Return Period from Rank (Weibull)

Before any distribution is fitted, return periods are counted rather than calculated, and this is how. Take the annual maximum series — the single largest event of each year — sort it from largest to smallest, and the event sitting at rank $m$ in an $n$-year record is assigned $T = (n+1)/m$ years. The largest event in a 30-year record plots at 31 years. The fifth largest in 49 years plots at 10. Those plotted points are what an IDF curve, a flood frequency curve and a design rainfall table are all drawn through, so this unglamorous fraction sits underneath every return period anyone quotes. Turn it round and it reads plainer still: the exceedance probability assigned to that event is $m/(n+1)$, and the return period is its reciprocal. The rank is simply a count of how many years equalled or beat it, so the whole estimate is a tally over a record length. No distribution has been assumed and none is needed, which is exactly why the plotted points are the evidence and the fitted curve through them is only the interpretation.

The $n+1$ rather than $n$ is the Weibull convention, and it exists for a reason worth knowing. With $n$ in the numerator the largest observed event would be assigned a return period exactly equal to the record length, which quietly asserts that the worst thing you happened to see is the worst thing that happens at that frequency. The $+1$ is the unbiased estimator of exceedance probability for any distribution, and it puts the largest event just outside the record instead of on its edge. Other conventions exist and are not wrong: Gringorten's $(m - 0.44)/(n + 0.12)$ is preferred for Gumbel-distributed extremes, Hazen's $(m - 0.5)/n$ turns up in older US work, and they disagree most at the extreme ranks — which is exactly where the design values live. If you are comparing two published curves that seem to disagree, check which plotting position each used before concluding anything.

The hard limit is the one that matters most and is ignored most. Plotting position cannot reach past $n+1$ years. Thirty years of record gives you a 31-year event, and that is the end of what counting can tell you — a 100-year value from a 30-year record is not a measurement, it is a fitted distribution extrapolated more than threefold beyond its data, with a confidence interval to match. That extrapolation is legitimate and it is what frequency analysis is for, but it should be labelled as what it is. Two smaller cautions. The series must be annual maxima and genuinely independent, so one storm cannot contribute to two years and a partial-duration series needs different treatment. And a record with a gap is shorter than it looks; $n$ is the number of years you actually have, not the span between the first and last.