Fig. 1

The rms \(({\mathcal {O}}\text {-}{\mathcal {F}})_{B}\) of the magnetic field in case II (dashed line) and case III (solid line). In both cases, \(({\mathcal {O}}\text {-}{\mathcal {F}})_{B} \ll 1\) and decays monotonically after the first three analysis cycles and then levels off in the last 20 years. This shows the continuing improvement in the forecast accuracies. In addition, the \(({\mathcal {O}}\text {-}{\mathcal {F}})\) results in case III (with the assimilation of \({b_{l}^{m(o)}}\) and \({\dot b}_{l}^{m(o)}\)) are in general more than 20 % smaller than in case II (with only the assimilation of \({b_{l}^{m(o)}}\)), showing a clear improvement in forecast accuracies