Talbot IDF Rainfall Intensity

Talbot's form is the three-parameter curve with the exponent nailed to one: $i = a/(t+b)$. Two constants instead of three, which matters more than it sounds when you are fitting by hand or by eye, because a two-parameter fit is stable on a short record where a three-parameter one wanders. With a = 5000 and b = 20 minutes, a 30-minute storm gives $5000/50 = 100$ mm/h. The shape is a rectangular hyperbola in duration, asymptotic to zero at long durations and finite at zero duration thanks to the offset — which is what the offset is for.

Fixing c at 1 does something the other IDF forms cannot do, and it is worth a moment. Multiply both sides by the duration and $a = i\,(t+b)$, which has the dimensions of a length. Talbot's $a$ is a depth. Convert it: 5000 mm/h per minute is 5000/60 = 83.3 mm, and that is the total accumulation the curve tends towards as the storm runs long, since $P = it$ approaches $a$ once $t$ dwarfs $b$. So a Talbot fit encodes a limiting storm depth in a single number, and the $b$ alongside it is the duration at which intensity has fallen to half its notional value at zero. That is a genuinely readable pair of parameters, which is more than can be said for most curve fits.

The cost of the simplicity is range. A single hyperbola cannot follow a real IDF curve from 5 minutes to 24 hours; the exponent that best fits the short end is not the one that fits the long end, which is precisely why the three-parameter form exists. Fitted over a narrow band — say 5 to 60 minutes, which is where most urban drainage lives — Talbot is excellent, and extended to 12 hours it under-predicts. Fitting one is nearly a two-point exercise, which is the other half of its appeal. Rearranged as $1/i = t/a + b/a$ it is linear in the reciprocal of the intensity, so plotting $1/i$ against $t$ gives a straight line whose slope is $1/a$ and whose intercept is $b/a$. That linearisation is why the form survived decades of hand computation, and it is still the quickest way to check whether a published pair of constants really reproduces the table it came from. Everything said about units on the sibling page applies here without softening: $a$ belongs to one region, one return period and one unit convention, and this page assumes mm/h against minutes. The anchor test on this formula deliberately enters an intensity in inches per hour to prove the point, because an $a$ that comes out 25.4 times too small is the single most common way this calculation goes wrong. And as always, the curve is a summary of a historical record. Where your authority has reissued its IDF table, the older constants are out of date rather than merely conservative, and the current publication is the design basis.