The 475-year earthquake is not geology
Seismic design maps carry two strange numbers: 475 years and 2,475 years. They sound like the memory of a fault. They are nothing of the kind.
Seismic design is built around two hazard levels: a 10% chance that the design shaking is exceeded in 50 years, and — for the structures that must not fail — a 2% chance in the same 50 years. Written as return periods, those become the 475-year earthquake and the 2,475-year earthquake, and in that form they take on the texture of geology, as though someone had counted the scars in a fault trench and found a 475-year rhythm.
Nobody counted anything. Both numbers are the output of one line of arithmetic applied to a decision that engineers, not faults, had to make.
The line
If an event has probability 1/T of happening in any given year, and years are treated as independent draws, then the chance of seeing at least one in n years is
risk = 1 − (1 − 1/T)n
Turn it around and ask the engineer's question instead. Given a building you intend to stand for 50 years, and a failure probability you are willing to tolerate over that life, what return period must you design for?
T = 1 / (1 − (1 − risk)1/n)
Put 10% and 50 years into it and out comes 475.1. Put 2% and 50 years into it and out comes 2,475.4.
That is the whole provenance of both famous numbers. Someone chose a design life of fifty years. Someone chose to accept a one-in-ten chance of exceedance across that life for ordinary buildings, and a one-in-fifty chance for the ones that must not fail. The geology arrives afterwards, and it answers a different question: given that return period, how hard does the ground shake at this particular site? The 475 is the policy. The shaking is the science.
The same line, in a river
The phrase that does the most damage is not on an engineering drawing at all. It is on an insurance letter: the 100-year flood.
It means a flow with a 1% chance of being exceeded in any year. It does not mean a schedule, and the arithmetic above says what it does mean over a span a person actually cares about:
- Over a 30-year mortgage: 26.03%
- Over 50 years: 39.50%
- Over a century: 63.40%
Read the last one the other way round, because that is the way it lands. The chance that a "100-year flood" does not arrive during a hundred years is 36.60% — better than one in three. The name suggests a metronome. The mathematics describes a die.
The tidiest number in the topic
Run any hazard for exactly its own return period — a 10-year event over 10 years, a 500-year event over 500 — and the answer barely moves:
- T = 10: 65.13%
- T = 100: 63.40%
- T = 500: 63.25%
It converges on 1 − 1/e = 63.21%. Waiting one full return period gives you about a 63% chance and never a certainty, and it hardly matters which hazard you picked. A slider that spans two orders of magnitude and refuses to move the answer is a rare thing to watch.
Why the misreading is expensive
Two "100-year floods" in a decade get reported as proof that the statistics are broken. They are proof of nothing at all: at one site the chance of two exceedances in ten years is small but unremarkable, and across the thousands of gauges in a country, somewhere gets a pair almost every year. That is the coincidence arithmetic, not a failure of hydrology.
The genuine caveat is elsewhere, and it deserves stating plainly rather than being used as a rhetorical escape hatch. The formula assumes each year is an independent draw from an unchanging distribution. Both halves are approximations. Floods cluster — a wet decade is a real thing — and a distribution fitted to a record from the last century may not describe the next one. When someone says the hundred-year flood is arriving more often, the substantive version of that claim is not that the arithmetic is wrong; it is that the 1% was estimated from a world that has since moved.
Two compressions, one habit
There is a family resemblance to the other number that gets misread on the same subject. A magnitude scale compresses energy logarithmically, so a 7 is about 31.6 times a 6 rather than one seventh more. A return period compresses probability, so 475 is a tolerance rather than a wait.
Both are compressions handed to a reader whose every previous scale was linear. The magnitude scale at least looks unfamiliar enough to invite the question. "Once every hundred years" sounds like plain language, which is exactly why it slips past.
References (1)
- the paper that replaced a single design earthquake with an annual probability, which is where return-period design comes from Cornell, C. A. (1968). Engineering seismic risk analysis. Bulletin of the Seismological Society of America, 58(5), 1583–1606. doi:10.1785/bssa0580051583