From comment #254 above, I was asking about coming up with an adjoint approach to inverting the sin(f(t)) model for ENSO. It's obvious that we can try arcsin to do the inversion, but this generates a solution only on a limited domain. I tried guessing what an additional phase shift can be, but this induces a correlation that is highly artificial (e.g. it's very easy to get a CC over 0.9 without trying hard). After normalizing the input ENSO to peaks of ±1 across the time-series range, this is what the fit looks like:
The level shifts derive from the harmonic folding frequency that was inferred from the forward sin(f(t)) fit. From that fit, there is a strong fundamental frequency shown below that reproduces several phase shifts in the fit above.
So I think it is interesting that an inversion might be possible, although it might not help in making the fit process more efficient as its parametric selectivity is low in comparison to the high sensitivity of the forward fit.