Thursday, September 15, 2011

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.


  1. Nick, big thanks for going to the trouble at my request. I don't understand much of what is going on here, but I did find it noticeable that the smoothed impulse graphs you generated using Spencer and Desslers chosen datasets: CERES and ERA re-analysis are pretty much the mirror image of each other. Did someone get their dataset upside down?

  2. TB,
    Firstly, a big Oops. I tend to write related posts on the same file, reusing some of the same html stuff. This time I included the older posts at the bottom, which got confusing. I've fixed that.

    An interesting observation, but I don't think it's a mistake. What happens is that the transfer functions (h) have a rather similar shape, because they include features relating to the data cut-off. But they can readily change sign. The thing I keep pointing out is that the figure held to be important, the area under h, is just the ratio of the trends of the two data sets. And it happens that the ERA set has a positive trend, while CERES is negative. The temp is positive in each case (starting early 2000). That's what flips the graphs. And if we cut out 2000, the trends in hadcrut3 at least would change and the CERES-based graph would flip (and the imputed feedback).

    So if h has to have the same shape, and area of different sign, that's what you get.

    Still, there are some interesting features. The peak at about 2 years makes sense - they do seem to have a delayed association. The peak at zero also makes some sense, but it certainly is spiky.

    What I'd really like to understand is why one side is much more oscillatory than the other.

  3. Hi Nick,
    The main difference between the data used by Spencer and Dessler is Spencer using CERES clear sky where Dessler uses ERA re-analysis. In your previous post you said:
    "col 5 is TOA flux measured by CERES, and col 8 is a notional clear sky value. The cloud effect dR is got by subtracting col 8 from col 5. Dessler points out that this is biased, because it in effect extrapolates from clear patches of sky to the whole atmosphere, and the air there is much drier than the global average. So there is less wv IR absorption. He used reanalysis data which corrects for this (but may bring other problems)."

    But isn't the whole point of subtracting clear sky from whole sky to get the cloudiness fraction where most of the moisture is? What does a global average moisture content have to do with it?

    "And it happens that the ERA set has a positive trend, while CERES is negative."

    Are we talking about the Cloud RF data here? Because I get a similar positive trend for both:

  4. Hi Nick,

    I'm afraid I can't interpret these charts but I do note you write, "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."

    So are you saying in your view neither analysis - Spencer nor Dessler - is telling us anything?

  5. Also with the temperature data, a similar positive trend for both. So the ratios making such a difference to the imputed feedback seems a bit hard to understand. Why should the model be so sensitive to small differences in the datasets used?

  6. Alex,
    No, I'm saying that Bart's system analysis is not telling us anything. Which is not to say that it isn't an interesting idea, and maybe could be made useful - probably with Laplace transforms. But it's failure has no implication for Spencer or Dessler.

  7. TB,
    The underlying query is why is the CERES-based nett flux trending down, while the ERA-based CRF is trending up? Well, as I said, Dessler gave a cogent reason for not just subtracting the CERES clearsky flux from the total, and I guess this reflects it.

    "Why should the model...,"
    If you mean Bart's model, then the reason is that it has the temp trend in the denominator, and that is small and of fluctuating sign, over that decade. That doesn't say anything about anyone else's model.

  8. TB #4,
    Dessler says that the CRF that he's looking at is what you would get if you removed clouds, leaving everything else fixed. Subtracting clear-sky from total means in effect that what you see in clear-sky regions is extrapolated to whole earth. Humidity included.

    So when it comes to the LW component of flux, you're subtracting a much drier air from the real air of the total model.

  9. "TB,
    The underlying query is why is the CERES-based nett flux trending down, while the ERA-based CRF is trending up?"

    You lost me Nick. The net radiative flux after I subtracted the clear sky from all sky CERS data is trending up, as my graph shows.

    Or are you referring to something else?

  10. Well, TB, that's a mystery. I get the same plot as you do for ERA, but not for CERES. My CERES looks somewhat similar, but definitely trends down. My numbers match Bart's.

    Did you use the file flux.csv from Steve's site? Subtract col 8 from col 5?

  11. Nick, Yes, I subtracted col8 from col5.

    I'll double check my result in a while and confirm.

  12. Nick, I got it wrong, but this error has led to an interesting dicovery. I'll blog about this later.

    Also, here is a comparison of the ERA and HADcru temperature series with trendlines: