Thanks for catching those typos, David. Almost everything you say looks fine to me!

But this formula

> $$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} \tilde{T}(\ell,y,d) \tilde{T}(r,y,d+\tau) $$

still has a serious typo in it, due to me - there should not be tildes on those $T$s. Here's the right formula:

$$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$

The point is that it would be silly to define a correlation between temperatures $\tilde{T}$ using the top formula; we need to subtract off their means and use $T$s instead: we don't want the correlation to be big just because it's hot all the time!

I have taken the liberty of rewriting history and fixing my [original comment](, taking into account your typo fixes and also this.

In general people write $\langle X \rangle $ for the mean of any random variable $X$, so the standard deviation of $X$, say $Std(X)$ in your notation, is

$$ Std(X)^2 = \langle (X - \langle X \rangle)(X - \langle X \rangle) \rangle $$

Given two random variables $X$ and $Y$ we compute their correlation by first working out

$$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$

and then normalizing it by dividing by $ Std(X) Std(Y)$: this makes the result lie between $-1$ and $1$.

I hope you see that

$$ X(\ell,r,y,\tau) = \frac{1}{365} \sum_{d = 1}^{365} T(\ell,y,d) T(r,y,d+\tau) $$

is almost an example of this

$$ \langle (X - \langle X \rangle)(Y - \langle Y \rangle) \rangle $$

concept: we've taken $ \tilde{T}(\ell,y,d)$ and $\tilde{T}(r,y,d+\tau) $, subtracted off their mean (average over all years), multiplied them, and then taken the mean of that (but now averaging over days of a given year). In the paper it looks like they use $\langle \rangle_d$ and $\langle \rangle_y$ to mean averaging over days of a given year and averaging over years.

More to say, but it's my bedtime!