A new calibrated sunspot group series since 1749: statistics of active day fractions

Although the sunspot-number series have existed since the mid-19th century, they are still the subject of intense debate, with the largest uncertainty being related to the "calibration" of the visual acuity of individual observers in the past. Daisy-chain regression methods are applied to inter-calibrate the observers which may lead to significant bias and error accumulation. Here we present a novel method to calibrate the visual acuity of the key observers to the reference data set of Royal Greenwich Observatory sunspot groups for the period 1900-1976, using the statistics of the active-day fraction. For each observer we independently evaluate their observational thresholds [S_S] defined such that the observer is assumed to miss all of the groups with an area smaller than S_S and report all the groups larger than S_S. Next, using a Monte-Carlo method we construct, from the reference data set, a correction matrix for each observer. The correction matrices are significantly non-linear and cannot be approximated by a linear regression or proportionality. We emphasize that corrections based on a linear proportionality between annually averaged data lead to serious biases and distortions of the data. The correction matrices are applied to the original sunspot group records for each day, and finally the composite corrected series is produced for the period since 1748. The corrected series displays secular minima around 1800 (Dalton minimum) and 1900 (Gleissberg minimum), as well as the Modern grand maximum of activity in the second half of the 20th century. The uniqueness of the grand maximum is confirmed for the last 250 years. It is shown that the adoption of a linear relationship between the data of Wolf and Wolfer results in grossly inflated group numbers in the 18th and 19th centuries in some reconstructions.

A Variance-Minimizing Filter for Large-Scale Applications

A truly variance-minimizing filter is introduced and its per for mance is demonstrated with the Korteweg– DeV ries (KdV) equation and with a multilayer quasigeostrophic model of the ocean area around South Africa. It is recalled that Kalman-like filters are not variance minimizing for nonlinear model dynamics and that four - dimensional variational data assimilation (4DV AR)-like methods relying on per fect model dynamics have dif- ficulty with providing error estimates. The new method does not have these drawbacks. In fact, it combines advantages from both methods in that it does provide error estimates while automatically having balanced states after analysis, without extra computations. It is based on ensemble or Monte Carlo integrations to simulate the probability density of the model evolution. When obser vations are available, the so-called importance resampling algorithm is applied. From Bayes’ s theorem it follows that each ensemble member receives a new weight dependent on its ‘ ‘distance’ ’ t o the obser vations. Because the weights are strongly var ying, a resampling of the ensemble is necessar y. This resampling is done such that members with high weights are duplicated according to their weights, while low-weight members are largely ignored. In passing, it is noted that data assimilation is not an inverse problem by nature, although it can be for mulated that way . Also, it is shown that the posterior variance can be larger than the prior if the usual Gaussian framework is set aside. However , i n the examples presented here, the entropy of the probability densities is decreasing. The application to the ocean area around South Africa, gover ned by strongly nonlinear dynamics, shows that the method is working satisfactorily . The strong and weak points of the method are discussed and possible improvements are proposed.