Green's Functions and Dessler's data

Green's functions? What's that?

Well, I want to talk about applying Bart's analysis to Dessler's ERA data. Tallbloke asked  for this, and so, a while ago did Steve McIntyre.

But the previous discussion at CA was couched in terms of impulse response, and that caused confusion. The word is correct, but to many people it means a causal response in time. You bang something, you get a response afterward. Not before.

To do such an analysis, you need Laplace transforms. But Bart used a FFT, which has no preferred direction. The effect of an impulse spreads in both directions, as in spatial diffusion, for example. And his response function is two-sided.

There doesn't seem to be a universal word conveying that. It's like a digital filter, but people think of that as smoothing, rather than a mapping. Or a weighted moving average.



The idea of a Green's function (GF) comes from the theory of differential equations, or differential operators. I won't explain in detail - for the moment, let's think of it as a two-sided impulse response.

So OK, back to data. Steve has compiled here the data that Dessler used corresponding to the CERES data. It has an explicit CRF function, and also its own ERA temperature. It covers 10 years 2000-2009. So I used that with Bart's algorithm. I did not use Hann tapers on the data, nor did I use Bart's truncation of the GF (impulse response).

Here are some graphs.


Because the GF is two-sided and the FFT output is periodic, I'm now showing the GF (still called impulse response in the titles) about t=0. That doesn't affect the magnitude and phase plots. So results? The one most quoted is the "DC gain" - the area under the GF. That comes out positive. It's just the area under the GF, which is also the zero value of the Fourier transform. That's easy, because the GF is obtained from an iFFT. And that is 5.12 W/m2/°C.

It comes out positive, and people want to interpret it as positive feedback. But there's no basis for that, and you can see why when it is shown to be a much simpler figure. The linear trend of CRF over the period was 0,0364 W/m/°C/yr, and the linear trend of surface temp was 0.0071 °C/yr. The quotient is 5.12 W/m2/°C - just as above. That's all that figure from the Fourier is.

Now if you interpret CRF as entirely determined by T then OK, this trend ratio does determine the factor. But no-one believes that.

The other figure of some interest is the delay of the response (if there is a response). Bart got this by thinking about a damped oscillator, but it's simpler than that. Thinking of the GF as a pdf, say, you just want its mean. Put another way, you want the first moment. And just as the area (0th moment) came from the zero value of the transfer function (FT of the GF), the 1st moment comes from the derivative. And it's 22 months - ie if CRF was determined by T, a characteristic time for the delay would be 22 months. But that's a big if.