ranges for this time interval (up to 6‰) but in 1053 CE, one year prior to the SN1054 visible light arrival to the Earth atmosphere, and attribute it to a possible SEP event (Eastoe et al. 2019; Brehm et al. 2021b). Brehm et al (2021b) suggested this designation for a 6‰ change from 1052 CE to 1053 CE (1 yr) or a ca. 8‰ variation from 1052 CE to 1055 CE (3 yr). This research points to the interval 1052–1055 CE that may have two separate ¹⁴C spikes causes by two different cosmic events. Interestingly, some tree-ring series did not show any significant rise in Δ ¹⁴C at that time (see Güttler et al. 2013) and others record a slight intensification of ¹⁴C production (Menjo et al. 2005). Dee et al. (2016) examined the Δ¹⁴C response in tree-ring series covering the timing of four historical supernovae and found no significant increase in Δ¹⁴C tree-ring series around 1054 CE. It appears that the SN signature in annual ¹⁴C series is more difficult to identify than SEP signatures.

In this paper, we employ a novel technique of dynamic time warping (DTW) to ¹⁴C annual series in order to investigate the temporal coherence of the published ¹⁴C series for 1053(52)–55CE signature from four tree-ring archives at three locations (Finland, California, and England) from published literature (Brehm et al. 2021b; Eastoe et al. 2019; Terrasi et al. 2020). The DTW algorithm performs matching of similar patterns in datasets that may not be coeval. The method allows stretching or compressing sub-sections of the time axis at different scales to minimize and quantify the dissimilarity between the compared time series, as discussed by Izakian et al. (2015).

METHODS

We evaluate the temporal coherence of five ¹⁴C series with 1052 CE/1054 CE signatures developed independently and published by Eastoe et al. (2019), Terrasi et al. (2020), and Brehm et al. (2021b) (Figure 1a). The tree rings are originated from four locations (Figure 1b). At one location we measured ¹⁴C in earlywood (EW) and latewood (LW) of the rings separately. Supplementary material includes the original ¹⁴C datasets used in this study. All details concerning the tree-ring sampling, ¹⁴C measurements and Δ¹⁴C calculation can be found in the relevant publications. The records of three tree species are positioned in the coordinate box 36°N, 68°N and 118°W, 28°E, with an elevational gradient between 191 m asl and 1890 m asl (Figure 1b). The wood is sourced from living trees (sequoia), buried subfossils (pine) and historical timber (oak). The tree-ring records of Sequoiadendron giganteum come from two nearby sites in the Sierra Nevada Mountains, California (36.56°N, 118.75°W, 1890 m asl and 36.75°N, 118.97°W, 1850 m asl), Pinus sylvestris from Finnish Lapland (68.31°N, 28.09°E, 191 m asl) and English oak from the UK historical buildings near Hertfordshire and Greater Manchester with unknown harvesting locations in the area (51.75°N, 0.34°W for STA-C34 specimen and 53.41°N, 2.15°W for STK-C10 specimen). Annually resolved ¹⁴C data in addition to one location with sub-annually earlywood and latewood series of sequoia were examined for the period 1037–1067 CE (30 yr) with DTW.

Figure 1 ¹⁴C dataset used in the study. Left: annual series of Δ ¹⁴C attributed to possible SN1054 or SEP events. Color code of the curves: red—Full Ring Sequoia (Seq) from Eastoe et al. (2019); blue—EW Seq, brown—LW Seq and magenta—Finnish Pine from Terrasi et al. (2020); green—English oak from Brehm et al. (2021b). Shaded area is the signature interval 1052–1055 CE. Right: map of the tree-ring data locations. (See online version for color figures.)

DTW is an approach of machine learning (ML) and artificial intelligence (AI) for data analysis that is gaining popularity with increasing computational power, although many of the ML and AI techniques were developed in the 1960s–1980s. DTW was introduced by Sakoe and Chiba (1978) for spoken-word recognition, where a warping time function is used for optimal dynamic programming. Paliwal et al. (1982) proposed an improvement of the method with removal of the warping function slope constraints. Recent improvements introduced weights for uncertainty accounting (see Zuo and Yan 2018). DTW is a generalization of a point-wise comparison of records based on Euclidean distance. Instead of “rigid” comparison of correlation, DTW allows “elastic” matching of patterns by means of minimizing the differences between time series with added time intervals, thus allowing variability in time by means of DTW alignment.

We use the Matlab built-in function for implementation of DTW (https://uk.mathworks.com/help/signal/ref/dtw.html) that performs the pairwise transformation and synchronization of time series as described in Aribas-Gil and Muller (2014). In our context, the pairs of time series are the tree-ring ¹⁴C series to be compared. Our goal is to achieve the alignment of the time series with optimized matching of the entire sequence (Aribas-Gil and Muller 2014). Given two time series $$A = \left( {{a_1},\; \ldots ,\;{a_n}} \right)$$ and $$B = \left( {{b_1},\; \ldots ,\;{b_m}} \right)$$ , we construct an nxm matrix of Euclidean distances of all pairs of (i,j) between the elements of A and B. The warping path W is a set the matrix elements $$W = \left( {{w_1},\; \ldots ,\;{w_k}} \right),\max \left( {m,n} \right) \le K \le m + n - 1$$ , satisfying the boundary condition and conditions of continuity and monotonicity required of a path (and there are multiple paths of this kind). The optimal path is obtained as

$$DTW\left( {A,B} \right) = argmi{n_{W = {w_1},\; \ldots ,\;{w_K}}}\sqrt {\mathop \sum \limits_{k \,=\,1}^K {{({a_i} - {b_j})}^2}} $$

where A and B are two ¹⁴C series, a and b refer to the ¹⁴C values, w represents time weights that account for the optimized alignment, k is the index of the alignment pairs. The DTW algorithm measures similarity between time series where oscillations vary in speed and power. In principle, this method can evaluate time-series variance over a period of several decades to several thousand years. We hypothesize that the cosmic excursion effect could be detected in Δ¹⁴C series from years with extreme variance such as year-to-year spikes. The DTW approach aligns the selected ¹⁴C series that register similar signal within different time-response window. Analyzing the dynamics of time series of interest with shape-based DTW distance determines a close relationship between the time series exhibiting a similar patterns in ¹⁴C production rate presented in tree rings at different years.

RESULTS

The studied signature in the annual ¹⁴C series appears as a rapid increase of Δ¹⁴C, the peak is sustained for the next one to two years and is followed by a 2-yr decay (Figure 1a). The range of Δ¹⁴C in the signature varies from –10.59‰ and –7.83‰ (1052–53 CE) and –2.9‰ and –0.91‰ (1054–55 CE). The standard deviation (1σ) for the sequoias and Finnish pine series is typical between 2‰ and 2.09‰. The English oak series has a much smaller measurement error of 1.13‰–1.6‰. The increment of Δ¹⁴C at the first-year increase is between 7.5‰ and 3.8‰, which is a significant change. However, the peak signature occurs at different year: 1054 CE or 1055 CE. Brehm et al. (2021b) named the signature “1052 CE event” observed in the English oak and Full-Ring Sequoia (Eastoe et al. 2019). Its attribution to a possible SEP occurred during the Oort solar minimum (1021–1060 CE) has yet not been confirmed, while Terrasi et al. (2020) attributed a similar-structured signature with 1-yr shift from three ¹⁴C series possibly to γ-rays from the Crab Nebula supernova (SN1054), an unusually weak solar minimum or a SEP.

We applied DTW to compare the 1054 CE event versa the 1052 CE event in pairing ¹⁴C tree-ring time series over the 30-yr interval: 1037–1067 CE. Firstly, the English oak ¹⁴C series was paired with four other ¹⁴C sets. Figure 2 shows the original series and globally aligned-signal series with the calculated Euclidean distance. Second, we aligned the ¹⁴C series of sequoia from California (EW, LW, and Full Ring). Figure 3 shows the best aligned series from this test. Finally, the sequoia and Finnish pine ¹⁴C series were processed in the same way (not shown). DTW is often used as a distance measure (not a norm, as the triangular inequality is not fulfilled) in cauterization of time series sets (Kate 2016). The DTW analysis suggests that the signatures of 1054 CE event and 1052 CE event are matching and manifest the same pattern in the studied time series, but they do not coincide in time. Moreover, the ¹⁴C time series from Finnish pine has the closest relation to all series, while the English oak and LW Sequoia are most remote time series (Table 1). Instead of a “rigid” comparison, DTW allows “elastic” matching of signals with minimization of the differences between time series via added time intervals, thus allowing the alignment of variance. Sequoia ¹⁴C series of EW and LW from the same ring demarcate two different Δ¹⁴C patterns that can be aligned by DTW (Figure 3). The difference in the ¹⁴C variance of these series possibly results from the different time scales. Early wood is mostly responsive to May–June moisture driving by the snowmelt, while the late wood formation occurs in July–August extremely dry season (Hughes and Brown 1992).

Figure 2 DTW-matching of 1054 CE event/1052 CE-event signal between the English oak (blue line) and other (red line) studied locations and AMS faculties: Isotope Climatology and Environmental Research Centre, Debrecen, Hungary (ICER), Center for Isotopic Research on the Cultural and Environmental heritage University of Naples, Italy (CIRCE), NSF-Arizona AMS facility, University of Arizona, USA (Arizona), Laboratory of Ion Beam Physics, Swiss Federal Institute of Technology in Zürich, Switzerland (ETHZ). In each of four blocks of the figure, the top panels display the original series to be compared for the interval 1038–1068 CE. The low panels show the best global alignment of the signal between ¹⁴C annual series with added time points. (See online version for color figures.)

Figure 3 DTW-matching of 1054CE event/1052 CE event signal between Sequoia series from two locations in California and two AMS labs. The top panel displays the original series, and the bottom plot denotes the best global alignment of the signal in the paired ¹⁴C annual series with added time points. (See online version for color figures.)

Table 1 Euclidean distance between DTW paired ¹⁴C series. Seq refers to Sequoia.

Analysis of the lagged cross-correlation for the ¹⁴C series suggests that there is a two-year lag between the English oak and California EW-LW sequoia series (Figure 4, right plot). Perhaps, this two-year offset is caused by cross-dating nonconformity between these two tree-ring chronologies. It is surprising that the 1052 CE event signal showed up in other California sequoia record since its chronology matched well with the EW and LW sequoia records with the 1054 CE signal. DTW analysis demonstrates that ¹⁴C variance amplification at both 1052 CE and 1054 CE is caused by the impact of a single event. The inconsistent fingerprint of the 1054 CE event in time across the space may result from cross-dating nonconformity, regional (local) flux of old carbon at the northern site (Finland) due to intensified permafrost melt and ancient CO₂ release when the tree grew during the Medieval Warm Epoch (Feng et al. 2013; Wild et al. 2019; Zhang et al. 2021) and/or large latitude gradient (e.g., Burr 2007; Büntgen et al. 2018; Uusitalo et al. 2018). Clearly, the 1052–1054 CE signature is a unique increase in ¹⁴C production rate and warrants further investigation.

Figure 4 Cross-correlation of considered time series at different lags. Left panel: cross-correlation between Finnish pine and EW Sequoia (blue line) and between Finnish pine and LW Sequoia (red line). Right panel: cross-correlation between English oak and EW Sequoia (blue line) and between English oak and LW Sequoia (red line). The sharp peaks denote the lags at which the time series have the highest cross-correlations confirming the DTW analysis and the shift in the timing of the ¹⁴C events. (See online version for color figures.)

CONCLUSIONS

Our first-time application of a DTW approach to ¹⁴C annual datasets has shown that it is possible to scale and match similar patterns in the Δ¹⁴C variance with the signal placed chronologically apart and originated by an astrophysical impact (γ-ray or solar protons). DTW analysis effectively measures the distance with respect of signal separated in time. Temporal separation of the signal can be attributed to complex cross-dating of various wood sources and continues progress in the development of multimillennial tree-ring chronologies. Another source of transient errors can arise from instrumental bias and occur during ring sampling, cellulose processing, and digitization of datasets.

Whatever the reason for the chronological divergence of signals, DTW can identify divergencies of the discording signals. Our DTW analyses demonstrates an effective way of determining the distance between individual ¹⁴C series (points in space) with focusing on the similarity of their temporal (not chronological) behavior. In this case study, we cannot attribute the observed ¹⁴C signature between 1052/53–1056 CE to either SN1054 or a possible SEP event. Moreover, this signature and the offset in the ¹⁴C production rate should be revisited and investigated with larger spatial datasets.