Should IterationSummary be done after the iteration?

[navidcy]

As it stands now there is a 0-th iteration summary that includes the initial free parameters.

But then, the first iteration repeats that, so the first iteration summary, to a great extent, same as the 0-th one. Thus, the figures, e.g.

https://clima.github.io/OceanTurbulenceParameterEstimation.jl/dev/literated/perfect_convective_adjustment_calibration/

have "Iteration 1" lying on top of "Initial ensemble".

I think that within iterate!. the iteration summary should come after the update_ensemble! step. Does that sound reasonable? Would that mean that we need to call

θ = get_u_final(eki.ensemble_kalman_process) # (N_params, ensemble_size) array
G = eki.inverting_forward_map(θ) # (len(G), ensemble_size)

both before and after update_ensemble!? That's no good... How do we avoid that?

[glwagner]

I think we need the concept of the "state" of the Kalman process. It might make sense to hold on to a reference to G, and also to calculate G when constructing the EKI object (by default at least). Then the IterationSummary / StateSummary is generated every time we advance to a new state.