> "Lunisolar Earth tides start as a three-body system, so chaos is predicted from the start."

You may be inadvertently supporting my lunar-induced Chandler wobble model. The 3-body system is intractable because of the mutual dynamics between the bodies. But what else is the moon's torque on the earth's axis due to its [declination cycle](https://en.wikipedia.org/wiki/Lunar_standstill) -- and a synchronization with the solar declination cycle (i.e. seasonal) -- but a mutual interaction? This is a slight effect of course and it is not chaotic on human-scales because it really is just a perturbation of a small body acting on a larger body.

> "I suspect both the inherent risk of overfitting from noisy data that you cite"

As far as overfitting is concerned, it is entirely possible, but I take pains to minimize the degrees of freedom (DOF) in the model. For the ENSO tidal model, the precise formulation of parameters has to fit into a narrow range so as to follow the orbital path.

![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Lunar_orbit.png/450px-Lunar_orbit.png)

This is a good recent reference that lays out the resolved tidal forcing http://phy.hk/PP/How_tidal_forces_cause_ocean_tides.pdf

The important projection is described in the figure below where it is critical to consider the horizontal tidal force, which then not only brings the lunar distance into the equation (the monthly anomalistic tidal cycle **Mm**) but also the lunar declination with respect to the equator (the nearly symmetric fortnightly draconic or nodal cycle **Mf**).

![](https://imagizer.imageshack.com/img922/2346/2B80I9.png)

So the simplest formulation I applied is a linear combination of the two applied to a 1/R^3 gravitational pull.

$$~\frac{1}{(1 + a Mf + b Mm)^3}$$

This minimal expression generates a simple waveform in the denominator

![](https://imagizer.imageshack.com/img924/8088/u05MbM.png)

which is large enough in amplitude to match the rich spectrum of tidal harmonics observed when the R^3 Taylor series expansion is applied. It also fits the ENSO time-series with the LTE modulation applied, along with a few secondary correction terms as shown above.

So essentially only 3 DOF -- 1 each for **Mm** and **Mf**, and a third for the LTE modulation is not overfitting at all to get an impressive fit for the ENSO time-series. It is only difficult because the slight differences in **Mm** and **Mf** give a nonlinearly modulated repeat cycle of over 180 years, so no wonder that the pattern has remained elusive for these many years. It's all straightforward to do but the results are unforgiving if you don't formulate the set of parameters precisely.

Good news is that my abstract for the upcoming EGU meeting was accepted today so that I will once again be able to present the findings to a wider audience.

You may be inadvertently supporting my lunar-induced Chandler wobble model. The 3-body system is intractable because of the mutual dynamics between the bodies. But what else is the moon's torque on the earth's axis due to its [declination cycle](https://en.wikipedia.org/wiki/Lunar_standstill) -- and a synchronization with the solar declination cycle (i.e. seasonal) -- but a mutual interaction? This is a slight effect of course and it is not chaotic on human-scales because it really is just a perturbation of a small body acting on a larger body.

> "I suspect both the inherent risk of overfitting from noisy data that you cite"

As far as overfitting is concerned, it is entirely possible, but I take pains to minimize the degrees of freedom (DOF) in the model. For the ENSO tidal model, the precise formulation of parameters has to fit into a narrow range so as to follow the orbital path.

![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Lunar_orbit.png/450px-Lunar_orbit.png)

This is a good recent reference that lays out the resolved tidal forcing http://phy.hk/PP/How_tidal_forces_cause_ocean_tides.pdf

The important projection is described in the figure below where it is critical to consider the horizontal tidal force, which then not only brings the lunar distance into the equation (the monthly anomalistic tidal cycle **Mm**) but also the lunar declination with respect to the equator (the nearly symmetric fortnightly draconic or nodal cycle **Mf**).

![](https://imagizer.imageshack.com/img922/2346/2B80I9.png)

So the simplest formulation I applied is a linear combination of the two applied to a 1/R^3 gravitational pull.

$$~\frac{1}{(1 + a Mf + b Mm)^3}$$

This minimal expression generates a simple waveform in the denominator

![](https://imagizer.imageshack.com/img924/8088/u05MbM.png)

which is large enough in amplitude to match the rich spectrum of tidal harmonics observed when the R^3 Taylor series expansion is applied. It also fits the ENSO time-series with the LTE modulation applied, along with a few secondary correction terms as shown above.

So essentially only 3 DOF -- 1 each for **Mm** and **Mf**, and a third for the LTE modulation is not overfitting at all to get an impressive fit for the ENSO time-series. It is only difficult because the slight differences in **Mm** and **Mf** give a nonlinearly modulated repeat cycle of over 180 years, so no wonder that the pattern has remained elusive for these many years. It's all straightforward to do but the results are unforgiving if you don't formulate the set of parameters precisely.

Good news is that my abstract for the upcoming EGU meeting was accepted today so that I will once again be able to present the findings to a wider audience.