> "How do you calculate the second derivative of ENSO? Generally speaking it's good to do it by smoothing with some kernel first. "
Thanks for the question. I think doing the second derivative correctly is part of the secret recipe to this analysis. I'm sure many people have taken the 2nd derivative of the raw ENSO and just about gagged when they noticed how noisy it is. Yet, there is a REAL signal inside that noise.
Nick Stokes at the Moyhu blog showed how one can either filter before calculating the 2nd derivative or delay the filtering step and do it after. I do it before using something that approximates a Gaussian filter -- a triple filter with staggered window widths, say 7+5+3 months. I try to keep that as minimal as possible, just enough to remove the seasonal noise.
I have a "kind of proof" that this works -- with the forcing solution it approximately finds, I then apply the result and do the double integration using Mathematica. It ends up with something that closely approximates the smoothed ENSO. This is actually a pretty cool pre-conditioning step in solving a DiffEq.
As a bottom-line, the assumption is that the real ENSO effects are on scales greater than 1 year, and those variations within a year are caused by weather and typhoon activity that is more-or-less random. The latter is all that we want to filter out. Yet, as a caveat, there may be something interesting at the monthly level that we can also eventually root out.