DA with the Kalman Filter. a) The true state q (black), prior mean q (blue), and posterior mean q (red) at the final assimilation time. The green curve is the solar forcing function, g, also at the final time. b) The state for the grid point at longitude = 0 as a function of time; same colour scheme as in (a). c) The maximum value of q as a function of time; same colour scheme as in (a). d) Solar forcing, F, as a function of time; same colour scheme as in (a). e) Spatially averaged error squared error with respect to the truth for q (solid) for the prior (blue) and posterior (red). The spatially averaged variance prediction from the Kalman filter (dashed) for the prior (blue) and posterior (red). f) Same as (e) but for the solar forcing parameter, F.