Monday, May 25, 2009

Climate and Morlet wavelet transform

I was kind of intrigued by some colorful pictures in a paper by
Basil Copeland and Anthony Watts
and decided to develop a more colorful version of their pictures.

I take some time series, e.g. GISS, and compute the inner products with three kinds of localized wavelets of different frequencies. I actually use the ground state and first two excited states of a quantum harmonic oscillator - so they're not quite Morlet wavelets (Gaussians multiplied by plane waves) but rather Motlet wavelets. ;-)

The frequencies are going to be logarithmically encoded on the y-axis: they determine the typical time resolution. The three inner products are rotated in a three-dimensional space so that a uniform warming (shift of the graph in the temperature direction) only changes the brightness.

Finally, they're squeezed into the standardized 3D cube by three Tanh functions and interpreted as RGB colors. The procedure is somewhat clear if you look at the Mathematica notebook. Here are some results:
Jump to the Picasaweb gallery...
Shift/click any picture below to zoom in...
This is how the "purple" global cooling with some white noise looks like:

On the contrary, the color of global warming with white noise is essentially green or cyan and looks like this:

This is how the image should look if the climate were dominated by "global warming". A minimum in the middle is blue:

White noise is still being added. The corresponding maximum would be yellow. It is more useful to look at a warming curve combined with sinusoidal waves (with a 30-year periodicity):

So far, all the datasets were fake. How does the real global mean temperature graph since 1850 look like? Here is the GISS curve:

The NCDC curve is similar but distinguishable. Note the beautiful balance of the Nature.

The figure has no color bias, proving that the variations (the second excited state of the harmonic oscillator) is equally important as the "trends" (contributing primarily to the first excited state). In particular, the real image looks nothing like the cyan "global warming" fingerprint discussed above. "Global warming" is an ill-defined term but in any sensible or moral sense, "global warming" doesn't exist.

In reality, various wiggles occur at all conceivable time scales. The picture resembles an X-ray of your teeth. And yes, the yellow spot in the right lower corner indicates that the temperatures have recently peaked.

Everyone who claims to understand the character of climate change and the type of "randomness" that contributes to it should produce some time series for average monthly temperatures and transform them using my program.

I claim that a vast percentage of the models would generate so bad data - so different from the Brownian-like noise that the temperatures seem to reproduce - that their qualitative difference from the real world would be obvious to the naked eye.

And that's the memo.

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