The physics-based reconstruction of solar activity (in terms of sunspot numbers) from cosmogenic proxy data includes several steps:

- Computation of the isotope’s production rate in the atmosphere from the measured concentration in the archive (Sections 3.2.2 and 3.3.2);
- Computation, considering independently-known secular geomagnetic changes (see Section 3.2.5) and a model of the CR-induced atmospheric cascade, of the GCR spectrum parameter quantified via the modulation potential (Section 3.4.2), some reconstructions being terminated at this point;
- Computation of a heliospheric index, whether of the open solar magnetic flux or of the average HMF intensity at the Earth’s orbit (Section 3.4.2)
- Computation of a solar index (sunspot number series), corresponding to the above-derived heliospheric parameter (Section 3.4.3).

Presently, all these steps can be completed using appropriate models. Although the uncertainties of the models may be considerable, the models allow a full basic quantitative reconstruction of solar activity in the past. However, much needs to be done, both theoretically and experimentally, to obtain an improved reconstruction.

Mathematical regression is the most apparent and often used (even recently) method of solar-activity
reconstruction from proxy data (see, e.g., Stuiver and Quay, 1980; Ogurtsov, 2004). The reconstruction of
solar activity is performed in two consecutive steps. First, a phenomenological regression (either linear or
nonlinear) is built between proxy data set and a direct solar-activity index for the available “training”
period (e.g., since 1750 for WSN or since 1610 for GSN). Then this regression is extended backwards to
evaluate SN from the proxy data. The main shortcoming of the regression method is that it depends on the
time resolution and choice of the “training” period. The former is illustrated by Figure 10, which
shows the scatter plot of the ^{10}Be concentration vs. GSN for the annual and 11-year smoothed
data.

One can see that the slope of the ^{10}Be-vs-GSN relation (about –500 g/atom) within individual cycles is
significantly different from the slope of the long-term relation (about –100 g/atom, i.e., individual cycles do
not lie on the line of the 11-year averaged cycles. Therefore, a formal regression built using the annual data
for 1610 – 1985 yields a much stronger GSN-vs-^{10}Be dependence than for the cycle-averaged data (see
Figure 10b), leading to a potentially-erroneous evaluation of the sunspot number from the ^{10}Be proxy
data.

It is equally dangerous to evaluate other solar/heliospheric/terrestrial indices from sunspot numbers, by extrapolating an empirical relation obtained for the last few decades back in time. This is because the last few decades (after the 1950s), which are well covered by direct observations of solar, terrestrial and heliospheric parameters, correspond to a very high level of solar activity. After a steep rise in activity level between the late 19th and mid 20th centuries, the activity remained at a roughly constant high level, being totally dominated by the 11-year cycle without an indication of long-term trends. Accordingly, all empirical relations built based on data for this period are focused on the 11-year variability and can overlook possible long-term trends (Mursula et al., 2003). This may affect all regression-based reconstructions, whose results cannot be independently (directly or indirectly) tested. In particular, this may be related to solar irradiance reconstructions, which are often based on regression-like models, built and verified using data for the last three solar cycles, when there was no strong trend in solar activity.

As an example let us consider an attempt (Belov et al., 2006) to reconstruct cosmic-ray intensity since 1610 from sunspot numbers using a (nonlinear) regression. The regression between the count rate of a neutron monitor and sunspot numbers, established for the last 30 years, yields an agreement at a 95% level for the period 1976 – 2003. Based on that, Belov et al. (2006) extrapolated the regression back in time to produce a reconstruction of cosmic-ray intensity (quantified in NM count rate) to 1560 (see Figure 11). One can see that there is no notable long-term trend in the reconstruction, and the fact that all CR maxima essentially lie at the same level, from the Maunder minimum to modern times, is noteworthy. It would be difficult to dispute such a result if there was no direct test for CR levels in the past. Independent reconstructions based on cosmogenic isotopes or theoretical considerations (e.g., Usoskin et al., 2002b; Scherer et al., 2004; Scherer and Fichtner, 2004) provide clear evidence that cosmic-ray intensity was essentially higher during the Maunder minimum than nowadays. This example shows how easy it is to overlook an essential feature in a reconstruction based on a regression extrapolated far beyond the period it is based on. Fortunately, for this particular case we do have independent information that can prevent us from making big errors. In many other cases, however, such information does not exist (e.g., for total or spectral solar irradiance), and those who make such unverifiable reconstructions should be careful about the validity of their models beyond the range of the established relations.

The modulation potential (see Section 3.1.1) is most directly related to cosmogenic isotope production in the atmosphere. It is a parameter describing the spectrum of galactic cosmic rays (see the definition and full description of this index in Usoskin et al., 2005a) and is sometimes used as a stand-alone index of solar (or, actually, heliospheric) activity. We note that, provided the isotope production rate is estimated and geomagnetic changes can be properly accounted for, it is straightforward to obtain a time series of the modulation potential, using, e.g., the relation shown in Figure 7. Several reconstructions of modulation potential for the last few centuries are shown in Figure 12. While being quite consistent in the relative changes, they differ in the absolute level and fine details. Reconstructions of solar activity often end at this point, representing solar activity by the modulation potential, as some authors (e.g., Beer et al., 2003; Vonmoos et al., 2006; Muscheler et al., 2007) believe that further steps (see Section 3.4.3) may introduce additional uncertainties. However, since is a heliospheric, rather than solar, index, the same uncertainties remain when using it as an index of solar activity. Moreover, the modulation potential is a model-dependent quantity (see discussion in Section 3.1.1) and therefore does not provide an unambiguous measure of heliospheric activity. In addition, the modulation potential is not a physical index but rather a formal fitting parameter to describe the GCR spectrum near Earth (Usoskin et al., 2005a) and, thus, does not seem to be a universal solar-activity index.

Modulation of GCR in the heliosphere (see Section 3.1.1) is mostly defined by the turbulent heliospheric
magnetic field (HMF), which ultimately originates from the sun and is thus related to solar
activity. It has been shown, using a theoretical model of the heliospheric transport of cosmic
rays (e.g., Usoskin et al., 2002b), that on the long-term scale (beyond the 11-year solar cycle)
the modulation potential is closely related to the open solar magnetic flux F_{o}, which is a
physical quantity describing the solar magnetic variability (e.g., Solanki et al., 2000; Krivova
et al., 2007).

Sometimes, instead of the open magnetic flux, the mean HMF intensity at Earth orbit, B, is used as a
heliospheric index (Caballero-Lopez and Moraal, 2004; McCracken, 2007). Note that B is
linearly related to F_{o} assuming constant solar-wind speed, which is valid on long-term scales. An
example of HMF reconstruction for the last 600 years is shown in Figure 13. In addition, the
count rate of a “pseudo” neutron monitor (i.e., a count rate of a neutron monitor if it was
operated in the past) is considered as a solar/heliospheric index (e.g., Beer, 2000; McCracken and
Beer, 2007).

The open solar magnetic flux F_{o} described above is related to the solar surface magnetic phenomena such as
sunspots or faculae. Modern physics-based models allow one to calculate the open solar magnetic flux from
data of solar observation, in particular sunspots (Solanki et al., 2000, 2002; Krivova et al., 2007). Besides
the solar active regions, the model includes ephemeral regions. Although this model is based on physical
principals, it contains one adjustable parameter, the decay time of the open flux, which cannot be measured
or theoretically calculated and has to be found by means of fitting the model to data. This free parameter
has been determined by requiring the model output to reproduce the best available data sets for the last
30 years with the help of a genetic algorithm. Inversion of the model, i.e., the computation of
sunspot numbers for given F_{o} values is formally a straightforward solution of a system of linear
differential equations, however, the presence of noise in the real data makes it only possible in a
numerical-statistical way (see, e.g., Usoskin et al., 2004, 2007). By inverting this model one can compute
the sunspot-number series corresponding to the reconstructed open flux, thus forging the final
link in a chain quantitatively connecting solar activity to the measured cosmogenic isotope
abundance. A sunspot-number series reconstructed for the Holocene using ^{14}C isotope data is shown
in Figure 14. While the definition of the grand minima (Section 4.3) is virtually insensitive
to the uncertainties of paleomagnetic data, the definition of grand maxima depends on the
paleomagnetic model used (Usoskin et al., 2007). Since the Y00 paleomagnetic model forms an
upper bound for the true geomagnetic strength (Section 3.1.2), the corresponding solar-activity
reconstructions may underestimate the solar-activity level. Accordingly, the grand maxima
defined using the Y00 model are robust and can be regarded as “maximum maximorum” (see
Section 4.4).

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