Return period
A return period, also known as a recurrence interval or repeat interval, is an
average time or an estimated average time between events such
as earthquakes, floods,[1] landslides,[2] or river discharge flows to occur.
It is a statistical measurement typically based on historic data over an extended period,
and is used usually for risk analysis. Examples include deciding whether a project
should be allowed to go forward in a zone of a certain risk or designing structures to
withstand events 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.
Estimating a return period
Recurrence interval
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][failed verification]
Return period as the reciprocal of expected
frequency.
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.
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. That 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 obtained will be
similar under both the Poisson and binomial interpretations.
Poisson
The probability mass function of the Poisson distribution is
where is the number of occurrences the probability is calculated for, the
time period of interest, is the return period and is the counting rate.
The probability of no-occurrence can be obtained simply considering the
case for . The formula is
Consequently, the probability of exceedance (i.e. the probability of an
event "stronger" than the event with return period to occur at least once
within the time period of interest) is
Note that for any event with return period , the probability of
exceedance within an interval equal to the return period (i.e. ) is
independent from the return period and it is equal to . 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.
Example
If the return period of occurrence is 243 years () then the probability
of exactly one occurrence in ten years is
Binomial
In a given period of n years, the probability of a given number r of
events of a return period is given by the binomial distribution as
follows.
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 in such a way that then
Take
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,
So the probability that such an event
occurs exactly once in 10 successive
years is:
Risk analysis.
Return period is useful for risk
analysis (such as natural, inherent,
or hydrologic risk of failure).[6] 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
where
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.
References.
1. ^ ASCE, Task
Committee on
Hydrology Handbook of
Management Group D
of (1996). Hydrology
Handbook |
Books. doi:10.1061/978
0784401385. ISBN 978
-0-7844-0138-5.
2. ^ Peres, D. J.;
Cancelliere, A. (2016-
10-01). "Estimating
return period of
landslide triggering by
Monte Carlo
simulation". Journal of
Hydrology. Flash floods,
hydro-geomorphic
response and risk
management. 541:
256–
Alluvial river
Braided river
Blackwater river
Channel
Channel pattern
Channel types
Rivers Confluence
(lists) Distributary
Drainage basin
Subterranean river
River bifurcation
River ecosystem
River source
Tributary
Arroyo
Bourne
Burn
Chalk stream
Coulee
Current
Streams Stream bed
Stream channel
Streamflow
Stream gradient
Stream pool
Perennial stream
Winterbourne
Estavelle/Inversac
Geyser
Holy well
Hot spring
list
Springs list in the US
(list) Karst spring
list
Mineral spring
Ponor
Rhythmic spring
Spring horizon
Abrasion
Anabranch
3. 271. Bibcode:2016JHyd..541..256P. doi:10.1016/j.jhydrol.2016.03.036.
4. ^ Kumar, Rajneesh;
Bhardwaj, Anil
(2015). "Probability
analysis of return period
of daily maximum
rainfall in annual data
set of Ludhiana,
Punjab". Indian Journal
of Agricultural
Research. 49 (2):
160. doi:10.5958/0976-
058X.2015.00023.2. IS
SN 0367-8245.
5. ^ National Resources
Conservation Service
(August
2007). "Chapter 5:
Stream
Hydrology". National
Engineering Handbook,
Part 654: Stream
Restoration Design.
Washington, D.C.: U.S.
Department of
Agriculture. Retrieved 7
February 2023.
6. ^ Anonymous (2014-11-
07). "Flood Estimation
Handbook". UK Centre
for Ecology &
Hydrology.
Retrieved 2019-12-21.
7. ^ Water Resources
Engineering, 2005
Edition, John Wiley &
Sons, Inc, 2005.
Categories:
Hydrology
Seismology
Time in science
Durations
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