@NiV: <i>A further improvement could to use a sinc-shaped filter to get a square-topped frequency response, passing through all the data for which signal is greater than noise. It depends.</i>
Quite right. The exact width and rolloff of each band determines the rate at which the side lobes decay. If you want a modestly square octave-wide frequency response you can get this using the <a href="https://en.wikipedia.org/wiki/Mexican_hat_wavelet" rel="nofollow">Mexican hat wavelet</a>, aka the Ricker wavelet, which decays the side lobes relatively quickly.
Although it's far from obvious from the way I described the filters in my AGU spreadsheet, this is in fact how the four filters (serving to separate four octaves) actually worked in cascade. To see this, download <a href="http://clim.stanford.edu/hadcrut3.xls" rel="nofollow">my spreadsheet</a> and set columns AN52:AN212 to zero. (Enter 0 in AN52, select the range AN52:AN212, and type ^D.) Now enter 0.4 in AN130. This will serve as the impulse (scaled to 0.4 so as not to require rescaling the plots), allowing you to observe the impulse response at each of the subsequent filters.
There are four outputs to observe, which can be seen graphically two at a time in respectively Figures 9 (SOL, accessed via the tabs at the bottom) and 12 (DEC). All four plots are of approximate Mexican hat wavelets of progressively decreasing width as you go up in frequency, an octave (more or less) at a time.
Nowadays I skip all this cascading stuff and just work directly with the wavelets needed for any given band.
@NiV: <i>But this application makes no sense unless you have clearly defined what you mean by ‘signal’ and ‘noise’. </i>
A point that I had hoped would be evident from my poster, but which in hindsight clearly wasn't, is that it made no distinction between signal and noise but treated all frequency bands in the signal as equally important. <i>I discarded none.</i> The last figure, <a href="http://clim.stanford.edu/Fig11.jpg" rel="nofollow">Figure 11</a>, collects all the bands into three main groups, namely MUL (multidecadal), SOL (solar), and DEC (decadal). Their sum is <i>exactly</I> HadCRUT3. I discarded nothing!
