The eVolv2k reconstruction provides a time series of volcanic stratospheric sulfur injection (VSSI) and latitudinally resolved stratospheric aerosol optical depth (SAOD). This reconstruction uses ice core records to estimate the timing and amount of VSSI and the Easy Volcanic Aerosol (EVA) forcing generator to reconstruct the spatiotemporal distribution of SAOD given the estimated eruption latitude, date and VSSI. Estimates of VSSI and SAOD from ice core sulfate measurements have a significant associated uncertainty . The eVolv2k reconstruction provides estimates of the 1-sigma random uncertainty in the VSSI estimates . These uncertainties represent uncertainties due to two sources: (1) the uncertainty in the ice sheet (Greenland or Antarctica) mean sulfate flux, based on the variation between the individual ice core records that make up the ice sheet composite, and (2) a representativeness error, corresponding to the error inherent in taking the ice sheet mean as a measure of the full hemispheric sulfate deposition. The latter source of uncertainty is estimated based on the variability in ice sheet average deposition for eruptions of fixed magnitude in aerosol model simulations . A total random error is computed by combining the measurement and representativeness error. Average estimated percent uncertainties in eVolv2k are 34 %, with values for eruptions of greater than 10 Tg S ranging from 14 % to 40 %. There is also uncertainty in the timing of eruptions for all but the relatively few ice core sulfate peaks that can be attributed to historically recorded eruptions. For unidentified eruptions, the year listed in the eVolv2k database represents the year in which excess sulfate is first detected in the ice cores, with an estimated uncertainty of ±2 years. This uncertainty includes the overall dating uncertainty of the ice core and uncertainty in the time lag between eruption and first deposition of sulfate flux to the ice sheets. To incorporate uncertainties in VSSI amount and eruption timing into the present work, we constructed an ensemble of VSSI time series from the default eVolv2k data. For each member of the constructed eVolv2k forcing ensemble (eVolv2k-ENS, Fig. S1 in the Supplement), the following steps were taken.

For each eruption individually, we perturbed the VSSI amount by a normally distributed random variable of mean zero and standard deviation of the reported VSSI uncertainty for that eruption. If the resulting value was negative, we set the value to zero.

Reconstructions of solar irradiance are based on cosmogenic isotope records of 14C and 10Be : 14C is oxidised to CO2 in the atmosphere and metabolised by trees to form tree rings (e.g. ), while 10Be attaches to atmospheric aerosols, which are removed from the atmosphere in different ways and subsequently deposited to ice sheets (e.g. ). They represent independent proxy records and are mostly consistent – especially on long timescales – but deviations during certain periods have been noted due to the different formation rates .

We include four different reconstructions of solar forcing in our analysis, resulting from either of these isotope records or combined versions, including the PMIP4 records PMOD, SAT10Be and SATC14 . The latter two are derived from the updated SATIRE-M model , with SAT10Be using 10Be isotope records as input, and SATC14 using 14C isotope records. PMOD uses only 14C as input , and represents an update from the older SEA model . PMIP3 includes seven different records , including, among others, the SBF record , which was used as the solar forcing record for running HadCM3, and SEA , which represents the maximum-amplitude solar forcing record. Apart from SEA and PMOD, all records represent small long-term solar forcing and share a similar amplitude. Due to the similarities in the low solar forcing records, we restrict our analysis to the PMIP4 records and, for completeness, include SEA, representative of the maximal (although unlikely) physically possible forcing (Fig. b). We thus sample two independent isotope records (10Be and 14C), combined with two independent reconstruction methods (SATIRE and SEA/PMOD).

The ensemble variance is dominated by the difference in the long-term amplitude (Fig. b) and is especially high during the Spörer Minimum (around 1390–1550 CE), the Wolf Minimum (around 1270–1340 CE) and the Maunder Minimum (around 1640–1720 CE) . Short-term changes are less uncertain and are caused by quasi-periodic oscillations ranging between 9 and 14 years , mainly as a result of discrepancies between the different isotope records . The variance is largest during the mid-15th century, coinciding with the large volcanic eruption in 1458 CE. hypothesised that large stratospheric volcanic eruptions may deplete the stratospheric 10Be isotope for around a decade, biassing the isotope records. This could partially explain the large amplitude of solar minima in the SEA record, a record based on 10Be only, which all coincide with periods of major volcanic activity.

We assess the model performance by comparing the residuals between the response model, tuned to HadCM3 and using HadCM3 forcings, and transient HadCM3 simulations to the standard deviation of HadCM3 control runs over 20- and 200-year sliding windows (Fig. ). For a perfect response model, which accurately reproduces the climate response to forcing in HadCM3, the residuals would represent the internal variability simulated by HadCM3 and thus match the spread of control variability. However, given that the response model is a simplistic simulation of large-scale climate and does not account for hemispheric transport, seasonal processes or non-linear mechanisms, we expect the residuals to also include model error and thus to exceed the control variability. Remaining internal variability in the HadCM3 ensemble mean due to the relatively small ensemble size could additionally increase the residuals.

At 20-year timescales the residuals mostly agree with the spread of control variability, especially for global simulations. However, in the NH, residuals are very large during the mid-15th century as well as around 1800, which are both periods of high volcanic forcing. The larger residuals around 1800 are also present in the global simulation. Both periods are subject to very large volcanic forcing, and thus it is likely that the response model may be too simplistic to accurately model the associated climate processes as simulated by HadCM3. We find a larger discrepancy for both global and NH simulations at the 200-year timescale, although the 90th percentile is within the range of control variability at most times. The residual clearly exceeds the control variability during the 17th century, which could be explained by a possible non-linearity in the solar and volcanic forcing during that period . As observed at the 20-year timescales, residuals are smaller for global simulations, indicating that model error is larger in the NH. This finding is unsurprising, given that regional and seasonal climate processes play a larger role for NH summer temperature as opposed to global annual temperature.

We compare the proxy reconstructions with response model simulations including the different solar forcing reconstructions. On centennial timescales the difference between simulations including SAT10Be and SATC14 is almost negligible, and both follow the proxy reconstructions closely after around 1300 CE (Fig. a and b). In contrast, the temperature response to PMOD and SEA forcing show very large variability, with particularly strong anomalies during the Wolf (around 1300 CE) and the Spörer (15th century) minima. Since the availability of proxy records decreases quickly before 1300 CE, and therefore reconstructions are much more uncertain , we use the period from 1300 to 1900 to calculate residuals between proxy reconstructions and response model simulations and compare to estimated observed and modelled internal variability. The residuals between the proxy reconstructions and the full response model ensemble show that even when accounting for volcanic and proxy reconstruction uncertainty the SATIRE forcings consistently generate the lowest residuals, and SEA consistently generates the highest (Fig. c and d), with very little overlap between them. For the Northern Hemisphere, the former shares a very similar distribution with the root-mean-square deviation of the CMIP6 control variability, although the distribution shows two peaks, generated by different proxy reconstructions. Given that the model error ranges around 0.03 K for NH summer simulations and 0.01 K for global annual simulations (Fig. ), we expect the response model to produce slightly higher residuals than the control variability. This suggests that SATIRE forcing is consistent with the reconstructions compared to the distribution of control variability. In contrast, both PMOD and SATIRE are inconsistent with the majority of CMIP6 control runs.

At the global scale, the CMIP6 control runs show a significantly lower variability than the residuals from all solar forcings. This is roughly consistent with the difference between HadCM3 and the response model run with HadCM3 forcing (Fig. ) and thus could be caused by model error or could indicate that global annual variability is slightly underestimated by CMIP6 models. PMOD is only consistent with the upper tail of the distribution, caused by the high-variability-type models, and residuals from SEA are pretty much inconsistent with CMIP6 control variability.

It is worth noting that the relative magnitude of residuals associated with the different solar forcings is independent of the temperature target and the reconstruction (Figs. S10 and S11). In addition, it is also independent of response model parameters such as climate sensitivity. While for the low climate sensitivity the differences between the temperature response to SATIRE, PMOD and SEA forcing decrease, SEA still produces consistently larger residuals than SATIRE forcing (Fig. S12). For the high climate sensitivity the differences increase (Fig. S13), but the order remains unchanged: SATIRE produces the smallest residuals, followed by PMOD and finally SEA forcing.

To quantify the effect of volcanic forcing uncertainty, we fix the solar forcing to SATC14, the default PMIP4 forcing . The volcanic ensemble generates a large variety of amplitudes in response to individual eruptions. At times, the spread between the ensemble members exceeds 1 K in the NH, indicating that large uncertainty of the volcanic forcing record progresses into the temperature response (Fig. a). The largest spread is found during the 1450s CE, and after the eruption of Laki (1783 CE). Accordingly, the magnitude of the residuals between response model simulations and proxy reconstructions also varies significantly throughout the millennium (Fig. b). In the NH, the 75th percentile of the residuals obtained from eVolv2k-ENS agrees with the spread of residuals obtained from eVolv2k, which represent the proxy reconstruction uncertainty, at most times. The proxy reconstruction uncertainty tends to be larger during years of low volcanic activity, and decreases around the eruptions, except for the eruption of Samalas in 1257 CE. The residuals of eVolv2k-ENS are largest and show the largest spread during periods of high volcanic activity. The residuals peak around the eruption of Samalas for both eVolv2k and eVolv2k-ENS, which also features the largest spread of results. This reflects the large uncertainty around this eruption with regard to both forcing and reconstruction. The mismatch between the model response to the eruption of Samalas and the response shown by proxy reconstructions is a well-known discrepancy. It reflects a mismatch between the signals in ice cores, the source data of the volcanic forcing in model simulations, and tree rings, the source data of the proxy reconstructions . Additionally, the response model and the proxy reconstructions match better going forwards in time, with a particularly poor agreement prior to ca. 1300 CE. This reflects the decreasing data availability and quality as we go back in time. Prior to 1300 CE only few high resolution MXD records are available, which reduces the confidence in the proxy reconstructions for the first 3 centuries and hence in the proxy response to the eruption of Samalas. Nevertheless, given the large ensemble spread between residuals from the eVolv2k-ENS members (around 0.2 K) around this eruption, volcanic forcing uncertainty could additionally explain a part of this discrepancy. A similar observation holds for the mid-1450s CE eruptions ; however, in this case the spread of reconstructions is smaller.

While volcanic forcing uncertainty provides a possible contribution to the discrepancy of these specific eruptions, it is unlikely that it can be fully explained by it – at least not with such a simple model. Proxy biases, reconstruction uncertainty and model errors may provide further explanations. Apart from Samalas and the mid-1450s eruptions, the lower range of residuals is well in line with the observed unforced variability. Thus, for most eruptions we can find an ensemble member that is in line with the uncertainty range of observed internal variability and control variability. It is likely that the lowest range overfits the data to some extent; however, it is well in line with observed unforced variability (Fig. a). For most of the instrumental period – except the 1920s – the 20-year variability of control runs exceeds the estimates of observed unforced variability. Based on these findings, we conclude that for many eruptions, the temperature differences resulting from volcanic forcing uncertainty are in line with the magnitude of internal variability on decadal timescale. This indicates that differences in the response to volcanic eruptions due to uncertainty in the eruption parameters may not emerge from the background of internal variability. This is particularly important for comparing model simulations to proxy reconstructions. For eruptions that exceed the control variability, it is likely that the proxy temperature reconstructions can constrain the volcanic forcing by identifying forcing ensemble members that produce temperature anomalies incompatible with the reconstructions. This includes, for example, the 1345 CE eruption, the Samalas eruption, the unknown mid-15th century eruption, the tropical eruptions in the 17th century (1600, 1640, 1695), the Laki eruption (1783) and the 19th century eruptions (1809 and Tambora 1815). However, residuals from most of eVolv2k-ENS and residuals from eVolv2k fully agree with the range of control variability after 1300 CE (Fig. a).

The ensemble spread for global temperature is less variable, and the biggest spread just about exceeds 0.3 K (Fig. c). This is partly caused by the smaller variability of global annual temperature in contrast to NH land summer temperature, and thus it could be an effect of the tuning. It could also reflect a possibility that random errors average out in the global reconstruction. However, note that high-frequency variability is strongly suppressed due to the bandpass filter that we applied to global annual temperature (compare Fig. S14).

At global scale, the 90th percentile of the uncertainty ensemble agrees at all times with the range of eVolv2k residuals. This implies that after removing the higher frequencies with the 15- to 300-year bandpass filter, spread due to volcanic forcing uncertainty is small compared to the proxy uncertainty range. We therefore concentrate on the analysis of the residuals from NH reconstructions and simulations. However, we note that the lower range of the eVolv2k-ENS spread perfectly matches the lower range of the control variability at all times for the global residuals and exceeds the upper range only during periods of very large volcanic forcing. Thus, eVolv2k lies well within the range of control variability, and at times of low volcanic forcing the residuals of PAGES 2k/eVolv2k, CMIP6 control runs and the residuals of HadCRUT5/CMIP6 match almost perfectly. However, it is likely that the lowest range is an overfit of the data (see Fig. b) due to due to a large number of degrees of freedom.

Our results suggest that internal variability and proxy reconstruction uncertainty may at times disguise the effects of volcanic forcing uncertainty, which range in the same order of magnitude. However, overall, they confirm the large-scale quality of proxy reconstructions, where the distribution of residuals between eVolv2k and proxy reconstructions overlaps well with the distribution of control variability and observed unforced variability (Fig. ).

Note that due to the way eVolv2k-ENS was compiled, it does not represent systematic uncertainty, but it instead randomly combines the uncertainties associated with individual eruptions. Thus, the distribution of the residuals over the whole millennium do not differ very much between the different ensemble members (Fig. ). Compared to eVolv2k, the main difference can be found at the upper tail of the distribution, which, for most ensemble members, is shifted towards larger residuals. In Fig. b and d, the lower range represents the minimum residual that can be generated by one ensemble member over 20 years. Due to the limited ensemble size of 1000 ensemble members, randomly combining 105 eruptions, the lower range does not represent one individual ensemble member but a combination of them over the whole millennium for each 20-year slice. While this may strongly overfit the data due to the large number of degrees of freedom, it can give information about the very lowest boundary of residuals that can be achieved by tweaking the volcanic response. Remarkably, the lower range of NH residuals fits the estimate of observed variability relatively well (Fig. a), although the left tail of the distribution indicates that a significant number of residuals approach zero. This is not consistent with any other data source, and it thus clearly indicates overfitting. For global data, the same observation can be made, and the distribution of the lower range is considerably smaller than control and estimated observed variability. However, the large discrepancy between the distribution of eVolv2k-ENS and the distribution of the lower range of residuals indicates that choice of the volcanic forcing record can influence the variability of last millennium temperature significantly, especially at (but not limited to) higher frequencies.

As a sensitivity study, we repeat our analysis with the highest and the lowest climate sensitivity presented by a CMIP6 model. We find that for very high climate sensitivity, the differences between the ensemble members are much elevated (Fig. S15), making the residuals overall less consistent with control variability (Fig. S16a and b). This concerns both residuals obtained from eVolv2k and eVolv2k-ENS, although the best fit considerately overlaps with control and observed variability. In contrast, very low climate sensitivity suppresses the response to forcing and thus decreases the spread of eVolv2k-ENS (Fig. S17), and the distribution of residuals is shifted towards zero (Fig. S16c and d). Consequentially, if climate sensitivity was very low, the effects of volcanic forcing uncertainty may be to a very large extent disguised by internal variability.

The exact effect of timing and magnitude uncertainty on the temperature response to individual eruptions can be quantified by focusing on the 11 eruptions with the largest eruption magnitudes (Fig. a–k). Due to the better high-frequency sensitivity of NH reconstructions, we limit our analysis to NH temperature. Furthermore, we consider only eruptions past 1300 CE due to the increased quality of proxy reconstructions . We compare the eVolv2k-ENS ensemble spread to the proxy reconstructions and find the forcing ensemble members that minimise the residuals between response model and each proxy reconstruction. For most eruptions we find at least one ensemble member that matches the proxy reconstruction very closely in the immediate aftermath of an eruption, in particular for the eruptions of 1345, 1640, 1695 and 1783, as well as all 19th century eruptions. This has a significant effect on the superposed epoch analysis that combines these eruptions (Fig. l) and brings the simulated volcanic response in line with the proxy reconstructions. Eruptions that still show a significant discrepancy include those of 1453, 1458 and 1600.

Comparing the eruption parameters of eVolv2k-ENS best fit and eVolv2k, we find that for most eruptions the sulfate injection magnitude favoured by the proxy reconstructions is significantly reduced (Table S2 and Fig. S18). Only for the 1453 eruption do all four reconstructions suggest a larger sulfate injection magnitude than eVolv2k, while for the 1835 eruption three reconstructions agree on a slightly larger one. For all other eruptions considered, the majority of proxy reconstructions suggest smaller eruption amplitudes, with a reconstruction average ranging from 32 % for Laki (1783) to 88 % for Tambora (1815). From the four reconstructions considered, GCS17 stands out regarding the magnitude, suggesting a higher magnitude for about half the eruptions considered (1453, 1695, 1809, 1815 and 1835).

For undated eruptions, eVolv2k-ENS accounts not just for the uncertainty of sulfate injection, but also for uncertainty in eruption month and year. Out of the 11 eruptions shown in Fig. , this concerns the eruptions of 1345, 1453, 1458, 1694 and 1809. While the eruption year plays a role regarding the spread of the temperature response, the eruption month does not seem to account for much variance (Fig. S19). It is likely that this is caused by the simple nature of our temperature response model, which does not take seasonal processes into account, and thus we cannot draw any significant conclusions regarding the role of the eruption season. The relative eruption year varies greatly between the different undated eruptions, as well as the different reconstructions, with no visible trend (Fig. S20). We find that only for the 1695 eruption do all reconstructions agree on a slightly later eruption date, ranging between +1 and +3 years. For the other eruptions results range between -2 and +3 years.

The wide range of volcanic forcing uncertainty found in the temperature response to the 11 individual eruptions leads to a similarly large range in the epoch analysis combining these eruptions (Fig. l). At the time of the maximum amplitude, 2 years after an eruption, the difference between the different ensemble members comprises around 0.8 K. The ensemble range encompasses the range of proxy reconstructions at all times, while the simulation using eVolv2k shows a larger response until around 4 years after the eruption. At this point we find a quick recovery from the cooling, while the proxy reconstructions show more persistent cooling, leading to an overlap between proxy reconstructions and eVolv2k response. In contrast, the weaker cooling and slower recovery is replicated by the best fitting eVolv2k-ENS members. Here, we find an almost perfect overlap within the proxy reconstruction range, although the epoch analysis still shows a slight discrepancy in the timing and magnitude of the peak cooling and a more prolonged response in the proxy reconstruction. The latter could be explained by biological memory processes in the tree ring data . Note that the response model does not account for internal variability, but that the volcanic response in the proxy reconstruction is affected by it. Therefore, we do not necessarily expect the best fitting ensemble members to be the true implementation of volcanic forcing over these eruptions. However, it should be noted that the overall response – including magnitude, peak cooling and recovery time – shown by the epoch analysis is heavily affected by volcanic forcing and can differ by up to 0.8 K during peak cooling. This exceeds the magnitude of the difference between PMIP3 models and proxy reconstructions .