Return period

Recurrence interval ${\displaystyle ={n+1 \over m}}$

n number of years on record;

m is the rank of observed occurrences when arranged in descending order[3]

For floods, the event may be measured in terms of m³/s or height; for storm surges, in terms of the height of the surge, and similarly for other events. This is Weibull's Formula.[4]^: 12[5]

The theoretical return period between occurrences is the inverse of the average frequency of occurrence. For example, a 10-year flood has a 1/10 = 0.1 or 10% chance of being exceeded in any one year and a 50-year flood has a 0.02 or 2% chance of being exceeded in any one year.

This does not mean that a 100-year flood will happen regularly every 100 years, or only once in 100 years. Despite the connotations of the name "return period". In any given 100-year period, a 100-year event may occur once, twice, more, or not at all, and each outcome has a probability that can be computed as below.

Also, the estimated return period below is a statistic: it is computed from a set of data (the observations), as distinct from the theoretical value in an idealized distribution. One does not actually know that a certain or greater magnitude happens with 1% probability, only that it has been observed exactly once in 100 years.

That distinction is significant because there are few observations of rare events: for instance if observations go back 400 years, the most extreme event (a 400-year event by the statistical definition) may later be classed, on longer observation, as a 200-year event (if a comparable event immediately occurs) or a 500-year event (if no comparable event occurs for a further 100 years).

Further, one cannot determine the size of a 1000-year event based on such records alone but instead must use a statistical model to predict the magnitude of such an (unobserved) event. Even if the historic return interval is a lot less than 1000 years, if there are a number of less-severe events of a similar nature recorded, the use of such a model is likely to provide useful information to help estimate the future return interval.

The probability mass function of the Poisson distribution is

${\displaystyle P_{t}(r)={(\mu t)^{r} \over r!}e^{-\mu t}={(t/T)^{r} \over r!}e^{-t/T}}$

where ${\displaystyle r}$ is the number of occurrences the probability is calculated for, ${\displaystyle t}$ the time period of interest, ${\displaystyle T}$ is the return period and ${\displaystyle \mu =1/T}$ is the counting rate.

The probability of no-occurrence can be obtained simply considering the case for ${\displaystyle r=0}$. The formula is

${\displaystyle P_{\text{no-occurrence}}(t)=e^{-\mu t}=e^{-t/T}}$

Consequently, the probability of exceedance (i.e. the probability of an event "stronger" than the event with return period ${\displaystyle T}$ to occur at least once within the time period of interest) is

${\displaystyle P_{\text{exceedance}}(t)=1-P_{\text{no-occurrence}}(t)=1-e^{-\mu t}=1-e^{-t/T}}$

Note that for any event with return period ${\displaystyle T}$, the probability of exceedance within an interval equal to the return period (i.e. ${\displaystyle t=T}$) is independent from the return period and it is equal to ${\displaystyle 1-\exp(-1)\approx 63.2\%}$. This means, for example, that there is a 63.2% probability of a flood larger than the 50-year return flood to occur within any period of 50 year.

In a given period of n years, the probability of a given number r of events of a return period ${\displaystyle \mu }$ is given by the binomial distribution as follows.

${\displaystyle P(X=r)={n \choose r}\mu ^{r}(1-\mu )^{n-r}.}$

This is valid only if the probability of more than one occurrence per year is zero. Often that is a close approximation, in which case the probabilities yielded by this formula hold approximately.

If ${\displaystyle n\rightarrow \infty ,\mu \rightarrow 0}$ in such a way that ${\displaystyle n\mu \rightarrow \lambda }$ then

${\displaystyle {\frac {n!}{(n-r)!r!}}\mu ^{r}(1-\mu )^{n-r}\rightarrow e^{-\lambda }{\frac {\lambda ^{r}}{r!}}.}$

Take

${\displaystyle \mu ={\frac {1}{T}}={m \over n+1}}$

where

T is return interval

n is number of years on record;

m is the number of recorded occurrences of the event being considered

• 100-year flood
• Cumulative frequency analysis
• Frequency of exceedance
• Residence time