Sherman IDF Rainfall Intensity

Sherman's form drops the offset entirely: $i = a/t^c$. Take logarithms of both sides and you get $\ln i = \ln a - c \ln t$, a straight line. That is the whole appeal. Plot a set of annual-maximum intensities against duration on log-log paper, lay a ruler along them, and the slope is $-c$ while the intercept is $\ln a$. No iteration, no solver, nothing to converge. With a = 800 and c = 0.7, a 20-minute storm gives $800/20^{0.7} = 98.3$ mm/h. Generations of drainage engineers fitted their local curves exactly this way, and a great many published tables are still Sherman constants whatever the surrounding text calls them.

Where it fails is the short end, and it fails in a specific direction. Without the offset, intensity goes to infinity as duration goes to zero, so the curve rises faster than real rainfall does below about 10 or 15 minutes and over-predicts there. That is not always the wrong side to be wrong on, but it is worth knowing you are on it. Adding $b$ is what fixes it, at the cost of a third parameter and a fit that no longer falls out of a ruler. The practical rule is that Sherman is fine when your durations of interest sit above roughly 15 minutes and the fitted band is narrow; below that, use the three-parameter form. There is a second failure at the far end, and it goes the other way. Extended past 24 hours the straight line keeps falling at the same rate while real depth-duration curves flatten, so a 48-hour intensity taken off a Sherman fit runs low. Neither end is a defect in the algebra. Both are what happens when a two-parameter line is asked to describe a record that curves.

Two properties of $c$ are worth carrying around. It is the elasticity of intensity with respect to duration — doubling the duration multiplies the intensity by $2^{-c}$, so at c = 0.7 a storm twice as long is 62 percent as intense — and it is remarkably stable within a climate, typically landing between 0.5 and 0.9, while $a$ moves with return period. That means a single $c$ often serves a whole family of curves with only $a$ changing, which is a useful sanity check on a table you have been handed. Everything said elsewhere about units holds here too, and the exponent makes it worse rather than better: because $a$ has dimensions of intensity times duration to the power $c$, changing the duration unit rescales $a$ by $60^c$, which is not a round number and leaves no fingerprint in the value. This page assumes mm/h against minutes. And the record behind the fit is historical: if your jurisdiction has published newer curves, those are the ones to use, and the honest place to get them is the authority itself rather than any number a calculator could bake in.