> John, leaving aside the fact that you think the definitions were not appropriate, were you able to parse them, in detail, and did you find them to be fundamentally well-defined?

They seemed well-defined to me.

They are using abbreviations and shorthands the way physicists and many other scientists do. For example, when they have a function of several variables, they don't bother to write the variables that they are uninterested in at that moment. The idea is that you're supposed to figure out what the notation means by thinking about what would be the only reasonable thing for it to mean. The idea is that writing things out in a completely unambiguous way makes for big ugly expressions which they'd rather avoid.

Since this seems to drive computer programmers crazy, I will modify their notation a bit.

> For instance, they didn’t define their angle brackets...

Angle brackets are a standard symbol for an arithmetic mean, or average. So is an overline or a capital letter $E$.

So, here is what they are doing. They start with a bunch of temperatures $\tilde{T}(\ell,y,d)$: one for each grid point $\ell$, year $y$ ($1 \le y \le N$) and day of the year $d$ ($1 \le d \le 365$). Then they consider

$$T(\ell,r,y,d) := \tilde{T}(\ell,y,d) - \frac{1}{N} \sum_{y = 1}^N \tilde{T}(\ell,y,d)$$

This tells you how much "hotter it is today than it usually is here at this time of year".

Then, for two grid points $\ell$ and $r$ and a year $y$ they work out a kind of correlation between $\tilde{T}(\ell,y,d)$ and $\tilde{T}(\ell,y,d+\tau)$ where $\tau = 1,2,3,\dots$) :

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

I assume that when $d+\tau$ exceeds 365 you go over to the next year, since that's the reasonable thing to do.

The point is that $X(\ell,y, \tau)$ tells you how much the temperature at grid point $\ell$ is correlated to the temperature at grid point $r$, $\tau$ days later, during year $y$.

Does this make sense so far?

So far this is what anyone would be tempted to do when studying how a randomly fluctuating quantity _here and now_ is correlated to the same quantity _somewhere else and a bit later_. The fun part starts later.