“If you’re a meteorologist, you have to decide weather or not…” — Mark Simmons

Are you ready to hear a tale of horror? Then settle down and listen closely. Especially you, meteorologists. You know what you did((Or at least, you will by the end of this.))

Everybody knows what a 50-year flood is, right? It’s a flood-level you expect to see once every fifty years. More precisely, it’s a level such that the probability of seeing an event as severe (or worse) in a random year is 0.02.

Or is it?

Well yes, it is. That’s the definition. But there’s a problem, especially with fairly common events.

JUMP SCARE! A one-year event isn’t the same thing as a 365-day event.

Once you’ve picked up all of the dropped popcorn and repaired the eardrums ruptured by the screaming, let’s look at that more closely.

A one-year event… isn’t really a thing. It’s a level that (according to the definition) is guaranteed to happen every year. If anything, it’s a non-event — I’m looking at hourly rainfall, so a one-year event is exactly zero, the only level of rainfall that’s guaranteed to happen every year.

A 365-day event is one that occurs with a probability of $p=\frac{1}{365}$ on any given day. That’s actually a 1.58-year event — the probability of it not occuring in a given year is $(1-p)^{365} \approx \frac{1}{e}$, so the probability of it happening in a year is $1 - \frac{1}{e}$, or $\frac{e-1}{e}$; the reciprocal of that is the (meterorological) return period in years.

I’m deeply unhappy about this: it feels profoundly wrong that a one-year event and a 365-day event should be different. I propose instead a more mathematically delightful Continuous Return Period.

If we assume events follow a Poisson process — they occur at random in continuous time with a parameter $\frac{1}{T}$ (in units of $\mathrm{time}-1$, so that the number of events $N$ in a time interval $t$ is distributed as $N \sim \mathrm{Poisson}\left(\frac{t}{T}\right)$ — then the CRP is $T$.

Our earlier (meteorological) 50-year event had a probability of 0.02 of occuring in a year, or (equivalently) a probability of 0.98 of not occuring in $t = 1 \mathrm{year}$. If $N \sim \mathrm{Poisson}\left(\frac{t}{T}\right)$, then $P(N=0) = e^{-\frac{t}{T}}$, so we have $e^-{\frac{t}{T}} = 0.98$. Taking logs and cranking the handle gives $T \approx 49.5$ years, which is certainly a workable approximation. For rare events, the CRP and the meteorological return period are generally close together.

What about our 365-day event? The same analysis gives $T \approx 364.5 \mathrm{days}$, which again feels like it’s what it ought to be.

We can readily convert the other way, too, in a similar way to before: if the Continuous Return Period is $T$ and $Y$ is one year, the meteorological return period is $\frac{ e^{\frac{Y}{T}}}{e^{\frac{Y}{T}}-1}$.

Don’t get me wrong: I don’t expect the entire field of meteorology to adopt this, however much more sensible and mathematically rigorous it is. But it definitely should.