Oh, I see. I thought U(t) was autocorrelation describing some solar cycles, or the like. Then I was going to test whether observed T(t) and the autocorrelation U(t) match. If they don't, we know that the solar cycles are not enough, and the hunt for a better model is on.
The essence of the question about linear trend remains the same. If we manage to accept T(t)=a + b*t + random noise, we should be using T(t) as the weather predictor, not a more complex function. So I am a bit at a loss why there is some hope of having a suitable "trend" defined after some 100 years, or ever.
A completely different way of looking at it is: the question of temperature having a trend is like a question of knowing the slope of a derivative. It has no predictive power. (Like, "what's the trend of a sine at 2019?")
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The essence of the question about linear trend remains the same. If we manage to accept T(t)=a + b*t + random noise, we should be using T(t) as the weather predictor, not a more complex function. So I am a bit at a loss why there is some hope of having a suitable "trend" defined after some 100 years, or ever.
A completely different way of looking at it is: the question of temperature having a trend is like a question of knowing the slope of a derivative. It has no predictive power. (Like, "what's the trend of a sine at 2019?")