Thats exactly what I’m saying.

And I’ve already said that A(t) is the albedo offset by a constant (and scaled).

I was raising the issue of the non-constant “restoring” term 0.95*T, a positive change for T < 0, which pushes T up regardless of A(t), but it turns out I didn't even need to make that point ….

Which doesn’t affect the correlation.

… because I’ve been looking at your linear fits to the first 100 points of T(t) and A(t) by running the code concatenated from #46 and #47, and I see a clear—I mean obvious—correlation between the fit slopes of the two quantities.

plot(Results$ATrnd,Results$TTrnd)
ccf(Results$ATrnd,Results$TTrnd) produces quite a spike in the cross-correlation between the two series!!
cor.test( Results$ATrnd, Results$TTrnd ) yields around 0.79 or 0.78 for each run !!

and going further, looking at the correlation between A(t) and the 1st derivative of T(t) by replacing
Results$ATCor[j] = cor(ModelData$Albedo,ModelData$Temperature)^2
with
Results$ATCor[j] = cor(ModelData$Albedo[1:99],ModelData$Temperature[2:100]-ModelData$Temperature[1:99])^2

gives a very different picture – note the R-squared histogram in the lower left.

So I didn’t even need to raise my concerns about the model—the temperature vs A(t) correlations are quite apparent in your model as it is. (Unless I’m doing something REALLY wrong, but darned if I can see what that might be)

And I’ve already said that A(t) is the albedo offset by a constant (and scaled). Which doesn’t affect the correlation.

I agree that my original proposal was carelessly worded. I assumed the relationship of the anomalies with actual albedo/temperature would be obvious, and oversimplified the terminology. It’s exactly the same as when people direct me to graphs of “global temperature rise” and omit to mention that it is actually the global mean monthly adjusted mid-diurnal temperature anomaly – even though the scale shows it centred on zero, such language normally passes without comment or dispute.

What I’m not sure of is whether you’re just being picky about it, or whether you really think it is a serious issue that invalidates the more general point.

I can, if you like, write out the energy balance as a differential equation, apply perturbation theory about the equilibrium point, linearise, and take discrete samples, and get exactly the equation above.

Not necessary, since the logic error has been shown with the model as presented in #26 and #46-48. To reiterate, you claimed, in #24

(1) The albedo causes the temperature to be what it is, but there is no trend in the albedo, while there is in the temperature.

and then stated in #26

(2) A(t) has no trend. (Check that.) But for a high percentage of cases, T(t) does.

It is now clear that A(t) as defined in (2) is not the albedo in (1), thus a fallacy of equivocation has occurred. That A(t) in (2) is uncorrelated with T (#26) does not imply at all that albedo (1) is uncorrelated with T.

You have thus not provided a valid example of this claim from #29

The point is, it is entirely possible to have an apparent trend in the output, with no trend in the input.

which I claim is incorrect.

If you subtract a constant from a set of data, the linear fit still has the same slope.

And as I’ve said repeatedly, it’s not supposed to be a complete model – it’s a toy example for the purposes of illustration.

I can, if you like, write out the energy balance as a differential equation, apply perturbation theory about the equilibrium point, linearise, and take discrete samples, and get exactly the equation above. But such lengthy details are not relevant to the point I’m making. I can do it if it’s the only way to stop you chasing off after distractions, but there’s no point if all you’re doing is clutching at straws to try to avoid thinking about the conclusion?

Scientist – Space and time are curved like a rubber sheet with weights placed on it.
Audience – What sort of rubber, exactly? Is it vulcanised? Your ‘curvature’ hypothesis must be wrong, because with rubber there would be more friction, right?

#56,

“An error source that does not actually produce erroneous results pretty much by definition cannot be called “significant.””

Sigh. As we’ve just pointed out, it does produce erroneous results. If you ask about the trend in the maxima, it produces the wrong answer. If you ask about the trend in the minima, it produces the wrong answer. If you ask about the absolute value, it produces the wrong answer. If you ask about your degree of uncertainty, it produces the wrong answer. If you ask about the diurnal range, it produces the wrong answer. If you mix stations together with a homogenisation adjustment, it will produce the wrong answer. If you ask about regional/local trends, it produces the wrong answer. If you ask about mean trends for different time intervals than the one considered here – since the cancelling errors occurred at different times – you get the wrong answer.

The data is wrong. It produces wrong results. You are doing the equivalent of reading chicken entrails, and claiming that because one of your predictions came out pretty close your method is totally validated and correct and to be trusted.

Getting the right answer by the wrong method is still wrong.

But you do help me immensely by your style and content of argument, which I appreciate. Thank you!

An error source that does not actually produce erroneous results pretty much by definition cannot be called “significant.” In much the same way, subsequent authors using different methods to reproduce the same originally-claimed temperature trend is pretty much by definition a “vindication.”

That yarn you spun about your precious parking lots and jet exhaust has been documentarily falsified. You may now cease talking about it. Don’t worry: you don’t have to admit you were wrong. That would be the part of a legitimate scientific character that even a good con artist would have the most trouble faking. And in any case it would be redundant. I’m just glad I could help. You’re welcome!

Thus A(t), or ModelData$Albedo, is not the albedo source term. The real albedo source is partly in A(t) and partly not, and that part is a quasi-randomly varying fraction.

AFit = lm(Albedo ~ Time, ModelData)

and AFit is therefore not a fit to the physical albedo. Thats where the mistake is.

Yep.

I didn’t express it very clearly. Code does make it much easier.

“However the 0.95T term, which might be interpreted as a cooling or loss term for positive T, is not such for negative T; it is a heating term for negative T.”

Yes. Both the temperature and the albedo are departures from equilibrium. The hidden energy source is the constant subtracted from the albedo to give a delta from zero. The buried energy source is the albedo that leads to equilibrium.

Ah I see what your problem is. You are allowing for negative temperatures and the albedo term to go negative (which is not in itself unreasonable if you are looking at departures from equilibium).

However the 0.95T term, which might be interpreted as a cooling or loss term for positive T, is not such for negative T; it is a heating term for negative T. When T is negative, 0.95T moves T upwards, and thus effectively adds energy automatically under those conditions regardless of the A(t) term. Your model therefore implicitly includes another energy source which is uncorrelated with the random albedo energy source. Essentially there are two conditions which can cause T to go up: 1) A(t) > 0 and also 2) T < 0. And conversely two independent ways for T to go down: 1) A(t) < 0 and 2) T > 0. Since the sign of A(t) is independent of the sign of T, its no surprise no correlation is seen with A(t). You’ve got something else BESIDES albedo built in that boosts the system energy when its below equilibrium.

So back to #24:

The point is admittedly subtle.

the math was, at least for me.

But this last statement does not follow from the earlier statements. There can be a (short-term) trend in the temperature, with no trend in the albedo, and no correlation between temperature and albedo. The albedo causes the temperature to be what it is, but there is no trend in the albedo, while there is in the temperature.

Thats because you implicitly buried a second energy source which is uncorrelated with the albedo. Thats cheating.

You only need 46 and 47.