Assessment of sunspot number cross-calibration approaches

Context. Group sunspot number data form the longest record of direct observations of solar activity and variability. However, the observations were conducted by many observers using different telescopes and at diverse locations, which necessitates their proper cross-calibration. Historically, such a cross-calibration was performed with a simple linear scaling. More recently some non-linear approaches have also been developed, as well as modifications of the classical linear scaling. This resulted in a number of new composite sunspot series, which diverge before the 20th century, thus also leading to an uncertainty in the past solar activity and variability. Aims. Our aim was to understand the causes of divergence between different sunspot series. To this end, we scrutinised the existing cross-calibration methods to identify the sources of their biases and uncertainties. Methods. We used synthetic data imitating observers with different observing capabilities to test the performance of different cross-calibration approaches, including both simple linear scaling and non-linear non-parametric techniques. Some of these methods require a direct overlap between the records of two observers, while others rely on statistical properties of sunspot groups. Results. We found that linear approaches generally overestimated and underestimated the maxima of strong and weak activity cycles, respectively, thus introducing a bias in the secular variability. In contrast, for typical characteristics of existing records of observers, non-parametric approaches returned more consistent results and lower errors. Out of these latter, methods relying on statistical properties of the records return worse results. Conclusions. Our analysis revealed limitations of the various approaches and identified the best approaches. For future recalibrations of sunspot number, we recommend using a direct non-linear calibration when the data coverage is sufficient. However, the errors returned by such daisy-chain methods accumulate when going further back in time, if a multi-step daisy-chain (backbone) calibration is needed. To bridge extensive data gaps, we therefore recommend using a statistical method (e.g. active-day fraction).