He used to frequent the Azimuth Forum, but not recently.

]]>In this two-part series, I’ll cover the Azimuth stochastic resonance example program, by Allan Erskine and Glyn Adgie. In the Azimuth blog article Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek give a nice explanation of the idea of stochastic resonance, which includes some clear and exciting graphs.

]]>This is the Southern Oscillation Index with connections to the Chandler Wobble:

http://contextearth.com/2014/04/05/the-chandler-wobble-and-the-soim/

Graham wrote:

So even with A=0, the system will have some kind of power peak at some frequency. Maybe best to understand this better first…

I agree that the ‘confining’ effect of the potential should damp out low-frequency noise—I don’t know this, but it seems plausible. Understanding these things analytically seems tough, though. Mathematically it’s a bit like nonlinear quantum field theory, which is pretty easy perturbatively but hard for nonperturbative questions, while I believe is what we’ve got here. So it might be easier for someone who likes programming to study this that way.

]]>The Wiener process (=Brownian motion) on its own would have a power spectrum proportinal to 1/frequency-squared according to http://en.wikipedia.org/wiki/Brownian_noise#Power_spectrum. With A=0, there is the ‘double dip’ which stops it wandering off, and that will, I think, make some sort of low frequency cutoff. So even with A=0, the system will have some kind of power peak at some frequency. Maybe best to understand this better first…

]]>When Glyn and Tim wrote this article, I became curious about this question. I didn’t make a lot of progress. But if you state the question more precisely, that’ll make it easier to answer.

In fact, if you state it precisely enough, that will *be* the answer!

Do you want to see how much a sinusioidal signal of any given frequency gets amplified? This is easy to make precise for *linear* time-invariant systems using Fourier transform techniques. But here we are dropping linearity, though we are keeping time-invariance (if we time translate the input, the output gets time translate by the same amount). This means the answer can depend in a complicated nonlinear way on the amplitude of the input as well as its frequency. So more information is needed to make the question precise.

I should have said “what are good candidates” for such a measure.

But maybe it’s more natural and clear to have two separate measures, one for how much of the input signal gets through, and how much is it the only thing that gets through (“recall”, and “precision”), For the former we could use the frequency component in the output, with a deduction for the component at that frequency in the noise (don’t want to give too much credit to the noise).

For the latter, we could use a measure of how much the entire spectrum is concentrated at the signal frequency. But does that involve choosing an arbitrary bandwidth.

I’d like to program this. If anyone can propose an appropriate, concrete spec, that would be great.

]]>John wrote:

While the total amount of solar radiation delivered to the Earth doesn’t change much…

I was referring to changes in total annual insolation due to Milankovitch cycles, since that is what David was asking about: we were talking about how Milankovitch cycles may trigger glacial cycles. I was not referring to the seasonal cycles in insolation due to the elliptical orbit of the Earth, nor to changes due to inherent changes in the luminosity of the Sun.

I calculated the changes due to Milankovitch cycles in Part 9 of the Mathematics of the Environment series: they cause a change of 0.167% in the total annual amount of solar energy delivered to the top of the Earth’s atmosphere.

How does the 1366±2 W/m

^{2}go together with the 6.9% [seasonal variation due to the elliptical shape of the Earth’s orbit]?

I believe the figure of 1366±2 W/m^{2} is the figure averaged over a year.

Is there a latitudinal dependence and if yes how big is it?

These figures refer to the amount of solar radiation a square meter directly facing the Sun at the top of the Earth’s atmosphere would receive. So, it has nothing to do with the slant of the Earth’s surface with respect to the rays of light from the Sun, the length of the daytime, or the light absorbed by the Earth’s atmosphere, all of which depend on the latitude (and other things).

All these other factors are incredibly important too, but they are not what is being discussed here!

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