Return period

A return period, also known as a recurrence interval (sometimes repeat interval) is an estimate of the likelihood of an event, such as an earthquake, flood or a river discharge flow to occur. It is a statistical measurement typically based on historic data denoting the average recurrence interval over an extended period of time, and is usually used for risk analysis (e.g. to decide whether a project should be allowed to go forward in a zone of a certain risk, or to design structures to withstand an event with a certain return period). The following analysis assumes that the probability of the event occurring does not vary over time and is independent of past events.

Equation

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

n number of years on record;

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

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.

Return period as "expected frequency"

The theoretical return period is the inverse of the probability that the event will be exceeded in any one year (or more accurately the inverse of the expected number of occurrences in a year). 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.

Note also that 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.

This 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.

Probability distributions

One would like to be able to interpret the return period in probabilistic models. The most logical interpretation for this is to take the return period as the counting rate in a Poisson distribution since it is the expectation value of the rate of occurrences. An alternative interpretation is to take it as the probability for a yearly Bernoulli Trial in the Binomial Distribution. This is disfavoured because each year does not represent an independent Bernoulli trial but is an arbitrary measure of time. This question is mainly academic as the results given obtained will be similar under both the Poisson and Binomial interpretations.

Poisson

Take the Poisson distribution as

$P_t(r)={(\mu t)^r \over r!} e^{-\mu t}$

where $r$ is the number of occurrences the probability is calculated for, $t$ the time period of interest and $\mu$ is the counting rate.

Example

If the return period of occurrence is 1 in 234 years ($\mu$ = 0.0043) then the probability of no occurrence in ten years is

$P_t(r)={(\mu t)^r \over r!} e^{-\mu t}$

$P_{10}(0)={(0.0043 \times 10)^0 \over 0!} e^{-0.0043 \times 10} = 0.958$

Binomial

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

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

In the limit of :$n \rightarrow \infty, \mu \rightarrow 0$, such that $n \mu \rightarrow \lambda$

then

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

Take

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

where

T is return interval

n is number of years on record;

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

Example

Given that the return period of an event is 100 years,

$p={1\over 100}=0.01$.

So the probability that such an event occurs exactly once in 10 successive years is:

$P(X = 1)=\binom{10}{1} \times 0.01^1 \times 0.99^9$

$\approx 10 \times 0.01 \times 0.914$

$\approx 0.0914 \,$

Risk analysis

Return period is useful for risk analysis (such as natural, inherent, or hydrologic risk of failure).^[1] When dealing with structure design expectations, the return period is useful in calculating the riskiness of the structure.

The probability of at least one event that exceeds design limits during the expected life of the structure is the complement of the probability that no events occur which exceed design limits.

The equation for assessing this parameter is

$\overline {R}=1-(1-{1\over T})^n=1-(1-P(X\ge{x_T}))^n$

where

${1\over T}=P(X\ge{x_T})$ is the expression for the probability of the occurrence of the event in question in a year;

n is the expected life of the structure.

See also

• Cumulative frequency analysis
• Frequency of exceedance
• Hundred year flood
• Residence time

References

1. Water Resources Engineering, 2005 Edition, John Wiley & Sons, Inc, 2005.