For #265, I reviewed an adjoint approach to unwinding a model from the data. In a related context, here is a recent paper based on the methods of Errico(1997).
> Wang, Qiang, Mu Mu, and Guodong Sun. “A Useful Approach to Sensitivity and Predictability Studies in Geophysical Fluid Dynamics: Conditional Nonlinear Optimal Perturbation.” National Science Review, 2019.
The one part I described in comments #254 & #265 is the arcsin() inversion, but another aspect is the forcing time integration which creates the erratic square wave basis. Since that is close to a strongly lagged time integration, the straight-forward adjoint of that is the time derivative!
After applying these two adjoint functions, it may be possible to linearize the problem and invert for the set of tidal factors. I think the only iteration term left is the arcsin transfer factor period.
This would provide the quickest path to an initial guess to the actual forcing, which can then be refined through the non-adjoint forward iteration optimization process. That is essentially the rationale for the work of Wang and before that Errico. The problem there (i.e. with GCMs) is to come up with a plausible set of initial conditions among the infinite number of possibilities. In contrast, for the ENSO model here, it is to efficiently estimate the tidal forcing factors from the much more limited set of possible factors -- and many of these are further constrained by known ephemeris.