Fig. 6

Time evolutions of \(\sqrt {{\text{d}}P}\) (Eq. 40) for the numerical experiments (see Table 1 for the summary). Left and right columns show results of 10-year and 5-year assimilation windows, respectively. All ordinates are \(\sqrt {{\text{d}}P}\), i.e., the metric for deviation from the MCM6 model. The 10-year window spans from 2004.50 to 2014.25, while the 5-year one from 2009.50 to 2014.25. \(\alpha '\left( { = 14} \right)\) in the left and \(\alpha ''\left( { = 7} \right)\) in the right indicate \(\alpha_{S}\) (Eq. 39) adopted in the final 5th iteration. a, b The weighted sum of ensemble members by Eq. (13) (the Ens. wei. sum) and MHD dynamo simulation starting from optimized initial conditions at the beginning (\(k^{\prime} = 0\)) or the end (\(k^{\prime} = 0\)) of the assimilation window (see Eqs. 14–16) (the MHD forecasts). c, d The dependence on the weight coefficient, \(\alpha_{S}\) (Eq. 39), where the red dashed lines show the results with 0.1 \(\alpha '\) and 0.1 \(\alpha ''\). e, f The ensemble-weighted sum, MHD dynamo, and KD simulations starting at the end of assimilation window (the MHD and KD forecasts with \(k^{\prime} = K\)) for 0.1 \(\alpha '\) and 0.1 \(\alpha ''\). In all the panels, the black solid lines indicate \(\sqrt {{\text{d}}P}\) calculated between the MCM6 model and IGRF-12. The release time for our forecast, assumed 0.5 year after the end of the assimilation window, is 2014.75, while 2015.00 for IGRF-12