Large explosive volcanic eruptions cover wide areas in thick pyroclastic deposits and release volatiles into the atmosphere that can perturb climate for years1,2,3. Although the impacts of large volcanic eruptions can be far reaching, they vary considerably. The impact on climate is linked to the amount of sulfur injected into the stratosphere, but models show that the latitude and timing of the injection have implications for the longevity of the sulfate in the stratosphere and thus also control their effects2,4. Despite a wealth of detailed models, there are only a few large eruptions for which there is enough data on the aspects thought to control climate impacts, and even still there are considerable gaps in understanding and large uncertainties5. Quantifying the volume of magma erupted, volatiles released and plume height of a variety of past eruptions is critical to provide empirical evidence to further interrogate the relationship between explosive eruptions and their climate effects, as well as testing the accuracy of dispersal models.

Sulfate layers preserved in the polar ice cores provide a detailed chronology of volcanic eruptions in the last few thousand years, many of which have been attributed to particular eruptions using glass compositions that act as a chemical fingerprint6. This time anchored history allows for the impact of various eruptions to be assessed using paleoenvironmental records (e.g., tree rings and pollen7) and written records for more recent events. It is important that these impacts are linked to eruption parameters to further inform our understanding on what controls the impact on climate. A notable eruption for which we have good constraints on both the date and climate response is the 946 CE Millennium eruption (ME) of Paektu, an active caldera situated on the border of the Democratic People’s Republic of Korea (DPRK) and People’s Republic of China (Fig. 1). The tephra erupted during this event is found over large areas, including over Greenland that is more than 7000 km from vent8, suggesting that it was a large magnitude eruption of ~M6.7, but the climatic impact appears limited9,10. This is at odds with what is observed from other large eruptions (e.g., the 1815 CE eruption of Tambora, Indonesia11) so it is not clear whether the eruption parameters for this event have been accurately constrained. Here we combine tephra fallout thickness measurements with a new ensemble-based method as part of a dual step inversion, which involves the FALL3D atmospheric tephra transport and deposition model12,13, to constrain the magnitude of the eruption and the eruption parameters such as mass eruption rate, duration, and plume height. For the first time we model the different phases of the eruption which are known from the proximal and distal deposits. By coupling these new volume constraints with published petrological data on the volatile content14 we can update the estimates of the amount of the volatiles released into the stratosphere and ultimately help understand the limited climatic impact.

Fig. 1: Studied area.
figure 1

Studied area and locations of tephra observations used for the dispersal modelling, compiled from terrestrial exposures and marine cores, for the entire deposit and for the points where the two magmatic phases can be distinguished (see Supplementary Table 1 for further details and references). The map was created using the QGIS open software ( Paektu caldera, around Lake Tianchi, in the inset is from astronaut photograph ISS006-E−43366 acquired April 4, 2003 (ISS Crew Earth Observations experiment and the Image Science & Analysis Group, Johnson Space Center).

A large eruption from Paektu

The discovery of a fine and widespread volcanic ash (tephra) layer in northern Japan was the first evidence to suggest there was a recent eruption from Paektu volcano15, also referred to as Baekdu (South Korean), Baegdusan (Japanese), or Changbaishan (Chinese). This visible tephra was widely identified in sequences in northern Japan, including southern Hokkaido, northern Honshu, and the Sea of Japan15. Based on the decrease in tephra thickness to the east and the presence of K-feldspar, consistent with observations from pyroclastic density current (PDC) deposits around the volcano16, Paektu was identified as the source by Machida and Arai15. This distal tephra layer was labelled B-Tm following Japanese tephrostratigraphy conventions, with B referring to Baegdusan (Paektu) and Tm relating to the type locality for the deposit at Tomakomai, Hokkaido15. Since these initial discoveries, the B-Tm tephra layer has been identified in numerous marine, lacustrine and archaeological sequences across northern Japan, northeast China, and coastal regions of Russia (see McLean et al.17 and references cited in Supplementary Table 1). In addition, glass shards with B-Tm geochemical compositions have been identified as a non-visible layer (cryptotephra) in ice cores from Greenland8, over 7000 km from its source. The glass compositions of the tephra deposits across Japan verify that the occurrences are indeed from the same eruption of Paektu volcano and have correlated the B-Tm tephra to the Millennium eruption (ME) deposits18.

Geological history of Paektu volcano

Volcanism at the intraplate Mt. Paektu (42.00°N, 128.05°E, 2750 m a.s.l.) is fuelled by the upwelling of mantle material. Initial basaltic fissure eruptions commenced at the location ~28 My ago and the volcanic edifice started to develop at ~4 Ma with activity divided into three stages. A basaltic shield volcano developed in Stage 1 and more explosive activity began in Stage 2 at around 1 Ma and built the trachytic-rhyolitic stratovolcano19. The Cheonji (Tianchi in Chinese) caldera-forming Stage 319,20 generated several large eruptions in the last 80 ka21. Deposits are found in distal records testifying to multiple explosive eruptions during this most recent stage22,23,24, which have spanned a range of magnitudes from small vulcanian eruptions, such as the 1903 event25, to the Plinian Millennium eruption26.

Precise date of the Millennium eruption

Radiocarbon dates of tree remains found with the B-Tm ash initially indicated the eruption occurred between 794–1192 CE15. The date of the tephra was shown to be consistent with that of the deposits close to the source, which was subsequently constrained further over the years using radiocarbon measurements of organic material recovered in the ME deposits around the vent. A precise wiggle-matched radio-carbon age of 946 ± 3 CE was obtained from a tree felled by the eruption that was recovered from the PDC on the western flank of the volcano9. Further confirmation that the eruption occurred in 946 CE was established by Oppenheimer et al.10 by obtaining radiocarbon measurements and identifying the 775 CE cosmogenic event27 across a similar tree stump recovered from the same location within the deposits. The B-Tm tephra is also found in Greenland ice cores and a date of 939–940 CE was assigned to the eruption using the Greenland Ice Core Chronology8. Subsequent research established the GICC05 ice core timescale was offset by −6 years28 so the ice core date is consistent with that obtained using the tree stump.

Written records from across the region document phenomena consistent with a large volcanic eruption in both 946 and 947 CE, with references such as the “roll of drums within the sky” in 946 CE (record from the Koryǒ dynasty, Korean Peninsula29,30), “white ash falls as snow in the night” in Nara, Japan, on 3rd November 946 CE10,30,31, and two records from 7th February 947 CE that noted “sound in the sky, like thunder”30,32. These historic accounts and the tree and ice records indicate that the eruption certainly started in 946 CE, with the further historical accounts suggesting there may have been another eruption phase or other activity in early 947 CE.

Proximal deposits of the Millennium eruption

The Millennium eruption stratigraphy around Paektu has been the focus of numerous studies over the last three decades21. The studies have been focussed on the deposits on the separate sides of the border, with seemingly no research group working on those from both China and DPRK. Various pyroclastic units are identified around the volcano but their attribution to a particular eruptive event had remained somewhat ambiguous. Both Horn and Schmincke33 and Pan et al.21 indicated the ME deposits had both a comendite and trachytic phase. However, Horn and Schmincke33 noted that the comenditic component was the most dominant and only ~5% of the melt erupted was trachytic in composition, found in mingled clasts within the deposit. Their ratio of comendite to trachyte is inconsistent with the similar thickness measurements of the comenditic and trachytic fallout units recorded by Pan et al.21, and at odds with the distal records that reveal that up to 50% of the glass shards are trachytic in composition17. Pan et al.21 interpreted that all the proximal units labelled Baguamiao, Millennium and some others, attributed by other researchers34,35 to later eruptions, were all phases of the ME. As such, Pan et al.21 labelled the deposits associated with the 946 CE eruption the Generalized Millennium Eruption, here referred to as ME, and note that the sequence includes a light grey and dark grey coloured comenditic fallout and PDC and a much darker coloured trachytic fallout and PDC. Both comenditic and trachytic fallout phases are decimetres in thickness in sections ~30 km east of the vent. The comenditic PDC is more than 10 m thick at sections <15 km east of the vent and it is partially welded in places21,33. At least 0.4 m of a trachytic PDC was deposited after a fallout deposit from the same magma, but it has been eroded in most sections suggesting that it is a minimum thickness for the deposit21. The observed deposit thicknesses both proximally and distally suggest that the comendite and trachyte fallout phases were similar in size, but the comenditic PDC was greater in volume relative to the trachytic PDC.

A lahar deposit was observed by Pan et al.21 between the two phases of the ME at a locality ~15 km east of the volcano. They suggested that this could reflect a hiatus of a few wet months, which is consistent with historic accounts that record phenomena commonly related to volcanic eruptions from 3rd November 946 through to 7th February 947 CE21. Although the distal sedimentary records seemingly indicate a single layer15, the sedimentation rates at the sites are not high enough to record detectable sedimentation between deposits of the different phases if they were indeed separated by months. In addition, current ice core sampling routines also mean that datasets are unlikely to show a clear separation in the different phases.

Tephra fallout volumes were recently reassessed by Yang et al.36 based on known proximal and distal thicknesses of the deposit. They dealt with the entire deposit thickness (both comenditic and trachytic) and estimated that the fallout equated to 13.4 to 37.4 km3 Dense Rock Equivalent (DRE) of magma36. The PDC volumes associated with the eruption were estimated by Horn and Schmincke33 to be between 5.3–7.6 km3 DRE (12.3 to 17.5 km3 bulk volume) based on assumptions of their extent in China. Recently, Zhao et al.37 mapped the PDC deposits in China and estimated they accounted for ~ 3 km3 DRE (7 km3 bulk volume). These previous estimates were reconciled by Yang et al.36 to constrain the total extra-caldera PDC volume to ~4.1–5.7 km3 DRE (9.3–13.0 km3 bulk) and estimated up to ~2.1 km3 DRE (maximum of 4.9 km3 bulk tephra) of intra-caldera PDC. These estimates indicate that the bulk volume of the entire ME deposit is between 40.2 and 97.7 km3, which equates to 17.5 to 42.5 km3 DRE magma assuming a tephra deposit density of 1000 kg/m3 and a magma density of 2300 kg/m3 that is appropriate for rhyolite. Nonetheless, these represent two different magma compositions and the ratio of comendite to trachytic magma was not quantified.


Proxy records of past climate such as the maximum Latewood Density of tree rings, which is sensitive to summer temperatures7, from a range of Northern Hemisphere locations, including China, do not show noticeable changes following the Millennium eruption10. Furthermore, the non-sea salt sulfur (nssS; i.e. that deposited from volcanic emissions) recorded in the Greenland ice core is ~30 ppb at 946 CE, roughly double the background levels, with no perturbation in nssS in Antarctic records28. These nssS are less than half that recorded in Greenland (80 ppb; 80–130 ppb in Antarctic ice cores) following the 1815 CE eruption of Tambora, Indonesia, but the latitudes are very different which would have affected sulfate deposition28. This muted climatic response indicates either low S volatile concentrations or tropospheric plume heights.

New insights into the Millennium eruption

Here we use all the published thickness and compositional data of the deposits that are found over an area >700,000 km2 east of the volcano (see Fig. 1 and Supplementary Table 1) with a dual-step inversion method to model the tephra dispersal from the two phases of the Millennium eruption. A new ensemble-based inversion modelling strategy38 was used to gain insights into the eruption dynamics (e.g., mass eruption rate and column height) and estimate the volume of the comenditic and trachytic magma erupted during the ME. We then use these total volume estimations with existing data on the volatile compositions of the melts14,33 to estimate the likely total volatile release. The probabilistic nature of the ensemble-based method38 used for solving this inverse problem allowed us to properly assess the uncertainties.

Results and discussion

Tracing the dispersal of the comenditic and trachytic phases

Although the comenditic and trachytic phases of the ME deposit are visually distinct in proximal locations within 40 km of the volcano, at distal locations (more than a few hundred kilometres from the vent) the two phases cannot be distinguished visually15. However, since the melts are compositionally distinct the glass chemistry can be used to establish which phase is present at a site. Many distal sites have deposits from both phases, and the percentage of which phase is present can be estimated based on the proportion of glass shards of each composition17. We used this published glass chemistry to attribute a thickness to each phase at distal sites where the tephra thickness is also reported. This assumes that the glass compositions analysed are representative, which may not be the case, but given that they are similarly easy to analyse there is no likely bias. The B-Tm ash in some distal records does not form a visible layer, with the layers (termed cryptotephra) being identified through density extraction techniques39. McLean et al.17,40 carried out detailed cryptotephra analysis of the Lake Suigetsu sediment cores (central Japan, ~950 km from Mt Paektu) and noted that the B-Tm is only visible in some core sections and has 1 mm in thickness. The tephra corresponds to shard concentrations of 25,000 shards/gram dry sediment in the 1 cm (core depth) sample40. Based on these measurements, we have assumed that 25,000 shards/gram of dry sediment equates to 1 mm in thickness and used this to assess likely tephra thickness where only shard concentrations are known (see Supplementary Table 1).

We have 23 locations for which we know thicknesses for both the comenditic (first) and trachytic phases of the ME deposit17,21,40. These data points were used for the inversion (details below), and the solutions related to each eruptive phase then used to compare and validate the results with the total thicknesses of the deposit for which more points are available (n = 83; see Fig. 1 and Supplementary Table 1).

Solving an inverse problem using an ensemble-based method

To estimate Eruption Source Parameters (ESPs) from observed deposit data we used a dual step inversion method. The first-step inversion used a fast semi-analytical code, such as PARFIT-2.3.1 (, to identify suitable wind conditions and ranges of ESPs for the subsequent ensemble-based inversion. For this purpose, we considered more than 18,000 daily-averaged wind profiles near the eruptive vent location (Supplementary Table 1) over a period of 50 years (1961 to 2010), which were obtained from the ERA5 Reanalysis41.

We solved three separate inverse problems, one for the comenditic phase, one for the trachytic phase, and another for the total deposit (two phases together; Supplementary Table 1). For each case we used the weighted least square method, considering both proportional and statistical weights (i.e. \({\theta }_{1,i}\propto 1/{{T}_{i}}^{2}\) and \({\theta }_{2,i}\propto 1/{T}_{i}\), being \(\theta\) the weights and \(T\) deposit loads42), and allowed wide ranges of ESP and meteorological conditions. In this step, we considered only the tephra load at distances >40 km from the vent where the comendite and trachyte fraction have been estimated (see Supplementary Table 1; proximal outcrops were excluded as the model is oversimplified at these distances43). The results of this step generated the ranges of input needed for the second step (see “Methods”).

The results of the first step inversion are reported in Table 1. For the comenditic phase, the total erupted mass (TEM), expressed as DRE, ranges from 16 to 20 km3 and column height is about 22–26 km above vent level (a.v.l.), whereas, for the trachytic phase, the volume ranges from 17 to 36 km3 DRE with a column height from 26 to 35 km (a.v.l.). Using the dataset consisting of all measurement points available at a distance from the volcano greater than ~40 km, we found an erupted magma volume of about 15–19 km3 DRE and a column height ranging from 38 to 45 km. The wide ranges obtained for the TEM and column height, and the fact that the total volume is smaller than each individual phases, reflects the intrinsic uncertainties in the solution of the inverse problem with simplified analytical models such as PARFIT, which are not well constrained due to the scarcity and sparsity of the data44,45. Furthermore, solving the column height and wind intensity with only thickness data is not ideal46,47. For these reasons, these first-step ranges were used as a starting point to find an inverse problem solution using the second ensemble-based method.

Table 1 Results of PARFIT tephra dispersal inversion.

In the second step, for each optimal meteorological configuration (Table 1) we performed six ensemble runs using the FALL3D-8.2 dispersal model38,48 considering an ensemble size of 128 members. The best meteorological configurations (start date reported in Table 1) were used to drive the dispersal model for each ensemble run. This approach provides an ensemble of deposit mass loading field estimations (first guess). Subsequently, observations are assimilated to reconstruct the deposit mass loading (analysis) using the GNC method described by Mingari et al.48. The deposit was reconstructed for the individual phases considering the six meteorological configurations and assimilating the 23 observations (assimilation dataset) corresponding to each phase. The total tephra deposit is obtained as the sum of the comenditic and trachytic deposits. The total deposit resulting from every combination (36 possible configurations) was evaluated using different metrics computed using the full dataset of 83 measurements of total deposit thickness (independent validation dataset).

The root-mean-square error (RMSE) was used to measure the differences between the values observed and the analyses interpolated at the sampling sites. For this purpose, we used the weighted version of RMSE and other statistical indicators, as explained in the Methods section. An overview of all used statistical parameters is reported in Supplementary Table 2.

The optimal configuration, that minimises the Confidence Ratio and Composite index (that is on average all the other metrics), corresponds to the deposit reconstructed using the meteorological datasets starting on 23 September 1986 for the comendite phase and 30 October 1992 for the trachyte phase (see Supplementary Table 2). Similarly optimal solutions for the are comendite phase were obtained with the meteorological conditions starting on 23 September 1986, and additional optimal metrological conditions for the trachytic phase were those observed on 21 April 1994 (RMSE and MAE) and 29 November 1978 (Bias and SMAPE) (see Supplementary Figures). In our approach these represent the most compatible meteorological conditions scanned across the ERA5 dataset. The reconstructed deposit fields for the comendite and trachyte phases and for the total deposit are shown in Fig. 2.

Fig. 2: Reconstructed tephra deposits.
figure 2

Reconstructed deposit thickness contours for the comendite (a), trachyte (b) eruptive phases, and for the total deposit (c) of the Millennium eruption (ME). The maps were created using the open software “Cartopy: a cartographic python library with a matplotlib interface” (

Figure 3 shows measurement and analysis data from the assimilation datasets (23 measurements for each eruptive phase) and for the validation dataset (83 measurements for the total deposit). Note that a RMSE close to 1, like in the presented case (Supplementary Table 2), means that deviations between measurements and analysis are similar to the observation errors, i.e. \(\left|{y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right|\sim {\epsilon }_{i}\). On the other hand, a positive bias (0.39–0.40) for the optimal configuration means that the analysis tends to underestimate the observations, but the bias is within the observation error range (\(\left|{{{{{{\rm{Bias}}}}}}}\right| < 1\)). Finally, more than 68% of the observations are within a factor 2.4 (that is one standard deviation for a lognormal distribution). The other two optimal solutions are reported in the Supplementary Figures.

Fig. 3: Comparison models and observations.
figure 3

Quantitative comparison between observed and modelled (analysis) deposit thickness of the comendite (a) and trachyte (b) eruptive phases of the Millennium eruption (ME) using the assimilation dataset (23 measurements for each eruptive phase). In addition, the comparison between the total deposit thickness and the validation dataset (83 measurements) is also shown (c). Note that zero-valued observations for the comendite (P82 and P83, Supplementary Table 1) and trachyte (P34, P68, P72, P74, Supplementary Table 1) phases are not shown in these log-log plots (a) and (b), respectively. The plot corresponding to the total deposit (c) includes the full dataset of 83 data points. Errors are estimated in accord to Mingari et al.38 assuming a minimum value of 30%. The plots were created using the open software Matplotlib (

The source term was inverted using the procedure based on the GNC method described by Mingari et al.48. This method provides a set of weight factors (\({w}_{i}\)) for every ensemble member (see “Methods” section). According to the inversion procedure, the first-guess emission source term for the ith ensemble member (\({s}_{i}\)) is then rescaled according to \({s}_{i}\to {w}_{i}{s}_{i}\). The resulting total source term (in kg s−1 m−3), defined as the weighted sum of the individual source terms:

$$s={\sum }_{i}{w}_{i}{s}_{i}$$

is represented in Fig. 4 in terms of the linear source emission strength \(\delta s\) (in kg s−1 m−1), i.e. \(\delta {sdA}\) being \({dA}\) the area of the cell grid. The emission rate (i.e. \(\delta s\) vertically integrated) is also represented in Fig. 4 (dash-dotted blue line) along with the cumulative erupted volume (solid black line). The total erupted volume for the fallout phases according to the inverted source term was ~7.2 and 9.3 km3 DRE for the comendite and trachyte phase, respectively (see Fig. 4 and Table 2), with a mean error of about a factor 2.4, that is a volume between 3 to 17 km3 for the comendite phase and between 4 and 22 km3 for the trachyte phase.

Fig. 4: Time evolution of the eruption columns.
figure 4

Time evolution of the emission source profiles derived for the comenditic first phase (a) and trachytic phase (b) of the Millennium eruption (ME). The plots were created using the open software Matplotlib (

Table 2 Eruption parameter model output.

Volume and eruption parameters

The results from this new approach indicate that fallout (Plinian and co-ignimbrite) from each of the compositionally distinct fallout phases erupted roughly the same volume of magma, that is about 7–9 km3 DRE (Table 2). This indicates that around 16 km3 of magma was dispersed by the two fallout phases, but, considering the uncertainty, such a value could have been as low as 7 km3 and up to 39 km3. For comparison, Horn and Schmincke33 calculated a volume of 23 ± 5 km3 DRE for the comendite phase, and recent re-estimates by Yang et al.36 for the fallout for both phases were estimated to be between 13.4 and 37.4 km3 DRE. The tephra fallout volumes need to be added to the PDC estimates, which are 4.1–5.7 km3 DRE for those deposited outside the caldera and an additional ~2.1 km3 DRE for PDC emplaced in the caldera36 to estimate the total volume erupted. Our new fallout volumes combined with those estimated for the PDCs indicate the total volume of magma erupted during the ME was about 23 km3 DRE, with at least 13 km3 and up to 47 km3 DRE erupted, corresponding to a M6.5 to M7.0 eruption. The average volume (23 km3) is similar to the amount of material that missing from edifice (22 km3) that would have collapsed into the void generated by evacuating the magma. The amount of material displaced was estimated using 2.8 km as the mean radius of the caldera, obtained averaging the maximum distances along the different directions, and ~900 m height based on the difference between the maximum elevation and the bottom of the lake. The agreement between these values suggests that the optimal estimations are more realistic than the upper bounds.

Our results indicate that each of the two phases lasted less than a day (~15 and 20 h; Fig. 4) and peak mass flow rates were estimated to be about 4 × 108 kg/s, which are typical for Plinian eruptions49. Similar values were estimated from observations of the climatic phase of the 1991 Pinatubo eruption50,51. These high mass flow rates would have fed umbrella clouds51 and sustained columns that, for both phases, reached ~30–40 km (Fig. 4 and Table 2), which are well above the tropopause height of ~12 km at this latitude. This indicates that almost all of the eruptive mixture, including volatiles released during the climatic phase of the eruption, were injected into the stratosphere.

Volatile release and impact

The petrological method52,53 was previously used with the comendite volumes33 (23 ± 5 km3 DRE) with volatile contents of the melt nclusions (MI) and matrix glasses to estimate the amount of S, Cl and F released. Measurements of the volatiles in MI and matrix glasses from the comenditic magma have been made by Horn and Schmincke33, Iacovino et al.14 and very recently re-analysed by Scaillet and Oppenheimer52.The F and Cl contents of MI and matrix glasses cover a similar range14,33, suggesting the melts were probably not saturated in F or Cl, and loss of these volatile phases could be negligible. Iacovino et al.14 modified the petrological approach used to also estimate the syn-eruptive degassing to quantify the total pre-eruptive volatile loss associated with the comenditic magma. This approached assumed the comendite erupted was linked to the trachyte erupted via fractional crystallisation, which is consistent with regression modelling of the major elements. They compared the observed volatile contents relative to incompatible elements (U in this case) to values estimated from fractional crystallisation models to estimate the volatiles lost from the melt during crystallisation14. Their data indicates that the pre-eruptive volatile release from the comenditic magma could have been substantial.

If we consider comendite magma volumes calculated with the new method presented here, which range from 3 to 17 km3 DRE, and the mean volatile contents measured by Iacovino et al.14 in the MI and matrix glass, we can rescale the volatiles that were removed from the melt prior to eruption to be between 5 and 30 Tg S, 6–32 Tg F, and 2–15 Tg Cl. While our scaled estimates for the syn-eruptive S released into the stratosphere is 0.5–2.6 Tg (1.0–5.2 Tg SO2). Nonetheless, these estimates do not consider the volatile release associated with the trachytic melt (4–22 km3 DRE) as there are no published volatile data for trachytic matrix glass or MI in crystals exclusively associated with the trachytic magma. The new volumes suggest the pre-eruptive volatile release would have been smaller than previous reported by Iacovino et al.14. The volatile elements released prior to the eruption are likely to have been sequestered by the aquifers and hydrothermal fluids or passively degassed, and it is expected that most of these volatiles did not make it into the stratosphere. Nonetheless, it appears that some of the pre-eruptive S loss from the 1991 Pinatubo magma made its way into the stratosphere (which is higher at the more tropical latitude; ~18 km) as the satellite measured S content in the stratosphere was greater than the syn-eruptive load estimated by the petrological method53.

The lack of recorded climatic impacts for the eruption7,10 suggests a low syn-eruptive volatile release, which is consistent with the above scaled syn-eruptive yields despite not including any release from the trachytic melt. The S yield is similar to the ~2 Tg estimated from the ice core non-sea salt sulfate record54, which represents both the comenditic and trachytic phases. The modelled climate impacts of high latitude eruptions with S loads of <5 Tg are shown to be negligible, especially those in winter as insolation affects the removal of from the stratosphere55. Nonetheless, recent simulations have shown that stratospheric sulfur injections in the extratropics spend longer in the stratosphere than those from low latitudes, and suggest that cooling is more pronounced, but confined to the hemisphere in which it is injected4. Stratospheric halogens also exacerbate the radiation forcing of the sulfate56, but measured values suggest the syn-eruption halogen release during the ME was low. In addition, the halogens are probably scavenged in the lower plume with only a few percent making it into the stratosphere57.

Despite the lack of any significant climate impact, there would have been quite a substantial regional impact linked to the deposition of the thick deposits, lahars, and gas release. It is likely that the eruption led to acid rain, which would have affected crops, livestock, and high F potentially poisoned local freshwater supplies in the downwind regions.


We applied a dual step strategy for solving an inverse problem aimed at reproducing the observed tephra deposit thicknesses generated by the ME of Paektu volcano. The first step uses a fast efficient code for tephra sedimentation based on an analytical solution (PARFIT). The second step considers a recently proposed ensemble-based inversion method with an High Performance Computing (HPC) numerical tephra dispersal model (FALL3D) to reconstruct the eruption dynamics and the volume of magma erupted during the comendite and trachyte phases. Such estimates of past eruption volumes and parameters, and their uncertainties, are key to understanding the impacts on climate. The new inversion method allows us, for the first time, to estimate the evolution of the mass eruption rate with time during the two main phases, and not only the time-averaged values, as commonly done. The total erupted mass of the two Plinian phases are similar, with best estimates of ~7 km3 DRE for the comendite phase and ~9 km3 DRE for the trachyte phase. Considering previous estimations for PDC deposits and intra-caldera filling, we can estimate a total magma volume of about ~23 km3 DRE for the eruption, which corresponds to a M6.7, consistent with other recent estimations. The determination of the contribution of the two different magmas could be used to better estimate the amount of released sulfur once volatile data from the trachyte phase (~9 km3 DRE) is available. The estimations for the comendite phase made with the proposed method are about one third smaller than the reference values used in the literature. A simple scaling suggests between 0.5 to 2.8 Tg syn-eruptive sulfur release, which is in agreement with S loads estimated from ice-core records, and consistent with negligible effects recorded in climate proxy records. This research highlights that in eruptions that tap compositionally distinct melts, with different volatile histories, it is important to understand the temporal relationship between changes in melt chemistry and eruption dynamics so that the likely impact of past eruptions can be accurately assessed.


Computational methodology

Tephra dispersal model

Tephra dispersal simulations were performed using the latest version (8.2) of the FALL3D-8.2 model12,38,48. FALL3D-8.2 is an Eulerian model for atmospheric transport and deposition based on the solution of the so-called the Advection-Diffusion-Sedimentation (ADS) equations12. This code has been significantly refactored over the last years, resulting in major improvements to the scalability and performance12. In addition, the model physics and numerics have been revised in this version12,58 to allow aerosol and radionuclide dispersal modelling. The updated model includes new coordinate mapping options, a more efficient and less diffusive solving algorithm, and better memory management and parallelisation strategy based on a full 3-D domain decomposition, suitable for HPC applications. A set of case studies that use the model, including the simulation of the dispersal of long-range fine volcanic ash and volcanic SO2, and the deposition and dispersal of tephra fallout and radionuclides, were recently presented by Prata et al.58.

In addition, the capabilities of FALL3D have been extended by making it possible to perform an ensemble of model runs. Ensemble modelling allows model uncertainties, associated with poorly constrained input parameters or errors in the underlying meteorological driver, to be characterised and quantified. Furthermore, it is possible to explore different ensemble-based data assimilation techniques or inversion modelling methods based on ensemble approaches in order to produce improved simulations of volcanic clouds and tephra deposits consistent with real observations.

Inversion method

We used a novel strategy to model the tephra dispersal of the Millennium Eruption that improves, from a computational point of view, the approach of Costa et al.59, which was based on a set of several hundreds of simulations of the FALL3D ash dispersal model12 and 3D time-dependent meteorological fields across the region of interest. This new approach involves two separate inversion steps. The first step uses a simpler analytical solution60,61 ( to identify compatible meteorological conditions that describe the general observed tephra dispersal and estimate the most likely ranges of the Eruption Source Parameters (ESP). These first inversion estimates are then used in the subsequent step built on the ensemble-based inversion algorithm recently proposed by Mingari et al.38 to further constrain dispersal and the ESPs. This dual inversion step method is computationally more efficient than that used by Costa et al.59 and thus, allows a very wide range of meteorological conditions and ESP to be explored and allows for more extensive analysis.

First step inversion

The first inversion is based on a semi-analytical solution of the mass conservation equation, which is not computationally intensive and allows tens of thousands of simulations to be explored on a simple desktop computer. We used the most recent version of the PARFIT software package (PARFIT-2.3.1 code: to solve an inverse volcanological problem, which consists of finding mean wind profile, column height, Total Erupted Mass (TEM), and Total Grain Size Distribution (TGSD), using thickness and grain-size measurements from available eruption deposit outcrops. PARFIT minimises the difference between the simulated and the real deposit loadings. The fitting is performed using a least-squares method which compares measured and calculated deposits thickness and grain-sizes. The function to be minimised42 is:

$${\chi }^{2}=\frac{1}{N-p}\mathop{\sum }\limits_{i \, = \, 1}^{N}{\theta }_{i}{\left({Y}_{{{{{{{\rm{obs}}}}}}},i}-{Y}_{{{{{{\mathrm{mod}}}}}},i}\right)}^{2}$$

where \({\theta }_{i}\) are weighting factors, N is the number of measurements, p is the number of free parameters, \({Y}_{{{{{{{\rm{obs}}}}}}},i}\) denote the observed ground loads (kg/m2) and \({Y}_{{{{{\mathrm{mod}}}}},i}\) the values predicted by the model. The choice of the weighting factors, \({\theta }_{i}\), depends upon the distribution of the errors42. To accurately constrain the input parameters, it is necessary to know thickness and grain-size values for a large number, N, of locations, such as Np. Without a large statistical set of observational data there is a risk of finding parameters that are incorrect, especially when considering limited portions of the deposit61 that represent the tail of the distribution relative to a particle class44,46. In order to reduce the degrees of freedom, avoiding these problems46,47, the TGSD was estimated following Costa et al.62.

Tephra dispersal simulations are performed using the HAZMAP model that is embedded in PARFIT. The HAZMAP program simulates the tephra deposit generated by the sedimentation of volcanic particles from a sustained column60,61. The user defines the ranges and resolution steps of the eruption parameters. This defines a multidimensional grid with combinations of all parameters. PARFIT performs a search on the grid for the minimum χ2 between the simulated deposit and the field data.

Real wind profiles are used in the PARFIT-2.3.1 model that can be assessed from an ensemble of daily winds over the investigated area. In this case, wind data were obtained from ERA5 Reanalysis dataset41 and converted to the required format using specific routines within the software package. Due to the enhanced computational efficiency, tephra dispersals driven by tens of thousands of daily-averaged wind profiles were investigated to identify the most compatible meteorological conditions within the range explored (50 years, from 1961 to 2010).

For this first-step inversion we considered thickness data associated with the comenditic and trachytic phases, reported in Supplementary Table 1, and searched for the best fit eruption parameters for each distinct phase (similarly to Marti et al.63) in the ranges of values shown in Table 3. Since there is extremely limited grain-size data, we estimated the TGSD using the Costa et al.58 parameterisation that is based on magma viscosity and column height. For this purpose, we considered a magma viscosity of about 107 Pa s for a rhyolitic composition14,17,19,64 and a plume height of 30 km, a typical value for Plinian eruptions49. We used the parameterisation proposed by Bonadonna and Phillips65 to characterise densities of particles from rhyolitic magma and used particle shapes that have been measured for other explosive eruptions66. Ash aggregation67 was considered using the parameterisation of Cornell et al.68. The resulting TGSD used at this step is reported in Table 4.

Table 3 Eruption parameter ranges for PARFIT.
Table 4 TGSD estimated according to Costa et al.62, and accounting for ash aggregation following Cornell et al.68, for the PARFIT first step inversion.

Second-step ensemble-based inversion

As mentioned above, the parameter ranges explored in this second step are derived from the results obtained in the first step. The second step inversion reconstructed the spatial distribution of tephra deposit thickness (i.e. mass loading) associated with the ME using the ensemble-based assimilation method proposed by Mingari et al.38. This algorithm provides a map of a tephra-fall deposit given a dataset of observations of a volcanic deposit at scattered points. To this purpose, an ensemble of model realisations is additionally required. Given a background ensemble of model states characterised by the state vectors \({x}_{i}^{b}\) (i = 1,…, m), representing m model realisations (ensemble members), along with a vector of observations \(\vec{{y}_{o}}\), the state estimate (analysis) is expressed as a linear combinations of the background ensemble:

$${{x}^{a}={w}_{1}x}_{1}^{b}+\ldots +{{w}_{m}x}_{m}^{b}$$

where the coefficients \({w}_{i}\ge 0\), i = 1,…, m are weight factors to be determined which depend on the observations. In our case, the model state vector \(x\) is assumed to be the two-dimensional deposit mass loading on a regular grid. In the limit case in which no observations are available, the ensemble members are uniformly weighted according to the constant factors: \({w}_{i}=\frac{1}{m}\) and, consequently, the state vector \({x}^{a}\) is simply the ensemble mean.

In order to determine the weight factors \({w}_{i}\), a cost function must be minimised in the subspace spanned by the ensemble members with the non-negative constraints \({w}_{i}\ge 0\). Let the vector model state \(\vec{x}\) be a linear combination of the corresponding ensemble member vectors expressed as in Eq. (3), i.e. \(\bar{x}={\sum }_{i}{{w}_{i}x}_{i}^{b}\). We look for the vector state minimising the following cost function:

$$J\left(y\right)\propto {\left(y-\bar{y}\right)}^{T}{P}^{-1}\left(y-\bar{y}\right)+{\left({y}_{O}-y\right)}^{T}{R}^{-1}\left({y}_{O}-y\right)$$

where the vectors \(y\) and \(\bar{y}\) are defined as \(y={Hx}\) and \(\bar{y}=H\bar{x}\), being \(H\) a linear observation operator which translates the model state into the observation space; \(P\) is the ensemble-based error covariance matrix computed in the observation space and we assume a Gaussian distribution with error covariance matrix \(R\) for the measurements. In this work, the observation operator interpolates the modelled mass loading into the sampling site and computes the deposit thickness assuming a typical bulk density for consolidated deposits44 of \({\rho }_{b}=1000\) kg/m3. This optimisation problem leads to a problem of non-negative quadratic programming that can be solved using an iterative approach. See Mingari et al.38 for further details.

The root-mean-square error (RMSE) was used to measure the differences between the values observed and the analyses interpolated at the sampling site:

$${{{{{{\rm{RMSE}}}}}}}=\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{{p}_{i}^{2}\left({y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right)}^{2}}$$

being \({y}_{i}^{{{{{{{\rm{obs}}}}}}}}\) the ith observation and \({y}_{i}^{{{{{{{\rm{an}}}}}}}}\) the corresponding analysis for each individual phase. Notice that if the non-weighted (\({p}_{i}=1\)) version of (1) is used, only very proximal data (i.e. largest deposit thickness measurements) makes a significant contribution to RMSE. Instead, in order to treat the deviations more evenly, the observation errors (\({\epsilon }_{i}\)) are used to define the weights according to \({p}_{i}=1/{\epsilon }_{i}\).

Similarly, we computed the bias using the following definition:

$${{{{{{\rm{Bias}}}}}}}=\frac{1}{n}\mathop{\sum }\limits_{i\, = \,1}^{n}{p}_{i}\left({y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right)$$

and the mean of absolute value of errors (MAE):

$${{{{{{\rm{MAE}}}}}}}=\frac{1}{n}\mathop{\sum }\limits_{i\, = \,1}^{n}{p}_{i}\left|{y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right|$$

The mean absolute percentage error (MAPE) is also useful:

$${{{{{{\rm{MAPE}}}}}}}=\frac{100}{n}\mathop{\sum }\limits_{i\, = \,1}^{n}\frac{\left|{y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right|}{{y}_{i}^{{{{{{{\rm{obs}}}}}}}}}$$

where yobs is a vector of 83 observations and yan the vector of the corresponding analysis interpolated to the sampling sites. However, such definition of the error is very sensitive to the observations having very small values (close to zero) and for this reason is preferable using the symmetric mean absolute percentage error (SMAPE), which is another accuracy measure based on relative errors used to evaluate the performance of the deposit reconstruction:

$${{{{{{\rm{SMAPE}}}}}}}=100\times \frac{2}{n}\mathop{\sum }\limits_{i=1}^{n}\frac{\left|{y}_{i}^{{{{{{{\rm{obs}}}}}}}}-{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right|}{\left|{y}_{i}^{{{{{{{\rm{obs}}}}}}}}\right|+\left|{y}_{i}^{{{{{{{\rm{an}}}}}}}}\right|}$$

We have also calculated the confidence ratio (CR), defined as the value discriminating the symmetric band on the log-log space of observations vs. analyses, containing 68% of the points represented in Fig. 3 (this percentage will correspond to a sigma if we assume a lognormal distribution). Finally, we considered a composite index (CI), defined as:


The values of the evaluation metrics described above are reported in Supplementary Table 2.

Ensemble modelling

The FALL3D computational domain was configured using a horizontal resolution of 0.23° (longitude) and 0.16° (latitude) and a domain size of \({n}_{x}\times {n}_{y}\times {n}_{z}\) = 150 × 150 × 60 grid cells. Numerical simulations used the meteorological fields estimated using the PARFIT-2.3.1 code. Among the different meteorological configurations selected in the first step procedure, the conditions corresponding to 25–29 January 1962, for the comendite phase, and 29 February–3 March 1976 for the trachyte phase, showed the best agreement with observations during the second step (i.e. when using the ensemble-based FALL3D; see Supplementary Table 2). The Total Grain Size Distribution (TGSD) was computed from the column height using the parameterisation proposed by Costa et al.62 considering 10 ash bins with particle diameters ranging between −2Φ and 8Φ. Moreover, an additional bin aggregate of diameter dagg = 200 μm was considered. Table 5 provides further details about the model configuration.

Table 5 FALL3D model configuration.

A background ensemble with 128 members was generated by perturbing Eruption Source Parameters (ESP), the horizontal wind components, and the aggregate density using either uniform or truncated normal distributions13,48. A Latin Hypercube Sampling (LHS)13,69 was used to efficiently sample the parameter space. The ensemble configuration requires defining a central value (i.e. results from the first inversion step), a perturbation range and a probability density function (PDF) for each parameter to be perturbed. The ensemble is built by randomly sampling the corresponding PDF. Table 6 provides a summary of the configuration used to define the background ensemble along with the list of the perturbed parameters and model inputs.

Table 6 Ensemble configuration.

The emission start time was uniformly sampled between t = 0 and t = 48 h, where t is the time after the simulation starts in hours, as detailed in Table 6. The emission source duration, \(\triangle t\), was sampled in the range from 4 to 10 h, which means that emission is only possible for \(0\le t\le 58\) h. Since we aim to simulate both phases of the ME (i.e. November 946 CE and February 947 CE events), we invert independently the two phases and consider the formation of the total tephra fallout deposit as the sum of the first and second eruptive event for validation purposes.

Source term inversion

FALL3D solves an almost linear problem with weak nonlinearity effects (e.g., due to gravity current, wet deposition, or aggregation). Consequently, a re-scaling of the emission source term according to \({s}_{i}\to {w}_{i}{s}_{i}\) leads to a deposit mass loading re-scaled correspondingly: \({{{x}_{i}^{b}\to w}_{i}x}_{i}^{b}\), where \({s}_{i}\) is the emission source term in kg m−3 s−1 for the ith ensemble member. This is the mass injected into the computational domain per unit time and depends on time and height above the vent. The analysed tephra deposit (mass loading) on the ground is described by Eq. (3) that gives:

$${x}^{a}={\sum }_{i}{x}_{i}^{b}$$

i.e., the analysed mass loading is a result from the contribution of multiple members of an ensemble defined by the first-guess emission source term \({s}_{i}\) rescaled by \({w}_{i}\).