Fig. 7

a Comparison of SA, the second time derivative of the geomagnetic field at the Earth’s surface, between the MCM6 model, and the MHD and KD forecasts using 10- and 5-year assimilation windows. The initial conditions for the MHD and KD forecasts are given at 2014.25 for all forecasts. b, c Comparison of the first time derivative (SV) and second time derivative (SA) of two Gauss coefficients, \(h_{32}\) and \(h_{33}\), among the MCM6 model (the black line), the weighted sum of the final ensemble members by Eq. (13) (the Ens. wei. sum; the red line), the MHD forecast (the blue line), and the KD forecast (the green line) for 5- and 10-year assimilation windows. The assimilation windows are indicated by gray arrows. Panel c shows that SV of \(h_{32}\) and \(h_{33}\) have large discrepancies between the KD forecasts (the green lines) and the other three lines (the blue, red, and black lines) with 5-year assimilation window. The monotonic decrease and growth of SV in the KD forecasts are caused by steady negative and positive SA after the assimilation window, respectively, while oscillatory variation of SA in the MHD forecasts and the Ens. wei. sum suppress the departure of SA lines from that of the MCM6 model