Cosmogenic radionuclides, produced in the atmosphere by high energy cosmic rays1 and measured in well-dated and high-resolution natural archives, can be used to reconstruct variations in solar activity and geomagnetic field on Holocene timescales and longer. The transport of galactic cosmic rays through the heliosphere is usually described by Parker’s transport equation and solved by the simplified force-field approximation that depends on a single solar modulation parameter ϕ2. The geomagnetic field modulation of incoming cosmic-ray particles takes place in space, and the atmospheric production rates are therefore mainly sensitive to the large-scale (dipole/quadrupole) field3,4.

After production 14C and 10Be follow different pathways due to differences in their geochemical behaviour. 14C oxidizes to 14CO2 and becomes incorporated into the global carbon cycle and is exchanged between the atmosphere, biosphere and ocean. Due to the relatively long atmospheric residence of 14CO2, estimated to 4–10 years (ref. 5), the latitudinal production rate differences (due to the geomagnetic shielding) can be assumed to be largely evened out. Variations in 14C deposited in tree rings are, therefore, often assumed to reflect the global average production rate, which is mainly sensitive to the dipole moment (DM) of the geomagnetic field6. In contrast, 10Be attaches to aerosols and is more rapidly removed from the atmosphere, with a mean atmospheric residence time of 1–2 years (ref. 7). Roughly two-thirds of 10Be are produced in the stratosphere and one-third in the troposphere. Traditionally, hemispherical production rate differences have not been considered, and the stratospheric production signal has been treated as a global average8. However, as recently shown through a study of 10Be transport using a general circulation model and a chemical transport model9, there is only limited hemispherical mixing of the 10Be between the Northern Hemisphere (NH) and Southern Hemisphere (SH). These results imply that there could be a substantial hemispherical dependency of the 10Be deposition that will be important for the interpretation of polar 10Be ice core records. Furthermore, incomplete mixing of the tropospheric production fraction may lead to an attenuation of the geomagnetic field signal in polar 10Be records due to the weak shielding at high latitudes, the so-called polar-bias effect10,11.

Separating geomagnetic field and solar modulation signals in radionuclide records is challenging and remains one of the biggest uncertainties in palaeo solar activity reconstructions12. In general, variations on millennial timescales are dominated by the geomagnetic field, and decadal to centennial variations are mainly attributed to changes in solar activity12. The variations in production rate due to geomagnetic field are typically removed using independent DM reconstructions13,14,15,16. However, Holocene DM reconstructions show significant differences17 that are probably related to the uneven geographical distribution of palaeomagnetic data, with most data concentrated to the NH. After correcting for the geomagnetic field, the residual ‘solar activity’ time series contain long-term variations that are in some cases similar to the Holocene DM18, suggesting either systematic problems with the geomagnetic field models or incorrect assumptions regarding the atmospheric mixing of the radionuclides. In addition, different radionuclide records also exhibit differences in the long-term (millennial-scale) variations, often speculated to be due to changes in the carbon cycle, which affect 14C, or atmospheric circulation patterns that affect the deposition of 10Be in the polar regions14,18,19.

To avoid ambiguities with the geomagnetic field correction, alternative methods are applied using differences in the expected rate of change to disentangle the geomagnetic field and solar signals12. Due to the overlapping frequency spectra, a straightforward separation using a hard cut-off frequency will inevitably lead to loss of information20. The expected statistical behaviour of the two signals can instead be encoded as prior information in a Bayesian model that co-estimates geomagnetic field and solar activity. With this approach it is possible to simultaneously extract decadal to centennial-scale variations in solar activity and generate a DM reconstruction consistent with independent palaeomagnetic data, as previously demonstrated using 14C data over the past 2,000 years (ref. 21).

A largely unexplored factor for solar activity reconstructions is the effect of north–south hemispherical asymmetries in the radionuclide production rates, which can potentially be preserved in 10Be polar records due to the above-mentioned incomplete atmospheric mixing. The difference between NH and SH average production rates, caused by large-scale asymmetries in the geomagnetic field, has been shown to vary substantially in calculations of cosmogenic radionuclide production rates over the past 100,000 years (refs. 22,23). These reconstructions, which use trajectory tracing methods to estimate the shielding effect of a full-scale representation of the geomagnetic field24,25, also show evidence for a persistent hemispherical asymmetry in the production rate, with on-average higher production rates due to weaker shielding in the SH23. Similar persistent hemispherical asymmetries in the variation of the geomagnetic field have been speculated to be linked to lower-mantle thermal heterogeneities influencing the geodynamo26,27. However, there is also a possibility that the observed hemispherical differences could be an artefact due to the uneven data distribution, with more data in the NH28,29.

Here we show that it is possible to reconcile several different Holocene cosmogenic radionuclide records and geomagnetic field models by accounting for hemispherical asymmetries in the production rates.

Hemispherical cosmic-ray shielding

A new detailed study of 10Be transport (in both the stratosphere and the troposphere) using two different global models shows that ~95% of the 10Be deposition in the polar ice caps arises from the production in the same hemisphere9. This suggests that the 10Be deposition in polar ice caps reflects average hemispherical (QNH, QSH) rather than global (QGL) production rates. In Fig. 1a we show the predicted variations of QNH, QSH and QGL based on cosmic-ray cut-off rigidity maps22 from the CALS10k.2 geomagnetic field model27 and assuming a constant average solar modulation23. Unless otherwise stated, we define ϕ based on the local interstellar spectrum of ref. 30 (Methods). From these model results, it is clear that variations in the hemispherical production asymmetry, defined as (QNH − QSH) / QGL, are predominantly traced to the axisymmetric quadrupole component of the geomagnetic field \({g}_{2}^{0}\) (ref. 23) (Fig. 1b). The relationship is well approximated by a linear function:

$$\frac{{Q}_{\mathrm{NH}}-{Q}_{\mathrm{SH}}}{{Q}_{\mathrm{GL}}}=\kappa \left(\phi \right){g}_{2}^{0}$$

where κ(ϕ) is a coefficient of proportionality expressed as a function of ϕ (Extended Data Fig. 1). Tests using different geomagnetic field models show that the values for κ(ϕ) determined using CALS10k.2 (Methods) are not model dependent (Extended Data Fig. 2). Consequently, there is a straightforward method to calculate and subsequently remove geomagnetic field modulation from polar 10Be records using existing production rate models31 to obtain QGL (dependent on DM and ϕ) and then through equation (1) derive the hemispheric production rates QNH and QSH (Extended Data Fig. 3).

Fig. 1: Predictions of 10Be production rates and their relationship to field asymmetry.
figure 1

a, Average global (QGL) and hemispherical (QNH, QSH) 10Be production rates based on cut-off-rigidity maps22 from the CALS10k.2 geomagnetic field model27 and a constant solar modulation. b, Scatter plot showing the relationship between the axisymmetric quadrupole field (\({g}_{2}^{0}\)) from CALS10k.2 and the hemispherical production asymmetry, defined as (QNH − QSH) / QGL, as shown in a. The red line shows a least-squares fit to the data (N = 201).

Model-data comparison of 10Be and 14C production rates

To test the hypothesis that polar 10Be records represent average hemispherical production rates, we compare the predictions of QNH and QSH based on three recent geomagnetic field models, pfm9k.228, ArchKalmag14k.r32 and COV-ARCH33, with 10Be records from Greenland and Antarctica (Fig. 2 and Methods). For this comparison we generally exclude geomagnetic field models, such as CALS10k.2, that are heavily reliant on sedimentary data. If not properly accounted for, such data will cause excessive smoothing of the recovered field28,34.

Fig. 2: Model-data comparisons of the geomagnetic field and 10Be production rates.
figure 2

ad, Dipole moment (DM) (a), average NH 10Be production rates (QNH) (b), axisymmetric quadrupole field (\({g}_{2}^{0}\)) (c) and average SH 10Be production rates (QSH) (d) inferred from the geomagnetic field models and the Greenland/Antarctic 10Be data (Methods). The geomagnetic field model predictions of QNH and QSH were calculated assuming ϕ = 550 MV and using the linear relationship defined in Fig. 1b. Four different Holocene palaeomagnetic field models (pfm9k.228, ArchKalmag14k.r32, COV-ARCH33 and CALS10k.227) are shown using the same colour coding in each plot. The 10Be data (grey lines) were scaled to obtain the same average hemispherical 10Be production rate as predicted by pfm9k.2 over the overlapping period and low-pass filtered (black lines) for better comparison with the geomagnetic field model predictions. Model predictions are represented by the mean (solid lines) and the 95% credible intervals (shaded areas).

The comparison highlights the large uncertainties in DM (Fig. 2a) based on the current geomagnetic field models, with significant differences between the model predictions apparent, for example, around 700 bce (ref. 28). On the contrary, a striking agreement is obtained between the models (apart from CALS10k.2, see discussion above) when comparing the predicted QNH (Fig. 2b), calculated assuming a constant ϕ = 550 MV and the production rate asymmetry derived from \(g_2^0\) (Fig. 2c) through equation (1). The model predictions also agree with the independent 10Be records from Greenland, supporting the hypothesis that the polar ice-core data represent a hemispherical production signal. To facilitate the comparison, we low-pass filtered the radionuclide data to preserve variations on timescales of 650 years and longer, previously noted to be important for the geomagnetic field28. This is a crude method to isolate geomagnetic field variations in the 10Be data with variations due to solar activity probably still present, for example around 1,500 ce due to the clustering of three so-called grand solar minima35. Still, the good overall agreement, primarily on millennial to multi-millennial timescales, between geomagnetic field model predictions and the 10Be measurements implies that there is no need to invoke long-term solar activity variations to explain the data.

A similar comparison of QSH shows large disagreements between different geomagnetic field model predictions (Fig. 2d), suggesting that the model differences in DM are related to differences in the SH field intensity, where the geomagnetic field models are less well constrained by data. In comparisons to evaluations with 10Be data from Greenland, none of the geomagnetic field models show particularly good agreement with the independent 10Be data from Antarctica. This suggests that the models are missing important field structures in the SH and that 10Be data can provide additional constraints to solve this problem.

A comparison between the global average 14C production rate \({{Q}_{\mathrm{GL}}}{{}^\mathrm{(C14)}}\) inferred from INTCAL2036 (Methods) with the prediction based on the pfm9k.2 DM and ϕ = 550 MV (ref. 6), shows that the 14C data exhibit larger amplitude millennial-scale variations than can be explained by current geomagnetic field models10 (Fig. 3). In addition, we note that neither the Greenland nor the Antarctic 10Be data from Fig. 2, converted to equivalent 14C production rates (Methods), are consistent with the 14C data (Extended Data Fig. 4). However, a good agreement is obtained if the 10Be records are averaged to obtain the global 10Be production rates (\({{{{Q}}}_{{\rm{GL}}}}{}^{({\rm{Be}}10)}=({{{Q}}}_{{\rm{NH}}}+{{{Q}}}_{{\rm{SH}}})/2\)) (Fig. 3). This further supports the hypothesis that the millennial-scale differences between the Greenland and Antarctic 10Be records are primarily due to hemispherical asymmetries in the production rate as opposed to climate effects, such as changes in atmospheric circulation patterns affecting 10Be deposition, unlikely to be removed by a simple average. Simultaneously, the comparison shows that the long-term variations in global 14C production rates, often thought to be due to changes in the carbon cycle20, are supported by 10Be data and therefore most likely reflect variations in the DM.

Fig. 3: Average global 14C production rates.
figure 3

The global average 14C production rate (QGL(C14)) determined from the INTCAL20 tree-ring 14C record36 (grey line) low-pass filtered (thick black line) and compared to the pfm9k.228 mean model prediction assuming ϕ = 550 MV (blue line) and the equivalent global average 14C production rate (red line) based on the low-pass filtered 10Be data shown in Fig. 2 (\({Q}_{\mathrm{GL}}^{(\mathrm{Be}10)}=({Q}_{\mathrm{NH}}+{Q}_{\mathrm{SH}})/2\)). The shaded area denotes the 95% credible interval.

Holocene variations in dipole moment and solar activity

On the basis of the model-data comparisons in Figs. 2 and 3, we make the following observations: (1) polar ice core 10Be records primarily reflect hemispherical production rates, (2) solar activity does not vary substantially on millennial timescales, apart from a potential clustering of grand solar minima, and (3) current Holocene geomagnetic field models (DM, \({g}_{2}^{0}\)) are probably biased due to a lack of SH data. This suggests that the geomagnetic field modulation of radionuclide production rates can currently only be determined with reasonable accuracy for the NH (that is, QNH). Conversely, the unknown systematic errors in geomagnetic field model predictions of \({Q}_{\mathrm{GL}}^\mathrm{(C14)}\) and QSH precludes the reliable extraction of solar activity variations from 14C records or Antarctic ice-core 10Be records without introducing additional assumptions.

To circumvent the problem with biased geomagnetic field models, we adapt an existing Bayesian method21 to simultaneously estimate DM, \({g}_{2}^{0}\) and ϕ using 10Be records from Greenland and Antarctica and INTCAL20 14C data and the pfm9k.2 and ArchKalmag14k.r geomagnetic field models (Methods, Extended Data Figs. 57 and Extended Data Table 1). Briefly explained the model defines DM, \({g}_{2}^{0}\) and ϕ as stochastic processes that are linked to the different datasets through production models6,31 and equation (1) (Extended Data Figs. 3 and 5). An ensemble of models, that is, samples from the joint posterior distribution, are generated using an efficient Hamiltonian Monte Carlo algorithm37. A key point of this approach is that the biased geomagnetic models (pfm9k.2 and ArchKalmag14k.r) are only used to generate time series of QNH that serve as data in the model (Extended Data Fig. 7h).

The model results show, on average, stronger DM before 500 ce compared with pfm9k.2 (Fig. 4a) but agree with previous DM reconstructions based on 14C data20. We obtain almost identical results when 14C data are excluded from the model (Extended Data Fig. 7), further supporting our previous observations. The resulting variations in \({g}_{2}^{0}\) (Fig. 4b), primarily constrained by the difference between Greenland and Antarctic 10Be records, indicate more positive values (less asymmetric field) compared with pfm9k.2 over the same time. Over the past two millennia the model predicts weaker intensities in the SH (negative \({g}_{2}^{0}\)), but overall, the new model shows less evidence for a persistent field asymmetry compared with pfm9k.2.

Fig. 4: Joint geomagnetic field and solar activity model results.
figure 4

ac, Variations in dipole moment (a), axisymmetric quadrupole field (\({g}_{2}^{0}\)) (b) and solar modulation (ϕ) (c) based on the new model (red lines) compared with pfm9k.228 model predictions (blue lines in a and b). Model predictions are represented by the mean (solid lines) and the 95% credible intervals (shaded areas).

Overall, the model output ϕ varies around a mean of 600 MV and occasionally drops to a minimum level of around 200 MV during grand solar minima (Fig. 4c), which is consistent with some solar cycle variability (in the range of ±200 MV) still being present38,39. Over the past 6,000 years, ϕ does not vary substantially on millennial timescales. Before 4000 bce, variations in ϕ are positively correlated with DM suggesting that the geomagnetic field signal has not been adequately separated, as no gross correlation between ϕ and DM is expected. Although an artefact of a weakly constrained magnetic field model, this shows that the prior distribution of ϕ is flexible enough to capture millennial variations if required, as would be expected from the power spectrum of the prior (Extended Data Fig. 6c).

Implications for solar activity

The good agreement between the low-pass filtered Greenland 10Be record and the average NH 10Be production rates QNH predicted by independent geomagnetic field models (Fig. 2b) suggests that solar activity does not vary substantially on millennial timescales. Our modelling result (Fig. 4c) strengthens this conclusion, explaining differences between different radionuclide records by hemispheric asymmetries in the geomagnetic field rather than changes in the carbon cycle for 14C or system effects for 10Be. Our results contrast with previous Holocene solar activity reconstructions14,15, determined using incomplete/biased global geomagnetic field descriptions, that exhibit variations on multi-millennial timescales (Extended Data Fig. 8). Separating the geomagnetic field signal in the radionuclide records reveals a bi-modality in solar modulation related to the occurrence of grand solar minima (Extended Data Fig. 8b), previously identified for the past 3,000 years (ref. 40) and hinted at in longer reconstructions15. The resulting solar modulation distribution, which is well approximated by two Gaussian distributions, suggests that the Sun spends about 9% of its time in the ‘grand solar minima’ mode, but there is no evidence for a grand solar maximum mode as previously proposed40.

A ~2,400-year quasi-periodic signal, often referred to as the Hallstatt cycle, has previously been identified in Holocene 14C records (Fig. 5a) and attributed to either climate41, solar activity18 or the geomagnetic field42,43. Whereas part of this signal is potentially present in our solar activity reconstruction, related to the clustering of grand solar minima (Fig. 5b), we show here that the observed ‘baseline’ variations on ~2,400-year timescales are well explained by the geomagnetic field (Fig. 5c). The variations in DM of ~2 × 1022 Am2, which are required to describe the observed production changes on these timescales18, are also supported by independent geomagnetic field reconstructions such as pfm9k.2. Such millennial-scale oscillations in the geomagnetic field could potentially be signatures of slow magnetic Rossby waves in Earth’s outer core44,45,46.

Fig. 5: The Hallstatt cycle in 14C production rates split into ϕ and DM.
figure 5

ac, The global average 14C production rate QGL (note inverted y axis) (a), solar modulation ϕ (b) and dipole moment DM (c), band-pass filtered to preserve variations on timescales shorter than 3,000 years and longer than 1,000 years: INTCAL2036 (black), the new model (red) and pfm9k.228 (blue). A 2,400-year sine wave, representative of the previously identified Hallstatt cycle, is shown for reference in all three plots. Model predictions are represented by the mean (solid lines) and the 95% credible intervals (shaded areas).

Hemispherical geomagnetic field asymmetry

Our model results suggest that the north–south geomagnetic field asymmetry, represented by \({g}_{2}^{0}\) (Fig. 4b), has varied substantially over the past 9,000 years with periods of both on-average stronger and weaker field intensities in the NH compared to the SH. These variations in the model are mainly driven by differences between the Greenland and Antarctic 10Be data and are, therefore, sensitive to possible uncertainties in these records, for example, due to variations in snow accumulation16,47. Despite these unaccounted-for sources of uncertainties, the good agreement between the global 10Be signal and 14C data (Fig. 3 and Extended Data Fig. 7) supports that the 10Be data mainly reflect variations in the production rate. The model also predicts a slight predominance of negative \({g}_{2}^{0}\), implying a persistent field asymmetry. This conclusion is, however, less robust given the uncertainties related to the normalization and the long-term trends of the 10Be data.

On the basis of our comparisons between Holocene geomagnetic field models and cosmogenic radionuclide records (Figs. 2 and 3), we conclude that the existing geomagnetic field models are probably biased due to the uneven data distribution. However, tests based on synthetic data do not show similar biases27,34,48, suggesting that the problem is not inherent to the modelling methodologies but probably due to more subtle effects, such as inaccurate assumptions regarding the error distribution49. Because variations in the DM are primarily driven by changes in the axisymmetric dipole field \({g}_{1}^{0}\), the model bias can be thought of as a shift of power between the axisymmetric dipole (\({g}_{1}^{0}\)) and quadrupole (\({g}_{2}^{0}\)) fields. It can be shown that the changes in \({g}_{2}^{0}\) implied by our new model compared to pfm9k.2, leading to higher intensities in the SH, are largely compensated by the associated changes in \({g}_{1}^{0}\) in the NH where most of the data are concentrated (Extended Data Fig. 9).


Solar modulation parameter conversions

The absolute values of the solar modulation parameter ϕ depends on which local interstellar spectrum (LIS) model is used. In this study we define solar modulation based on the LIS of ref. 30, which we denote ϕHE17 or simply as ϕ. The 10Be and 14C production rate models that we use in this study6,31 were based on the LIS as given in ref. 50 in the form parametrized by ref. 51, which we denote ‘US05’. For the sake of consistency, we therefore use ϕUS05 in all our calculations. As has previously been shown30, it is possible to convert from one LIS-dependent solar modulation to another by means of linear regression functions. To convert from ϕVP15, based on the LIS model of ref. 52 and used by refs. 23,53, we use the following relationship

$${\phi }_{\mathrm{VP}15}=1.012\times {\phi }_{\mathrm{US}05}-26.16\,({\rm{MV}})$$

derived following the approach of ref. 30. Before presenting our final model results, we convert from ϕUS05 to ϕHE17 as it is based on the more recent and accurate LIS model of ref. 30

$${\phi }_{\mathrm{HE}17}=1.025\times {\phi }_{\mathrm{US}05}+24.18\,({\rm{MV}})$$

Hemispherical 10Be production rate asymmetry

As shown in Extended Data Fig. 1a, the proportionality constant κ of equation (1), that is, the slope of the linear relationship between (QNH − QSH) / QGL and \({g}_{2}^{0}\), is dependent on ϕ. To account for this dependency, we fit a third-order polynomial (Extended Data Fig. 1b):

$$\begin{array}{l}\kappa \left({\phi }_{\mathrm{US}05}\right)=-1.06\times {10}^{-15}\times {\phi }_{\mathrm{US}05}^{3}+6.11\times {10}^{-12}\\\qquad\qquad\,\times {\phi }_{\mathrm{US}05}^{2}-1.99\times {10}^{-8}\times {\phi }_{\mathrm{US}05}+6.07\times {10}^{-5}.\end{array}$$

This definition of κ is valid for \({g}_{2}^{0}\) expressed in nT.

Cosmogenic radionuclide data

To reconstruct variations in average hemispherical 10Be production rates (QNH, QSH), we select the longest continuous Holocene records from Greenland and Antarctica, respectively: the Greenland Ice Core Project (GRIP) record13,54,55 (with timescale synchronized to the international radiocarbon calibration curve INTCAL1356) and the European Project for Ice Coring in Antarctica, Dronning Maud Land (EDML) record14. Both records show relatively small variations in accumulation rates during the Holocene and we therefore use 10Be concentrations rather than calculating 10Be fluxes. The long records were augmented with shorter ones to bridge the gap to the present. For Greenland we used short records from the North Greenland Ice Coring Project (NGRIP) record57, Dye-358 and Milicent59, and for Antarctica, we used short records from Dome Fuji60, Siple Dome61 and two different records from South Pole62,63,64. All records from Greenland were normalized to the NGRIP record for the overlapping time periods, and all Antarctic records were normalized similarly to the newest South Pole record. To produce the final Greenland and Antarctic stacks, all the data from either location were averaged over consecutive 22-year windows, consistent with the resolution of the only available continuous long-term record from Antarctica (EDML).

To reconstruct global average 14C production rates (\({Q}_{\mathrm{GL}}^\mathrm{(C14)}\)) we use the inferred atmospheric 14C concentrations from Northern Hemisphere tree-ring measurements compiled in INTCAL2036. Data after 1950 ce are heavily influenced by anthropogenic 14C produced during the nuclear bomb tests and were therefore not used. The 14C production rates were calculated using a box-diffusion carbon cycle model, assuming a constant carbon cycle during the Holocene and accounting for the Suess effect from 1850 ce38,65. The derived 14C production data were averaged over the same consecutive 22-year windows used for the 10Be data.

All three records were scaled to absolute values so that the mean production rate over the past 9,000 years is the same as that predicted by pfm9k.2 and assuming a constant ϕHE17 = 550 MV (refs. 6,31).

To convert the low-pass filtered 10Be production rates in Fig. 2 to equivalent 14C production rates in Fig. 3 and Extended Data Fig. 4, we first assume that the 10Be data represent global average production rates and that solar modulation is constant (ϕHE17 = 550 MV). We then calculate the equivalent DM using the 10Be production rate model31 and convert this back to equivalent global 14C production rates using the 14C production rate model6.

Bayesian model

The method to construct a joint model of variations in DM and ϕUS05 based on cosmogenic radionuclide data, which this work is based on, is described in detail in ref. 21. In addition to DM and ϕUS05, we here also model variations in the axisymmetric quadrupole field \({g}_{2}^{0}\), which is related to the hemispherical 10Be production rate through equation (1). The model is constrained by data from the three cosmogenic radionuclide records described above and two geomagnetic field models predictions of QNH (Extended Data Fig. 5). We use a Hamiltonian Monte Carlo algorithm37 to generate M = 1,000 samples from the joint posterior probability distribution.

Following ref. 21, the time series of DM, \({g}_{2}^{0}\) and ϕUS05 are parameterized in terms of Gaussian Processes66, defined entirely by their mean and covariances, and the model is evaluated at fixed intervals every 22 years from −6999 ce to 1999 ce, the same intervals used to calculate the data averages (above). The prior distributions of DM and \({g}_{2}^{0}\) were defined in the same way as for pfm9k.2 (Extended Data Table 1) and attached to the COV.OBS.x2 mean model67 from 1840 ce and onwards, as was also done for the pfm9k.2 model28. For DM we use the covariance function for the dominant axisymmetric \({g}_{1}^{0}\) Gauss coefficient defined by the variance σ2 and two timescales χ−1 and ω−1 (ref. 68)

$${k}_{DM}\left(r\right)=\frac{{\sigma }^{2}}{2\xi }\left(\left(\chi +{\rm{\xi }}\right)\mathrm{e}^{-\left(\chi -\xi \right)r}-(\chi -\xi )\mathrm{e}^{-\left(\chi +\xi \right)r}\right)$$

where \(r=\left|t-t^{\prime} \right|\) is the absolute difference between two input times and \({\xi }^{2}={\chi }^{2}-{\omega }^{2}\). The prior mean μ and variance σ2 for DM were re-scaled accordingly. For \({g}_{2}^{0}\) we use a two-parameter covariance function with a single timescale ω−1 (ref. 68)

$${k}_{{g}_{2}^{0}}\left(r\right)={\sigma }^{2}\left(1+{\rm{\omega }}{{r}}\right)\mathrm{e}^{-\omega r}.$$

The prior distribution for ϕUS05 is defined to only capture the variations on 22-year timescales and longer that are preserved in the cosmogenic radionuclide data described above. We use a covariance function of the form:

$${k}_{\phi }\left(r\right)={\sigma }^{2}\mathrm{e}^{-{\left(\omega r\right)}^{2}}$$

which is the same as a squared exponential kernel but using the notation of equations (5) and (6). A priori values of μ, σ2 and ω are estimated from a reconstruction of solar modulation ϕGSN based on group sunspot numbers (GSN)21,69 filtered the same way as the cosmogenic radionuclide data and converted to equivalent ϕUS05 through equation (3). To correct for a slight underestimation of solar modulation compared to reconstructions based on neutron monitor data (below), the ϕGSN data were re-scaled/corrected by a factor of 1.06 (Extended Data Fig. 6). To illustrate the uncertainties in the conversion of sunspot numbers to solar modulation, we show an alternative reconstruction based on the international sunspot numbers (ISN)70 and using a more updated conversion ϕISN71,72, similarly corrected using a factor of 1.17 (Extended Data Fig. 6). We note that using ϕGSN to define the prior amounts to a conservative choice due to the larger variance of this time series (less informative prior). The first two parameters were calculated as the sample mean and the sample variance of the ϕGSN time series and the a priori correlation time was determined as

$${\omega }^{-1}=\sqrt{{\sigma }_{\mathrm{GSN}}^{2}/{\dot{\sigma }}_{\mathrm{GSN}}^{2}}$$

where \({\sigma }_{\mathrm{GSN}}^{2}\) and \({\dot{\sigma }}_{\mathrm{GSN}}^{2}\) are the sample variance of ϕGSN and its time derivative. Finally, the prior distribution is attached through Gaussian Process regression34 to an independent record of ϕ based on neutron monitor data30,73 going back to 1951 ce, converted from ϕVp15 to ϕUS05 and then filtered the same way as the cosmogenic radionuclide data (Extended Data Fig. 6a).

Because of the limited length of the ϕGSN time series, the prior shows essentially no variability on timescales longer than a few hundred years (Extended Data Fig. 6c). However, this does not preclude the possibility to recover variations on longer timescales if these are required to explain the variations in the cosmogenic radionuclide data. This could for example be large amplitude changes in production rates on multi-centennial to millennial timescales that are inconsistent with the expected range of variability of the hemispheric geomagnetic field shielding.

To account for the uncertainties in the scaling of the cosmogenic radionuclide data to absolute production rates, we introduce a scaling parameter to each record with a Gaussian prior distribution \({\theta }_{i} \sim N({\mu }_{\theta },{\sigma }_{\theta }^{2})\), where μθ = 1 and σθ = 0.05. All model parameters, including the time series of DM, \({g}_{2}^{0}\) and ϕUS05 and the scaling parameters {θ1, θ2, θ3}, are stored in vector x.

The INTCAL20 14C data (re-scaled by θ1) are assumed to represent \({Q}_{\mathrm{GL}}^\mathrm{(C14)}\), which is related to DM and ϕUS05 through the 14C production rate model of ref. 6, approximated using a quadratic polynomial surface21

$$\begin{array}{l}{Q}_{\mathrm{GL}}^{\left({\mathrm{C}}14\right)}({DM},\,{\phi }_{{\mathrm{US}}05})=\left(9.06\times {10}^{-2}+3.12\times {10}^{-2}\right.\times\,{DM}+2.62\times {10}^{-4}\\\qquad\qquad\qquad\qquad\;\,\times {\phi }_{{\mathrm{US}}05}-1.02\times {10}^{-4}\times{DM}^{2}+1.91\times {10}^{-5}\\\qquad\qquad\qquad\qquad\;\,\left.\times\, {\phi }_{{\mathrm{US}}{05}}\times{DM}+1.07\times {10}^{-8}\times {\phi }_{{\mathrm{US}}05}^{2}\right)^{-1}.\end{array}$$

The Greenland and Antarctic 10Be data (re-scaled by θ2 and θ3) are assumed to represent QNH and QSH, respectively, which are related to DM, \({g}_{2}^{0}\) and ϕUS05 through

$${Q}_{\mathrm{NH}}\left(DM,\,{g}_{2}^{0},\,{\phi }_{\mathrm{US}05}\right)=\left(1+\frac{\kappa \left({\phi }_{\mathrm{US}05}\right){g}_{2}^{0}}{2}\right){Q}_{\mathrm{GL}}(DM,\,{\phi }_{\mathrm{US}05})$$
$${Q}_{\mathrm{SH}}\left(DM,\,{g}_{2}^{0},\,{\phi }_{\mathrm{US}05}\right)=\left(1-\frac{\kappa \left({\phi }_{\mathrm{US}05}\right){g}_{2}^{0}}{2}\right){Q}_{\mathrm{GL}}(DM,\,{\phi }_{\mathrm{US}05})$$


$$\begin{array}{l}{Q}_{\mathrm{GL}}(DM,\,{\phi }_{\mathrm{US}05}) = \left(5.58+1.90\times DM+1.38\times{10}^{-2}\right.\\\times\,{\phi}_{\mathrm{US}05}-1.30\times{10}^{-2}\times DM^{2}+1.49\times {10}^{-3}\\\left.\times\,{\phi}_{\mathrm{US}05}\times DM-2.85\times{10}^{-7}\times{\phi}_{\mathrm{US}05}^{2}\right)^{-1}\end{array}$$

is the 10Be production rate model of ref. 31, approximated using a quadratic polynomial surface.

The geomagnetic field model predictions of QNH, based on pfm9k.2 and ArchKalmag14k.r, were calculated according to equation (10) with a fixed solar modulation ϕHE17 = 550 MV and are related to the modelled DM and ϕUS05 in the same way. The QNH input data, defined as the ensemble mean of each model (N = 1,000 and N = 250, respectively), were linearly interpolated to a 44-year resolution timescale to be consistent with the model timescale while roughly preserving the original number of data based on the native 50-year resolution of both geomagnetic field models.

The data from the three cosmogenic radionuclide records and the two geomagnetic field model predictions of QNH are stored in vector y. The data are related to the model parameters x through


where H(x) is a nonlinear forward operator based on equations (9)–(12) and e is the data error vector, whose statistical properties are characterized by the covariance matrix \({C}_{\mathrm{ee}}=E(\bf{e}\bf{e}^T)\). For the parts related to the cosmogenic radionuclide data Cee is diagonal and filled with conservatively assigned uncertainty estimates of 5% and 10% for 14C and 10Be data, respectively, based on typical measurement uncertainties. For the geomagnetic field model predictions of QNH, Cee is filled with the sample covariances calculated from the ensemble model predictions.