Climate scientists as a general rule don’t understand crystallography deeply (I do). They also don’t understand cryptography (that, I don’t understand deeply either). Yet, as the last post indicated, knowledge of these two scientific domains is essential to decoding dipoles such as the El Nino Southern Oscillation (ENSO). Crystallography is basically an exercise in signal processing where one analyzes electron & x-ray diffraction patterns to be able to decode structure at the atomic level. It’s mathematical and not for people accustomed to existing outside of *real space*, as diffraction acts to transform the world of 3-D into a *reciprocal space* where the dimensions are inverted and common intuition fails.

Cryptography in its common use applies a *key *to enable a user to decode a scrambled data stream according to the instruction pattern embedded within the key. If diffraction-based crystallography required a complex unknown key to decode from reciprocal space, it would seem hopeless, but that’s exactly what we are dealing with when trying to decipher climate dipole time-series -— we don’t know what the decoding key is. If that’s the case, no wonder climate science has never made any progress in modeling ENSO, as it’s an existentially difficult problem.

The breakthrough is in identifying that an analytical solution to Laplace’s tidal equations (LTE) provides a crystallography+cryptography analog in which we can make some headway. The challenge is in identifying the decoding key (an unknown forcing) that would make the reciprocal-space inversion process (required for LTE demodulation) straightforward.

According to the LTE model, the forcing has to be a combination of tidal factors mixed with a seasonal cycle (stages 1 & 2 in the figure above) that would enable the last stage (Fourier series a la diffraction inversion) to be matched to empirical observations of a climate dipole such as ENSO.

The forcing key used in an ENSO model was described in the last post as a predominately *Mm*-based lunar tidal factorization as shown below, leading to an excellent match to the NINO34 time series after a minimally-complex LTE modulation is applied.

Critics might say and justifiably so, that this is potentially an over-fit to achieve that good a model-to-data correlation. There are too many degrees of freedom (DOF) in a tidal factorization which would allow a spuriously good fit depending on the computational effort applied (see **Reference 1** at the end of this post).

Yet, if the forcing key used in the ENSO model was **reused as is **in fitting an independent climate dipole, such as the AMO, and this same key required little effort in modeling AMO, then the over-fitting criticism is invalidated. What’s left to perform is finding a distinct low-DOF LTE modulation to match the AMO time-series as shown below.

This is an example of a *common-mode cross-validation* of an LTE model that I originally suggested in an AGU paper from 2018. Invalidating this kind of analysis is exceedingly difficult as it requires one to show that the erratic cycling of AMO can be randomly created by a few DOF. In fact, a few DOFs of sinusoidal factors to reproduce the dozens of AMO peaks and valleys shown is virtually impossible to achieve. I leave it to others to debunk via an independent analysis.

addendum: LTE modulation comparisons, essentially the wavenumber of the diffraction signal:

This is the *forcing *power spectrum showing the principal * Mm *tidal factor term at period 3.9 years, with nearly identical spectral profiles for both ENSO and AMO.

According to the precepts of cryptography, decoding becomes straightforward once one knows the key. Similarly, nature often closely guards its secrets, and until the key is known, for example as with DNA, climate scientists will continue to flounder.

** References **

- Chao, B. F., & Chung, C. H. (2019). On Estimating the Cross Correlation and Least Squares Fit of One Data Set to Another With Time Shift.
*Earth and Space Science*, 6, 1409–1415. https://doi.org/10.1029/2018EA000548

“*For example, two time series with predominant linear trends (very low DOF) can have a very high ρ (positive or negative), which can hardly be construed as an evidence for meaningful physical relationship. Similarly, two smooth time series with merely a few undulations of similar timescale (hence low DOF) can easily have a high apparent ρ just by fortuity especially if a time shift is allowed. On the other hand, two very “erratic” or, say, white time series (hence high DOF) can prove to be significantly correlated even though their apparent ρ value is only moderate. The key parameter of relevance here is the DOF: A relatively high ρ for low DOF may be less significant than a relatively low ρ at high DOF and vice versa.*“

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