There are two key technical approaches that I am using in doing these DiffEq fits.

One is to use the NINO34 data to smooth out the SOI data for calculating the 2nd-derivative *only*. The SOI data is so noisy that the NINO34 is smooth enough that it removes the significant amount of noise that two successive differentiations will introduce.

Second is to apply a hybrid goodness-of-fit metric. The hybrid nature of the metric is that it combines a correlation coefficient (which emphasizes shape of the time series) and absolute error minimization (which reduces the relative error).

The hybrid is essentially $ \frac{CC}{\Delta Err}$

So my process is to apply the hybrid metric first to the Solver. Once that converges to get the scale right, I run a correlation coefficient goal to emphasize the shape. And when that converges, I run the hybrid again to reduce the relative error, which is really an energy minimization -- in that a good correlation needs to be balanced by minimum energy.

With the Excel Solver, the process takes a few hours total and I let it run in the background while I do other stuff. The linear multiple regression solver only takes a second but it can't handle the nonlinear Mathieu modulation, so that is tweaked manually to get a good fit. But the Solver approach is great in that I can start with a completely blank slate and find a largely reproducible solution in just a few hours.

So this is the starting point, with a zeroed model:

![](http://imageshack.com/a/img924/8898/grhQfE.png)

and this is the finishing point:

![](http://imageshack.com/a/img922/677/l0If8t.png)

What the biennial Mathieu modulation does is exaggerate the peaks and valleys until the forcing RHS matches the DiffEq LHS. The Solver iterates on the two sides until it converges to achieve an optimal metric.

And always the caveat in that there is a biennial phase inversion of ENSO between the years 1980-1996. Without that premise, the fit would not work, and which is again the likely reason that the underlying model has escaped notice all these years. Try doing any kind of fit on recent data of the last 50 years without inverting the phase over that 16-year interval and all you will find is anti-correlations and will soon give up. But as Ronald Coates said in the early 1960's *"if you torture the data enough, nature will confess"*.

One is to use the NINO34 data to smooth out the SOI data for calculating the 2nd-derivative *only*. The SOI data is so noisy that the NINO34 is smooth enough that it removes the significant amount of noise that two successive differentiations will introduce.

Second is to apply a hybrid goodness-of-fit metric. The hybrid nature of the metric is that it combines a correlation coefficient (which emphasizes shape of the time series) and absolute error minimization (which reduces the relative error).

The hybrid is essentially $ \frac{CC}{\Delta Err}$

So my process is to apply the hybrid metric first to the Solver. Once that converges to get the scale right, I run a correlation coefficient goal to emphasize the shape. And when that converges, I run the hybrid again to reduce the relative error, which is really an energy minimization -- in that a good correlation needs to be balanced by minimum energy.

With the Excel Solver, the process takes a few hours total and I let it run in the background while I do other stuff. The linear multiple regression solver only takes a second but it can't handle the nonlinear Mathieu modulation, so that is tweaked manually to get a good fit. But the Solver approach is great in that I can start with a completely blank slate and find a largely reproducible solution in just a few hours.

So this is the starting point, with a zeroed model:

![](http://imageshack.com/a/img924/8898/grhQfE.png)

and this is the finishing point:

![](http://imageshack.com/a/img922/677/l0If8t.png)

What the biennial Mathieu modulation does is exaggerate the peaks and valleys until the forcing RHS matches the DiffEq LHS. The Solver iterates on the two sides until it converges to achieve an optimal metric.

And always the caveat in that there is a biennial phase inversion of ENSO between the years 1980-1996. Without that premise, the fit would not work, and which is again the likely reason that the underlying model has escaped notice all these years. Try doing any kind of fit on recent data of the last 50 years without inverting the phase over that 16-year interval and all you will find is anti-correlations and will soon give up. But as Ronald Coates said in the early 1960's *"if you torture the data enough, nature will confess"*.