The math of seasonal aliasing

/ Paul Pukite (@whut)

We use seasonal aliasing of the lunar gravitational pull to generate the incredible fits to the QBO.

How does one get the harmonics of the aliasing?

The starting premise is that a known lunar tidal forcing signal is periodic

L(t) = k \cdot \sin(\omega_L t + \phi)

The seasonal signal is likely a strong periodic delta function, which peaks at a specific time of the year.  This can be approximated as a Fourier series of period 2\pi.

s(t) = \sum\limits_{i=1}^n a_i \sin(2 \pi t i +\theta_i)

For now, the exact form of this doesn’t matter, as what we are trying to show is how the aliasing comes about.

The forcing is then a combination of the lunar cycles L(t) amplified in some way by the strongly cyclically peaked seasonal signal s(t).

f(t) = s(t) L(t)

Multiplying this out, and pulling the lunar factor into the sum

f(t) = k \sum\limits_{i=1}^n a_i \sin(\omega_L t + \phi) \sin(2 \pi t i +\theta_i)

then with the trig identity

\sin(x) \sin(y) = \frac{1}{2} (\cos(x-y)-\cos(x+y))

Expanding the lower frequency difference terms and ignoring the higher frequency additive terms

f(t) = k/2 \sum\limits_{i=1}^n a_i \sin((\omega_L - 2 \pi i)t +\psi_i) + ...

Now you can see how the high frequency \omega_L term gets reduced in frequency by multiples of 2\pi, until it nears the period of the seasonal cycle. And those are the factors that feature in the multiple regression fit. What the regression does is determine the weighting of the a_i terms, across the set of lunar \omega_L terms.

This is a link to the interactive QBO app on the dynamic context server: http://entroplet.com/context_qbo/navigate

12 thoughts on “The math of seasonal aliasing”

    • Keith, $$\psi$$ is the phase angle, which is a constant and required to set the alignment of the sinusoids over time. So that $$\phi_i$$ and $$\theta$$ will combine in some way (not derived because not important) to get $$\psi$$.

      An alternative way to look at this of course is to chart the values on a spreadsheet and sample the data on seasonal boundaries.

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  3. Hi Paul,

    I tried the QBO on the entroplet server, very nice!

    Using only 3 parameters and matching 1953 to 1980, I get a CC of 38.2 vs 44 with 18 parameters. It is not clear that the additional parameters give you a lot, probably better to keep the number of parameters as low as possible to avoid overfitting. I might have this wrong as I am assuming RMS=0 means the parameter was not used (coefficient set to zero in regression). Couldn’t post chart, but the 2.72, 2.37, and 0.5 parameters had RMS>0.

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