Clearly separate fit and update steps for all ensemble assimilation algorithms

[Blunde1]

Is your feature request related to a problem? Please describe.
All ensemble based assimilation algorithms involve a fitting procedure of ensemble to some parametric or semi-parametric model.
After the fit, it will employ an update to the realizations, based on the fit.
All statistical issues for ensemble based methods reside within the fit part of the algorithm.
To allow us to

• target the statistical issues Implement adaptive localization #4411 Use Bonferroni (or similar) correction to the correlation threshold #5557 Implement simple configuration of distance based localization #5553
• write an analysis report Create analysis report for quality control of history match / sensitivity analysis #5556
• perform testing Use heat equation for testing #5529
  will become much simpler if we clearly separate the fit and the update.

Describe the solution you'd like
Create the API so that we at least have fit and update, similar to fit and predict in sklearn.
Note, in the iterative_ensemble_smoother we almost have this, but the fit and predict reside within the iterative procedure, while I would argue it should be outside.

Describe alternatives you've considered
Trying to solve the listed problems above without performing this separation is a difficult task.

[tommyod]

Can you elaborate a bit on this? What do you propose as signatures for fit and update in IES?

[Blunde1]

fit inputs ensemble and outputs and reports on statistical quantities in the estimation
update inputs ensemble and statistical quantities and updates the realizations.

Thinking the standard EnKF, ES, or similar I would like the fit to output (and report) on the estimated covariance matrix. Note that is often not explicitly calculated. The update would be the actual MV-Normal conditional mean update used to update realizations.

Considering Alg. 1 For the IES in Evensen 2019, this would mean the fit calculates and reports on W_i (the "i" is missing in the paper, but step 8, and Ws should be indexed) and related statistical quantities. update performs step 9, but outside of the loop.

The fit + update steps are usually entangled in the data assimilation literature, and combined called "the analysis step". The statistical estimation (that we should report the quality on) is often hidden both notationally and in the numerical linear algebra. I would like to be more explicit about this, and at least be able to draw out (a report on) the estimates.

[dafeda]

Can we close or update this issue now that we have a new API in the iterative_ensemble_smoother package?

[tommyod]

I think we can close it, yes. In the future there could be further discussions on API, but if so they should be written as new concrete issues :)