Re-scaling the AD1000 PC1

I spent quite a few hours (again) last weekend trying to figure that out. I did not succeed. Although it does not matter the slightest bit (the “fix” is arbitrary anyhow), it really annoys me also. Maybe some of the new readers here have better luck?

]]>It would make sense to me to regress temperature on tree rings and CO2 (or its log and/or its square), and then reconstruct pre-instrumental temperature by inputing CO2 from ice cores. There might also be some way to pre-adjust the TR series for CO2 before running temperature on it by itself.

I’ve since evolved (under UC’s influence on the subsequent UC on CCE thread) to think it’s a mistake to regress Temp on Treerings (“ICE”), since this unjustifiably flattens out the response. It’s better to regress Treerings (TR) on Temp and invert the relationship (“CCE”).

In order to adjust for CO2, one would then estimate something like

TR = c + a Temp + b CO2

during the calibration period, and see if b is significantly different

from 0. Then to estimate Temp’ from TR’ and CO2′ during the reconstruction period, solve (in the uniproxy case) for

Temp’ = (Tr’ – c – b CO2′)/a

The estimation error on b will somewhat increase the s.e. of Temp’, but that’s the way it goes when there’s multicorrelation. In the uniproxy case, at least, adding b and CO2 to the equation doesn’t conceptually complicate confidence intervals, since b CO2 just acts like an addition to c.

]]>I have no problem with Manns contention that whetever fertilizing CO2 may have may saturate after some point.

On the other hand, I have a huge problem. There is no measurement confirmation of it.

The anomaly, if it is real, could just as well be caused, for all we know, from selenium dust in the air generated from increased wear of Xerox drums pumping out copy papers on dendrothermometry.

]]>However, it is pretty meaningless as a CO2 adjustment.

This adjustment has a meaning. Without it you wouldn’t see any substantial secular peak (significant to red noise), related to astronomical forcing, which is thought to have driven long-term temperatures downward since the mid-Holocene. Dr. Thompson’s Thermometer verifies this pre-industrial cooling.

]]>In the first thread, you waste a lot of time trying to figure out how they scale and shift their CO2 series, but in fact it doesn’t matter, since they don’t use its numerical value for anything. In fact, all they do is compare it visually to their “residual” and decide that the “residual” is a good proxy for CO2 fertilization. The scale of the CO2 graph is only relevant insofar as it allows the line to go off the top of the chart around 1900 (just when actual CO2 is starting to grow more quickly), thereby disguising how poorly the two series n fact fit one another. Anyway, the “secular trend residual” flattens out around 1900, which apparently serves their purposes better than any actual function of CO2.

Let y be the “unadjusted” PC1, and x be the Jacoby and D’Arrigo NT series. Let sy represent the filtered value of the series y using their 50-year filter s, whatever its details are, and sx represent the filtered value of the series x. They define their residual R = sy – sx, but since this is presumably a linear filter, this is equivalent to s(y-x), ie applying the filter s to the differences directly. They then apply a 50-year filter to R, and then a further trend extraction that retains timescales longer than 150 years after 1700 and simply flattens the series before 1700. Call the combined effect of these two filters S, so that the resulting adjustment is SR = Ss(y-x) = Ssy – Ssx. Their “adjusted” PC1 is then y* = y – SR = Ssx + (y – Ssy). In other words, after 1700 they just replace PC1 with the doubly smoothed NT series, and then add the high frequency variation in PC1 back in to make it look like PC1 even though it has the trend of NT. Before 1700 the procedure does nothing.

Note that they would get exactly the same result by applying the compound (triple, in fact) filter Ss directly, either to x and y individually or to their difference. The intermediate series sx and sy are just red herrings that complicate and obscure (one might say obfuscate) the calculation. In any event, the adjusted series has absolutely nothing to do with the numerical values of their CO2 series (whatever it is).

The precise result they get does depend on how they normalize the two series. From the fact that their “secular trend” dies out smoothly to exactly 0 at 1700, I would assume that they are taking 1700 as a breakpoint of sorts, and subtracting the mean of 1400-1700 from both series to give them the same (0) average there. Then they probably chose the vertical scale so as to equate their variances or some such.

Both series do in fact have interesting big synchronous dips around 1460 and 1820, plus some smaller ones. 1820 would be Tambora (Volcanic Explosivity Index level 7, 1815, dixit Wiki) and 1460 perhaps the smaller Kuwae explosion (VEI 6, 1452 or 53), so evidently they are both actually picking up some kind of signal. But is this signal temperature, or sunshine per se? It could be either.

Although this adjustment weakens the contrast between present and past temperatures, it apparently makes the adjusted series fit the instrumental data (which isn’t already on the rise in 1850 the way the raw PC1 is) better than the raw series, and so was cherry picked. However, it is pretty meaningless as a CO2 adjustment.

I have no problem with Mann’s contention that whetever fertilizing CO2 may have may saturate after some point. Using log(CO2) for starters, and then maybe trying even a nonlinear function of that (constrained to be monotonic, perhaps) would be reasonable. It would make sense to me to regress temperature on tree rings and CO2 (or its log and/or its square), and then reconstruct pre-instrumental temperature by inputing CO2 from ice cores. There might also be some way to pre-adjust the TR series for CO2 before running temperature on it by itself. If the effect is nonlinear, that is fine, but the possibility that it might be is no excuse for substituting thie “R” for CO2 itself.

]]>I agree that the White’s get less precip. Even further east though where I was living in 2004 on the eastern slopes of Mt. Charleston near Las Vegas, the tree rings there show dramatic increases in growth rates within the last 40-50 years. I have pictures of stumps cut in the summer of 04 where this is clearly shown at a location in Lee Canyon at about 8560 ft altitude. There are rain gauges up there as well with fairly long records of rain and snowfall, especially on Mt. Charleston. This should bracket the White mountains to give a measure of control studies over what happens up there. I’ll dump a couple of them into Google earth so that you can look at the rings. They did not show up very well but you should be able to get an indication of the recent growth spurt.

]]>I did consider doing a short-time fourier transform to investigate the end effects, but once I took the view that the filtering was done in the frequency domain, I started to investigate that more. I did try earlier a DFT on the centre half (which means bin alignment is closely related between the curves), but couldn’t see anything significantly different jumping out at me. The magnitude and angle of the first bins (reduced to 5 in this case) still didn’t really tell much of a story; they tracked, but frustratingly imperfectly, with no obvious relationship.

]]>That is pretty much the conclusion I came to – with one small difference. I don’t think the smooth curve is a reduction of the data at those frequencies – I think the mid-to-high frequencies have just been zero’d out. The smooth curve is just an artefact – probably spectral leakage either from a strong point in the data, or some post-processing operation on the data.

My evidence for this is looking at the phase angle of the complex result – the angle, much like the amplitude, is smooth. If it were just a magnitude reduction of the original data, it would have retained the phase. Phase angle plot below. Once again it tracks closely to the original data for the first 9 bins (imperfectly with no obvious simple relationship), then just follows a smooth curve. Here, though, the last ten bins show some curious results – so maybe there is something more in these; not sure what it is though. It doesn’t seem to be rounding error, as the magnitude is similar to those of the mid bins (see #57), so if it was rounding, they would be affected too.

Figure: complex angle plot of jacoby raw (blue) and smooth (red)

]]>Try with middle parts of data, I guess there’s some end-point operation that is done separately

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