Flood Flow for a Return Period (Gumbel Frequency Analysis)

The flow a river usually carries and the flow it reaches in its worst year are different things, and this formula is about the second. A flood with return period $T$ is one that the river reaches or passes, on average, once in $T$ years, and $Q_T$ is the flow of that flood. You supply the mean $\mu$ and the standard deviation $\sigma$ of the annual peak flows, meaning the highest flow in each year of the record, and the formula assumes those peaks follow the Gumbel (extreme value type I) distribution.

Read the answer as the mean plus a number of standard deviations, $Q_T = \mu + K_T\,\sigma$. All the dependence on the return period sits in the frequency factor $K_T = -\frac{\sqrt{6}}{\pi}\left[\gamma + \ln\ln\frac{T}{T-1}\right]$, where $\gamma$ is the constant 0.5772. Nothing about the river enters $K_T$, only $T$, so one short table of factors serves every gauge: about 1.3 for 10 years, 2.0 for 25, 2.6 for 50 and 3.1 for 100. The factor can be negative. For 2 years it is $-0.164$, so the two-year flood comes out a little under the mean.

Take a record with a mean of 1,000 and a standard deviation of 300 m³/s, so that the flood is 1,000 plus 300 times the factor. For 10 years the factor is 1.305 and the flood is 1,391.5 m³/s. For 100 years the factor is 3.137 and the flood is 1,941.1 m³/s. The page also runs the other way: a flow of 1,941.1 m³/s on that same record comes out as a flood expected once in 100 years. Solving for $\mu$ or $\sigma$ rearranges the same relation.

The method is Gumbel's, from his 1941 paper on the return period of flood flows, and the general form with a frequency factor is the one Chow set out in 1951 for hydrologic frequency analysis. The factors above are the textbook Gumbel values for an infinite record. A real record is finite, and $\mu$ and $\sigma$ are estimated from the years you have, so a flood with a long return period asks the record to say more than its years directly show: the number is only as good as the record is long. If the peaks at a gauge do not follow the Gumbel distribution, the factors do not apply. To put a return period on a flood that has already happened, using only its rank in the record, use the plotting-position page on return period from rank instead.