Use Hydrology for Return Period Estimation of Rainfall, Like a 100 Year Flood

Most people have heard the terms, 100 year flood, 50 year storm, or 200 year flood, as a description of the magnitude of a storm or flood. We understand that the larger the number before 'year flood', the greater will be the effect on river levels and on anything out on the river's flood plain. Also, we understand from the words '100 year flood' that it has a flood return period of 100 years, or in other words, it should only happen every 100 years on the average, but questions remaining are 'Could a 100 year flood occur the next year after one has just occurred?' and 'What is the likelihood of its occurring within any given time period?'

Frequency Analysis provides a systematic approach for using historical data to relate the magnitude of a naturally occurring event (e.g. river levels at 5 ft above flood stage) to the probability of its occurring in a given time period or to its recurrence interval. Frequency analysis is typically included in the study of hydrology. This article is an introduction to frequency analysis as used for return period estimation of rainfall, and in connection with hydrologic events such as storms or floods, including terminology and basic concepts.

In order to discuss the concepts of frequency analysis as used in hydrology, and present some of the fundamental relationships, it's necessary to be familiar with a few terms that are used in this component of hydrology. Following are definitions for those few essential terms.

1. Return Period (T) - The average length of time in years for an event (e.g. flood or river level) of given magnitude to be equalled or exceeded. For example, if the river level with a 50 year return period at a given location is 8 ft above flood stage, this is just another way of saying that a river level of 8 ft above flood stage, or greater, should occur at that location on the average only once every 50 years.
2. Probability of Occurrence (p) (of an event of specified magnitude) - The probability that an event of the specified magnitude will be equalled or exceeded during a one year period.
3. Probability of Nonoccurrence (q) (of an event of specified magnitude) - The probability that an event of the specified magnitude will not be equalled or exceeded during a one year period.
4. Probability of Occurrence within a period of N years (p_N) - The probability that an event of specified magnitude will be equalled or exceeded within a period of N years.
5. Probability of Nonoccurrence within a period of N years (q_N) - The probability that an event of specified magnitude will not be equalled or exceeded within a period of N years.

A fundamental relationship is that between flood return period (T) and probability of occurrence (p). These two variables are inversely related to each other. That is p = 1/T and T = 1/p. For example, the probability of a 50 year storm occurring in a one year period is 1/50 or 0.02.

The probability of occurrence and probability of nonoccurrence are related by the fact that something must either occur or not occur, so p + q = 1 and p_N + q_N = 1.

From basic probability theory, q_N = q^N. Substituting to get an equation relating p_N and p: 1 - p_N = (1 - p)^N. This can be rearranged to: p_N = 1 - (1 - p)^N.

Question #1: Could a 100 year flood occur in the next year after a 100 year flood has taken place.

Solution: The answer is yes. A 100 year flood has a return period of T = 100, so the probability of a flood of equal or greater magnitude occurring in any one year period is p = 1/T = 1/100 = 0.01. Thus there is a probability of 0.01 or 1 in 100 that a 100 year flood will occur in any given year. It is not likely, but it is possible. The fact that a 100 year flood occurred in one year has no effect on the probability of its occurring in the next year.

Question #2: What is the probability that a 50 year river level will occur within the next 10 years at any given location?

Solution: The answer to this can be found using the last equation in the previous section, p_N = 1 - (1 - p)^N. Since we are considering a 10 year period of time, N = 10. Also, for a 50 year river level, p = 1/50 = 0.02. Substituting into the equation: P₁₀ = 1 - (1 - 0.02)¹⁰ = 1 - 0.98¹⁰ = 0.183. Thus the probability of a 50 year river level occurring in a 10 year period is about 18%.

The three rainfall variables, intensity (in/hr), duration (min or hr), and frequency (return period) are interrelated. If any two of these variables are specified the third is fixed. Intensity-Duration-Frequency (IDF) graphs or equations are available for many U.S. locations, typically through a state agency. For more information about IDF relationships see the article, "Calculating Design Rainfall Intensity for Use in the Rational Equation."

For More Information:

1. McCuen, Richard H., Hydrologic Analysis and Design, 2nd Ed., Upper Saddle River, NJ, Prentice Hall, 1998

2. Flood Return Period Calculator - http://www.srh.noaa.gov/epz/?n=wxcalc_floodperiod

3. USGS site - http://ga.water.usgs.gov/edu/100yearflood.html