The motivation for modeling the past variations in solar irradiance arises due to the Sun playing a crucial role in influencing Earth's climate, while direct measurements of TSI from space cover only a limited period (since 1978). Consequently, various models have been devised over the years to reconstruct past TSI variations. These models vary in their approach, where some are more physics-based and others more empirical. Regardless of their methodologies, all these models rely on historical data of solar activity, typically the sunspot number series or data on cosmogenic radioisotopes. Therefore, regular updates of the models are essential to integrate any updates or corrections made to the input data.

Direct measurements of TSI also serve as a benchmark for evaluating all models. The longer the records of these direct measurements, the better the models can get refined. Recent research, benefiting from improvements in both data quality and modeling techniques, generally indicates that TSI variations over the past four centuries have been relatively modest[1]. However, the uncertainty remains regarding their precise values, and some scientific literature still suggests a significant long-term trend in TSI variations.

It is noteworthy that most such claims regarding the high magnitude of solar irradiance variations rely on old and outdated TSI series, but mostly on a single TSI dataset, the one by HS93. This model has also been used to argue that the Earth’s temperature increase over the last decades was almost entirely due to the Sun.

However, the accuracy of this model was not evaluated, and its creators did not provide any update since the original publication in 1993. Considering that studies attempting to replicate published results are relatively uncommon, this left the claims made based on this model without a thorough understanding of its quality. Furthermore, the cryptic description of the model contributed to hindering its replication. These sparked my curiosity for this model and, initially at my spare time, I undertook the challenge of replicating this model and deciphering its methodology.

The HS93 TSI model takes a purely empirical approach to reconstruct TSI utilizing five sunspot activity indices:

1. Sunspot number series (annual values & 11-year averages)
2. Equatorial rotation rate
3. Fraction of "penumbral spots"
4. Lengths of solar cycles
5. Rate of decay of solar cycles

I started by trying to identify the source data and replicate their methodologies in producing each of these 5 indices. This required consulting additional sources from the literature which might contain hints about this model as well as digitizing the indices from the HS93 paper figures to reveal their characteristics. The sunspot number series was the most straightforward to derive, and that was already sufficient to reveal significant flaws in HS93's analysis. Specifically, they incorrectly shifted the 11-year smoothed sunspot number series 11 years into the past, resulting in an artificial spike in solar activity over the recent decades due to estimated future values.

 Inconsistencies with the production of the other indices were realized too. In particular the incorrect combination of data from different sources without properly accounting for archive differences and thus leading to artefacts and spurious trends in the indices.

Having replicated HS93’s methodologies to produce the five indices, it was possible to gain insights in the process HS93 used to reconstruct TSI. HS93 linearly scaled the previously described five indices and then averaged them to reconstruct TSI. Effectively, the large amplitude of TSI variations of this model was not an intrinsic characteristic, but a free parameter of the model. Another limitation of the model is its inability to offer real-time TSI reconstructions; it can only provide annual TSI values and is restricted to a reconstruction extending only up to 12 years before the present day.

Using the model with updated versions of the five indices and through a comparison with direct TSI measurements, I found that the large variations in TSI suggested by the original HS93 TSI series did not align with direct measurements of TSI over recent decades. In particular it appears that HS93's imposed variations were exaggerated by approximately a factor of five. Overall, the variations suggested by the HS93 model align with other high-quality models from the literature, such as e.g. the Spectral And Total Irradiance (SATIRE) model[2]. It is important to note that while the original TSI series by HS93 was recently used to question the solar forcing used by the Intergovernmental panel on climate change (IPCC)[3], updating the model with recent data brings it in line with the dataset used by the IPCC.

Lastly, when comparing the TSI reconstruction with Earth's temperature data, it becomes evident that the model cannot support the argument that TSI correlates with temperature fluctuations on Earth in recent decades.

 If these issues had been recognized earlier, it is improbable that the Hoyt and Schatten (1993) TSI series would have garnered the attention it did.

References:

[1] Chatzistergos, T., Krivova, N.A., Yeo, K.L., 2023. Long-term changes in solar activity and irradiance. Journal of Atmospheric and Solar-Terrestrial Physics 252, 106150. https://doi.org/10.1016/j.jastp.2023.106150

 [2] Krivova, N.A., Solanki, S.K., Fligge, M., Unruh, Y.C., 2003. Reconstruction of solar irradiance variations in cycle 23:  Is solar surface magnetism the cause? Astronomy and Astrophysics 399, L1–L4. https://doi.org/10.1051/0004-6361:20030029

[3] https://www.ipcc.ch/report/ar6/wg1/